Question

Question 3

Claire consumes 𝑐1 and 𝑐2 in period 1 and period 2 respectively, and her

intertemporal utility function is 𝑈(𝑐 , 𝑐 ) = 2𝑐2𝑐2. Her income in period 1 is 𝑚 = 1212 1

$1,500 and period 2 is 𝑚2 = $2,000. Assume that the interest rate is 10% for both borrowing and saving. [25%]

a. Find the intertemporal budget constraint for Claire.

b. Find the optimal consumption.

c. Assume now that the interest rate for saving is only 5%. Find the new

intertemporal budget constraint.

d. Would Claire be better off at the new interest rate in (c)? Discuss.

Answer 1

(a)

U(c,c)=2c2c2

The budget constraint for period 1 is:

$c_1+b=m_1$

The budget constraint for period 2 is:

$c_2=m_2+(1+r)b$

(b)

The consumer has a utility function over $c_1$ and $c_2$ .

$U(c_1,c_2)=u(c_1)+\beta u(c_2)$

$\beta$ is the time discount rate.

(c)

From the first period budget constraint,

$b=m_1-c_1$

Plugging this into the second period budget constraint yields:

$c_2=m_2+(1+r)(m_1-c_1)$

$\implies (1+r)c_1+c_2=(1+r)m_1+m_2$

$c_1+(\frac{1}{1+r})c_2=m_1+(\frac{1}{1+r})m_2$

(d)

Claire would be better off at the new interest rate.