Number of units produced monthly,N
FIGURE 9. Incremental-cost curves.
monthly, P = monthly profit, and the subscript refers to the article. If production is re-
stricted to one article, the firm can produce 1000 units of A, 1500 units of B, and 1800
units of C per month. What monthly production of each article will yield the maximum
profit?
Calculation Procedure:
- Express the constraint imposed on production
Let T = number of months required to produce TV units of an article, with a subscript to
identify the article. Then TA = TVyiOOO; T 8 = AT 5 ASOO; and Tc = TVC/1800. Since 1 month
is available, TVyiOOO + TV1500 + TVC/1800 = 1, or 1.87V 4 + .2NB + Nc = 1800, Eq. a.
- Determine how the values of N can vary
Assume for simplicity that NA is restricted to integral values but NB and NB can assume
nonintegral values. Equation a reveals that if NA increases by 1 unit, N 8 must decrease by
1.8/1.2 = 1.5 units, or NB must decrease by 1.8 units. Expressed formally, the partial de-
rivatives are SNBldNA = -1.5 and dNcldNA = -1.8.
- Write the equations of incremental profits
If TV units of an article have been produced, the incremental profit at that point is the prof-
it that accrues from producing the (TV+ l)th unit. Let /= incremental profit. If TV is large, 7
— dPldN. By differentiating the foregoing expressions, the incremental profits are IA -
475 - 0.010OTV^, I 8 = 2.60 - 0.0028TV 5 , and Ic = 2.25 - 0.002OTV 0 where the subscript
refers to the article. Also, if TV is large, the profit from the TVth unit =* profit from the (TV+
l)th unit.
- Establish the condition at which the total profit is maximum
Arbitrarily set TV 4 = 300, TV 5 = 400, TVC = 780, satisfying Eq. a. The incremental profits are
Incremental
cost
C, $
