Question

Question 1. (Consumption-Saving Problem): Suppose that a consumer lives for two periods. The utility function of the consumer is given by with u> 0 where c and c2 are consumption in period 1 and period 2 respectively. Sup- pose that consumer has income y in the first period, but has no income in the second period. Consumer has to save in the first period in order to consume in the second period. Let s be the savings in the first period and R be the gross market rate of interest. I Write the inter-temporal budget constraint of the consumer and interpret it. [Marks 0.5 II Set up the consumer optimization problem and illustrate its optimal choices using the indifference curve and inter-temporal budget constraint. [Marks III Derive first order conditions and interpret them in terms of marginal cost and marginal benefit of optimal choices. [ Marks 1 IV Derive optimal choices of the consumer. Now suppose that y changes by one unit. How does it affect optimal consumption pattern and saving? Marks 1.5

Answer 1

Intertemporal budget constraint: c_1 = y - s: c_2 = Rs c_1 + c_2/R = y - s + R_s/R = y where c_1 and c_2 are the consumptions in time 1 and 2 respectively. The consumption at time 2 is divided by r to get the discounted value of future consumption at time 1. The income is y in period 1. The budget constraint says that discounted value of life - time consumption today equals the lifetime income. 2) the consumer aims to maximize his utility by carefully selecting consumption in period 1 and period 2 given the fixed income. Therefore, the optimization problem as follows: m a x_c_1, c_2 U(c_1, c_2) s. t c_1 + c_2/R = y setting up the Lagrange: L(c_1, c_2, lambda) = max_c_1, c_2 lambda U(c_1, c_2) - lambda (c_1 + c_2/R - y) = max_c_1 c_2 lambda c^1 - mu_1/1 - mu + c^1 - mu_2/1 - mu - lambda (c_1 + c_2/R - y) 3) First order conditions w. r. t. c_1, c_2, lambda