Derive the degree of homogeneity of the production function
Business Economics
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) = x11/2 + x21/2. (1)
where
f: R2++ → R++
Consider the set
D = {(x1, x2) ∈ R2++,[x11/2 + x21/2 ≥ 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 = (x1, x2) ∈ D and v = (v1, v2) ∈ D.
and use Definition 1.31.
(d) Let
S = {(x1, x2) ∈ R2++|x11/2 + x21/2 = 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
X11/2 + x21/2= 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.
