Extra Credit

How does this problem differ from the one we solved in HW6? (Besides from the fact that we are using two demand curves in the present problem, while we used three in HW6. This is NOT the answer.)

In particular, how does the ATC function in this problem make the solution more "simultaneous" than the ATC function in HW6.

In case you forgot, in HW6, the ATC function was ATC = 20/Q + 15, and the demand curves were 63 -4Q1 105 - 5Q2 and P3 = 75 - 6Q3THE PROBLEM:

1) A monopolist sells a product in two separate markets at different prices; i.e., he price discriminates. The demand curves in these markets are:

100 - QA and PB = 60 - 0.5QB

His average cost function (ATC) is: ATC = Q + 100/Q where Q = QA + QB

In your calculations, let be defined as profit.

(Hint: Find an equation form. Then replace Q with QA+ QB. Then, maximize with respect to QA and QB.)

a)Use Cramer's Rule to find the profit maximizing quantities QA* and QB*.

(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,

c)Find where QA*+ QB*

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

\fPAGE 3 oo SUPPLY VALUES FOR EACH OFF (LA) CURVE PLEASE SUPPLY THE NUMBERED POINTS VALUES FOR EACH OF THE QLABEL ALL NUMBERED POINTS. LABEL ALL CURVESExample 4 To make the above example more concrete, let us now give a numerical version. Suppose that our monopolistic firm has the specific average-revenue functions P1=634Q1 so that R1 = P101=6301'Qf epicsso; R2 = P202 =10502SQ 93:75-60; Rs: P303 =7SQs-6Q and that the total-cost function is C = 20 + 150 Then the marginal functions will be R3263801 R'z=1051OQZ R=751203 C'=15 When each marginal revenue R:- is setequal to the marginal cost C' of the total output. the equilibrium quantities are found to be 07:6 {33:9 and 03:5 3 Thus Q'=ZQ'{=20 r'=1 Substituting these solutions into the revenue and cost equations, we get at = 679 as the total prot from the triple~mar|