Project #45649 - Mathematics and Economics

There are two questions. Do them both. These are tricky questions, so expect to spend some time on them. Don't bid if you don't have a strong mathematical background.

 

Hints:

 

In question 1, the two equations are P = D(Q) (the demand function) and PQ - C(Q) = F (the zero profit condition). Your mathematical analysis will be easier if you do the total differentials using the form I have written, rather than dividing both sides of the zero profit condition by Q. However, your graph is easier if you draw the demand curve and then the P=A(Q) curve. You should assume that the limit as Q approaches zero of A(Q) is infinity (while I have told you that the demand curve has D(0) finite), so that A(Q) dips below the demand curve at Q1>0 and then crosses it again a second time at Q2>Q1. (Note that A(Q) may be rising at Q2 or it may be still falling at Q2; which it is does not affect the results.) Assume for simplicity that P=A(Q) only crosses the demand curve P=D(Q) twice. The key to this question is understanding that the Jacobian matrix sign depends upon the relative slopes of the demand and average cost funtions. This is why I want you to tell me about the slope dP/dQ of the demand function and of the zero profit condition in part (a). Thus, your answer to part (b) is different from your answer to part (c). You should also prove that dP/dF > 0 holding Q constant, which shows you how to shift the zero profit condition. 

Hints on question 1: You should plot C'(Q) relative to A(Q) to understand how they are related, as that plays a role in the Jacobian matrix and the slopes of the two functions.

In question 2, you should get that the best response functions in part (a) implicitly solve dProfit(i)/dq(i) = D(q1+q2) + qi*D'(q1+q2) - ci = 0, for  i=1,2. The function P=D(Q) is known only to have a negative first derivative, so that D'(Q)<0 I have not told you anything about D''(Q). You may assume it is not infinite, but other than that, it must satisfy the second order condition that d^2Profit(i)/dq(i)^2 < 0 (where the ^2 means 2 is the exponent). In part (b), you need to find what the second order condition implies about the second partial derivative of the first-order-necessary condition with respect to own output. Then, using that information, you can use total differentials and the implicit function to determine the slope of the best-response functions, dq1/dq2, and then find what additional condition has to hold in order that they are downward sloping. Then part (c) asks you to take the Hahn condition on the relative slopes of the best-response functions and to determine what that implies about the determinant of the Jacobian matrix. Part (d) then uses Cramer's Rule to to comparative statics with respect to c1 (you should show how the best-response functions shift as a result of an increase in c1 to check your intuition). Part (e) uses the fact that Q=q1+q2 to use the Cramer's Rule results to determine how Q changes when c1 increases, given that dQ/dc1 = dq1/dc1 + dq2/dc1. Finally, P = P(Q), so changes in P are the opposite as changes in Q. 

Hint on question 2: The total differential of firm 1's best-response curve should be:  [2P'(Q) + q1*P''(Q)]dq1 + [P'(Q) + q1*P''(Q)]dq2 = 1*dc1, where I have written Q each time the quantities are summed. Figure out why what I have written is correct, and derive the equavalent total differential for firm 2. You might find it useful to check your logic with a demand function which is linear, i.e., P(Q) = a - bQ, but you must do the general case where P''(Q) is not necessarily equal to zero to get marks on Part (b).

Subject Mathematics
Due By (Pacific Time) 11/02/2014 12:00 am
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