[Answer] Consider a consumer with preferences: U(x_1,x_2 - #80141
Problem 1:
Consider a consumer with preferences: \(U\left( {{x}_{1}},{{x}_{2}} \right)=\ln \left( {{x}_{1}} \right)+{{x}_{2}}\) and income and prices p1, p2, m.
a. Formulate the consumer's choice problem.
b. Solve for the demand functions \({{x}_{1}}\left( {{p}_{1}},{{p}_{2}},m \right),\,\,{{x}_{2}}\left( {{p}_{1}},{{p}_{2}},m \right)\)
c. Suppose p1 = p2 = 2. Depict the Engel curve.
d. Depict the demand curve for good 1. Assume that m = 10, p2 = 5.
a Check if good 1 and 2 are normal or inferior, if they are ordinary or giffen goods and
whether they are complements or substitutes (this part may not have a straightforward
answer but you should discuss it.)
Problem 2: Redo question 1 for \(U({{x}_{1}},{{x}_{2}})=\min \{{{x}_{1}},2{{x}_{2}}\}\)
Problem 3: Redo question 1 for \(U({{x}_{1}},{{x}_{2}})=\ln ({{x}_{1}})+2\ln \left( {{x}_{2}} \right)\)
Problem 4: Consider an investor choosing between a safe bond and an investment in a risky stock, The safe bond earns an interest of rs percent per dollar invested. Investment of a dollar in the risky stock yields rl dollars with probability 0.5 and with probability of
0.5 the return is rh. The investor has a wealth W to invest.
a. Suppose that \({{r}_{s}}<{{r}_{l}}<{{r}_{h}}\). What would be the solution to the consumer's portfolio
problem?b. Suppose that \({{r}_{l}}<{{r}_{h}}<{{r}_{s}}\). What would be the solution to the consumer's portfolio
problem?c. Suppose that preferences are given by U = In(c). Find conditions under which
the consumer invests in the risky asset.
Problem 5: Sheila has wealth of 40 dollars She is only considering investment in two risky stocks: M and N. Each stock (of both M and N) costs 1 dollar. Each stock of M will
be worth 10 cents with probability 1/2 and 10 dollars with probability 1/2. Each stock
of N will be worth 10 cents with probability 1/2 and 10 dollars with probability 1/2 as
well. The value of the M stock and the N stock, however, are independent draws. This
implies that with probability 0.25 value (in dollars) of M stock is 0.1 and the value of
stock N is 0.1. With probability 0.25 value of M stock is 10 and the value of stock N is
0.1. With probability 0.25 value of M stock is 0.1 and the value of stock N is 10, and
with probability 0.25 the value of stock N is 10 and the value of stock M is 10. Denote
the value of consumption when both stocks are worth 0.1 dollars by \({{c}_{ll}}\), the value of
consumption when M is worth 10 and N is worth 0.1 by \({{c}_{hl}}\), the value of consumption
when M is worth O1 and N is worth 10 by \({{c}_{lh}}\), and the value of consumption when both
stocks are worth 10 by \({{c}_{hh}}\). Sheila's expected utility is given by:
\[\frac{1}{4}\sqrt{{{c}_{hh}}}+\frac{1}{4}\sqrt{{{c}_{hl}}}+\frac{1}{4}\sqrt{{{c}_{lh}}}+\frac{1}{4}\sqrt{{{c}_{ll}}}\]
a. Suppose Sheila invested all her wealth in stocks of company M. calculate \({{c}_{hh}},{{c}_{hl}},{{c}_{lh}},{{c}_{ll}}\)
and Sheila's expected utility.
b. Suppose Sheila invested all her wealth in N. calculate \({{c}_{hh}},{{c}_{hl}},{{c}_{lh}},{{c}_{ll}}\) and Sheila's
expected utility.
c. Suppose Sheila invested 20 dollars in each company. Calculate \({{c}_{hh}},{{c}_{hl}},{{c}_{lh}},{{c}_{ll}}\) and
Sheila's expected utility. Is it possible that it is optimal for Sheila to invest all her
wealth in a single asset? Explain.
d. Formulate the relevant optimization problem for Sheila from which the optimal
investment strategy can be derived. (you do not need to solve the problem.)
Problem 6: Consider a worker/consumer with non-labor income of 2 dollars. Time endowment is \(\bar{L}\) = 24, and denote labor supply by L. Suppose the hourly wage is 5 and the consumption good's price is 1. Suppose that the optimal labor supply (given the prices) is L* = 8.
a. Suppose that consumption and leisure are both normal goods and that the hourly wage rate is now 10. Show carefully in a graph how the change in labor supply can be decomposed into income and substitution effects.
b. Suppose that the worker is offered the following contract: For every one of the first 8 hours the wage per hour is 5. If the worker chooses to work for more than 8 hours the wage rate is 10 for each extra hour (above 8). In a new graph depict the budget set. Can you determine how labor supply changes? (comparing it to the case in which the wage rate is fixed at 5 dollars per hour).
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