The Government Wage Bill and Private Activity

We estimate the macroeconomic effects of public wage expenditures in U.S. data by identifying shocks to public employment and public wages using sign restrictions. We find that public employment shocks are mildly expansionary at the federal level and strongly expansionary at the state and local level by crowding in private consumption and increasing labor force participation and private sector employment. Similarly, state and local government wage shocks lead to increases in consumption and output, while shocks to federal government wages induce significant contractionary effects. In a stylized DSGE model we show that the degree of complementarity between public and private goods in the consumption bundle is key for explaining the observed heterogeneity.


Introduction
The last …nancial crisis and the subsequent Great Recession still take their toll on many advanced economies. They have posed a serious threat on output and the labor markets, leading to an unusually slow recovery. This fact has revived the debate on the e¤ectiveness of discretionary …scal policy as a tool to stimulate private activity, establish sustainable growth and recover lost jobs. Another relevant question that naturally arises in this context is which …scal instruments are the most e¤ective for fueling economic activity.
Most of the empirical VAR literature on the macroeconomics of …scal policy does not distinguish between di¤erent types of government spending, and treats total government spending as a single …scal instrument. Needless to say, not all types of government spending are expected to induce the same e¤ects on the macroeconomy. Furthermore, most of the literature interprets the empirical e¤ects of this total government spending instrument as if they were the result of changes in government consumption of goods and services. However, government spending is not only consumption of goods and services. Wage and salary payments account for a large share of public expenditure in the U.S. During the postwar period, government wage and salary expenditure has accounted for about 50% of government expenditure (See Figure 1(b)). In the aftermath of the Great Recession, concern about the government budget has focused greater attention on the costs that the government incurs to compensate its employees.
Given the weight of wage expenditures in total government spending, the purpose of this paper is to estimate the e¤ects of public wage bill policies on output and the labor market of the private sector, and draw policy implications that could be useful in the aftermath of the crisis.
Using U.S. data over the period 1979-2007, we identify exogenous shocks to public employment and public wages. Following Mountford and Uhlig (2009), we adopt an agnostic identi…cation that sets a minimum set of sign restrictions to the …scal shocks identi…ed. In particular, we identify shocks to government employment that raise government employment for four quarters and shocks to government wages that induce an increase in the average hourly wage rate for four quarters after the shock. We also ensure that the identi…ed shocks to the government wage bill are orthogonal to business cycle, monetary policy, and government revenue shocks.
In a spirit similar to Ramey (2012), we ask whether the two shocks di¤er in their ability to stimulate private activity, raising employment and lowering unemployment. Our …ndings indicate that the e¤ects of government employment shocks are clearly expansionary at the state and local (S&L henceforth) level, and only mildly expansionary at the federal level, while shocks to the government wage rate can be contractionary at the federal level and expansionary at the (ii) disaggregating the e¤ects by government level; (iii) examining the e¤ects on the labor force participation and unemployment rates.
In order to explain the empirical …ndings, we develop a Dynamic Stochastic General Equilibrium (DSGE) model with sticky prices augmented with public good production, allowing for both productive and utility-enhancing services for the public good, search and matching frictions, and endogenous labor force participation Our theoretical model matches qualitatively the empirical evidence for all shocks considered in the empirical exercise. More speci…cally, public employment shocks are expansionary by crowding in private consumption and increasing labor force participation and employment in the private sector. In the standard neoclassical growth model, increases in public employment should reduce private consumption and private employment as the additional labor supply spurred by the …scal shock's negative income e¤ect is entirely absorbed by the public sector (see Finn (1998)). We show that the complementarity of the public good with private consumption in the aggregate consumption bundle of the household can overturn the negative wealth e¤ect of the shock and lead to an increase in private consumption. Our …ndings con…rm, in a di¤erent framework, the results of Linnemann (2009) who shows that if public services are complementary to private consumption goods in the household's utility function, an increase in public employment raises private consumption and private sector employment. 1 Also in a similar framework to ours, Forni et al. (2009) demonstrate that shocks to public employment can lead to increases in private consumption A C C E P T E D M A N U S C R I P T in a model with rule of thumb consumers. 2 In their model, there is also a positive covariation between public and private employment, since the additional consumption demand makes private sector …rms, which are demand constrained, expand their labor input to meet the increase in aggregate demand. Here, we study the e¤ects of both public employment and public wage shocks and provide an alternative mechanism which rests on the complementarity channel rather than liquidity constrained households, and it is able to explain the transmission of both types of shocks.
Our model also explains how government wage shocks can be contractionary or expansionary at the di¤erent government levels. More speci…cally, wage shocks lead to public-private wage spillovers, inducing a negative labor demand e¤ect, a sharp fall in private employment, and an increase in unemployment. At the same time, they can lead to a crowd in of private consumption given the complementarity of the latter with the public good in the aggregate consumption bundle of the household. These two opposite channels can help explain the empirical results. For su¢ciently high degree of complementarity between the public good and private consumption in the aggregate consumption bundle, our model predicts positive e¤ects of government wage shocks on private activity, as found for S&L government wages in the data.
On the other hand, when the complementarity channel is weaker, the wage spillover e¤ect in the private sector dominates, leading to a substantial fall in private employment and a short-run contraction in private activity. To examine the sensitivity of our conclusions, we perform a sort of prior predictive analysis (see e.g., Leeper et al. (2015)). The analysis reports the probability distribution of impulse responses that the model can produce. The model with low degree of complementarity between public and private goods can generate negative output responses for government wage shocks and small multipliers for government employment shocks, while for strong complementarity between the private and the public goods the model can generate positive responses for wage shocks and higher multipliers for employment shocks. We conclude that the degree of complementarity between private and public goods is key for explaining the observed heterogeneity of responses in the di¤erent government levels.
Our analysis therefore suggests that the public good provided at the federal level may exhibit a di¤erent degree of complementarity with private consumption than that at S&L level. 2 The response of private consumption following total government spending shocks has received much attention in the literature. Deep habits or rule-of-thumb consumers have been shown to generate consumption crowding in (e.g., Ravn et al., 2006 andGali et al., 2007), whereas another class of models includes government investment as part of the production function (Leeper et al., 2010, Drautzburg andUhlig, 2015). Monacelli et al. (2010) show that a combination of consumption-leisure complementarity in household's preferences and New Keynesian features can generate consumption crowding in a model with search and matching frictions.

A C C E P T E D M A N U S C R I P T
This might be justi…ed by the di¤erent nature of the public good provided in each case. For instance, federal government employees largely comprise military and defense employees, while S&L government employees provide mainly education, health care and transportation services.
Research by Fiorito and Kollintzas (2004) using European data has indeed shown that the degree of complementarity between government and private consumption is not homogeneous over types of public expenditures. In particular, 'merit' goods, including health and education, complement private consumption, while 'public' goods, referring to defense, public order and justice, are substitutes with private consumption. This idea is in line with recent work by Perotti (2014) who shows that defense spending shocks in a SVAR generate contractionary responses, while civilian government spending shocks generate large expansionary responses. The theoretical explanation provided in that paper is based on the assumption that civilian spending exhibits Edgeworth complementarity with private consumption, while defense spending is not utility enhancing. In a similar vein, Pieroni and Lorusso (2015) present VAR estimates for the U.S. economy showing that civilian expenditure induces a positive response of private consumption, whereas military spending has a negative impact. Our results square well also with the evidence presented in Bouakez and Rebei (2007) who, using a maximum-likelihood estimation with U.S. data, …nd a strong Edgeworth complementarity between the two types of consumption goods. Also, Fève et al. (2013) show that government spending multipliers obtained in the literature may be downward biased because the standard approach does not allow for complementarities between private consumption and government spending in the utility function.
Our work has a number of useful policy implications in the aftermath of the crisis and the slow recovery in advanced countries. In particular, increases in public employment can stimulate the private sector's employment, encourage labor force participation and private demand. On the other hand, public wage policies could be expansionary only if the increases in wages are associated with the production of those public goods that strongly complement private consumption. Wage increases should target, for instance, employees that work in public education or the public health system. The outline of the paper is as follows. The next section describes the data on the U.S. government wage bill and public employment, the estimated VAR model and empirical …ndings. Section 3 presents our theoretical model which matches qualitatively the empirical evidence.

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A C C E P T E D M A N U S C R I P T 2 Empirical analysis

Data
As shown in Figure 1(a), since the 1970s public wage expenditures have accounted for around 50% on average of government expenditures in the U.S. and around 5% of GDP. Although the literature has looked extensively at the macroeconomic e¤ects of certain components of U.S. government spending, such as public investment, research on the e¤ects of the public wage bill has been surprisingly limited, despite the fact that it represents the largest component of spending, as shown in Figure 1(b). Looking at a decomposition of public wage expenditures by government level, we see a shift over time towards states and localities, with the federal share amounting to between 20% and 30% from the 2000s (Figure 1(c)). In 1980, federal civilian employees made up 2.3% of the workforce, while they accounted for 1.7% of the workforce in 2010 ( (Falk (2012)). According to the same authors, for the past 30 years, the number of civilians employed by the federal government has ‡uctuated around 2 million people. Besides federal civilian workers, the armed services include steadily more than 2 million uniformed personnel.
In order to take a view of the variation in the government wage bill, in Figure 2 we plot the quarterly growth changes in the two basic components of the wage bill: government employment and the average real hourly wage in the public sector. Government employment is de…ned as the number of government employees per capita, including both civilian and military employees. Data on civilian employees comes from the Bureau of Labor Statistics, while the military employees series is constructed by Ramey (2011). The average real hourly wage is constructed using the NBER extracts of the CPS Merged Outgoing Rotation Groups. We construct quarterly series for hourly wages at the federal and the S&L government level, and also for the private sector, by regressing for the repeated (monthly) cross section the log of hourly wage -separately for each of the two categories -on socio-demographic variables (age group, gender, race, and marital status) and dummies for each state in order to adjust for worker quality and composition of the workforce. Monthly residuals are transformed to quarterly observations using the mid-quarter residuals estimates and are de-seasoned and de ‡ated using the GDP de ‡ator series. We exclude from the sample workers that have never been employed full time and self-employed individuals. 3

A C C E P T E D M A N U S C R I P T
As can be readily seen, the various …scal episodes (i.e. unusually large changes in the spending components) are not correlated. Those …scal episodes can be related to several policy episodes in history. In particular, the data point to a signi…cant increase in federal employment in 1990, when President Bush increased government employment for defense in the face of the German reuni…cation, and to a fall in public employment in 1980, after Reagan won the presidential election and cut the Comprehensive Employment and Training Act of 1974.

The VAR model
In this subsection, we formalize the econometric framework in order to estimate the short-run e¤ects of public employment and wage shocks on private activity. We consider a VAR model of eleven endogenous variables. We …rst include the four main items of government spending: the log of government employment per capita, the log of average real hourly public wage, the log of real per capita government expenditure in goods purchases, de…ned as government consumption minus compensation of government employees, and the log of real per capita gross …xed investment. The second set of seven variables included in the VAR are: the log of real per capita net (of transfers) tax revenue, the log of real per capita private GDP, private and we were not able to harmonize the data. Also, we would have preferred to de ‡ate using state-level data on GSP de ‡ators, but we were constrained to use aggregate GDP de ‡ator series due to data availability. 4

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A C C E P T E D M A N U S C R I P T consumption and private investment, the in ‡ation rate, a measure of short-term interest rate and a labor market variable. The latter alternates between (i) the log of private employment per capita, (ii) the real private wage rate, (iii) the unemployment rate, and (iv) the labor force participation rate. Finally, we include in the VAR an exogenous war dummy variable with several lags to control for strong anticipation e¤ects (see Ramey (2011)). Following Uhlig (1994) and Mountford and Uhlig (2009), we do not include any constant or time trend. 5 The type and number of variables included in the VAR is mainly dictated by the identi…cation scheme we use in order to identify government employment and wage shocks, as described in the next subsection. The fact that we seek to uncover the e¤ects of …scal shocks on the private economy is another reason that orientated us towards considering private sector's measures of most variables. The output variable, for instance, refers to an approximation of the value added produced by the private sector, which equals total GDP net of the government wage bill (according to the de…nition of 'Private Sector Production' in Ramey (2012), Figure   1). The exclusion of the government wage bill also allows us to isolate the second-round e¤ects of public wage expenditures on output, net of the direct impact of the public wage bill on GDP.
According to information criteria, we set the lag length of the VAR to two. Following Uhlig (2005), we carry out a Bayesian estimation using ‡at priors on the coe¢cients of the model and the covariance matrix of the shocks (see Appendix B). We use quarterly, seasonally adjusted data for the U.S. from 1979 to 2007. The starting period is mainly dictated by the availability of the public wage series, while the sample stops in 2007 in order for our results not to be biased from extraordinary economic conditions (e.g. interest rates close to the zero lower bound, …nancial crisis etc.). We estimate the e¤ects of spending policies by government level: federal and S&L. Hence, the VAR exercise is repeated twice, using government expenditure series for each government level. The series come from the Bureau of Economic Analysis, the Bureau of Labor Statistics and other sources. A detailed description of the data is provided in Appendix A.

Identifying the shocks
We base the identi…cation of the …scal shocks on the sign restriction approach (Arias et al.  5 As the authors discuss, this is important to obtain more robust results because of the interdependencies in the speci…cation of the prior between these terms and the roots in the autoregressive coe¢cients. However, we have also estimated a VAR including a constant and a quadratic trend. The results remain qualitatively robust and are available in the online Appendix. 8

A C C E P T E D M A N U S C R I P T
The use of sign restrictions avoids, in principle, typical problems associated with the identi…cation of economically meaningful …scal shocks. In particular, problems concerning the scarceness of reasonable zero-identifying restrictions are to a large extent avoided. Hence, we opt for an agnostic identi…cation that sets a minimum set of sign restrictions on the responses to the …scal shocks and, at the same time, controlling for the business cycle or other macroeconomic shocks.
Similarly to Mountford and Uhlig (2009) and Arias et al. (2014), we perform a sequential shock identi…cation. We …rst identify a generic business cycle shock that implies a positive comovement of output, private consumption, private investment and government revenue for four quarters k = 0, 1, 2, 3. According to some methodological advances in the empirical literature (Arias et al. (2015)), we identify a monetary policy shock by imposing two sets of restrictions on the structural representation coe¢cients of the interest rate equation, which can be thought as an approximated Taylor rule equation. In particular, we require that (i) the federal funds rate is the monetary policy instrument and it only reacts contemporaneously to output and prices, and (ii) the contemporaneous reaction of the federal funds rate to output and in ‡ation is nonnegative. 6 We also require the monetary policy shock to be orthogonal to the business cycle shock.
After identifying the business cycle and monetary policy shock, we turn to the identi…cation of the …scal shocks. In particular, we recover sequentially a government revenue shock, a government investment shock, a government (non-wage) consumption shock, a government wage shock, and a government employment shock. All …scal shocks are identi…ed by requiring that (i) they are orthogonal to the business cycle, the monetary policy and the rest …scal shocks, and (ii) the respective …scal instrument increases for quarters k = 0, 1, 2, 3 after the shock. 7 For imposing both zero and sign restrictions we apply the methodology of Arias et al. (2014). This is the most appropriate for our aim for two reasons; …rstly, it o¤ers a faster procedure of identifying shocks using both zero and sign restrictions comparably to other methodologies; secondly, other sign restriction methodologies implicitly rely on conditionally agnostic priors (not trully agnostic priors), something avoided by the present approach (for detailed discussion see Arias et al. (2014)).
The sign restrictions are summarized in Table 1, while the identi…cation algorithm is detailed 6 In a subsequent study (Arias et al. (2016)) the authors discuss the risk of using both sign and zero restrictions to get conditionally agnostic priors. To check whether our analysis draws safe results we have also estimated impulse responses based on an identi…cation of the monetary policy shock without zero restrictions (Uhlig (2005)). The results of this robustness exercise are qualitatively similar and available upon request. 7 We have experimented with di¤erent number of restricted horizons from 1 to 4 periods. Since the implications of the analysis are similar and in order to follow the existing literature (Mountford and Uhlig (2009)), we use an horizon of four quarters as benchmark. Sensitivity analysis results are available upon request. 9 A C C E P T E D M A N U S C R I P T in Appendix C. The estimation of the VAR and the identi…cation of shocks are based on N=500 draws from the posterior distribution of the VAR parameters and M=3000 draws of orthonormal matrices.

The identi…ed shocks to the government wage bill
Following the strategy outlined in the previous section, we identify shock series for government wages and employment at the federal and the S&L level. Figure 3 displays the median and the 68% con…dence bands of the two structural shocks calculated from the VAR residuals at the di¤erent government levels. The values in the vertical axes correspond to standard deviations.
It can be easily seen that the volatility of the shocks at the di¤erent government level is of similar magnitude. Notice that the peaks of the series at each government level do not coincide. For example, we observe a surge in federal government employment in 1990 and 2000, while S&L public employment is, if anything, increasing slightly in the 1990s and falls signi…cantly in 2000.
The fall in government employment around 1983 re ‡ects cuts in the Reagan presidency era. The 1982 …scal budget that was proposed by Reagan represented indeed a reduction of $44 billion, or 5.7%, and all categories of public employment, except national defense, were reduced. Over half of the $44 billion budget reduction came from two areas: income security; and education, training employment and social services that seemed to have a¤ected more severely the S&L governments.
Big wage cuts occur in 1985, 1990 and 2001 at the S&L level, while in 1996 at the federal level. It seems that the cuts in public wages at the S&L level could be associated with the need of local governors to adjust the budget. For example, Governor Wilson of California pressed 21 government employee unions in 1991 to accept a 5% pay cut because the state was especially hard hit by the 1990-91 recession. 8 As long as this reaction does not constitute a systematic response to recessions, we conclude that confronting the identi…ed shock series and real events, our identi…cation strategy is successful in extracting meaningful public employment and wage shocks.
Additionally, we calculate the forecast error variance decomposition of private output and the public wage bill at the di¤erent government levels. The two public wage bill shocks at any government level explain together approximately 20-23% of private output at business cycle 8 See Bureau of National A¤airs (1991).

A C C E P T E D M A N U S C R I P T
frequencies. On the other hand, the public wage bill shocks at the S&L level explain around 24% of public wage bill ‡uctuations, while the federal wage bill shocks explain 29% of public wage bill ‡uctuations. Moreover, public wage and employment shocks share equal roles in inducing business cycle ‡uctuations at any government level. 9

Impulse response functions and multipliers
In Figure 4 we present the responses of output and its components, employment and the real wage in the private sector, as well as the unemployment and labor force participation rates, and also responses of tax revenues and the other spending components to the two …scal shocks under investigation for the two government levels considered. For comparability purposes, employment and wage shocks are scaled to represent a 1% of GDP increase in the government wage bill. 10 Each graph presents median estimates (solid line) and pointwise 68% credible bands (dashed lines).
According to Figure 4, the shock to government employment signi…cantly increases private output at both the federal and the S&L level. The timing of responses seems di¤erent: output increases signi…cantly in the short run for shocks to government employment at the federal level and with a lag at the S&L level. Also, private consumption increases signi…cantly after an increase of government employment at the S&L level, while it does not move after the same shock at the federal level. Similarly, private investment does not react signi…cantly in the short run after a sudden increase in federal employment, while it falls and then increases with a considerable lag after an increase in S&L government employment. Private employment responses track both quantitatively and qualitatively the responses of private output, while private wages increase in a more pronounced way to shocks to S&L government employment. As before, the combined increase in private employment and the labor force participation renders the responses of unemployment insigni…cant. On the other hand, all …scal variables seem to react insigni…cantly to the public employment shock at the federal level, but they may react with one or more lags at the S&L level. 11 Above all, government employment shocks have 9 In the online Appendix, we present graphs for the forecast error variance decomposition of private output and the public wage bill. Our estimates are in line with the ones reported in the existing literature. 10 As in Perotti (2014), the responses to government employment (wage) shocks are divided by the initial response of government employment (wage), and further divided by the sample mean ratio of the government wage bill to GDP. In other words, this scaling refers to an increase in the wage bill induced only by the shocked wage bill component, as if the other component was kept …xed. 11 The fact that we have orthogonalized the shocks does not mean that the respective shocked …scal instruments cannot react to each other contemporaneously or with a lag. In the next subsection, we investigate the sensitivity of our results in other VAR speci…cations that result to insigni…cant reactions of the rest …scal variables to the The di¤erence in the impulse responses translates into di¤erences in the …scal multipliers. Table 2 presents point estimates of the impact output multipliers and the present-value cumulative multipliers up to …ve years after the shock. Multipliers are computed according to Mountford and Uhlig (2009). The exact formula is presented in Appendix C. In parenthesis we report 68% credible sets for the computed multipliers. A 1% of GDP increase in the government wage bill induced by a shock to federal government employment implies a signi…cant expansion of private output by 0.59% and 0.92% on impact and one year after the shock, respectively. On the contrary, a 1% of GDP increase in the government wage bill induced by a shock to federal government wages implies a contraction in private production by -1.22% and -0.59% one and two years after the shock, respectively. Finally, at the S&L level, multipliers are positive and signi…cant for both shocks on impact and at later horizons. As noted earlier, multipliers at the S&L level for government employment shocks are signi…cantly higher than at the federal level.
In the online Appendix, we include the impulse responses following shocks to 'total' government employment, de…ned as the sum of federal and S&L government employment, and to the 'total' government wage, de…ned as the average of the federal and S&L government wage rates. Given the important di¤erences observed at the two di¤erent government levels, it does not come as a surprise that adding the two results in mostly non-signi…cant e¤ects. At the aggregate level, government wage shocks imply insigni…cant output multipliers, while a 1% of GDP increase in the government wage bill induced by a shock to government employment raises output by 1.34% on impact.

Controlling for all government levels
When identifying shocks to the federal or S&L public wage component one may worry that such shocks are correlated. Increases in the wages of federal employees might correlate with increases in the wages of public employees at the S&L level, for instance. Our benchmark results con…rm this for some of the VARs considered. To check the sensitivity of our results to the possible correlation of shocks to federal and local government wage bill spending, we repeat the estimation now controlling for the co-existence of federal and S&L shocks in the same VAR. In other words, when identifying federal (S&L) government shocks we further require the shocks to be orthogonal to a generic S&L (federal) government spending shock. We use the same VAR model enhancing it with an extra variable that stands for either the federal or S&L government expenditure. The extra shock to federal (S&L) government spending is identi…ed by making it orthogonal to all the rest of shocks, and further requiring federal (S&L) government spending to increase for quarters k = 0, 1, 2, 3 after the shock. Figure 5 shows the impulse responses to federal and S&L government spending shocks, while the middle panel of Table 2 presents the respective output multipliers. As can be easily seen, results remain unchanged: government employment shocks remain robustly expansionary at both government levels, and have higher e¤ects at the S&L level. S&L government wage increases also expand output and employment in the private sector, while federal wage increases have contractionary e¤ects. Multipliers are also comparable.

An alternative identi…cation scheme
Another robustness exercise is related to the identi…cation scheme used to extract the …scal shocks. In particular, we repeat our VAR analysis extracting the …scal shocks using a simple recursive (Choleski) identi…cation (Blanchard and Perotti (2002)). We keep the same ordering of the variables as it is stated in the benchmark VAR speci…cation. Impulse responses to federal and S&L shocks are presented in Figure 6. As in our benchmark speci…cation, a public employment increase leads to a signi…cant expansion of private output, consumption and employment, and a signi…cant increase in the participation rate. Those e¤ects hold across any government level and are stronger at the S&L level, as before. Federal wage shocks induce contractionary e¤ects on output and employment in the private sector, while S&L government wage shocks are clearly expansionary, thus further con…rming our benchmark conclusions. As demonstrated in Table 2 (bottom panel), the ranking and sign of the multipliers are similar to the ones obtained when we use sign restrictions to recover the shocks. 12

Theoretical analysis
In this section we develop a New Keynesian model with a public sector, search and matching frictions, and endogenous labor force participation. We assume that a public good produced in the economy provides both productive services to private sector …rms and utility-enhancing services to the representative household. There are three types of …rms in the economy: (i) a public …rm that produces the public good, which is provided for free (ii) private competitive  In each period, jobs in each sector j = p, g (i.e. private/public) are destroyed at a constant fraction σ j and a measure m j of new matches are formed. The evolution of employment in each sector is thus given by: where we assume that matches become productive in the same period. The government, upon forming a match, makes a take-it-or-leave-it o¤er to matched workers. We also assume that σ p > σ g in order to capture the fact that, relatively speaking, the public sector is characterized by greater job security. 13 We consider search as being random and so there is one matching function that has unemployment, u t , and the total number of vacancies, υ p t and υ g t , as inputs: where the matching e¢ciency is given by ρ m . We also assume equal vacancy …lling probabilities in the two sectors:

Households
The representative household consists of a continuum of in…nitely lived agents. The members of the household derive utility from leisure, which corresponds to the fraction of members that are out of the labor force, l t , and a consumption bundle, cc t , de…ned as: where n t = n p t + n g t . The household owns the private capital stock, which evolves over time according to: k p t are adjustment costs. The intertemporal budget constraint is given by: The problem of an intermediate …rm consists of choosing k p t and υ p t to maximize: where x t is the relative price of intermediate goods, κ is a utility cost associated with posting a new vacancy, and Λ t,t+1 = β s Uc t+s Uc t is a discount factor. The maximization takes place subject to the private employment transition equation: The …rst-order conditions are: According to (10) and (11) where ǫ > 1 is the constant elasticity of demand for retail goods. The …nal good is sold at a The demand for each intermediate good depends on its relative price and on aggregate demand: Following Calvo (1983), we assume that in any given period each retailer can reset its price with a …xed probability (1 − χ). Hence, the price index is given by: Firms that are able to reset their price choose p * it so as to maximize expected pro…ts given by: The resulting expression for p * it is:

Wage bargaining
Wages are determined by ex post (after matching) Nash bargaining. Workers and …rms split rents and the part of the surplus they receive depends on their bargaining power. If we denote by ϑ ∈ (0, 1) the …rms' bargaining power, the Nash bargaining problem is to maximize the weighted sum of log surpluses: where V H n p t and V F n p t have been de…ned above. The optimization problem leads to the following solution for w p t :

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Hence, the equilibrium wage is the sum of the value of the marginal product of employment and the value to the …rm of the marginal job multiplied by the hiring probability, weighted by the worker's bargaining power, and the outside option of being unemployed, weighted by the …rm's bargaining power.

Government
The government sector produces the public good using public capital and labor: where we assume that TFP shocks, ε A t , are not sector speci…c and is the share of public capital. The government holds the public capital stock. Similar to the case of private capital, the government capital stock evolves according to: Government expenditure consists of government consumption, modelled as a waste, public investment, public wage payments, public vacancy costs, unemployment bene…ts, and lumpsum transfers, while revenues come from the capital and labor income. The government de…cit is therefore de…ned by: where T R t ≡ (w p t n p t + w g t n g t ) + τ k (r p t − δ p )k p t − T t denotes tax revenues net of transfers. The government budget constraint is given by: To ensure determinacy of equilibrium and a non-explosive solution for debt (see e.g. Leeper (1991)), we assume a debt-targeting rule of the form:

A C C E P T E D M A N U S C R I P T
where ß is the steady state level of debt to GDP ratio, ß t = Bt yt . If Ψ g = υ g , w g , c g denotes the di¤erent …scal instruments, we assume …scal rules of the form: where ε ψ g t is a zero-mean, white-noise disturbance, and ρ ψ g determines the persistence of the di¤erent processes.

Monetary policy
There is an independent monetary authority that sets the nominal interest rate as a function of current in ‡ation according to the rule: where ε R t is a monetary policy shock and π t measures in ‡ation in deviation from the steady state.

Resource constraint
Private output must equal private and public demand. The resource constraint is given by:

Calibration
We solve the model by linearizing the equilibrium conditions around a non-stochastic steady state in which all prices are ‡exible, the price of the private good is normalized to unity, and in ‡ation is zero. We calibrate the model for the U.S. at a quarterly frequency. Table 3 shows the key parameters and steady-state values targeted in our calibration.
We calibrate the labor force participation and unemployment rate to match the observed average values. Thus, we set labor force participation, 1-l ≡ n + u, equal to 65% and the unemployment rate to 6.5%. We …x the separation rate in the public sector σ g = 0.045 and in the private sector σ p = 0.05, which is comparable with the estimates for the job separation rate in Hobijn and Sahin (2009). We …x the probability of …lling a vacancy and then we use it to derived steady-state vacancies in each sector (υ j = m j /ψ f j ). We use the value ψ f p = 0.4, which allows us to obtain meaningful values for vacancies (υ p = 0.064, υ g = 0.011). For the matching elasticity with respect to vacancies we use a = 0.6. Hiring probabilities for each sector are computed as the ratio of sector-speci…c matches to unemployed jobseekers.
The capital depreciation rates, δ g and δ p , are set equal to 0.025. Following the literature, we set the discount factor β = 0.99, which implies a quarterly real rate of interest of approximately 1%. The elasticity of demand for retail goods, ǫ, is set such that the gross steady state markup, ǫ ǫ−1 , is equal to 1.25, and the price of the …nal good is normalized to one. The TFP parameter, A, is normalized to one. For the capital share in the private sector production function we assume a standard value ψ = 0.36, and in the public sector production function we use = 0.1.
We set the capital ratio k g /k p = 0.31 using data from Kamps (2006).
We set the replacement rate Finally, the model's steady state is independent of the degree of price rigidities, the monetary policy rule, and the size of the capital adjustment costs. Capital adjustment costs are included to moderate the response of investment with respect to …scal shocks. We set the in ‡ation targeting parameter in the Taylor rule ζ π = 1.5, the capital adjustment costs parameter ω = 1, and the price-stickiness parameter χ = 0.75.

Results
In Figure 7 we present impulse response functions to a 1% of steady state output increase in the public wage bill induced by an increase in public vacancies (top panel) and in public wages (bottom panel). 15 All responses are expressed in percentage deviations from respective steady state values, with the exception of the unemployment and labor force participation rates that are expressed in absolute percentage points. We …rst report the results of our benchmark parameterization (solid lines) for which public wage shocks have contractionary e¤ects on private sector production in the short run. We then investigate which are the key elements of the model that can account for the case of positive output e¤ects, as found for S&L government wage shocks.
The predictions of our theoretical model match well the empirical evidence for public employment shocks (see Figure 7 (top panel)). It can be readily seen that this type of shocks to the government wage bill is expansionary for the private sector by crowding in consumption and increasing labor force participation and employment. In particular, the complementarity of the public good with private consumption in the aggregate consumption bundle of the household overturns the negative wealth e¤ect of the shock and leads to an increase in private consumption (after the impact period). The unemployment rate rises due to the increase in labor market participation.
Our model can also explain how government wage shocks can be contractionary or expansionary, as found in the data, depending on the relative magnitude of the forces at play. More speci…cally, wage shocks lead to public-private wage spillovers, inducing a negative labor demand e¤ect, which is re ‡ected in the response of private vacancies, and a fall in employment in the private sector, as well as an increase in unemployment. At the same time, there is a boost in the production of the public good as labor supply and employment in the public sector increase. Consequently, public wage shocks can lead to a crowd in of private consumption given the complementarity of the latter with the public good in the aggregate consumption bundle of the household. These two opposite channels can help explain the empirical results. As we can see in Figure 7 (bottom panel), with our benchmark calibration, we observe a short run contraction in private-sector production and a rise in the unemployment rate, which matches the empirical evidence found for federal government wage shocks. In this case, the complementarity channel is not su¢ciently strong to overturn the wage increase and the negative labor demand 15 In the Online Appendix, we present responses for the other three shocks in the model, namely a government consumption, a monetary policy, and a business cycle shock. The model does well in matching qualitatively the responses in the empirical model.

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e¤ect in the private sector. We next examine whether increasing the degree of complementarity between the public good and private consumption can generate an expansion in the private sector, as observed in the data for S&L government wage shocks.

The complementarity between public and private goods
As already emphasized, the degree of complementarity between the public good and private consumption in the aggregate consumption bundle of the household is key for determining the e¤ects of government wage bill shocks. In this subsection, we investigate how varying the degree of this complementarity a¤ects the transmission of both types of shocks to the public wage bill. The dashed lines in Figure 7 (top panel) represent responses to a shock in government vacancies when we increase the degree of complementarity between public and private goods (by setting α 2 = −3.9). As we can see, the e¤ects of government vacancy shocks are signi…cantly more pronounced than in our benchmark calibration. This is line with the empirical evidence for public employment shocks exhibiting stronger e¤ects at the S&L level relative to the federal level. Turning to the public wage shocks (see Figure 7, bottom panel), with a higher complementarity between public and private goods, the increase in private consumption becomes larger, and the fall in private employment becomes smaller and short-lived, and so does the rise in the private wage. The increase in public employment and output leads now to an expansion in private-sector production.
Our theoretical analysis therefore suggests that the degree of complementarity with private consumption at the S&L and federal level is key for explaining the observed heterogeneity to wage bill shocks. 16 The di¤erential macroeconomic e¤ects of public wage shocks can be justi-…ed by the di¤erent nature of the public good provided in each government level. For instance, federal government employees largely comprise military employees, and even one-third of the federal civilian workforce are employed in the Department of Defense. 17 On the other hand, S&L government employees provide mainly education, health care and transportation services. 18 16 As pointed out by a referee, preferences with complementarity of hours and consumption would in principle allow us to …t the increase in private activity after an increase in government employment, but this would go against the federal and S&L distinction. 17 Falk (2012) provides detailed information on the occupational tasks of the federal civilian workforce: 57% of this workforce worked at three departments in 2010: (i) the Department of Defense employs more than onethird; (ii) the Department of Veterans A¤airs employs 14%; (iii) the Department of Homeland Security employs 8%. Another 40% of federal civilian employees work for the other departments and agencies of the executive branch, while the remaining 3% is employed by the legislative and judicial branches of government.

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Research by Fiorito and Kollintzas (2004) with European data has indeed shown that the degree of complementarity between government and private consumptions is not homogeneous over types of public expenditures. In particular, they …nd that while some categories of public spending seem to be substitutable with private consumption, there are also public expenditure categories which are complements to private spending. More importantly, they report that the latter case of complementarity seems to be the stronger relation, such that overall government and private consumption are complements in the aggregate. 'Merit goods', including health and education, complement private consumption while 'public goods', referring to defense, public order and justice, are substitutes with private consumption. Bouakez

Sensitivity analysis: The importance of complementarity against other possible hypotheses
The previous subsection has highlighted that the model can explain the heterogeneity of responses of the empirical model at the di¤erent government levels if one accepts the idea that public goods at the S&L level are 'merit' goods, while the goods produced at the federal level are 'public' goods, according to the terminology adopted in Fiorito and Kollintzas (2004). Yet, there are many characteristics in the model that could potentially explain the di¤erences in results at the di¤erent government levels. For example, one might ask whether the presence of the public good in the production function of the private sector could be an alternative channel that can explain the mixed sign of the output response for public wage shocks. Also, vacancy …lling probabilities in the public sector at the S&L level might di¤er from those at the federal employment are protective services (including police o¢cers and …re …ghters), higher education, health care, and transportation (including road maintenance workers and bus drivers). Formally speaking, let h(y t (θ|x t ))) be a J × 1 vector of functions of the data y t produced by the model, when the N × 1 vector of structural parameters θ is employed, conditional on the shock x t . We let θ be uniformly distributed over Θ , where Θ = Q i Θ i is the set of admissible parameter values and Θ i is an interval for each parameter i. We draw θ l i , i = 1, . . . , N from each Θ i , construct h(y t (θ l |x t ))) for each draw l = 1, . . . , 10000 and order them increasingly. Then we report the 84 and 16 percentiles of the simulated distribution of h(y t (θ|x t ))), where h(.) stands for the model's impulse responses when we vary the related parameters. Since the 68% bands of the IRFs track well the responses in the empirical model and are signi…cantly di¤erent when we change only the complementarity degree of the public good with private consumption in the utility speci…cation, we conclude that the value of the degree of complementarity is key for explaining the empirical results.
We restrict the range of Θ i on the basis of theoretical and practical considerations and draw uniformly from these ranges. This way our approach also formalizes, via Monte Carlo methods, standard sensitivity analysis conducted in many calibration exercises. We split the parameter vector θ = (θ 1 , θ 2 ), where θ 1 represents the parameters which are …xed to a particular value, either to avoid indeterminacies or because of steady state considerations, while θ 2 are the parameters which are allowed to vary between the two government levels. Table 3 gives the ranges for the parameters in θ 2 .
The vector θ 1 includes the discount factor, the steady state TFP level, the debt to GDP ratio and the risk aversion parameter. The intervals for the parameters that vary in our robustness exercise are centered around the benchmark calibrated values. For example, we allow the The share of public employees in total employment is much smaller in the federal level than in the benchmark calibration. For that reason when constructing our priors we allow this papemeter to vary in the [0.5,0.18] interval. We assume also di¤erences in the unemployment rate, the replacement rates, the job destruction probabilities of jobs at the S&L and federal level and di¤erences in the matching e¢ciencies varying this parameter at the [0.45,0.65] interval.
The labor supply elasticity, which is crucial in determining the labor market responses to …scal shocks and may di¤er for federal versus S&L jobs, varies in the interval [0. 25,1], which covers well the range of existing estimates. The capital adjustment costs parameter shapes investment responses to shocks and therefore indirectly a¤ects labor market dynamics. The chosen range allows for small (̟=0.1) and large (̟=3) adjustment costs. We also vary private and public capital depreciation rates. The parameter ν controls the interactions between public and private goods in production. Depending on its value, an increase in government capital has large, or small e¤ects on private output. Aschauer (1989) estimates the elasticity of output with respect to public capital in a range from 0.39 to 0.56. Evidence from more recent studies, however, suggests positive, but lower, values for ν both at national and at the regional level (see, e.g., Garcia-Mila and McGuire (1992)). In our model we use public output rather than public capital in the private production function, we choose for ν the range [0, 0.2], which covers both the case of unproductive government output and most of the estimates for the elasticity of output to changes in public capital in the literature. We vary the productivity of public capital at the The degree of price stickiness and the coe¢cient on in ‡ation in the policy rule determine both the shape and the size of output and in ‡ation responses following a …scal shock. Since at the state level nominal price rigidities might be less severe than at the federal level and their interaction with monetary policy might operate di¤erently, we post intervals both for the degree of price stickiness as well as the in ‡ation coe¢cient in the Taylor rule. Notice that the determinacy of equilibrium depends nonlinearly on the values of χ, ζ π and ζ b , for that reason the ranges of these parameters are truncated so as to guarantee determinacy. Finally, the persistence of the government wage bill shocks is allowed to vary between [0,1,0.9].
We report the 68% con…dence of the model IRFs in Figure 8   is willing to accept that the public goods produced at the S&L level are stronger complements with private goods relative to public goods produced at the federal level.
Our analysis therefore suggests that the public good provided at the federal level may exhibit a di¤erent degree of complementarity with private consumption than that at S&L level. This might be justi…ed by the di¤erent nature of the public good provided in each case. Government hourly wage rate: Series have been constructed using the NBER extracts of the CPS Merged Outgoing Rotation Groups to generate the following: national-level, quarterly data for two wage categories: (a) federal government, (b) S&L governments (excluding from the sample workers that have never been full time and self-employed). Hourly wages were obtained for all workers using weekly wages and dividing by hours worked. We regressed for the repeated (monthly) cross section the log of hourly wage -separately for each of the two categories -on socio-demographic variables (age group, gender, race, and marital status). We also included dummies for each state. We ran the regression at the monthly level and then calculated residuals for each month. We averaged residuals by month and used the mid-quarter month to obtain quarterly data. We seasonally adjusted the quarterly data using a seasonal ARIMA model and de ‡ated using the U.S. CPI quarterly in ‡ation series.

B VAR estimation method
We consider a N-variable VAR model of the following form: The Maximum Likelihood Estimator for (B, Σ) is given by: We assume that the prior and posterior distributions of (B, Σ) belong to the Normal-Wishart family: Uhlig (1994) states that if the prior is described by B o , N 0 , S 0 and ν 0 , then the posterior is described by B T , N T , S T and ν T , where: Following Uhlig (2005), we use a weak prior that sets N 0 = 0, ν 0 = 0, S 0 and B 0 arbitrary. Flat priors give results that are robust to the reordering of the variables in the VAR.

C Algorithm for shock identi…cation, derivation of IRFs and multipliers
The shock identi…cation follows the approach of Arias et al. Let n be the number of endogenous variables in the VAR and F (φ −1 (B, Σ)) any function from the base VAR parameterization with dimensions nr × n that satis…es the condition 1 (B, Σ)) can be a set of impulse response functions or the structural representation matrices. In addition, let Z j be a matrix z j × nr so that Z j F (φ −1 (B, Σ, Q))e j = 0, where z j is a number of zero restrictions needed to be imposed on F (φ −1 (B, Σ)), and e j is the j th column of the identity matrix I n . Then, the algorithm is summarized as follows: 1. Draw (B, Σ) from the posterior distribution of the reduced-form parameters as speci…ed in the previous section.
2. Draw (x 1,..., x n ) independently from a standard normal distribution on R n .

Set
where the columns of matrix N j form an orthonormal basis for the null space of the If we denote by λ ct , λ n p t , λ n g t , λ ut the Lagrange multipliers, the …rst-order conditions of the household's optimization problem are: According to (A6), V H n p t has the following components: …rst, the increase in utility given by the real after-tax wage; second, the decrease in utility from lower leisure; third, the continuation utility value, which depends on the separation probability: a private employee will continue having the same job next period with probability 1 − σ p .

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