Real Wage Cyclicality and Monetary Policy

1.1 Related Literature

This paper is related to different strands of the literature that focus on the transmission mechanism of monetary policy in the presence of incomplete markets. Auclert (2019) argues that redistribution channels including earnings heterogeneity channels play an essential role in accounting for monetary policy effects on aggregate consumption. Kaplan et al. (2018) consider a New Keynesian model with incomplete financial markets in which two types of assets are introduced with different degrees of liquidity and returns. Having a model’s ability to reproduce empirically realistic wealth distributions across liquid and illiquid assets and a distribution of the marginal propensities to consume (MPC), they show that the indirect channels from general equilibrium effects dominate the direct effects, which are mainly from intertemporal substitution effects. Werning (2015) also studies the monetary transmission mechanism in the presence of an incomplete market and finds that indirect channels offset direct effects. Gornemann et al. (2016) consider a heterogeneous-agent New Keynesian economy where households are different in their employment status and wealth. They show that contractionary monetary policy shocks increase inequality, and, importantly, a majority of households prefer substantial stabilization of unemployment even though this implies deviations from stable price dynamics. The main contribution in this paper relative to the previous studies in this literature is that i) the current study incorporates wage rigidities into an incomplete markets model in the context of New Keynesian economies to study the cyclicality of real wages conditional on monetary policy shocks, and ii) it also reconciles the puzzling mismatch between the data and the model’s predictions.

This paper can complement the recent discussions on the importance of sticky wages in the monetary transmission channel. Perhaps most closely related to this paper is Broer et al. (2019), who study the interaction between inequality and monetary policy by using a tractable heterogeneous-agent version of the New Keynesian model. The key finding in Broer et al. (2019) is that whether nominal frictions arise from price or wage rigidity matters greatly for the transmission mechanism for monetary policy: wage rigidities are key to accounting for the plausible monetary transmission channel. Nekarda and Ramey (2020) empirically document that the price markup, which is based on the inverse of the labor share, increases in response to expansionary demand shocks, inconsistent with the standard sticky-price New Keynesian model’s prediction. Nekarda and Ramey (2020) conclude that one possible way to solve the mismatch would be incorporating sticky wages as in the old Keynesian models. The main finding in this paper can back up this literature by showing that model-consistent real wages are countercyclical conditional on a monetary policy shock, which implies that the monetary transmission mechanism in New Keynesian models with sticky wages is in accordance with the data. Related, Kaplan et al. (2018) and Qiu and Rios-Rull (2021) provide alternative theories that solve the puzzle independently on the movement of real wages.

Another important related work is Cantore et al. (2020), who show robust cross-country evidence that the labor share is countercyclical and that real wages are procyclical conditional on a monetary policy shock. They argue that this empirical evidence cannot be consistent with any medium-scale New Keynesian models under consideration even if these models possibly break the close link between the labor share and the inverse markup. Accordingly, Cantore et al. (2020) cast doubts on the standard monetary transmission mechanism in New Keynesian macroeconomic models where the labor share plays a key role. The current paper revisits this view. When taking household heterogeneity into account, the perceived mismatch can be addressed, and the disparity arises from an inconsistent measure for real wages between the data and the model. Moreover, in contrast to Cantore et al. (2020)’s argument, the heterogeneous-agent New Keynesian model with sticky wages is able to replicate the joint behavior of real wages and the labor share if one uses the data-equivalent measure for real wages in the model.

This paper is also closely related to the empirical literature which measures the price of labor using average hourly earnings. The seminal contribution by Bils (1985) highlights differences in real wages behavior using disaggregated data.^[1] Haefke et al. (2013) show that a better measure is the wage of the new hires.^[2] Particularly, Basu and House (2016) discuss precisely why wages are hard to measure, present several theoretically consistent measures of wages and their implications for New-Keynesian models with price and wage rigidities. However, Gertler et al. (2020) argue that procyclical match improvement for workers making job-to-job transitions may generate the excess wage cyclicality among new hires. In this paper, I also empirically document that a composition bias associated with cyclical changes in match quality for new hires may be procyclical conditional on monetary policy shocks, which is consistent with the predictions of New Keynesian models that incorporate sticky wages.

The paper is organized as follows. Section 2 presents a quantitative New Keynesian model economy with heterogeneous households. Section 3 presents the primary results from the benchmark model, mainly focusing on the cyclical behavior of the labor share and real wages. Section 4 concludes the paper.

2.1 Households

The model economy is populated by a continuum of infinitely lived households. Each household maximizes its expected lifetime utility by choosing consumption, c _t , and hours worked, h _t :

subject to

and

where 0 < β < 1, σ, Ξ > 0, and ν denote the time discount factor, the degree of relative risk aversion, a parameter for disutility from working, and labor supply elasticity, respectively. Each household is endowed with a unit of time in each period, allocated between hours worked and leisure. When a household supplies h _t units of labor, it earns w _t z _t h _t as labor income, where w _t is the wage rate per effective unit of labor, and z _t denotes its labor productivity. Each household earns profit income, ξ _t , from firms and pays taxes, T _t , to the government. Labor productivity, z, is assumed to follow an AR(1) process in logs:

Households can invest financial assets, a _t , on government bonds for a real rate of return, r _t . Following Huggett (1993) and Aiyagari (1994), the asset markets are incomplete: a _t is the only asset available to households to insure against idiosyncratic risks, z. A household faces a borrowing constraint that limits the fixed amount of debt: the assets holding, a_t+1, cannot go below a ̲ for all t.

The household’s problem can be recursively written as follows. Dene x and X as the vectors of individual and aggregate state variables, respectively: x ≡ (a, z) and X ≡ (μ, η), where μ(x) is the type distribution of households, and η denotes monetary policy shocks.^[3] The value function for an individual household, denoted by V(x, X), is defined as:

subject to

and

where T denotes the law of motion for μ, time subindices are suppressed to simplify notation, and primes denote variables in the next period.

2.2 Labor Union

It is assumed that nominal wages are rigid in the economy. To incorporate sticky wages, I follow Erceg et al. (2000) and assume that a union sells a different type of labor services, z _t (k)h _t (k), provided by each household k, to a representative employment agency in a competitive market. The employment agency uses the differentiated labor services to produce aggregate effective labor input, N _t , according to the constant elasticity of substitution (CES) technology given by:

where ϵ _w is the elasticity of substitution across labor services.

The employment agency maximizes profits given aggregate effective labor, N _t :

The profit maximization implies the labor demand curve for household k:

where W _t is the aggregate nominal wage, which can be defined as:

Wage adjustment costs are introduced similar to a Rotemberg (1982)’s mechanism: the adjustment costs are given by a quadratic function of the change in nominal wages, governed by the parameter, θ _w ≥ 0, and are proportional to individual productivity, z _t (j). The union sets the nominal wage W t * for an effective unit of labor such that W t ( k ) = W t * and h t ( k ) = h t * by solving the following problem:

subject to

where g ( h ) = Ξ h 1 + 1 / ν 1 + 1 / ν , C _t is the aggregate consumption, and Π w ̄ is the steady-state gross wage inflation. The first-order condition under W t * = W t and h t * = H t implies the standard wage Phillips curve:

where w t = W t P t , Π t w = W t W t − 1 , z ̄ = ∫ z t d μ t ,^[4] and H _t is aggregate hours.^[5] Following Eggertsson and Singh (2019) and Hagedorn et al. (2019), I assume that wage adjustment costs are perceived by the union, and are taken into account only in their maximization problem; however the wage adjustment costs do not involve any physical costs.^[6]

2.3 The Representative Final Goods-Producing Firm

It is assumed that the representative finished goods-producing firm operates in a competitive sector. The final good firm uses y _t (j) units of each intermediate good j ∈ [0, 1] to produce a homogeneous output, Y _t , according to the CES technology given by:

where ϵ _p > 1 is the elasticity of substitution for intermediate goods. The final good firm in this sector takes the final-goods price, P _t , as given and purchased at the nominal price p _t (j) for each of its inputs, where p _t (j) is the price of the jth intermediate input. The profit maximization problem of the representative finished goods-producing firm is given by:

subject to Equation (9). The first-order condition for the final-goods firm’s problem, together with the zero-profit condition, implies that the demand for intermediate good j is given as:

2.4 Intermediate Goods-Producing Firm

There is a continuum of monopolistically competitive firms indexed by j ∈ [0, 1], each of which produces a different type of intermediate good y _t (j). Intermediate goods-producing firms use n _t (j) units of effective labor in order to produce y _t (j) units of intermediate good j, following a decreasing returns to scale (DRS) production function:

where 1 − α is the degree of decreasing returns to labor, and Ω ≥ 0 is the fixed cost of production.^[7] It should be noted that fixed costs in production break the direct link between the labor share and marginal costs and help generate the countercyclical behavior of the labor share.^[8] It is assumed that nominal prices are rigid in the economy, and price adjustment is subject to a Rotemberg (1982)’s price setting mechanism: each intermediate goods firm j faces a quadratic cost of adjusting their price governed by the parameter, θ _p ≥ 0. An intermediate goods-producing firm j maximizes its expected discounted profit by choosing its price p _t (j):

subject to

where mc_t+s is the real marginal cost of a unit of intermediate good, and Π p ̄ is the steady-state gross inflation. Equation (13) is the demand for intermediate good j, driven by the final good firm’s optimization. In the symmetric equilibrium conditions (i.e. p _t (j) = P _t and y _t (j) = Y _t ),^[9] the first-order condition associated with the optimal price implies:

where Π t p = P t P t − 1 . Similar to the wage adjustment process, I follow Hagedorn et al. (2019) and assume that price adjustment costs do not correspond to resource costs of price changes, but firms behave as if they were in the maximization problem. Accordingly, the aggregate dividend, D _t , paid by the intermediate goods-producing firms is defined as:

2.5 Government and Central Bank

The government plays three roles in this economy: they i) collect taxes from households, ii) supply public bonds, and iii) distribute profits from intermediate goods-producing firms to households. I follow McKay et al. (2016) and assume that the government levies taxes according to household’s labor productivity, z _t :

where τ _t is a tax rate, which is proportional to z _t .^[10] Since individual labor productivity follows an exogenous process, this assumption does not distort households’ optimal choices. The government issues bonds with real face value B _t , and they change taxes to finance interest payments on this public debt. Finally, following McKay et al. (2016), I assume that the government collects taxes to run a balanced budget keeping a constant level of public debt each period (i.e. B t = B ̄ ):

The government is also in charge of distributing monopoly profits from intermediate goods-producing firms to households. As in Kaplan et al. (2018), I assume that dividend distribution is proportional to both asset holdings and productivity:

where ψ is the fraction of profits for asset holdings, Δ t a = a t ∫ a t d μ t , and Δ t z = z t ∫ z t d μ t . How profits are distributed across households is not important in this model economy since rigid adjustments of wages make movements in profits small, so they do not generate large distributional effects.

The gross nominal interest on risk-free bonds, R _t , is assumed to follow a conventional Taylor rule:

where R ̄ is the deterministic steady-state value of the gross nominal interest rate, and ϕ _π > 1 is a coefficient on the inflation gap. η _t is monetary policy shocks, which follow an AR(1) process:

Finally, the relationship between the real interest rate, the nominal interest rate, and inflation satisfies the Fisher relation:

2.6 Definition of Equilibrium

A recursive competitive equilibrium is a value function V(x, X), a transition operator T ( X ) , a set of policy functions c ( x , X ) , a ′ ( x , X ) , h ( x , X ) , n j ( X ) , p j ( X ) , y j ( X ) , h k ( X ) , W k ( X ) , and a set of prices {w(X), r(X), R(X), P(X), W(X)} such that:

1. Individual households’ optimization: given w(X) and r(X), optimal decision rules c(x, X), a′(x, X), and h(x, X) solve the Bellman equation, V(x, X).

2. Intermediate goods firms’ optimization: given w(X), r(X), and P(X), the associated optimal decision rules are n _j (X) and p _j (X).

3. Final good firm’s optimization: given a set of prices P(X) and p _j (X), the associated optimal decision rules are y _j (X) and Y(X).

4. The union’s optimization: given w(X), r(X), and W(X), the associated optimal decision rule is W _k (X).

5. The employment agency’s optimization: given a set of prices W(X) and W _k (X), the associated optimal decision rules are h _k (X) and N(X).

6. The gross nominal interest rate, R(X), satisfies the Taylor rule (Equation (19)).

7. The Fisher relation holds: E 1 1 + r ( X ) R ( X ) Π p ( X ′ ) = 1 .

8. Balanced budget of the government: r ( X ) B ̄ = ∫ T ( x , X ) d μ .

9. For all Ω,

   1. (Labor market) N(X) = ∫zh(x, X)dμ = ∫n _j (X)dj

   2. (Bond market) B ̄ = ∫ a ′ ( x , X ) d μ

   3. (Goods market) Y(X) = C(X) where Y(X) = N(X)^1−α − Ω and C(X) = ∫c(x, X)dμ.

10. Consistency of individual and aggregate behaviors: for all A 0 ⊂ A and Z 0 ⊂ Z ,

where Γ _z (z′|z) is a transition probability distribution function for z.

2.7 Parameterization

I choose parameters in the model economy, based on the values of structural parameters from literature. The parameter values used in the model economy are summarized in Table 1.

I use a value for the risk aversion parameter, σ, equal to one. The Frish elasticity of labor supply, γ, is set to one as well. These are standard values in the business cycle and New Keynesian literature. The disutility parameter of working, Ξ, is set so that the steady-state hours are 0.233.^[11] The time discount factor, β, is chosen to obtain a steady state value of the real interest rate of 2 % annualized, as in McKay et al. (2016). Regarding individual labor productivity shocks, I follow Debortoli and Gali (2018) and set ρ _z = 0.978 and σ _z = 0.193. This set of parameters implies that individual wages display an autoregressive coefficient of 0.914 and an innovation standard deviation of 0.258 at an annual frequency, which are similar to the estimates of Floden and Linde (2001). The shock process is approximated with a 17 states Markov chain approximated using the Tauchen (1986) method. The borrowing limit, a ̲ , is simply chosen to be zero following McKay et al. (2016) and Hagedorn et al. (2019).

The degree of decreasing returns to labor is set to 0.7, implying that α = 0.3. The production fixed cost, Ω, is set for intermediate goods firms to have zero profit in the steady state. The elasticities of substitution across households and intermediate goods, ϵ_w and ϵ_p, are set equal to 10. The Rotemberg adjustment cost parameters for nominal wages and price, θ_w and θ_p, are chosen for the corresponding Calvo stickiness parameters to be 0.75 and 0.65, respectively.^[12] These parameter values are consistent with the estimates of Smets and Wouters (2007).

Following McKay et al. (2016), I set debt to annual GDP to 1.4, and the tax rate τ _t is chosen for the government to run a balanced budget every period. As in Kaplan et al. (2018), I assume that the fraction of profits for asset holdings, ψ, is the same as α, i.e. ψ = α = 0.3. As mentioned earlier, wage rigidity generates relatively small movements in profits, so how they are distributed across households does not significantly affect the main findings in this paper. The Taylor rule coefficient of inflation, ϕ _π , is chosen to be 1.5. The steady-state gross inflations for wages and price, Π w ̄ and Π p ̄ , are assumed to be one. Regarding the monetary policy shocks, I choose ρ _η = 0.7 and σ _η = 0.0025. Given the value of standard deviation, σ _η , the annualized size of a typical monetary policy shock is 100 basis points.

3.1 Flexible versus Sticky Wages

This subsection explores whether sticky wages lead to more plausible responses compared to their flexible-wage counterparts. To investigate this, I compare two model economies: one with sticky prices and flexible wages (referred to as the flexible-wage model) and the benchmark model, which includes both sticky prices and sticky wages (referred to as the sticky-wage model). Note that both models share the feature of sticky prices but differ in their assumptions about wage stickiness. In the flexible-wage model, I assume that θ _w = 0, and real wages are determined by the equilibrium condition in the frictionless labor market.^[13]

I compare the dynamics of the two models in Figure 1, which shows the impulse responses of aggregate variables to a 100-basis-point (annualized) expansionary monetary policy shock for 20 quarters. In the flexible-wage model, the expansionary monetary policy increases the demand for the labor input and real wages. Accordingly, households increase hours by 0.1 %, and output or consumption rises by 0.2 %. Finally, a 100-basis-point (annualized) monetary expansion increases annualized inflation by 0.88 % point.

Figure 1:

Aggregate responses: flexible-wage model vs. sticky-wage model. Note: Impulse response to a 100-basis-point (annualized) monetary policy shock. For output, consumption, hours, the labor share, and real wages, the y axis shows percent changes, while the y axis shows changes in annualized percentage points for inflation. The x-axis shows quarters after the shock. “Flexible-wage model” refers to a sticky-price model with flexible wages, while “sticky-wage model” refers to a sticky-price model with rigid wages.

As is well-known, the labor share (or the price markup) plays a crucial role in the monetary transmission mechanism in standard New Keynesian economies. In this sense, monetary policy shocks in models should affect the cyclical behavior of the labor share in ways that are consistent with the data. As the dashed line in Figure 1 shows, when prices are sticky but wages are not, the labor share rises in response to expansionary monetary policy shocks. However, this prediction is inconsistent with empirical findings in the literature, such as Cantore et al. (2020) and Nekarda and Ramey (2020) among others.^[14] Thus, the monetary transmission mechanism in the flexible-wage model is not consistent with the data.

In response to the perceived mismatch between the data and predictions of the flexible-wage model, Nekarda and Ramey (2020) suggest that one possible way to solve the mismatch would be incorporating sticky wages as in the old Keynesian models.^[15] As shown in Figure 1 (solid lines), when wage rigidities are incorporated in the sticky-price model economy, the transmission mechanism for monetary policy shocks is different from that in the flexible-wage model. Since the baseline calibration implies that prices are more flexible than wages, a monetary expansion decreases the real wage rate in the sticky-wage model. The fall in real wages leads firms to increase their labor demand significantly,^[16] which increases output (or consumption) relative to the flexible-wage model. Sticky wages also make marginal costs faced by firms less affected by monetary policy given the production function, which strongly dampens the response of inflation relative to the flexible-wage model.

The responses of aggregate variables in the sticky-wage model are in line with the data from a quantitative perspective: empirical evidence shows that i) monetary policy shocks have a more considerable impact on output in the calendar periods in which wages are relatively rigid (Olivei and Tenreyro 2007, 2010; Bjorklund et al. 2019), and ii) inflation adjusts very little following a monetary policy shock (Christiano et al. 2005; Alpanda and Zubairy 2019; Cantore et al. 2020). Notably, the monetary policy transmission mechanism in the sticky-wage model is empirically realistic from a qualitative perspective: sticky wages successfully generate the observed countercyclical labor share. According to Figure 1, when wages are sticky, the labor share decreases while profits increase in response to a monetary policy easing. These results are in accordance with empirical findings in the aforementioned literature.

However, one of the critical limitations in the models with a standard sticky wage setting is that this class of models fails to jointly match the response of the labor share and real wages to monetary policy (Cantore et al. 2020). In the model economy with sticky wages, as the solid line in Figure 1 shows, real wages decrease in response to a monetary easing, which is not consistent with the data. Many papers in empirical literature provide robust evidence that the real wage rate is procyclical conditional on monetary policy shocks (Christiano et al. 2005; Coibion et al. 2017; Cantore et al. 2020).^[17]

In the next subsection, I argue that once heterogeneity is taken into account, the sticky-wage model aligns with the data, as it successfully generates procyclical real average hourly earnings. This distinction forms the primary focus of this paper.

3.2 Cyclicality of Real Wages

Replicating the response of real wages is essential in studying the effects of monetary policy shocks since real wages are a crucial component of the labor share, through which a key monetary transmission channel in New Keynesian models works, as discussed above. The main emphasis in this paper is on the cyclicality of real wages conditional on monetary policy shocks. In this subsection, I employ the quantitative model economy to revisit the cyclicality of real wages by comparing the dynamics of real wages (wage rate per effective unit of labor) and that of a data-consistent measure for real wages (average hourly earnings).

In the model economy, the labor share, LS, can be defined as the ratio between real wages, w, and labor productivity, Y/N:

where N denotes aggregate effective labor. Accordingly, real wages and productivity jointly determine the dynamics of the labor share. Importantly, it should be noted that w is defined as the wage rate per effective unit of labor in the model, as shown in the households’ budget constraint (Equation (1)). In practice, the model-consistent real wages may not be observable since effective labor is not an observable measure. In other words, in the data, one can observe total labor compensation, wN, but cannot identify w and N separately. Accordingly, the definition of real wages in the data cannot help being inconsistent with the model. Indeed, empirical literature uses an alternative definition of real wages. For example, Cantore et al. (2020) construct the real wage measure for the U.S. economy as real Wages and Salaries from National Income and Product Accounts (NIPA) 1.12^[18] divided by total hours worked in the economy from the Bureau of Labor Statistics (BLS). Recall that, in general, real wages in the data (denoted by ω) are defined as average hourly earnings – total labor compensation divided by total hours worked:

where H is aggregate hours in the economy. Based on Equation (22), data-equivalent real wages can be defined in the model economy as well. Of course, in the representative-agent model where there are infinitely many identical households, the dynamics of effective labor, N, will be the same as that of the aggregate hours H, so w has the same dynamics as ω all the time. In contrast, in heterogeneous-agent economies where households are different in their productivity, the dynamics of aggregate effective labor will be different from hours dynamics. Accordingly, the responses of w and ω may be very different quantitatively and/or qualitatively. The question is then, are data-consistent real wages also countercyclical conditional on monetary policy in the heterogeneous-agent New Keynesian model with sticky wages? This is an important question considering the previously-mentioned dependence of New Keynesian models on specific transmission channels of monetary policy shocks.

Figure 2 plots the responses of the two different real wages (w and ω) and other labor-related variables to a monetary policy shock in the model with benchmark calibration where both sticky price and wages are incorporated. An important finding is that in the benchmark model, even if the wage rate per effective unit of labor, w, falls following a monetary expansion, real average hourly earnings, ω, indeed increases, which is consistent with the empirical findings in the literature discussed before. Accordingly, the monetary transmission mechanism in the sticky-wage model is not inconsistent with the data since real average hourly earnings are procyclical conditional on a monetary policy shock.

Figure 2:

Response of labor-related variables: sticky-wage model. Note: impulse response to a 100-basis-point (annualized) monetary policy shock. It is generated by the “sticky-wage model,” which represents a sticky-price framework with rigid wages. The y axis shows percent changes while the x-axis shows quarters after the shock.

This result contrasts with recent work by Cantore et al. (2020). From rich empirical and theoretical analysis, Cantore et al. (2020) conclude that medium-scale New Keynesian models commonly used for monetary policy study cannot generate empirically realistic responses of the labor share and real wages simultaneously. However, the benchmark economy considered in this study is able to reproduce the joint behavior of real wages and the labor share if one uses average hourly earnings for real wages.

It is also important to note that household heterogeneity matters. As briey discussed above, the representative-agent Keynesian counterpart predicts that both w and ω decrease following a monetary expansion. In this sense, the heterogeneous-agent New Keynesian model with wage rigidities can reconcile the puzzling mismatch between the data and predictions of the representative-agent model with sticky wages.^[19] This finding is in line with Broer et al. (2019). They suggest that a simple heterogeneous-agent model with sticky price and wages is a more relevant benchmark setting for monetary policy analysis since it provides better understanding of the micro-founded monetary transmission relative to a representative-agent counterpart.

Lastly, the model with benchmark calibration does an excellent job of accounting for the underlying behavior of the labor share dynamics. Empirical evidence provided by Christiano et al. (2005) and Cantore et al. (2020) shows that productivity should increase more than real wages to have the countercyclical labor share conditional on monetary policy shocks. It follows that the benchmark model can successfully reproduce the response of productivity, defined as output divided by total hours worked (i.e. Y/H), from the quantitative perspective. The labor share can be redefined as the ratio between ω (=wN/H) and productivity, Y/H:

According to Equation (23), Y/H should increase more than ω for the labor share to respond countercyclically to a monetary policy shock. Indeed, as shown in Figure 2, productivity increases by 0.25 %, while ω increases by less (0.15 %).

3.3 What Explains Procyclical Real Wages in Data?

At this point, the reader may wonder what explains procyclical real wages in the data. I answer this question by further decomposing the data-consistent measure for real wages, ω, in the model economy. According to Equation (22), given the negative response of w, effective labor per hours (N/H) should increase enough so that ω responds procyclically to a monetary policy shock. Indeed, it follows that N/H increases by around 0.2 % (not shown in Figure 2), but ω increases by only 0.15 %.

The positive response of effective labor per hours worked implies that skilled households increase hours of work more than unskilled households following expansionary monetary policy shocks. Figure 3 confirms this hypothesis: in the benchmark economy, relative hours between households in the highest productivity quintile (5th quintile) and the lowest productivity quintile (1st quintile) increase by 0.6 % in the wake of a monetary expansion. Therefore, the heterogeneous responses of hours worked between high- and low-skilled households can account for the cyclicality of real wages in the data: an expansionary monetary shock increases average hourly earnings, ω, since productive households relatively work more, which leads to total labor compensation, wN, to increase more than total hours, H.

Figure 3:

Response of relative hours and consumption: sticky-wage Model. Note: Impulse response to a 100-basis-point (annualized) monetary policy shock. It is generated by the “sticky-wage model,” which represents a sticky-price framework with rigid wages. The y axis shows percent changes while the x-axis shows quarters after the shock.

The question is then, why do productive households work more following a monetary expansion? This can be explained by the intratemporal optimality condition where households determine optimal hours and consumption. Since low-productivity households tend to a have higher marginal propensity to consume (MPC), they increase consumption more than households with higher productivity. Accordingly, as shown in Figure 3, the relative consumption between high- and low-productivity households decreases in response to expansionary monetary policy shocks.^[20] Finally, given the same responses of real wages (w) and the labor markup across all households,^[21] smaller wealth effects from the little movement in consumption lead productive households to increase hours worked by more. Specifically, in the sticky-wage model with household heterogeneity, while aggregate effective labor is determined by the demand-side, individual households still optimally choose their consumption and labor supply based on the real wage rate and labor markup. Since the labor markup is determined by labor unions and follows the wage Phillips curve (as specified in Equation (8)), it remains identical for all households. As a result, the wealth effect generates a different response in hours compared to consumption, even within the sticky-wage framework.

As mentioned above, different hours responses across the productivity distribution affect real average hourly earnings. To explore this further, I consider a counterfactual scenario to examine how the share of the lowest-productivity households affects the dynamics of real average hourly earnings. Specifically, using the decision rule for labor supply for each household in the benchmark economy, I hypothetically adjust the share of the lowest-productivity households in the economy (which is 0.2 in the benchmark case), while the remaining shares are equally distributed among the other households.^[22] Figure 4 illustrates the response of real average hourly earnings to monetary policy shocks under different shares of the lowest-productivity households. As expected, real average hourly earnings become acyclical as the share increases. For example, when the share of the lowest-productivity households is 0.9, the impact response is less than 0.04 %, significantly smaller than the 0.15 % observed in the benchmark case (the share is 0.2).

Figure 4:

Low-productivity share and average hourly earnings: sticky-wage model. Note: impulse response to a 100-basis-point (annualized) monetary policy shock. It illustrates the response of real average hourly earnings under different shares of the lowest-productivity households. It is generated by the “sticky-wage model,” which represents a sticky-price framework with rigid wages. The y axis shows percent changes while the x-axis shows quarters after the shock.

3.4 Empirical Evidence

The model’s prediction – productive households work more following a monetary expansion – has empirical support from recent work by Dolado et al. (2021). By using monthly data from the Current Population Survey (CPS), Dolado et al. (2021) find that a monetary expansion increases employment of skilled workers relative to unskilled workers, and this result is robust to different empirical specifications. I replicate their results in Figure 5, which shows the responses of relative employment between high and low-skilled workers and other macro variables. High-skilled and less-skilled workers are classified based on education status: whether they have experienced some college or not. Employment is the number of workers in each skill category. Following Mertens and Ravn (2013), I employ a proxy Vector Autoregression (VAR) model for this analysis, using measures developed by Romer and Romer (2004) as an external instrument to capture monetary policy shocks.^[23] The analysis uses monthly data covering the period from 1983:1 to 2007:12. As shown in Figure 5, in response to a monetary policy shock leading to a decrease in the federal funds rate (FFR), unemployment initially rises but begins to fall after 20 months, while inflation increases after the same period. These macroeconomic responses are well-aligned with findings from much of the empirical literature. Notably, consistent with findings in Dolado et al. (2021), high-skilled workers increase employment more than low-skilled workers in response to expansionary monetary policy shocks. The employment rate ratio increases by around 2 % points.

Figure 5:

Responses of key aggregate variables and relative employment: proxy VAR. Note: impulse response to a 100-basis-point (annualized) monetary policy shock. The y axis shows percent point changes while the x-axis shows months after the shock. The shaded regions are the 68 % confidence bands generated by a block bootstrap.

While Dolado et al. (2021) find that expansionary monetary policy increases earnings inequality, other studies suggest different findings. For instance, using quarterly data from the Consumer Expenditure Survey (CEX), Coibion et al. (2017) find that expansionary monetary policy shocks tend to increase earnings inequality. Using high-frequency changes in interest rates around FOMC announcements and speeches by the Federal Reserve Chair, Graves et al. (2023) empirically document that expansionary monetary policy shocks increase low-skill employment more than high-skill employment. There remains ongoing debate about the cyclical nature of earnings inequality in response to monetary policy shocks, with mixed results across different studies and time periods.

3.4.1 Composition Bias

One may argue that the empirical evidence presented above is inconsistent with the results of Basu and House (2016), who showed that the composition-bias corrected wage for new hires is considerably more procyclical than the average wage in the data.^[24] In principle, the aggregate data are subject to “a composition bias” that makes the cyclicality of aggregate (or average) wage rates different from the wages of individual workers. However, an influential paper by Gertler et al. (2020) argues that this excess wage cyclicality among new hires might be attributed to procyclical match improvement for workers making job-to-job transitions. They find no evidence of excess wage cyclicality for new hires coming from unemployment. More importantly, as Gertler et al. (2020) discuss, wage cyclicality among new hires identified in the existing empirical literature cannot perfectly control for a composition bias associated with match quality. To be more specific, consider the measurement equation for wages given by

(24) log w ijt = Ψ X ijt + α i + e ijt ,

where w _ijt is the wage of individual i in job j at time t, X _ijt is worker level characteristics such as education and job tenure as well as a time trend, α _i is a worker fixed effect, and e _ijt is an error term. If workers find better matches following a monetary expansion, then the error term e _ijt will be correlated with monetary policy shocks. Moreover, a degree of match improvement can be heterogeneous across different jobs. As a consequence, the resulting estimate for the cyclicality of newly hired workers’ wages would be biased. Introducing fixed effects at the worker-job level would be a possible way to control for match-specific effects that differ across jobs (Gertler et al. 2020). However, it is almost impossible to observe workers on specific jobs for long periods enough to identify the individual-job fixed effect in the data. Therefore, the prototypical regression in the literature conflates the possible wage cyclicality of new hires with procyclical improvements of match quality for new hires.^[25]

Next, I provide empirical evidence that a composition bias associated with cyclical variations in match quality for new hires can be procyclical conditional on monetary policy shocks. In other words, I empirically document that a monetary expansion still increases a wage gap between high-skilled and less-skilled workers even if one corrects the composition bias of wages using the standard regression method in the literature. To this end, I use data on composition-bias corrected measures of wages constructed by Haefke et al. (2013). They use individual-level data from the CPS and construct composition-bias corrected wages using the standard approach: they regress wages on observable worker characteristics including time-invariant individual fixed effects.^[26] Due to data availability, I define the wage gap (or wage inequality) as a difference between the mean and median wages for newly hired workers from non-employment.

The left panel of Figure 6 exhibits the wage gap between mean and median wages for new hire.^[27] There are interesting points that should be discussed. First, as observed in the figure, the mean worker earns 6.17 % more than the median worker on average. Second, the wage gap is largely volatile over the business cycles: the standard deviation of the wage gap is 1.82 %. Third, the wage gap tends to decrease during economic recessions: the gap falls significantly during the early 1980s and 2000s recessions. This can be suggestive evidence for the procyclical variations in match quality over the business cycles.

Figure 6:

Wage gap between mean and median new hires. Note: the left panel shows the wage gap between mean and median wages for new hires. The wages are composition-bias corrected measures of wages constructed by Haefke et al. (2013). The right panel shows the impulse response to a 100-basis-point (annualized) monetary policy shock. The y axis shows percent points changes while the x-axis shows quarters after the shock. The shaded regions are the 68 % confidence bands generated by a block bootstrap. (A) Wage Gap over 1979–2006. (B) Responses of Wage Gap to MP shocks.

To investigate how monetary policy shocks affect the wage gap, I again employ the proxy VAR. I use quarterly data covering the period of 1983:I to 2006:I^[28] and use measures developed by Romer and Romer (2004) as an external instrument for monetary policy shocks. The right panel of Figure 6 shows the response of the wage difference between the mean and median wages for newly hired workers. Overall, the wage gap increases in response to an expansionary monetary policy shock. This result implies that a composition bias associated with cyclical changes in match quality for new hires may be procyclical conditional on monetary policy shocks. This finding is in line with Graves et al. (2023), who show that real wages measured from the Employment Cost Index (ECI) are conditionally countercyclical and free from the type of composition bias that, as shown in the analysis above, drives the conditional procyclicality of average real wages.

3.4.2 Relative Employment Responses Across Industries

A measure of wages adjusted for occupation or industry effects is crucial for understanding real wage cyclicality. For example, Black and Figueiredo (2022) find that the wages of new hires who remain in the same occupation are no more cyclical than those of workers who stay in their positions, but wages for workers who switch occupations, either across or within firms, exhibit significant cyclical variation. Even if Haefke et al. (2013) control for occupation and industry effects in their wage data, fully accounting for match-specific composition effects remains challenging due to the inability to observe job matches over a long panel period.

I address this indirectly by using data from Dolado et al. (2021) to investigate how employment responses to monetary policy shocks differ between skilled and low-skilled workers. Based on Dolado et al. (2021)’s data, I consider six industries: (A) Manufacturing, (B) Education and Health Services, (C) Agriculture, Mining, and Transportation, (D) Wholesale and Retail Trade, (E) Professional Services, and (F) Financial and Informational Services. These categories collectively represent about 80 % of the labor force.

Figure 7 displays the relative employment response to expansionary monetary policy shocks. Two key results emerge from this figure. First, the benchmark empirical results remain robust across different industries. Overall, expansionary monetary policy shocks increase employment among skilled workers more than among low-skilled workers. Second, there is considerable heterogeneity in the relative employment responses across industries. For instance, the response is muted in the “Manufacturing” industry, while it decreases in the “Financial and Informational Services” sector. This heterogeneity suggests that the cyclical nature of wages for occupation or industry switchers may differ from that of stayers.

Figure 7:

Relative employment responses across 6 industries. Note: impulse response to a 100-basis-point (annualized) monetary policy shock. The y axis shows percent point changes while the x-axis shows months after the shock. The shaded regions are the 68 % confidence bands generated by a block bootstrap.

3.5 Other Possible Theories

Without having further complicated features, the model economy in this paper can successfully generate the fact that high-skilled households increase hours worked by more in response to expansionary monetary policy shocks. Any sticky-wage models producing this composition effect – productive households work more following a monetary expansion – would come to the same conclusion. In this sense, this paper does not argue that the key mechanism of the model is the right source of the heterogeneous responses of hours worked between high- and low-productivity households. The main focus of this paper is not on the causes of the composition effect but on the effect itself. In this subsection, I discuss some possible theories that can provide sources of the composition effect.