Question

Assume a consumer who has current-period income y=200, future period income y’=150, current taxes t = 40, and future taxes t’= 50, and faces a market interest rate of r=5 percent or .05. The consumer would like to consume such that c’=c*(1+r) if possible. However, this consumer is faced with a credit market imperfection, in that no borrowing is allowed. That is s must be greater or equal to zero.  

a. Show the consumer’s lifetime budget constraint and indifference curves in a diagram.

b. Calculate the optimal c and c’ for this consumer and show this in your diagram.

c. Suppose that everything stays the same except that t = 20 and t’ = 72. Calculate the effects on c, c’, and s. Show this in your diagram.

d. Now, suppose that y = 100. Repeat the previous 3 parts and explain any differences.

Answer 1

Consumer lifetime budget constraint

C'=C*(1+r), So that the lifetime budget constraint is C+C/1+r

Then C+C'/1.05=200 -40 + 150-50/1.058

C+0.95 C = 255.2

El= (160,100) initial endowment point and BEID=Initial budget constraint. Thus a kink at El because the consumer cannot borrower and cannot consume more than 160 (which is y-t) in the first period

Now we calculate optimal C and C' for the consumer.

C=C',C+0.95 c = 255.2

C=C' =130.7 and

S=y-t-C

S=160-130.7=29.3

In this situation, the fact that the consumer cannot borrow does not matter for the consumer's choice, as the consumer decides to be a lender.

Suppose that everything unchanged, except the now t=20andt'=72 then we calculate the effect of C+C'/1.05 = 200-20 + 150-72/1.05

Therefore, the lifetime consumer's wealth remains constant at 255.2 and budget constraint shift to BE2F.

The new endowment point is E2 = 80, 79). This change does not matter for the consumer again because he chooses to at lender. C=C'=130.7 and saving (s) = 49.3