Compensating Wage DifferentialsA. Hedonic Wage Theory – Labor Supply1. General issuesa. assume that goods, and more specifically jobs, arevalued for a variety of utility-bearing qualitiesb. how do we find shadow prices and demands for theseparate attributes when only the bundled good (job)and its total price (wage) are observed?2. Assumptionsa. a job can be described by a vector of N characteristicsZ = (Z1, Z2, ..., ZN)1) these possible job attributes include the risk ofinjury, location (e.g., lots of urban amenities),working conditions, risk of layoff, required hours2) separate from “fringe benefits” such as insurancecoverage, pension, or vacation; these typically haveseparate explicit pricesb. for convenience, we will describe all of the Zs asgoods (e.g., job safety vs. risk of injury)c. there are a sufficiently large number of available jobsthat people face a continuum of Zd. workers and firms are atomistic and take the availablewage as givene. characteristics of jobs cannot be unbundled; jobs areoffered on a “take it or leave it” basisf. will consider only a single job per individual()[]Max ,,, ,,,,,,,ZZ ZNNNUfZZ Z ZZ Z1212 12KKK∂∂UZUf UUUfiNiWi ZZWiii=+= =− =01or for ,g. people have preferences defined over the wage rate(general consumption) and the attributes such thatU = U(W, Z1, Z2, ..., ZN) = U(W, Z)h. wages are a function of the attributes such thatW = f(Z1, Z2, ..., ZN) = f(Z); we define MW'MZi = fi3. Worker’s maximization problema. with these assumptions, we can write the worker’sproblem asb. the first-order condition for an interior solution isc. graphically, 1) the worker’s budgetconstraint is depicted asW(Z)2) preferences can bedescribed in terms ofindifference curvesbetween consumption andattribute i3) slope at the tangencybetween the budget constraint and indifferencecurve can be interpreted as the “virtual price” ofthe attribute4) can extend the budget line from the tangency todefine a “virtual full income”~~~()pfandywpf fZii iiiN=− = + = −=∑ZZ1Max ( , )~~,WUWyWpZZZsubject to =+Zgpy gff ff fZWhpy hff ff fZiNiiiNNiiiN== −== −==∑∑(~,~)(,,,,() )(~,~)(,,,,() )121121KKZZd. rewriting the problem in terms of virtual prices andincomes1) at the solution, let2) worker’s problem becomes3) holding and constant, the solutions can be~y~pwrittenwhile this system is clearly simultaneous in the Zs,the properties of g(A) and h(A) are well-known4) this approach is useful for describing behavior atthe maximum (at a single point); however, it doesnot address selectivity of W and Z combinationsB. Labor Demand1. Firm’s problema. assume that the only factor of production is laborb. assume that job attributes can be provided at a costC = C(Z)()()MaxLNNPQ L WL CPQ L f Z Z L C Z Z,() ()() ,, ,,ZZ−−=− −11KKLPQWZfCLLiii::==−c. firm’s maximize profits; so, the firm’s problem can bewrittend. for an interior solution, the first-order conditions are2. Offer functionsa. define an offer function N(Z, B) which satisfiesB = P Q(L) ! N L ! C(Z) and P QL = Nb. the offer function represents all of the combinations ofZ which the firm can offer and maintain the sameprofit level; this is similar to an iso-cost linec. graphically,1) for any given firm, we candefine a family of offerfunctions2) firm will minimize theoffer curves subject to theprevailing wage rateC. Wage Determination1. Assumea. that there are several firms with offer curves N1,N2,...,NKb. and several workers with indifference curves I1, I2,...,IJ2. The equilibrium wage will be an envelop between theoffer and indifference curves3. Graphically,a. there are a family ofindifference curvesb. there are a family of offercurvesc. and assuming a continuum ofpeople and firms, there is asmooth wage functionD. Estimation Issues1. General issuesa. want to identify how differentiated qualities affectwagesb. also want to identify demand and supply functions forthe individual attributes2. Epple’s critiquea. let prices be given by p(Z)b. for each attribute Zi, there is1) a demand function: pi(Z) = fi(Z, XD, y, p(Z))pZZZpZAZ H XpZAZ HXkk kkkkkkkkkkkkkkkk=+′+′+=+ +=+ +γψπζ∂∂ν∂∂ν2demandsupply:DDDDSSSS:()ZAHXkk DkDkk=− +−′−πυψDD12) a supply function: pi(Z) = gi(Z, XS, p(Z))3) where XD and XS are observable demand andsupply shiftersc. assume that there are K distinct markets with lineardemand and supply relationshipsd. consider linear attribute demand and supply functionse. OLS in the attribute demand and supply equations isnot consistent1) to see this in the demand equation, solve for Zk2) either Zk or XDk depends on <Dk and OLS applied tothe attribute demand equation is inconsistentf. consistent estimation of the price equation requiresE(.k <Dk) = 0 and E(.k <Sk) = 03. Goddeeris (JPE 1988)a. examines differences between public-interest andprivate attorneysb. hedonic characteristic is a dummy variable (similar tointroduction in Handbook chapter by Rosen); greatlysimplifies choice problemc. uses endogenous dummy variable model to estimatethe wage premium associated with choice of law firmd. actual estimation procedure is complicated by the useof a choice-based sample4. Moffitt (JoLE, 1984)a. examines relationship between labor supply and awage function which depends on hours worked (wagefunction in which hours are an endogenousdeterminant)b. motivation for endogenous wages1) labor imposes quasi-fixed costs to firm; thus, lowhours are associated with a low marginal rate (Oi,1962)2) marginal productivity declines at high hoursleading to lower wages (Barzel, 1973)3) result is an S-shaped budget constraintc. considers1) linear labor supply function2) quadratic (in hours) earnings functiond. this specification leads to a nonlinear budgetconstrainte. solves estimation problem by estimating the budgetconstraint at 15 discrete points1) sensitivity check with different numbers of points2) also compares results with a linear budgetconstraintf. results1) rejects linear wage relationship2) use of a nonlinear hours function leads to lowerwage elasticities3) nonlinear model provides a much better fit to