Claire consumes π1 and π2 in period 1 and period 2 respectively, and her
intertemporal utility function is π(π1 , π2 ) = 2π1^2 c2^2. Her income in period 1 is π1= $1,500 and period 2 is π2 = $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.
(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
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