### Section - A

1. Answer all of the following six parts in about 100 words each.

Marks: 5 × 7 = 35

#### (a) Show the conditions for a Cobb-Douglas production function under: (i) increasing returns to scale (ii) consant returns to scale (iii) dimishing returns to scale Are the law of returns compatible?

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Solution video:

#### (c) What is monopoly power? What factors determine the amount of monopoly power?

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#### (d) Explain the difference between Bandwagon effect and Snob effect.

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#### (e) What is meant by deadweight loss? Why does a price ceiling usually result in a deadweight loss

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#### (f) State the fundamental theorems of Welfare Economics.

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#### (g) Public goods are non-rival and non-exclusive. Explain each of these terms and show clearly how they differ from each other.

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### Section - B

Answer any five out of the following seven questions.

Marks: 18 × 5 = 90

#### 2 (a). Explain the meaning of Nash Equilibrium. How does it differ from the equilibrium in dominant strategies? (8 Marks)

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.
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.
All dominant strategy equilibrium is a Nash Equilibrium but all Nash Equilibrium are not dominant strategy equilibrium.

#### 2 (b). Let market demand faced by duopolist be $P = 100 - 0.5Q$ $Q = {Q_1} + {Q_2}$ and their respective cost function as: ${C_1} = 5{Q_1}$ and ${C_2} = 5{Q_2}$ Find out Cournot-Nash Equilibrium. (10 Marks)

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: $\frac{{\partial {\pi _1}}}{{\partial {Q_1}}} = 0$ $95 - 0.5{Q_1} - {Q_2} = 0$ ${Q_2} = 95 - 0.5{Q_1}$

Substituting the value of q2 in q1 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.
Similarly, quantity sold by the second duopolist will be: ${Q_2} = \frac{{190}}{3}$

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.
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.

Solution: Video:

#### 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?

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#### 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. (i) Show that cost minimisation requires $$\frac{{rK}}{\alpha } = \frac{{wL}}{\beta }$$. What is the slope of the expansion path for this firm? (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 $$\beta$$. (iii) Show that if $$\alpha + \beta = 1$$, total cost (TC) is proportional to Q. (iv) Calculate the firm's marginal cost curve.

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#### 5 (a). Distinguish between economic rent and transfer earnings. Can economic rent exist in the long run? Justify your answer.

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#### 5 (b). Explain graphically the role of elasticity of supply of a factor determining the economic rent.

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#### 6 (a). Why do externalities prevent markets from being efficient? How does Coase theorem correct an externality?

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#### 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?

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#### 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.

ABCDEFGH
First Judge52814637
Second Judge45732816

Solution Video

#### 8. Discuss Social Choice theory in Economics. Distinguish between the views of Amartya Sen and Kenneth Arrow in making choices for social welfare.

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### Section - C

Answer any three of the following five questions.

Marks: 25 × 3 = 75

#### 9 (a). Consider a two-variable linear regression model $\begin{array}{l}{Y_t} = \alpha + \beta {X_t} + {U_t}\\and\,\,{U_t} = {}_\rho {U_{t - 1}} + {\varepsilon _t};\,\left| \rho \right| < 1\end{array}$Find Mean, Variance and Covariance of random disturbance term $$\left( {\,{U_t}} \right)$$.

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#### 9 (b). Consider the model of wage determination: ${Y_t} = {\beta _1} + {\beta _2}{X_t} + {\beta _3}{Y_{t - 1}} + {U_t}$ where Y = wages X = productivity $${U_t} = {}_\rho {U_{t - 1}} + {\varepsilon _t};\,\,\, - 1 < \rho < 1$$ Discuss the method of testing with the help of appropriate test statistic.

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#### 9(c). Cosider a model $\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,{Y_t} = {\beta _1} + {\beta _2}{X_t} + {U_t}\\and\,\,\,{U_t} = {}_\rho {U_{t - 1}} + {\varepsilon _t}\end{array}$ Discuss the process of the removal of autocorrelation when (i) $$\rho$$ is known (ii) $$\rho$$ is unknown (using Cochrane-Orcutt iterative method)

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#### 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, (i) Calculate the gross output of the two commodities and the total labour required. (ii) Write down technology matrix. (iii) Do you think that the system is viable? (iv) Determine the equilibrium prices, if the wage rate is ₹ 10 per man-day.

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#### 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. (i) Write down the pay-off matrix of the above problem. (ii) Whom do you consider in the better situation?

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#### 11 (a). Compare the distribution theory of Marx with that of Ricardo.

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#### 11 (b). Explain when Kaldor's theory of distribution beomes more appropriate.

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#### 11 (c). Narrate the areas where Kaldor's distribution model fails.

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#### 12 (a). The kinked demand curve describes price rigidity. Explain how the model works. Why does price rigidity occur in oligopolistic market

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#### 12 (b). State and prove Product Exhaustion Theorem. How does it differ from Clark-Wicksteed-Walras Theorem?

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#### 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 $$\beta$$.

1. $${\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } }$$ is unbiased estimator of $$\beta$$.
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 Xi 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 $$\beta$$

2. $${\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over \beta } }$$ is inefficient estimator of $$\beta$$.
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.

#### 13 (b). Consider a three variable linear regression model ${Y_i} = {\beta _1} + {\beta _2}{X_{2i}} + {\beta _3}{X_{3i}} + {\varepsilon _i}$ and suppose that (i) $$\sigma _i^2 = {\sigma ^2}Z_i^2$$ (ii) $$\sigma _i^2 = {\sigma ^2}{X_{1i}}$$ (iii) $$\sigma _i^2 = {\sigma ^2}X_{_{1i}}^2$$ Discuss Generalised Least Squares (GLS) method to overcome the heteroscedasticity problem under three cases (i, ii and iii).

(Comment for solution.)

1. Sir, in the cournot nash equilibrium question I could't understand the last a few steps. When you use the profit maximisation condition, how did 190 come after derivation? After derivation wont 95 be left and the market equilibrium quantity therefor be 95? pls help! Thank you.

1. You're right. There is something wrong with this. Let me check it.

2. The answer has been corrected. Do you still have any confusion?

3. no confusion! thank you so much sir!

2. Sir any source/reading material through which I can solve more quesitons like these and also understand the theoretical foundation behind this?

1. Which question are you talking about?
For Microeconomics - Nicholson and for statistics SC Gupta. I have told about the references in the video.
Microeconomics is available in the Google Drive folder.