Managerial Economics (ARE) 100A University of California, Davis Fall Quarter, 2012 Instructor: John H

Managerial Economics (ARE) 100A

University of California, Davis

Fall Quarter, 2012

Instructor: John H. Constantine

Homework 3: Due Monday october 22, 2012

Problem1)Joe has $2,500 dollars.  His preferences as defined for goods X and Y are given by:

                        U(X, Y) = 50X^4/5Y^1/5.

In addition, the price of X is 10 and the price of Y is 15.

(a)        Solve for the optimal (numeric) values of X and Y.  You are to use the Lagrange method.

2. What are the values for MU_X and MU_Y when evaluated at the optimal solutions?

3. Verify that MRS_XY = P_X/P_Y at the optimal solution.  You are to verify MRS_XY^* with numbers.

4. What is the utility value at the optimal solution?

(e)        What is the value of the marginal utility of income?  Interpret this result.

Suppose the government sets a consumption quota of 150 units X—that is, Joe can consume up to 150 units, but not beyond.

(f)        What is the new optimal solution?

(g)        What is the value of utility given the restriction?  Does this result make sense?

8. Does the MRS_XY = P_X/P_Y at your solution in part (g)?

Suppose the government sets a consumption quota of 225 units X—that is, Joe can consume up to 225 units, but not beyond.

1. What is the new optimal solution?

10. What is the value of utility given the restriction?  Does this result make sense?
11. Does the MRS_XY = P_X/P_Y at your solution in part (j)?

Problem 2:

Suppose that the only two goods you consume are wine and roses.  On Tuesday the price of wine goes up, and at the time your income increases by just enough so that you are equally as happy as you were on Monday.

1. What happens to the quantity of wine that you will consume?  Illustrate this with indifference curves.
2. On Tuesday would you still be able to afford the same basket that you were buying on Monday?  How do you know?

On Wednesday there are no new price changes (so the Tuesday prices are still in effect), but your income changes to the point where you can just exactly afford Monday’s basket.

3. Are you happier on Wednesday or Monday? 
4. Is it possible to say with certainty whether you buy more wine on Wednesday than on Monday?  If not, on what would your answer depend?
5. Is it possible to say with certainty whether you buy more wine on Wednesday than on Tuesday?  If not, on what would your answer depend?

Problem 3 (A table is given at the end of the assignment to assist you):

The following table shows total utility, measured in utils of satisfaction, which Mr. Johnson would get by purchasing various amounts of products A, B, C, and D and by saving.  The prices of A, B, C, and D are $5, $6, $8, and $20, respectively.  Mr. Johnson has a money income of $95 to spend in the current period.

(a)        What is Mr. Johnson’s utility-maximizing consumption bundle.  You must show your work to receive full credit.

(b)        How much total utility does Mr. Johnson receive.

(c)        Suppose Mr. Johnson is currently consuming nothing.  If he is allowed to choose an item for free, which item will he choose first?  Second?  Third?  Explain.

(d)        Suppose now Mr. Johnson is currently consuming nothing but must pay for the goods.  Which item will he choose first?  Second?  Third?  Explain.

Problem 4:

Consider the following demand function for good S:

            Q_S = 500 – 2P_S – 10P_E + 10P_H – 5I.

P_S, P_E, and P_H are the respective prices of goods S, E, and H, and I is income.  The following information is known: P_S = $5, P_E = $2.50, P_H = $0.75, and I = 80.

Calculate all relevant demand elasticities, that is, the own-price, cross-prices, and income demand elasticities.