### Section - A

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

Marks: 5 × 10 = 50

#### (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.

(Comment for solution.)

#### (b) Show that if the consumer is free from money illusion, the demand function is homogeneous of degree zero. (Marks - 5)

Solution video:

#### (c) Given the different views of equity and use utility possibility frontier to show that efficieny does not necessarily imply equity.

(Comment for solution.)

#### (d) State the assuptions of Classical Linear Regression Model. Why are the regressors (X) assumed to be non-stochastic in repeated samples?

(Comment for solution.)

#### (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.

(Comment for solution.)

#### (f) Define level of significance. How is this level decided for a given problem? Can we take it as 2% or 6%? Explain.

(Comment for solution.)

### Section - B

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

Marks: 15 × 6 = 90

#### 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.

(Comment for solution.)

#### 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."

(Comment for solution.)

#### 4. Distinguish between point estimation and interval estimation of a populaation parameter. State the small sample properties of a good estimator.

(Comment for solution.)

#### 5 (a). Derive the long run supply function under perfect competition when there are external economies or external diseconomies.

(Comment for solution.)

#### 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 \({a_1} > 0,\,\,{a_2} > 0\).

Derive the aggregate supply function of the industry when there are (i) external economies (b<0), and (ii) external diseconomies (b<0).

(Comment for solution.)

#### 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.

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.

(Comment for solution.)

#### 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.

(Comment for solution.)

#### 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.

(Comment for solution.)

### Section - C

Answer any ** four** of the following six questions in about 300 words each.

Marks: 20 × 4 = 80

#### 9 (a). Distinguish between a cooperative and a non-cooperative game. (Marks - 5)

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.

While in a non-cooperative game players compete to optimize their payoffs. Here, game theory focuses on finding Nash Equilibrium.

#### 9 (b). In a non-cooperative game, find:
- saddle point in a pure strategy game. (Marks - 5)
- maximim expected payoff in a mixed strategy game (Marks - 5)
- solution of a sequential game in an 'extensive form' (Marks - 5)

- 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.

This video will help you to understand this point well - Game Thery Basics - 2: Saddle point - 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.
**Maximum expected payoff**is a set of maximum payoffs from each column of the expected payoff matrix. This is used in minimax strategy. - There are two methods of solution of sequesntial game in extensive form:

**1. Subgame Perfect Equilibrium**

**2. Backward Induction**

In the sumgame perfect equilibrium, a game can be divided into many sub gamesand each game has an equilibrium decision point.

In the backward induction method,the process of solution starts from the the last point of extensive form.

For detailed explanation, see Chapter - 8: Microeconomics - Basic Principles and Extensions by Walter Nicholsan and Christopher Snyder

#### 10 (a). Define heteroscedasticity.

(Comment for solution.)

#### 10 (b). Explain:

(i) Consequences of heteroscedasticity on OLS estimates

(ii) Detection of heteroscedasticity in a model

(iii) Estimation procedure in the presence of heteroscedasticity

(Comment for solution.)

#### 11. Given the Classical Linear Regression model with usual assumptions
\[{Y_i} = {\beta _0} + {\beta _1}{X_i} + {U_i}\,\,\,\,\,\,\,\,\,\,\,\,i = 1,\,2,\,...\,\,n\]

(a) Examine the goodness of fit of the model using ANOVA.

(b) If the value of \({\bar R^2}\) is low, how can it be improved?

(Comment for solution.)

#### 12. Distinguish between basic feasible solution, feasible solution and optimal solution of a Linear Programming
Problem (LPP). Solve the following LPP graphocally:

Maximize Y = q_{1} + 2q_{2}

subject to

q_{1} + 3q_{2} ⋜ 18

q_{1} + q_{2} ⋜ 8

2q_{1} + q_{2} ⋜ 14

q_{1}, q_{2} ⋝ 0

Solution video:

#### 13. Examine the situation of market-equilibrium when;

(a) Supply and demand are not equal at a non-negative price-quantity combination.

(b) Supply and demand are equal at more than one non-negative price-quantity combination.

(Comment for solution.)

#### 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.

(Comment for solution.)

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