Question

5. A consumer who lives for two periods has a standard Cobb-Douglas utility func- tion: u(C1,C2) = ccm, where Ct = consumption in period t and a + b = 1. Her income in period one is 1 = 500 and 12 = 400, and she can lend or borrow at interest rate r = 0.2. (a) Find the optimal consumption demand. (b) What values of a, if any, makes the consumer a borrower? Interpret this result. (c) Suppose now that a = but that r is no longer 0.2. What values of r, if any, makes the consumer a borrower? Interpret this result.

Answer 1

a. The consumer's problem is:

$\dpi{80} \fn_phv \small max \ c_1^\alpha c_2^\beta \ s.t. \ x_1+\frac{c_2}{1.2}=833.33$

At equilibrium, marginal rate of substitution is equal to the 'price ratio'.

$\dpi{80} \fn_phv \small \frac{\alpha c_1^{\alpha-1}c_2^\beta}{\beta c_1^\alpha c_2^{\beta -1}}=\frac{\alpha c_2}{\beta c_1}=1.2\rightarrow c_2=\frac{1.2\beta c_1}{\alpha}$

Substituting this into the consumer's budget equation:

$\dpi{80} \fn_phv \small c_1+\frac{\frac{1.2\beta c_1}{\alpha}}{1.2}=833.33\rightarrow c_1=\frac{833.33}{\frac{\alpha+\beta}{\alpha}}\rightarrow c_1=833.33\alpha \because \alpha+\beta=1$

$\dpi{80} \fn_phv \small c_2=1000*\frac{1-\alpha}{\alpha}$

b. For the consumer to be a borrower, consumption in period one has to be greater than income in period one:

$\dpi{80} \fn_phv \small 833.33\alpha>500\rightarrow \alpha>0.6$

c. The MRS is equal to 'price ratio'.

$\dpi{80} \fn_phv \small \frac{4/7 c_2}{3/7 c_1}=1+r\rightarrow c_2=\frac{c_1(1+r)}{4}$

Substituting this into the consumer's budget equation:

$\dpi{80} \fn_phv \small c_1+\frac{\frac{c_1(1+r)}{4}}{1+r}=500+\frac{400}{1+r}\rightarrow c_1=\frac{720+400r}{1+r},c_2=180+100r$

For the consumer to be a borrower, consumption in period one is greater than income in period one:

$\dpi{80} \fn_phv \small c_1=\frac{720&plus;400r}{1&plus;r}>500\rightarrow r<2.2$