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Past Year Entrance Exams with solutions
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∗Source †Updated: July 30, 2021
1 ISI PEB 2006
Li, i = 1, 2 , find the allocation of labour between the two sectors. (b) Suppose that prices of all variable factors and output double. What will be its effect on the short-run equilibrium output of a competitive firm? Examine whether the short-run profit of the firm will double. (c) Suppose in year 1 economic activities in a country constitute only production of wheat worth Rs. 750. Of this, wheat worth Rs. 150 is exported and the rest remains unsold. Suppose further that in year 2 no production takes place, but the unsold wheat of year 1 is sold domestically and residents of the country import shirts worth Rs. 250. Fill in, with adequate explanation, the following chart:
Year GDP = Consumption + Investment + Export − Import 1 2
L − 2 L. He has 4 units of labour in his family and he cannot hire labour from the wage labour market. He does not face any cost of employing family labour. (a) Find out his equilibrium level of output. (b) Suppose that the government imposes an income tax at the rate of 10 per cent. How does this affect his equilibrium output? (c) Suppose an alternative production technology given by: F (L) = 11
L − L − 15 is available. Will the farmer adopt this alternative technology? Briefly justify your answer.
(a) Show that θ represents the elasticity of marginal utility with respect to consumption in each period. (b) Write down the agent’s optimization problem, i.e., her problem of maximizing utility subject to the budget constraint. (c) Find an expression for s as a function of w and r. (d) How does s change in response to a change in r? In particular, show that this change depends on whether θ exceeds or falls short of unity. (e) Give an intuitive explanation of your finding in (d)
2 ISI PEB 2007
Year GDP = Consumption + Investment + Export − Import 1 2 (b) Consider an IS-LM model for a closed economy with government where investment (I) is a function of rate of interest (r) only. An increase in government expenditure is found to crowd out 50 units of private investment. The government wants to prevent this by a minimum change in the supply of real money balance. It is given that dIdr = −50, slope of the LM curve, drdy (LM ) = 2501 , slope of the IS curve, dr dy (IS) =^ −^
1 125 ,^ and all relations are linear.^ Compute the change in^ y^ from the initial to the final equilibrium when all adjustments have been made.
x 2 x 3.
average propensities to consume and import are 0.8 and 0. 3 , respectively. The investment (I) function and the level of export (X) are given by I = 100 + 0. 4 Y and X = 100. (a) Compute the aggregate demand function if the maximum possible level of imports is 450. Can there be an equilibrium for this model? Show your result graphically. (b) How does your answer to part a change if the limit to import is raised to 615? What can you say about the stability of equilibrium if it exists?
u (C 1 , C 2 ) =
C 11 −θ− 1 1 − θ
1 + ρ
C^12 −θ− 1 1 − θ
, 0 < θ < 1 , ρ > 0
where the first term represents utility from consumption during youth. The second term represents discounted utility from consumption in old age, 1/(1 + ρ) being the discount factor. During the period, the agent has a unit of labour which she supplies inelastically for a wage rate w. Any savings (i.e., income minus consumption during the first period) earns a rate of interest r, the proceeds from which are available in old age in units of the only consumption good available in the economy. Denote savings by s. The agent maximizes utility subjects to her budget constraint. (a) Show that θ represents the elasticity of marginal utility with respect to consumption in each period. (b) Write down the agent’s optimization problem, i.e., her problem of maximizing utility subject to the budget constraint. (c) Find an expression for s as a function of w and r. (d) How does s change in response to a change in r? In particular, show that this change depends on whether θ exceeds or falls short of unity. (e) Give an intuitive explanation of your finding in d
3 ISI PEB 2008
(b) Suppose there is a temporary increase in nominal money supply by 2%. Find the new equilibrium income and the rates of interest. (c) Now assume that the 2% increase in nominal money supply is permanent leading to a 2% increase in the expected future price level. Work out the new equilibrium income and the rates of interest.
(b) What is the set of perfectly competitive (Walrasian) outcomes? You may use dia- grams for parts (i) and (ii). (c) Are the perfectly competitive outcomes Pareto optimal? Does this result hold generally in all exchange economies?
[5,5,5,5] 5. A monopoly sells its product in two separate markets. The inverse demand function in market 1 is given by q 1 = 10 − p 1 , and the inverse demand function in market 2 is given by q 2 = a − p 2 , where 10 < a ≤ 20. The monopolist’s cost function is C(q) = 5q, where q is aggregate output. (a) Suppose the monopolist must set the same price in both markets. What is its optimal price? What is the reason behind the restriction that a ≤ 20? (b) Suppose the monopolist can charge different prices in the two markets. Compute the prices it will set in the two markets. (c) Under what conditions does the monopolist benefit from the ability to charge dif- ferent prices? (d) Compute consumers’ surplus in cases a and b. Who benefits from differential pricing and who does not relative to the case where the same price is charged in both markets?
[3,5,12] 6. Consider an industry with 3 firms, each having marginal cost equal to 0. The inverse demand curve facing this industry is p = 120 − q, where q is aggregate output. (a) If each firm behaves as in the Cournot model, what is firm 1 ’s optimal output choice as a function of its beliefs about other firms’ output choices? (b) What output do the firms produce in equilibrium? (c) Firms 2 and 3 decide to merge and form a single firm with marginal cost still equal to 0. What output do the two firms produce in equilibrium? Is firm 1 better off as a result? Are firms 2 and 3 better off post-merger? Would it be better for all the firms to form a cartel instead? Explain in each case.
[2,6,6,6] 7. Suppose the economy’s production function is given by
Yt = 0. 5
Kt
Nt (1)
Yt denotes output, Kt denotes the aggregate capital stock in the economy, and N denotes the number of workers (which is fixed). The evolution of the capital stock is given by;
Kt+1 = sYt + (1 − δ)Kt (2)
where the savings rate of the economy is denoted by, s, and the depreciation rate is given by, δ. (a) Using equation (2), show that the change in the capital stock per worker, Kt+1 N− Kt, is equal to savings per worker minus depreciation per worker. (b) Derive the economy’s steady state levels of KN and YN in terms of the savings rate and the depreciation rate.
(c) Derive the equation for the steady state level of consumption per worker in terms of the savings rate and the depreciation rate. (d) Is there a savings rate that is optimal, i.e., maximizes steady state consumption per worker? If so, derive an expression for the optimal savings rate. Using words and graphs, discuss your answer.
[5,5,2,8] 8. Suppose there are 10 individuals in a society, 5 of whom are of high ability, and 5 of low ability. Individuals know their own abilities. Suppose that each individual lives for two periods and is deciding whether or not to go to college in period 1. When individuals make decisions in period 1, they choose that option which gives the highest lifetime payoff, i.e., the sum of earnings and expenses in both periods. Education can only be acquired in period 1. In the absence of schooling, high and low ability individuals can earn yH and yL respectively in each period. With education, period 2 earning increases to (1 + a)yH for high ability types and (1 + a)yL for low ability types. Earnings would equal 0 in period 1 if an individual decided to go to college in that period. Tuition fee for any individual is equal to T. Assume yH and yL are both positive, as is T. (a) Find the condition that determines whether each type of person will go to college in period 1. What is the minimum that a can be if it is to be feasible for any type of individual to acquire education? (b) Suppose yH = 50, yL = 40, a = 3. For what values of T will a high ability person go to college? And a low ability person? Which type is more likely to acquire education? (c) Now assume the government chooses to subsidise education by setting tuition equal to 60. What happens to educational attainment? (d) Suppose now to pay for the education subsidy, the government decides to impose a x% tax on earnings in any period greater than 50. So if an individual earns 80 in a period, he would pay a tax in that period equal to x% of 30. The government wants all individuals to acquire education, and also wants to cover the cost of the education subsidy in period 1 through tax revenues collected in both periods. What value of x should the government set?
[5,5,5,5] 9. Consider the goods market with exogenous (constant) investment I,¯ exogenous govern- ment spending, G¯ and constant taxes, T. The consumption equation is given by,
C = c 0 + c 1 (Y − T )
where C denotes consumption, c 0 denotes autonomous consumption, and c 1 the marginal propensity to consume. (a) Solve for equilibrium output. What is the value of the multiplier? (b) Now let investment depend on Y and the interest rate, i
I = b 0 + b 1 Y − b 2 i
5 ISI PEB 2010
[3+8+4,5] 1. (a) Imagine a closed economy in which tax is imposed only on income. The government spending (G) is required (by a balanced budget amendment to the relevant law) to be equal to the tax revenue; thus G = tY , where t is the tax rate and Y is income. Consumption expenditure (C) is proportional to disposable income and investment (I) is exogenously given. i. Explain why government spending is endogenous in this model. ii. Is the multiplier in this model larger or smaller than in the case in which government spending is exogenous? iii. When t increases, does Y decrease, increase or stay the same? Give an answer with intuitive explanation. (b) Consider the following macroeconomic model with notation having usual meanings: C = 100 + 1. 3 Y (Consumption function), I = (^500) r (Investment function), M D^ = 150 Y + 100 − 1500 r (Demand for money function) and M S^ = 2100 (Supply of money). Do you think that there exists an equilibrium? Justify your answer using the IS-LM model.
[8,5,7] 2. Consider a market with two firms. Let the cost function of each firm be C(q) = mq where q ≥ 0. Let the inverse demand functions of firms 1 and 2 be P 1 (q 1 , q 2 ) = a − q 1 − sq 2 and P 2 (q 1 , q 2 ) = a − q 2 − sq 1 , respectively. Assume that 0 < s < 1 and a > m > 0 (a) Find the Cournot equilibrium quantities of the two firms. (b) Using the inverse demand functions P 1 (q 1 , q 2 ) and P 2 (q 1 , q 2 ) , derive direct demand functions D 1 (p 1 , p 2 ) and D 2 (p 1 , p 2 ) of firms 1 and 2. (c) Using the direct demand functions D 1 (p 1 , p 2 ) and D 2 (p 1 , p 2 ) , find the Bertrand equilibrium prices.
[12,4+4] 3. (a) A monopolist can sell his output in two geographically separated markets A and B. The total cost function is T C = 5 + 3 (QA + QB ) where QA and QB are quantities sold in markets A and B respectively. The demand functions for the two markets are, respectively, PA = 15 − QA and PB = 25 − 2 QB. Calculate the firm’s price, output, profit and the deadweight loss to the society if it can get involved in price discrimination. (b) Suppose that you have the following information. Each month an airline sells 1500 business-class tickets at Rs. 200 per ticket and 6000 economy class tickets at Rs. 80 per ticket. The airline treats business class and economy class as two separate markets. The airline knows the demand curves for the two markets and maximizes profit. It is also known that the demand curve of each of the two markets is linear and marginal cost associated with each ticket is Rs. 50. i. Use the above information to construct the demand curves for economy class and business class tickets. ii. What would be the equilibrium quantities and prices if the airline could not get involved in price discrimination?
[15,5] 4. Consider an economy producing two goods 1 and 2 using the following production func- tions: X 1 = L
(^12) 1 K^
(^12) and X 2 = L
(^12) 2 T^
(^12) , where X 1 and X 2 are the outputs of good 1 and 2, respectively, K is capital used in production of good 1, T is land used in pro- duction of good 2 and L 1 and L 2 are amounts of labour used in production of good 1 and 2 , respectively. Full employment of all factors is assumed implying the following: K = K, T¯ = T , L¯ 1 + L 2 = L¯ where K,¯ T¯ and L¯ are total amounts of capital, land and labour available to the economy. Labour is assumed to be perfectly mobile between sectors 1 and 2. The underlying preference pattern of the economy generates the rel- ative demand function, D D^12 = γ
p 1 p 2
, where D 1 and D 2 are the demands and p 1 and p 2 prices of good 1 and 2 respectively. All markets (both commodities and factors) are competitive. (a) Derive the relationship between X X^12 and p p^12. (b) Suppose that γ goes up. What can you say about the new equilibrium relative price?
[9,6,5] 5. Consider the IS-LM representation of an economy with the following features:
∗ P −^ mY,^ where^ Y^ is GDP and^ β^ and^ m^ are positive parameters,^ m being the marginal propensity to import. (a) Taking into account trade balance equilibrium and commodity market equilibrium, derive the relationship between Y and the interest rate (r). Is it the same as in the IS curve for the closed economy? Explain. Draw also the LM curve on the (Y, r) plane. (b) Suppose that the government spending is increased. Determine graphically the new equilibrium value of Y. How does the equilibrium value of e change? (c) Suppose that P ∗^ is increased. How does it affect the equilibrium values of Y and e?
[14,2+2+2] 6. (a) A firm can produce its product with two alternative technologies given by Y = min
L 2
and Y = min
L 3
. The factor markets are competitive and the marginal cost of production is Rs.20 with each of these two technologies. Find the equation of the expansion path of the firm if it uses a third production technology given by Y = K
2 (^3) L 1 (^3). (b) A utility maximizing consumer with a given money income consumes two com- modities X and Y. He is a price taker in the market for X. For Y there are two alternatives:
[8,6+3+3]10. (a) Consider an economy with two persons (A and B) and two goods (1 and 2). Utility functions of the two persons are given by UA (xA 1 , xA 2 ) = xαA 1 + xαA 2 with 0 < α < 1; and UB (xB 1 , xB 2 ) = xB 1 + xB 2. Derive the equation of the contract curve and mention its properties. (b) i. A firm can produce a product at a constant average (marginal) cost of Rs. 4. The demand for the good is given by x = 100 − 10 p. Assume that the firm owner requires a profit of Rs. 80. Determine the level of output and the price that yields maximum revenue if this profit constraint is to be fulfilled. ii. What will be the effects on price and output if the targeted profit is increased to Rs. 100? iii. Also find out the effects of the increase in marginal cost from Rs. 4 to Rs. 8 on price and output.
6 ISI PEB 2011
[30] 1. A monopolist sells two products, X and Y. There are three consumers with asymmetric preferences. Each consumer buys either one unit of a product or does not buy the product at all. The per-unit maximum willingness to pay of the consumers is given in the table below. Consumer No. X Y
The monopolist who wants to maximize total payoffs has three alternative marketing strategies:
However, the monopolist cannot price discriminate between the consumers. Given the above data, find out the monopolist’s optimal strategy and the corresponding prices of the products.
[6,18,6] 2. Consider two consumers with identical income M and utility function U = xy where x is the amount of restaurant good consumed and y is the amount of any other good consumed. The unit prices of the goods are given. The consumers have two alternative plans to meet the restaurant bill. Plan A: they eat together at the restaurant and each pays his own bill. Plan B: they eat together at the restaurant but each pays one-half of the total restaurant bill. (a) Find equilibrium consumption under plan A. (b) Find equilibrium consumption under plan B. (c) Explain your answer if the equilibrium outcome in case b differs from that in case a.
[7,16,7] 3. Consider a community having a fixed stock X of an exhaustible resource (like oil) and choosing, over an infinite horizon, how much of this resource is to be used up each period. While doing so, the community maximizes an intertemporal utility function U =
t=0 δ
t (^) ln Ct where Ct represents consumption or use of the resource at period t and δ(0 < δ < 1) is the discount factor (a) Set up the utility maximization problem of the community and interpret the first order condition.
