Suppose a firm faces the total benefit function:
B(Q) = 18Q - Q^2
and the total cost function:
C(Q) = 2 - 2Q +Q^2
a. What quantity should the firm select?
b. What is the amount of total benefits, total costs and total net benefits at the selected quantity?
c. If the first term in the cost equation changed from 2 to 5, how would that impact the solution you provided above in part (a)?
d. Explain the impacts in part (c) regarding the change to the optimal output quantity. How do you unambiguously know the quantity selected is the "optimal" one?
a) To find the optimum quantity, you have to find the Marginal Benefit (MB) and Marginal Cost (MC) functions. The maximum net benefit occurs when MB = MC.
B = 18Q - Q^2
MB is the derivative of B:
MB = 18 - 2Q
C = 2 - 2Q + Q^2
MC is the derivative of C:
MC = ...
This is a detailed, step-by-step solution that calculates a firm's optimal Quantity when its Benefit and Cost functions are both known. The correctness of the solution is demonstrated by a spreadsheet.