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.
In a college Residence A there are 320 students. In Residence A there are 1720 canned drinks consumed per week. In another residence, Residence B, there are 260 students who consume 1480 canned drinks per week.
a.Calculate the linear function (equation) if drink consumption is a linear function of the number of students.
b.Calculate the number of canned drinks required to stock Residence C per week knowing that the number of students in Residence C is 600.
The XYZ Company can sell 10,000 units of steel for $500,000. The cost of these units is $600,000. The company can sell 25,000 units for $1,250,000 and these cost $1,050,000. Assuming there is a linear relationship between these variables:
a.Find the revenue, cost and profit functions.
b.Find the breakeven point and graph the revenue and cost functions for 0 ≤ X ≤ 30000.
An auto parts company requires a large number of gaskets which they currently buy for .50 each. A recent feasibility study has indicated that if they produced them internally, their annual fixed costs (loan payments on equipment, equipment maintenance, etc.) would total $10,000 and the material and labour costs would be .40 per gasket.
a.They currently require 70,000 gaskets per year. Should they begin to produce their own gaskets? Explain.
b.They estimate their gasket requirements will increase by 10,000 a year. In how much time will it become profitable to manufacture the gaskets internally assuming that the costs remain the same.