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The Production Function and Cost Analysis

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1. The attached table summarizes the short-run production function for your firm. Your product (Q) sells for $50 per unit, labor (L) costs $5 per unit, and the rental price of capital (K) is $25 per unit. Lastly, assume capital and labor are used only in whole units, i.e. it is not possible to use 1.37 units of capital or labor, but instead 1,2, 3, etc. units are used. Complete the following table, and then answer the accompanying questions.
a. Fill in the missing values in the above table.
b. Which inputs are fixed inputs? Which are the variable inputs? Explain.
c. How much are your fixed costs at an output of 10? Show your calculations.
d. What is the cost of producing 2 units of output? Show your work.
e. How many units of the variable input should be used to maximize profits? Explain.
f. What are your maximum profits? Show your calculations.

2. You hire an intern from Southern University to help you analyze your production process, and she uses your historical cost records to estimate that your total cost function is C(Q) = 100 + 2Q + 3Q^2. Using this equation, answer the following and show your logic and calculations.
a. How much is your firm’s profit or loss at an output of zero? Explain.
b. With the above cost equation, average total costs (ATC), first decline as output increases, and then begin to rise. Assuming output occurs in whole units, i.e. 1, 2,3,… at what output does the low point of ATC occur and what is the ATC at that output? Show your calculations.
c. What is the variable cost of an output of 50?
d. What are Average Fixed Costs (AFC) at a production level of 10?

3. Suppose the new leadership in Congress decides to repeal some of the tax breaks granted to large businesses during the past several years. If investment tax credits granted earlier are either repealed or eliminated, what impact will the repeal have on the exporting of jobs to foreign countries? Explain using isoquant/icocost theory.

4. You are the manager of USA 800, a small telemarketing company. Your company owns its own building and rents part of it as temporary storage for $10,000 per month. USA 800 has been asked to do a new telemarketing campaign for Ford motor company, but accepting this campaign will require it to use all the rented space. What is the opportunity cost of accepting the offer to work for Ford?

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LINEAR PROGRAMMING: COMPUTER SOLUTION AND SENSITIVITY ANALYSIS

LINEAR PROGRAMMING: COMPUTER SOLUTION AND SENSITIVITY ANALYSIS

TRUE/FALSE

1. When the right-hand sides of 2 constraints are both increased by 1 unit, the value of the objective function will be adjusted by the sum of the constraints' prices.

2. Sensitivity analysis is the analysis of the effect of parameter changes on the optimal solution.

3. The sensitivity range for an objective coefficient is the range of values over which the current optimal solution point (product mix) will remain optimal.

4. For a maximization problem, the range of feasibility for a resource is from 8 lbs. to 20 lbs. Assume that the original amount of the resource available to the company is 12 lbs, then it can be concluded that if up to 8lb. of this resource is added, neither the product mix, nor the total profit will change.

5. Decision variables must be clearly defined after constraints are written.

FILL IN THE BLANK

6. The reduced cost (shadow price) for a positive decision variable is_____.

7. The sensitivity range for a _____________ is the range of values over which the quantity values can change without changing the solution variable mix, including slack variables.

8. The production manager for Beer etc. produces 2 kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. He can get at most 4800 oz of malt per week and 3200 oz of wheat per week. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. What is the objective function?

9. The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. The company operates one "8 hour" shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the constraint for production time available?

10. Mallory Furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. This problem was formulated as a linear programming problem, and the solution was obtained using a computer software package. At the optimal solution point, the maximum profit is $45,000. In order to obtain the maximum profit, Mallory Furniture should purchase 150 big shelves and no medium shelves. The sensitivity range for the profit of the big shelf is from 250 to . The sensitivity range for the profit of the medium shelf is from to 180. If the Mallory Furniture is able to increase the profit per medium shelf to $200, would the company purchase medium shelves. What would be the new product mix and the total profit?

11. The linear programming problem whose output follows is used to determine how many bottles of fire red nail polish (x1), bright red nail polish (x2), basil green nail polish(x3), and basic pink nail polish(x4) a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Note that green nail polish does not require any time to prepare its display. Constraints 3 and 4 are marketing restrictions. Constraint 3 indicates that the maximum demand for fire red and green polish is 25 bottles, while constraint 4 specifies that the minimum demand combined for bright red, green and pink nail polish bottles is at least 50 bottles.

MAX 100x1 + 120x2 + 150x3 + 125x4

Subject to 1. x1 + 2x2 + 2x3 + 2x4 108
2. 3x1 + 5x2 + x4 120
3. x1 + x3 25
4. x2 + x3 + x4 50
x1, x2 , x3, x4 0

Optimal Solution:
Objective Function Value = 7475.000

Variable Value Reduced Costs
X1 8 0
X2 0 5
X3 17 0
X4 33 0

Constraint Slack / Surplus Dual Prices
1 0 75
2 63 0
3 0 25
4 0 -25

Objective Coefficient Ranges

Variable Lower Limit Current Value Upper Limit
X1 87.5 100 none
X2 none 120 125
X3 125 150 162
X4 120 125 150

Right Hand Side Ranges

Constraint Lower Limit Current Value Upper Limit
1 100 108 123.75
2 57 120 none
3 8 25 58
4 41.5 50 54

How many bottles of fire red nail polish, bright red nail polish, basil green polish and pink nail polish should be stocked? What is the maximum profit?

MULTIPLE CHOICE

13. For a maximization problem, assume that a constraint is binding. If the original amount of a resource is 4 lbs., and the range of feasibility (sensitivity range) for this constraint is from
3 lbs. to 6 lbs., increasing the amount of this resource by 1 lb. will result in the:
a. same product mix, different total profit
b. different product mix, same total profit as before
c. same product mix, same total profit
d. different product mix, different total profit

14. The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. The company operates one "8 hour" shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. For the production combination of 135 regular cases and 0 diet cases, which resource is completely used up (at capacity)?
a. only time
b. only syrup
c. time and syrup
d. neither time nor syrup

15. Use the constraints given below and determine which of the following points is feasible.
(1) 14x + 6y 42
(2) x - y 3
a. x = 1; y = 5
b. x = 2; y = 2
c. x = 2; y = 8
d. x = 2; y = 4
e. x = 3; y = 0.5

16. Explain the connection between reduced costs and the range of optimality, and between dual prices and the range of feasibility.

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