On the cyclicality of real wages and wage differentials

Christopher Otrok; Panayiotis M. Pourpourides

doi:10.1515/bejm-2017-0047

Published by De Gruyter September 20, 2017

On the cyclicality of real wages and wage differentials

Christopher Otrok and Panayiotis M. Pourpourides

From the journal The B.E. Journal of Macroeconomics

https://doi.org/10.1515/bejm-2017-0047

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Abstract

Previous empirical literature suggests that estimated wage cyclicality depends on the structure of the relationship between real wages and an observed indicator of the business cycle that econometric models impose prior to estimation. This paper, alleviates the problem of imposing such structure by searching directly for the largest common cycles in longitudinal microdata using a Bayesian dynamic latent factor model. We find that the comovement of real wages is related to a common factor that exhibits a significant but imperfect correlation with the national unemployment rate. Among others, our findings indicate that the common factor explains, on average, no more than 9% of wage variation, it accounts for 20% or less of the wage variability for 88% of the workers in the sample and roughly half of the wages move procyclically while half move countercyclically. These facts are inconsistent with claims of a strong systematic relationship between real wages and business cycles.

Keywords: Bayesian analysis; business cycles, wage differentials; wages

JEL Classification: C11; C13; C22; C23; C81; C82; J31

Acknowledgement

We thank Julie Yates (Bureau of Labor Statistics) and Steve McClaskie (National Longitudinal Surveys user services) for information and help with the data.

Appendix

A The factor model and two theoretical models

A.1 A neoclassical model

The dynamic factor model of Section 4 can be motivated by a standard real business cycle model augmented with a model of measurement error induced by the agency gathering data. This motivation follows directly from the work of Sargent (1989). We start with a ‘textbook’ real business cycle model, that of King, Plosser, and Rebelo (1988), which specifies preferences, technology and budget constraints. Using standard parametric functional forms for preferences and technology the model can be log-linearized and solved.^[20] As is well known the solution of this model takes the form of a state law of motion and set of decision rules for observable variables:

(7)St+1=ΦSt+Et+1

(8)Yt=HSt

The first system of equations describes the dynamic evolution of the vector of state variables and exogenous shocks, such as capital and technology. The second system of equations are the decision rules, linking the vector of endogenous choices, Y_t, to the current state vector, S_t. Typical decision variables are labor effort and consumption. Of course, the real wage would appear in Y_t as well.

As is well known, the real wage of the representative agent in this model is highly procyclical as the wage is equal to the marginal product of labor. To clarify this implication we follow the conventional way to decentralize the Pareto optimal equilibria of the model by assuming spot-competitive labor markets. Let the utility of agent i, Uⁱ, be defined over consumption, C_it, and work effort, H_it such that UCi>0, UCCi<0, UHi<0 and UHHi<0, where subscripts denote derivatives. Let θ_t be an exogenous state variable which is the driving force of business cycle fluctuations. Without loss of generality, since in RBC models, θ_t commonly denotes the level of neutral technology, we will simply call θ_t technology.^[21] Let ψ_i(θ_t) denote the agent’s marginal productivity which is an increasing function of technology θ_t. The intratemporal efficiency condition derived from an RBC model is

(9)−UHi(Cit,Hit)UCi(Cit,Hit)=ψi(θt)

This condition results from the agent equating his marginal rate of substitution between consumption and leisure to the real wage, while firms choose labor such that the marginal product of labor equals the real wage. The spot-market equilibrium then implies that real wages equal marginal productivities. Consequently, under spot-competitive labor markets we expect that over the business cycle there is a common (macro) component, θ, driving the real wages of all agents, and that these wages move in the same direction.

Our extension of this model assumes that we do not get to observe the ‘true’ real wage. Instead, we have many noisy observations on individual wages from this competitive spot labor market. The noise is induced by a data-gathering agency which must survey individuals to find out their wages. These survey data are riddled with errors, both recall errors from the agents and statistical errors from the agency itself. Our second system of equations then becomes:

(10)Yt=HSt+Ut

where U_U represents the measurement error and the Y_t vector contains the full set of indivudals surveyed.

The empirical model we will use in this paper, a dynamic factor model, is motivated directly from equations 7 and 10. These equations take the same general form as a dynamic factor model. To make this link concrete consider the dynamic factor representation for a vector of wage data y_t:

(11)yt=bft+εt

where b is a N × K matrix of factor loadings. The factor f_t is assumed to follow an autoregressive process:

(12)ft=ϕf(L)ft−1+utf

It is clear from comparing equations 7 and 10 with equations 11 and 12 that the dynamic factor model takes the same form as the linearized solution to the real business cycle model with measurement error. Were one to simulate data from the RBC model and estimate a factor model on the simulated data, the estimated dynamic factor would then be the common state variable (e.g., technology shock) in the business cycle model. When we turn to actual data, if the neoclassical labor market embodying this model is largely correct, then when we estimate the factor model on wage data we should have two key results. First, as long as the wage data are not dominated by measurement errors, the common factor should be quantitatively important for explaining real wage dynamics. Second, the wages should all respond with the same sign to this common factor since in the business cycle model all wages respond positively to changes in productivity.

A.2 A wage contracting model

Our second labor market model is based on an alternative way to decentralize the Pareto optimal equilibria by considering a model where agents trade labor contracts. In such a model, wages and employment are specified in a contract which is the outcome of dynamic bargaining between workers and firms. The contract, {wⁱ(θ_t), Hⁱ(θ_t)}, consists of an hourly wage rate and hours of work that are contingent on the future state of technology. The contract is such that the efficiency condition 9 holds, but the hourly wage rate is not necessarily equal to ψ_i(θ_t). The hourly wage not only responds to changes in productivity but also provides insurance to risk averse agents against business cycle fluctuations.^[22] Contrary to the spot market case, under reasonable assumptions, in equilibrium the wage will not be strongly correlated with productivity. This is due to the fact that the wage embodies an insurance component which minimizes their fluctuations. Furthermore, a given change in θ may induce the wages of some agents to increase while others to decrease. Hence, responses of different signs to a given change in the common component are consistent with the theory of implicit contracts. To illustrate these two points, we provide a simple example where consumption equals labor earnings that is, Cit=wtiHit, and the agents differ in terms of their aversion toward risk. Assuming separable CRRA preferences, condition 9 can be solved for the equilibrium wage (see Boldrin and Horvath (1995)):^[23]

(13)wti=δi[ψi(θt)]1λi[1−HitHit]

where δ_i > 0 and λ_i is the agent’s coefficient of risk aversion. (Note that the linearized version of equation 13 would enter the decision rules 8 or 10 in the state space system describing the model dynamics.) In this case, the equilibrium wage is comprised of two components, productivity and insurance (which is the ratio of leisure to labor). Productivity is strongly procyclical whereas the insurance component is countercyclical because hours of work are procyclical. The latter offsets the increases (decreases) in productivity and thus, wages do not appear to respond strongly to technology shocks. Notice that parameter λ_i controls the elasticity of the hourly wage to the marginal product of labor fluctuations. Depending on the value of λ_i, for some individuals the effect of the insurance component may dominate the effect of productivity and thereby, the change in their wage, in response to an increase in θ, will have a negative sign. The more risk averse an agent is the more likely she/he is to have a negative wage response to an increase in θ. To summarize, the contracting model first implies that real wages will not exhibit a strong commmon cycle, implying that any common dynamic factor should have little explanatory power for real wage fluctuations. Second, if there is heterogeneity in preferences then the model predicts that the factor loading coefficients in the dynamic factor model will have both positive and negative signs.^[24]

B Factor models with unbalanced panels

In this appendix we describe the procedure for estimating the missing observations. This procedure forms one block of our Gibbs sampler. In block one we draw the parameters conditional on factors and missing data. In block two we draw the factors conditional on parameters and missing data. In block three we draw the missing data conditional on parameters and factors. In essence, we fill in the missing observations of the unbalanced panel using information in both the model and available data. It is this last block that we describe in this appendix. The first two blocks are described in Otrok and Whiteman (1998) .

Let ξi,t=ϕi,1ξi,t−1+⋯+ϕi,piξi,t−pi+ui,t where ξi,t=yi,t−bc,iftc−bs,ifts. Then, the following state space system is obtained:

(14)yi,t=Ai′xt+H′ ξi,t+wi,t

(15)ξi,t+1=Fiξi,t+vi,t+1

where

yi,t=yi,t, ξi,t=[ξi,tξi,t−1⋯ξi,t−pi+1]′, xt=[1ft−1c⋯ft−qicft−1s⋯ft−qisk]′wi,t=bc,i uc,tf+bs,ius,tf, vi,t=[ui,t01x(pi−1)]′, Ai′=Bi⋅Φ, Bi=[bc,ibs,i]

H′=[101x(pi−1)], Φ=[ϕc,1⋯ϕc,qi0⋯00⋯0ϕs,1⋯ϕs,qi], Fi=[ϕi,1⋯ϕi,pi−1ϕi,piI(pi−1)x(pi−1)0(pi−1)x1]

The variance matrix of v_i,t is

E(vi,tvi,τ′)=Qi={[σi20⋯00⋯⋯⋮⋮⋯⋯⋮0⋯⋯0]for t=τ0pi×piotherwise

Consequently, the system 14–15 satisfies the following conditions:

E(wi,t2)=bc,i2σf,c2+bs,i2σf,s2=Ri
E(wi,twi,τ)=0, and E(vi,twi,τ)=0 for all t and τ

Equations 14 and 15 are the observation and state equations, respectively. The recursion of the Kalman filter begins with ξ^i,0∣0 which denotes the unconditional mean of ξ_i,1, where ξ^i,0∣0=E(ξi,1)=0, The asssociated Mean Square Error (MSE) is Pi,0∣0=Σ=E(ξi,1ξi,1′) where Σ=FΣF′+Q. To enable the recursion steps we replace missing observations with values drawn from the distribution of the data,^[25]

L(yi,t/ϕ,ξi,t,…,ξi,t−p)=(2πσi2)−1/2exp⁡{−12σi2(yi,t−y^i,t/t−1)2}

where y^i,t/t−1=yi,t−ξi,t+ϕi,1ξi,t−1+⋯+ϕi,pξi,t−p. The transition from ξ^i,t−1∣t−1 and Pi,t−1∣t−1 to ξ^i,t∣t and Pi,t∣t is given by the following set of equations^[26]

ξ^i,t∣t−1=Fiξ^i,t−1∣t−1

Pi,t∣t−1=FiPi,t−1∣t−1Fi′+Qi

y^t∣t−1=Ai′xt+H′ξ^i,t∣t−1

ξ^i,t∣t=ξ^i,t∣t−1+Pi,t∣t−1H(H′Pi,t∣t−1H+Ri)−1(yt−y^t∣t−1)

Pi,t∣t=Pi,t∣t−1−Pi,t∣t−1H(H′Pi,t∣t−1H+Ri)−1H′Pi,t∣t−1

Since our goal is to form an inference about the value of ξ_i,t based on the full set of time series we compute the smoothed estimate ξ^i,t∣T and the corresponding MSE, Pi,t∣T, by conditioning on next period’s observation that is, ξ^i,t∣T=ξ^i,t∣t+Jiτ(ξ^i,t+1∣T−ξ^i,t+1∣t) and Pi,t∣T=Pi,t∣t+Jit(Pi,t+1∣T−Pi,t+1∣t)Jit′ where Jit=Pi,t∣tFi′Pi,t+1∣t−1.^[27] Wherever there is a missing observation, in each loop of the Markov chain, we replace it with yi,t∗=ξi,t∗1+bc,iftc+bs,ifts where ξi,t∗1 is the first element of the drawing ξi,t∗ from N(ξ^i,t∣T,Pi,t∣T). The values for the missing observations are drawn right after the completion of steps 1 and 2 of the estimation procedure.

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Published Online: 2017-09-20

On the cyclicality of real wages and wage differentials

Abstract

Acknowledgement

Appendix

A The factor model and two theoretical models

A.1 A neoclassical model

A.2 A wage contracting model

B Factor models with unbalanced panels

References

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