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Economics Homework Seven - Model (view source)
Revision as of 20:28, November 1, 2009
, 20:28, November 1, 2009better, but still not solved
11. (Challenging, with extra credit) Prove mathematically that MC>MR for all Q>0 in the honors discussion above. (Hint: define MC in terms of the change in AVC, and then regroup the terms and draw conclusions about them to show MC>MR).
11. (Challenging, with extra credit) Prove mathematically that MC>MR for all Q>0 in the honors discussion above. (Hint: define MC in terms of the change in AVC, and then regroup the terms and draw conclusions about them to show MC>MR).
:No student attempted this! Perhaps I made it too scary. Using the hint, I define MC in terms of AVC: MC for the "Q+1" unit is: MC(Q+1)=TC(Q+1)-TC(Q). Next I can write TC in terms of AVC: TC=AVC*Q. Thus MC(Q+1)=AVC(Q+1)*(Q+1) - AVC(Q)*Q. Thus MC(Q+1)=AVC(Q+1)*Q + AVC(Q+1) - AVC(Q)*Q. Now regroup as the hint suggests: MC(Q+1)=Q*(AVC(Q+1)-AVC(Q)) + AVC(Q+1) <in progress>
:No student attempted this! Perhaps I made it too scary.
:From above, the problem is "Suppose the price of a widget produced by a firm in perfect competition is less than the minimum average variable cost (AVC) in the market. What will the firm do next in the short run?" We are being asked to prove "mathematically that MC>MR for all Q>0."
Using the hint, I define MC in terms of AVC: MC for the "Q+1" unit is: MC(Q+1)=TVC(Q+1)-TVC(Q). Next I can write TVC in terms of AVC: TVC=AVC*Q. Thus MC(Q+1)=AVC(Q+1)*(Q+1) - AVC(Q)*Q. Thus MC(Q+1)=AVC(Q+1)*Q + AVC(Q+1) - AVC(Q)*Q. Now regroup as the hint suggests: MC(Q+1)=Q*(AVC(Q+1)-AVC(Q)) + AVC(Q+1) <in progress>