Econ 407, Oct.8, 1999, Friday's Minitest

Examine the following Total Revenue (TR) function for a firm that produces two products: q1, and q2. Find the combination of q1 and q2 that will result in maximum total revenue. Please demonstrate using first, second, and cross derivatives that your soulutions give the optimum values for q1 and q2. You will have several values for q1 and q2, but remember that only positive values of q1 and q2 are acceptable in this case. Please provide the numerical value of the TR at the optimum solution.

>

> TR:=-5*q1^2 -8*q2^2-2*q1*q2+14*q1+34*q2;

> plot3d(TR,q1=0..10,q2=0..20);

> TRq1:=diff(TR,q1);

> TRq2:=diff(TR,q2);

> solve({TRq1,TRq2},{q1,q2});

> TRq11:=diff(TRq1,q1);

> TRq22:=diff(TRq2,q2);

> TRq1q2:=diff(TRq1,q2);

> TRq11*TRq22;

> (TRq1q2)^2;

> eval(TR,{q1=1,q2=2});

>

Since the products of the two second derivatives = 160, which is bigger than the square of the cross derivatives--4, we have established a relative maximum.

Please note TRq1q1=-10, TRq2q2=-16, and TRq1q1*TRq2q2=160>(TRq1q2)=4. So we have a maximum. For further clarification see the lecture on the saddle point.

please read page 130 of the text, do questions 5 and 6. Also Example 7 on page 131, and 132.