Thomas has income of $1500 today and $1000 tomorrow. He can lend and borrow at a

Question

Thomas has income of $1500 today and $1000 tomorrow. He can lend and borrow at an interest rate of 10%. There is 10% inaation. His preferences for intertemporal consumption are represented by the following utility function U (c1 ; c2 ) = c1 + c2 .

(a) Write down the equation for his intertemporal budget constraint (c1 is his consumption today and c2 is his consumption tomorrow), graph it and label his endowment and the intercepts on each axis.

(b) What is his optimal consumption bundle?

(c) If interest rates increases to 20%, what will be his new optimal consumption bundle? Is he better or worse o§?

Explanation / Answer

c1 + c2 /1 + r = m1 + m2/ 1 + r .

There is no inflation, the interest rate is 10%, and the consumer has income 100 in period 1 and 121 in period 2. Then the consumer’s budget constraint c1+c2/1.1 = 100 + 121/1.1 = 210. The ratio of the price of good 1 to the price of good 2 is 1 + r = 1.1. The consumer will choose a consumption bundle so that MU1/MU2 = 1.1. But MU1 = c2 and MU2 = c1, so the consumer must choose a bundle such that c2/c1 = 1.1. Take this equation together with the budget equation to solve for c1 and c2. The solution is c1 = 105 and c2 = 115.50. Since the consumer’s period-1 income is only 100, he must borrow 5 in order to consume 105 in period 1.

there happened to be an inflation rate of 20%, and suppose that the price of period-1 goods is 1. Then if you save $1 in period 1 and get it back with 10% interest, you will get back $1.10 in period 2. But because of the inflation, goods in period 2 cost 1.06 dollars per unit. Therefore the amount of period-1 consumption that you have to give up to get a unit of period-2 consumption is 1.06/1.10 = .964 units of period-2 consumption. If the consumer’s money income in each period is unchanged, then his budget equation is c1 + .964c2 = 210. This budget constraint is the same as the budget constraint would be if there were no inflation and the interest rate were r, where .964 = 1/(1 + r). The value of r that solves this equation is known as the real rate of interest. In this case the real rate of interest is about .038. When the interest rate and inflation rate are both small, the real rate of interest is closely approximated by the difference between the nominal interest rate, (10% in this case) and the inflation rate (6% in this case), that is, .038 .10 .06.

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