Question

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

intertemporal utility function is 𝑈(𝑐1 , 𝑐2 ) = 2𝑐1^2 c2^2. Her income in period 1 is 𝑚1= $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)

Utility function is given by:

$U(C_1,C_2)=2C_1^2C_2^2$

where $C_1$ is consumption in period 1.

$C_2$ is consumption in period 2.

and income in period 1, $m_1=1500$

income in period 2, $m_2=2000$

Interest rate, r=10%=0.1

(a)

Intertemporal budget constraint for Claire is:

$C_1(1+r_1)+C_2=m_1(1+r_1)+m_2$

$1.1C_1+C_2=1500(1+0.1)+2000$

$1.1C_1+C_2=3650$

(b)

For optimal consumption,

Slope of IC= slope of BL.

For $MRS=\frac{MU_{c_1}}{MU_{c_2}}=\frac{\frac{\delta u}{\delta c_1}}{\frac{\delta u}{\delta c_2}}$

$=\frac{4c_1c_2^2}{4c_1^2c_2}=\frac{c_2}{c_1}$

$\frac{c_2}{c_1}=1+0.1$

$\frac{c_2}{c_1}=1.1\implies c_2=1.1c_2$

Putting value of $c_2$ in budget constraint equation:

$1.1c_1+1.1c_1=3650$

$2.2c_1=3650$

$c_1=1659.09$

$c_2=1508.26$

(c)

If interest rate for saving is only 5%, r=5%. Since Claire is a borrower in period 1, so new slope of budget line is 1.05.

$m_1+\frac{m_2}{1+r_2}=c_1+\frac{c_2}{1+r_2}$

$1500+\frac{2000}{1.05}=c_1+\frac{c_2}{1.05}$

$3404.76=c_1+\frac{c_2}{1.05}$

(d)

Since Claire was a borrower in period 1, with a fall in interest rate for saving will make Claire better off