Question

Afirm faces an (inverse) demand function p(y)=10 –y, where p is price,yis quantity.The cost function of the firm is given by c(y) = y^2+1. Dont copy other guys solution!!!

(1) Draw thecurves of(inverse) demand function and marginal revenue. Show your detailed work such as slope, intercept.

(2) What is the optimal choice of output and corresponding profitof the firm?Show each of your steps clearly.(Results might not be integers. Do NOTround your answer)

Answer 1

1)

Given

p(y)=10-y

Slope of demand curve=dp(y)/dy=-1

x-intercept (quantity axis) is attained where p(y)=0, i.e.

10-y=0

y=10

So, x-intercept of demand curve is (10,0)

y-intercept (price axis) is attained where y=0, i.e.

10-p=p(0)

p(0)=10

So, y-intercept of demand curve is (0,10)

Total Revenue=TR=p(y)*y=(10-y)*y=10y-y^2

Marginal Revenue=MR=dTR/dy=10-2y

Slope of Marginal Revenue curve=dMR/dy=-2

x-intercept (quantity axis) is attained where MR=0, i.e.

10-2y=0

y=5

So, x-intercept of MR curve is (5,0)

y-intercept (price axis) is attained where y=0, i.e.

10-2*0=MR

MR=10

So, y-intercept of MR curve is (0,10)

b)

c(y)=y^2+1

Marginal cost=MC=dc/dy=2y

Set MR=MC for profit maximization

10-2y=2y

4y=10

y=2.50 units (profit maximizing output)

p=10-y=10-2.50=$7.50 (profit maximizing price)

Total Revenue=TR=p*y=7.50*2.50=$18.75

Total cost=TC=y^2+1=(2.50)^2+1=$7.25

Maximum profit=TR-TC=18.75-7.25=$11.50