Question

the demand equiation of a goof is given by P+2Q=20 abd the total cost function is Q3-8Q^2+20Q+2

a) find the level of output that maximizes total revenue

b) find the maximum profit and the value of Q at which it is achieved. verify that, at this value of Q, MR=MC

Answer 1

A).

Consider the given problem here the demand function is given by, => P=20-2*Q, => TR=P*Q.

=> TR = (20-2*Q)*Q = (20*Q-2*Q^2), => at the optimum FOC require “dTR/dQ = 0”.

=> 20 -4*Q = 0, => Q = 20/4 = 5, => Q*=5. So, the revenue maximizing production is given by, => “Q*=5”.

B).

The profit function is given by, => A=TR-TC, => A = (20*Q-2*Q^2) - (Q^3 – 8*Q^2 + 20*Q + 2). So, at the optimum “dA/dQ = 0”.

=> (20-4*Q) - (3*Q^2 – 16*Q + 20) = 0, => 20-4*Q - 3*Q^2 + 16*Q - 20 = 0, => 12*Q - 3*Q^2 = 0.

=> 3*Q^2 = 12*Q, => Q=12/3=4, => Q=4. So, the profit maximizing level of production is “Q*=4”. Now, the maximum profit is given by, => A =TR-TC, => A = (20*Q-2*Q^2) - (Q^3 – 8*Q^2 + 20*Q + 2).

=> A = (20*4-2*4^2) - (4^3 – 8*4^2 + 20*4 + 2) = (80-32) - (64 – 128 + 80 + 2) = 48-18 = 30, => A*=30.

Now, given the “TR” the “MR” is given by, => MR = 20 - 4*Q, => at “q=4”, => MR = 20-4*4 = 4. Now, the “MC” is given by, => MC = 3*Q^2 – 16*Q + 20 = 48-64+20=4, => at “q=4” the “MR=MC”.