Problem 4 Part 2 [15] 0 points possible (ungraded) For a monopolistic company, the demand curve is given by the function P = 160 – 8Q, where P is the price per unit and Q is the total amount of units produced. The cost per unit, C, also depends the total amount of units produced and the relation is given by C = 120 – 6Q. That is, the price per unit decreases linearly as the number of units produced increases. a. Find the total profit U as a function of Q. Remember that Total profit = Total Revenue – Total Cost, where Total revenue is equal to price per unit times the total amount of units and Total Cost = cost per unit times the total amount of units. [3] b. Find the value of which maximizes the total profit U. Round your answer to the nearest integer. [8] c. Find the maximum value of the profit. [4]

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EXPERT ANSWER

Demand : P = 160 – 8Q
Cost : c = 120 – 6Q

a) Profit = Revenue – Cost
Revenue = P*Q = 160Q – 8Q2
Profit : U = 160Q – 8Q2 – 120 – 6Q
= 154Q – 8Q2 – 120
Total profit function : U = 154Q – 8Q2 – 120  

b) Maximizing total profit :
dU/dQ = 154 – 16Q = 0
or, 16Q = 154
or, Q = 154/16
or, Q = 9.62 or 10 (rounded)

d2U/dQ2 = -16 <1
Hence for Q = 10, profit will be maximum.

c) Maximum profit will be obtained by the value U at of Q=10

Maximum Total profit : U = 154Q – 8Q2 – 120  
= (154*10) – (8*10*10) – 120
= 1540 – 800 – 120
= 620

Hence maximum value of profit = 620