A firm in a competitive market has the following cost functi

A firm in a competitive market has the following cost function C(q,beta)=q^B, where beta greaterthan1 Assume that the firm sells its product at a piece p find q(p, b)and the indirect profit function (p, b)what is the economic interpretation of this sing? Find Determine its sing. what is the economic interpretation of this sing? Find Determine its sing. What is the economic interpretation of this sing?

Solution

a) MC = Bq^B-1

R = p*q

MR = p

p = Bq^B-1

(p/B)^(1/(B-1)) = q

Profit =(p/B)^(B+1/(B-1)) - (p/B)^(1/(B-1))*p A positive sigh here will mean profit and negative sign loss

b) dq/dp = - 1/((B-1)p(p/B)^1/(B-1). The negative sign means that the function is decreasing i.e. with increase in price quantity will decrease.

c) dq/dB = - (p/B)^1/(B-1)((ln(p/B) + 1)B -1) / ((B-1)^2*B). The negative sign means that the function is decreasing i.e. with increase in B will decrease.

d) d (profit)/dB = - (p/B)^(1/(B-1)((p/B)^(B/(B-1)*B^2 + (2ln(p/B)(p/B)^(B/(B-1) -P ln(p/B) - p)*B - (p/B)^(B/(B-1)*p) / (B-1)^2*B. The negative sign means that the function is decreasing i.e. with increase in B will decrease.