To illustrate, consider the case of Micromachines, Inc., in Chapel Hill, North Carolina. Micromachines assembles and markets Lilliputian-size machines: tiny gears and cranks the size of large specks of dust. The firm plans to introduce a new microscopic motor with the following demand conditions:
Motor parts are purchased from a number of independent subcontractors and put together at Micromachines' assembly plant. Each unit of output is expected to require 2 hours of labor. Total costs for parts acquisition before assembly labor costs are as follows:
To assemble this product, the firm will need to hire and train new staff. Given tight labor market conditions, Micromachines expects that an increase in employment will be possible only at higher wage rates. The firm projects the following labor supply curve in the highly competitive local labor market:
Based on this information, it is possible to derive Micromachines' demand curve for labor. Because 2 hours of labor are required for each unit of output, the company's profit function is n = TR - TCn
'PARTS ± ASSEMBLY
= ($120 - $0.0004Q)Q - $1,810,000 - $24Q - 2PLQ = -$0.0004Q2 + $96Q - 2PLQ - $1,810,000
and 2PlQ is the total cost of assembly.
To find Micromachines' labor demand curve, it is necessary to determine the firm's optimal level of output. The profit-maximizing level of output is found by setting marginal profit equal to zero (Mn = An/AQ = 0), where
This implies a direct relation between the price of labor, PL, and the firm's optimal level of output:
This expression can be used to indicate the optimal employment level. In setting Mn = MR - MC = 0, the firm has also implicitly set MR = MC. In terms of employment, this means that MRPl = PL for each and every input at the profit-maximizing activity level. Therefore, Micromachines' marginal revenue product of labor is MRPL = $48 - $0.0004Q.
To identify Micromachines' optimal level of employment at any given price of labor, simply determine the amount of labor required to produce the profit-maximizing level of output. Because each unit of output requires two units of labor, L = 2Q and Q = 0.5L. By substitution, the firm's demand curve for labor is
At any given wage rate, this expression indicates Micromachines' optimal level of employment. At any given employment level, this expression also indicates Micromachines' optimal wage rate. The equilibrium wage rate and employment level are determined by setting the demand for labor equal to the supply of labor:
Labor Demand = Labor Supply
15,000Pl = 240,000
To calculate the equilibrium employment level, set labor demand equal to labor supply at a wage rate of $16:
Labor Demand = Labor Supply
160,000 = 160,000 (worker hours)
Individual workers are typically employed 2,000 hours per year. This implies Micromachines should hire 80 full-time workers. This also implies that Micromachines has a profit-maximizing activity level of 80,000 micromotors (units of output) because Q = 0.5L = 0.5(160,000) = 80,000 units.
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