Chapter 22 LABOR SUPPLY Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved. MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON

Allocation of Time Individuals must decide how to allocate the fixed amount of time they have We will initially assume that there are only two uses of an individual’s time –engaging in market work at a real wage rate of w –leisure (nonwork)

Allocation of Time Assume that an individual’s utility depends on consumption (C) and hours of leisure (H) utility = U(C,H) In seeking to maximize utility, the individual is bound by two constraints L + H = 24 C = wL

Allocation of Time Combining the two constraints, we get C = w(24 – H) C + wH = 24w An individual has a “full income” of 24w –may spend the full income either by working (for real income and consumption) or by not working (enjoying leisure) The opportunity cost of leisure is w

Utility Maximization The individual’s problem is to maximize utility subject to the full income constraint Setting up the Lagrangian L = U(C,H) + (24w – C – wH) The first-order conditions are  L/  C =  U/  C - = 0  L/  H =  U/  H - = 0

Utility Maximization Dividing the two, we get To maximize utility, the individual should choose to work that number of hours for which the MRS (of H for C) is equal to w –to be a true maximum, the MRS (of H for C) must be diminishing

Income and Substitution Effects Both a substitution effect and an income effect occur when w changes –when w rises, the price of leisure becomes higher and the individual will choose less leisure –because leisure is a normal good, an increase in w leads to an increase in leisure The income and substitution effects move in opposite directions

Income and Substitution Effects U1U1 U2U2 Leisure Consumption A B C The substitution effect is the movement from point A to point C The individual chooses less leisure as a result of the increase in w The income effect is the movement from point C to point B substitution effect > income effect

Income and Substitution Effects U1U1 U2U2 Leisure Consumption A B C The substitution effect is the movement from point A to point C The individual chooses more leisure as a result of the increase in w The income effect is the movement from point C to point B substitution effect < income effect

A Mathematical Analysis of Labor Supply We will start by amending the budget constraint to allow for the possibility of nonlabor income C = wL + N Maximization of utility subject to this constraint yields identical results –as long as N is unaffected by the labor- leisure choice

A Mathematical Analysis of Labor Supply The only effect of introducing nonlabor income is that the budget constraint shifts out (or in) in a parallel fashion We can now write the individual’s labor supply function as L(w,N) –hours worked will depend on both the wage and the amount of nonlabor income –Since leisure is a normal good,  L/  N < 0

Dual Statement of the Problem The dual problem can be phrased as choosing levels of C and H so that the amount of expenditure (E = C – wL) required to obtain a given utility level (U 0 ) is as small as possible –solving this minimization problem will yield exactly the same solution as the utility maximization problem

Dual Statement of the Problem A small change in w will change the minimum expenditures required by  E/  w = -L –this is the extent to which labor earnings are increased by the wage change

Dual Statement of the Problem This means that a labor supply function can be calculated by partially differentiating the expenditure function –because utility is held constant, this function should be interpreted as a “compensated” labor supply function Lc(w,U)Lc(w,U)

Slutsky Equation of Labor Supply The expenditures being minimized in the dual expenditure-minimization problem play the role of nonlabor income in the primary utility-maximization problem L c (w,U) = L[w,E(w,U)] = L(w,N) Partial differentiation of both sides with respect to w gives us

Slutsky Equation of Labor Supply Substituting for  E/  w, we get Introducing a different notation for L c, and rearranging terms gives us the Slutsky equation for labor supply:

Cobb-Douglas Labor Supply Suppose that utility is of the form The budget constraint is C = wL + N and the time constraint is H = 1 – L –Note that we have set maximum work time to 1 hour for convenience

Cobb-Douglas Labor Supply Combining these functions, we can express utility as a function of labor supply only U 2 = CH = (wL + N)(1 – L) = wL – wL 2 + N – NL Differentiation of U 2 with respect to L yields the first-order condition for utility maximization  U 2 /  L = w – 2wL – N = 0 L = ½ - N/2w

Cobb-Douglas Labor Supply Note that if N = 0, the person will work ½ of each hour no matter what the wage is –the substitution and income effects of a change in w offset each other and leave L unaffected

Cobb-Douglas Labor Supply Using the income effect of the Slutsky equation and substituting in for L, we get If N = 0, the income effect becomes An increase in w will reduce L because leisure is a normal good

Cobb-Douglas Labor Supply Calculation of the substitution effect in the Slutsky equation is more messy One must derive the indirect utility function and then use this to eliminate N from the optimal labor supply choice

Cobb-Douglas Labor Supply This constant utility labor supply function shows that if only substitution effects are allowed,  L c /  w > 0 Replacing U with the indirect utility function, we get

Cobb-Douglas Labor Supply If N = 0, and the Slutsky equation shows that the substitution and income effects are precisely offsetting in this Cobb-Douglas case

Effects of Nonlabor Income If N  0, the precise offsetting of the substitution and income effects in the previous example would not occur –The individual will always choose to spend half of his nonlabor income on leisure –Since leisure costs w per hour, an increase in w means that less leisure can be bought with N

Market Supply Curve for Labor LLL w ww sAsA sBsB LA*LA* LB*LB* w*w* To derive the market supply curve for labor, we sum the quantities of labor offered at every wage Individual A’s supply curve Individual B’s supply curve Total labor supply curve L*L* S L A * + L B * = L*

Market Supply Curve for Labor LLL w ww sAsA sBsB Note that at w 0, individual B would choose to remain out of the labor force Individual A’s supply curve Individual B’s supply curve Total labor supply curve S w0w0 As w rises, L rises for two reasons: increased hours of work and increased labor force participation

Other Uses of Time Allocation Models Job Search Theory –time must be invested to find suitable employment the higher an individual’s wage, the more likely he will use search methods that economize on time The Economics of Childbearing –opportunity costs of time play a role in the decision to have children some economists have suggested this is why birth rates have fallen in the United States

Other Uses of Time Allocation Models Transportation Choices –in choosing between alternative transportation modes, individuals will take both time and dollar costs into account some studies have suggested that individuals are quite sensitive to time costs (especially those involving waiting or walking)

Labor Unions If association with a union was wholly voluntary, we can assume that every member derives a positive benefit With compulsory membership, we cannot make the same claim –even if workers would benefit from the union, they may choose to be “free riders”

Labor Unions We will assume that the goals of the union are representative of the goals of its members In some ways, we can use a monopoly model to examine unions –the union faces a demand curve for labor –as the sole supplier, it can choose at which point it will operate this point depends on the union’s goals

Labor Unions Labor Wage D MR S The union may wish to maximize the total wage bill (wL). This occurs where MR = 0 L 1 workers will be hired and paid a wage of w 1 L1L1 w1w1 This choice will create an excess supply of labor

Labor Unions Labor Wage D MR S The union may wish to maximize the total economic rent of its employed members This occurs where MR = S L 2 workers will be hired and paid a wage of w 2 L2L2 w2w2 Again, this will cause an excess supply of labor

Labor Unions Labor Wage D MR S The union may wish to maximize the total employment of its members This occurs where D = S L 3 workers will be hired and paid a wage of w 3 L3L3 w3w3

Modeling a Union A monopsonistic hirer of coal miners faces a supply curve of L = 50w Assume that the monopsony has a MRP L curve of the form MRP L = 70 – 0.1L The monopsonist will choose to hire 500 workers at a wage of $10

Modeling a Union If a union can establish control over labor supply, other options become possible –competitive solution where L = 583 and w = $11.66 –monopoly solution where L = 318 and w = $38.20

A Union Bargaining Model Suppose a firm and a union engage in a two-stage game –first stage: union sets the wage rate its workers will accept –second stage: firm chooses its employment level

A Union Bargaining Model This two-stage game can be solved by backward induction The firm’s second-stage problem is to maximize its profits:  = TR(L) – wL The first-order condition for a maximum is TR’(L) = w

A Union Bargaining Model Assuming that L* solves the firm’s problem, the union’s goal is to choose w to maximize utility U(w,L) = U[w,L*(w)] and the first-order condition for a maximum is U 1 + U 2 L’ = 0 U 1 /U 2 = L’

A Union Bargaining Model This implies that the union should choose w so that its MRS is equal to the slope of the firm’s labor demand function The result from this game is a Nash equilibrium

Wage Variation It is impossible to explain the variation in wages across workers with the tools developed so far –we must consider the heterogeneity that exists across workers and the types of jobs they take

Wage Variation Human Capital –differences in human capital translate into differences in worker productivities –workers with greater productivities would be expected to earn higher wages –while the investment in human capital is similar to that in physical capital, there are two differences investments are sunk costs opportunity costs are related to past investments

Wage Variation Compensating Differentials –individuals prefer some jobs to others –desirable job characteristics may make a person willing to take a job that pays less than others –jobs that are unpleasant or dangerous will require higher wages to attract workers –these differences in wages are termed compensating differentials

Important Points to Note: A utility-maximizing individual will choose to work that number of hours for which the MRS (of H for C) is equal to w An increase in w creates income and substitution effects that operate in different directions in their effect on labor supply –this result can be shown using a Slutsky- type equation

Important Points to Note: The theory of time allocation is relevant to a number of other economic decisions in addition to the labor supply decision –because most activities require time to complete them, the notion that they have both market prices and time prices has far- reaching consequences for economic theory

Important Points to Note: Unions can be treated analytically as monopoly suppliers of labor –labor market equilibrium in the presence of unions will depend on what goals the union chooses to pursue in its supply decision and in the bargaining between unions and firms