Question

The company produces the Type A pencils and the Type B pencils. Suppose that x is the number of thousands of the Type A pencils. The demand function for the Type A pencils is P1 = 100 - x, where P1 is in dollars. Suppose that y is the number of thousands of the Type B pencils. The demand function for the Type B pencils is given by P2 = 120 - 2y, where P2 is also in dollars. This company’s joint cost function is C = 5xy (in thousands of dollars).

Note that this is an unconstrained optimization problem.

i) To maximize profit, how many of each type of pencils should be sold?
ii) Find the maximum profit.
iii) Applying the second-order condition with the use of the Hessian determinant, confirm that your answer in part ii) is a maximum.

Answer 1

i) profit is can be defined as the difference between total revenue and total cost.
TR from type A : TR(A) = 100x - x^2
TR from type B: TR(B) = 120y - 2y^2
Cost function: 5xy
In formal notation we can write,
P = 100x-x^2 + 120y-2y^2 -5xy
To find the optimal amount of x and y to be sold that gives maximum profit, we use the first order conditions equated to 0 in order to maximize P.
When P is differentiated with respect to x, we have,
100 -2x -5y = 0 ---- (1)
When P is differentiated with respect to y, we have,
120 - 4y -5x = 0 ---- (2)
Solving equations (1) and (2) simultaneously, we get ,

2x + 5y = 100 and 5x + 4y = 120.
or, x = 200/17 and y= 260/17

ii) maximum profit = 100x-x^2 + 120y-2y^2 -5xy , where x= 200/17 and y = 260/17.
So we have Pmax = 1739.80 (approx)

iii) The hessian determinant is given by the second order differentials which are
d^P/dx2 = -2
d^P/dydx = -5
D^P/dy2 = -4
|H| = 8-25 = -17 <0
This confirms that P is maximized using second order conditions.