tag:blogger.com,1999:blog-13683879201590695142022-06-24T04:22:59.826+05:30Indian Economic Service | NET/JRF Economics | RBI Grade B DEPRSantosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.comBlogger16125tag:blogger.com,1999:blog-1368387920159069514.post-42052509564414808402022-06-12T18:58:00.077+05:302022-06-19T17:33:52.838+05:30RBI Grade B DEPR Practice Question Set - 1<ol> <li>Find the slope of the following isoquant: ${P_K}K + {P_L}L = E$</li> <ol type="A"> <li>$$\frac{E}{{{P_K}}}$$</li><br/> <li>$$\frac{{{P_L}}}{{{P_K}}}$$</li><br/> <li>$$- \frac{E}{{{P_K}}}$$</li><br/> <li>$$- \frac{{{P_L}}}{{{P_K}}}$$</li><br/><hr/> </ol> <li>Find the intercept of the following isoquant: ${P_K}K + {P_L}L = E$</li> <ol type="A"> <li>$$\frac{E}{{{P_K}}}$$</li><br/> <li>$$\frac{{{P_L}}}{{{P_K}}}$$</li><br/> <li>$$- \frac{E}{{{P_K}}}$$</li><br/> <li>$$- \frac{{{P_L}}}{{{P_K}}}$$</li><br/><hr/> </ol> <li>Find the equilibrium quantity demanded and the equilibrium price for the following supply and demand function: $\begin{array}{l}{Q_s} = - 5 + 3P\\{Q_d} = 10 - 2P\end{array}$</li> <ol type="A"> <li>Q = 3 &#38; P = 3</li><br/> <li>Q = 4 &#38; P = 4</li><br/> <li>Q = 4 &#38; P = 3</li><br/> <li>Q = 3 &#38; P = 4</li><br/><hr/> </ol> <li>Suppose there is a simple two sector economy given as followes: $\begin{array}{l}Y = C + I\\C = {C_o} + bY\\I = {I_o}\end{array}$ where, <br/>$${C_o} = 85$$ <br/>$$b = 0.9$$ <br/>$${I_o} = 55$$ <br/>which of the following option is incorrect about the econonomy:</li> <ol type="A"> <li>Y = 1400</li><br/> <li>MPC = 0.9</li><br/> <li>MPS = 0.1</li><br/> <li>C = 1400</li><br/><hr/> </ol> <li>Consider the following two sector economy model: $\begin{array}{l}Y = C + I\\{M_d} = {M_z} + {M_t}\\C = 48 + 0.8Y\\I = 98 - 75i\\{M_s} = 250\\{M_t} = 0.3Y\\{M_z} = 52 - 150i\end{array}$ Find the equilibrium interest rate:</li> <ol type="A"> <li>0.8</li><br/> <li>0.08</li><br/> <li>8.8</li><br/> <li>8.08</li><br/><hr/> </ol> <li>A person has ₹ 120 to spend on two goods (X,Y) whose respectiue prices are ₹ 3 and ₹ 5. What happens to the original budgel line if the budget falls by 25%?</li> <ol type="A"> <li>The budget line shifts parallel to the left</li><br/> <li>The budget line shifts parallel to the right</li><br/> <li>The budget line becomes steeper</li><br/> <li>The budget line becomes flatter</li><br/><hr/> </ol> <li>What happens to the budget line in the if the price of x doubles?</li> <ol type="A"> <li>The budget line shifts parallel to the right left</li><br/> <li>The budget line shifts parallel to the right</li><br/> <li>The budget line becomes steeper</li><br/> <li>The budget line becomes flatter</li><br/><hr/> </ol> <li>What happens to the budget line in the if the price of Y fall to 43</li> <ol type="A"> <li>The budget line shifts parallel to the right left</li><br/> <li>The budget line shifts parallel to the right</li><br/> <li>The budget line becomes steeper</li><br/> <li>The budget line becomes flatter</li><br/><hr/> </ol> <li>Find the equilibrium national income for the following model:</li> $\begin{array}{l}Y = C + I\\C = 100 + 0.6Y\\{I_0} = 40\end{array}$ <ol type="A"> <li>275</li><br/> <li>375</li><br/> <li>250</li><br/> <li>350</li><br/><hr/> </ol> <li>Find the equilibrium national income for the following model:</li> $\begin{array}{*{20}{l}}{Y = C + I}\\{C = 100 + 0.6Y}\\{{I_0} = 40}\\{{Y_d} = Y - T}\\{T = 50}\end{array}$ <ol type="A"> <li>275</li><br/> <li>375</li><br/> <li>250</li><br/> <li>350</li><br/><hr/> </ol> <li>Find the equilibrium national income aggregate demand from the following income determination model:</li> $\begin{array}{*{20}{l}}{Y = C + I}\\{C = 85 + 0.75{Y_d}}\\{{I_0} = 30}\\{{Y_d} = Y - T}\\{T = 20 + 0 \cdot 2Y}\end{array}$ <ol type="A"> <li>275</li><br/> <li>375</li><br/> <li>250</li><br/> <li>350</li><br/><hr/> </ol> <li>Given the following set of simultaneous equations for two related markets, beef(B) and pork(p), Find the equilibrium quantities $$({Q_B}\,\,\&#38; \,\,{Q_P})$$ for each market: <br/> Beef Market: $\begin{array}{l}{Q_{dB}} = 82 - 3{P_B} + {P_P}\\{Q_{sB}} = - 5 + 15{P_B}\end{array}$ Pork Market: $\begin{array}{l}{Q_{dP}} = 92 - 2{P_B} + 4{P_P}\\{Q_{sB}} = - 6 + 32{P_P}\end{array}$ </li> <ol type="A"> <li>$${Q_B} = 70\,\,\&#38; \,\,{Q_P} = 70$$</li><br/> <li>$${Q_B} = 90\,\,\&#38; \,\,{Q_P} = 90$$</li><br/> <li>$${Q_B} = 90\,\,\&#38; \,\,{Q_P} = 70$$</li><br/> <li>$${Q_B} = 70\,\,\&#38; \,\,{Q_P} = 90$$</li><br/><hr/> </ol> <li>Find the equilibrium quantity for two complementory goods, slacks (s) and jackets (J) for the following model: <br/> Slack Market: $\begin{array}{l}{Q_{ds}} = 410 - 5{P_s} - 2{P_J}\\{Q_{ss}} = - 60 + 3{P_s}\end{array}$ Jacket Market: $\begin{array}{l}{Q_{dJ}} = 295 - {P_s} - 3{P_J}\\{Q_{sJ}} = - 120 + 2{P_J}\end{array}$</li> <ol type="A"> <li>$${Q_S} = 60\,\,\,\&#38; \,\,\,{Q_J} = 30$$</li><br/> <li>$${Q_S} = 30\,\,\,\&#38; \,\,\,{Q_J} = 60$$</li><br/> <li>$${Q_S} = 30\,\,\,\&#38; \,\,\,{Q_J} = 30$$</li><br/> <li>$${Q_S} = 60\,\,\,\&#38; \,\,\,{Q_J} = 60$$</li><br/><hr/> </ol> <li>Find the equilibrium price for the following demand and supply function: <br/> Demand function: $P + {Q^2} + 3Q - 20 = 0$ Supply function: $P - 3{Q^2} + 10Q = 5$</li> <ol type="A"> <li>2</li><br/> <li>4</li><br/> <li>6</li><br/> <li>8</li><br/><hr/> </ol> <li>Given: $Y = C + I + G$$C = {C_0} + bY$$I = {I_0}$$G = {G_0}$$C = 135$$b = 0.8$${I_0} = 75$${G_0} = 30$ find the equilibrium consumption level:</li> <ol type="A"> <li>1095</li><br/> <li>1295</li><br/> <li>1000</li><br/> <li>1200</li><br/><hr/> </ol> <li>Given: <br/> Demand function: $3P + {Q^2} + 5Q - 102 = 0$ Supply function: $P - 2{Q^2} + 3Q + 71 = 0$ find the equilibrium price:</li> <ol type="A"> <li>2</li><br/> <li>4</li><br/> <li>6</li><br/> <li>8</li><br/><hr/> </ol> <li>Find the numerical value of the equilibrium national income for the following model where investment is also function of national income: $Y = C + I$$C = {C_0} + bY$$I = {I_0} + aY$${C_0} = 65$${I_0} = 70$$b = 0.6$$a = 0.2$</li> <ol type="A"> <li>600</li><br/> <li>625</li><br/> <li>675</li><br/> <li>700</li><br/><hr/> </ol> <li>Find the equilibrium level of national income for the following national income determination model in the foreign sector: $Y = C + I + G + (X - Z)$$C = {C_0} + bY$$Z = {Z_0} + zY$$I = {I_0} = 90$$G = {G_0} = 65$$X = {X_0} = 80$${C_0} = 70$${Z_0} = 40$$b = 0.9$$z = 0.15$</li> <ol type="A"> <li>2060</li><br/> <li>2000</li><br/> <li>1060</li><br/> <li>1000</li><br/><hr/> </ol> <li>Given $C = 102 + 0.7Y$$I = 150 - 100i$${M_s} = 300$${M_t} = 0.25Y$${M_z} = 124 - 200i$ find equilibrium level of interest rate:</li> <ol type="A"> <li>0.04</li><br/> <li>0.08</li><br/> <li>0.16</li><br/> <li>0.12</li><br/><hr/> </ol> <li>Find the equilibrium national income for the following IS-LM model: $C = 89 + 0.6Y$$I = 120 - 150i$${M_s} = 275$${M_t} = 0.1Y$${M_z} = 240 - 250i$</li> <ol type="A"> <li>500</li><br/> <li>525</li><br/> <li>575</li><br/> <li>600</li><br/><hr/> </ol> <li></li> <ol type="A"> <li></li><br/> <li></li><br/> <li></li><br/> <li></li><br/><hr/> </ol> <li></li> <ol type="A"> <li></li><br/> <li></li><br/> <li></li><br/> <li></li><br/><hr/> </ol> <li></li> <ol type="A"> <li></li><br/> <li></li><br/> <li></li><br/> <li></li><br/><hr/> </ol> <li></li> <ol type="A"> <li></li><br/> <li></li><br/> <li></li><br/> <li></li><br/><hr/> </ol>$$({Q_B}\,\,\& \,\,{Q_P})$$ </ol>Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com0tag:blogger.com,1999:blog-1368387920159069514.post-76413557447485996752022-06-11T11:55:00.003+05:302022-06-11T17:48:55.867+05:30Previous Year Paper Solution | Indian Economic Service Exam 2021 | General Economics - IISantosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com0tag:blogger.com,1999:blog-1368387920159069514.post-49848229003798595872022-06-11T09:13:00.015+05:302022-06-11T17:47:25.299+05:30Previous Year Paper Solution | Indian Economic Service Exam 2021 | General Economics - II<h4>Q. No. - 1.(a) Differentiate between Adam Smith's concept of subsistence wage and Ricardo's concept of natural wage.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 1.(b) Is the underground economy beneficial or subversive?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 1.(c) Why is saving in Keynesian system a function of disposable income but in classicla system a function of rate of interest?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 1.(d) Distinguish between (i) bonds and debentures, (ii) bonds and non-convertible debentures.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 1.(e) How can you justify the absence of saving ratio in the quantitative speed of convergence towards steady state equilibrium in the neo-classical growth model?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 1.(f) How far is the 'Forex Index' an index of economic strenth?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 2. Assess and justify the analytical merits and historical relevance of mercantilist views on foreign trade.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 3. What are the approaches to compute 'Green' GDP? How would you make GDP more inclusive in the light of Index of Sustainable Economic Welfare (ISEW)?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 4. (a) Show that MPC is less than APC in the Keynesian analysis.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 4. (b) Why is consumption function considered to be an epoch-making contribution in the Keynesian theory?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 5. State the basic differences between Futures and Options. Explain in detail the working of the futures and options derivatives.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 6. Both Marx and Schumpeter believe in the downfall of capitalism but for very different reasons. Elucidate.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 7. "Mundell-Fleming model brings about internal and external balance through the equality between IS-LM-BP schedules." Explain.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 8. How is the Human Development Index (HDI) computed? Why is HDI required to be adjusted for inequality?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 9. Highlight the dimensions and implications of Eurozone crisis.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 10. Assess the performance of WTO in achieving its mission of ending extreme poverty and promoting shared prosperity.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 11. State the policy implications of Philips curve and explain the dilemma faced by the policy-makers. Discuss how the government can contribute to economic stability.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 12. Discuss the 'development of underdevelopment' thesis by Frank and Amin. Is the thesis seeded in the Marxian framework of capitalism? Substantiate.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 13. (a) What was the rationale behind the development of endogenous growth models?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 13. (b) Use a basic model of endogenous growth to exhibit its superiority over the exogenous model.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 14. Examine the importance of the latest issues addressed by G-20. What role and responsibilites does G-20 have in tody's world?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 14. Examine the importance of the latest issues addressed by G-20. What role and responsibilities does G-20 have in today's world?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 15. State and explain the H-O theorem and factor-price equalisation theorem. Explain how the prevalence of Factor Intensity Reversal would result in the rejection of both theorems.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 16. (a) Following Baumol-Tobin approach, find out the transaction demand for money and examine its interest rate sensitivity.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. - 16. (b) Show and interpret that such a transaction demand for money is homogeneous of degree zero in real income and rate of interest.</h4><p>(Comment for solution.)</p><hr/>Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com0tag:blogger.com,1999:blog-1368387920159069514.post-16138257465429527752022-06-05T09:30:00.033+05:302022-06-06T19:30:05.643+05:30Previous Year Paper Solution | Indian Economic Service Exam 2021 | General Economics - I<h4>Q. 1(a) - The demand and supply functions are $${P_d} = {\left( {6 - x} \right)^2}$$ and $${P_s} = 14 + x$$ respectively. Find the consumer surplus under pure competitive market. </h4><p>For solution, See Q. No. 15.29 and 15.30, Chapter - 15, Shaum's Outline Introduction to Mathematical Economics by Edward Dowling, 3rd Edition</p><hr/> <h4>Q. 1(b) - Distinguish between Economies and Diseconomies of scope. How can the degree of Economies of scope be determined?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 1(c) - How is the inter-temporal pricing a form of price discrimination? Give example.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 1(d) - For a monopsonist, what is the relationship between the supply of an input and the marginal expenditure on it?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 1(e) - Explain divergence between private and social costs and mislocation of resources in a perfectly competitive system.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 1(f) - Describe the methods used in isolating secular trend in a time series.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 1(g) - Distinguish between pure strategies and mixed strategies of a game.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 2(a) - Derive Slutsky equation and interpret it.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 2(b) - Given the utility fuction as $12y = 36 - {x^2}$ and budget line as $M = 12x + 24y$ Determine the utility maximizing basket of the two goods. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. 3(a) - Definde cost-output elasticity. Show how it can be used to explain existence or absence of economies of scale in production. Verify your answer on the following: <br/>(i) AC = 20 and MC = 10 <br/> (ii) AC = MC = 15 <br/> (iii) AC = 20 and MC = 30</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 3(b) - Given total cost (TC) $$= a + bQ + c{Q^2}$$ <br/>Show that $$MC = AC = b + 2\sqrt {(ac)} \,at\,Q = \sqrt {\left( {\frac{a}{c}} \right)}$$ <br/> where AC is minimum.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 4(a) - What do the Cournot and Bertrand models have in common? What are the differences between these two models?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 4(b) - Why is it possible for a monopolist to earn supernormal profit in the long-run?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 5(a) - State and explain the modern theory of rent. Show how it can be applied to other factors of production</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 5(b) - Explain in terms of the marginal productivity theory how a 'monopolist-monopsonist' firm exploits the society.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 6(a) - Distinguish between Egalitarian society, Utilitarian society Market-oriented society and Rawlsian society.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 6(b) - Using Bergson's welfare contours and grand utility possibility frontier, determine the optimal point of social welfare.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 7(a) - Consider two variable linear regression model: $Y = \alpha + \beta x + u$ The following results are given below: <br/>$$\sum {{X_i}} = 228,\,\sum {{Y_i}} = 3121,\,{\sum {{X_i}Y} _i} = 38297,\,\,\sum {X_i^2} = 3204$$ and <br/>$${\sum {{x_i}y} _i} = 3347 \cdot 60,\,\,\sum {x_i^2} = 604 \cdot 80$$ and $$\,\,\sum {y_i^2} = 19837$$ and n = 20 <br/> Using this data, estimate $$\alpha$$ and $$\beta$$ and the variances of your estimates. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. 8(a) - Given the technology matrix $A = \left[ {\begin{array}{*{20}{c}}{0 \cdot 1}&{0 \cdot 3}&{0 \cdot 1}\\0&{0 \cdot 2}&{0 \cdot 2}\\0&0&{0 \cdot 3}\end{array}} \right]$ and final demands are $${F_1}$$, $${F_2}$$ and $${F_3}$$. <br/> Find the output levels if $${F_1} = 20,\,{F_2} = 0$$ and $${F_3} = 100$$. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. 9(a) - Given the production function as:$Q = A{L^\alpha }{K^\beta };\,\alpha > 0,\,\beta > 0,\,A > 0$ Find the shape of isoquant from the above function. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. 9(b) - Defind Elasticity of Substitution of factors. What will be the shape of the isoquant when elasticity of substitution is zero and infinity?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 9(c) - Write down the Constant Elasticity of Substitution (CES) production function and show that Cobb-Douglas (CD) production function is a special case of CES function</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 10 - Given the production and cost functions as: $Q = 500{L^{\frac{1}{4}}}{K^{\frac{3}{4}}}$ $C = wL + rK$ <br/>(a) Derive the demand curve for labour and capital with a view to maximizing the output when the cost is limited to ₹10,000. Would your answer change if the objective shifts to cost minimization with a desired level of output? Give reasons in support of your answer. <br/>(b) Determine the equilibrium levels of employment of the factors in each case given: <br/>w = 10 and r = 75</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 11(a) - Discuss the notion that the 'bargaining solution' to environmental problems results in the same outcome whether the polluter compensates the sufferers or the sufferers pay the polluter to reduce their levels of emissions.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 11(b) - Why is there a social cost to monopsony power? If the gains to buyer from monopsony power could be redistributed to sellers, would the social cost of monopsony power be eliminated? Explain.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 12 - A local business firm is planning to advertise a special sale on radio and television. Its weekly advertising budget is ₹ 16,000. A radio commercial costs ₹ 800 per 30 - second slot while a television commercial costs ₹ 4,000 per 30 - second slot. Radio slots cannot be bought less than 5 in number while TV slots available are at the most 4 per week. Given that a TV slot is 6 times as effective as a radio slot in reaching consumers, how should the firm allocate its advertising budget to attract the largest number of them? How will the optimal solution be affected if the availability of the television slots is no longer constrained? <br/>(i) Formulate LPP model. <br/>(ii) Solve the above using graphical method.</h4><p>(Comment for solution.)</p><hr/> <h4> Q. 13. a(i) - Explain the concept of direct regression and reverse regression in presence of two variables X and Y. <br/>(ii) - Suppose a two variable linear regresssion model without intercept term. Estimate the slope parameter of such a model and show that it is an unbiased estimator. <br/> (iii) - Find the value of $${r^2}$$ when the intercept term is absent in the two variable linear regression model. </h4><p>(Comment for solution.)</p><hr/> <h4> Q. 13. b - Consider Cobb-Douglas production function: $Q = {b_0}{L^{{b_1}}}{K^{{b_2}}}$ Test the hypothesis at 5% level of significance ${H_0}:{b_1} + {b_2} = 1$ again ${H_0}:{b_1} + {b_2} = 1$ <p>(Comment for solution.)</p><hr/></h4> Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com0tag:blogger.com,1999:blog-1368387920159069514.post-20162780828067965122021-06-14T13:33:00.077+05:302022-06-04T19:19:31.408+05:30Previous Year Paper Solution | Indian Economic Service Exam 2020 | General Economics - I<h4>Q. No. - 1 (a) - Show the conditions for a Cobb-Douglas production function under: <ul style="list-style-type: none;"> <li>(i) increasing returns to scale</li> <li>(ii) consant returns to scale</li> <li>(iii) dimishing returns to scale</li> Are the law of returns compatible? </ul></h4><p>(Comment for Solution.)</p><hr /> <h4>Q. No. 1 (b) - Define homothetic preferences. Explain the common characteristics of such preferences with the help of necessary diagrams.</h4><p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwgLi-4B1eR81RPWXv1SkxLkG2qHHevVHYNAizpCuYMNlfCtROLDPBKW-G-EtR2eN5ckuUpVUA1Nxqxit3n2A' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr /> <h4>Q. No. 1 (c) - What is monopoly power? What factors determine the amount of monopoly power?</h4><p>(Comment for Solution.)</p><hr/> <h4>Q. No. 1 (d) - Explain the difference between Bandwagon effect and Snob effect.</h4><p>(Comment for Solution.)</p><hr/> <h4>Q. No. 1 (e) - What is meant by deadweight loss? Why does a price ceiling usually result in a deadweight loss</h4><p>(Comment for Solution.)</p><hr/> <h4>Q. No. 1 (f) - State the fundamental theorems of Welfare Economics.</h4><p>(Comment for Solution.)</p><hr/> <h4>Q. No. 1 (g) - Public goods are non-rival and non-exclusive. Explain each of these terms and show clearly how they differ from each other.</h4><p>(Comment for Solution.)</p><hr/> <h4>Q. No. 2 (a) - Explain the meaning of Nash Equilibrium. How does it differ from the equilibrium in dominant strategies? (8 Marks)</h4><p>Nash Equilibrium is a strategy profile such that each player's equilibrium strategy is the best response to other players' equilibrium strategy and no player can benefit from deviating to other strategies.<br />While dominant strategy equilibrium is a strategy profile such that each player's equilibrium dominant strategy is the best response to other players' equilibrium dominant strategy and no player can benefit from deviating to other strategies. Dominant strategy equilibrium exists if only if both players have dominant strategies to play. All games do not hve dominant strategy equilibrium.<br /> All dominant strategy equilibrium is a Nash Equilibrium but all Nash Equilibrium are not dominant strategy equilibrium.</p><hr/> <h4>Q. No. 2 (b) - Let market demand faced by duopolist be<br />$P = 100 - 0.5Q$ $Q = {Q_1} + {Q_2}$ and their respective cost function as: ${C_1} = 5{Q_1}$ and ${C_2} = 5{Q_2}$ Find out Cournot-Nash Equilibrium. (10 Marks) </h4><p>Let profit function of the first duopolist be: ${\pi _1} = P{Q_1} - {C_1}$ Substituting the value of P (inverse demand function): ${\pi _1} = \left( {100 - 0.5Q} \right){Q_1} - 5{Q_1}$ Substituting the value of Q: ${\pi _1} = \left( {100 - 0.5{Q_1} - 0.5{Q_2}} \right){Q_1} - 5{Q_1}$ ${\pi _1} = 100{Q_1} - 0.5Q_1^2 - 0.5{Q_1}{Q_2} - 5{Q_1}$ ${\pi _1} = 95{Q_1} - 0.5Q_1^2 - 0.5{Q_1}{Q_2}$ Applying the first order condition for maximization: $\frac{{\partial {\pi _1}}}{{\partial {Q_1}}} = 0$ $95 - {Q_1} - 0.5{Q_2} = 0$ ${Q_1} = 95 - 0.5{Q_2}$ Similarly, let profit function of the second duopolist be: ${\pi _2} = P{Q_2} - {C_2}$ Substituting the value of P (inverse demand function): ${\pi _2} = \left( {100 - 0.5Q} \right){Q_2} - 5{Q_2}$ Substituting the value of Q: ${\pi _2} = \left( {100 - 0.5{Q_1} - 0.5{Q_2}} \right){Q_2} - 5{Q_2}$ ${\pi _2} = 100{Q_2} - 0.5{Q_1}{Q_2} - 0.5Q_2^2 - 5{Q_2}$ ${\pi _1} = 95{Q_2} - 0.5{Q_1}{Q_2} - 0.5Q_2^2$ Applying the first order condition for maximization: $\frac{{\partial {\pi _1}}}{{\partial {Q_1}}} = 0$ $95 - 0.5{Q_1} - {Q_2} = 0$ ${Q_2} = 95 - 0.5{Q_1}$ </p><p>Substituting the value of q<sub>2</sub> in q<sub>1</sub> and simplifying: ${Q_1} = 95 - 0.5\left( {95 - 0.5{Q_1}} \right)$ ${Q_1} = 95 - 47.5 + 0.25{Q_1}$ ${Q_1} - 0.25{Q_1} = 47.5$ $0.75{Q_1} = 47.5$ ${Q_1} = \frac{{47.5}}{{0.75}} = \frac{{190}}{3}$ This is the quantity sold by the first duopolist at P.<br /> Similarly, quantity sold by the second duopolist will be: ${Q_2} = \frac{{190}}{3}$ </p><p>Note that $$\frac{{190}}{3}$$ is not the competitive market equilibrium quantity in the given market. We can find the competitive equilibrium quantity using D = S. We know the demand curve and supply curve which is also the Average Revenue (AR) curve is the upward sloping portion portion of Marginal Cost (MC) curve. Since the demand curve is linear, AR = P. Thus, we can say P = MC is the equilibrium condition in a competitive market. Now, we can find the equilibrium quantity.<br /> We first need to calculate the MC. We know that: $TC = {C_1} + {C_2}$ $TC = 5{Q_1} + 5{Q_2}$ $TC = 5\left( {{Q_1} + {Q_2}} \right)$ $TC = 5Q$ Taking the first derivative: $MC = \frac{{d\left( {TC} \right)}}{{dQ}} = 5$ Now, equilibrium condition is: $D = S$ $P = MC$ $100 - 0.5Q = 5$ $0.5Q = 100 - 5$ $Q = \frac{{95}}{{0.5}} = 190$ The market equilibrium quantity (Q) is 190. Cournot Nash equilibrium quantity of duopolists is $$\frac{{190}}{3}$$, that is, each firms sell one-third of total output which can be sold in a perfectly competitive market.</p><hr /> <h4>Q. No. 3 (a) - Suppose the utility function for the consumer take one of the following forms: $\left( i \right)U = 50x + 20y$ $\left( {ii} \right)U = 20x + 50x$ $\left( {iii} \right)U = 80x + 40x$ The budget of the consumer is ₹10,000. The price of good X and good Y are ₹50 and ₹20 per unit respectively. Determine the possibility of determination of the equilibrium basket in each case using diagram and comment on the nature of the solution</h4><p>Solution: Video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dxZAKymm0-P09ZMfWDTZBOLSYlJp13wik36r2q2xWVGWQSoreZOFhOBVLswEVk_c_O2Yzh0Ut6oYNRfjOGqHw' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr /> <h4>Q. No. 3 (b) - Outline how the production possibility frontier can be used to explain the concept of opportunity cost. Why is the production possibility frontier concave to the origin? </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 4 - Suppose that a firm's production function is given by the Cobb-Douglas function:$$Q = {K^\alpha }{L^\beta }(where\,\,\alpha ,\beta > 0)$$.The firm can purchase all the K and L it wants in competitive input markets at rental rates of r and w respectively.<br/> (i) Show that cost minimisation requires $$\frac{{rK}}{\alpha } = \frac{{wL}}{\beta }$$. What is the slope of the expansion path for this firm? <br/> (ii) Assuming cost minimisation, show that total costs can be expressed as a function of Q, r and w of the form $TC = B{Q^{\frac{1}{{\alpha + \beta }}}}.{w^{\frac{\beta }{{\alpha + \beta }}}}.{r^{\frac{\alpha }{{\alpha + \beta }}}}$ where B is a constant depending on $$\alpha$$ and $$\beta$$. <br/> (iii) Show that if $$\alpha + \beta = 1$$, total cost (TC) is proportional to Q. <br/> (iv) Calculate the firm's marginal cost curve. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5 (a) - Distinguish between economic rent and transfer earnings. Can economic rent exist in the long run? Justify your answer.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5(b) - Explain graphically the role of elasticity of supply of a factor determining the economic rent.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 6 (a) - Why do externalities prevent markets from being efficient? How does Coase theorem correct an externality?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 6(b) - Using a particular industry, explain what is meant by economies of scale and economies of scope. How do these affect the industry you have identified?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 7 (a) - In a contest, two judges ranked eight condidates A, B, C, D, E, F, G and H in order of their preference as shown in the following table. Find the rank correlation coefficient.</h4> <table> <tbody><tr><th></th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th><th>F</th><th>G</th><th>H</th></tr> <tr><th>First Judge</th><td>5</td><td>2</td><td>8</td><td>1</td><td>4</td><td>6</td><td>3</td><td>7</td></tr> <tr><th>Second Judge</th><td>4</td><td>5</td><td>7</td><td>3</td><td>2</td><td>8</td><td>1</td><td>6</td></tr> </tbody></table> <h4>Q. No. 7 (b) - The regression equation of the variables x and y are $8x - 10y + 66 = 0$ and $40x - 18y = 214$ The variance of x is 9. Identify the two regression lines. Find the simple correlation coefficient between the two variables and variance of y. </h4><p>Solution Video</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dxiVSLNBX1T1_0d2u-Ps6Hm-LOWt5gWC9JdE5xte-Mcj5WI6XtyIMEUsN2P7Jqrk_WFtiWnMdhA2lQ6_WKm9Q' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr /> <h4>Q. No. 8 - Discuss Social Choice theory in Economics. Distinguish between the views of Amartya Sen and Kenneth Arrow in making choices for social welfare.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9(a) - Consider a two-variable linear regression model $\begin{array}{l}{Y_t} = \alpha + \beta {X_t} + {U_t}\\and\,\,{U_t} = {}_\rho {U_{t - 1}} + {\varepsilon _t};\,\left| \rho \right| &#60; 1\end{array}$Find Mean, Variance and Covariance of random disturbance term $$\left( {\,{U_t}} \right)$$. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9(b) - Consider the model of wage determination: ${Y_t} = {\beta _1} + {\beta _2}{X_t} + {\beta _3}{Y_{t - 1}} + {U_t}$ <br/> where <br/> Y = wages <br/> X = productivity <br/>$${U_t} = {}_\rho {U_{t - 1}} + {\varepsilon _t};\,\,\, - 1 &#60; \rho &#60; 1$$ <br/> Discuss the method of testing with the help of appropriate test statistic.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9(c) - Cosider a model $\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,{Y_t} = {\beta _1} + {\beta _2}{X_t} + {U_t}\\and\,\,\,{U_t} = {}_\rho {U_{t - 1}} + {\varepsilon _t}\end{array}$<br/> Discuss the process of the removal of autocorrelation when <br/>(i) $$\rho$$ is known <br/> (ii) $$\rho$$ is unknown (using Cochrane-Orcutt iterative method)</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10(a) - An economy produces only coal and steel. The two commodities serve as intermediate inputs in each other's production. 0.4 tonne of steel. Similarlly, 0.1 tonne of steel and 0.6 tonne of coal are required to produce a tonne of coal. No capital inputs are needed. 2 and 5 labour days are required to produce a tonne of coal and steel respectively. If the economy needs 100 tonnes of coal and 50 tonnes of steel, <br/> (i) Calculate the gross output of the two commodities and the total labour required. <br/> (ii) Write down technology matrix. <br/> (iii) Do you think that the system is viable? <br/> (iv) Determine the equilibrium prices, if the wage rate is ₹ 10 per man-day.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10(b) - Mohan is paid ₹ 8 if two coins turn both heads and ₹ 1 if two coins turn both tails. Ram is paid ₹ 3 when the two coins do not match. <br/> (i) Write down the pay-off matrix of the above problem. <br/> (ii) Whom do you consider in the better situation?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11(a) - Compare the distribution theory of Marx with that of Ricardo.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11(b) - Explain when Kaldor's theory of distribution beomes more appropriate.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11(c) - Narrate the areas where Kaldor's distribution model fails.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12(a) - The kinked demand curve describes price rigidity. Explain how the model works. Why does price rigidity occur in oligopolistic market</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12(b) - State and prove Product Exhaustion Theorem. How does it differ from Clark-Wicksteed-Walras Theorem? </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 13 (a): Consider the two variable regression model: ${Y_i} = \alpha + \beta {X_i} + {U_i}$ and $Var\left( {{U_i}} \right) = E\left( {U_i^2} \right) = \sigma _i^2$ Show that $${\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } }$$ is unbiased and inefficient estimator of $$\beta$$. </h4> <p>1. $${\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } }$$ is unbiased estimator of $$\beta$$.<br /> Proof: We can use deviation form formula for the estimator which is given as follows: $\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } = \frac{{\sum {{x_i}{y_i}} }}{{\sum {x_i^2} }}$ Let: $\frac{{{x_i}}}{{\sum {x_i^2} }} = {k_i}$ Now, we can write: $\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } = \sum {{k_i}{y_i}}$ Since, ${y_i} = {Y_i} - \bar Y$ We can also write: $\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } = \sum {{k_i}\left( {{Y_i} - \bar Y} \right)}$ $\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } = \sum {{k_i}{Y_i} - \bar Y\sum {{k_i}} }$ Let us find the value of $$\sum {{k_i}}$$ $\sum {{k_i}} = \frac{{\sum {{x_i}} }}{{\sum {x_i^2} }}$ The sum of deviation from mean is always equals zero. So, $\sum {{x_i}} = \sum {\left( {{X_i} - \bar X} \right)} = 0$ It means: $\sum {{k_i}} = 0$ Now, we can write: $\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } = \sum {{k_i}{Y_i}}$ Substituting the regression model in the above equation: $\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } = \sum {{k_i}\left( {\alpha + \beta {X_i} + {U_i}} \right)}$ $\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } = \alpha \sum {{k_i}} + \beta \sum {{k_i}{X_i}} + \sum {{k_i}{U_i}}$ We have already proved that: $\sum {{k_i}} = 0$ Now, let us find the value of $$\sum {{k_i}{X_i}}$$: $\sum {{k_i}{X_i}} = \frac{{\sum {{x_i}{X_i}} }}{{\sum {x_i^2} }}$ $\sum {{k_i}{X_i}} = \frac{{\sum {{X_i}\left( {{X_i} - \bar X} \right)} }}{{\sum {x_i^2} }}$ $\sum {{k_i}{X_i}} = \frac{{\sum {X_i^2 - \bar X\sum {{X_i}} } }}{{\sum {x_i^2} }}$ $\sum {{k_i}{X_i}} = \frac{{\sum {X_i^2 - \bar X\sum {{X_i}} - \bar X\sum {{X_i}} + \bar X\sum {{X_i}} } }}{{\sum {x_i^2} }}$ $\sum {{k_i}{X_i}} = \frac{{\sum {X_i^2 - 2\bar X\sum {{X_i}} + n{{\bar X}^2}} }}{{\sum {x_i^2} }}$ $\sum {{k_i}{X_i}} = \frac{{\sum {\left( {X_i^2 - 2\bar X{X_i} + {{\bar X}^2}} \right)} }}{{\sum {x_i^2} }}$ $\sum {{k_i}{X_i}} = \frac{{\sum {{{\left( {{X_i} - \bar X} \right)}^2}} }}{{\sum {x_i^2} }}$ $\sum {{k_i}{X_i}} = \frac{{\sum {x_i^2} }}{{\sum {x_i^2} }} = 1$ Now, we can write: $\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } = \beta + \sum {{k_i}{U_i}}$ Taking expectation keeping in mind that X<sub>i</sub> are fixed values: $E\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = E\left( \beta \right) + \sum {{k_i}E\left( {{U_i}} \right)}$ Expectation of a constant, here $$\beta$$, is the constant itself and by assumption, it is know that: $E\left( {{U_i}} \right) = 0$ Therefore, $E\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = \beta$ The mean (expectation) of the estimator is equal to the population parameter. Therefore, $${\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } }$$ is the unbiased estimator of $$\beta$$ </p> <p>2. $${\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } }$$ is inefficient estimator of $$\beta$$.<br /> Proof: An estimator is called inefficient when it does not has minimum variance property. We can calculate variance as follows: $Var\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = E{\left[ {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } - E\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right)} \right]^2}$ $Var\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = E{\left[ {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } - \beta } \right]^2}$ From: $\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } = \beta + \sum {{k_i}{U_i}}$ We can write: $Var\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = E{\left[ {\sum {{k_i}{U_i}} } \right]^2}$ Removing summation operatou: $Var\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = E{\left[ {{k_1}{U_1} + {k_2}{U_2} + ... + {k_n}{U_n}} \right]^2}$ $Var\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = E\left[ {k_1^2U_{_1}^2 + k_{_2}^2U_{_2}^2 + ... + k_{_n}^2U_{_n}^2 + 2{k_1}{k_2}{u_1}{u_2} + .... + 2{k_{n - 1}}{k_n}{u_{n - 1}}{u_n}} \right]$ $Var\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = k_1^2E\left( {U_{_1}^2} \right) + k_{_2}^2E\left( {U_{_2}^2} \right) + ... + k_{_n}^2E\left( {U_{_n}^2} \right) + 2{k_1}{k_2}E\left( {{u_1}{u_2}} \right) + .... + 2{k_{n - 1}}{k_n}E\left( {{u_{n - 1}}{u_n}} \right)$ From the assumption of no serial correlation: $E\left( {{u_{i}}{u_j}} \right) = 0$ Now, we can write: $Var\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = k_1^2E\left( {U_{_1}^2} \right) + k_{_2}^2E\left( {U_{_2}^2} \right) + ... + k_{_n}^2E\left( {U_{_n}^2} \right)$ From the question: $E\left( {U_i^2} \right) = \sigma _i^2$ So, we can write: $Var\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = k_1^2\sigma _1^2 + k_2^2\sigma _2^2 + ... + k_n^2\sigma _n^2$ $Var\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = \sum {k_i^2\sigma _i^2}$ Substituting the value of k: $Var\left( {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } } \right) = \frac{{\sum {{x_i}\sigma _i^2} }}{{{{\left( {\sum {x_i^2} } \right)}^2}}}$ This variance of estimator $${\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } }$$ does not follows minimum variance property because there are different values of variance for different samples. Therefore, it is an inefficient estimator. </p> <hr /> <h4>Q. No. 13(b) Consider a three variable linear regression model ${Y_i} = {\beta _1} + {\beta _2}{X_{2i}} + {\beta _3}{X_{3i}} + {\varepsilon _i}$ <br/> and suppose that <br/>(i) $$\sigma _i^2 = {\sigma ^2}Z_i^2$$ <br/>(ii) $$\sigma _i^2 = {\sigma ^2}{X_{1i}}$$ <br/>(iii) $$\sigma _i^2 = {\sigma ^2}X_{_{1i}}^2$$ <br/>Discuss Generalised Least Squares (GLS) method to overcome the heteroscedasticity problem under three cases (i, ii and iii).</h4><p>(Comment for solution.)</p><hr/>Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com6tag:blogger.com,1999:blog-1368387920159069514.post-4861019237391078122021-06-14T13:31:00.059+05:302021-07-04T09:07:21.138+05:30Previous Year Paper Solution | Indian Economic Service Exam 2019 | General Economics - I <h4>Q. No. 1 (a) - In a two commodity framework, the marginal rate of substitution is everywhere equal to 2. The price of the two goods are equal. Draw a diagram to identify the utility maximizing equilibrium. (Marks - 5)</h4><p>Solution Video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwT55hO2h9FaNk8FAQGEOyUIg9wzrCwbxsFS1izcjBezXHlvLu_2HlnMW3i2ofxUaMbWe9HlLkug6HP5TKzjA' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/> <h4>Q. No. 1 (b) - The cost-minimizing demand for labour is: $L = \frac{Q}{{50}}\sqrt {\frac{r}{w}}$ and that for capital is: $K = \frac{Q}{{50}}\sqrt {\frac{w}{r}}$ where w and r denote wage and prices of capital respectively. Find the production function. </h4><p>Let: $v = \sqrt {\frac{r}{w}}$ Now, the equations in the questions can be written as: $L = \frac{{Qv}}{{50}}$ $v = \frac{{50L}}{Q}$ and $K = \frac{Q}{{50v}}$ $v = \frac{Q}{{50K}}$ Equating results derived above: $\frac{Q}{{50K}} = \frac{{50L}}{Q}$ ${Q^2} = 2500LK$ $Q = \sqrt {2500LK}$ $Q = 50\sqrt {LK}$ $Q = 50{L^{\frac{1}{2}}}{K^{\frac{1}{2}}}$ This is the production function we want know. </p><hr/> <h4>Q. No. 1 (c) - Explain the principle of average cost pricing in the contect of natural monopoly.</h4><p>See Chapter - 7, Section - <b>VI. Government-Regulated Monopoly</b>, P. No. - 200, Microeconomics by A. Koutsoyinnis. The book is available in the Google Drive Folder.</p><hr/> <h4>Q. No. 1 (d) - Find the monopolist's demand function when labour market is perfectly competitive.</h4><p>See P. No. 456 Section: <b>(c) The market demand for and supply of labour</b>, Microeconomics by A. Koutsoyinnis. You can use the figure given therein.</p><hr/> <h4>Q. No. 1 (e) - Explain the concept of external economies in context of marginal social benefits and marginal social costs.</h4><p>External economies (positive externalities) occurs when marginal social costs are less than marginal social benefits. For example, vaccination save an individual from viruses but a vaccinated person breaks the chain of infection so as others are also safe from her. The cost of vaccine is marinal social cost and also private cost here. Safety from viruses is private benefit for which one pays for. Breaking the chain of infection is social benefit which is additional benefit for the society.</p><hr/> <h4>Q. No. 1 (f) - Suppose that Leontief input-output coefficient matrix is: $A = \left[ {\begin{array}{*{20}{c}}{0.1}&{0.4}\\{0.2}&{0.5}\end{array}} \right]$ and the final demand vector is: $\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]$ Find the total direct and indirect requirement of the second input to satisfy the final demand. </h4><p>Let requirements of inputs be $${x_1^*}$$ and $${x_2^*}$$: ${x^*} = {\left( {I - A} \right)^{ - 1}}d$ $\left[ {\begin{array}{*{20}{c}}{x_1^*}\\{x_2^*}\end{array}} \right] = {\left[ {\begin{array}{*{20}{c}}{1 - 0.1}&{0 - 0.4}\\{0 - 0.2}&{1 - 0.5}\end{array}} \right]^{ - 1}}\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}}{x_1^*}\\{x_2^*}\end{array}} \right] = {\left[ {\begin{array}{*{20}{c}}{0.9}&{ - 0.4}\\{ - 0.2}&{0.5}\end{array}} \right]^{ - 1}}\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}}{x_1^*}\\{x_2^*}\end{array}} \right] = \frac{1}{{0.37}}\left[ {\begin{array}{*{20}{c}}{0.5}&{0.4}\\{0.2}&{0.9}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}}{x_1^*}\\{x_2^*}\end{array}} \right] = \frac{1}{{0.37}}\left[ {\begin{array}{*{20}{c}}{0.5(1) + 0.4(1)}\\{0.2(1) + 0.9(1)}\end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}}{x_1^*}\\{x_2^*}\end{array}} \right] = \frac{1}{{0.37}}\left[ {\begin{array}{*{20}{c}}{0.9}\\{1.1}\end{array}} \right]$ $x_2^* = \frac{{1.1}}{{0.37}} = 2.97$ </p><hr/> <h4>Q. No. 1 (g) - Show that in the regression model $${Y_i} = \alpha + \beta {X_i} + {U_i};i = 1,2,...,n$$, the covariance between the regressor and the error term is zero under ordinary least squares method of estimation. </h4><p>${\mathop{\rm cov}} \left( {{X_i},{U_i}} \right) = E\left\{ {\left[ {{X_i} - E\left( {{X_i}} \right)} \right]\left[ {{U_i} - E\left( {{U_i}} \right)} \right]} \right\}$ By assumption of OLS: $E\left( {{U_i}} \right) = 0$ Now, we can write: ${\mathop{\rm cov}} \left( {{X_i},{U_i}} \right) = E\left\{ {\left[ {{X_i} - E\left( {{X_i}} \right)} \right]{U_i}} \right\}$ ${\mathop{\rm cov}} \left( {{X_i},{U_i}} \right) = E\left[ {{X_i}{U_i} - E\left( {{X_i}} \right){U_i}} \right]$ ${\mathop{\rm cov}} \left( {{X_i},{U_i}} \right) = E\left( {{X_i}{U_i}} \right) - E\left( {{X_i}} \right)E\left( {{U_i}} \right)$ ${\mathop{\rm cov}} \left( {{X_i},{U_i}} \right) = {X_i}E\left( {{U_i}} \right)$ Since X<sub>i</sub> are fixed values: ${\mathop{\rm cov}} \left( {{X_i},{U_i}} \right) = {X_i}E\left( {{U_i}} \right)$ ${\mathop{\rm cov}} \left( {{X_i},{U_i}} \right) = 0$ </p><hr/> <h4>Q. No. 2 (a) - An individual has the ulility function <strong>U = XY</strong> and her budget equation is <strong>10X + 10Y = 1000</strong>. Find the maximum utility that she can attain. (Marks - 6)</h4><p>Solution Video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dw3Z1MxVHnp_ia3GDMb8yT12Mc3UC-5lCJzMUYgq5ppFrXvbn1UL--l2PrGUhVIoOinn_6TJKrUA3UCt9va-w' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/> <h4>Q. No. 2 (b) - If the price of good X decreases to 5 find the compensating variation in income in order to maintain her level of satisfaction in part (a) </h4><p>Compensating variation in income refers how much income the consumer shhould be given after price change so that she can achieve the level of satisfaction without price change. This can be both negative or positive. We need to solve dual of the problem in part (a), that is, expenditure minimization when utility level will be fixed at what we calculated in part (a). Problem can be mathematically stated as follows: Minimize: $I = 5X + 10Y$ Subject to: $2500 = XY$ Taking Lagrangian: $L = 5X + 10Y + \lambda \left( {2500 - XY} \right)$ Applying the first order condition: $\frac{{\partial L}}{{\partial X}} = 0$ $5 - \lambda Y = 0$ $\lambda = \frac{5}{Y}$ and $\frac{{\partial L}}{{\partial Y}} = 0$ $10 - \lambda X = 0$ $\lambda = \frac{{10}}{X}$ Equating both results: $\frac{5}{Y} = \frac{{10}}{X}$ $\frac{X}{Y} = 2$ $X = 2Y$ Substituting this value of X in the constraint: $2500 = \left( {2Y} \right)Y$ $2500 = 2{Y^2}$ $Y = \sqrt {\frac{{2500}}{2}}$ $Y = 25\sqrt 2$ Again substituting this value of Y in the constraint: $2500 = X\left( {25\sqrt 2 } \right)$ $X = \frac{{2500}}{{25\sqrt 2 }}$ $X = 50\sqrt 2$ Substituting both values in the objective function: $I = 5\left( {50\sqrt 2 } \right) + 10\left( {25\sqrt 2 } \right)$ $I = 250\sqrt 2 + 250\sqrt 2$ $I = 500\sqrt 2$ $I = 707(aprrox.)$ Consumer needed ₹1000 to obtain 2500 units utility before decrese in the price. Now, she needs ₹707 (approx.). Therefore, compensating variation (CV) is: $CV = 1000 - 707 = 293$ </p><hr/> <h4>Q. No. 2 (c) - An individual buys two goods X and Y at prices P<sub>X</sub> and P<sub>Y</sub>. Check whether her behaviour satidfies the Weak Axiom of Revealed Prference, given the following information: When (P<sub>X</sub>, P<sub>Y</sub>) = (1, 2), (X, Y) = (1, 2)<br/>When (P<sub>X</sub>, P<sub>Y</sub>) = (2, 1), (X, Y) = (2, 1) </h4><p>See Chapter - 7, Section - <b>7.5 Checking for WARP</b>, P. No. - 125 in Intermediate Microeconomics by Hal & Varian. Same problem has been explained there. The book is available in the Google Drive Folder.</p><p>The consumer doesn't satifies the weak axiom of revealed preference because when she revealed prefered the first bundel at intial price second bundle was also affordable. After change in price, the first bundle is still affordable so she should choose first bundle even after price change to be consistent with weak axiom of revealed preference. </p><hr/> <h4>Q. No. 3 (a) - Show that $$q = \gamma {\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){K^{ - \alpha }}} \right]^{ - \frac{1}{\alpha }}}$$ is a production function that represents the average of two inputs L and K for different values of $$\alpha$$, given that $$\gamma \prec 0$$ and $$0 \prec \delta \prec 1$$. </h4><p><strong>(Note - This question is quite confusing. I seems that want to ask elasticity of substitution $$\left( \sigma \right)$$ for different value of $$\alpha$$.)</strong></p> <p>The formula for elasticity of substitution is: $\sigma = \frac{{d\left( {\frac{K}{L}} \right)}}{{\frac{K}{L}}} \div \frac{{d\left( {\frac{w}{r}} \right)}}{{\frac{w}{r}}}$ $\sigma = \frac{{d\left( {\frac{K}{L}} \right)}}{{d\left( {\frac{w}{r}} \right)}} \times \frac{{\frac{w}{r}}}{{\frac{K}{L}}}$ where w and r are wage and rent respectively. At level of equilibrium: $MRT{S_{L,K}} = \frac{w}{r}$ $\frac{{M{P_L}}}{{M{P_K}}} = \frac{w}{r}$ $\frac{{\frac{{\partial q}}{{\partial L}}}}{{\frac{{\partial q}}{{\partial K}}}} = \frac{w}{r}$ Where, $\frac{{\partial q}}{{\partial L}} = - \frac{1}{\alpha }\gamma {\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){L^{ - \alpha }}} \right]^{ - \frac{1}{\alpha } - 1}}\left( { - \alpha \delta {L^{ - \alpha - 1}}} \right)$ $\frac{{\partial q}}{{\partial L}} = \gamma {\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){L^{ - \alpha }}} \right]^{ - \frac{1}{\alpha } - 1}}\left( {\delta {L^{ - \alpha - 1}}} \right)$ and, $\frac{{\partial q}}{{\partial K}} = - \frac{1}{\alpha }\gamma {\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){K^{ - \alpha }}} \right]^{ - \frac{1}{\alpha } - 1}}\left( { - \alpha \left( {1 - \delta } \right){K^{ - \alpha - 1}}} \right)$ $\frac{{\partial q}}{{\partial K}} = \gamma {\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){K^{ - \alpha }}} \right]^{ - \frac{1}{\alpha } - 1}}\left\{ {\left( {1 - \delta } \right){K^{ - \alpha - 1}}} \right\}$ Now, it can be written: $\frac{{M{P_L}}}{{M{P_K}}} = \frac{{\gamma {{\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){L^{ - \alpha }}} \right]}^{ - \frac{1}{\alpha } - 1}}\left( {\delta {L^{ - \alpha - 1}}} \right)}}{{\gamma {{\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){K^{ - \alpha }}} \right]}^{ - \frac{1}{\alpha } - 1}}\left\{ {\left( {1 - \delta } \right){K^{ - \alpha - 1}}} \right\}}}$ $\frac{{M{P_L}}}{{M{P_K}}} = \frac{{\delta {L^{ - \alpha - 1}}}}{{\left( {1 - \delta } \right){K^{ - \alpha - 1}}}}$ $\frac{{M{P_L}}}{{M{P_K}}} = \frac{\delta }{{1 - \delta }} \cdot {\left( {\frac{K}{L}} \right)^{1 + \alpha }}$ Now, equilibrium condition is: $\frac{\delta }{{1 - \delta }} \cdot {\left( {\frac{K}{L}} \right)^{1 + \alpha }} = \frac{w}{r}$ $\frac{K}{L} = {\left( {\frac{{1 - \delta }}{\delta }} \right)^{\frac{1}{{1 + \alpha }}}} \cdot {\left( {\frac{w}{r}} \right)^{\frac{1}{{1 + \alpha }}}}$ Taking derivating with repect to input price ratio: $\frac{{d\left( {\frac{K}{L}} \right)}}{{d\left( {\frac{w}{r}} \right)}} = \frac{1}{{1 + \alpha }} \cdot {\left( {\frac{{1 - \delta }}{\delta }} \right)^{\frac{1}{{1 + \alpha }}}} \cdot {\left( {\frac{w}{r}} \right)^{\frac{1}{{1 + \alpha }} - 1}}$ Now, elasticity of substitution is: $\sigma = \frac{1}{{1 + \alpha }} \cdot {\left( {\frac{{1 - \delta }}{\delta }} \right)^{\frac{1}{{1 + \alpha }}}}{\left( {\frac{w}{r}} \right)^{\frac{1}{{1 + \alpha }} - 1}} \times \frac{{\frac{w}{r}}}{{{{\left( {\frac{{1 - \delta }}{\delta }} \right)}^{\frac{1}{{1 + \alpha }}}} \cdot {{\left( {\frac{w}{r}} \right)}^{\frac{1}{{1 + \alpha }}}}}}$ $\sigma = \frac{1}{{1 + \alpha }}$ </p> <p>$$\sigma$$ is constant and its magnitude depends on the value of $$\alpha$$</p> <p><i><b>Reference: See page no. 396 to 399 of Mathematical Economics book by A. C. Chiang available in the google drive folder.</b></i></p><hr/> <h4>Q. No. 3 (b) - Find the marginal rate of technical substitution for the production function given in part (a).</h4><p>It has already been calculated in the answer of part (a). $MRT{S_{L,K}} = \frac{{M{P_L}}}{{M{P_K}}}$ $MRT{S_{L,K}} = \frac{{\frac{{\partial q}}{{\partial L}}}}{{\frac{{\partial q}}{{\partial K}}}}$ where, $\frac{{\partial q}}{{\partial L}} = - \frac{1}{\alpha }\gamma {\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){L^{ - \alpha }}} \right]^{ - \frac{1}{\alpha } - 1}}\left( { - \alpha \delta {L^{ - \alpha - 1}}} \right)$ $\frac{{\partial q}}{{\partial L}} = \gamma {\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){L^{ - \alpha }}} \right]^{ - \frac{1}{\alpha } - 1}}\left( {\delta {L^{ - \alpha - 1}}} \right)$ and $\frac{{\partial q}}{{\partial K}} = - \frac{1}{\alpha }\gamma {\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){K^{ - \alpha }}} \right]^{ - \frac{1}{\alpha } - 1}}\left( { - \alpha \left( {1 - \delta } \right){K^{ - \alpha - 1}}} \right)$ $\frac{{\partial q}}{{\partial K}} = \gamma {\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){K^{ - \alpha }}} \right]^{ - \frac{1}{\alpha } - 1}}\left\{ {\left( {1 - \delta } \right){K^{ - \alpha - 1}}} \right\}$ Now, we can write: $MRT{S_{L,K}} = \frac{{\gamma {{\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){L^{ - \alpha }}} \right]}^{ - \frac{1}{\alpha } - 1}}\left( {\delta {L^{ - \alpha - 1}}} \right)}}{{\gamma {{\left[ {\delta {L^{ - \alpha }} + \left( {1 - \delta } \right){K^{ - \alpha }}} \right]}^{ - \frac{1}{\alpha } - 1}}\left\{ {\left( {1 - \delta } \right){K^{ - \alpha - 1}}} \right\}}}$ $MRT{S_{L,K}} = \frac{{\delta {L^{ - \alpha - 1}}}}{{\left( {1 - \delta } \right){K^{ - \alpha - 1}}}}$ $MRT{S_{L,K}} = \frac{\delta }{{1 - \delta }} \cdot {\left( {\frac{K}{L}} \right)^{1 + \alpha }}$ </p><hr/> <h4>Q. No. 4 (a) - A firm with market power faces the demand curve given by: $P = 100 - 3Q + 4\sqrt A$ where P, Q and A denote price, quantity and expenditure on advertising respectively. The total cost is given as: $C = 4{Q^2} + 10Q + A$ Find the firm's profit-maximizing price. (Marks - 8) </h4><p>$TR = PQ$ $TR = Q\left( {100 - 3Q + 4\sqrt A } \right)$ $TR = 100Q - 3{Q^2} + 4Q\sqrt A$ It is known that: $\Pr ofit\left( \pi \right) = TR - C$ $\pi = 100Q - 3{Q^2} + 4Q\sqrt A - \left( {4{Q^2} + 10Q + A} \right)$ $\pi = 90Q - 7{Q^2} + 4Q\sqrt A - A$ Applying the first order condition: $\frac{{\partial \pi }}{{\partial Q}} = 0$ $90 - 14Q + 4\sqrt A = 0$ and $\frac{{\partial \pi }}{{\partial A}} = 0$ $\frac{{2Q}}{{\sqrt A }} - 1 = 0$ $\frac{{2Q}}{{\sqrt A }} = 1$ $\sqrt A = 2Q$ $A = {\left( {2Q} \right)^2}$ $A = 4{Q^2}$ Substituting the value of A : $90 - 14Q + 4\sqrt A = 0$ $90 - 14Q + 4\sqrt {4{Q^2}} = 0$ $90 - 14Q + 8Q = 0$ $6Q = 90$ $Q = \frac{{90}}{6}$ $Q = 15$ Substituting the value of Q: $A = 4{Q^2}$ $A = 4{\left( {15} \right)^2}$ $A = 900$ Substituting the value of Q and A in the inverse demand function: $P = 100 - 3Q + 4\sqrt A$ $P = 100 - 3 \times 15 + 4\sqrt {900}$ $P = 175$ </p><hr/> <h4>Q. No. 4 (b) - Suppose that the demand and supply function in a market are given as: ${Q_D} = 100 - P$ $and$ ${Q_S} = 200 - 5P$ Analyse whether the equilibrium is Walrasian stable or Marshallian stable. (Marks -10) </h4><p>It can easility be obseved from the equations that both demand and supply are negatively sloped. Taking Q on vertical axis and P on horizontal axis, it can also be observed that absolute slope of supply curve is 5 and that of the demand curve is 1. As a thumb rule, equilibrium is <b>Walrasian unstable </b> but <b>Marshallian stable</b>if absolute slope of supply curve is greater than the absolute slope of demand curve when both demand and supply curve are negatively sloped (Q should be on the vertical axis and P on horizontal axis). </p><p>Equilibrium price and quantity for a market is as follows: $100 - P = 200 - 5P$ $5P - P = 200 - 100$ $P = 25$ Substituting this value in either demand or supply equation, we will get: ${Q_D} = {Q_S} = 75$ </p><p><b>Walrasian Stability Analysis:</b> In the Walrasian system, if there is excess demand then price increses. As a result of increase in price, demands falls and supply increases. This simultaneous process brings equilibrium in the market price. We can analyse the equations in the same framework. Let price be 30. At this price, demand and quantity will be: ${Q_D} = 100 - 30 = 70$ ${Q_S} = 200 - 5 \times 30 = 50$ As it can be observed that there is excess demand at price 30. So, the prices should increase but the equilibrium price is 25 so increase in price will result in further disequilibrium. It means that equilibrium cannot be achieved in Walrasian syatem for this model. In other words, equilibrium is Walrasian unstable for this model. </p><p><b>Marshallian Stability Analysis:</b> In the Marshallian system, if demand price is greater than supply price for a given quantity then the sopplier will increase the supply. Due to incresed supply prices will fall. As a result of fall in prices, demand will increse. This will bring equilibrium in the market. Let quantity be 70. At this quantity level, demand price is: $70 = 100 - P$ $P = 30$ and the supply price is: $70 = 200 - 5P$ $P = \frac{{200 - 70}}{5}$ $P = 26$ It can be observed that the demand price is greater than supply price. So, supply should increase. As we have calculated, if supply is incresed to 75, market will be at the equilibrium level with price 25. It means equilibrium can be achieved in Marshallian system for this model. In other words, this equilibrium is Marshallian stable. </p><p>Also See: <a href="https://www.microeconomicsnotes.com/price-quantity-solutions/stability-of-the-equilibrium-2-approaches-microeconomics/14576" target="_blank">Stability of Equilibrium</a></p><hr/> <h4> Q. No. 5 (a) and (b) [See the question paper from Google Drive Folder]</h4><p> See P. No. - 458, Section: <b>Model B. The firm bas monopolistic power in the commodity market and monopsonistic power in the factor market</b>, Microeconomics by A. Koutsoyinnis. the book is available in the Google Drive Folder. Both question has been answered in this section.</p><hr/> <h4>Q. No. 6 (a) [See the question paper from Google Drive Folder]</h4><p>Following Section of Microeconomics by A Koutsoyiannis ( P. N0. - 497) exactly explains the answer.</p><ul> <li>2. STATIC PROPERTIES OF A GENERAL EQUILIBRIUM STATE (CONFIGURATION)</li> <ul> <li>(a) Equilibrium of productioo (efficiency in factor substitution)</li> <li>(b) EquiHbrium of consumption (efficiency in distribution of commodities): You can also call it. Equilibrium in exchange</li> <li>(c) Simultaneous equilibrium of production and consumption (efficiency in product-mix)</li> </ul></ul><hr/> <h4>Q. No. 6 (b) [See the question paper from Google Drive Folder]</h4><p> Private Optimal Price and quantity $Q = 24 - P$ $P = 24 - Q$ Private equilibrium condition is: $P = MC$ $24 - Q = 2 + Q$ $2Q = 22$ $Q = \frac{{22}}{2} = 11$ This is the private optimal quantity. Since $P = 24 - Q$ So $P = 24 - 11 = 13$ This is the private optimal price. </p><p>Social OPtimal Quatity and Price: Let Marginal Externality Cost be MEC<sub>E</sub>: $MEC = \frac{d}{{dQ}}\left( { - 2Q + \frac{{{Q^2}}}{2}} \right)$ $MEC = - 2 + Q$ Let Marginal Social Cost be: $MSC = MC + MEC$ $MSC = 2 + Q - 2 + Q$ $MSC = 2Q$ Private equilibrium condition is: $P = MSC$ $13 = 2Q$ $Q = \frac{{13}}{2} = 6.5$ This is the Social Optimal Quantity. Since $P = 24 - Q$ So $P = 24 - 6.5 = 17.5$ This is the social optimal quantity. </p><p>Comparision: Social optimal quantitity is less than private optimal quantity and the social optimal price is greater than private optimal price.</p> <p>For its theoretical explanation, see Microeconomics by A. Koutsoyiannis, P. No. - 542, Section: <b>A. Externalities in production</b></p><hr/> <h4>Q. No. 7 - Consider the following data on two variables:</h4> <table> <tr><th>Y</th><td>-5</td><td>-4</td><td>-3</td><td>-2</td><td>-1</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr> <tr><th>Y</th><td>25</td><td>16</td><td>9</td><td>4</td><td>1</td><td>0</td><td>1</td><td>4</td><td>9</td><td>16</td><td>25</td></tr> </table> <h4>(a) Find the product-moment correlation coefficient between X and Y. (Marks - 9)<br/> (b) Can you develop a suitable linear regression model to explain Y with the help of X? (Marks - 9)</h4><p>Solution Video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dySW25lKlkhQb9qpXTAi7guN6BmmF4Dhq6_KVRiRSdCNsyy88OZviT9kcHF1DUntDcmRPNV7fc6ZI9y14TQ0Q' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/> <h4>Q. No. 8 (a) [See the question paper from Google Drive Folder]</h4><p><a href="https://commons.wikimedia.org/wiki/File:Economics_Gini_coefficient2.svg#/media/File:Economics_Gini_coefficient2.svg"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/5/59/Economics_Gini_coefficient2.svg/320px-Economics_Gini_coefficient2.svg.png" alt="Economics Gini coefficient2.svg"></a><br>By Reidpath - The original file was on WikiMedia Commons (&lt;a class="external free" href="https://en.wikipedia.org/wiki/File:Economics_Gini_coefficient.svg"&gt;http://en.wikipedia.org/wiki/File:Economics_Gini_coefficient.svg&lt;/a&gt;). I have edited the file., Public Domain, <a href="https://commons.wikimedia.org/w/index.php?curid=7114030">Link</a></p><p>Gini Coefficient is the ration of area A and the total area of the rectangle. A is above the Lorez curve and B is below the Lorez curve. Let G represent the Gini coefficient. Now, the formula for Gini coefficient is: $G = \frac{A}{{A + B}}$ It is assumed that A + B = 0.5, that is half of the area of square of area unit 1. So, we can write: $G = \frac{{0.5 - B}}{{0.5}}$ $G = \frac{{0.5}}{{0.5}} - \frac{B}{{0.5}}$ $G = 1 - 2B$ Which is exactly:<br> Gini Coefficient = 1 - 2 $$\times$$ Area below the Lorenz curve. </p><hr/> <h4>Q. No. 8 (b) [See the question paper from Google Drive Folder]</h4><p>If &#x3B8; amount is added to all the person then Gini Coefficient will decrease meaning decline in inequality. &#x3B8; amount will result in greater percentage of increase in lower income group as compared to higher income group. For example, if A earns ₹10,000 and B earns ₹100,000 and both are given addition ₹1000 then this ₹1000 is 10% for A but only 1% for B. </p><hr/> <h4>Q. No. 8 (c) [See the question paper from Google Drive Folder]</h4><p>Calculation of Gini coefficients entails arranging population in accending order in terms of income and see how much percentage of income is received by how much percentage of population. it has nothing to do with the distribution of population. Therefore, Gini Coefficient is distribution insensitive.<br/> (Note: Mathematical proof is very lenghty so there is no possibility that it will be asked in any examination.) </p> <hr/> <h4>Q. No. 9 (a) [See the question paper from Google Drive Folder]</h4><p>See Equation 5.40, Pg. No. - 161 Microeconomics by Nicholson and Snyder. The book is available in Google Drive Folder.<br/>Slutsky Equation in elasticity form: ${e_{x,{p_x}}} = {e_{{x^c},{p_x}}} - {s_x} \cdot {e_{x,I}}$ [Ordinary Demand Elasticity] = [Compensated Demand Elasticity] - [Share of Expenditure on X][Income Elasticity]<br/>Ordinary Demand Elasticity and Compensated Demand Elasticity will have negative sign for Normal Goods. All negative sign can be cancelled after calculation. After cancelling negative sign it can be noticed that Ordinary Demand Elasticity is addition of Compensated Demand Elasticity and Share of Expenditure multiplied by Income elasticity. It simply means that ordinary demand elasticity is greater than compensated demand elasticity. </p><p>For inferior goods, income elasticity of demand has negative sign so it should be subtracted from compensated demand elasticity. It means compensated demand elasticity is greater than ordinary demand elasticity in case of inferior goods. </p><hr/> <h4>Q. No. 9 (b) [See the question paper from Google Drive Folder]</h4><p> $MC = \frac{{d\left( {TC} \right)}}{{dQ}}$ $MC = \frac{d}{{dQ}}\left( {5Q + 20} \right)$ $MC = 5$ Let Marginal Revenue in the first market be MR<sub>1</sub> and in the second market be MR<sub>2</sub>: ${Q_1} = 55 - {P_1}$ ${P_1} = 55 - {Q_1}$ $T{R_1} = {P_1}{Q_1} = 55{Q_1} - Q_1^2$ $\therefore M{R_1} = 55 - 2{Q_1}$ and ${Q_2} = 70 - 2{P_2}$ $2{P_2} = 70 - {Q_2}$ ${P_2} = 35 - \frac{{{Q_2}}}{2}$ $T{R_2} = {P_2}{Q_2} = 35{Q_2} - \frac{{Q_2^2}}{2}$ $\therefore M{R_2} = 35 - {Q_2}$ </p><p>(i) Equilibrium condition in the first market: $M{R_1} = MC$ $55 - 2{Q_1} = 5$ $2{Q_1} = 55 - 5$ ${Q_1} = \frac{{50}}{2}$ ${Q_1} = 25$ This is the equilibrium quantity for the first market.<br/><br/>Equilibrium condition in the second market: $M{R_2} = MC$ $35 - {Q_2} = 5$ ${Q_2} = 35 - 5$ ${Q_2} = 30$ This is the equilibrium quantity for the second market. </p> <p>(ii) Equilibrium price in the first market: ${P_1} = 55 - {Q_1}$ ${P_1} = 55 - 25$ ${P_1} = 30$ This is the equilibrium price for the first market.<br/><br/> Equilibrium price in the second market: ${P_2} = 35 - \frac{{{Q_2}}}{2}$ ${P_2} = 35 - \frac{{30}}{2}$ ${P_2} = 35 - 15 = 20$ This is the equilibrium price for the second market. </p> <p>(iii) Elasticity of demand in the first market at the point of equilibrium: ${e_{{Q_1},{P_1}}} = \frac{{d{Q_1}}}{{d{P_1}}} \times \frac{{{P_1}}}{{{Q_1}}}$ ${e_{{Q_1},{P_1}}} = - 1 \times \frac{{30}}{{25}}$ ${e_{{Q_1},{P_1}}} = - 1.2$ $\left| {{e_{{Q_1},{P_1}}}} \right| = 1.2$ <br/><br/> ${e_{{Q_2},{P_2}}} = \frac{{d{Q_2}}}{{d{P_2}}} \times \frac{{{P_2}}}{{{Q_2}}}$ ${e_{{Q_2},{P_2}}} = - 2 \times \frac{{20}}{{30}}$ ${e_{{Q_2},{P_2}}} = - 1.67$ $\left| {{e_{{Q_2},{P_2}}}} \right| = 1.67$ The monopolist charges higher price in the market where elasticity of demand in absolute term is lower and lower price in the market where elasticity of demand is low. </p><hr/><h4>Q. No. 10 (a) [See the question paper from Google Drive Folder]</h4><p>This question is easy. Google it. <br/>Public goods have two main charateristics - (i) Non-rivalry and (ii) Non-excudability. Non-ravalry means consumption by one individual will not reduce its quantity for others while Non-excludability means one cannot be excluded from consuming it. For example, air is both non-rival and non-excludable in consumption </p> <hr/> <h4>Q. No. 10 (b) [See the question paper from Google Drive Folder]</h4><p>Willingness to pay of individual - 1: ${P_1} = 100 - Q$ Willingness to pay of individual - 1: ${P_2} = 200 - Q$ Total willing to pay after horizontal summation: $WTP = 300 - 2Q$ </p><p> (i) Optimal Provision for public good when the Marginal Cost is 240: $WTP = MC$ $300 - 2Q = 240$ $Q = 30$ When the price is zero, consumption will be 300 units and the provider of public goods, say, government need to spend 300(240) = 72,000. However, government can limit the quantity at 30 units through taxation for which individual 1 is willing to pay 70(30) = 2100 and individual 2 is willing to pay 170(30) = 5100, The cost of government will be 240(30)= 7200 which is exactly equal to 2100 + 5100 = 7200 which two individuals are collectly ready to pay. </p><div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-Ky44Sj6yz2Q/YOBCXwpE6yI/AAAAAAAAk0c/Y8p-HHgiycgstackp2UDlzYBlP5HuRa5QCLcBGAsYHQ/s1280/30.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="720" data-original-width="1280" src="https://1.bp.blogspot.com/-Ky44Sj6yz2Q/YOBCXwpE6yI/AAAAAAAAk0c/Y8p-HHgiycgstackp2UDlzYBlP5HuRa5QCLcBGAsYHQ/s320/30.png"/></a></div><p> (ii) Optimal Provision for public good when the Marginal Cost is 240: $WTP = MC$ $300 - 2Q = 50$ $Q = 125$ Analysise in the same way as above. In this case, individual 1 is not willing to pay anything. Instead, he need a compensation of 25 per unit of over production. </p><div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-Skf4Nz2zTck/YOBC6iLSMVI/AAAAAAAAk0k/jEJA0BE-JOw6Zb_KFKjgqAz6cB9HBD_kACLcBGAsYHQ/s1280/125.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="720" data-original-width="1280" src="https://1.bp.blogspot.com/-Skf4Nz2zTck/YOBC6iLSMVI/AAAAAAAAk0k/jEJA0BE-JOw6Zb_KFKjgqAz6cB9HBD_kACLcBGAsYHQ/s320/125.png"/></a></div><hr/> <h4>Q. No. 10 (c) [See the question paper from Google Drive Folder]</h4><p>You can simply Google the definition of Coase theorem. It's commonly available.</p><p>(i) If the factory has right to pollute then the fisherman has two options:</p><ol><li> He can accept the damage of ₹1 lakh. In this case, he do not need to pay anything. His loss is ₹1 lakh.</li><li> He can pay to the factory ₹2 lakh to construct a pollution treatment plant. Thus, he willl avoid the damage of ₹1 lakh. However, his loss is still ₹1 lakh because he has paid ₹2 lakhs to avoid the damage of ₹1 lakh. </li></ol> <p>(i) If the fisherman has right to clean water then the factory has two options:</p> <ol> <li> Factory can construct the pollution treatment plant so that the fisherman avoid the damage of ₹1lakh. The benefit of the fisherman is ₹1 lakh after construction of the plant. </li> <li> The factory can pay the fisherman ₹2 lakhs and do not construct the pollution treatment plant. Thus, the fisherman will accept the damage of ₹1lakhs because he is being paid ₹2 lakh. His benefit is again ₹1 lakh. </li> </ol> <hr/> <h4>Q. No. 11 (a) - Consider the multiple regression model: ${Y_i} = \alpha + {\beta _2}{X_{2i}} + {\beta _3}{X_{3i}} + {U_i}$ $i = 1,2,...,n$ where U<sub>i</sub>s are independent and normally distributed with mean zero and variance $${\sigma ^2}$$. Also consider the auxiliary regressions: ${X_{2i}} = \hat a + \hat b{X_{3i}} + {{\hat V}_{2i}}$ ${X_{3i}} = \hat c + \hat d{X_{2i}} + {{\hat V}_{3i}}$ where $${{\hat V}_{2}}$$ and $${{\hat V}_{2i}}$$ are error terms. Show that $${{\hat \beta }_2}$$, the ordinary least squares estimate of $${\beta _2}$$, can be interpreted as a simple regression of Y on $${{\hat V}_{2}}$$. (Marks - 8)</h4><p>We can write the first auxiliary regression equation in devition form as follows: ${x_{2i}} = \hat b{x_{3i}} + {{\hat v}_{2i}}$ This equation implies that there exists multicollinearity in the original regression equation because $${X_{2i}}$$ is the function of $${X_{3i}}$$. <br/> The formula for the least square estimator in absense of multicollinearity is given as follows: ${{\hat \beta }_2} = \frac{{{{\sum {{y_i}x} }_{2i}}\sum {x_{3i}^2} - \sum {{y_i}{x_{3i}}} \sum {{x_{2i}}{x_{3i}}} }}{{\sum {x_{2i}^2} \sum {x_{3i}^2} - {{\left( {\sum {{x_{2i}}{x_{3i}}} } \right)}^2}}}$ Now, we can substitute the value of deviation form of multicollinear variable, that is, $${x_{2i}}$$: ${{\hat \beta }_2} = \frac{{\left\{ {\sum {{y_i}\left( {\hat b{x_{3i}} + {{\hat v}_{2i}}} \right)} } \right\}\sum {x_{3i}^2} - \sum {{y_i}{x_{3i}}} \sum {{x_{3i}}\left( {\hat b{x_{3i}} + {{\hat v}_{2i}}} \right)} }}{{{{\sum {\left( {\hat b{x_{3i}} + {{\hat v}_{2i}}} \right)} }^2}\sum {x_{3i}^2} - {{\left\{ {\sum {{x_{3i}}\left( {\hat b{x_{3i}} + {{\hat v}_{2i}}} \right)} } \right\}}^2}}}$ ${{\hat \beta }_2} = \frac{{\hat b\sum {{y_i}{x_{3i}}} \sum {x_{3i}^2} + \sum {{y_i}{{\hat v}_{2i}}} \sum {x_{3i}^2} - \hat b\sum {{y_i}{x_{3i}}} \sum {x_{3i}^2} - \sum {{y_i}{x_{3i}}} \sum {{x_{3i}}{{\hat v}_{2i}}} }}{{{{\left( {\hat b\sum {x_{3i}^2} } \right)}^2} + \sum {\hat v_{2i}^2} \sum {x_{3i}^2} - {{\left( {\hat b\sum {x_{3i}^2} } \right)}^2} - {{\left( {\sum {{x_{3i}}{{\hat v}_{2i}}} } \right)}^2}}}$ By the assumption of no serial correlation or autocorrelation in the auxiliary regressions: $\sum {{x_{3i}}{{\hat V}_{2i}}} = 0$ As a result: $\sum {{y_i}{x_{3i}}} \sum {{x_{3i}}{{\hat v}_{2i}}} = 0$ ${\left( {\sum {{x_{3i}}{{\hat v}_{2i}}} } \right)^2} = 0$ Now, we are remained with: ${{\hat \beta }_2} = \frac{{\hat b\sum {{y_i}{x_{3i}}} \sum {x_{3i}^2} + \sum {{y_i}{{\hat v}_{2i}}} \sum {x_{3i}^2} - \hat b\sum {{y_i}{x_{3i}}} \sum {x_{3i}^2} }}{{{{\left( {\hat b\sum {x_{3i}^2} } \right)}^2} + \sum {\hat v_{2i}^2} \sum {x_{3i}^2} - {{\left( {\hat b\sum {x_{3i}^2} } \right)}^2}}}$ Subtracting similar terms: ${{\hat \beta }_2} = \frac{{\sum {{y_i}{{\hat v}_{2i}}} \sum {x_{3i}^2} }}{{\sum {\hat v_{2i}^2} \sum {x_{3i}^2} }}$ Removing similar terms from nominator and denominator: ${{\hat \beta }_2} = \frac{{\sum {{y_i}{{\hat v}_{2i}}} }}{{\sum {\hat v_{2i}^2} }}$ This is the similar result which is found in simple linear regression model. Therefore, we can say that $${{\hat \beta }_2}$$, the ordinary least squares estimate of $${\beta _2}$$, is a simple regression of Y on $${{\hat V}_{2}}$$. </p><p><i><b>For reference, See Chapter - 10, Section - 10.3 of Basic Econometrics by Gujarati.</b></i></p><hr/> <h4>Q. No. 11 (b) - Consider the regression: ${Y_i} = \gamma + {{\hat \delta }_2}{{\hat V}_{2i}} + {{\hat \delta }_3}{{\hat V}_{3i}} + {W_i}$ where $${{\hat V}_{2}}$$ and $${{\hat V}_{3}}$$ are as defined in part (a). Find the relationship between the ordinary least squares estimate $${{\hat \delta }_2}$$ and the estimate $${{\hat \beta }_2}$$ as defined in part (a). (Marks - 5) </h4><p>$${{\hat \beta }_2}$$ explains regression of Y on only one variable, that is, V<sub>2</sub> while $${{\hat \delta }_2}$$ will explain regression of Y on both V<sub>2</sub> and V<sub>3</sub>. So, the relationship can be explained by specification bias.<br/>One of the normal equation derived from the multiple regression model in this question is: $\sum {{y_i}{{\hat v}_{2i}}} = {{\hat \delta }_2}\sum {\hat v_{2i}^2} + {{\hat \delta }_3}\sum {{{\hat v}_{2i}}{{\hat v}_{3i}}}$ Dividing both sides by $${\sum {\hat v_{2i}^2} }$$, we get: $\frac{{\sum {{y_i}{{\hat v}_{2i}}} }}{{\sum {\hat v_{2i}^2} }} = {{\hat \delta }_2} + {{\hat \delta }_3}\frac{{\sum {{{\hat v}_{2i}}{{\hat v}_{3i}}} }}{{\sum {\hat v_{2i}^2} }}$ We have already calculated that: $\frac{{\sum {{y_i}{{\hat v}_{2i}}} }}{{\sum {\hat v_{2i}^2} }} = {{\hat \beta }_2}$ $$\frac{{\sum {{{\hat v}_{2i}}{{\hat v}_{3i}}} }}{{\sum {\hat v_{2i}^2} }}$$ is simple regression of V<sub>3</sub> on V<sub>2</sub>. Let us denote it by a : $\frac{{\sum {{{\hat v}_{2i}}{{\hat v}_{3i}}} }}{{\sum {\hat v_{2i}^2} }} = a$ Now, we can write: ${{\hat \beta }_2} = {{\hat \delta }_2} + {{\hat \delta }_3} \cdot a$ or ${{\hat \beta }_2} - {{\hat \delta }_2} = {{\hat \delta }_3} \cdot a$ where, $${{\hat \beta }_2} - {{\hat \delta }_2}$$ is the specification bias. This specification bias is due to removal of one variable in $${{\hat \beta }_2}$$. There is no reason to assume that $${{\hat \delta }_3}$$ is zero. a is zero if there is perfect multicollinearity. So, $${{\hat \delta }_3} \cdot a$$ is positive. It implies that $${{\hat \beta }_2} \succ {{\hat \delta }_2}$$ if multicollinearity is imperfect and $${{\hat \beta }_2} = {{\hat \delta }_2}$$ if multicollinearity is perfect. </p><hr/> <h4>Q. No. 11 (c): <br/>(i) - In the regression model of part (a), what will be the variance of ordianary least squares estimate $${{\hat \beta }_2}$$?<br/> (ii) - Develop the F-test statistic for the goodness of fit of the regression model of part (a). </h4><p>(i) Variance in multiple regression setup is: ${\mathop{\rm var}} \left( {{{\hat \beta }_2}} \right) = \sigma _u^2\frac{{\sum {x_{3i}^2} }}{{\sum {x_{2i}^2} \sum {x_{3i}^2} - {{\left( {\sum {{x_{2i}}{x_{3i}}} } \right)}^2}}}$ But we know that the model is multicollinear where, ${x_{2i}} = \hat b{x_{3i}} + {{\hat v}_{2i}}$ So, by substituting the value of above equation we get: ${\mathop{\rm var}} \left( {{{\hat \beta }_2}} \right) = \sigma _u^2\frac{{\sum {x_{3i}^2} }}{{{{\sum {\left( {\hat b{x_{3i}} + {{\hat v}_{2i}}} \right)} }^2}\sum {x_{3i}^2} - {{\left\{ {\sum {{x_{3i}}\left( {\hat b{x_{3i}} + {{\hat v}_{2i}}} \right)} } \right\}}^2}}}$ ${\mathop{\rm var}} \left( {{{\hat \beta }_2}} \right) = \sigma _u^2\frac{{\sum {x_{3i}^2} }}{{{{\left( {\hat b\sum {x_{3i}^2} } \right)}^2} + \sum {\hat v_{2i}^2} \sum {x_{3i}^2} - {{\left( {\hat b\sum {x_{3i}^2} } \right)}^2} - {{\left( {\sum {{x_{3i}}{{\hat v}_{2i}}} } \right)}^2}}}$ ${\mathop{\rm var}} \left( {{{\hat \beta }_2}} \right) = \sigma _u^2\frac{{\sum {x_{3i}^2} }}{{{{\left( {\hat b\sum {x_{3i}^2} } \right)}^2} + \sum {\hat v_{2i}^2} \sum {x_{3i}^2} - {{\left( {\hat b\sum {x_{3i}^2} } \right)}^2}}}$ ${\mathop{\rm var}} \left( {{{\hat \beta }_2}} \right) = \sigma _u^2\frac{{\sum {x_{3i}^2} }}{{\sum {\hat v_{2i}^2} \sum {x_{3i}^2} }}$ ${\mathop{\rm var}} \left( {{{\hat \beta }_2}} \right) = \frac{{\sigma _u^2}}{{\sum {\hat v_{2i}^2} }}$ </p> <p>We have reduced multiple regression in part (a) in a simple linear regression model which can be expressed as follow: ${y_i} = {{\hat \beta }_2}{{\hat v}_{2i}} + {e_i}$ ${y_i} = {{\hat y}_i} + {e_i}$ where, ${{\hat y}_i} = {{\hat \beta }_2}{{\hat v}_{2i}}$ For this equation, the measure of goodness of fit R<sub>2</sub> is given by: ${R^2} = \frac{{\sum {\hat y_i^2} }}{{\sum {y_i^2} }}$ Summing up and squaring the second equation we get, $\sum {y_i^2} = \sum {\hat y_i^2} + \sum {e_i^2}$ Dividing both sides by $$\sum {y_i^2}$$: $1 = \frac{{\sum {\hat y_i^2} }}{{\sum {y_i^2} }} + \frac{{\sum {e_i^2} }}{{\sum {y_i^2} }}$ $1 = {R^2} + \frac{{\sum {e_i^2} }}{{\sum {y_i^2} }}$ $\frac{{\sum {e_i^2} }}{{\sum {y_i^2} }} = 1 - {R^2}$ F-statistic for a two variable regression is: $F = \left( {N - 2} \right)\frac{{\sum {\hat y_i^2} }}{{\sum {e_i^2} }}$ Dividing both nominator and denominator by $${\sum {y_i^2} }$$, $F = \left( {N - 2} \right)\frac{{\frac{{\sum {\hat y_i^2} }}{{\sum {y_i^2} }}}}{{\frac{{\sum {e_i^2} }}{{\sum {y_i^2} }}}}$ $F = \left( {N - 2} \right)\frac{{{R^2}}}{{1 - {R^2}}}$ The is the F-statistic of goodness of fit for the multicollinear regression model of part (a). </p><hr/> <h4>Q. No. 12 (a) - Suppose that the relationship to be estimated is: ${Y_i} = {b_0} + {b_1}{X_{1i}} + {b_2}{X_{2i}} + {U_i}$ and X<sub>1</sub> and X<sub>2</sub> are related with the exact relation ${X_2} = k{X_1}$ where k is the arbitrary constant. Show that the estimates of the coefficients are indeterminate and standard errors of these estimates become infinitely large. (Marks - 10) </h4><p>1. The estimates of the coefficients are indeterminate.<br/>Proof: Let $${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_0}$$, $${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1}$$ and $${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2}$$ be the estimates of $${b_0}$$, $${b_1}$$ and $${b_2}$$ respectively and it is known that th formulae for estimation of coefficients are a follows: ${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_0} = \bar Y - {{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1}{{\bar X}_1} - {{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2}{{\bar X}_2}$ ${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1} = \frac{{\left( {\sum {{x_{1i}}{y_i}} } \right)\left( {\sum {x_{2i}^2} } \right) - \left( {\sum {{x_{2i}}{y_i}} } \right)\left( {\sum {{x_{1i}}{x_{2i}}} } \right)}}{{\left( {\sum {x_{1i}^2} } \right)\left( {\sum {x_{2i}^2} } \right) - {{\left( {\sum {{x_{1i}}{x_{2i}}} } \right)}^2}}}$ ${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2} = \frac{{\left( {\sum {{x_{2i}}{y_i}} } \right)\left( {\sum {x_{1i}^2} } \right) - \left( {\sum {{x_{1i}}{y_i}} } \right)\left( {\sum {{x_{1i}}{x_{2i}}} } \right)}}{{\left( {\sum {x_{1i}^2} } \right)\left( {\sum {x_{2i}^2} } \right) - {{\left( {\sum {{x_{1i}}{x_{2i}}} } \right)}^2}}}$ Substituting kX<sub>1</sub> for X<sub>2</sub> in $${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1}$$, we get: ${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1} = \frac{{\left( {\sum {{x_{1i}}{y_i}} } \right)\left\{ {\sum {{{\left( {k{x_{1i}}} \right)}^2}} } \right\} - \left\{ {\sum {\left( {k{x_{1i}}} \right){y_i}} } \right\}\left\{ {\sum {{x_{1i}}\left( {k{x_{1i}}} \right)} } \right\}}}{{\left( {\sum {x_{1i}^2} } \right)\left\{ {\sum {{{\left( {k{x_{1i}}} \right)}^2}} } \right\} - {{\left\{ {\sum {{x_{1i}}\left( {k{x_{1i}}} \right)} } \right\}}^2}}}$ Taking k<sup>2</sup> as common: ${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1} = \frac{{{k^2}\left( {\sum {{x_{1i}}{y_i}} } \right)\left( {\sum {x_{1i}^2} } \right) - {k^2}\left( {\sum {{x_{1i}}{y_i}} } \right)\left( {\sum {x_{_{1i}}^2} } \right)}}{{{k^2}{{\left( {\sum {x_{1i}^2} } \right)}^2} - {k^2}{{\left( {\sum {x_{_{1i}}^2} } \right)}^2}}}$ ${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1} = \frac{0}{0}$ Simularly, for $${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2}$$, we obtain: ${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2} = \frac{{k\left( {\sum {{x_{1i}}{y_i}} } \right)\left( {\sum {x_{1i}^2} } \right) - k\left( {\sum {{x_{1i}}{y_i}} } \right)\left( {\sum {x_{_{1i}}^2} } \right)}}{{{k^2}{{\left( {\sum {x_{1i}^2} } \right)}^2} - {k^2}{{\left( {\sum {x_{_{1i}}^2} } \right)}^2}}}$ ${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2} = \frac{0}{0}$ Thus, $${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1}$$ and $${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2}$$ are indeterminate. Since $${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_0}$$ depends on the values of $${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1}$$ and $${{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2}$$ , it is also indeterminate. <br/><br/>2. Standard errors of these estimates are infinitely large.<br/>Proof: Variances of the estimates are given as follows: ${\mathop{\rm var}} \left( {{{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1}} \right) = \sigma _u^2\frac{{\sum {x_{2i}^2} }}{{\left( {\sum {x_{1i}^2} } \right)\left( {\sum {x_{2i}^2} } \right) - {{\left( {\sum {{x_{1i}}{x_{2i}}} } \right)}^2}}}$ ${\mathop{\rm var}} \left( {{{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2}} \right) = \sigma _u^2\frac{{\sum {x_{1i}^2} }}{{\left( {\sum {x_{1i}^2} } \right)\left( {\sum {x_{2i}^2} } \right) - {{\left( {\sum {{x_{1i}}{x_{2i}}} } \right)}^2}}}$ Substituting kX<sub>1</sub> for X<sub>2</sub> in $${\mathop{\rm var}} \left( {{{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1}} \right)$$: ${\mathop{\rm var}} \left( {{{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1}} \right) = \sigma _u^2\frac{{{k^2}\sum {x_{_{1i}}^2} }}{{{k^2}{{\left( {\sum {x_{1i}^2} } \right)}^2} - {k^2}{{\left( {\sum {x_{_{1i}}^2} } \right)}^2}}}$ ${\mathop{\rm var}} \left( {{{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_1}} \right) = \frac{{\sigma _u^2\sum {x_{_{1i}}^2} }}{0}$ Similarly for $${\mathop{\rm var}} \left( {{{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2}} \right)$$: ${\mathop{\rm var}} \left( {{{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over b} }_2}} \right) = \frac{{\sigma _u^2\sum {x_{_{1i}}^2} }}{0}$ Any positive number devided by zero is infinitely large. Therefore, variances of both estimates are infinitely large. </p><hr/><h4>Q. No. 12 (b) - Assume three explanatory variables like X<sub>1</sub>, X<sub>2</sub> and X<sub>3</sub>, which are found to be highly collinear. The following three principal components (PC) and their corresponding eigenvalues (&lambda;) are reported as $P{C_1} = {a_{11}}{x_1} + {a_{12}}{x_2} + {a_{13}}{x_3}$ $P{C_2} = {a_{21}}{x_1} + {a_{22}}{x_2} + {a_{23}}{x_3}$ $P{C_3} = {a_{31}}{x_1} + {a_{32}}{x_2} + {a_{33}}{x_3}$ where a<sub>ij</sub> be the factor loading of the ith principal component of jth factor, and &lambda;<sub>1</sub>, &lambda;<sub>2</sub> and &lambda;<sub>3</sub> are the egenvalues of first, second and third principal components respectively. How are factor loadings (a<sub>ij</sub>) related to egenvalues in each PC? Do you think that &lambda;<sub>1</sub> > &lambda;<sub>2</sub> > &lambda;<sub>3</sub>? What is the sum of &lambda;s ? (Marks - 10) </h4><p>Also see Q. No. 13 (b) <a href="https://www.studentsofeconomics.com/2021/06/general-economics-1-2011.html" target="_blank">General Economics - 1 Previous Year Paper Solution 2011</a>.</p><p>Eigenvalue in each PC is the summation of square of factor loading of ith principal component. Here, the three egenvalues are as follows: ${\lambda _1} = \sum\limits_i^3 {a_{1i}^2} = a_{11}^2 + a_{12}^2 + a_{13}^2$ ${\lambda _2} = \sum\limits_i^3 {a_{2i}^2} = a_{21}^2 + a_{22}^2 + a_{23}^2$ ${\lambda _3} = \sum\limits_i^3 {a_{3i}^2} = a_{31}^2 + a_{32}^2 + a_{33}^2$ </p><p>a<sub>ij</sub> are a form of correlation coefficients. They are derived from a correlation table of the set of k variables. The formula is: ${a_{ij}} = \frac{{\sum\limits_j^k {{r_{{x_i}{x_j}}}} }}{{\sqrt {\sum\limits_i^k {\sum\limits_j^k {{r_{{x_i}{x_j}}}} } } }}$ </p><p>By the design of the principal component method, the first principal components has the higher egeinvalue than the second; the second principal component has a higher egenvalue than the third and so on. To be precise, the values of the egenvalues become smaller and smaller for subsequent principal component. </p><p>The of the egenvalues of all principal components is eual to the number of independent variables Xs: $\sum\limits_i^k {{\lambda _i}} = k$ where k is the number of independent variables. </p><hr/><h4>Q. No. 12 (c) - How do you use principal component analysis to tackle the problem of multicollinearity in regression analysis? (Marks - 5)</h4><p>In the principal component analysis, the set of new variable, PC<sub>i</sub> is constructed out of theset of original variables, X<sub>i</sub>. The set of PC<sub>i</sub> are called principal components and they are also linear combination of X<sub>i</sub>. X<sub>i</sub> are followed by new sets of coefficients, a<sub>ij</sub> which are called factor loadings.<br/> Factor loadings are chosen such that the constructed principal components are uncorrelated. Three such principal components has been given in the example. Now, regression of Y on these principal components can be done to find the coefficients of original regression equation. Since principal components have no multicollinearity, the coefficients are also free from the effects of multicollinearity.</p><hr/> <h4>Q. No. 13 (a) [See the question paper from Google Drive Folder]</h4><p>At p = 10: ${Q_1} = 50 - {p_1}$ ${Q_1} = 50 - 10$ ${Q_1} = 40$ At the point of intersection: ${Q_1} = {Q_2} = 40$ ${p_1} = {p_2} = 10$ According to the question at the point of equilibrium: $6{e_{{Q_1},{p_1}}} = {e_{{Q_2},{p_2}}}$ $6 \cdot \frac{{d{Q_1}}}{{d{p_1}}} \cdot \frac{{{p_1}}}{{{Q_1}}} = \frac{{d{Q_2}}}{{d{p_2}}} \cdot \frac{{{p_2}}}{{{Q_2}}}$ $6 \cdot \frac{d}{{d{p_1}}}\left( {50 - {p_1}} \right) \cdot \frac{{10}}{{40}} = \frac{{d{Q_2}}}{{d{p_2}}} \cdot \frac{{10}}{{40}}$ $\frac{{d{Q_2}}}{{d{p_2}}} = - 6$ Takinf anti-derivative: ${Q_2} = \int {\left( { - 6} \right)d} {p_2}$ ${Q_2} = - 6{p_2} + c$ At the point of intersection: $40 = - 6\left( {10} \right) + c$ $c = 60 + 40$ $c = 100$ Therefore, the demand function for Q<sub>2</sub> is: ${Q_2} = 100 - 6{p_2}$ </p><hr/> <h4>Q. No. 13 (b) [See the question paper from Google Drive Folder]</h4><p>$p = {\left( {6 - X} \right)^2}$ $p = 36 - 12X + {X^2}$ $TR = pX = {X^3} - 12{X^2} + 36X$ $MR = 3{X^2} - 24X + 36$ Equilibrium Condition for a monopolist is: $MR = MC$ $3{X^2} - 24X + 36 = 14 + X$ $3{X^2} - 25X + 22 = 0$ $3{X^2} - 3X - 22X + 22 = 0$ $3X\left( {X - 1} \right) - 22\left( {X - 1} \right) = 0$ $\left( {X - 1} \right)\left( {3X - 22} \right) = 0$ $X - 1 = 0$ $X = 1$ and $3X - 22 = 0$ $X = \frac{{22}}{3}$ Both value for X are positive so second derivative test is needed to check for the valid value of x. Here, the second derivative test is to check whether the second derive of profit function is negative or not. second derivative of profit function should be negative because at the maximum point of profit, negative second derivative means profit will decline after maximum. First derivative of profit function is: $\frac{{d\pi }}{{dX}} = MR - MC$ $\frac{{d\pi }}{{dX}} = 3{X^2} - 25X + 22$ Second derivate: $\frac{{{d^2}\pi }}{{d{X^2}}} = 6X - 25$ Value of second derivative for X = 1: $\frac{{{d^2}\pi }}{{d{X^2}}} = 6\left( 1 \right) - 25$ $\frac{{{d^2}\pi }}{{d{X^2}}} = - 19$ <br/>Value of second derivative for X = 22/3: $\frac{{{d^2}\pi }}{{d{X^2}}} = 6 \times \frac{{22}}{3} - 25$ $\frac{{{d^2}\pi }}{{d{X^2}}} = 44 - 25$ $\frac{{{d^2}\pi }}{{d{X^2}}} = 19$ <br/>Since second derivative has minus sign for X = 1, it is a valid value of X because it is the point of maximum. For X = 1, price is: $P = {\left( {6 - X} \right)^2}$ $P = {\left( {6 - 1} \right)^2}$ $P = 25$ Now, Consumer Surplus can be calculated using following formulas: $CS = \int_0^X {f\left( Q \right)dQ} - QP$ $CS = \int_0^1 {{{\left( {6 - X} \right)}^2}dQ} - 1 \times 25$ $CS = \int_0^1 {\left( {36 - 12X + {X^2}} \right)dQ} - 25$ $CS = \left[ {36X - 6{X^2} + \frac{{{X^3}}}{3}} \right]_0^1 - 25$ $CS = 36\left( 1 \right) - 6{\left( 1 \right)^2} + \frac{{{1^3}}}{3} - 25$ $CS = \frac{{16}}{3} = 5.33$ </p><hr/> <h4>Q. No. 13 (c) - A firm is producing two goods A and B. It has two factories that jointly produce the two goods in the following quantities (per hour):</h4> <table> <tr><th></th><th>Factory 1</th><th>Factory 2</th></tr> <tr><th>Good A</th><td>10</td><td>20</td></tr> <tr><th>Good B</th><td>25</td><td>25</td></tr></table> <h4>The firm receives an order of 300 units of A and 500 units of B. The cost of operating the two factories are 10,000 and 8,000 per hour. Formulate the linear programming problem of minimizing the total cost of meeting this order. Also find the minimum cost.</h4><p>Solution Video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dx02RJC2omJEazNlT0meB8g-43RiiuDzn0cxk8Fy9nCfhRmSyBmsILrjoeBXg9fGr9eJQj63iUBoS_4ofuLXg' class='b-hbp-video b-uploaded' frameborder='0' /></div>Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com1tag:blogger.com,1999:blog-1368387920159069514.post-52108575510387019282021-06-14T13:30:00.081+05:302022-06-23T21:34:40.462+05:30Previous Year Paper Solution | Indian Economic Service Exam 2018 | General Economics - I<h4>Q. No. 1 (a) Economic rent is not earned when the supply of a factor is perfectly elastic. Elucidate. Use a diagram.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1(b) Show that the elasticity of substitution is contant in a Cobb-Douglas production function. Find its value and interpret.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (c) - Consider the optimization problem:<br/> Maximize u(x<sub>1</sub>, x<sub>2</sub>)<br/> subject to M = p<sub>1</sub>x<sub>1</sub> + p<sub>2</sub>x<sub>2</sub><br/> where, M, p<sub>1</sub> and p<sub>2</sub> are positive constants.<br/> Write down the . NLangrangian for this problem and explain why you need to assume that an interior solution exists before using the Langrangian method to solve the problem. </h4> <p>Lagrangian for this problem can be written as follows: $L = u\left( {{x_1},{x_2}} \right) + \lambda \left( {M - {p_1}{x_1} - {p_2}{x_2}} \right)$ Our aim is to find a unique value value of variables so that we can get calculate the optimal value of the function. So, it is important to assume that Lagrangian solution exist because we cannot find a unique value for the concerned variable if the solution doesn't exist. </p> <p>Solution video:</p> <div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dw4Zb8a9Oy6RKAuakRGoP1GFlRjv6M0Mx1DCI7p_lypnTS081y_UEyjGs2Cv1JrHJ1FSzAV2-rgMUC2xODxnQ' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/><h4>Q. 1(d) - An economy has 10 slave owners and 500 slaves. Slave owners like having slaves more than not having slaves, annd slaves would rather be free than remain as slaves. Explain why the institution of slavery is Pareto optimal in this case.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 1(e) Explain with a diagram why the compensated demand curve is vertical if the consumer's utiliy function is of the form <br/> v(x,y) = min[x,y]</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 1(f) Pharmacies often give senior citizens discounts on medicines. Explain why this may be profit maximizin behaviour as opposed to pure generosity on the part of the phrmacy owners.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 1(g) Suppose that the Government as a monopoly firm produces electricity and sells it to the people at a price p per unit. The demand (q) function for electricity is $$q = \alpha {p^{ - \beta }}$$. If the price elasticity of demand fr electricity in an absolute sense is found to be 0.894, should the Government reduce the price per unit to increase the revenue? Justify your answer.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 2 - A price taking consumer consumes two goods X and Y. Let x and y denote the quantities of goods X and Y respectively, and let P<sub>X</sub> and P<sub>Y</sub> be the respective prices of the two goods. Assume that:<br/> (i) the consumer's budget is given by M, &#x221E; > M > 0; and <br/> (ii) P<sub>X</sub> and P<sub>Y</sub> are finite and positive.<br/> (a) Let the consumer's utility function be given by $U(x,y) = \min [x,y]$ Define Indirect Utility Function and derive this consumer's Indorect Utility Function. (Marks 10) <br/> (b) Suppose instead that his utility function is given by $U(x,y) = xy$ Define expenditure function and derive this consumer's compensated demand for good X using his expenditure function (Marks 10). </h4> <p>Solution Video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwDDCSpU9gaMYAzMePxNR68JO5mbeLntj3YaMzncq7OIcGYJtoT5lHDx0fxr01uyAUPzgA3sgB33_xidecCHg' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/><h4>Q. No. 3 - Consider a one shot simultaneous move game with two players, player 1 and Player 2. Let s<sub>i</sub>, i = 1, 2 designate a pure strategy of player i. Let s<sub>i</sub> &#8800; 0 be the pure strategy set of player i, &#x213C;<sub>i</sub>(s<sub>1</sub>, s<sub>2</sub>) be the payoff function for player i, i = 1,2.</h4><h4>(a) Define the Nash Equilibrium in pure strategies for this game.</h4><h4>Consider the following game:</h4><table><tbody> <tr><th colspan="2" rowspan="2"></th> <th colspan="2">Player 2</th></tr> <tr><td>$$\mathop s\nolimits_2^1$$</td><td>$$\mathop s\nolimits_2^2$$</td></tr> <tr><th rowspan="2">Player 1</th><td>$$\mathop s\nolimits_1^1$$</td> <td>10, 10</td><td>0, 12</td></tr> <tr><td>$$\mathop s\nolimits_2^1$$</td> <td>12, 0</td><td>3, 3</td></tr></tbody></table><h4>Show that the unique pure strategy Nash Equilibrium is not Pareto Optimal.</h4><h4>(c) Consider two firms - Firm 1 and Firm 2 producing a homogeneous good Q. The output of the two firms is given by q<sub>1</sub> and q<sub>2</sub> respectively. The market inverse demand cureve is given by: $P = A - bq$ Where A > 0, P is the price of the good and $q = {q_1} + {q_2}$ Suppose that there is no fixed cost and the average cost for each firm is c, &#x221E; > c > 0. Find the unique pure strategy Nash Equilibrium for this game.</h4><p>(a) Let Player 1 plays strategy $${s_1^*}$$ and Player 2 plays strategy $${s_2^*}$$ in pure strategy. In this two player game, $$\left( {s_1^*,s_2^*} \right)$$ is Nash Equilibrium if $${s_1^*}$$ and $${s_2^*}$$ are mutual best response against each other: Symbolically, ${\pi _1}\left( {s_1^*,s_2^*} \right) \ge {\pi _1}\left( {{s_1},s_2^*} \right)$ and ${\pi _2}\left( {s_1^*,s_2^*} \right) \ge {\pi _2}\left( {s_1^*,{s_2}} \right)$ where, for all $${s_1} \in {S_1}$$ and $${s_2} \in {S_2}$$</p><p>(b) Here, the unique pure strategy Nash Equilibium is (3, 3) but the Pareto optimal strategy is (10, 10) because it can be achieved without making each other worse off from the Nash Equilibrium payoff profile (3, 3) if both player cooperate. It is evident that the Pareto Optimal strategy, that is, (10,10) is not the Nash Equilibrium strategy.</p><p>(c) Pure strategy Nash equilibrium in a dupoly market means that both firm will compete to maximize their profit but price is not the function of their quantity produced by each firm seperately. Price is the function of total quantity produced by both firms because both firms are selling homogeneous product. The profit maximization rule is as usual.</p><p>Let profit function of Firm 1 be as follows: ${\pi _1} = P{q_1} - c{q_1}$ Substituting the value of P (inverse demand function): ${\pi _1} = \left( {A - bq} \right){q_1} - c{q_1}$ Substituting the value of q: ${\pi _1} = \left( {A - b{q_1} - b{q_2}} \right){q_1} - c{q_1}$ ${\pi _1} = A{q_1} - bq_1^2 - b{q_1}{q_2} - c{q_1}$ Applying the first order condition for maximization: $\frac{{\partial {\pi _1}}}{{\partial {q_1}}} = 0$ $A - 2b{q_1} - b{q_2} - c = 0$ $2b{q_1} = A - b{q_2} - c$ ${q_1} = \frac{{A - b{q_2} - c}}{{2b}}$ </p><p>Similarly, let profit function of Firm 2 be as follows: ${\pi _2} = P{q_2} - c{q_2}$ Substituting the value of P (inverse demand function): ${\pi _2} = \left( {A - bq} \right){q_2} - c{q_2}$ Substituting the value of q: ${\pi _2} = \left( {A - b{q_1} - b{q_2}} \right){q_2} - c{q_2}$ ${\pi _2} = A{q_2} - b{q_1}{q_2} - bq_2^2 - c{q_2}$ Applying the first order condition for maximization: $\frac{{\partial {\pi _2}}}{{\partial {q_2}}} = 0$ $A - b{q_1} - 2b{q_2} - c = 0$ $2b{q_2} = A - b{q_1} - c$ ${q_2} = \frac{{A - b{q_1} - c}}{{2b}}$ </p><p>Substituting the value of q<sub>2</sub> in q<sub>1</sub> and simplifying: ${q_1} = \frac{A}{{2b}} - \frac{b}{{2b}}\left( {\frac{{A - b{q_1} - c}}{{2b}}} \right) - \frac{c}{{2b}}$ ${q_1} = \frac{A}{{2b}} - \frac{A}{{4b}} + \frac{{{q_1}}}{4} + \frac{c}{{4b}} - \frac{c}{{2b}}$ ${q_1} - \frac{{{q_1}}}{4} = \frac{A}{{4b}} - \frac{c}{{4b}}$ $\frac{{3{q_1}}}{4} = \frac{{A - c}}{{4b}}$ ${q_1} = \frac{{A - c}}{{4b}} \times \frac{4}{3}$ ${q_1} = \frac{{A - c}}{{3b}}$ This is the demand function of Firm - 1. </p><p>Similarly, we can find the demand function of Firm - 2 by the same process of substitution, which will be: ${q_2} = \frac{{A - c}}{{3b}}$ </p><p>It is also known that MR = MC at profit maximizing level. Here, MR = P and MC = c. So, we can write P in place of c in both the demand function as follows: ${q_1} = \frac{{A - P}}{{3b}}$ ${q_2} = \frac{{A - P}}{{3b}}$ </p><p>We can derive the market demand function from inverse function as follows: $P = A - bq$ $bq = A - P$ $q = \frac{{A - P}}{b}$ This is market demand function. Observe that this demand function is a part of Firm's demand functions. So, the demand function for both firms can be modified as follows: ${q_1} = \frac{{A - c}}{{3b}} = \frac{q}{3}$ ${q_2} = \frac{{A - c}}{{3b}} = \frac{q}{3}$ It means that each firm sell one-thrid of total output at the market price in a pure strategy Nash Equilibrium.. </p><hr/> <h4>Q. 4 Explain the concept of Social Welfare function. Does perfect competition ensure maximum social welfare? Analyse critically</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 5(a) How is quasi-rent diferent from rent?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 5(b) How can you get the wage offer curve and the supply curve of labour? In a flourishing econom there is every posibility that the labour supply curve will be backward bending. Do you agree? Justify your answer</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 6(a) Show that even if the production function is not linear homogeneous, the expansion path can be a straight line passing through the origin.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 6(b) Do you think that the Cobb-Douglas production function can analyse both the returns to a factor and returns to scale? Explain logically.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 6(c) Show that the concept of marginal product is implicit in the definition of the marginal rate of technical substitution.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 7(a) How is the monopoly power measured? State Lerner's measure of degree of monopoly power. Show that the degree of monopoly power is the inverse of the price elasticity of demand.</h4><p>(Comment for solution.)</p><hr/> <h4>Q.7 (b) A monopoly firm's demand curve is given by $$q = \frac{A}{P}$$, where q is the quantity demanded, P is the price of the good, and A is a positive constant. There are no fixed costs. The average cost curve is given by C(q) = cq, where $$\infty > c > 0$$. Using a diagram, show that this firm does not have a profit maximizing output. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 8 - Heights of fathers (X) and sons (Y) in inches are given in following tables:</h4> <table> <tr><th>X</th><td>65</td><td>66</td><td>67</td><td>67</td><td>68</td><td>69</td><td>70</td><td>72</td></tr> <tr><th>Y</th><td>67</td><td>68</td><td>65</td><td>68</td><td>72</td><td>72</td><td>69</td><td>71</td></tr> </table> <h4>(a) Calculate the correlation coefficient between the heights of fathers and those of sons.<br/> (b) Obtain the equations of lines of regression and the estimate of X for Y = 70.<br/> (c) Given that, X = 4Y + 5 and Y = kX + 4 are the ines of regression of X on Y and Y on X respectively. Show that 0 < 4k < 1. </h4> <p>Solution Video</p> <div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dzIJ27aukALQBc43hbSfDzG-bu5lsIEfFCIkRqP518TWRM-GpetR9a2MwyLT9lDsxNmU6ViMv8BOngfhf3bEA' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 9 (a) - Consider the utility function $U = {x^\alpha }{y^\beta }$ where, $U = {x^\alpha }{y^\beta }$ which is to be maximized subjcect to the budget containt: $m = {p_x}x + {p_y}y$ where m = income (nominal) and p<sub>x</sub> and p<sub>y</sub> are the prices respectively per unit of the goods X and Y.<br/> Derive the demand function for X and Y. Show that these demand functions are homogeneous of degree zero in prices and income. </h4> <p>Note: This question has derivation similar to Q. No. 2 of <a href="https://www.studentsofeconomics.com/2021/06/general-economics-1-2009.html" target="_blank">Previous Year Paper 2009</a>. So, the dervation is written their.</p> <p>Solution Video</p> <div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dyOBabeTnL4t5W8J-j6sRKh5GZCSbsYN6imIwOHk69R1ZIWOnojajypLEC8WlwEOFXodBPD8kxWmshKeVjw-g' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 9 (b) Given the production function of a commodity $$q = 40L + 3{L^2} - \frac{{{L^3}}}{3}$$, where q = output, L = labour inpur. Verify that when the average is maximum, it is equal to marginal product. Plot AP and MP on the graph paper. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9 (c) Assume that there are three sectors. The input coefficient matrix A and the final demand vector d is given as follows: $A = \left[ {\begin{array}{*{20}{c}}{0 \cdot 3}&{0 \cdot 4}&{0 \cdot 2}\\{0 \cdot 2}&0&{0 \cdot 5}\\{0 \cdot 1}&{0 \cdot 3}&{0 \cdot 1}\end{array}} \right]\,and\,d = \left[ {\begin{array}{*{20}{c}}{100}\\{30}\\{30}\end{array}} \right]$ Would the amount of the primary input required be consistent with what is available in the economy?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10 (a) State briefly the asumptions of Kaldor's model of income distribution. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10 (b) What do you mean by 'Widow's cruse'? Distinguish between the two phrases 'saving according to the classes of income' and 'savings according to the income of the classes'.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10 (c) Show that in Kaldor's model of income distribution 'the rate of profit' and 'the share of profit' are uniquely determined at equilibrium.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11 Define Pareto's law of income distribution and state its applications. How is the Pareto distribution related to the Log-normal distribution? For the Pareto distribution, calculate the Lorenz curve and the Gini coefficient. Explain their meanings.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12 (a) The ordinary least squares estimate of $$\beta$$ in the classical linear regression model ${Y_i} = \alpha + \beta {X_i} + {U_i},\,\,{\rm{i}} = 1,2,...,n$ is $$\hat \beta = \sum\limits_{i = 1}^n {{W_i}{Y_i}}$$, where $${W_i} = \frac{{{x_i}}}{{\sum\limits_{i = 1}^n {x_i^2} }}$$ <br/> and $${x_i} = {X_i} - \bar X,\,\,\bar X = \frac{1}{n}\sum\limits_{i = 1}^n {{X_i}} .$$ Show that if $$Var\left( {\hat \beta } \right) = \frac{{\hat \sigma _u^2}}{{\sum\limits_{i = 1}^n {x_i^2} }}$$, no other linear unbiased estimator of $$\beta$$ can be constructed with a smaller variance. (All symbols have their usual meaning) </h4><p>See Appendix - 3A.6 of Basic Econometrics 5th edition by Damodar N. Gujarat and Dawn C. Porter</p><hr/> <h4>Q. No. 12 (b) Consider the regression model ${Y_i} = \alpha + \beta {X_i} + {U_i}$ where Y is the quantity demanded of bread and X is the price of butter, and $${U_i}$$ is a random term that is distributed normally with mean zero and unknown variance $$\sigma _u^2$$. A sample of 20 observations yields the folllowing information: <br/>$$\sum\limits_{i = 1}^{20} {{Y_i}} = 21 \cdot 9$$ <br/>$$\sum\limits_{i = 1}^{20} {{{({Y_i} - \bar Y)}^2}} = 86 \cdot 9$$ <br/>$$\sum\limits_{i = 1}^{20} {{X_i}} = 186 \cdot 2$$ <br/>$$\sum\limits_{i = 1}^{20} {{{({X_i} - \bar X)}^2}} = 215 \cdot 4$$ <br/>$$\sum\limits_{i = 1}^{20} {({X_i} - \bar X)\left( {{Y_i} - \bar Y} \right)} = 106 \cdot 4$$ <br/> (i) Set up the null and alternative hypotheses to test if the price of butter as a determinant of the quantity demanded of bread is significant. <br/>(ii) How would you test your hypotheses? <br/>[Given that $${t_{0 \cdot 05;18}} = 1 \cdot 734,\,\,{t_{0 \cdot 01;18}} = 2 \cdot 552,\,\,{t_{0 \cdot 025;18}} = 2 \cdot 101\,{\rm{ and }}{t_{0 \cdot 005;18}} = 2 \cdot 878,\,{\rm{ where }}0 \cdot 025 = \mathop \smallint \limits_{{t_{0 \cdot 025}}}^\infty \,f(t)dt$$ </h4><p>See Chapter - 5 of Theory of Econometrics 2nd edition by A. Koutsoyiannis</p><hr/> <h4>Q. 13 (a) Define autocorrelation and state what are the possible sources of autocorrelation.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 13 (b) Suppose that the time series data follows the auto-regressive scheme of order one, that is, AR(1). Show that an AR(1) process is simply an $${\rm{MA(}}\infty {\rm{)}}$$ process (that is, moving average scheme of order infinity).</h4><p>(Comment for solution.)</p><hr/> <h4>Q 13 (c) Find the mean and variance if the tinme series data are modell by the process ${Y_t} = a + {Y_{t - 1}} + {\varepsilon _t}$ where $${\varepsilon _t}$$ is a pure white noise. Find out also, the auto-correlation coefficient of $${{\rm{s}}^{th}}$$ order. Interpret your results. How do you test stationarity in this case? </h4><p>(Comment for solution.)</p><hr/>Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com3tag:blogger.com,1999:blog-1368387920159069514.post-23831554009305449572021-06-14T13:29:00.071+05:302022-06-10T19:08:43.030+05:30Previous Year Paper Solution | Indian Economic Service Exam 2016 | General Economics - I <h4>Q. No. 1 (a) - Define substitution effect. Separate income effect from subtitution effect from substitution effect for a fall in the price of a Giffen type good using a suitable diagram.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (b) - Show that if the consumer is free from money illusion, the demand function is homogeneous of degree zero. (Marks - 5) </h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dyN1MwDDgliLxcAcGpCbAxy9GQuDsDQ6MpYsiredFgss3bflvD8lEBGTyWXJF93IvfQ09E2rracYsZpU3MbPQ' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 1 (c) - Given the different views of equity and use utility possibility frontier to show that efficieny does not necessarily imply equity. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (d) - State the assuptions of Classical Linear Regression Model. Why are the regressors (X) assumed to be non-stochastic in repeated samples?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1(e) - For the Cobb-Douglas production function $$Q = A{L^\alpha }{K^\beta }$$ (where symbols have usual meaning), calculate the input elasticities of output and also derive an expression for the expansion path of the firm.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1(f) - Define level of significance. How is this level decided for a given problem? Can we take it as 2% or 6%? Explain.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 2 - Derive the demand functions from the utility function $$U = f({q_1},\,{q_2},...\,{q_n})$$ subject to budget constraint $$y = {p_1}{q_1} + {p_2}{q_2} + ...\, + {p_n}{q_n}$$ and if the demand function for a commodity i (i = 1, 2, ... n) is homogeneous of degree zero in prices and income, then show that the sum of own and cross price elasticities of demand for the commodity equals its income ellasticity of demand with negative sign.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 3 - Show that, "If the second order condition is satisfied, every point of tangency between an isoquant and an isocost line is the solution of both a constrained maximum and a constrained minimum."</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 4 - Distinguish between point estimation and interval estimation of a populaation parameter. State the small sample properties of a good estimator.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5 (a) - Derive the long run supply function under perfect competition when there are external economies or external diseconomies.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5 (b) - Consider an industry representedd by two competitive firms with the total cost functions as follows: ${C_1} = {a_1}q_1^2 + b{q_1}q$ ${C_2} = {a_2}q_2^2 + b{q_2}q$ where $${q_1} + {q_2} = q$$ and $${a_1} > 0,\,\,{a_2} > 0$$. <br/>Derive the aggregate supply function of the industry when there are (i) external economies (b&#60;0), and (ii) external diseconomies (b&#60;0).</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 6 - Consider a duopoly with product differentiation in which the demand and cost functions are $\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,{q_1} = 88 - 4{p_1} + 2{p_2}\\\,\,\,\,\,\,\,\,\,\,\,\,{C_1} = 10{q_1}\\{\rm{and}}\,\,\,{q_2} = 56 + 2{p_1} - 4{p_2}\\\,\,\,\,\,\,\,\,\,\,{C_2} = 8{q_2}\end{array}$for for firm I & II respectively. <br/> Derive the price reaction functions for each firmm on the assumption that each maximises its profits with respect to its own price. Determine the equilibrium values of price, quantity and profit for each firm.</h4><p>(Comment for solution.)</p><hr/> <h4>Q No. 7 - "Pareto optimal allocation is contingent upon the assumption that there are no external effects on consumption and production." Examine what happens if there are external effects. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 8 - What is stationarity in a time series analysis? Show that a random work model is non-stationary. Discuss the Dickey-Fuller test for stationarity.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9 (a) - Distinguish between a cooperative and a non-cooperative game. (Marks - 5)</h4><p>A cooperative game is a game in which competitive players can reach an agreement for competitive behaviour. Here, the game theory focuses on which coalition will be formed from all possible coalition.<br/>While in a non-cooperative game players compete to optimize their payoffs. Here, game theory focuses on finding Nash Equilibrium.</p><h4>Q. No. 9 (b) - In a non-cooperative game, find: <ol> <li>saddle point in a pure strategy game. (Marks - 5)</li> <li>maximim expected payoff in a mixed strategy game (Marks - 5)</li> <li>solution of a sequential game in an 'extensive form' (Marks - 5)</li> </ol></h4><ol><li>Saddle Point in a pure strategy game is the payoff which is both the maximin and minimax of a zero-sum-game. In other words, a game has a saddle point when maximin and minimax are equal.<br/>This video will help you to understand this point well - <a href="https://youtu.be/O7mMb4xX43o" target="_blank">Game Thery Basics - 2: Saddle point</a></li> <li>In the uncertainty model of zer-sum-game of mixed strategy, expected payoffs of all strategies need to be calculated. This converts the payoff matrix into a expcted payoff matrix. <b>Maximum expected payoff</b> is a set of maximum payoffs from each column of the expected payoff matrix. This is used in minimax strategy.</li> <li>There are two methods of solution of sequesntial game in extensive form:<br/> <b>1. Subgame Perfect Equilibrium</b><br/> <b>2. Backward Induction</b><br/> In the sumgame perfect equilibrium, a game can be divided into many sub gamesand each game has an equilibrium decision point.<br/>In the backward induction method,the process of solution starts from the the last point of extensive form.</li></ol><p> For detailed explanation, see Chapter - 8: Microeconomics - Basic Principles and Extensions by Walter Nicholsan and Christopher Snyder</p><hr/> <h4>Q. No. 10 (a) - Define heteroscedasticity.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10 (b) - Explain: <br/> (i) Consequences of heteroscedasticity on OLS estimates <br/> (ii) Detection of heteroscedasticity in a model <br/> (iii) Estimation procedure in the presence of heteroscedasticity</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11 (a) Given the Classical Linear Regression model with usual assumptions ${Y_i} = {\beta _0} + {\beta _1}{X_i} + {U_i}\,\,\,\,\,\,\,\,\,\,\,\,i = 1,\,2,\,...\,\,n$ <br/> (a) Examine the goodness of fit of the model using ANOVA. <br/> (b) If the value of $${\bar R^2}$$ is low, how can it be improved?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12 - Distinguish between basic feasible solution, feasible solution and optimal solution of a Linear Programming Problem (LPP). Solve the following LPP graphocally:<br/> Maximize Y = q<sub>1</sub> + 2q<sub>2</sub><br/> subject to <br/> q<sub>1</sub> + 3q<sub>2</sub> &#x22DC; 18<br/> q<sub>1</sub> + q<sub>2</sub> &#x22DC; 8<br/> 2q<sub>1</sub> + q<sub>2</sub> &#x22DC; 14<br/> q<sub>1</sub>, q<sub>2</sub> &#x22DD; 0 </h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwYk8rXolXRcCLawonDchkhstlsfhP8lZX1v5SFsaAoLHObdqlD91NGLhfUYf6lMO5L_voCp_Gl_LhEOvaIYA' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 13 - Examine the situation of market-equilibrium when; <br/> (a) Supply and demand are not equal at a non-negative price-quantity combination. <br/> (b) Supply and demand are equal at more than one non-negative price-quantity combination.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 14 - How is distributional inequality of various kinds measured with the help of income as a resource? Name some common inequality measures and state their properties.</h4><p>(Comment for solution.)</p><hr/>Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com0tag:blogger.com,1999:blog-1368387920159069514.post-82450933773024084822021-06-14T13:29:00.054+05:302022-06-10T10:41:53.086+05:30Previous Year Paper Solution | Indian Economic Service Exam 2017 | General Economics - I<h4>Q. No. 1 (a) - Why do we need constancy assumption of marginal utility of money in Cardinal Ulitity Analysis? Justify your answer.</h4><p>In the Cardinal Utility Analysis, money is used as a unit of measurement of utility. It is eaasential for a unit of measurement to be constant. For example, the size of a ruler remains constant so that we can compare the length of different objects on the basis of the length of the ruler. If scale would be elastic than it is useless to compare length of two objects measured with that scale. Similarly, if marginal utility of money changes with change in income then it would not be possoble to use it as a unit of measurement.</p><hr/> <h4>Q. No. 1(b) - Define the method of Compensating Variation of Income and the method of Cost Difference. Why is the latter method superior to the former one?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1(c) - Distinguish between laws of variable proportions and laws of returns to scale. Find out the elasticity of substitution in the case of fixed coefficient type production function.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1(d) - Find out the cost elasticity of output at the minimum point of the average cost curve in the short-run.</h4><p>(Comment for solution.)</p><hr/> <h4> Q. No. 1(e) - Define Peak- load pricing. How does it differ from third degree price discrimination? Analyse graphically.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1(f) - Consider the equilibrium of a firm under perfect competition. Find out the condition for normal profit, or supernormal profit or loss (whichever is applicable for the firm) without using the average cost curve. Explain only diagrammatically.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1(g) - Explain the concept of divergence in the context of social and private welfare.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 2(a) - How can you measure the price elasticity of demand at any point on a straight line demand curve?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 2(b) Compare between price elasticity at a given price and also at a given quantity for a set of parallel demand functions. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 3(a) Write down the form of CES production functionn and interpret its parameters. Show that the Cobb-Douglas production function is a special case of CES function. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 3(b) Find out the elasticity of substitution of the CES production function.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 4 What do you mean by price discrimination? Under what circumstances is price discrimination profitable? Trace out the equilibrium situation under price discrimination.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5 State and explain the Kaldor-Hicks compensation priciple. How does Scitovsky provide an improvement of Kaldor-Hicks compennsation principle?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 6 State, prove and give an economic interprtation of Euler's theorem. Show that at the minimum point of the long-run average cost, the total product is exhausted.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 7:<br/> (a) Consider two regression equations of y on x and x on y. Show that the arithmatic mean of two regression coefficients is greater than the correlation coefficient, provided the correlation coefficient, provided the correlation coefficient is positive. Under which condition do the two regression lines coincide? (Marks - 12)<br/> (b) In a bivariate distribution, a researcher gets two lines of regression as $2x - y + 1 = 0$ $3x - 2y + 7 = 0$ Identify the two regression lines and find the mean of x and y. </h4> <p>Solution Video:</p> <div class="separator" style="clear: both; text-align: center;"><object class="BLOG_video_qclass" contentid="7e858e3d68d8db69" width="320" height="266" id="BLOG_video-7e858e3d68d8db69" aria-label="Upload video"></object></div> <hr/> <h4>Q. No. 8 If a single buyer focuses on a single seller, what are the outcomes likely to appear? Do you think that the exploitation of labour will emerge? Justify in favour of your arguments. Find out the equilibrium condition of a firm in the presence of perfect competition in both the prodduct and input market.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9(a) Let the demand function for a good be $q = A{p^\alpha }{y^\beta }$ where q = the quantity demanded, p = the price per unit and y = the income. What do the parameters $$\alpha$$ and $$\beta$$ imply and what is the sum of $$\alpha$$ and $$\beta$$? Interpret your result.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9(b) consider the utility function $$u = \log {x_1} + {x_2}$$ which is to be maximized subject to the budget constraint $$m = {p_1}{x_1} + {p_2}{x_2}$$, where $${p_1}$$ and $${p_2}$$ are the prices per unit of the goods $${x_1}$$ and $${x_2}$$ respectively, and m is the income of the consumer. Derive the demand for $${x_1}$$ and $${x_2}$$ and interpret your results.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9(c) Given the demand function and total ost function of a perfectly competitive firm as p = 32 - X, $$c = {X^2} + 8X + 4$$, p being price, c being cost and X = output. <br/> Find out the output, price, profit and total revenue corresponding to maximization of total profit.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10(a) Describe the Leontief static open input-output model along with its assumptions.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10(b) State the Hawkins-Simon conditions and explain their economic meaning and significance.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10(c) Find out the total demand for industries 1, 2 and 3 if the coefficient matrix A and the final demand vector B are as follows: <br/>$$A = \left( {\begin{array}{*{20}{c}}{0 \cdot 3}&{0 \cdot 4}&{0 \cdot 1}\\{0 \cdot 5}&{0 \cdot 2}&{0 \cdot 6}\\{0 \cdot 1}&{0 \cdot 3}&{0 \cdot 1}\end{array}} \right)\,$$ and $$B = \left( {\begin{array}{*{20}{c}}{20}\\{10}\\{30}\end{array}} \right)$$</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11(a) What do you mean by multicollinearity?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11(b) How does it affect the precision of estimates?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11(c) Consider a simple model ${y_i} = {\beta _2}{x_{2i}} + {\beta _3}{x_{3i}} + {u_i};\,i = 1,\,2,\,...,\,n$ and the variables are in deviation form. The disturbance term $${u_i}$$ satisfies all the classical assumptions. Suppose $${x_{2i}}$$ and $${x_{3i}}$$ are multicollinear. Should you drop either $${x_{2i}}$$ or $${x_{3i}}$$ to have precise estimates of the remainiing parameters? If so, under what condition are you permitted to do so?</h4><p>(Comment for solution.)</p><hr/> <h4> Q. No. 12(a) Explain the meaning of spurious regression.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12(b) How are the values of Durbin-Watson d static and $${R_2}$$ indicative of spurious regression?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12(c) Show that in the case of spurious regression between $${Y_t}$$ and $${X_t}$$, where both $${Y_t}$$ and $${X_t}$$ are generated by random walks, (i) the errors have a permanent effect; (ii)the jvariance of the errors is infinitely large. What should you interpret from your result? </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 13(a) Explain what do you mean by heteroscedasticity.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 13(b) Given the model $${Y_i} = {\beta _1} + {\beta _2}{X_{2i}} + {u_i}$$, where $$E(u_i^2) = {\sigma ^2}X_i^2$$ and i = 1, 2, 3, ..., n, find out the OLS and GLS variance of the regression slope.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 13(c) Show that if $${X_i}$$ takes the values 1, 2, 3, 4, 5, $$Var({\hat \beta _2}) > Var({b_2})$$, where $${\hat \beta _2}$$ is the OLS estimator of $${\beta _2}$$ and $${b_2}$$ is the GLS estimator of $${\beta _2}$$.</h4><p>(Comment for solution.)</p><hr/>Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com0tag:blogger.com,1999:blog-1368387920159069514.post-19954595060236305882021-06-14T13:28:00.027+05:302022-06-23T19:57:21.626+05:30Previous Year Paper Solution | Indian Economic Service Exam 2015 | General Economics - I<h4>Q. No. 1 (a) - State and explain Kaldor - Hicks compensation principle.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (b) - The demand function $${Q_1} = 50 - {P_1}$$ intersects another demand function $${Q_2}$$ at price P = 10. The elastcity of demand for $${Q_2}$$ is six times larger than that of $${Q_1}$$ at that point. Find out the demand function for $${Q_2}$$. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (c) - Suppose the Government as a monopolist firm produces electricity and sells it to people at price per unit 'p'. The demand function for the electricity, of the people is $$q = \alpha {p^{ - \beta }}$$. If the elasticity of demand for electricity in absolute sense is found to be 0.894, should the Government increase the price per unit to increase the revenue? Justify your answer.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (d) - Suppose that there are two goods, namely, chocolate cake and ice cream, such that there might well be some optimal amount of chocolate cake and ice cream that a consumer may want to eat per week. Any less than that amount would make her worse off, but any more than that amount would also make her worse off. Find the shape of the Indifference Curve and justify your answer.</h4><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dxCraprHIP5y4_TvV3i1wQIwuDTpClcxDBkFsPDjVbOECO5_x_Lp70tLGkhHw5zGGEu0oaKJvB8hpHMqO0izA' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/> <h4>Q. No. 1 (e) - Define consumer's and producer's surplus Given the demand function $$P = 113 - {q^2}$$ and the supply function $$P = {(q + 2)^2}$$ under perfect competition, find out the consumers' surplus and produers' surplus. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (f) - Elucidate the statement that no economic rent is earned when the supply of a factor is perfectly elastic.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (g) - Explain the concept of social welfare. Does perfect competition ensure maximum social welfare?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (h) - Show that in a translog production function, elasticity of substitution is not constant.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (i) - Define and distinguish between level of significance and confidence inserval. What do you mean by 'power of the test'?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (j) - Find out the total demand for industries 1, 2, and 3, if the coefficient matrix A and final vector B are given as $A = \left[ {\begin{array}{*{20}{c}}{0 \cdot 3}&{0 \cdot 4}&{0 \cdot 1}\\{0 \cdot 5}&{0 \cdot 2}&{0 \cdot 6}\\{0 \cdot 1}&{0 \cdot 3}&{0 \cdot 1}\end{array}} \right]{\rm{ and }}\,\,\,B = \left[ {\begin{array}{*{20}{c}}{20}\\{10}\\{30}\end{array}} \right]$</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (k) - Explain the distinction between the parametric and non-paraetric tests.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 2 - Consider the utility function as $$U = \sqrt {{q_1}{q_2}}$$, where q<sub>1</sub> and q<sub>2</sub> are two commodities on which the consumer spends his entire income of the month. Let the price per unit of q<sub>1</sub> and q<sub>2</sub> be ₹40 and ₹16 respectively and the monthly income of the consumer be ₹4,000. Find out the optimal quantities of q<sub>1</sub> and q<sub>2</sub>. </h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dz6YXb6G_PmpYLps5jDJbr_U43zKSjxQO9bU6xwM__tDcjcXguBcP0G43KbrRWdbpTnYEjl2LMqCes9xiOcZw' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 3 - Define Linear homogenous production function and give an example. Show that in the case of the linear homogenous production function the expansion path must be a straight line passing through the origin.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 4 - How can you graphically derive the long-run marginal cost curve from the short-run marginal cost curves?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5 - What is meant by excess capacity? Why is it bad? Are there any benefits of the excss capacity associatedd with monopolistic competition?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 6 - How is the monopoly power measured? State Lerner's measure of degree of monopoly power. Show that the degree of monopoly power is te inverse of the price elasticity of demand.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 7 - Derive the long-run supply curve in the constant cost industry under perfect competition. Under what conditions can the long-run supply curve of a competitive industry slope downward?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 8 - Consider the competitive market with excise tax such that the suppler receives the price netted of tax. Answer the following questions. <br/> (i) What is the equilibrium price in the presence of tax? <br/> (ii) Under which condition will the price be undefined? <br/> (iii) Show that the market price is totally unaffected in the case of perfectly inelastic supply curve. <br/> (iv) If the tax yield (T) is a fraction (t > 0, which is the rate of tax) of quantity (q), find out the tax yield and the conditions under which tax yield varies directly with the rate of tax (t). <br/> (v) Find out the value of the rate of tax such that the tax yield is maximum.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9 - What do you mean by collusive oligopoly? Distinguish between cartel and price-leadership with respect to the determination of price and quantity. Write a note on barometric price-leadership model.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10 - Consider a simple model of classical regression as $${Y_i} = \beta {X_i} + {u_i},$$ where $${u_i}$$ stands for random disturbance term with the standard assumptions and $${u_i} \sim N(0,\,\,{\sigma ^2})$$, and $${X_i}$$ is non-stochastic and i = 1, 2,..., n. <br/> (a) Find out the OLS estimator for $$\beta$$, say $${\hat \beta _{OLS}}$$. <br/> (b) Show that the OLS estimator for $$\beta$$ is BLUE. Prove ab-initio. <br/> (c) Prove that $$\bar \beta = \frac{{\bar Y}}{{\bar X}}$$, where $$\bar Y$$ and $$\bar X$$ are means respectively, is unbiases but less efficient estimator of $$\beta$$ than $${\hat \beta _{OLS}}$$.</h4><p>See Section 6.1 of Chapter - 6 and Appendix - 6A of Basic Econometrics 5th edition by Damodar N. Gujarat and Dawn C. Porter</p><hr/> <h4>Q. No. 11 (a) - Consider the Leontief static input-output model along with its assumptions. How can you confirm that the model is either open or closed? State the fundamental objective of Leontief static open input-output model.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11 (b) - State the Hawkinsk-Simon condition and explain its economic meaning and significance.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11 (c) - Derive the consumption possibility locus.</h4><p>(Comment for solution.)</p><hr/> Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com0tag:blogger.com,1999:blog-1368387920159069514.post-4547962176947627002021-06-14T13:27:00.024+05:302022-06-11T18:09:59.732+05:30Previous Year Paper Solution | Indian Economic Service Exam 2014 | General Economics - I <h4>Q. No. 1 (a) - Is the following statement <strong>true</strong> or <strong>false</strong>? Explain. <blockquote>"If a consumer's utility function is of the form = $$x_1^{\frac{1}{3}}x_2^{\frac{1}{3}}$$, she faces prices p<sub>1</sub> and p<sub>2</sub> and her income is I, then her indirect utility function is $$V = \frac{{{I^3}}}{{3{p_1}{p_2}}}$$."</blockquote></h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dxJlVRIAygUqmDWzlgBoPNf-FwNmInJzzjd7alrFAaIZuezgn6YbIS0gMiHku1IlUxbVW46LIbfosrjC2pisQ' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/> <h4>Q. No. 1 (b) - Define complements and substitutes. In the two-commodity case, can the commodities be complements? Explain. Is your answer valid in the case of gross substitutes and complements? Explain</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (c) - Other things equal, what happens to consumer surplus if the price of a good falls? Why? Illustrate using a demannd curve.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (d) - What is meant by "internlizing" an externality? How can a negative externality be internalized?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1(e) - What is productivity principle? How can this be achieved through market mechanism?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (f) - What is Nash Equilibrium? Do all games have Nash Equilibrium? Can a game have more than one equilibrium?</h4><p>Nash Equilibrium is a strategy profile such that each player's equilibrium strategy is the best response to other players' equilibrium strategy and no player can benefit from deviating to other strategies.<br /> All games do not have Nash Equilibrium in pure strategy. For instance, Rock, Paper, Scissors game does not have any Nash Equilibrium in pure strategy.<br />A game can have more than one Nash Equilibrium. For instance, the Battle of Sexes game has multiple Nash Equilibriums.</p><hr/> <h4>Q. No. 1 (g) - List out the sources of monopoly power.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (h) Explain the concept of co-integration in a time series analysis.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 2 - Hrishita likes sandwiches (S) and Coffee (C). Here indifference curve is bowed in towards the origin and do not intersect each other. The price of a sandwich is ₹5 and price of a cup of coffee is ₹3. She is spending all her income at the basket she is currently consuming, and her marginal rate of substitution of sandwiches for coffee is 2.<br/><br/>Is she at an optimum? If so, show why. If not, should she buy fewer sanwiches and more coffee, or the reverse? Argue in favour of your opinion.</h4><p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dzzZkT-KRHNgf6xfoicDuR4ncSFPNt2XCAJ6cm59KXxc_7N0sidiBpMnNoukWvD5aJsZW6KNyCjmrrynMP4hA' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/> <h4> Q. No. 3 - The demand for good X is estimated to be $$Q = 250,000 - 500P - 1 \cdot 5M - 240{P_R}$$, where M is the (average) consumer income and $${P_R}$$ is the price of a related good Y. The values of P, M and $${P_R}$$ are expected to be ₹200, ₹60,000 and ₹100 respectively. <br/> (a) Calculate the price elasticity of demand, income elasticity of demand and cross price elasticity. <br/> (b) Is the demand for X elastic, inelastic or unit-elastic? How would a small increase in P affect total revenue? <br/> (c) Is the good X normal or inferior? Are the goods X and Y substitutes or complements?</h4><p>(Comment for solution.)</p><hr/> <h4>Q.No. 4 - Assume that a monopolist sells a product with the cost function C = F + 20Q, where C is total cost, F is a fixed cost, and Q is the level of output. The inverse demand function is P = 60 - Q, where P is the price in the market. (i) How much profit does the firm earn when it charges the price that maximizes profit? (ii) At what price will the firm earn zero economic profit? </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5 - Distinguish between Differentiation and Integration.. Explain their application in economies with suitable examples.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 6 - There are only two firms in an industry, firm 1 and firm 2. The market demand curve is given by the equation $$P = 12 - ({q_1} + {q_2})$$ are the (total) cost functions facing the firms are $${C_i} = 4{q_i}$$, where $$i = 1,2$$. If firm 1 acts as a leader and firm 2 as a follower, what are the quantities that the two firms will produce in the equilibrium? What profits will they earn?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 7 - Consider a manufactured good whose production process generates pollution. The annual demand for the good is given by $${Q^d} = 100 - 3P$$. The annual market supply is given by $${Q^s} = P$$. In bot equations, P is the price in rupees per unit. For every unit of output produced, the industry emits one unit of pollution. The marginal damage from each unit of pollution is given by 2Q. <br/> (a) Find the equilibrium price and quantity in a market with no government intervention. <br/> (b) Find the socially optimal quantity of the good. What is the socially optimal market price?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 8 - What is autocorrelation? How can we detec it? How can it be removed from a single equation model?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9 - Consider the production function $$Q = {({K^{0 \cdot 5}} + {L^{0 \cdot 5}})^2}$$. <br/> (a) What is the name of this type of production function? <br/> (b) What is the elasticity of substitution for this production function? <br/> (c) Does this production function exhibit increasing,decreasing, or constant returns to scale? <br/> (d) Suppose that the production function took the form $$Q = {(100 + {K^{0 \cdot 5}} + {L^{0 \cdot 5}})^2}$$. Does this production function exhibit increasing, decreasing or constant returns to scale? </h4><p>(Comment for solution.)</p><hr/> <h4>Q. 10 - Consider a two-person, two-commodity, pure-exchange, competitive economy. The consumers' utility functions are $${U_1} = {q_{11}}{q_{12}} + 12{q_{11}} + 3{q_{12}}$$ and $${U_2} = {q_{21}}{q_{22}} + 8{q_{21}} + 9{q_{22}}$$ respectively (where $${q_{ij}}$$ denotes the consumption of commodity $${Q_j}$$ by consumer i, with i = 1, 2 and j = 1,2). Consumer 1 has initial endowments of 8 and 30 units of $${Q_1}$$ and $${Q_2}$$ respectively; consumer 2 has 10 units of each commodity. <br/> Determine the excess demand function for the two consumers. Determine an equilibrium price ratio for this economy.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. 11 - What is the problem of multicollinearity in a regression model? What is its plausibility? Explain Farrer - Glauber method to detect it. How can it be removed?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12 - What is optimization problem in Economics? How does linear programming technique help in assigning optimal solution in given resourse use? Explain. (Marks - 25)</h4><p>Solution video:</p> <div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwr6HcJhbVOPufsHGpRUXH5MVAZbRtZORKJVgoNHR0mV0a_VHKgIkZ-68ib1ZhTsmVtFYSRLP5p1dwZcr_2_g' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/>Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com1tag:blogger.com,1999:blog-1368387920159069514.post-14839638724738533002021-06-14T13:26:00.022+05:302022-06-11T19:01:38.709+05:30Previous Year Paper Solution | Indian Economic Service Exam 2013 | General Economics - I<h4>Q. No. 1 (a) - If the law of demand is $$x = a{e^{ - bp}}$$, where p is price and x is quantity demanded. Express price elasticity of demand, total revenue and marginal revenue as function of x. </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (b) - Explain 'Leontief Inverse' in the input-output model suggested by W.W. Leontief.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (c) - Graphically explain the expansion path of a firm taking labour and capital as inputs.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (d) - What is adverse selection in insurance markets? How the problem can be solved?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (e) - Describe Gini's coefficient as a measure of inequality.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (f) - Show that Cobb-Douglas production function $$Q = A{L^\alpha }{K^{1 - \alpha }}$$, where symbols have usual meaning, exhibits constant returns to scale but diminishing returns to a factor of production.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (g) - What is monopoly power? Give an expression for measuring it.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (h) - Why does a perfectly competitive firm keep on producing in the short-run even when it is incurring losses? Explain also when the firm will shut down. Use suitable diagram.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (i) WHat are type I and type II errors in testing of a hypothesis?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (j) - Given utility function U = q<sub>1</sub>q<sub>2</sub> and budget constraint<br/> Y = p<sub>1</sub>q<sub>1</sub> + p<sub>2</sub>q<sub>2</sub><br/> derive the indirect utility function. (Marks - 5)</h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dze8huY-2Jl4NkapZoFKpTUGgVjjZtsAk1qa7M0zrzwb1_MItdZKRgoEWNe8vn1D3bTGijxraUVBFcuQhvlaw' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/> <h4>Q. No. 1 (k) State the causes of market failure.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 2 - Cardinal utility approach and ordinal utility approach to demand suggest the same decision rule for the optimizing consumer (which one?). Yet, the latter approach is preferred over the former. Why?</h4><p>Solution Video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dyPyBfvPEDLERawmoRtKIoTrn3zGs3RuAVdNxDVUDo66N59fiX4G0bBFKTh_1ELMBOBmZyOgpAlJGe9WQI1gw' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/> <h4>Q. No. 3 - Describe Von Neuman and Morgenstern utility index. Is this index unique? Explain.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 4 - Define elasticity of goods substitution and distinguish it from cross-price elasticity of demand. Which one is a better measure of substitution and why?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5 - Write dual of the following linear programme problem and solve the obtained dual graphically: <br/> Minimize : Z = 3x<sub>1</sub> + 3x<sub>2</sub><br/> subject to:<br/> x<sub>1</sub> + 2x<sub>2</sub> &#x22DD; 1<br/> 2x<sub>1</sub> + x<sub>2</sub> &#x22DD; 1<br/> x<sub>1</sub> &#x22DD; 0, x<sub>2</sub> &#x22DD; 0<br/> (Marks - 15) </h4><p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dydVxP49AaAC_fAU1zZwR6S65fpPhrmKskN0wAsCoKQwPJbUgSSUi4uaAlgalFMv89JBSZ0KE0xLKu_ST4YOQ' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/> <h4>Q. No. 6 - Critically examine Hicks-Kaldor criterion of compensation. Give Scitovsky's improvement over this criterion.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 7 - State and explain the assumptions applying ordinary least square method (OLS) method to two variable linear regression model: ${Y_t} = {b_0} + {b_1} + {u_t}$ $t = 1,2,...,n$ (Marks - 15) </h4><p>Following assumptions apply to ordinary least square (OLS) method to two variable linear regression model: </p><ol> <li><b>Linearity in parameters:</b> The regression model is <b>linear in theparameters</b>. In the regression model in question, b<sub>0</sub> and b<sub>1</sub> are parameters to be estimated, they need to be linear else OLS will not be applicable. However, the regression model need not to be linear in variables, that is, X and Y here.</li> <li>Values of X are fixed: When values of X are fixed, the disturbance term, u<sub>t</sub> is not correlated with the independent variable, X<sub>t</sub>. It means X<sub>t</sub> and u<sub>t</sub> are independent. Symbolically: ${\mathop{\rm cov}} \left( {{u_t},{X_t}} \right) = 0$</li> <li><b>u<sub>t</sub> is a random variable:</b> This means the values of u<sub>t</sub> in any one period depends on chance and each value has a probability of occurance. It values may be positive, negative or zero.</li> <li><b>Zero Mean of the Disturbance term:</b> The disturbance term u<sub>t</sub> has zero mean in any particular period given the values of X<sub>t</sub>. For each value of X, u has a random value. Some are positive, some negative or 0. Positive and negative values cancel each other and the sum of u<sub>t</sub> is equal to zero. As a result, the mean is also zero. Symbolically, we can write: $E\left( {{u_t}|{X_t}} \right) = 0$ Since, X is not a random variable (stochastic), we can also write: $E\left( {{u_t}} \right) = 0$ </li> <li><b>Homoscedasticity:</b> Homoscedasticity means that the variance of the disturbance term u<sub>t</sub> is constant in each period regardless of the values of X<sub>t</sub>. We can derive this as follows: ${\mathop{\rm var}} \left( {{u_t}} \right) = E{\left[ {{u_t} - E\left( {{u_t}} \right)} \right]^2}$ Since, $$E\left( {{u_t}} \right) = 0$$ ${\mathop{\rm var}} \left( {{u_t}} \right) = E\left[ {u_t^2} \right] = \sigma _u^2$ </li> <li><b>The disturbance term, u<sub>t</sub> is normally distributed.</b> This assumption and the above assumption can be sumarized as follows: $u \sim N\left( {0,\sigma _u^2} \right)$ This means that u is normally distributed with zero mean and constant variance. </li> <li><b>No Autocorrelation:</b> Correlation between any two series, u<sub>t</sub> and u<sub>s</sub> is zero given that the corresponding series of X<sub>t</sub> and X<sub>s</sub> are non-stochastic. This is possible when observations are sampled independently. Symbolically: ${\mathop{\rm cov}} \left( {{u_t},{u_s}} \right) = 0$ Covariance is zero means that correlation is also zero and zero correlation means that there no linear relationship between concerned variables. </li> <li>The number of observations (n) must be greater than the number of explanatory variables. It is not possible to estimate parameters with a single pair or observation.</li> <li>Nature of Independent variable: Here, all values of the independent variable, X, must not be same. In technical terms, the variance of X<sub>t</sub> must be a positive number. The should also not be outliers, that, very small or large values of X compared to other data set. </li></ol> <h4>Q. No. 8 - "In the long-run competitive equilibrium rewarding each input according to its marginal physical product precisely exhausts the total physical product." Critically examine the above statement.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9 - Consider the following duopoly. Demand is given by $$P = 10 - Q$$, where $$Q = {Q_1} + {Q_2}$$. The firm's cost functions are: ${C_1}({Q_1}) = 4 + 2{Q_1}\,\,{\rm{and}}\,\,{C_2}({Q_2}) = 3 + 3{Q_2}\,\,$ <br/> (a) Suppose both firms have entered the industry. What is joint profit maximising level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry? <br/> (b) What is each firm's equilibrium output and profit if they behave non-co-operatively?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10 - Can the threat of a price war deter entry by potential competitors? What actions might a firm take to make this threat credible? Give example.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11 - For statistically estimated demand function for the commodity X, ${D_x} = \frac{{1547P_x^{0 \cdot 2}P_y^{0 \cdot 3}{A^{0 \cdot 4}}}}{{P_z^{0 \cdot 5}{B^{0 \cdot 3}}}}$ (where x, y, z are goods, A stands for advertisement outlay, B for budget of the consumer and $${P_x},\,{P_y},\,{P_z}$$ are prices of goods x, y, z respectively). <br/> Answer the following: <br/> (a) How are x, y and z related? <br/> (b) Whether x is an inferior, normal or Giffen type good? <br/> (c) What would be the percentage change in demand for $$x\,\,(i.e.\,\,\,\,{D_x})$$ and in which direction if advertisement outlay increases by 50 percent? </h4><p>(Comment for solution.)</p><hr/> Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com0tag:blogger.com,1999:blog-1368387920159069514.post-58293226487641228772021-06-14T13:25:00.013+05:302022-06-12T11:04:17.611+05:30Previous Year Paper Solution | Indian Economic Service Exam 2012 | General Economics - I<h4>Q. No. 1 (a) - Distinguish between Marshallian and Walrasian stability analysis.</h4><p>(Comment for solution.)</p><hr/> <h4> Q. No. 1 (b) - Discuss "Nash Equilibrium" for non-collusive firms.</h4><p>Nash Equilibrium for non-collusive firm is likely to be a perfect competition price because all firms may fear that other firms will cut their prices to appropriate more market share. This price war will utlimately result in the lowest possible price which is obviously the perfect competition price.</p><hr/> <h4>Q. No. 1 (c) - What are the basic features and the limitations of Leontief's input-output model?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (d) - How can you measure income inequality by using Lorenz curve method?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (e) - Suppose you have a demand function for milk of the form $${x_1} = 100 + \frac{m}{{100{p_1}}}$$ and your weekly income (m) is ₹ 12,000 and the price of milk $$\left( {{p_1}} \right)$$ is ₹ 20 per litre. Now suppose the price of milk falls from ₹ 20 to ₹ 15 per litre, then what will be the substitution effect? </h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (f) - Explain 'dead-weight' loss in a monopoly situation.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (g) - Define the terms 'white noise' and 'random walk' in time series analysis.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (h) - Show graphically on your answer-book that if a consumer buys only two goods, both cannot be inferior at the same time.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (i) - Highlight the role of market signalling when there is asymmetric information.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 2 - Separate income effect from substitution effect for a price change using (i) Hicks' method (ii) Slutsky's method. Hence explain the difference between the two compensated demand curves.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 3 - Assume that the market demand is <br/>$$P = 100 - 0 \cdot 5({X_1} + {X_2})$$ and the two collusive firms have costs given by $${C_1} = 5{X_1}$$ and $${C_2} = 0 \cdot 5X_2^2$$. Calculate the joint profit of the firms.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 4 - Compare different methods of measuring risk aversion.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5 - What are 'ridge lines'? What are their implications in the theory of the firm?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 6 - Distinguish between compensating variation and equivalent variation of the budget line. How can you measure consumer's surplus using these two concepts?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 7 - Derive an expression for elasticity of factor substitution for C.E.S. production function and use it to establish that Cobb-Douglas production function is a special case of C.E.S. production function.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 8 - Why is the convexity assumption so importnt in indifference curve analysis? In particular, would a consumer equilibrium exist, if indifferece curve were concave? Explain.</h4><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dybAVk7urtSgdqpMMZYe_izflbXJ9v4dh-P2IPVkbVLKUAv54vKhhFXXCaD0hRaR_BdDeWEq9Xvihr8ehfnQQ' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr/> <h4>Q. No. 8 - Why is the convexity assumption so important in indifference curve analysis? In particular, would a consumer equilibrium exist, if indifferece curves were concave? Explain. </h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dzgE4pS7B112yA_KJTPpPKJ1l7n3CZ_m0LG_aY05tg3iXNIFLFj95AjsHm9HOIc3h6Js2MNLNiJ3l1zxF_zFw' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 9 - What is the dual problem in Linear Programming? Explain its use with suitable examples.</h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dzvkAXQ5XGZYqoNa42QRNVUuc_DvR-Tgickf3kDtOsKKczaM0jyUzpsrgmBz8jl3o39wy2qpOJOrDHwcIC62A' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 10 - Explain the relationship between slope and elasticity of a straight line demand curve.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11 - "In the long run, a perfectly competitive firm will be earning just normal profit." Discuss.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12 - What is 'Prisoner's Dilemma'? Discuss its importance and implications in Game theory.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 13 - What is 'moral hazard' problem? How does it lead to inefficient allocation of resources? Suggest remedial measures.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 14 - In a discriminating monopoly, the total demand function is P = 100 - 2X and demand function of segmented markets are ${P_1} = 80 - 2 \cdot 5{X_1}\,\,\,{\rm{and}}$ ${P_2} = 180 - 10{X_2}$ The cost function is $C = 50 + 40({X_1} + {X_2});\,\,\,{\rm{where}}\,\,\,\,\,{X_1} + {X_2} = X$ Calculate the profit of the monopolist; (i) with discrimination and (ii) without discrimination.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 15 - Compare and contrast the theories of social choice as propounded by PRofessor A.K. Sen and Professor K.J. Arrow.</h4><p>(Comment for solution.)</p><hr/> Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com0tag:blogger.com,1999:blog-1368387920159069514.post-63259882417826044572021-06-14T13:21:00.023+05:302022-06-12T12:08:54.632+05:30Previous Year Paper Solution | Indian Economic Service Exam 2011 | General Economics - I<h4>Q. No. 1 (a) Define the compensated demand curve. How does it differ from the uncompensated demand curve?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (b) - What do you mean by corner solution? In the case of perfect complementory goods, where do you get the corner solution?</h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwis5l19k8CWkENgWpA8wEOaGHYr5L5CRNbvE6AOw6JZ4u7qRA6HUy26KRBZ0UInN6wbrzOcyG_TcN9kv_2Yw' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 1 (c) Given that the average revenue curve is a rectangular hyperbola, what will be the shape of the marginal revenue curve? Comment briefly.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (d) - Suppose that a monopolist faces a demand curve with price elasticity less than one should the monopolist adopt the policy of price-increase in order to increase revenue? Justify your answer.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (e) - Define cross elasticity of demand. Based on such definition, how can you distinguish between the substitute goods and the complementary goods?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (f) - Using graphical approach, solve the following linear programming problem:<br/> Minimize : C = 80X + 60Y<br/>Subject to : <br/>2X + 2Y &#x22DD; 3<br/>2X + 0Y &#x22DD; 1<br/>X, Y &#x22DD; 0</h4> <p>Solution video:</p> <div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwrjVvpAGNnaH0VO0wxkLJNqWx1D70lrW7iwckhmUZZjycAhOuw-xeMlnQg4lcLwgC974c8YV0ER2VQSbF2Ww' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 1 (g) - Consider a Cobb-Douglas production function $$Y = A{K^\alpha }{L^\beta }$$ where K and L are respectively the capital and labour to produce output Y. Show that if all the factors are paid according to their marginal products, the total product will be exhausted if $$\alpha + \beta = 1$$.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (h) - Consider a linear demand function q = a -bp, where q = quantity demanded, p = price per unit and a,b > 0. Find out the average and the marginal revenue and draw the diagram.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (i) - Establish mathematically the relationship between average cost (AC) and marginal cost (MC).</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (j) - Define and distinguish between the normal profit and the super-normal profit under perfect competition. In the short run, find out graphically the amount of profit corresponding to the equilibrium without using the average cost curve.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. N0. 1. (k) - In the game theory comment on the terms 'maxi-min' and 'mini-max'.</h4><p><b>Maxi-min</b> is the maximum payoff in the row-minimum in a zero-sum-game where raw minimum is a set of minimum payoffs from each raw in a payoff matrix.<br/><b>Mini-max</b> is the minimum payoff in the column-maximum in a zero-sum-game where column maximum is a set of maximum payoffs from each column in a payoff matrix.</p><p>Suggested vido to understand these concepts more clearly - <a href="https://youtu.be/O7mMb4xX43o" target="_blank">Game Theory Basics - 2</a></p><hr/> <h4>Q. No. 2 - Define income effect, substitution effect and price effect of any change in price Show that the price effect can be decomposed into the income effect and the substitution effect.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 3 - The demand function $${Q_1} = 50 - {P_1}$$ intersects another linear demand function $${Q_2}$$ at P = 10. The elasticity of demand for $${Q_2}$$ is six times larger than that of $${Q_1}$$ at that point. Find the demand function for $${Q_2}$$.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 4 - Consider the utility function $$u = \sqrt {{q_1}{q_2}}$$ where q<sub>1</sub> and q<sub>2</sub> are the quantities of two commodities on which the consumer spends his monthly income ₹5,000. If the price per unit of q<sub>1</sub> and q<sub>2</sub> be ₹50 and ₹20 respectively, find out the optimal quantities of q<sub>1</sub> and q<sub>2</sub>. (Marks - 15) </h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dylDjAfZkLBUpvRklD2xpIAYcyaOvStvZsYwfPQrCE_PgiLrZoD2AdjCzIalPaJljvGB6Y6dLfVste7B3P1PQ' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 5 - Define linear homogeneous production function with the help of CES production function. Also establish that CES production function is strictly quasi-concave for positive L and K, where L, K are labour and capital respectively.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 6 - What do you mean by price discrimination? Under which condition is the price discrimination profitable? Trace out the equilibrium situation under price discrimination.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 7 - How can you get the wage offer curve and the supply curve of labour? How can you justify the backward bending supply curve of labour?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 8 - What is meant by excess capacity? Why is it bad? Are there any benefits of excess capacity associated with monopolistic competition?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9 - If D = 250 - 50p and S = 25p + 25 are the demand and supply functions repectively, calculate the equilibrium price and the quantity. Hence calculate both consumer's and producer's surpluses under equilibrium.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10 - Define and distinguish between rent and quasi-rent. What do you mean by 'transfer earnings? Elucidate the statement that no economic rent is earned when the supply of a factor is perfectly elastic.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11 - How does Lorenz curve exxplain income inequality? Explain with one suitable example. Define Gini coefficient with the help of Lorenz curve and show that $Gini = [1 - 2 \times (Area\,below\,Lorenz\,curve)].$</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12 (a) - In computing the correlation coefficient between two variables X and Y from 25 pairs of observations, the intermediate results are: $n = 25$ $\sum\limits_i {{X_i}} = 125$ $\sum\limits_i {X_i^2} = 650$ $\sum\limits_i {{Y_i}} = 100$ $\sum\limits_i {X_i^2} = 460$ $\sum\limits_i {{X_i}{Y_i}} = 508$ Later on at the time of checking it was found that two pairs of observations which should be <table style="display:inline;"> <tr><td>X</td><td>8</td><td>12</td></tr> <tr><td>Y</td><td>6</td><td>8</td></tr> </table> had been incorrectly recorded us <table style="display:inline;"> <tr><td>X</td><td>6</td><td>14</td></tr> <tr><td>Y</td><td>8</td><td>6</td></tr> </table>. Calculate the correct value of correlation coefficient. (Marks - 15) <br/><br/> (b) - Find out the variance of numbers 1, 2, 3, ...., 50 and the coefficient of variation. What is the advantage of computing the coefficient of variation over the variance? (Marks - 5+3+2=10) </h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dw4yajYLTQyXRQDFLqY3f0-EmadJspnp6Z9rIMxlsoaRJdBV0KF3G-_hqyhusP2w2nMKLIU3AaZ0EIvu5p2rg' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 13 (a) - Explain the terms as follows and their importance in the context of inference analysis: Degree of freedom, Level of significance, and Power of the test.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 13 (b) Briefly discuss the principal component analysis and the rationale for its use.</h4> <p>A pdf related to principal component analysis has been available in the Google Drive Folder. See the Econometrics file of the folder.<br/> Also see Q. No. 12 (b) and (c) <a href="https://www.studentsofeconomics.com/2021/06/general-economics-1-2019.html" target="_blank">General Economics - 1 Previous Year Paper Solution 2019</a>.</p> <hr/>Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com0tag:blogger.com,1999:blog-1368387920159069514.post-323375317434831462021-06-14T13:20:00.030+05:302022-06-12T12:33:14.921+05:30Previous Year Paper Solution | Indian Economic Service Exam 2010 | General Economics - I <h4>Q. No. 1 (a) - Define consumer's surplus. Derive an expression for it using integral calculus.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (b) - Why is short-run average cost curve U-shaped? Show that marginal cost curve intersects the average cost curve at latter's minimum point.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (c) - Compare long-run equilibrium of the firm under perfect competition with that under monopolistic competition using suitable diagram.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (d) - What is a social welfare function? State the underlying assumption in its formulation.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (e) - State and explain the assumptions of two variable linear regression model.</h4> <p>Answer to this question is available in Q. No.<a href="https://www.studentsofeconomics.com/2021/06/general-economics-1-2013.html" target="_blank"> General Economics - I Previous Year Paper Solution 2013</a>. Assumption of regression and OLS are same because regression is generally done using OLS method.</p> <hr/> <h4>Q. No. 1 (f) What is log-normal distribution? Where is it used in economic analysis?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 2 - Derive consumer's expenditure function by minimizing total expenditure; $$y = {p_1}{x_1} + {p_2}{x_2}$$ subject to utility constraint $$\bar u = {q_1}{q_2}$$. (Marks - 15)</h4><div><p>Expenditure funtion can be derived if we know either indirect utility function or the compensated demand curve of the two goods. Let us first derive the the compensated demand curve for the two goods.</p> Here objective is to minimize total expenditure subject to utility constraint. We can write Lagrangian function as follows: $L = {p_1}{q_1} + {p_2}{q_2} + \lambda (\bar u - {q_1}{q_2})$ First order condition for minimization is: $\frac{{\partial L}}{{\partial {q_1}}} = \frac{{\partial L}}{{\partial {q_2}}} = \frac{{\partial L}}{{\partial \lambda }} = 0$ Taking partial derivative of the Lagransian function with respect to q<sub>1</sub> and equating to zero, we get: $\frac{{\partial L}}{{\partial {q_1}}} = 0$ ${p_1} - \lambda {q_2} = 0$ $\lambda = \frac{{{p_1}}}{{{q_2}}}$ Similarly, taking partial derivative of the Lagransian function with respect to q<sub>2</sub> and equating to zero, we get: $\frac{{\partial L}}{{\partial {q_2}}} = 0$ ${p_2} - \lambda {q_1} = 0$ $\lambda = \frac{{{p_2}}}{{{q_1}}}$ We can equate the values of $$\lambda$$: $\frac{{{p_1}}}{{{q_2}}} = \frac{{{p_2}}}{{{q_1}}}$ Rearranging the equation, we get: ${q_2} = \frac{{{p_1}{q_1}}}{{{p_2}}}$ Sustituting this value of $${q_2}$$ in the utility constraint, we get: $\bar u = {q_1} \times \frac{{{p_1}{q_1}}}{{{p_2}}}$ Rearranging: $\bar u = q_1^2 \times \frac{{{p_1}}}{{{p_2}}}$ ${q_1} = \sqrt {\bar u\frac{{{p_2}}}{{{p_1}}}}$ This is the compensated demand function for Good - 1. Sustituting this value of $${q_1}$$ in the utility constraint, we get: $\bar u = \sqrt {\bar u\frac{{{p_2}}}{{{p_1}}}} \times {q_2}$ Rearranging: ${q_2} = \sqrt {\bar u\frac{{{p_1}}}{{{p_2}}}}$ This is the compensated demand function for Good - 2. </div><div>Now, we can find expenditure function by substituting the compensated demand functions in the objective function as follows: $y = {p_1}{q_1} + {p_2}{q_2}$ $y = {p_1} \times \sqrt {\bar u\frac{{{p_2}}}{{{p_1}}}} + {p_2} \times \sqrt {\bar u\frac{{{p_1}}}{{{p_2}}}}$ $y = \sqrt {\bar u{p_1}{p_2}} + \sqrt {\bar u{p_1}{p_2}}$ $y = 2\sqrt {\bar u{p_1}{p_2}}$ This is the expenditure function, we want to find. It can also be written us: $y = 2{{\bar u}^{\frac{1}{2}}}p_1^{\frac{1}{2}}p_2^{\frac{1}{2}}$ </div> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwSWfz8RPn7w3iTDkB_BAL3hXZDLuaeduz-LsnXP4ybDJKaPD26JcPVBUFEVIpCXxngNbGWJgsR-tnKWlj5Vw' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 3 - Draw consumer's indifference curve from revealed Preference Theory.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 4 - Separate income effect from substitution effect of a price change for a Giffen type good. Use suitable diagram.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5 - What is elasticity of factor substitution? Give various forms of production function based on this concept.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 6 - "Asymmetric or incomplete information leads to market failure." Examine lemons' problem in the above context with the help of pricing of used cars.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 7 - What is Hicks-Kaldor criterion of compensation? What are its weaknesses? Give Scitovsky's suggestion for improvement.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 8 - Distinguish between positive and negative externalities and explain with examples. Why does government provide some goods which are not public goods?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 9 - What are type I and type II errors? Why is probability of type I error fixed in a hypothesis testing problem?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10 - Four products A, B, C, D are to be bought to satisfy minimum requirements of calories and Vitamin (which are 18 and 10 units respectively) at minimum cost. Formulate linear programming problem using additional information given below: <table> <tr><th>Product Type</th><td>A</td><td>B</td><td>C</td><td>D</td></tr> <tr><th>Calorie content</th><td>2</td><td>0</td><td>1</td><td>3</td></tr> <tr><th>Vitamin content</th><td>0</td><td>3</td><td>1</td><td>4</td></tr> <tr><th>Price per unit</th><td>5</td><td>10</td><td>12</td><td>15</td></tr> </table> </h4> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwXwLa7kFayArm0vhTeHD__cPvL4GXum39uAksNeggun3Xsa6TiaXJi3E4xxWYFq3MzMhkv-_wL6BV57w-PEg' class='b-hbp-video b-uploaded' frameborder='0' /></div> <hr/> <h4>Q. No. 11 - What is Peak-load Pricing? How is it different from third degree price discrimination? Give diagrams to illustrate your answer.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12 - Define production function. The production function for a product is given by Q = 100KL. <br/> If price of capital (K) is $120 per day and that of labour (L) is$ 30 per day, what is the minimum cost of producing 400 units of output?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 13 - </h4><p>(Comment for solution.)</p><hr/>Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com2tag:blogger.com,1999:blog-1368387920159069514.post-70021919738213330082021-06-14T13:19:00.058+05:302022-06-12T13:07:02.812+05:30Previous Year Paper Solution | Indian Economic Service Exam 2009 | General Economics - I<h4>Q. No. 1 (a) - How do you draw a Lorenz curve? Explain its use.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (b) - What is 'shadow price'? Why are shadow prices used in project analysis as against market prices?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (c) - What is Engel's law? Which sector/product(s) of an economy operate under this law?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (d) - State the first and second fundamental theorems of welfare economics, and comment on their usefulnesses.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (e) - State the Kuhn - Tucker conditions.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1(f) - Explain how Pareto's law of distribution is useful in measuring income distribution.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 1 (g) - Explain total factor productivity and mention any two popular measures of the same.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 2 - Show how a demand function may be derived from the Cobb-Douglas utility function.</h4><div>The Cobb-Douglas utility function can be written as follows: $U = {x^\alpha }{y^\beta }$ Where x and y are quantity demanded of the two goods, say, Good X and Good Y. A consumer's objective is to maximize utility with the given budget constraints which can be written as: $I = {p_x}x + {p_y}y$ Equlibrium Condition in this two commodity is given by: $MR{S_{x,y}} = \frac{{{p_x}}}{{{p_y}}}$ Where, $MR{S_{x,y}} = - \frac{{dy}}{{dx}} = \frac{{M{U_x}}}{{M{U_y}}}$ We can derive the MU of x and y from the utility function as follows: $M{U_x} = \frac{{\partial U}}{{\partial x}}$ $M{U_x} = \alpha {x^{\alpha - 1}}{y^\beta }$ $M{U_x} = \frac{{\alpha {x^\alpha }{y^\beta }}}{x}$ $M{U_x} = \frac{{\alpha U}}{x}$ Similarly, $M{U_y} = \frac{{\beta U}}{y}$ Now, we can derive MRS from the above result: $MR{S_{x,y}} = \frac{{M{U_x}}}{{M{U_y}}} = \frac{{\alpha U}}{x} \times \frac{y}{{\beta U}}$ $MR{S_{x,y}} = \frac{{\alpha y}}{{\beta x}}$ From the results derived above, we can write equlibrium condition for Cobb-Douglas as: $\frac{{\alpha y}}{{\beta x}} = \frac{{{p_x}}}{{{p_y}}}$ Rearranging the equation above, we can write: $y = \frac{\beta }{\alpha }\frac{{{p_x}x}}{{{p_y}}}$ Substituting this value of y in the budget constraint, we get: $I = {p_x}x + {p_y}\frac{\beta }{\alpha }\frac{{{p_x}x}}{{{p_y}}}$ We can remove p<sub>y</sub> from the equation above: $I = {p_x}x + \frac{\beta }{\alpha }{p_x}x$ Rearranging the equation above, we get: $I = {p_x}x\left( {\frac{{\alpha + \beta }}{\alpha }} \right)$ Let $${\alpha + \beta = 1}$$, $x = \frac{{\alpha I}}{{{p_x}}}$ this is the demand function for Good X, which shows a negtaive relationship between quantity demanded and the price of the commodity. Similarly, by substituting the value of this demand function in the budget constraint, we can derive the demand function for Good Y as follows: $y = \frac{{\beta I}}{{{p_y}}}$</div> <p>Solution video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dw9UND_AOOu7ojTbgZkevDhG7ZXX4MhpsJz_rZy0zK5YqvWDIX_eS_P3aQOEYgq8VGaM_44x0gWeweIA5ot1g' class='b-hbp-video b-uploaded' frameborder='0' /></div><hr /><h4 id="098"> <h4>Q. No. 3 Formulate a translog cost function and show how the elasticity of input substitution may be obtained.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 4 - What is "moral hazard" in economic theory? Discuss a situation that would describe a moral hazard problem.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 5 - What is "free-rider" problem? Discuss the possible solutions ot this problem.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 6 - State and explain the Coase theorem in the context of pollution control.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No 7 - Explain Leontief's static input-outut model and describe its limitations.</h4><p>(Comment for solution.)</p><hr/> Q. No. 8 - What is "prisoner's dilemma"? How is it related to strictly dominant strategy?</h4><p><b>Prisoner's Dilemma</b> is a hypothetical situation used in Game Theory to explain Nash Equilibrium and Dominance Rule. The hypothetical situation is:<br /> Two suspects, involved in a crime together, are taken into police custody. The police do not have sufficient evidence to convict them so they can be convicted if only if they confess. The police interrogate them in separate cells so that they couldn't communicate and the police put some conditions before them which can be explained with the following pay-off matrix:</p><table> <tbody><tr><th colspan="2" rowspan="2"></th> <th colspan="2">Suspect B</th></tr> <tr> <td>Confess</td><td>Not Confess</td></tr> <tr><th rowspan="2">Suspect A</th><td>Confess</td> <td>(-3, -3)</td><td>(0, -6)</td></tr> <tr> <td>Not Confess</td> <td>(-6, 0)</td><td>(-1, -1)</td></tr> </tbody></table><p>The payoffs in the payoff matrix represent years of jail as punishment. Minus sign implies that jail is a loss, not profit. So, suspects would like to minimize year of jail. It exhibits the following conditions:</p><ul><li>Payoffs (-3, -3) implies that both will get three year jail if both confess.</li><li>Payoffs (0, -6) implies that Suspect A will get zero year jail, that is, no punishment and Suspect B will get six years of jail if the former confesses and the latter does not confess.</li><li>Similarly, payoffs (-6, 0) implies that Suspect B will get zero year jail, that is, no punishment and Suspect A will get 6 years of jail if the former confesses and the latter does not confess.</li> <li>Lastly, payoffs (-1, -1) implies that both will get only one year jail if both do not confess.</li></ul><p>Let us first look at the matrix from A's point of view. Given the conditions above, it is beneficial for A to confess. If he confesses, he will get either three year jail (in case B also confesses) or zero year jail(in case B does not confess). If he does not confess, he will get either six year jail (in case of B confesses) or one year jail (in case B does also not confess). A would choose to confess as zero to three year jail is better than one to six years jail.<br /> Similarly, if you look at the payoffs from B's point of view, you will find the same result, that is, B will also confess. It means that the optimum strategy for both is to confess which represented payoffs (-3, -3).<br/> The situation described above can be found in many type of real life situation where two persons are interdependent for optimization of their respective payoffs. This is what we call <b>Prisoner's Dilemma</b>.</p><p><b>Strictly Dominant Strategy:</b> It is called a <b>dominant strategy</b> if a player in a game can play its optimal strategy regardless of what other player will play. Here, A and B both have the dominant strategy. A's optimal stategy is to confess where he can expect that he will get either 0 or 3 year jail which is better than the the alternative strategy, that is, not to confess where he can get 1 or 6 year jail. He can easily choose his best strategywithout worrying about what B will play. Similar situation is faced by B. The best stategy for B is also to confess without worrying about what A will play.<br/> It is called strictly dominant strategy when the alternative stategy always makes worse off. Here, we can see that alternative strategy (Not Confess) of A makes him better off than the optimal strategy if B does not confess. Same situation is faced by B. Therefore, neither of them has <b>strictly dominant strategy</b>.</p> <h4>Q. No. 9 - What is a superlative index number? How is it related to the theory of aggregation?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 10 - Define a quadratic form, and state the conditions under which it is (i) positive definite, (ii) positive sem-definite, and (iii) negative definite.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 11 - Explain the principl of least squares as a basis for multiple regression analysis. Also state the underlying standard assumptions of ordinary least Squares estimation. Explain further the consequences of violation of such assumptions.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 12 - In the case of a pure exchange econom how do you characterize Walrasian equilibrium? Also establish the conditions under which such an equilibrium exists.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 13 (a) - State and prove Euler's theorem. Is i relevant in the context of a firm?</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 13 (b) - Distinguish between technical and allocative efficiency in the context of a firm. Give an example.</h4><p>(Comment for solution.)</p><hr/> <h4>Q. No. 13 (c) - Distinguish between parametric and non - parametric tests in testing of hypotheses.</h4><p>(Comment for solution.)</p><hr/> Santosh Kumarhttp://www.blogger.com/profile/01469625530059844533noreply@blogger.com4