Suppose the production of airframes is characterized by a CES production function: Q = (L½ + K½)2. The marginal products for this production function are MPL = (L½ + K½)L−½ and MPK = (L½+ K½)K−½. Suppose that the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital for an airframe manufacturer that wants to produce 121,000 airframes.
The tangency condition implies
Given that and , this implies
Returning to the production function and assuming yields
Since , . The cost minimizing quantities of capital and labor to produce 121,000 airframes is and .
- the processing of payroll for the 10,000 workers in a large firm can alternatively be done using 1 hour of computer time (denoted by K) and no clerks or with 10 hours of clerical time (denoted by L) and no computer time. Computers and clerks are perfect substitutes; for example, the firm could also process its payroll using 1/2 hour of computer time and 5 hours of clerical time.
- a) Sketch the isoquant that shows all combinations of clerical time and computer time that allows the firm to process the payroll for 10,000 workers.
- b) Suppose computer time costs $5 per hour and clerical time costs $7.50 per hour. What are the cost-minimizing choices of L and K? What is the minimized total cost of processing the payroll?
- c) Suppose the price of clerical time remains at $7.50 per hour. How high would the price of an hour of computer time have to be before the firm would find it worthwhile to use only clerks to process the payroll?
K and L are perfect substitutes, meaning that the production function is linear and the isoquants are straight lines. We can write the production function as Q = 10,000K + 1000L, where Q is the number of workers for whom payroll is processed.
- b) If and , the slope of a typical isocost line will be . This is steeper than the isoquant implying that the firm will employ only computer time () to minimize cost. The cost minimizing combination is and . This outcome can be seen in the graph below. The isocost lines are the dashed lines.
The total cost to process the payroll for 10,000 workers will be .
- c) The firm will employ clerical time only if MPL / w > MPK / r. Thus we need 0.1 / 7.5 > 1/r or r >