PROBLEM SET 3

1. (Silberberg and Suen, Chapter 8, Problem 6) Consdier the production function y =

xα1

1 xα2

2 . Show that the constant-output factor demand functions have the form

x∗

i = kiw−αj/(α1+α2)i w

αj/(α1+α2)j y1/(α1+α2) i 6= j

Show that the cost function has the form

C∗ = (k1 + k2)wα1/(α1+α2)1 w

α2/(α1+α2)2 y1/(α1+α2)

and that ∂C∗/∂wi = x∗

i .

2. (Silberberg and Suen, Chapter 8, Problem 9) Derive an expression analogous to Eq.

(8-49) showing the relationship between the profit-maximizing and cost-minimizing

cross-effects ∂xpi /∂wj and ∂xy

i /∂wj , i 6= j. If xi and xj are both normal factors,

which cross effect is larger? Can these cross effects have different signs?

3. (Varian, Ex. 4.4) A firm has two plants. One plant produces output according to

the production function xa1x

1−a2 . The other plant has a production function xb

1x1−b2 .

What is the cost function of this technology?

4. For what parameter (a, b, c, d, e, f, g, h) are the following equations a valid system of

conditional factor demand functions:

x1 = (a + bwc1w

d2)y

x2 = (e + fwg1w

h2 )y

5. (Jehle and Reny, Ex. 1.3) Suppose the preference relation � satisfies all the axioms

about consumer preferences. Consider the binary relation � and ∼ derived from �.

Give a proof or convincing argument for each of the following claims.

(a) Neither � nor ∼ is complete.

(b) For any x1 and x

2 in X, only one of the following holds: x1 � x

2, x2 � x

1, or

x1 ∼ x

2.

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