Practice Questions for Mid-Term Examination - I. In answering questions just consider symmetric and stationary allocations!

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1 Practice Questions for Mid-Term Examination - I In answering questions just consider symmetric and stationary allocations! Question 1. Consider an Overlapping Generation (OLG) model. Let N t and N t 1 denote the population of the young and the old respectively at time t. The population is growing at a constant rate, N t = nn t 1, t 1. Suppose that each young individual receives an endowment of a perishable good y 1. Each old receives an endowment of the same perishable good y 2 (< y 1 ). Let the utility function of an individual be U(, ) = ln + ln where and are consumption as young and consumption as old respectively. I. Define the Golden Rule allocations. [Marks 1] Suggested Answer The Golden Rule allocations are consumption pattern, (, ), chosen by the social planner, which imizes the utility of future generations subject to the resource constraint. II. Derive and illustrate the Golden Rule allocations. [Marks 9] Suggested Answer The social planner problem is subject to, log + log + n = y 1 + y 2 n. (1) (1) and (2) together give the Golden rule allocations: = n. (2) c 1 = 1 2 [y 1 + y 2 n ] & c 2 = n 2 [y 1 + y 2 1

2 Question 2. Now consider a monetary economy. Suppose that the government distributes M units of money among the initial old. Rest of the structure of the economy remains identical to question 1. I. Define the competitive monetary equilibrium. [Marks 1] Suggested Answer (,, m t,, +1 ) together constitute the competitive equilibrium such that given and +1, the individual choice variables (,, m t ) imize his/her utility subject to his/her budget constraint, and markets clear. II. Set up an individual s utility imization problem and show that the first order condition yields = +1. [Marks 5] Suggested Answer The individual imizes his/her utility subject to the budget constraints,,m t log + log + m t = y 1 (1) The first order condition yields = m t +1 + y 2. (2) = +1. (3) III. Show that the rate of return on money, +1 = n. [Marks 2] Suggested Answer Using (1) and the money market clearing condition M = N t m t, we can derive (3) and (4) imply that +1 = n. (4) = n. (5) IV. Derive the allocations,, and m t q where is the purchasing power of money and m t is the demand for money by an individual. Illustrate the optimal choices. [Marks 7] Suggested Answer The goods market clearing condition is 2

3 From (5) and (6), we have + n = y 1 + y 2 n. (6) c 1 = 1 2 [y 1 + y 2 n ] & c 2 = n 2 [y 1 + y 2 Using the first period budget constraint, we get m v t = y 1 c 1 = 1 2 [y 1 y 2 Question 3. Consider an Overlapping Generation (OLG) model. Let N t and N t 1 denote the population of the young and the old respectively at time t. Suppose that each young individual receives an endowment of a perishable good y. The old do not receive any endowment. Let the utility function of an individual be U(,t,,t+1 ) where,t and,t+1 are consumption as young and consumption as old respectively. The consumption function is strictly increasing and concave in both the arguments. I. Define the Golden Rule allocations. [Marks 3] Answer: The Golden Rule allocations are the consumption pattern, {, }, chosen by the central/social planner, which imizes the utility of future generations subject to the feasibility constraint. II. Suppose that the population is constant, N t = N, t. Let the utility function has the following specific form U(,t,,t+1 ) = ln,t ln,t+1. Derive and illustrate the Golden Rule allocations. [Marks 19] Answer: subject to the feasibility constraint, ln +.9 ln + = y. By putting the feasibility constraint in the objective function, we get 3

4 ln(y ) +.9 ln. 1 y =.9 1 which implies that =.9y 1.9. From the feasibility constraint then we get = y 1.9. III. illustrate the Golden Rule allocations. [Marks 5] Draw the diagram showing tangency between the indifference curve and the feasibility constraint. Question 4. Now consider a monetary economy with no population growth, N t = N, t. Suppose that the government distributes M units of money among the initial old. Rest of the structure of the economy remains identical to question 3. I. Define the competitive monetary equilibrium. [Marks 3] Answer: The competitive monetary equilibrium consists of allocations,,t,,t, m t, and prices,, p t for all t, such that individual agents imize their utility subject to their budget constraint and the goods and the money markets clear. II. Let the utility function has the following specific form U(,t,,t+1 ) = ln,t ln,t+1. Derive the allocations,, and m t q where is the purchasing power of money and m t is the demand for money by an individual. [Marks 20] Answer: subject to the budget constraints,t,,t+1,m t ln,t ln,t+1 and,t = y m t,t+1 = +1 m t. By putting the budget constraints in the utility function, we have m t ln(y m t ) ln(+1 m t ). 4

5 = 0.9 y m t m t which in the stationary environment implies that q = m t = y. Then from the first-period budget constraint, we have = y q = y 1.9. The goods market clearing condition, + = y, implies that = y. III. Illustrate the optimal choices. [Marks 5] Draw the diagram depicting tangency between the indifference curve and the individual budget constraint. IV. Does competitive monetary equilibrium deliver golden rule allocations? If yes, what is the intuition. [Marks 5] The comparison of the allocations under the competitive monetary equilibrium with the Golden rule allocations (question 1) shows that both allocations coincide. The reason is that the introduction of money allows young and old to trade, which was not possible under the barter trade. 5