Economics

Intermediate Microeconomic Theory Fall 2015

Homework #1 - Part B

Choice:

Exercise 7: 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 �nding (x�1;x � 2) :

e. Graph the budget set, a couple of indi¤erence curves and the optimal choice.

Exercise 8: Assume preferences can be represented by the following utility function:

u(x1;x2) = �x12 +150x1 �2x22 +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 �nding (x�1;x � 2) :

Exercise 9: Assume preferences can be represented by the following utility function:

u(x1;x2) = 4ln(x1)+ x2

a. Is the utility function monotonic? Justify. b. Set up the consumer�s 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) :

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Demand:

Exercise 10: You are the owner of a supermarket that wants to understand your client�s

preferences so that you can optimally price your products. You record a client�s purchases of two products x1 and x2 in 8 di¤erent occasions. The following table summarizes the results (for a similar exercise see Varian, ch.5, sec.4):

Obs. p1 p2 m x1 x2 1 3 2 94 12 29 2 2 2 106 15 38 3 2 3 95 10 25 4 1 3 170 20 50 5 3 1 83 15 38 6 2 1 135 30 75 7 4 1 143 22 55 8 1 2 168 28 70

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 individual�s

behavior (Cobb-Douglas, Perfect Complements or Perfect Substitutes)? d. Write down a utility function that represents these preferences fairly well.

Exercise 11: You record a client�s purchases of two products x1 and x2 in 8 di¤erent

occasions. The following table summarizes the results:

Obs. p1 p2 m x1 x2 1 2 2 20 4 6 2 2 2 40 4 16 3 2 2 30 4 11 4 1 3 51 12 13 5 1 3 48 12 12 6 1 3 63 12 17 7 2 1 70 2 66 8 2 1 50 2 46

Notice that for observations 1,2 and 3 even though prices did not change for di¤erent 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.

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b. The preferences underlying this individual�s 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 client�s purchases for the given prices and income:

Obs. p1 p2 m x1 x2 9 2 2 50 10 2 4 100 11 3 3 45 12 4 1 20

Slutsky Equation:

Exercise 12: 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 consumer�s utility maximization problem for prices p1; p2 and

income m (the general case) c. Solve theproblem. Youwill obtaindemand functionsx�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 indi¤erent 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).

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Exercise 13: For the following demand function:

x�1 (p1;p2;m) = 4

p1 (3+m) ;

for values of m > 1: 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 14: Assume preferences can be represented by the following utility function:

u(x;y) = �x12 +100x1 +20x2

a. Is the utility function monotonic? Justify. b. Set up the consumer�s 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 p01 = (B +7)=2; where B is the last digit of your ASU ID#. f. Again, assuming the price of good 1 increases to p01 = (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.

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Intermediate Microeconomic Theory Fall 2015

StudentName:___________________ Student ID________

Homework 1 - Part B

Front Page

Selected Answers:

Exercise 7.d.

x�1 = ________ x � 2 = ________

Exercise 9.c.

x�2 = ___________

Exercise 12.d.

The income required to purchase the old bundle at the new prices is:

m0 = ___________

Exercise 13.e.

The income e¤ect with Slutsky compensation in terms of good one is:

x1(p 0 1;p2;m)�x1(p

0 1;p2;m

0) = ___________

Exercise 14.e.

The change in Consumer Surplus is:

�CS = ___________

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