微观经济学

Microeconomic Theory

Midterm Exam

1.(0

2.05.08.09.11)Mike ’s expenditure function is given by:

1212

(,)p p u E p u p p =+ (1) Derive the ordinary and compensated demands for commodity 1.

(2) Is Mike ’s preference homothetic?

(3) Mike ’s boss is thinking of sending him to another town where the price of good 1

increase by 10%. The boss offers no raise in pay. Mike, who understands compensating and equivalent variation perfectly, complains bitterly. He says he wouldn ’t mind moving if when he moved he got a raise of $A. What is A equal to?

2.(04) Mike purchases two goods, 1X and 2X , at prices 1P and 2P respectively. The compensated demand curves are given by

1/2211(,)P h P u u P ⎛⎫= ⎪⎝⎭ and 1/2

122(,)P h P u u P ⎛⎫= ⎪⎝⎭

. (1) Derive the ordinary demands for good 1 and good 2.

(2) Are these two goods gross substitutes or gross complements?

(3) Is Mike ’s preference homothetic?

(4) Mike ’s boss is thinking of sending him to another town where the price of 1X is 1.441P , and the price of 2X is 2P . The boss offers no raise in pay. Mike, who

understands compensating and equivalent variation perfectly, complains bitterly. He says having to move is as bad as a cut in pay of $A. He also wouldn ’t mind moving if when he moved he got a raise of $B. What are A and B equal to?

3.(06) The Stone-Geary utility function is given by:

1()ln()N

i i i i U x x k α==-∑,

Where 11N i i α

==∑, and i α>0, i k >0.

(1) Derive the ordinary demand function for good i.

(2) Derive the compensated demand function for good i.

4.(06) An individual ’s preferences for housing and other goods can be respresented by

the utility function 111212(,)U X X X X --⎡⎤=-+⎣⎦ where 1X denotes the consumption

of housing and 2X denotes the consumption of other goods. The ordinary demand functions for housing and for other goods are given by:

1112Y

X P PP =+ and 2212Y X P PP =+,

Where Y denotes income, 1P denotes the price of housing, and 2P denotes the price

of other goods.

(1) Derive the expenditure function, the compensated demand function for housing and the compensated demand function for other goods.

(2) Are these two goods gross substitutes or gross complements?

(3) Are these two goods net substitutes or net complements?

(4) Are the preferences homothetic?

(5) Suppose the government pays one-half of the cost of housing for low-income individuals and this reduces the price of housing for low-income individuals to 112

P . An alternative plan of providing assistance to a low-income individual is to give the individual a direct cash grant. What level of cash grant would be equivalent to the payment of one-half the cost of low-income individual ’s housing?

5.(05)For a simple two-period overlapping-generations model, utility depends on consumption in the two periods, 1C and 2C , and can be represented by a strictly

quasi-concave function, 12(,)U C C . Assume that income is earned only in the first period, in amount I. So the consumption in the second period is possible only by saving S in the first period. That is, 1C =I – S ，2C =S(1+r), where r is the exogenous

interest rate.

(1) Formulate this problem as an unconstrained optimization problem. Derive the first order condition for an interior solution.

(2) Formulate this problem as a constrained optimization problem. Derive the first order condition for an interior solution.

(3) It has been argued that reducing interest rate can increase consumption in the first period, and the Chinese government is doing this way. Use both the diagram and comparative statics to determine the effect of a reduction in the interest rate to the consumption in the first period (substitution effect, income effect, total effect). Assume both 1C and 2C are normal goods.

6.(06) A parent allocates the family ’s exogenous income, Y , between personal consumption, C, and private investment in a child ’s education, I, so that Y = C + I. The child ’s future income, C Y , depends upon the amount of private investment, I, and