(Solved): 2. (50 points) Set up a utility maximization model to think about the optimal debt of a consumer. A ...

2. (50 points) Set up a utility maximization model to think about the optimal debt of a consumer. Assume that the consumer lives for two periods and the utility function is $U$ $=u(c1)+u(c2)$ where $c1$ and $c2$ are the consumption in period 1 and 2 respectively. The consumer has 0 dollars income in period 1 and 200 dollars in period 2 . He can borrow as much as he wants in period 1 as long as he can repay the principle and interest in period 2. Let $d$ be the debt in period 1 and $r$ be the interest rate. Then $c1=d$ and $c2$ $=200?d?(1+r)$ a) Derive the present value inter-temporal budget constraint of the consumer: $c1+\frac{c2}{1+r}=\frac{200}{1+r}$ using the conditions: $c1=d$ and $c2=200?d?(1+r)$. Show your steps. b) Draw the budget constraint on coordinates where y-axis is $c2$ and $\mathrm{x}$-axis is $c1$. What is the slope of the budget line? c) Demonstrate the utility maximizing choice of the consumer as a tangency point of the budget constraint and an indifference curve. Label the optimal debt on period 1. d) Demonstrate the new utility maximizing choice when the interest rate $(r)$ decreases by $50\%$, and decompose the substitution effect and income effects by drawing an auxiliary budget line. e) Should the consumer borrow less in period 1 when the interest rate increases? Discuss the substitution and income effects to answer the question.