Consider a firm which produces a good, y, using two inputs or factors of production, x1 and x2. The firm's production function describes the mathematical relationship between inputs x1 and x2 and output y, and is y=f(x1,x2)=x1^1/2+x2^1/2, where f:R²₊₊->R++

consider the set D={(x1,x2) ?R²₊₊|x1^1/2+x2^1/2 ?y0}

(a) Determine the degree of homogeneity of the production function given by(1).Show all steps in deriving your answer. No marks will be awarded for an unsupported answer.

(b)Prove that the production function

y =x1^1/2+x2^1/2

is strictly concave on R++

(c)Prove that the set

D={(x1,x2) ?R²₊₊|x1^1/2+x2^1/2 ?y0}

is a convex set.

Hint 1: Assume that x=(x1,x2)?D and v=(v1,v2)?D

and prove that z=?x+(1-?)v?D

for any 0

(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 y0.

The equationx1^1/2+x2^1/2 =y0,

implicitly defines x1 as a function of x2.

i) Derive the slope of the isoquant for y0,

dx2/dx1

ii) Derived²x2/dx1²

iii) What do you conclude regarding the slope and curvature of the isoquant for y0? Briefly explain.

Question 2 (25 marks) Consider a rm which produces a good, y, using two inputs or factors of production, x1 and X2. The rm's production function, which describes the mathematical relationship between the inputs x1 and X2 and output y, is y =f(x1,x2) = xi\" +xi'2, (1) where f: RL > RH. Consider the set D = {(x1,x2) 6 RAM/2 +x/2 2 yo}. (2) That is, D is the set of all (x1,x2) 6 RL which, given (1), produces at least output level yo. D is known as the upper contour set associated with output yo. (a) Determine the degree of homogeneity of the production function given by (1). Show all steps in deriving your answer. No marks will be awarded for an unsupported answer. (4 marks) (b) Prove that the production function y = xi/z + x/Z is strictly concave on R++. (6 marks) (c) Prove that the set D = {(x1,x2) e Repel\" +x'2 2 yo} (2) is a convex set. (7 marks) Hint 1: Assume that x = (x1,x2) e D andv = (v1,vz) e D andprove that z=iLx+(1Z)veD for any 0