Question

4. a.       For the following total profit function of a firm, where X and Y are two goods sold by the firm:

Profit = 196X – 6X2 –XY -5Y2 +175Y -270

(a)          Determine the levels of output of both goods at which the firm maximizes total profit.

(b)          Calculate the profit. (20 points)

To work problem 4 you will need to

                a. Find the partial derivative (see p. 106) with respect each of the variables. Remember that when taking the partial derivative with respect to a variable that the other variable is treated as a constant; e.g., the partial derivative with respect to X of the expression V = 20XY is 20Y         

                b. You want to find the quantities at each partial derivative (i.e., each marginal profit) are simultaneously zero. You solve these by simultaneous equations (pp, 107-108) or using substitution (pp. 108-109). Either way works but the substitution is usually the simplest.

4. b.       Do problem 4. a. again, but this time with a constraint of X + Y =25. (20 points)

a.            To work problem 4. b. use the constraint equation to find an equation for X in terms of Y (X = 40 – Y) or an equation for Y in terms of X.

b.            Substitute the equation into the total profit equation at every place you find X (or Y)

c.             Set the derivative equal to zero and solve for X (or Y)

d.            Plug the result from part c into the constraint equation to get the other variable.

Answer 1

4a:

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4b.