A commodity has a demand function modeled by p = 30 − 0.5x, and a total cost function modeled by C = 9x + 37.

Question

What price yields a maximum profit?

When the profit is maximized, what is the average cost per unit?

Answer 1

P=30- x/2

C=9x+37

Revenue = Px = 30x -(1/2)x^2

Cost = 9x +37

Profit = Revenue minus Cost = 30x-(1/2)x^2 - 9x - 37

= -(1/2)x^2 +27x -37

take the derivative and set it equal to zero, solve for x

(R-C)' = -x +21 = 0

x=21

plug it back into the Profit equation

-(1/2)21^2 +21(21) -37 = -220.5 + 441 -37 = 220.5 - 37 = 183.5 = max profit

plug x= 21 into the demand equation to get profit maximizing price

P=30-x/2 = 30=21/2 = 19.5 = price that maximizes profit

AC = C/x = Average Cost = (9x+37)/x = 9+37/x = 9 +37/21 = 10 16/21= 10.76

Another way to view this is the profit maximizing output level is when marginal cost = marginal revenue

MR=MC

MR = 30-x

MC= 9

30-x=9

x = 30-9 = 21