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Eco 600E Advanced Microeconomics I Term: Spring (1st), 2009

Lecturer: Yosuke Yasuda

Problem Set 2 Due in class on June 4

1. Question 1 (15 points)

Suppose the utility function u(x) is continuous and strictly increasing. Then, prove the following properties of the indirect utility function v(p; !):

(a) Strictly increasing in !. (b) Decreasing in p.

(c) Quasi-convex in (p; !). 2. Question 2 (15 points)

A good is called normal (resp. inferior) if consumption of it increases (resp. declines) as income increases, holding prices constant. Show the following claims. You can use the Slutsky equation (without proof) if it is needed.

(a) A decrease in the own price of a normal good will cause quantity demanded (Marshallian demand) to increase.

(b) If an own price decrease causes a decline in quantity demanded (known as Gi¤en’s paradox), the good must be inferior.

(c) In a two-good case, if one good is inferior, the other good must be normal. 3. Question 3 (20 points)

Consider the consumer problem where the utility function and the budget constraint are given by

u(x¹; x2_{) = x}

1 2

1x

1 3

2, and

!= p¹x1_{+ p}2x2_,

respectively. Calculate (i) the Marshallian demand, (ii) the indirect utility function, (iii) the Hicksian demand, and (iv) the expenditure function.

4. Queston 4 (15 points)

Consider the indirect utility function given by v(p¹; p2_{; !) =}

! p1_{+ p}2

. (a) What are the (Marshallian) demand functions?

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(b) What is the expenditure function?

(c) Find a direct utility function which is consistent with the above indirect utility function.

5. Question 5 (10 points)

Suppose the production function f satis…es (i) f (0) = 0, (ii) increasing, (iii) con- tinuous, (iv) quasi-concave, and (v) constant returns to scale. Then, show that f must be a concave function of x.

6. Question 6 (15 points)

A real-valued function f ( ) is called superaddittive if f_(x¹_{+ x}²₎ f_(x¹_{) + f (x}²_).

Show that every cost function is superadditive in prices. Use this property to prove that the cost function is nondecreasing in input prices.

7. Qusetion 7 (10 points)

Let c(w; y) be the cost function generated by the production function f and suppose the following two maximization problems (1) and (2) have solutions y and x 0,respectively.

(1) : max

y ⁰

py c_{(w; y)} (2) : max

x2R^{K 1}₊

pf(x) wx

(a) Show that b^y= f () solves (1).

(b) Show that if c(w; y ) = wb^xand y = f (bx), then b^xsolves (2).

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