Economics Problems: Utility Function, Taxes, College Education, Costless Magical MacGuffin, and Rica

Problem 1 - Utility function

Problem 1 - Utility function
1. What is a utility function?
2. What are the indifference curves?
3. Plot a family of indifference curves for U (x, y) = x
4. Do indifference curves depend on the budget constraint?
5. Can indifference curves intersect? Explain.

Problem 2 - Taxes
A worker receives a wage rate w and has L hours of leisure every day (the total endowment of hours is 24 hours per day). The government taxes his income at the constant rate T. The worker spends all his income.
1. Write a budget constraint of this individual and plot it.
2. Display graphically what is the optimal consumption-leisure choice for this worker.
3. Imagine that the government increases the tax rate to T ′. What is the new budget constraint? Display on the same picture. In the new optimum is the consumption higher? Explain the answer in terms of wealth and substitution effects.

Problem 3 - Should I go to school?
Suppose a consumer who has to decide if she wants to go to college or not. If she does NOT go, she will get a low income in both periods, Y1, Y2. If she attends college, she will get a higher wage in the second period Y ′2 > Y2. In the first period, she will have NO income (Y ′1 = 0) and, in addition, she will have to pay tuition S in the first period. She has increasing and concave preferences over consumption in the two periods (C1 and C2).

Consider first that she does not go to school
1. Write down the dynamic budget constraints
2. Derive the intertemporal budget constraint
3. Show graphically the budget constraint and the optimal consumption point in period 1 and 2.If she decides to go to school,
4. Write down the dynamic budget constraints
5. Derive the intertemporal budget constraint
6. In your previous graph, draw the new budget constraint and the new optimal consumption point.
7. Is it good for her to go to school? Under what conditions will she be better off by going to school rather than not going to school? Explain.
8. If the government subsidizes the cost of going to school in the first period, and the consumer has to pay back the subsidy in the second period (including the interest rate), it is more likely that she optimally chooses to go to school?. Discuss the validity of this statement.

Problem 4 - Costless Magical MacGuffin
Consider a consumer that lives only for two periods. He works in period 1 (and gets income Y1) and retires in period 2 (and gets income Y2 < Y1).

Problem 2 - Taxes

This consumer has the usual preferences over time: u(C1) + βu(C2)
1. Assume this consumer cannot save. What is the consumption in period 1 and period 2? Display graphically. Show the corresponding utility curve.
2. Assume that now the consumer is allowed to save or borrow. Write down the new budget constraint What is the consumption in period 1 and period 2? Display graphically. Could the consumer be worse of? Could the consumer be better of? Draw budget constraints such that for one of them consumer prefers to borrow and for the other - prefers to save.
3. Assume once again that a consumer cannot save, but can buy some MacGuffins, which have no consumption value, but can be transported to the next period and sold by the next period price. Assume that MacGuffin costs P1 > 0 in the first period and is expected to cost  ?P2 in the second period.Write down the new budget constraint. Would a consumer buy a MacGuffin? What is the condition on the  ?P2? Is  ?P2 a fair price of a MacGuffin? Could the consumer be better off with a MacGuffin?

Problem 5 - Ricardian Equivalence
There is a consumer who lives for two periods. His income is given by Y1 and Y2. He has access to the credit market with the interest rate r.The government collects lump-sum taxes T1 and T2 (note that T1 and T2 might be negative meaning that the government makes a transfer). The government can run a surplus or a deficit, but must borrow (or save) in the credit market at the interest rate r.

1. Write down the government intertemporal budget constraint. Note that the government also has the access to the credit market. Write down the consumer’s budget constraint. Show the consumption choice graphically.
2. Imagine that the government increases the taxes in period 1 and introduces a tax cut in period 2 to pay back any debt and interest from the previous period. This is anticipated by consumers. Show the new consumption choice. How does that compare to the result from the previous section.
3. Assume that the government is not constraint by the balanced budget and can have deficit in both periods. Now the government is being generous and pays transfers −T1 > 0 and −T2 > 0. Show the new consumption choice.
4. Now assume that consumers believe the government will behave as in part 3, but in the second period the government is forced to settle the deficit (note that the government has to pay back the interest rate). What will happen to the consumption choice? Are the consumers better off.