Chapter 22Allocation of TimeSlide 3Slide 4Utility MaximizationSlide 6Income and Substitution EffectsSlide 8Slide 9A Mathematical Analysis of Labor SupplySlide 11Dual Statement of the ProblemSlide 13Slide 14Slutsky Equation of Labor SupplySlide 16Cobb-Douglas Labor SupplySlide 18Slide 19Slide 20Slide 21Slide 22Slide 23Effects of Nonlabor IncomeMarket Supply Curve for LaborSlide 26Other Uses of Time Allocation ModelsSlide 28Labor UnionsSlide 30Slide 31Slide 32Slide 33Modeling a UnionSlide 35A Union Bargaining ModelSlide 37Slide 38Slide 39Wage VariationSlide 41Slide 42Important Points to Note:Slide 44Slide 45Chapter 22LABOR SUPPLYCopyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONSEIGHTH EDITIONWALTER NICHOLSONAllocation 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 constraintsL + H = 24C = wLAllocation of Time•Combining the two constraints, we getC = 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 wUtility Maximization•The individual’s problem is to maximize utility subject to the full income constraint•Setting up the LagrangianL = U(C,H) + (24w – C – wH)•The first-order conditions areL/C = U/C -  = 0L/H = U/H -  = 0Utility Maximization•Dividing the two, we get) for ( //CHMRSwHUCU•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 diminishingIncome andSubstitution 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 directionsIncome andSubstitution EffectsU1U2LeisureConsumptionABCThe substitution effect is the movementfrom point A to point CThe individual chooses less leisure as a result of the increase in wThe income effect is the movementfrom point C to point Bsubstitution effect > income effectIncome andSubstitution EffectsU1U2LeisureConsumptionABCThe substitution effect is the movementfrom point A to point CThe individual chooses more leisure as a result of the increase in wThe income effect is the movementfrom point C to point Bsubstitution effect < income effectA Mathematical Analysisof Labor Supply•We will start by amending the budget constraint to allow for the possibility of nonlabor incomeC = wL + N•Maximization of utility subject to this constraint yields identical results–as long as N is unaffected by the labor-leisure choiceA Mathematical Analysisof 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 < 0Dual 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 (U0) is as small as possible–solving this minimization problem will yield exactly the same solution as the utility maximization problemDual 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 changeDual 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 functionLc(w,U)Slutsky Equation ofLabor Supply•The expenditures being minimized in the dual expenditure-minimization problem play the role of nonlabor income in the primary utility-maximization problemLc(w,U) = L[w,E(w,U)] = L(w,N)•Partial differentiation of both sides with respect to w gives uswEELwLwLcSlutsky Equation ofLabor Supply•Substituting for E/w, we getNLLwLELLwLwLc•Introducing a different notation for Lc , and rearranging terms gives us the Slutsky equation for labor supply:NLLwLwLUU0Cobb-Douglas Labor Supply•Suppose that utility is of the formCHU •The budget constraint isC = wL + N and the time constraint isH = 1 – L–Note that we have set maximum work time to 1 hour for convenienceCobb-Douglas Labor Supply•Combining these functions, we can express utility as a function of labor supply onlyU2 = CH = (wL + N)(1 – L) = wL – wL2 + N – NL•Differentiation of U2 with respect to L yields the first-order condition for utility maximizationU2/L = w – 2wL – N = 0L = ½ - N/2wCobb-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 unaffectedCobb-Douglas Labor Supply•Using the income effect of the Slutsky equation and substituting in for L, we get244121221wNwwwNNLL •If N = 0, the income effect becomeswNLL41•An increase in w will reduce L because leisure is a normal goodCobb-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 choicewNwU22wUUwLc1),(Cobb-Douglas Labor Supply•This constant utility labor supply function shows that if only substitution effects are allowed, Lc/w >