Question

Question 1: Changes in the real interest rate and borrowing constraints Consider the problem of an agent that has an endowment of y- 2 apples in period 1 and y2 in period 2, and takes the real interest rate r as given. Preferences are given by: 4 apples U (c1,c2)log (ci) + log (c2) where cı and c2 denote consumption of apples in period 1 and period 2, respectively. a) Solve for c1,c2 and savings when the real interest rate is r 0 (i.e. 1+r1). Use a figure to illustrate your answers (include an example indifference curve in your figure and make sure to label the axes) b) Solve for ci, c2 and savings when the real interest rate is r -1 (i.e. 12). On the same figure used in part 1or using a new figure, illustrate how cı, c2 and savings responds to the change in r. c) Is the consumer better-off when r = 0 or when r = 1? Provide an intuitive explanation for your answer. (Hint: is the consumer initially a saver or borrower?) d) Back to the original situation: r = 0, Now, suppose the consumer faces a borrowing limit: she is only allowed to borrow a maximum of 1 apple in period 1. Solve for ci,c2 and savings. Does the borrowing constraint bind or not? Use a figure to illustrate your answers.

Answer 1

a) The utility function is given as u = log (c_1) + log (c_2) The budget line: c_1 + c_2/1 + r = y_1 + y_2/1 + r Therefore c_1 + c_2 = 6 At equilibrium: Mu (c_1)/Mu(c_2) = 1 + r or, 1/c_1/1/c_2 = 1 Therefore c_1 = c_2 From budget constraint: c_1 = c_2 = 6/2 = 3 Therefore s = -1 The consumer is borrower, she borrows lapple from future. b) At equilibrium: Mu(c_1)/Mu(c_2) = 1 + v = 2 Therefore c_1 = 2c_2 From budget constraint: c_1 + c_2/2 = 4 Therefore c_1 = 2, c_2 = 4 c) The consumer now consumes at endowment. As be agent is net borrower increase in interest rate makes him worse off. The consumer takes current consumption in terms of future too expensive.

d) After the borrowing constraint, the budget line becomes AE_aC'. In this case with r=0, the consumer consumes c1=3 in period 1. Thus, with the constraint, the consumer still consumes the same amount as he would have been without any constraint on borrowing. Then The constraint is not binding.