1. Determine whether the following production functions exhibit constant, increasing, or decreasing returns to scale. Show all your calculations and explain what you are doing and why.
A. Q = 25X + 0.5Y + 8Z
B. Q = 9L + 5K - 400
C. Q = 10A + 7B - 4AB
D. Q = 6L2 + 3LK + 2K2
E. Q = 2L0.4K0.6
2. A firm uses two variable inputs, labor (L) and capital (K), in producing its output. At its current level of output:
Price (or cost) of L = $10 per unit
Price (or cost) of K = $2 per unit
Marginal product of L = 25
Marginal product of K = 4
A. Determine whether the firm is operating efficiently, given that its objective is to minimize the cost of producing the current level of output. Explain your work, what you are doing, and why.
B. Determine what changes, if any, in the relative proportions of labor and capital needed to be made to operate efficiently. Explain your answer well.
3. Cost Concepts. Answer each of the following as true or false and explain your answers briefly. No credit will be given for just a true/false answer if no explanation is provided.
A. If ec > 1, increasing returns to scale and decreasing average costs are indicated.
B. Average cost exceeds marginal cost at the minimum efficient scale of plant.
C. When total fixed cost and price are held constant, a reduction in average variable cost will typically cause a reduction in the breakeven activity level.
D. An increase in total fixed cost will always increase the degree of operating leverage for firms making a positive net profit.
E. When long-run average cost is decreasing, it can pay to operate smaller plants at their peak efficiency rather than larger plants with some excess capacity.
4. San Francisco's Pier 9, Inc., sells souvenir T-shirts at a price of $25. Of this amount, $15 is profit contribution. Pier 9 is considering differentiating its product from several other competitors by using higher quality T-shirts. Doing so would increase unit cost by $2.50 per shirt. Current monthly profits are $10,000 on 2,500 unit sales.
A. Assuming average variable costs are constant at all output levels, what is Pier 9's total cost function before the proposed change?
B. What will the total cost function be if higher quality t-shirts are used?
C. Assume shirt prices remain stable at $25. What percentage increase in sales would be necessary to maintain current profit levels if they use the higher quality T-shirts rather than their current ones?
5. DynaLinear, Ltd., produces digital-to-analog converters for compact disk players used by radio stations and audio enthusiasts. It is contemplating an expansion into the moderately-priced home audio market by producing a converter that would sell at a price of $300. The production of each converter would require $100 in materials, and 3.75 hours of labor at the rate of $20 per hour for wages and fringe benefits plus variable overhead tied to labor. Energy, supervisory and other variable overhead costs would amount to $50 per unit. The accounting department has derived an allocated fixed overhead charge of $25 per converter (at a projected volume of 14,000 units) to account for the expected increase in fixed costs.
A. What is DynaLinear's breakeven sales volume (in units) for converters?
B. Calculate the degree of operating leverage at a projected volume of 14,000 units and explain what the DOL means.