Question 3 Elasticity of Substitution The elasticity of subs

Question 3: Elasticity of Substitution The elasticity of substitution is a measure of how the ratio of inputs changes as the marginal rate of substitution changes. Formally, the elasticity of substitution of F at (Lo, Ko) is given by:1 It can be shown that for a CES production function: r = 1/(1-aj is the elasticity of substitution of F (a) What does the production function F become when 1, what is the elasticity of substitution in this case? (b) What does the production function F become when 0, what is the elasticity of substitution in this case? (Hint: Take natural logarithm and use l\'Hôspital\'s law) (c) (Bonus) What does the production function F become when -oo, what is the elasticity of substitution in this case?

Solution

a. As sigma tends to 1 we will have the case that the production function collapses to F=aL+(1-a)K. Where a here is alpha. So when the sigma tends to 1, the CES production function collapses to a linear combination of inputs and so we have perfect substitutes. As sigma tends to 1, the CES production function tends to perfect substitutes.

b. When sigma tends to 0 the CES production function will become a constant 1. So inside the brackets we have 1, while the power will explode but as sigma tends to 0 the production function will tends to 1.

c. When sigma tends to infinity the production function tends to 1 as the exponent will go to 0.