Question 1:

Consider a firm which produces a good y, using two inputs or factors of production, x₁ and x₂. The firm's production function is

y = f(x₁, x₂) = x₁^1/2 + x₂^1/2.          (1)

where

f: R²₊₊ → R₊₊

Consider the set

D = {(x₁, x₂) ∈ R²₊₊,[x₁^1/2 + x₂^1/2 ≥ Y₀} (2)

That is, D is the set of all (x₁, x₂) ∈ R²₊₊ which, given (1), produces at least output level y₀. D is known as the upper contour set associated with output y₀.

(a) Derive the degree of homogeneity of the production function given by (I). Show all steps in deriving your answer.

(b) Prove that the fiumtion in (1) above is strictly concave on R₊₊.

(c) Prove(2) above is a convex set.

Hint 1: Assume that

x = (x_1, x₂) ∈ D and v = (v₁, v₂) ∈ D.

and use Definition 1.31.

(d) Let

S = {(x₁, x₂) ∈ R²₊₊|x₁^1/2 + x₂^1/2 = y₀}.

That is, S is the set of all combinations of (x₁, x₂) that produce output level y₀. Economists call S the isoquant associated with output level), The equation

X₁^1/2 + x₂^1/2= y₀

implicitly defines xt as a function of x₂.

i) Use implicit differentiation to derive

dx₂/dx₁

ii) Use implicit differentiation to derive

d²x₂/d²x₁

iii) What conclusions do you derive regarding the slope and curvature of the isoquant? Briefly explain.