Suppose That You Are Given a Cost Function C(w,r,x)=2w^1/2r^1/2x^3/2 Where

Question

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The Answer is a. The marginal cost is

a. The marginal cost is and the derivative of this with respect to output is positive -- i.e. the MC curve is upward sloping. This implies decreasing returns to scale. b. We derived it above as (Part (a) could be answered without making explicit reference to the MC function.) c. The supply function can be derived by setting MC equal to price and solving for output. (Recall the supply curve is the part of the MC curve that lies above AC -- and since we know this production process to have decreasing returns to scale throughout, the MC lies above AC everywhere.) Solving gives the supply function d. In that case, the production process has increasing returns to scale (with downward-sloping MC). As a result, the firm's profit maximization problem does not have an interior solution -- the firm would produce an infinite amount. Of course this does not make sense -- because it does not make sense to assume price-taking firms can have production technologies that have increasing returns to scale throughout.

and the derivative of this with respect to output is positive -- i.e. the MC curve is upward sloping. This implies decreasing returns to scale.
b. We derived it above as

(Part (a) could be answered without making explicit reference to the MC function.)
c. The supply function can be derived by setting MC equal to price and solving for output. (Recall the supply curve is the part of the MC curve that lies above AC -- and since we know this production process to have decreasing returns to scale throughout, the MC lies above AC everywhere.) Solving

gives the supply function

d. In that case, the production process has increasing returns to scale (with downward-sloping MC). As a result, the firm's profit maximization problem does not have an interior solution -- the firm would produce an infinite amount. Of course this does not make sense -- because it does not make sense to assume price-taking firms can have production technologies that have increasing returns to scale throughout.

Suppose That You Are Given a Cost Function C(w,r,x)=2w1/2r1/2x3/2 Where

Definitions:

Suppose That You Are Given a Cost Function C(w,r,x)=2w^1/2r^1/2x^3/2 Where