PROBLEMS AND SHORT CASES

6-1. Donald K. Brown and Company operates a pearl-diving operation in the North Pacific Ocean. Mr. Brown owns a large trawler with all the required equipment. He hires local divers from the nearby islands and pays each of them on the basis of the weight of oysters recovered. He sells the pearls and the oyster meat separately. Over the past month he has been out pearling eight times in the same general area, taking all the divers who showed up for each trip. The particulars are as follows:

Trip Number	Divers Employed	Oysters Recovered (kg)
1	6	38
2	17	76
3	9	56
4	5	32
5	12	74
6	3	15
7	14	80
8	15	78

(a) Over what ranges do there appear to be increasing, constant, and diminishing returns to the variable factor?

(b) What number of divers appears to be the most efficient in terms of output per diver?

6-2. Taras Panache is the owner-manager of Panache Shirts Enterprises, which manufactures shirts by using rented space and equipment in a large warehouse. Because of the technical aspects of shirt production and the available equipment, separate production centers are used, each con-

sisting of one cutting machine, two sewing machines, and three operators. Six months ago Mr. Panache had only one such production center, but recently he doubled, then tripled, and finally quadrupled the number of production centers by renting more space and equipment and by hiring more operators. Throughout the expansion Mr. Panache has personally supervised all the operators and has handled all other aspects of the business. He kept a record of the average daily output from the entire plant for each of the four situations, as follows:

One production center: Two production centers: Three production centers: Four production centers:

20.6 shirts/day 42.4 shirts/day 60.8 shirts/day 76.3 shirts/day

Each production center costs $3,000 per month in fixed and variable costs. Mr. Panache pays himself $3,000 per month, and the remaining fixed costs are $1,000 per month. Assume there are 20 working days in a month.

(a) Can the expansion of Panache Shirts be regarded as a case of an increase in the scale of operations or simply an increase in the size of operations? Why?

(b) Are there economies and/or diseconomies of scale/size evident? Explain.

6-3. The Superhealth Exercise Company has estimated its short-run production function to be as follows:

Q = 421 + 0.6/, 2 - 0.08T 3

where Q represents output per hour and L represents units of the variable inputs, which cost $200 per unit. The firm’s fixed inputs cost $100 per hour.

(a) Derive the firm’s total product and marginal product schedules for L = 1 through/, = 10.

(b) Transform the TP schedule into a total variable cost (TVC) schedule and calculate the average variable cost (AVC) and average cost (SAC) values.

(d) At what point do diminishing returns set in? Explain.

6-4. Silver Star Corporation has estimated its production function as follows:

Q = 38.6K + 3.2/C 2 - 1.8K 3 + 16.31 + 2.8L 2 - 0.85 V

where K represents units of the capital input (in $1,000 units) and L represents units of the labor input (in hundreds of labor hours).

(a) Construct the total product and marginal product curves for the case of K = 5.

(b) At what level of labor input do diminishing returns become evident?

(c) If labor were available at no cost (students wishing to gain work experience and willing to work without wages), what input level would you choose? Why?

6-5. Gewurz Fabricators Limited manufactures and assembles small aluminum buildings suitable for garden toolsheds, garages, and children’s playhouses. Stephen Gewurz, the owner, is considering opening a new plant to diversify into the production of luxury dog kennels for the expanding large dogs’ market. He has carefully considered the labor and capital requirements and the substitutability between these inputs at various output levels and has summarized the production function as follows:

242 Production and Cost Analysis

Table captionGewurz Fabricator—Production Function (output in units per year)

				LABOR INPUTS (person-years)
1	2	3	4	5	6	7	8
Capital Inputs	1	30	52	80	110	130	145	155	162
(machine-years)	2	50	80	120	164	200	220	235	248
	3	80	124	175	226	260	274	282	287
	4	100	160	218	272	302	320	335	345

(a) Supposing that the cost of each unit of capital is $20,000 and the cost of each unit of labor is $10,000, derive the SAC curve for each of the four plant sizes indicated.

(b) What conclusions can you draw about the returns to increasing plant size in this example?

(c) Which of the four plants should be selected if demand is expected to be (1) 125 units? (2) 250 units? (3) somewhere within the range of 200-300 units? (Explain and defend your decision fully.)

6-6. The newly formed Beaudry Automobile Corporation plans to produce an expensive sports car and has asked your consulting firm for advice on the size of plant to construct. Because of the union contract and technical features of automobile production, labor must be paid $12,000 per person per annum, and each incremental change in plant size involves $900,000 in annual expenses for depreciation, interest, and other fixed costs. The maximum the firm will have available for expenditure on capital and labor is $9 million per annum. BAC has supplied the following details of its production function, meticulously derived by its chief engineer. (The data in the body of the table represent automobiles produced, in units.) Labor can be varied virtually continuously; the table shows units of 50 persons for convenience. All other variable expenses are constant at $2,500 per vehicle produced.

Capital (units of $900,000)		LABOR (units of 50 persons)
1	2	3	4	5	6
3	40	90	140	170	180	185
4	60	120	180	220	230	236
5	100	170	230	250	260	268
6	170	200	240	270	280	289

BAC s market research indicates that the new vehicle should be sold at around $50,000 per unit and that the expected demand situation is as follows:

Units Demanded (annually)	Probability
100	0.20
150	0.50
200	0.20
250	0.10

(a) Plot the SAC curves suggested by the production function and input cost figures.

(b) Comment upon the economies and diseconomies of plant size (if any) which are evident in your graph.

6-7. The Peachy Cosmetics Company has established the following relationship between the variable inputs and the output level for its production of its face cream products. Each unit of the variable product includes one person working 40 hours weekly, the necessary utilities, and a variety of ingredients in the required proportions, including the small glass jars and cardboard boxes in which the cream is packed. These variable inputs cost $600 per unit. Overhead (fixed) costs are $60,000 per week.

Variable Input (units)	Output (units)
100	6,500
200	14,300
300	20,200
400	24,400
500	27,800
600	30,000

(a) Use these data to derive the firm’s AVC and SAC schedules for the output levels shown.

(b) Graph the AVC and SAC curves and sketch in your best estimate of the marginal cost curve.

(c) At what output level do you think diminishing returns first start? Explain with reference to the graph and reconcile this result with the input-output data.

(d) What is the full capacity output level? What is average cost at that level? Under what conditions would Peachy produce in the overfull capacity area of its cost curves?

6-8. Suppose that one of the very expensive ingredients in Peachy Cream (see problem 6-7) was suddenly found in abundance, and the cost of the variable units fell to $500 each.

(a) What is the impact of this discovery on the firm’s AVC, SAC, and MC curves?

(b) Does it change the point where diminishing returns set in?

Now suppose that after the reduction in the price of that ingredient, worker productivity increases as a result of improvements in morale, such that output goes up by 10 percent at each input level.

(d) What is the impact of both changes on the firm’s full capacity output level?

6-9. Paxtronix Corporation is considering the installation of plant and equipment to manufacture a nonlethal stun gun for use by law enforcement agencies only. It has a choice of three plant sizes, A, B, and C. The average cost schedules for these plants are shown on page 244. Also shown is the estimated probability distribution of demand for each of the first two years.

(a) Plot the average cost curves for plants A, B, and C on the same graph.

(b) Calculate the expected value of average cost for each plant, given the probability distribution of demand.

(d) Given the plant size you have recommended, how much excess capacity will Paxtronix have?

244 Production and Cost Analysis

Output Level (thousands)	Demand	AVERAGE COST LEVEL ($)
Probability	Plant A	Plant B	Plant C
1	0.10	110	140	180
2	0.20	90	100	110
3	0.35	80	70	80
4	0.20	80	60	50
5	0.10	90	70	40
6	0.05	110	90	50

6-10. Magnocrunch Computer Company is currently producing at full capacity. This firm is fortunate to have fully divisible fixed inputs, such that its average variable cost is constant out to the full capacity level. Demand for the new Magnum computer is increasing, and the firm faces a backlog of orders. It cannot build inventories and fears that it will lose 20 percent of the buyers on the backlist if it cannot deliver within another month. The firm is considering two alternative plans. Plan A is to increase the rate of production to an overfull capacity situation, which would raise marginal cost from the present level of $1,500 by $300 for every 100 units beyond the current output rate of 500 computers per month. Alternatively, plan B is to contract out for the manufacture of various components, and then have these assembled into the finished unit by another firm. The firm estimates that these units will end up costing $2,200 per unit, regardless of volume. The present price of the computer is $2,995, and the firm’s fixed costs per month are $250,000 per month. The firm’s fixed costs are not expected to change, regardless of which plan is selected.

(a) Supposing that the backlist is 1500 buyers, calculate the cost of solving the firm’s immediate problem, under each plan taken separately.

(b) What strategy would you suggest to help Magnocrunch eliminate its backlist and begin to build up inventories?