Lecture 2Labor DemandEquilibriumNoah WilliamsUniversity of Wisconsin - MadisonEconomics 312Spring 2014Williams Economics 312An ExampleA representative worker has preferences over consumption Cand leisure l given by:u(C , l) = 2√C − a(1 − l)1.51.5Here a is a taste parameter governing the relative preference forleisure and consumption. The worker has no unearned incomeso his budget constraint is:C = w(1 −l),where 1 is the hours in the day, so 1 − l is labor supply.1Find the household labor supply function.Williams Economics 312What is a firm?A firm uses capital K and labor N to produce output Yvia a production function F :Y = zF (K , N )z is the level of technology or total factor productivity(TFP).The main example we’ll use is Cobb-Douglas productionfunction, with α ∈ (0, 1):Y = zKαN1−αWe will start with a static model: K is constant.Williams Economics 312Properties of the TechnologyWe’ll make several assumptions on the technology F , all ofwhich are satisfied by Cobb-Douglas.1. Inputs are essential.F (0, N ) = F (K , 0) = 02. Constant returns to scale:F (λK , λN ) = λF (K , N )Doubling inputs doubles output. Compared to decreasing(increasing) returns to scale where doubling inputs leads toless (more) than double output.F (K , N ) = KαN1−αF (λK , λN ) =Williams Economics 3123. Marginal productivities of capital and labor are positiveand decreasing.MPK = FK> 0, FKK< 0MPN = FN> 0, FNN< 0Increasing each factor gives more output, but at adecreasing rate.F (K , N ) = KαN1−αFK=FKK=FN=FNN=Williams Economics 312Copyright © 2005 Pearson Addison-Wesley. All rights reserved.4-15Figure 4.14 Production Function, Fixing the Quantity of Capital and Varying the Quantity of LaborWilliams Economics 3124. Marginal productivity of each factor increases in the other.∂MPK∂N=∂∂NFK= FKN> 0∂MPN∂K=∂∂KFN= FKN> 0Note one implies other since FKN= FNK.Additional capital makes workers more productive: spreadworkers among more machines (and vice versa).FK= αKα−1N1−αFKN=FN= (1 − α)KαN−αFNK=Williams Economics 312Copyright © 2005 Pearson Addison-Wesley. All rights reserved.4-19Figure 4.18 Total Factor Productivity IncreasesWilliams Economics 312Problem of the Firm ICompetitive firm rents capital at rate r , hires labor atwage w.Profits: output minus costsπ = zF (K , N ) − rK − wNNote everything in real terms – same as setting price ofoutput to 1.Firm maximizes profits by hiring labor and capital.We take first order conditions for choice of K and N :zFK(K , N ) = rzFN(K , N ) = w (1)Factors are paid their marginal products.(1) can be solved to give the labor demand: Nd(w)Williams Economics 312Copyright © 2008 Pearson Addison-Wesley. All rights reserved.4-40Figure 4.20 The Marginal Product of Labor Curve Is the Labor Demand Curveof the Profit-Maximizing FirmWilliams Economics 312Problem of the Firm I: Special CaseWith Cobb-Douglas production, we have:π = zKαN1−α− rK − wNWe take first order conditions for K and N := r (2)= w (3)(3) can be solved to give the labor demand:Nd(w; K ) = MPN−1(w; K )=Nddecreasing in w. Increases in z, K shift labor demand.Williams Economics 312Problem of the Firm IITo fully solve the example, we want to solve for K and N .We begin dividing (2) by (3):αzKα−1N1−α(1 − α) zKαN−α=rworα1 − αNK=rworK =wrα1 − αN (4)This gives us the optimal capital/labor ratio of the firm.Depends on relative prices w/r, relative productivities α, 1 − α.Williams Economics 312Problem of the Firm IIIBut if we substitute (4) in (1), trying to solve for N , we get:(1 − α) zKαN−α= w(1 − α) zwrα1 − αNαN−α= w(1 − α) zwrα1 − αα= wN disappears!You can check that the same happens with K if wesubstitute (4) in (2).What is wrong?Williams Economics 312Problem of the Firm IVWe have constant returns to scale.The size of the firm is indeterminate: we can have just one!Can show constant returns equivalent toF (K , N ) = FKK + FNN .This implies profits are always zero if firm maximizes, sincer = MPK , w = MPN .π = zKαN1−α− rK − wN= zKαN1−α− MPK · K − MPN · N=So the firms really only pick the labor-capital ratio givenrelative prices:NK=1 − ααrwWilliams Economics 312General EquilibriumIn equilibrium, markets clear so:Nd(w, K ) = Ns(w)Kd(r , N ) =¯K i.e., fixedr = αzKα−1N1−αw = (1 − α) zKαN−αWe have a system of four equations in four unknowns(K , N , r , w).In other words, firms choose K /N ratio, but by assumptionK is fixed at¯K in short run, which gives labor demand Nd.We will return to discussing general equilibrium shortly.Williams Economics 312Return to the ExampleA representative worker has preferences:u(C , l) = 2√C − a(1 − l)1.51.5The budget constraint is:C = w(1 −l),where 1 is the hours in the day, so 1 − l is labor supply. Capitalis fixed at 1, and the representative firm technology is:Y = zN0.51Find the household labor supply function.2Find expressions for the equilibrium values of the laborinput and the wage.3Suppose that a increases but z is unchanged. What is theeffect of this change on labor and the wage?Williams Economics 312What are we missing?Wages are often different from marginal productivity.Labor market not completely a spot market.Some reasons:1Long term contracting. Sticky wages: Taylor (1980).2Search frictions in finding a job. Wages determined bybargaining: Nash (1950).3Workers may exert effort which influences output, difficultto observe. Efficiency wages: Shapiro-Stiglitz (1984).We will return to some of these frictions later in the class. Fornow continue with competitive, spot labor market.Williams Economics 312Adding a GovernmentOperational definition: takes in taxes T and spends G.We’ll assume a balanced budget. No debt in this staticmodel.Also assume household does not value governmentspending. Not crucial here.We’ll also assume proportional labor income taxes onhouseholds. Later will consider lump sum.G = TT = τwNWilliams Economics 312New Problem of the HouseholdSet unearned income equal to capital income rK .Incorporate labor income taxes.Problem for household is now:maxc,Nu (c, l )s.t. c = (1 − τ) wN + rKFirst order conditions:Household now equates MRS to after-tax wage.Taxes affect labor supply!Williams Economics 312Labor Supply Effects in an ExampleConsider the log example from last time:u(c, l ) = log c + γ log lThen we get:γc∗h − N∗= (1 − τ )wc∗= N∗(1 − τ )w + πTherefore