Question

4. Steve's utility function over leisure and consumption is given by NLY) - min (31.7. Wage rate is w and the price of the composite consumption good is p=1. (a) Suppose w = 5. Find the optimal leisure consumption combination. What is the amount of hours worked? (b) Suppose the overtime law is passed so that every worker needs to be paid 1.5 times their current wage for hours worked beyond the first 8 hours, Will this law induce Steve to work more hours? If so, how many? If not, explain 5. A consumer who lives for two periods has a standard Cobb-Douglas utility fune ca) - where = consumption in period tanda+B= 1. Her income in period one is 1, 500 and l400, and she can lend or borrow at interest rater 02. (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 is no longer 02 What values of rif any, makes the consumer a borrower? Interpret this result. 6. Consider an individual whose utility function over money is = 1+2 (a) is the individual risk-averse, risk-neutral, or risk-loving? Does it depend (b) Suppose the individual has initial wealth YW and faces the possible less of The probability that the loss will occur is. Suppose insurance is available at price p, where p is not necessarily the fair price. Find the optimal amount of insurance the individual should buy You may assume that the solution is interior (c) Is there a price at which the individual will not want to buy any insurance? If so, find it. If no, explain 7. Suppone, as al, Elmos utility function over gambles satisfies the expected utility property. Consider two gambles g and such that Ell>[A] (a) Suppose Elmo is risk-averse. Will Elmo necessarily prefer to h? Explain. (b) What if Elmo is risk-neutral? Explain. (c) What if Elmo is risk-loving? Explain.
intermediate micro

Answer 1

4. a. The consumer's problem is:
$\dpi{80} \fn_phv \small max \ min \ (3L,Y) \ s.t. \ Y+5L=120$
The consumer treats leisure and consumption as perfect complements and consumes them in a fixed ratio: $\dpi{80} \fn_phv \small Y=3L$ . Substituting this into the budget equation:
$\dpi{80} \fn_phv \small 8L=120\rightarrow L=15, Y=45$

b. The consumer's new problem is:
$\dpi{80} \fn_phv \small max \ min \ (3L,Y) \ s.t. \ Y&plus;5L=120 \ if \ L\geq 15, Y&plus;7.5L=152.5 \ if \ L<15$
Since the consumer chose to work for 8 hours at the original wage rate and this new law would affect the consumer if he were working for more than 8 hours, therefore, this new law does not affect the consumer's choice of leisure and consumption. The consumer will continue to work for 8 hours and consume 45 units.

5. a. The consumer's problem is:
$\dpi{80} \fn_phv \small max \ c_1^\alpha c_2^\beta \ s.t. \ x_1&plus;\frac{c_2}{1.2}=833.33$
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:
$\dpi{80} \fn_phv \small c_1&plus;\frac{\frac{1.2\beta c_1}{\alpha}}{1.2}=833.33\rightarrow c_1=\frac{833.33}{\frac{\alpha&plus;\beta}{\alpha}}\rightarrow c_1=833.33\alpha \because \alpha&plus;\beta=1$
$\dpi{80} \fn_phv \small c_2=1000*\frac{1-\alpha}{\alpha}$

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'.
$\dpi{80} \fn_phv \small \frac{4/7 c_2}{3/7 c_1}=1&plus;r\rightarrow c_2=\frac{c_1(1&plus;r)}{4}$
Substituting this into the consumer's budget equation:
$\dpi{80} \fn_phv \small c_1&plus;\frac{\frac{c_1(1&plus;r)}{4}}{1&plus;r}=500&plus;\frac{400}{1&plus;r}\rightarrow c_1=\frac{720&plus;400r}{1&plus;r},c_2=180&plus;100r$
For the consumer to be a borrower, consumption in period one is greater than income in period one:
$\dpi{80} \fn_phv \small c_1=\frac{720&plus;400r}{1&plus;r}>500\rightarrow r<2.2$