Exercise 10. Cobb-Douglas vs. CES

Income is m=100, price of good 2 is normalized to p₂=1, while price on good 1, p₁ is a variable. The budget constraint is 100 = p₁x₁ + x₂.

a)With a Cobb-Douglas utility, u=x₁x₂, derive the demand for good 1, x₁=x(p₁).

b)Demand elasticity is = %x₁/%p₁ = (x₁/x₁)/(p₁/p₁) = (x₁/p₁)/(x(p₁)/p₁). Calculate the demand elasticity. Does elasticity change at different price levels? Using the midpoint method, calculate the percentage change from p₁=1 to p₁'=2 and compare it to the percentage change from x₁=x(p₁) to x₁=x(p₁').

c)With a CES utility, v=(x₁+x₂)², derive the demand for good 1, x₁=x(x₂,p₁).

d)What is the demand elasticity for this CES? Which utility best represent an elastic good?