A note on the unemployment volatility puzzle

Abstract

1. Introduction

Understanding labour market fluctuations is an important issue in economics. Shimer (2005) argues that for common calibrations of the canonical equilibrium search model of unemployment, vacancies and unemployment are almost insensitive to productivity shocks and the model cannot account for the key cyclical movements in labour market activity (the unemployment volatility puzzle). Following Shimer’s contribution, various modifications to the canonical model have been proposed to resolve the puzzle—notable contributions include Hagedorn and Manovskii (2008), Hall and Milgrom (2008), and Pissarides (2009). However, Chodorow-Reich and Karabarbounis (2016) show that the opportunity cost of employment is procyclical in the US data and that this cyclicality poses a challenge to models relying on constant opportunity cost to solve the puzzle.

The contribution of this paper is to show that big responses of unemployment to productivity changes are possible even with Nash bargaining wage contracting and strongly procyclical opportunity cost. My approach incorporates into the canonical search model of unemployment two features: cyclical opportunity cost of employment and rationing of suppliers in the goods market, that is, demand-determined output. I show that the fundamental surplus (FS) (Ljungqvist and Sargent, 2017) is inversely related to price markups and that strongly countercyclical markups amplify the impact of aggregate shocks on the FS, and as a consequence, on labour market outcomes.

Models that feature real wage rigidities, for example, Hall (2005), can attain big responses of unemployment to productivity even with cyclical opportunity cost. This is because wages do not reflect the cyclicality of the worker’s outside option and the opportunity cost has no impact on anything. Instead, Chodorow-Reich and Karabarbounis (2016) focus their critique on four models in which the opportunity cost affects market outcomes. These include Mortensen and Pissarides (1994) and Hagedorn and Manovskii (2008), featuring Nash bargaining, Hall and Milgrom (2008), featuring alternative-offer-bargaining, and environments pioneered in Moen (1997) with directed search and wage posting. In that respect, retaining the assumption of Nash bargaining, hardwired into the canonical model, allows me to address the critique and differentiates my paper from the literature on wage rigidities.

The economy features frictions in the labour market and a two-sector firm structure. Workers search for jobs; wholesale firms open costly vacancies to hire workers and produce intermediate output; and monopolistically competitive retail firms produce final goods with intermediate output, facing preset prices. The modelling of the retail sector follows the approach in Korinek and Simsek (2016). Wages are determined via Nash bargaining. The canonical model, augmented with cyclical opportunity cost, obtains when final good prices are flexible and monopolistic distortions are corrected.

Productivity changes impact wholesalers technology and induce changes in the aggregate demand for final goods. Aggregate demand externalities spill-over to hiring decisions through changes in the relative price of intermediate output (real marginal cost of retailers). The relative price channel induces a substantial adjustment of hirings in response to productivity changes, despite the fact that the significant adjustment of wages, due to the cyclicality of the opportunity cost, absorbs part of the response.

To generate big responses of unemployment to movements in productivity, matching models require a high elasticity of market tightness with respect to productivity. The channel through which economic forces generate that high elasticity is the FS. It is an upper bound on what the ‘invisible hand’ can allocate to vacancy creation. If productivity changes translate into large percentage changes in the FS, then a given percentage change in productivity translates into a much larger percentage change in resources used for vacancy creation and hence big responses of unemployment to movements in productivity.

In the current framework, the FS is equal to the relative price of intermediate output times productivity (value of an additional worker for wholesalers) less the opportunity cost of employment. Under the flexible prices benchmark, relative prices are constant and the response of the FS to productivity is muted because of the procyclicality of the opportunity cost. In contrast, under demand-determined output, relative prices are procyclical, amplifying the response of the FS to productivity changes, even with procyclical opportunity cost.

I compare the elasticity of market tightness to productivity changes between two economies (regimes) in steady state: the ‘benchmark’ regime of flexible prices and the ‘demand-determined output’ regime. The elasticity is decomposed in two terms: the first is determined by common calibrations and has limited influence on the elasticity and the second is the elasticity of the FS to productivity. I conduct a calibration exercise where I evaluate the elasticity numerically for each regime.

The relative price of intermediate output can be associated to the markup of the price over marginal cost of retailers. The latter is equal to the inverse of the relative price of intermediate output, so that procyclicality of relative prices is equivalent to countercyclicality of the retailer’s price markup.

My paper is not the first to study the effects of countercyclical price markups on labour market outcomes. Rotemberg (2008) introduces large, imperfectly competitive firms into an otherwise standard search model of unemployment. Countercyclical markups have a big impact on employment by weakening the worker’s bargaining power and making wages less procyclical. Changes in market power are a source of fluctuations and, importantly, the opportunity cost of employment is constant. I depart from these assumptions.

Another paper investigating the importance of price markups in explaining business cycle fluctuations is Bils et al. (2018). This paper decomposes the ‘labour wedge’ into product market and labour market distortions. Using US data, they measure that price markup movements are at least as cyclical as wage markup movements. They conclude that countercyclical price markups should play a central role in explaining fluctuations alongside labour market frictions. The predictions of my model are consistent with their results. Specifically, the cyclical behaviour of the FS is controlled by two forces: countercyclical markups amplify the impact of shocks on the FS, while procyclical opportunity cost dampens that impact. In the absence of wage rigidities, the dampening force can be strong, so it is not immediate that an equilibrium model can generate big responses of the FS to productivity. My contribution is to build an equilibrium model consistent with such big responses.

Following Shimer’s contribution, the literature has advanced three main approaches: (1) specify a low value of forming a match, (2) introduce real wage rigidities, and (3) incorporate labour market institutions.

Ljungqvist and Sargent (2017) review approaches (1)–(3) and identify a common channel through which economic forces that generate high elasticity of market tightness to productivity must operate. This channel, as we already noted, is the FS. In the class of model they consider, productivity changes translate into large percentage changes in the FS when the FS is small.¹ Similarly, Mortensen and Nagypal (2007) combine several forces in a way that is consistent with the argument in Ljungqvist and Sargent (2017).

The first approach feature models where primitive parameters are calibrated to yield low FS. Hagedorn and Manovskii (2008) propose an alternative calibration of the canonical model that places a high value on the (constant) opportunity cost of employment.²Hall and Milgrom (2008) replace Nash bargaining with alternative-offer-bargaining. The opportunity cost of a match is equal to the value derived by Hagedorn and Manovskii (2008).

The second approach confronting the unemployment volatility puzzle features models with real wage rigidities. Hall (2005) introduces sticky wages into the standard matching model in a way consistent with low FS; Gertler and Trigari (2009) adopt a staggered version of Nash bargaining with infrequent wage changes; Blanchard and Galí (2010) specify a real wage function that partially responds to productivity changes; and Michaillat (2012) specify a real wage function that responds as much to productivity changes as wages of new hires in the data.

The third approach enriches the structure of the matching models with a more detailed description of the labour market, including training costs, a welfare state, and various labour market policy instruments. Mortensen and Nagypal (2007) and Pissarides (2009) show that the inclusion of training costs allows the model to match the cyclical movements in labour market activity; Ljungqvist and Sargent (2017) show that generous welfare states or layoff taxes amplify the impact of shocks by lowering the FS; and Zanetti (2011a) shows that the inclusion of labour instruments, like unemployment benefits, firing costs, and income taxes, into the canonical search model of unemployment, amplify the impact of shocks on labour market variables.³

Finally, there exists a literature that incorporates labour market frictions into the New Keynesian monetary framework (see a review in Blanchard and Galí, 2010). The specification of monetary policy has nontrivial implications for the impact of shocks on aggregate outcomes. In the current paper, I abstract entirely from policy issues.

Section 2 presents the environment; Section 3 presents the main argument; Section 4 presents a numerical example; Section 5 offers various extensions; and Section 6 concludes. Derivations are delegated into the Online Appendix.

2. Environment

2.1 Agents and markets

Time is discrete and infinite (⁠$t=0,1,…$⁠). The economy is populated by a large household with a continuum of members of unit measure,⁴ and a two-sector firm structure with a continuum of wholesale firms, $j∈[0,1]$⁠, and a continuum of retail firms, $i∈[0,1]$⁠. Market participants trade differentiated consumption goods, $i∈[0,1]$⁠, an intermediate good, labour, and money. Money is the numeraire. Household members search for jobs in frictional labour markets. The wholesale sector is competitive, with firms opening costly vacancies to hire workers in order to produce intermediate output; while the retail sector is monopolistically competitive, with firms buying intermediate output to transform them into differentiated (final) consumption goods.

The consumption index is given by $c≡(∫01c(i)1−1ϵdi)ϵϵ−1$⁠, with $ϵ>1$ (Dixit–Stiglitz aggregator), while the price index and the demand of each differentiated good, resulting from minimizing consumption expenditure, are given by $p≡(∫01p(i)1−ϵdi)11−ϵ$ and $c(i)=(p(i)/p)−ϵc$⁠, respectively, with p(i) denoting the price of each differentiated good.

Here, $Δt+1,t≡(ct+1/ct)−ζ$ and z denotes the opportunity cost of employment in terms of consumption, $zt=ctζψ′(nt)$⁠. Condition (2) states that along an optimal path, the utility cost of holding $1.00 at t must be equal to the utility gain from holding that dollar for a period and converting it back into consumption at t + 1. Condition (3) states that along an optimal path, the marginal value to the household of an additional worker at t in terms of consumption, $StH$⁠, is equal to the flow value plus the present discounted marginal value at t + 1. The flow value consists of the flow gain from wages and a flow loss associated with moving an individual from unemployment to employment.

All wholesalers are identical and produce the intermediate good according to the technology $yW=an,$ where a is productivity. They post vacancies to hire new employees and lose employees at the rate s. The cost of each additional vacancy is equal to $pγ$⁠, with $γ>0$ denoting recruitment costs. Since job postings are homogeneous to final goods, each wholesaler j buys final goods v(j, i) from each i retail firm so as to minimize expenditure, subject to $v(j)≡(∫01v(j,i)1−1ϵdi)ϵϵ−1$⁠, with v(j) denoting the number of vacancies posted by wholesaler j. The demand by wholesalers for final goods produced by retail firm i is $v(j,i)=(p(i)/p)−ϵv(j)$⁠. Vacancies are filled at the rate $q(θ)∈(0,1)$⁠, which denotes the vacancy-filling rate as function of market tightness, so that new hirings of each wholesaler j are $h(j)=q(θ)v(j)$⁠. In the sequel, I will work with hirings rather than vacancies. The number of workers available for production in wholesale firm j are $nt(j)=(1−s)nt−1(j)+ht(j).$ At the aggregate level, workers available for production at t equal $nt=(1−s)nt−1+ht,$ with $nt=∫01nt(j)dj$ and $h=∫01h(j)dj$⁠.

Condition (5) states that along an optimal path, the marginal cost of hiring an extra employee is equal to the flow value plus the present discounted value of the marginal cost at t + 1. The flow value consists of the marginal benefit of an additional worker, $(pW/p)a,$ and the flow loss associated with the wage cost of an additional worker. I assume free entry in the vacancy creation process so that the marginal value to the wholesale firm of an additional employee, S^W, is equal to the marginal cost, that is, $StW=γ/q(θt)$⁠. Any current worker can be immediately replaced with someone who is unemployed by paying the hiring cost.

Retail firms are monopolistically competitive and transform one-to-one the intermediate good into differentiated (final) goods, $yt(i)=xt(i)$⁠, where x(i) is the quantity of the intermediate good demanded by retailer i. I distinguish between two regimes with respect to retailer decisions. In the first regime, retail firms can set prices freely (flexible prices) and this regime will serve as a benchmark; while in the second regime, I assume that retailers have preset nominal prices that are equal to each other and that never change, $pt(i)=p$ for all t. This implies that the final good price is also constant, p_t = p for all t.

Each retailer is constrained by an isoelastic demand for its good i, $y(i)=(p(i)/p)−ϵy˜$⁠, where $y˜$ denotes the aggregate demand for final goods. Here, $τ(i)$ captures linear subsidies to retailer i, which are financed by lump-sum taxes $Tt=(ptW/pt)∫01τ(i)yt(i)di$⁠, levied on households. I introduce subsidies in order to correct the distortions that arise from monopolistic competition. This serves the purpose that the benchmark regime corresponds to the canonical search model of unemployment, augmented with cyclical opportunity cost. I set $τ(i)=1/ϵ$ for all $i∈[0,1]$⁠, so that the optimality conditions of Equation (6) yields $p(i)/p=pW/p$ (zero price markups).

Matches are produced by a constant return to scale matching technology, $h(u,v),$ where u are job searchers and $v=∫01v(j)dj$ are aggregate vacancies. The matching function is differentiable and strictly increasing in both arguments and satisfy $h(u,v)≤min(u,v)$⁠. Conditions in the labour market are summarized by the labour market tightness $θ≡v/u$⁠. An unemployed household member finds a job with probability $f(θ)≡h(u,v)/u$ and a vacancy is filled with probability $q(θ)≡h(u,v)/v.$ A useful property is $f(θ)=θq(θ).$

2.2 Steady-state equilibria

A benchmark equilibrium is a path of prices, ${pt(i),pt,ptW,wt}$⁠, allocations, ${ct,ct(i),mt,nt,ytW(j),ht(j),vt(j,i),vt(j),nt(j),yt(i),θt,ut,y˜t}$⁠, such that (i) households minimize consumption expenditure and solve Equation (1), (ii) wholesalers minimize vacancies expenditure and solve Equation (4), (iii) retailers solve Equation (6), (iv) Nash solution is satisfied, and (v) all markets clear. An equilibrium under the demand-determined regime is similarly defined but final goods prices are preset for all t, which implies that retailers solve Equation (7) instead of Equation (6). I restrict attention to symmetric non-stochastic steady-state equilibria, thus all labels i, j are dropped. (See Online Appendix A for detailed derivations of the equilibrium under each regime.)

Expression (9) is the key equilibrium relation to determine market tightness θ under the benchmark regime. Combining symmetry with retailer’s optimality yields $pW/p=1$⁠. The LHS denotes the FS, a−z, which is equal to productivity less the worker’s opportunity cost of employment in terms of consumption. The opportunity cost is a function of aggregate household demand c and aggregate employment n, recall that $z=cζ·ψ′(n)$⁠. To complete the characterization under the benchmark equilibrium, we express z as function of tightness, substitute it into Equation (9) and solve for equilibrium tightness.

If firms post more vacancies, market tightness increases, which raises the job-finding rate $f(θ)$ and increases employment n. In turn, aggregate hirings are equal to outflows from unemployment (equal to inflows in steady state), that is, $h(θ)=s·n(θ)$⁠.

The RHS denotes output net of hiring costs and the LHS household consumption demand.

Taken together, Equations (10) and (11) allow me to express the opportunity cost as function of productivity and tightness, $z(a,θ)=(c(a,θ))ζ·ψ′(n(θ))$⁠, and substituting it into Equation (9), I solve for equilibrium market tightness. The price level p adjusts to satisfy Equation (11) residually. Observe that household demand at equilibrium affects the FS through its impact on the opportunity cost z; however, aggregate demand externalities are muted since relative prices are acyclical $pW/p=1$⁠. This is because demand c accommodates any supply expressed by sellers through adjustments of the price level, placing no additional restriction on the equilibrium set, which imply that market tightness is determined independently from factors affecting household demand $c(m¯/p)$⁠.

It is useful for the subsequent analysis to provide a graphical interpretation of Equation (11). Figure 1 plots net output (equal to household consumption in equilibrium) as a function of tightness. Assumptions about primitive parameters (see Online Appendix A.1) ensure that it starts from the origin, reaches a maximum at $θ=θ*$ (⁠$∂c/∂θ=0$⁠), and cuts the x-axis at $θ=θ^$⁠. Equilibrium under the benchmark regime does not place a priori any restrictions as to whether equilibrium tightness should be below, equal, or above $θ*$⁠; however, equilibrium under the demand-determined regime does. This is what we turn to next.

Fig. 1

Expression (11).

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I introduce aggregate demand externalities by assuming that prices p are preset. Expressions (9)–(11) are still relevant to describe equilibrium, but the logic of the previous argument does not apply. At preset prices, household demand is pinned down by real balances which, focusing on Equation (11), implies that equilibrium market tightness cannot be determined independently from demand conditions. In other words, Equation (11) imposes additional non-trivial restrictions to the equilibrium set. Simultaneously, relative prices $pW/p$ are a function of market tightness in order for Equation (9) to be satisfied. The closed loop of interactions is initiated with the impact of consumption on tightness, which spillover to hiring decisions via its impact on relative prices which, in turn, affect aggregate demand via hiring costs and so on.

Cyclicality of relative prices translates to cyclicality of price markups. Denote the retailer’s price markup by $M.$ It is defined as the inverse of the real marginal cost, $M≡1/[(1−ϵ−1)(pW/p)]$⁠. In turn, the FS is equal to $FS≡[(1−ϵ−1)M]−1a−z$⁠. Cyclicality of markups have a direct impact on the FS.

Construction of a demand-determined equilibrium is explained with the help of Fig. 1. Suppose an equilibrium under the benchmark case is somewhere in the graph, say at point B, then, I fix the (final good) price close to the equilibrium price under the benchmark case. It follows that one of the solutions in Equation (11) is in the neighbourhood of point B. Simultaneously, relative prices adjust to satisfy Equation (9) and existence require price markups to be positive, $M>1$⁠. A demand-determined equilibrium in the neighbourhood of the benchmark implies that frictions due to aggregate demand externalities are small (or even negligible). This is an appealing property given that demand externality spillovers persist in the long-run and the analysis has abstracted from policy.

However, at given preset prices, satisfaction of Equation (11) is consistent with two equilibria (see Fig. 1). All demand-determined equilibria to the left of the peak (⁠$0<θ<θ*$⁠) imply that improvements in productivity, a, generate a contraction in the labour market⁸ and depress the opportunity cost, z (countercyclicality). While equilibria to the right (⁠$θ*<θ<θ^$⁠) are consistent with procyclical opportunity cost. (This is shown in Section 3 where I analyse comparative statics.) To that end, I restrict the analysis to demand-determined equilibria in the neighbourhood of the benchmark (small frictions) and to the right of the peak in Fig. 1.

2.3 Discussion of the assumptions

I discuss three assumptions of the model: money in the utility function, separation between final and intermediate good production, and preset prices.

The presence of money in the utility function is a convenient short-cut to obtain an interesting concept of aggregate demand in a model without savings; otherwise, households would spend mechanically all their income on the produced good (Say’s law). To see this, let me abstract from money in the utility. Then, consumption adjust to satisfy budget balance and Equation (3) is the only relevant optimality condition. In turn, this implies that Equation (11) does not impose any restrictions to the equilibrium set since c adjusts mechanically to satisfy any supply expressed by producers. We uncover the property of the benchmark equilibrium that market tightness is determined independently from demand. To avoid this problem, households must have the choice between consumption, labour, and something else. The assumption of money in the utility function has been employed, among many others, by Blanchard and Kiyotaki (1987) and more recently by Michaillat and Saez (2015), in order to avoid Say’s Law.

The separation between retailers with monopoly power and perfectly competitive intermediate good producers is done to avoid interactions between price setting and wage bargaining at the firm level. This assumption has been employed by Ravenna and Walsh (2008) and Blanchard and Galí (2010), among others.

The modelling of retailer decisions, namely, Equations (6) and (7), is borrowed from Korinek and Simsek (2016).⁹ The assumption of preset prices is made for analytical tractability, which allows a clean characterization of comparative statics. To check the robustness of the results, I develop a version of the model in which prices are partially sticky (see Section 5 for details). Specifically, a fraction of retailers reoptimize their price level every period. These firms solve problem (6). The remaining fraction has preset prices in the sense that they do not reoptimize their price level every period. These firms solve problem (7). Also, I extend the previous version to include constant inflation and price indexation to long-run inflation. After reviewing the microeconomic evidence on price adjustments (see Section 5 for related literature), I perform various quantitative exercises to show the robustness of my results.

3. Elasticities

This section characterizes the elasticity of market tightness with respect to productivity changes, denoted by $εθ,a$⁠. In steady state, the unemployment rate is $u=1−(1−s)·n(θ)$⁠, where the elasticity of unemployment to productivity changes is given by $εu,a=−(1−u)·εf,θ·εθ,a.$ High values of $εθ,a$ produce big responses of unemployment to productivity changes. I compute the elasticity of market tightness to productivity changes for each regime in steady state, namely, the benchmark regime of flexible prices and the demand-determined output regime. 

• (i) (Benchmark)

  $εθ,a=ωf(θ)+1−β(1−s)β(1−s)ωf(θ)+1−β(1−s)β(1−s)εq,θ·a−zεz,aa−z;$
• (ii) (Demand-determined output)

  $εθ,a=ωf(θ)+1−β(1−s)β(1−s)ωf(θ)+1−β(1−s)β(1−s)εq,θ·{(pW/p)a−zεz,a(pW/p)a−z+(pW/p)a(pW/p)a−zεpW/p,a}.$

In the remainder of this section, I demonstrate the proof of Proposition 1.

The term $ϒ$ has limited influence on the elasticity (⁠$ϒ≈1.15$ in our calibration). If $εz,a=0$⁠, which obtains with linear utility (ζ = 0) and infinite elastic labour supply (⁠$εψ′,n=0$⁠), Equations (13) and (14) coincide with the formulas of the canonical model. In that case, low values of a−z generate large values of the elasticity (Hagedorn and Manovskii, 2008; Ljungqvist and Sargent, 2017). Instead, augmenting the canonical model with procyclical opportunity cost (⁠$εz,a≈1$ in our calibration) weakens substantially the impact of $εFS,a$ on the elasticity and the amplification mechanism. This is the Chodorow-Reich and Karabarbounis critique.

The elasticity of the FS to productivity is affected by two opposing forces. Procyclical relative prices (⁠$εpW/p,a>0$⁠) (or countercyclical markups) amplify the impact of productivity shocks on the FS by generating high values in Equation (19), while procyclical opportunity cost (⁠$εz,a>0$⁠) dampens that impact. The latter elasticity is determined only by cyclical fluctuations in the disutility cost of employment, given that household consumption is pinned down by real money balances.

Observe that Equations (18) and (19) link the elasticity of market tightness to the elasticity of relative prices. To complete the characterization, I differentiate Equation (17) with respect to productivity to determine the elasticity of tightness and then, I substitute it back to Equation (18) to determine the elasticity of relative prices.

The first term is bounded above by a and has limited influence on the elasticity. The second term is the inverse of the slope of net output and becomes unboundedly large as the slope goes to zero in the neighbourhood of $θ*$ (Fig. 1). If net output is increasing in tightness (⁠$θ<θ*$⁠), then the second term is positive and productivity increases induce a contraction in the labour market, and the opportunity cost is countercyclical (⁠$εz,a<0$⁠); I do not focus on this type of equilibrium. Instead, I restrict the analysis to equilibria in a neighbourhood of $θ*$⁠, where net output is decreasing in tightness (⁠$θ>θ*$⁠), the elasticity $εθ,a$ attains large values, and the opportunity cost is procyclical.

It follows that large values of $εθ,a$ (and of $εz,a$⁠) induce strongly procyclical relative prices.

An intuitive explanation goes as follows. A small fall in productivity induces a large fall in hiring costs and aggregate demand for final goods $y˜$⁠. This translates into large falls in the intermediate inputs demand, which require large falls (increases) in relative prices (price markups) to clear the intermediate good’s market. Aggregate demand externalities spill-over to hiring decisions via lower relative prices. In turn, the marginal value of an additional worker falls more than the initial fall in productivity. Wages fall, reflecting both the lower opportunity cost and the lower value of an additional worker for the firm, but cannot absorb the fall in the marginal value of an additional worker. Thus, hirings adjust substantially. Compare with the mechanism under the benchmark case. Namely, reductions in the marginal value of an additional worker are absorbed almost entirely by wages, so that amplification of shocks is weak. 

4. Numerical example

I compute a numerical example to illustrate the mechanism discussed in the previous section. As a benchmark against which to evaluate the results, I use the estimate of Chodorow-Reich and Karabarbounis (2016) for the elasticity of unemployment to productivity in the data, $εu,a≈−9.5$⁠.

I assume a Cobb–Douglas matching function $h(u,v)=κuαv1−α$⁠, with $κ>0, α∈(0,1)$⁠, and $ψ(n)=χn1+ξ/(1+ξ),$ with $ξ>0$ the inverse Frisch elasticity of labour supply and $χ>0$⁠.

I set $δ=ζ=1$ (log-utility in consumption and real balances) and normalize $m¯=a=1$⁠. Furthermore, following Chodorow-Reich and Karabarbounis (2016), I set externally β = 0.99, s = 0.045, α = 0.6;¹⁰ and using common calibrations in the business cycle literature, I set ξ = 1 and ϵ = 6. The remaining parameters ${κ,ω,γ,χ}$ are used to hit four target.

Consider the benchmark equilibrium. Following the (monthly) estimates in Chodorow-Reich and Karabarbounis, I hit the following three targets: $f˜=0.704, q˜=0.71$⁠, and $z˜=0.74$⁠,¹¹ which imply $θ˜=0.9915$ and $u˜=0.062$⁠. Consistent with my analysis, I restrict the equilibrium to the right of the peak in Fig. 1. This is achieved by targeting a negative value for the slope of equilibrium net output. A convenient value to target is $εc,θ=−0.013$⁠, which places the benchmark equilibrium close to maximum net output.

The numerical value of parameters used to target the above aggregates is $κ=0.7064, ω=0.16, χ=0.82$⁠, and $γ=0.95$⁠.¹² The numerical value of the endogenous variables are $c˜=0.9223, hiring costs:=γh˜/q˜=0.059, p˜=0.010842$⁠, and $n(θ˜)=0.981431$⁠.

The elasticity of tightness to productivity is decomposed in two terms: $ϒ$ multiplied by the elasticity of FS to productivity. It follows that $ϒ=1.15, εz,a=1.07$⁠, which imply $εθ,a=0.9175.$ The elasticity of unemployment to productivity is $εu,a=−(1−u˜)(1−α)·εθ,a=−0.345.$ The upshot is that the elasticity of unemployment to productivity under the benchmark case is almost 30 times below its empirical counterpart.

Notice that the elasticity of the opportunity cost (15) is a function of $εc,θ$⁠. Performing robustness checks, it turns out that small values of $εc,θ$ have a negligible impact on the amplification mechanism under the benchmark case.

Next, consider the demand-determined equilibrium. The parameters in Ξ are equal to the values assigned above. The equilibrium is indexed by (preset) prices p and existence requires positive markups, $M>1$⁠. The only parameter to calibrate here is p. Assigning a numerical value should satisfy two goals. First, the price is calibrated close to $p˜$⁠, so that allocations and relative prices under the demand-determined regime are close to the benchmark equilibrium values (small frictions arising from demand externalities). Second, the calibrated parameter should be consistent with price markups in the 10–20% range.¹³ Attaining both goals is made possible because of the assumption that monopoly subsidies are set as in the benchmark regime, so that $M$ is a function of relative prices and ϵ. I set the price at $p=0.0108425$ (⁠$≈p˜$⁠) consistent with $M=1.19854$ (slightly below 20% markups).

The elasticity of tightness to productivity is computed from Equation (21) and is equal to $εθ,a=79$⁠.¹⁴ The opportunity cost is strongly procyclical with the relevant elasticity equal to $εz,a=1.975$⁠. Combining these results, we obtain $εpW/p,a=24$⁠. Finally, the elasticity of unemployment to productivity is $εu,a=−29$⁠, which is three times higher, in absolute terms, than its empirical counterpart. Evidently, aggregate demand spillovers via relative prices produce a strong amplification mechanism.

It is important to note that the elasticity of tightness computed from Equation (21) is inversely related to the slope of net output. Low values of the latter induce very strong amplification, while higher values moderate the responses of the model, at the expense of higher matching costs as percentage of GDP. The targeted value of $εc,θ=−0.013$ is a reasonable compromise.

5. Extensions

5.1 Intensive margin

I extend the model to include variation in hours per worker (intensive margin). The disutility cost of the large family is $χnt(et1+ϕ/(1+ϕ)),$ where $χ,ϕ>0$⁠, n denotes variations along the extensive margin and e denotes variations of hours per worker. Linearity with respect to n implies that variations along the extensive margin do not affect the opportunity cost. This assumption allows me to isolate the impact of variation along the intensive margin on the opportunity cost. I continue assuming Nash bargaining contracting, but now workers and firms bargain over wages and hours. (All derivations and proofs are in Online Appendix B.1.) 

5.2 Money growth

I extend the model to include constant, nonzero, growth rate of money. Let $mt=μ·mt−1$⁠, where $μ>0$ is the gross growth rate and $m−1=m¯>0$ is the initial condition. The government budget is given by $mt−mt−1=Ht$⁠, where H are nominal transfers that accrue to the household budget constraint. Stationarity requires $mt/pt=mt−1/pt−1=⋯=m0/p0=(μ·m−1)/p0$⁠, starting from an initial condition, say $m−1$⁠; constant inflation (or deflation) $Πt=pt/pt−1=μ$⁠; real transfers pinned down by real balances $Ht/pt=(mt/pt)(μ−1)/μ$⁠; and relative prices and real variables are constant. Note that nominal preset prices p_t are indexed by calendar time; however, what is important for the analysis is that real variables and relative prices are constant in the steady state.

The analysis in the baseline model assumed μ = 1. The growth rate affects the equilibrium only through its impact on c. However, the analysis in Sections 2.2 and 3 did not depend on the specific functional form of household’s consumption demand; and in the calibration exercise, the initial condition was normalized to unity without loss of generality. All the theoretical results follow intact as long as $μ>β$ and given a suitable numerical value for μ, a re-normalization of initial conditions attains exactly the same calibration.

5.3 Partially sticky prices

The baseline model features an extreme form of nominal price stickiness. I develop a version of the model in which prices are partially flexible and conduct various quantitative exercises. (Derivations of formulas and details pertaining to the quantitative exercise can be found in Online Appendix B.2.)

The equilibrium reduces to four unknowns, ${psticky/p,pflex/p,pW/p,θ}$⁠, in four equations, (9) and (22)–(24). Existence of a demand-determined equilibrium with partial stickiness requires positive markups, $M>1$⁠.