Question

Question #5: (10 points) Suppose that a monopolist produces an identical product in three plants and face an inverse demand function P = 40 - QF = Q, + Q, +Q,. The output from the three plants is where total quantity in all three plants is Q produced at the costs c = Q + Q C₂ = 30, z = 2Q - Qz where C refers to the total cost required to produce Q, units from each facility, i = 1, 2, 3 . (a) Assuming that the total revenue TR = PQ, and the total cost TC = C, +C, +C, calculate the profit function. that maximize this firm's profits. Check the (b) Determine the levels of output, Q, Q, and second-order conditions.

Answer 1

.(W) Given P = 4o-ar where or sit azt az Now TR[Total Revenue =.P.O = P.Or But P = 40-y .huiven) Hence (40-Or] Dr. Now Total cost - And given values of Cat Cat (a c= 0, +9, 2 C = 30 y = 26² - Q3 we have Pulling values Co Now profit function by defination putting values we have 40-49, +62+)-(3,+82 +0,)- [a,+8, 2+382 + 283 - 03]

(6) For maximum profit df (Total Revenue - Total cost) o a dr or putting values we have [40 (0, +8 27 ₂) - (a + Q ₂ ta. 0,+8, 2+30+20,4-2,] 2 (B1+027 03) On partial differentiation we have O HITR-TC) +40-2681+01+03,) 8-20,20 1.(TR-TE) I ) 40-2 (0,+Q, +h)-3 =0 do. af (TR-TC) - 40-2(81 + 2 + 2y) - 4Q 2+1=0 And solving au, linear equation we have 7:03 = 1., B1=1 & 2, = 33 |

We basically have to use the following formula

1. Profit function is the function of Total Revenue and Total cost

2. For solving for Q1, Q2, Q3 we have used marginal revenue- marginal cost equal to zero and then solved.

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