Question

PLEASE SHOW YOUR WORK!

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

Profit = 226X – 5X2 –XY -2.5Y2 +150Y -210

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

Calculate the profit.

To work part A, you will need to

i. Find the partial derivative 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

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

B) Do question 3 Part A again, but this time with a constraint of X + Y =40.  

i.To work 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.

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

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

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

Answer 1

The total profit function of a firm is Profit = 226X – 5X2 –XY -2.5Y2 +150Y -210

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

Calculate the profit.

Find the first order partial derivatives of profit function with respect to X and Y

Π’(X) = 0

226 – 10X – Y = 0…….(1)

– X – 5Y + 150 = 0………..(2)

Solve the two equations using the value of Y from first equation (1) as Y = 226 – 10X and place it in (2)

– X – 5(226 – 10X) + 150 = 0

– X – 1130 + 50X + 150 = 0

X* = 20 and Y* = 26

The profit is Π = 226*20 – 5*20^2 –20*26 – 2.5*26^2 + 150*26 – 210 = 4000

B) Do question 3 Part A again, but this time with a constraint of X + Y =40.  

Now that X is equal to 40 – Y, profit function becomes

Π =226(40 – Y) – 5(40 – Y)^2 –(40 - Y)Y – 2.5Y^2 +150Y – 210

= 9040 – 226Y – 8000 – 5Y^2 + 400Y – 40Y + Y^2 – 2.5Y^2 + 150Y – 210

= 830 – 284Y – 6.5Y^2

Profit is maximized when Π’(X) = 0

284 – 13Y = 0

Y = 21.85 and X = 40 – 21.85 = 18.15

Profit is 3932.154