Technology

This article is to provide insights to exercise problems of Chapter 1 “Technology” of Hal Varian’s Microeconomic Analysis (3rd Edition). Due to the length of the article, I will only give solutions to first three problems. Hope this article will help you understand the materials better.

Problem 1

True or false? If V(y) is a convex set, then the associated production set Y must be convex.

Answer:

False.

If Y is a convex set, then for x and x’ that makes and (y,-x’) in Y, there must be in Y. i.e. in Y.

If x and x’ in V(y), then tx+(1-t)x’ also in V(y), then V(y) is convex.

Counterexample that shows convex input set doesn’t mean convex production set.

Consider production function. Production set is not convex. But input set is convex.

Notes:

We need to raise a counterexample to tackle this proof.

Problem 2

What is the elasticity of substitution for the general CES technology when ?

Answer:

To calculate elasticity of substitution, we need to calculate substitution of technology.

Take log

Note:

This problems tests the concept of elasticity of substitution (), and its relationship with TRS.

The function of TRS is , then you can get the sigma.

Problem 3

Define the output elasticity of a factor i to be

If , what is the output elasticity of each factor?

Answer:

, so the elasticity of production of factor 1 is:

the elasticity of production of factor 2 is:

Note:

This is an interesting thing. Output elasticity is denoted by