Question

3) A monopolist sells a product in two separate markets at different prices; i.e., he price discriminates. The demand curves in these markets are: PA= 100 - QA and P8 = 60 - 0.5 QB His average cost function (ATC) is: ATC = Q + 100/Q where Q = QA + QB In your calculations, let it be defined as profit. (Hint: Find an equation for . Then replace Q with QA + Qs. Then, maximize with respect to QA and Qs.) a) Use Cramer's Rule to find the profit maximizing quantities Quand Qs". (If you absolutely do not remember Cramer's rule, do it any way you can.) You do NOT have to check second derivative conditions for a maximum. b) Find the maximum profit, i c) Find ATC(Q) where Q = QA+ Qo' d) Graph the solution, as we did in HW6. You should have two graphs of demand curves and one graph of cost curves. The following page has a sample set of graphs that you can use as a guide, with all labels and coordinates missing. Label the curves and lines in each graph. On the sample, I have labeled 19 points with plain numbers. Replace these numbers with the values from your solution. NOTICE that the graphs line up horizontally. For example, numbers 4, 11, and 16 should have the same value. Similarly, 3, 10, and 15 should have the same value. Show the profit in each market on the graphs. (It is a rectangle.) Calculate the profit in each market (show your calculations) and verify that it equals

Answer 1

The dagram as mentioned in the question is missing. Hope it is similar to the one I have given here.

TC = -100 Q ATC) = Q (Q+lecom] TC = q* +100 = Qu+Qa)*100 = Qž +Q6 +22nQrt100 → T = P QA+PQC-TC = (100-QA) QA +60-0's QB) Qo-Q+Q8 +2 QA Q3 +100) = 100 QA - Q +60Q6-05 Q6 -QA - Q3 -2.QA QB- - 100 Qa+60 Q8 - 2 QA - 1.5Q8-9 QAQB-100 Foc|8T --0->100-4QA-208-0 → 4 QA+2QB = 100 -> 2QA Qe=50 AQA an 060-396-20APO => 20A+3Qp = 60 226 a) Equation 0 and can be rewritten in matrix forem as le :][%][C] 151 +12 31-4 , balle -90 ,1021 -1 3 = => QA = D. and Q8 - 10 21 = 20 = 5 D Hence, profit maxemizing quantities and QI =22:5 and (b) ** Since QA-22.5, Pa* = 100-22.5 = 77.5 Q* - 5 PT-60-055) - 57.5 Ta Q = QarQ -> TC = Q+100 (27.5) HOO - 856.25 =* - Pa Qu + post QTC - (+7.5) (22.5) + (57.5) (5) -856-25 = 1175. 6 From (b) we find TC=856.25 = 31.14 (rup to 2 decimal pororts) - = 20 -22.5 -5 -22.5 +5 = 27.5 → ATC - 856.25 27.5 PA Market A Market B Cast MC 77.5 AC 60 55 1 AR MR. AR 100 MR1 60 he 21.5 5 120