1. Elizabeth has the following utility function for goods X and Y:   U =X² Y.   Her income is

$300 per unit of time, the price of  X equals  $10 per unit, and the price of good Y equals  $2 per unit.   

a. Find the MRS.

b. Calculate and sketch the budget constraint.

c. What is the utility-maximizing consumption bundle for Elizabeth?

d. How would your answer to part (c) change if the price of X increased to $20 per unit?

e. Derive Elizabeth's demand curve for good X.

f. Suppose Elizabeth's utility function took the general form : U =X^a Y^b . Derive the demand curve for goods X and Y.

g. Using your answer from part (f) and assuming a=b=1, find the indirect utility function.

2. Tom likes Xs, hates Ys, and is completely indifferent to Zs. Draw his indifference curves between  (a) Xs and Ys, (b) Xs and Zs, and (c) Ys and Zs.

3. Find the  MU_x , MU_y , and MRS equations for each of the following utility functions.

a.       U = x^0.6y^0.4.

b.       U = x² + y²        x,y>0.

c.        U = 2x + 4y.

d.       U = x²y².

e.        U = x^ay^b.

4.For (a)-(d), which of the  utility functions exhibit diminishing marginal utility for good X? Hint: Using your equation for MU_x, determine if MU_x falls as X rises.

5. Which of the above utility functions exhibit diminishing MRS? (That is, which of the above yield convex indifference curves?).

Consumer Choice Model

6. Do you think diminishing marginal utility is a necessary condition to get diminishing MRS? Use your answers for (a), (d), and (e) to justify your answer.

7. Consider the CES utility function:   U = a(X^d /d) + b(Y^d /d) if d ¹ 0 and  U= a lnX + b lnY if d=0.

a. Assuming d ¹ 0,  derive the equation for MU_x and MU_y. Find MRS.

b.  What sort of preferences are exhibited when d = 1? ...when d = 0? ...when 0d?

8. Assume that U =xy and that P_x =$10, P_y=$5, and I=$100. Use the Lagrange method to find the first-order conditions and the optimal values (i.e., utility maximizing values) of  x and y.

9. For each case below, draw a graph of the budget line.  Indicate the values of X and M at the kinks and intercepts.  Assume I = $100 and P_x = $1.

a. The government provides a per-unit subsidy of $0.50/X (so the consumer faces a price of  $0.5/unit) but only beyond the first 10 units of X.

b. The government provides a per-unit subsidy of $0.50/X (so the consumer faces a price of  $0.5/unit) but only up to the first 10 units of X.

c. The government introduces a program in which the first 5 units of  X are free . After that, the government provides a per-unit subsidy of $0.50/X (so the price of X to consumers is $0.50 per unit) but only up to 10^th unit of X.  The government does not provide any subsidy for units purchased beyond the 10^th unit. (The consumer still retains the "gift" for the earlier units.)

 d. Same as in part (c ), except  that in this case if  the consumer consumes more than 10 units of  X, the government takes away ALL subsidies-i.e., the consumer no longer gets the first 5 units for  free,etc.

      e. The government imposes a price ceiling on good  X at $0.5/unit. The consumer is only allowed to  purchase a maximum amount of  50 units at this price.   

      10.  A consumer faces the following utility function:  U=xM, with M representing dollars spent on  all  goods   other than good x (therefore  P_M  º 1). Assume that  P_x =$1 and  I = $100. 

   a. Find the optimal consumption bundle and the level of utility at that bundle. Show the result from this part on a graph. Place x on the horizontal axis and M on  the vertical axis.

   b. Suppose the government provides the consumer with $20 worth of X-stamps. Find the new optimal consumption bundle. HINT: To find the solution you should assume that the consumer received a gift of

  $20 cash. (QUESTIONS  TO PONDER: Why can make we make this assumption-after all, the consumer received food stamps not cash? Can we always make this assumption?).  Show this result on the same graph as used in part (a).

       c. Suppose the government replaces its food stamp program with a per-unit subsidy program. The per-unit subsidy is selected so as to allow the consumer to achieve the same level of utility as under the food stamp program. Using the indirect utility function, find the per-unit subsidy that would be required to achieve this  result.  (NOTE: The per-unit subsidy equals $1 minus  price of X under the per-unit subsidy. Notice that we are implicitly assuming that the supply of X is perfectly elastic and therefore  the entire subsidy is  passed on to consumers).

       Find x, M, and the cost to the government of providing this subsidy. Show this outcome on the same graph as used in parts (a) and (b).  On your graph, indicate the cost to the government of each  program.