Part A
Choice: Exercise 1:
For this exercise replace A with the last digit and B with the second-to-last digit of your ASU ID#.
Assume preferences can be represented by the following utility function: u(x1; x2) = (A + 1) ln (x1) + ln (x2);
a. Is the utility function monotonic? Justify.
b. Determine the set of bundles that are ranked higher than the bundle (x1; x2) = (10; 10)
c. Set up the utility maximization problem for the consumer, when facing prices p1 = 6; p2 = B + 1 and income m = 2520(A + 2):
d. Solve the problem by Finding (x 1 ; x 2 ).
e. Graph the budget set, a couple of indifference curves and the optimal choice.
Exercise 2:
Assume preferences can be represented by the following utility function: u(x1; x2) = x1 2 + 150x1 2x2 2 + 100x2 + x1 x2
a. Is the utility function monotonic? Justify.
b. Obtain a bundle that is ranked higher than the bundle (x1; x2) = (100; 100)
c. Set up the utility maximization problem for the consumer, when facing: prices p1 = 2; p2 = 1 and income m = 30:
d. Solve the problem by finding (x 1 ; x 2 ):
Exercise 3:
Assume preferences can be represented by the following utility function: u(x1; x2) = 4 ln (x1) + x2
a. Is the utility function monotonic? Justify.
b. Set up the consumers utility maximization problem for prices p1; p2 and income m (the general case)
c. Solve the problem. You will obtain solutions x 1 (p1; p2; m); x 2 (p1; p2; m) in terms of the parameters of the model (p1; p2; m): 1
Demand:
Exercise 4:
You are the owner of a supermarket that wants to understand your clients preferences so that you can optimally price your products.
You record client's purchases of two products x1 and x2 in 8 different occasions.
The following table summarizes the results (for a similar exercise see Varian, ch.5, sec.4):
a. Notice observations number 2 and 5. Quantities purchased are the same but prices are not. What does this mean in terms of the
marginal rate of substitution at those quantities?
b. Plot the 8 bundles purchased by the individual in a graph.
c. Which type of preferences comes closer to describing this individualism behavior (Cobb-Douglas, Perfect Complements or Perfect
Substitutes)?
d. Write down a utility function that represents these preferences fairly well.
Exercise 5:
You record a clients purchases of two products x1 and x2 in 8 different occasions.
The following table summarizes the results:
Notice that for observations 1,2 and 3 even though prices did not change for different amounts of income spent by the client, he/she
still purchased the same amount of good one in all three occasions. The same can be said for observations 4,5 and 6 and observations
7 and 8.
a. Draw the Engel curves for good 1 and the income expansion paths for the three sets of prices in the table. 2
b. The preferences underlying this individualís behavior can be represented by one of the three utility functions in Exercises 7,8 and 9.
Find which one it is and explain why.
c. Based on your answer to part b,
complete the following table with your predictions on the clients purchases for the given prices and income:
Slutsky Equation:
Exercise 6:
Assume preferences can be represented by the following utility function: u(x1; x2) = x1 x2 2
a. Is the utility function monotonic? Justify.
b. Set up the consumers utility maximization problem for prices p1; p2 and income m (the general case)
c. Solve the problem. You will obtain demand functions x 1 (p1; p2; m); x 2 (p1; p2; m) in terms of the parameters (p1; p2; m): Obtain
price elasticity of demand for good one. Obtain income elasticity of demand for good 2.
d. Assume that, originally, the consumer faces: prices p1 = 2; p2 = 5 and income m = 30(A + 1); where A is the last digit of your ASU
ID#. Now assume the price of good 1 increases to p ; 1 = 3: Obtain the income and substitution e§ects for good 1 with Slutsky compensation (that is, compensating the individual so that it can still buy the old bundle at the new prices).
e. Find the amount of compensation needed for Hicks compensation (that is, compensating the individual so that he is indi§erent to his old bundle). To do this plug the old bundle into the utility function to obtain the level of utility you want to acheive.
Then plug the demand functions into the utility function. Then replace prices with new prices and equate the two utilities. By now you should have a function of income equal to a number. Solve for the appropriate income level. That is the compensation needed to make the individual indifferent to the old bundle. The amount of compensation needed should be lower than with Slutsky compensation, but because the price change is very small, there should be barely any di§erence between the two. f. Graph your results in (e) by plotting the old and new indi§erence curves, the old, compensated and new budget sets and the old, compensated and new choices (quantities demanded). 3
Exercise 7 :
For the following demand function:
a. Obtain Income elasticity of demand. Plot the Engel curve for p1 = 1:
b. Is this a normal good?
c. Assuming that preferences are monotonic (then the individual always spends all its income), use the budget constraint to solve for x 2 (p1; p2; m).
d. The consumer faces the following prices and income level: prices p1 = 1; p2 = 1:5 and income m = 5: Calculate the quantity demanded for goods 1 and 2 at these prices and this income level. e. Obtain income and substitution e§ects with Slutsky compensation
when the price of good 1 drops to p ; 1 = 0:5
Exercise 8:
Assume preferences can be represented by the following utility function: u(x; y) = x1 2 + 100x1 + 20 x2
a. Is the utility function monotonic? Justify.
b. Set up the consumerís utility maximization problem for prices p1; p2 and income m (the general case)
c. Solve the problem. You will obtain demand functions x 1 (p1; p2; m) and x 2 (p1; p2; m) in terms of the parameters (p1; p2; m):
d. Graph the demand function for good 1 when the price of good 2 is p2 = 2 and income is m = 200:
e. Obtain the change in consumer surplus when the price of good 1 goes from p1 = 2 to p 0 1 = (B + 7)=2; where B is the last digit of
your ASU ID#.
f. Again, assuming the price of good 1 increases to p 0 1 = (B + 7)=2: Find the Compensating and the Equivalent Variations
g. For the same price increase, obtain the income and substitution e§ects on good 1, both with Slutsky and Hicks compensations. 4 Intermediate Microeconomic Theory Fall 2015 Student
Name:___________________ Student ID________ Part A
Front Page Selected Answers:
Exercise 1.d. x 1 = ________ x 2 = ________ Exercise3.c. x 2 = ___________ Exercise 6.d.
The income required to purchase the old bundle at the new prices is: m0 = ___________ Exercise 7.e.
The income effect with Slutsky compensation in terms of good one is: x1(p 0 1 ; p2; m) x1(p 0 1 ; p2; m0 ) = ___________ Exercise
8.e. The change in Consumer Surplus is: CS = ___________ 5