Costs And Cost Minimization

Chapter 4. Costs and Cost Minimization

Problem Set

1. 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 andcapital for an airframe manufacturer that wants to produce 121,000 airframes.

The tangency condition implies

[pic]
Given that [pic] and [pic], this implies

[pic]

Returning to the production function and assuming [pic] yields

[pic]

Since [pic], [pic]. The cost minimizing quantities of capital and labor to produce 121,000 airframes is [pic] and [pic].



2. The processingof 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 showsall 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 priceof an hour of computer time have to be before the firm would find it worthwhile to use only clerks to process the payroll?

a)
[pic]

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 [pic]and [pic], the slope of a typical isocost line will be [pic]. This is steeper than the isoquant implying that the firm will employ only computer time ([pic]) to minimize cost. The cost minimizing combination is [pic] and [pic]. This outcome can be seen in the graph below. The isocost lines are the dashed lines.
[pic]

The total cost to process the payroll for 10,000 workers will be [pic].c) The firm will employ clerical time only if MPL / w > MPK / r. Thus we need 0.1 / 7.5 > 1/r or r > 75.



3. A firm produces an output with the production function Q = KL, where Q is the number of units of output per hour when the firm uses K machines and hires L workers each hour. The marginal products for this production function are MPK = L and MPL = K. The factor price of K is 4 andthe factor price of L is 2. The firm is currently using K = 16 and just enough L to produce Q = 32. How much could the firm save if it were to adjust K and L to produce 32 units in the least costly way possible?

Currently the firm must be using L = Q/K = 32/16 = 2 units of labor. Let the factor prices of capital and labor be, respectively, r and w.
Its total expenditure is C = wL + rK = 2(2) +4(16) = 68.
If it were to minimize cost, it would hire L and K so that (1) MPK/r = MPL/w, or L/4 = K/2, or L = 2K and (2) Q = LK.
(1) and (2) imply that Q = 2K2, or 32 = 2K2, and thus K = 4 and L = 8.
So Q = 32 can be produced efficiently with a cost of C = wL + rK = 2(8) + 4(4) = 32.
The firm could save 68 – 32 = 36 by producing efficiently.


3. A firm operates with the productionfunction Q = K2L. Q is the number of units of output per day when the firm rents K units of capital and employs L workers each day. The manager has been given a production target: Produce 8,000 units per day. She knows that the daily rental price of capital is $400 per unit. The wage rate paid to each worker is $200 day.
a) Currently the firm employs at 80 workers per day. What is the firm’s daily...