Demand and cost function

Question

1. A firm has estimated its demand function to be:

QD=2000-1000P

TC= 150 0.25Q

Find the optimal price, output and profit.

2. The demand and cost function for a companyare estimated to be as follows:

P:100-8Q

TC=50 80Q-10Q(square 2nd power) 0.6Q(cube 3rd power)

Sry, dont know how to get the powers at the top

A)Find the price that the firm should charge if it wishes to maximise profits in the short run.

b) The price if it wishes to maximise revenue in the short run,

Is there any website I can use, Econs is not my thing so pls help!!

Answer 1

For the cost functions, are there supposed to be plus signs?

If so...

1. Q=2000 - 1000P

But we should get this in terms of Q, because the cost function is in terms of Q.

1000P=2000-Q

P=2-Q/1000

Now we can maximize the profit.  Profit equals total revenue minus total cost, and revenue equals P*Q.

Pr=P*Q-TC

Pr = (2-Q/1000)*Q - (150+.25Q)

Take the derivative and set it equal to zero to maximize the function.

Pr'=2-2*(Q/1000) -.25=0

Pr'=1.75-Q/500=0

1.75=Q/500

Q=875

P=2-875/1000

P=1.125

Pr=875*1.125-(150+218.75)

Pr=9849.375-368.75

Pr=9480.625

2. Same thing as above for finding profit maximizing price

Pr=Q*(100-8Q)-(50+80Q-10Q^2+.6Q^3)

Pr'=100-16Q-80+20Q-1.8Q^2=0

Pr'=20+4Q-1.8Q^2=0

Q=~-2.4 or 4.625

A negative quantity is impossible, so it's 4.625

P=100-8(4.625)=63

But I'm not sure if this answer is correct, so wait for someone else to answer it as well.

b) Now you just maximize the revenue part.

TR=Q*(100-8Q)

TR'=100-16Q=0

100=16Q

Q=6.25

P=100-8(6.25)

P=50

Added: 3. Where the demand curve is inelastic, if the firm lowers their price, total revenue falls.  This means that at that point, marginal revenue is negative.  Since a firm sets marginal revenue equal to marginal cost, they will never choose a quantity where marginal revenue is negative, since marginal cost is never negative.