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

Lecturer: Yosuke Yasuda

Problem Set 2: Solution 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 !. Answer:

Let x 2 B(p; !) be the optimal consumption bundle given (p; !). Suppose the initial wealth is increased from ! to !⁰(> !), the budget set strictly expands. This implies that there exists some x⁰ 2 B(p; !⁰) such that x⁰ x . Then, we obtain

v(p; !) := max

x2B(p;!⁰)^{u(x) u(x}

0) > u(x ) = v(p; !) where the strict inequality comes from monotonicity of u. (b) Decreasing in p.

Answer:

For any two price vectors p^{0 p}, and for any consumption bundle x ⁰, p⁰x px holds. This implies

x2B(p⁰; !) ) x 2 B(p; !)

and for any ! 0. Speci…cally, the optimal consumption bundle under p⁰ (and

!) is always feasible under p. Thus, the corresponding utility never decreases. (c) Quasi-convex in (p; !).

Answer:

Let B¹, B² and B^t be the budget sets available when prices and income are (p¹; !¹), (p²; !²) and (p^t; !^t), respectively, where

p^t:= tp¹+ (1 t)p²

!^t:= t!¹+ (1 t)!². Then, we will show that

v(p^t; !^t) max[v(p¹; !¹); v(p²; !²)] for all t 2 [0; 1]. (1) This relation trivially holds when t = 0 or 1. So, in what follows we focus on t 2 (0; 1). Suppose (1) is not satis…ed. Then, there must exist some consumption

bundle x⁰ such that

x⁰ 2B^t; x⁰ 2= B¹ and x⁰ 2= B². By de…nition of budget sets, we must have

p¹x⁰ _{> !}¹ _{and p}²x⁰ _{> !}²_. For any t 2 (0; 1), the following also holds:

tp¹x⁰ > t!¹ and (1 t)p²x⁰ > (1 t)!². Adding, we obtain

(tp¹+ (1 t)p²)x⁰ > t!¹+ (1 t)!² , p^tx > y^t,

which contradicts our assumption that x⁰ 2 B^t. Therefore, (1) holds, i.e., v(p; !) is 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.

Answer:

The Slutsky equation says

@xi(p; !)

@pi

= ^@x

h i^{(p; u )}

@pi

xi(p; !)^{@xi(p; !)}

@! ^{. (2)}

Since the good i is assumed to be normal, ^@xi_@!^(p;!) > 0. So, the second term must be negative, which combined with the fact that ^@xhi_@p^{(p;u )}

i 0 implies that the right hand side must be negative. Thus, the own price e¤ect for a normal good is always negative.

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

Answer:

Suppose the right hand side of (2). Since the …rst term is always non-positive, the second term must be strictly positive. Thus ^@xi_@!^(p;!) < 0, i.e., the good should be inferior.

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

Suppose both goods are inferior, and let x be the optimal cumsumption bun- dle under (p; !). If the initial wealth is increased from ! to !⁰(> !). Then, by de…nition of inferior goods, the optimal consumption bundle under (p; !⁰), denoted by x⁰, satis…es x⁰ x . This implies

px⁰ px = ! < !⁰, which contradicts Walras’ law.

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

1^x

1 3

2^{, and}

! = p1x1+ p2x2,

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

Answer is given in class. 4. Queston 4 (15 points)

Consider the indirect utility function given by v(p1; p2; !) = ^!

p1+ p2

. (a) What are the (Marshallian) demand functions? (b) What is the expenditure function?

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

Answer is given in class. 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.

Answer:

Take any x¹ 0 and x² 0, and let y¹ = f (x¹) and y² = f (x²). Then, y¹; y² > 0 because f (0) = 0 and f is strictly increasing. Therefore, because f is homogeneous of degree one,

f (^x

1

y^{1) = f (} x² y^{2) = 1.}

Because f is quasi-concave, f (^tx

1

y^{1 +}

(1 t)x²

y^{2 )} 1 for all t 2 [0; 1]. (3)

Now choose t = _y1^y_+y¹ 2 (and1 t = _y1^y_+y² 2). Then (3) becomes

f ( ^x

1

y¹+ y^{2 +} x²

y¹+ y^{2) 1.} Again invoking the homogeneity of f gives us

f (x¹+ x²) y¹+ y² = f (x¹) + f (x²). (4) By continuity, (4) holds for all x¹; x² 0 (not only x¹; x² 0). Note that for any two vector x⁰; x⁰⁰ 0 and for any t 2 [0; 1], we have

f (tx⁰) = tf (x⁰) and f ((1 t)x⁰⁰) = (1 t)f (x⁰⁰).

Substituting x¹ = tx⁰ and x² = (1 t)x⁰⁰ into (4), we conclude that f (tx⁰+ (1 t)x⁰⁰) tf (x⁰⁰) + (1 t)f (x⁰⁰), as desired.

6. Question 6 (15 points)

A real-valued function f ( ) is called superaddittive if f (x¹+ x²) f (x¹) + f (x²).

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

Answer:

We have to show that for any y > 0, the following inequality holds:

c(w¹+ w²; y) c(w¹; y) + c(w²; y) (5) Let x¹²; x¹; x² be the optimal conditional input demand to produce y for each price vector. Then, we obtain

c(w¹+ w²; y) = (w¹+ w²)x¹² = w¹x¹²+ w²x¹². (6) By de…nition of c( ),

w¹x¹² c(w¹; y) and w²x¹² c(w²; y)

must hold. Substituting them into (6), we achieve (5). 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 by = f (x ) solves (1). Answer:

Suppose not. Then, there must exist y⁰ 6_{= b}y(= f (x )) and x⁰ such that

py⁰ c(w; y⁰_{) > pb}y _{c(w; b}y) (7) and

y⁰ = f (x⁰). Note that (7) can be written as

pf (x⁰) wx⁰ > pf (x ) w(x ),

which contradicts to the assumption that x is an optimal solution of (2). (b) Show that if c(w; y ) = w_bxand y = f (_bx), then _bxsolves (2).

Answer:

Suppose not. Then, there must exist x⁰⁰6=_b^xsuch that pf (x⁰⁰) wx⁰⁰ > pf (_bx) w_bx. Let y⁰⁰ = f (x⁰⁰). Then, this inequality can be written as

py⁰⁰ c(w; y⁰⁰) > py c(w; y ),

which contradicts to the assumption that y is an optimal solution of (1).