Question

Solve the following:

1. The market demand curve is D (p) = 100 - 0.5p, what is the inverse demand curve?

2.In the following demand functions, say what price maximizes income

a) D (p) = 12−2p

b) D (p) = 100/p

Answer 1

1)

Let Q=D(p)=100-0.5p

On rearranging we get

0.5p=100-Q

Multiply by 2 both sides, we get

p=200-2Q

It represents inverse demand curve.

2)

In absence of any cost information, we will say that income maximization and revenue maximization are equivalent

a)

Q=D(p)=12-2p

First we find the inverse demand function. On rearranging we get

2p=12-Q

p=6-0.5Q

Total Revenue=p*Q=(6-0.5Q)*Q=6Q-0.5Q²

Marginal Revenue=MR=dTR/dQ=6-Q

Set MR=0 for revenue maximization,

6-Q=0

Q=6

p=6-0.5Q=6-0.5*6=$3

Income is maximized at a price of $3.

b)

Q=D(p)=100/p

First we find the inverse demand function. On rearranging we get

p=100/Q

Total Revenue=TR=p*Q=(100/Q)*Q=100

We find that Total Revenue is constant. It does not depend any output level.

Hence, Total Revenue does not depend upon price in this case.

Income is same for all p>0