### Section - A

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

Marks: 5 × 7 = 35

#### (a) Economic rent is not earned when the supply of a factor is perfectly elastic. Elucidate. Use a diagram.

(Comment for solution.)

#### (b) Show that the elasticity of substitution is contant in a Cobb-Douglas production function. Find its value and interpret.

(Comment for solution.)

#### (c) Consider the optimization problem:

Maximize u(x_{1}, x_{2})

subject to M = p_{1}x_{1} + p_{2}x_{2}

where, M, p_{1} and p_{2} are positive constants.

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.

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.

Solution video:

#### (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.

(Comment for solution.)

#### (e) Explain with a diagram why the compensated demand curve is vertical if the consumer's utiliy function is of the form

v(x,y) = min[x,y]

(Comment for solution.)

#### (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.

(Comment for solution.)

#### (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.

(Comment for solution.)

### Section - B

Answer any **five** out of the following seven questions in about 200 words each.

Marks: 18 × 5 = 90

#### 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_{X} and P_{Y} be the respective prices of the two goods. Assume that:

(i) the consumer's budget is given by M, ∞ > M > 0; and

(ii) P_{X} and P_{Y} are finite and positive.

(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)

(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).

Solution Video:

#### 3. Consider a one shot simultaneous move game with two players, player 1 and Player 2. Let s_{i}, i = 1, 2 designate a pure strategy of player i. Let s_{i} ≠ 0 be the pure strategy set of player i, ℼ_{i}(s_{1}, s_{2}) be the payoff function for player i, i = 1,2.

#### (a) Define the Nash Equilibrium in pure strategies for this game.

#### Consider the following game:

Player 2 | |||
---|---|---|---|

\(\mathop s\nolimits_2^1 \) | \(\mathop s\nolimits_2^2 \) | ||

Player 1 | \(\mathop s\nolimits_1^1 \) | 10, 10 | 0, 12 |

\(\mathop s\nolimits_2^1 \) | 12, 0 | 3, 3 |

#### Show that the unique pure strategy Nash Equilibrium is not Pareto Optimal.

#### (c) Consider two firms - Firm 1 and Firm 2 producing a homogeneous good Q. The output of the two firms is given by q_{1} and q_{2} 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, ∞ > c > 0. Find the unique pure strategy Nash Equilibrium for this game.

(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 \({s_2} \in {S_2}\)

(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.

(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.

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: \[\frac{{\partial {\pi _1}}}{{\partial {q_1}}} = 0\] \[A - 2b{q_1} - b{q_2} - c = 0\] \[2b{q_1} = A - b{q_2} - c\] \[{q_1} = \frac{{A - b{q_2} - c}}{{2b}}\]

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: \[\frac{{\partial {\pi _2}}}{{\partial {q_2}}} = 0\] \[A - b{q_1} - 2b{q_2} - c = 0\] \[2b{q_2} = A - b{q_1} - c\] \[{q_2} = \frac{{A - b{q_1} - c}}{{2b}}\]

Substituting the value of q_{2} in q_{1} 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.

Similarly, we can find the demand function of Firm - 2 by the same process of substitution, which will be: \[{q_2} = \frac{{A - c}}{{3b}}\]

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: \[{q_1} = \frac{{A - P}}{{3b}}\] \[{q_2} = \frac{{A - P}}{{3b}}\]

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..

#### 4. Explain the concept of Social Welfare function. Does perfect competition ensure maximum social welfare? Analyse critically

(Comment for solution.)

#### 5 (a). How is quasi-rent diferent from rent?

(Comment for solution.)

#### 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

(Comment for solution.)

#### 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.

(Comment for solution.)

#### 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.

(Comment for solution.)

#### 6 (c). Show that the concept of marginal product is implicit in the definition of the marginal rate of technical substitution.

(Comment for solution.)

#### 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.

(Comment for solution.)

#### 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.

(Comment for solution.)

#### 8. Heights of fathers (X) and sons (Y) in inches are given in following tables:

X | 65 | 66 | 67 | 67 | 68 | 69 | 70 | 72 |
---|---|---|---|---|---|---|---|---|

Y | 67 | 68 | 65 | 68 | 72 | 72 | 69 | 71 |

#### (a) Calculate the correlation coefficient between the heights of fathers and those of sons.

(b) Obtain the equations of lines of regression and the estimate of X for Y = 70.

(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.

Solution Video

### Section - C

Answer any ** three** of the following five questions in about 300 words each.

Marks: 25 × 3 = 75

#### 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_{x} and p_{y} are the prices respectively per unit of
the goods X and Y.

Derive the demand function for X and Y. Show that these demand functions are homogeneous of degree zero
in prices and income.

Note: This question has derivation similar to Q. No. 2 of Previous Year Paper 2009. So, the dervation is written their.

Solution Video

#### 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.

(Comment for solution.)

#### 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?

(Comment for solution.)

#### 10 (a). State briefly the asumptions of Kaldor's model of income distribution.

(Comment for solution.)

#### 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'.

(Comment for solution.)

#### 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.

(Comment for solution.)

#### 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.

(Comment for solution.)

#### 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 \({W_i} = \frac{{{x_i}}}{{\sum\limits_{i = 1}^n {x_i^2} }}\)

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)

See Appendix - 3A.6 of Basic Econometrics 5th edition by Damodar N. Gujarat and Dawn C. Porter

#### 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:

\(\sum\limits_{i = 1}^{20} {{Y_i}} = 21 \cdot 9\)

\(\sum\limits_{i = 1}^{20} {{{({Y_i} - \bar Y)}^2}} = 86 \cdot 9\)

\(\sum\limits_{i = 1}^{20} {{X_i}} = 186 \cdot 2\)

\(\sum\limits_{i = 1}^{20} {{{({X_i} - \bar X)}^2}} = 215 \cdot 4\)

\(\sum\limits_{i = 1}^{20} {({X_i} - \bar X)\left( {{Y_i} - \bar Y} \right)} = 106 \cdot 4\)

(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.

(ii) How would you test your hypotheses?

[Given that \({t_{0 \cdot 05;18}} = 1 \cdot 734,\,\,{t_{0 \cdot 01;18}} = 2 \cdot 552,\,\,{t_{0 \cdot 025;18}} = 2 \cdot 101\,{\rm{ and }}{t_{0 \cdot 005;18}} = 2 \cdot 878,\,{\rm{ where }}0 \cdot 025 = \mathop \smallint \limits_{{t_{0 \cdot 025}}}^\infty \,f(t)dt\)

See Chapter - 5 of Theory of Econometrics 2nd edition by A. Koutsoyiannis

#### 13 (a). Define autocorrelation and state what are the possible sources of autocorrelation.

(Comment for solution.)

#### 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).

(Comment for solution.)

#### 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?

(Comment for solution.)

Sir In Question No. 1 (c) the langrangian that you have written is Lamda(M - P1X1 + P2X2). But usually its written as Lambda(M - P1X1 - P2X2). Why is there a plus sign beside P2X2 in your langrangian??

ReplyDeleteYou're correct. It should be minus sign.

DeleteCorrect solution has been written.

DeleteHello sir.. sir please can you give solution of question 1 part 'e' ??

ReplyDelete