{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Rubinstein2005-page103

# Rubinstein2005-page103 - = a k w y(the k th input commodity...

This preview shows page 1. Sign up to view the full content.

October 21, 2005 12:18 master Sheet number 101 Page number 85 Problem Set 7 Problem 1. ( Easy ) Assume that technology Z and the production function f describe the same producer who produces commodity K using inputs 1, . . . , K 1 . Show that Z is a convex set if and only if f is a concave function. Problem 2. ( Boring ) Here is a very standard exercise (if you have not done it in the past, it may be “fun” to do it “once in a lifetime”): Calculate the supply function z ( p ) and the profit function π( p ) for each of the following production functions: f ( a ) = a α 1 for α 1. g ( a ) = α a 1 + β a 2 for α > 0 and β > 0. h ( a ) = min { a 1 , a 2 } . i ( a ) = ( a α 1 + a α 2 ) 1 for α 1. Problem 3. ( Easy ) Consider a producer who uses L inputs to produce K L outputs. Show the following: C w , y ) = λ C ( w , y ). C is nondecreasing in any input price w k . C is concave in w . Shepherd’s lemma: If C is differentiable, dC / dw k ( w ,
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ) = a k ( w , y ) (the k th input commodity). • If C is twice continuously differentiable, then for any two commodi-ties j and k da k / dw j ( w , y ) = da j / dw k ( w , y ). Problem 4. ( Moderately difFcult. Based on Radner (1993). ) It is usually assumed that the cost function C is convex in the output vec-tor. Much of the research on production has been aimed at investigating whether the convexity assumptions can be induced in more detailed mod-els. Convexity often fails when the product is related to the gathering of information or data processing. Consider, for example, a Frm conducting a telephone survey immediately following a TV program. Its goal is to collect information about as many...
View Full Document

{[ snackBarMessage ]}