How to calculate profit maximization?

Question

Here r my queries:

prob 1: Qd=490-7P

MC=10

Prob 2: find P = ? and Q = ?

P=85-0.5Q

MC=10

I am getting confused on these qts and i require more practice to solve this. You can provide me similar example to the problems (as these r my quiz qts) and i want to understand the solving these probs.

cheers

Answer 1

Prob1:

Profit maximization is when MC=MR

MR=(TR)'

TR=P*Q

7P=490-Q

P=70-Q/7

TR=P*Q=(70Q-Q²/7)

MC=(70Q-Q²/7)' = 70-2Q/7

70-2Q/7 = 10

2Q/7=70-10=60

Q=42/2=21

P=490-7*21=490-147=343

Prob2:

MC=MR

MC=10

MR=(TR)'

TR=Q*P=85Q-0.5Q²

MR=85-Q

85-Q=10

Q=75

P=85-75/2=47.5

Added:

For your second example:

P=1000-5Q

MC=3Q

Well actually marginal revenue is first order derivative by quantity from total revenue, so MR=(TR)' or:

MR=δTR/δQ

And TR=P*Q = (1000-5Q) * Q = 1000Q - 5Q²

TR= 1000Q - 5Q²

MR= δTR/δQ = (TR)' = (1000Q - 5Q²)' = 1000-2*5Q= 1000 - 10Q

Now MR=MC

1000 - 10Q = 3Q

1000=3Q+10Q = 13Q

Q = 1000/13 ≈ 76.92

Now put Q into demand equation P=1000-5Q to get price:

P=1000-5*76.92=1000 - 384.615 = 615.385