Answer to Question #269238 in Microeconomics for lilu

Question #269238

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

intertemporal utility function is π‘ˆ(𝑐1 , 𝑐2 ) = 2𝑐1^2 c2^2. Her income in period 1 is π‘š1= $1,500 and period 2 is π‘š2 = $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.


1
Expert's answer
2021-11-23T11:11:08-0500

(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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