Reforming the public sector wage policy

Reforming the public sector wage policy ^∗

Pedro Gomes

Universidad Carlos III de Madrid February 27, 2014

Abstract

I set up a DSGE model with search and matching frictions and heterogeneous workers to evaluate the public sector wage policy in steady-state and over the business cycle. I calibrate the model to the UK economy based on Labour Force Survey data. A review of the pay of all public sector workers to align the distribution of wages with the private sector reduces steady-state unemployment by 1.5 percentage points. Implementing a procyclical simple rule to determine the yearly growth rate of public sector wages reduces unemployment volatility by 5 to 8 percent. In a sample of 29 OECD countries for the pre-crisis period of 1995-2007, countries that deviated more from the rule had a larger increase in the unemployment rate and a higher volatility of unemployment relative to GDP.

JEL Classification: E24; E62; J45.

Keywords: Public sector employment; public sector wages; unemployment; fiscal rules.

∗An earlier version of the manuscript was prepared for the workshop Government wage bill: determinants, interactions and effects, organized by the European Commission (DG ECFIN). I would like to thank Iacopo Morchio, Matthias Burgert, Matteo Salto, Gernot M¨uller and the participants of the workshop for comments and suggestions. .

†Department of Economics, Universidad Carlos III de Madrid, Calle Madrid 126, 28903 Getafe, Spain.

1 Introduction

There are three sets of stylized facts about public sector employment and wage policy related to their size, cyclicality and heterogeneity across skills. First, they always stand out as major components, whether one looks at the labour market or the government budget.

Governments of OECD countries take hold of 18 percent of total employment and their wage bill represents more than half of government consumption expenditures. Regarding their cyclicality, public sector wages fluctuate less than the private sector and are less procyclical.¹ Perhaps less known, is the heterogeneity across the skill dimension. The public sector predominantly hires skilled workers. In the United Kingdom, for instance, the government employs 36 percent of college graduates and only 16 percent of workers with lower qualifica- tions. The payment it offers also varies across workers. Researchers estimate that the public sector wage premium, although it is positive on average, differs across education groups.

More educated individuals receive a lower premium, while less educated individuals are paid more in the public sector.² Finally, adding to the wage compression across education levels, there is a wage compression within education categories with the bottom quantiles having higher premium and the top quantiles having a lower or even negative premium.³

This paper aims to build a quantitative macro model that incorporates these three sets of stylized facts and use it to evaluate the government wage policy and a reform that strengths the link with private sector wages.

Given the heterogeneity across skills, its surprising that most of the theoretical literature on public employment has ignored this dimension by assuming homogeneous workers. Exam- ples include: Finn (1998), Algan, Cahuc, and Zylberberg (2002), Ardagna (2007), Quadrini and Trigari (2007) or more recently Michaillat (2014), Gomes (2014) and Afonso and Gomes (2014). Two reasons motivate me to address the gap in the literature.

As a payment for a factor of production, identical workers should have the same value of working in the private or the public sector. In a simple RBC model, as in Finn (1998), even if the productivity differ across sectors, identical workers receive the same wage because

1This was found using aggregate data by Quadrini and Trigari (2007) for the United States, by Lamo, P´erez, and Schuknecht (2013) and Lane (2003) for several OECD countries and by Devereux and Hart (2006) using UK microdata.

2This was found in the United States by Katz and Krueger (1991), in the United Kingdom by Postel- Vinay and Turon (2007) or Disney and Gosling (1998) and in several European countries by Christofides and Michael (2013), Castro, Salto, and Steiner (2013) and Giordano et al. (2011).

3This was found in Poterba and Rueben (1994) for the United States, Postel-Vinay and Turon (2007) or Disney and Gosling (1998) for the United Kingdom or Mueller (1998) for Canada.

of arbitrage. With frictions, the labour market tolerates different wages. Gomes (2014), building on Quadrini and Trigari (2007), examines the optimal wage policy in the context of a two-sector search and matching model. If the government sets a high wage, it induces too many unemployed to queue for public sector jobs and raises private sector wages, thus reducing private sector job creation and increasing unemployment. Conversely, if it sets a lower wage, few unemployed want a public sector job and the government faces recruitment problems. The heterogeneous public sector wage premium suggests that we might have the two inefficiencies simultaneous, with long queues and high unemployment for unskilled workers and recruitment problems of skilled workers.

The second reason stems from the recent experience of European countries subject to austerity packages. The first chart in Figure 1 shows the government wage bill as a fraction of the private sector wage bill and the size of government employment relative to the private sector employment, for OECD countries in 2008. Six countries stand out for a high public sector wage bill relative to their level of public employment: Greece, Cyprus, Ireland, Por- tugal, Italy and Spain. These countries were later in the center of the Euro area crisis for their poor public finance and sclerotic labour markets. The implemented austerity measures naturally included public sector wage cuts. However, many governments opted for asymmet- ric cuts, centered on the highest earners, instead of a reform aligning the wage distribution with that of the private sector. Although the cuts reduced spending, they did not correct the inefficiencies at the bottom and probably exacerbated the inefficiencies on the top.

The motivation for looking at the dynamic side of the government wage policy lies in Figure 1b. It shows the evolution of the ratio between the two variables, which is simply the ratio of average wages in the two sectors. How could government wages grew so much relative to the private sector, in so many countries? To understand it we must recognize that public sector wages are simultaneously a payment to a factor of production and a transfer from society to a specific group of citizens. As a consequence, they are vulnerable to be manipulated for electoral reasons, in the spirit of Nordhaus (1975) political cycles.

For instance, Borjas (1984) finds that, in the United States, the pay increases in federal agencies are 2 to 3 percent higher on election years. Matschke (2003) also finds evidence of systematic public wage increases of about 2 to 3 percent prior to federal elections in Germany. In Portugal in 2009 - year of crisis and three elections - public sector workers saw their real wage increase by 4 percent. If we want to avoid these situations in the future, we should design institutions that limit the scope of politicians to manipulate public sector wages, while maintaining a certain degree of optimality.

Figure 1: Government wage bill and employment, OECD countries

Portugal

Cyprus

Greece

Italy Ireland

Spain

.1.2.3.4.5Government wage bill (% of private sector wage bill)

.1 .2 .3 .4

Government employment (% of private sector employment)

(a) Government wage bill and employment in 2008 Source: EUROSTAT, AMECO and OECD.

Greece Cyprus Ireland Portugal Italy Spain

Mean

11.522.5

1995 2000 2005 2010

Year

(b) Ratio of gov. wage bill (% private wage bill) over gov. employment (% of private employment)

In this paper I extend the model of Gomes (2014) by introducing worker heterogeneity along two dimensions: education and ability. I consider heterogeneous ability for two rea- sons. First, because the public sector wage premium also varies within education groups.

Second, to acknowledge the common argument that public sector wage cuts limit the scope of governments to hire high-ability workers. Nickell and Quintini (2002) document the fall in relative pay of British public sector workers during the 1980’s and find that men entering the public sector had significant lower test score positions compared to the precious decade.

With this extra dimension I can examine the effects on the average quality of public sector workers. The model also includes capital stock assuming capital-skill complementarity in the private sector, nominal rigidities and distortionary taxes. The model is calibrated for the United Kingdom. I use the Labour Force Survey from 1996-2006 to calibrate the parameters related with the worker heterogeneity, labour market and wages.

The objective of the model is twofold. First, it aims to understand the steady-state effect of a pay review of different types of public sector workers on: the equilibrium unemployment rate, the quality of the pool of public sector workers and its composition, total government spending and welfare. Wage cuts of unskilled government employees reduce the unemploy- ment rate and government spending. A 7 percent cut reduces the unemployment rate by 1.5 percentage points. Wage cuts of skilled workers can reduce spending, but up to a limit. If the cuts are too severe, they actually increase government spending and reduce welfare. As the government lowers the pay of skilled workers too much, it faces recruitment problems.

It spends more to recruit a skilled worker or substitutes hiring towards unskilled workers. In

our calibration, wage cuts above 13 percent of skilled wages are welfare reducing.

Second, it quantifies how the volatility of unemployment, consumption and inflation de- pend on the government wage policy, when the economy is subject to technology, government employment and cost-push shocks. I propose a simple rule to determine the public sector wage growth, that aims to stabilize an aggregate ratio as in Figure 1b. This procyclical rule reduces the volatility of unemployment by 5 to 8 percent and of private consumption by 2 to 9 percent, relative to the benchmark rule estimated for the United Kingdom.

Using data from European Commission and OECD, I compute the evolution of the ratio of average public to private wages of 29 countries for the pre-crisis period of 1995-2007. Nordic countries and France seem to implicitly follow such a simple rule. I show that countries that deviate more this rule had larger increases in the unemployment rate and higher volatility of unemployment relative to GDP.

In the final section I draft a roadmap for a reform of public sector wages. The key idea, that follows from the discussion, is to use private sector wages as the benchmark when deciding the pay in the public sector, across workers and over time. The first pillar consists of a review of pay of all public sector workers, having as benchmark the wages of equivalent workers in the private sector. The second pillar is to establish a rule to guide the annual increase of public sector wages. The idea behind the rule is that the average public sector wage should target the average private wage. This reform offers several advantages:

i) it guarantees the parity between the two sectors for all workers and that this parity is maintained over the business cycle; ii) it avoids the use of the public sector wage by politicians as an electoral tool; iii) it requires low tax burden in recessions and iv) it is simple and offers more predictability in one of the most important decisions governments take every year.

Beside the aforemention literature on public employment with homogeneous workers, there are two papers that also focus on the skill dimension. Domeij and Ljungqvist (2006) document a divergence of the skill premium between the United States and Sweden and argue that it can be explained by the expansion of the public sector employment of low- skilled workers in Sweden. Bradley, Postel-Vinay, and Turon (2013) study the effects of public employment on the private wage distribution, in a setting with on-the-job search and transitions between the two sectors.

2 Model

The dynamic stochastic general equilibrium model extends Gomes (2014) by adding hetero- geneous workers, capital accumulation, distortionary taxation,and nominal rigidities.

2.1 General setting

The economy has two sectors j ∈ {p, g}. Public sector variables are denoted by the super- script g and private sector variables by p. Time is discrete and denoted by t. The economy is populated by a measure one of workers. Workers differ ex-ante from one another. There are four types of workers i ∈ {¯h, h, ¯µ, µ}, with two dimensions of heterogeneity. The first dimension is education with skilled workers (college degree) denoted by h and the unskilled (bellow college degree) denoted by µ. Within each group there are workers with higher ability, (¯h, ¯µ), and others with lower ability (h, µ). The productivity of workers of type i is denoted by zⁱ, with z^¯h > z^h and z^µ¯ > z^µ. The mass of workers of type i is ωⁱ, with P

iωⁱ = 1.

For each type, a fraction of workers are unemployed (uⁱ_t), while the remaining are working either in the public (l_t^g,i) or in the private (l^p,i_t ) sectors.

1 = l_t^p,i+ l^g,i_t + uⁱ_t, ∀i. (1) Total unemployment is denoted by u_t =P

iωⁱuⁱ_t. The presence of search and matching frictions prevents some unemployed from finding jobs. The evolution of employment of type i in sector j depends on the number of new matches m^j,i_t and on the job separations. In each period, jobs are destroyed at rate λ^j,i, potentially different across sectors and types.

l_t+1^j,i = (1 − λ^j,i)l_t^j,i+ m^j,i_t , ∀ji. (2) I assume the markets are segmented and are independent across types. This assumption is worth discussing. While the length of education is easily observable by the employers from the CV, it is not necessarily the case with ability. We have to take a stand whether it is observable ex-ante by the employer or it is private information. If ability is unobservable, low- ability workers can apply to high-ability jobs breaking down an equilibrium with segmented markets. I want to abstract from the complications arising from asymmetric information.

I rely on previous papers on asymmetric information with labour market frictions, such as Guerrieri, Shimer, and Wright (2010) or Fern´andez-Blanco and Gomes (2013). These papers argue that firms can design mechanisms such that workers self-select into the correct

segment.⁴ Section 2.4 explains why is not a problem to assume observable types.

I assume the unemployed can direct their search to the private or the public sector.

This assumption finds support in micro-econometric evidence and was discussed in length in Gomes (2014). Together with the assumption of segmented markets it allows is to express the new matches with the following matching functions:

m^j,i_t = m^j(u^j,i_t , v^j,i_t ), ∀ji. (3) I assume the unemployed choose the sector in which they concentrate their search; thus, u^j,i_t represents the number of unemployed of type i searching in sector j. Vacancies in each segment are denoted by v_t^j,i. An important part of the analysis focuses on the behaviour of those unemployed specifically searching for a public sector job, defined as: sⁱ_t≡ ^u_u^g,iti

t . We also define q_t^j,i as the probability of filling a vacancy of type i is sector j and f_t^j,i as the job-finding rate of an unemployed of type i conditional on searching in sector j:

q_t^j,i= m^j,i_t

v_t^j,i , f_t^j,i = m^j,i_t

u^j,i_t , ∀ji.

2.2 Households

Following Merz (1995), I assume all the income of the members is pooled so the private consumption is equalized across members. This is a common assumption on the literature to maintain a representative agent framework in the presence of unemployment. Without this risk sharing assumption, risk-averse workers with different employment histories would accumulate different levels of wealth. As the wealth distribution is not relevant to our problem, I prefer to simplify and keep the representative agent framework. The household is infinitely-lived and has the following preferences:

E₀

∞

X

t=0

β^t[u(c_t) + ν(u_t)], (4)

4In the case of Guerrieri, Shimer, and Wright (2010) this is done by contracts specifying the hours worked.

Assuming that high-ability workers have lower disutility of work, firms can post a contract specifying a higher wage but more hours, which excludes the low-ability type. In this paper I follow the setting of Fern´andez- Blanco and Gomes (2013). They assume that the output of a match depends on the capital supplied by firms and that wages are bargained over between firms and workers. Firms specify a capital plan ex-ante. With capital-skill complementarity, the low-ability worked does not have the incentive to apply to the high-ability job, as it implies having too much capital and hence lower wages.

where

c_t≡

Z 1 0

c

ξ−1 ξ

n,t dn



ξ ξ−1

(5) is the Dixit-Stiglitz basket of consumption goods produced by the final good retail sector.

The household also derives utility from the members that are unemployed ν(u_t), which captures the value of leisure and home production. β ∈ (0, 1) is the discount factor. The budget constraint in period t is given in real terms by:

c_t+B_t+1

p_t +K_t+1= (1 + i_t−1)B_t

p_t +(1−δ)K_t+(1−τ_t) r_t

p_tK_t+X

j

X

i

ωⁱw_t^j,i p_t l_t^j,i

! +X

i

ωⁱχ^gu_t+Π_t, (6) where it−1 is the nominal interest rate from period t − 1 to t and Bt are the holdings of one period bonds. The households can also save by accumulating capital stock K_t. The capital stock depreciates at a rate δ and can be rented to firms at a nominal rental rate of r_t. The second source of income is labour income with w^j,i_t being the nominal wage rate from the members of type i working in sector j. The unemployment members collect unemployment benefits χ^g. The household pays a tax τ_t on both its labour and capital income. Finally, Π_t encompasses the lump sum taxes or transfers from the government and possible net profits from the private sector firms. The aggregate price level,p_t, is given by

p_t ≡

Z 1 0

(pⁿ_t)^1−ξdn

_1−ξ¹

. (7)

The household chooses the sequence of {c_t, K_t+1, B_t+1^c }^∞_t=0 to maximize the expected utility subject to the sequence of budget constraints, taking taxes and prices as given. The solution is the Euler equation and an arbitrage condition between capital and bonds:

u_c(c_t) = β(1 + i_t)E_t[ p_t

p_t+1u_c(c_t+1)], (8)

1 + i_t= E_t[p_t+1 pt

(1 − δ + ˜r_t+1(1 − τ_t+1)], (9) where ˜r_t= ^r_p^t

t is the real rental rate of capital. The second condition compares the nominal interest rate paid by a one-period bond with the expected nominal return on a unit of investment. Notice that, in this specification, the income tax introduces an extra burden on investment in capital relative to bonds.

2.3 Workers

The unweighted value of each member of type i to the household depends on their current state. The values of being employed are:

W_t^p,i = (1 − τ_t) ˜w_t^p,i+ E_tβ_t,t+1[(1 − λ^p,i)W_t+1^p,i + λ^p,iU_t+1^p,i ], ∀i, (10) W_t^g,i = γ_tⁱ+ (1 − τ_t) ˜w_t^g,i+ E_tβ_t,t+1[(1 − λ^g,i)W_t+1^g,i + λ^g,iU_t+1^g,i ], ∀i, (11) where β_t,t+k = β^{k u}_u^c(ct+k)pt

c(ct)p_t+k is the stochastic discount factor and ˜w_t^j,i = ^w

j,i t

pt is the real wage.

γ_tⁱ is a random variable with cumulative distribution Γ, that represents a idiosyncratic pref- erence for the public sector. The value of being employed in a specific sector depends on the current wage, as well as the continuation value of the job, which depends on the separation probability. Under the assumption of direct search, unemployed are searching for a job in either the private or the public sector, with value functions given by:

U_t^p,i = ν_u(u_t)

u_c(c_t) + χ^b+ E_tβ_t,t+1[f_t^p,iW_t+1^p,i + (1 − f_t^p,i)U_t+1^p,i], ∀i, (12)

U_t^g,i = γ_tⁱ+ν_u(u_t)

u_c(c_t) + χ^b+ Etβt,t+1[f_t^g,iW_t+1^g,i + (1 − f_t^g,i)U_t+1^g,i ], ∀i. (13) As in Hall and Milgrom (2008), the unemployed collect unemployment benefits χ^b and con- tribute to home production (marginal utility from unemployment relative to the marginal utility of consumption). The idiosyncratic preference, γ_tⁱ, also enters the value of unemploy- ment for the public sector. The continuation value of being unemployed and searching in a particular sector depends on the probability of finding a job and the value of working in that sector. Optimality implies that movements between the two segments guarantee no additional gain for searching in one sector vis-`a-vis the other:

U_t^p,i = U_t^g,i = U_tⁱ, ∀i. (14) The idiosyncratic preference for the public sector is important quantitatively. Without it, as in Gomes (2014), small changes in relative wages generate implausible large swings in the fraction of unemployed searching in the public sector. With a distribution of preferences, even if the government pays low wages, the workers with strong preferences for the public sector would still apply for jobs there. Γ puts discipline on the fluctuations on s_t, that are given in equilibrium by

sⁱ_t = 1 − Γ(γ_t^i,∗), ∀i, (15)

where γ_t^i,∗is the cut-off point of the distribution for type i at time t. All unemployed members with preferences above the cut-off search in the public sector, while the ones bellow search in the private. This threshold is given by

γ_t^i,∗ = f_t^p,iE_tβ_t,t+1[W_t+1^p,i − U_t+1ⁱ ] − f_t^g,iE_tβ_t,t+1[W_t+1^g,i − U_t+1ⁱ ], ∀i. (16) An increase in the value of employment in the public sector, driven by either wage increases or a decrease in the separation rate, raises st until no extra gain exists for searching in that sector. However, the marginal searcher will have a lower preference for the public sector.

2.4 Intermediate good producers

There is a large continuum of firms that produce one of four types of intermediate goods xⁱ_t that is sold at price p^x,i_t . Firms open vacancies in a given sub-market i. If the vacancy is filled, the firm is matched to a type-i worker and produce f (a_t, zⁱ, k_tⁱ), where a_t is an aggregate productivity that is stochastic and k_tⁱ is the capital used in the match. The production technology f (·, ·, ·) is increasing and concave in all its arguments with a positive cross partial derivative of capital and skill. The value of a job in real terms is given by

J_tⁱ = max

kⁱ_t

[˜p^x,i_t fⁱ(a_t, zⁱ, k_tⁱ) − ˜w_t^p,i− ˜r^p,i_t k_tⁱ+ E_tβ_t,t+1[(1 − λ^p,i)J_t+1ⁱ ], ∀i. (17)

For each match, the firm chooses how much capital it wants to rent to provide to the worker.

The optimal level of capital k_t^∗i is solves the first-order condition

˜

p^x,i_t f_kⁱ(at, zⁱ, k^∗i_t ) = ˜rt, ∀i. (18) So we can write the value of a job as

J_tⁱ = [˜p^x,i_t fⁱ(a_t, zⁱ, k^∗i_t ) − ˜w_t^p,i− ˜r_t^p,ik_t^∗i+ E_tβ_t,t+1[(1 − λ^p,i)J_t+1ⁱ ], ∀i. (19) The value of opening a vacancy for type i is given by:

V_tⁱ = −κ^p,i+ E_tβ_t,t+1[q_t^p,iJ_t+1ⁱ + (1 − q_t^p,i)V_t+1ⁱ ], ∀i, (20) where κ^p,i is the cost of posting a vacancy. The number of firms is determined in equilibrium by free entry

V_tⁱ = 0, ∀i. (21)

The surplus from the match is shared by the firm and the worker as wages are the outcome of a Nash bargaining

˜

w_t^p,i = arg max

˜ w^p,i_t

(W_t^p,i− Uⁱ_t)^b(J_tⁱ)^1−b, ∀i. (22) where b denote the worker’s bargaining power. The solution is given by:

(W_t^p,i− U_tⁱ) = b(1 − τt)

1 − bτ_t (W_t^p,i− U_tⁱ+ J_tⁱ), ∀i. (23) With distortionary taxes, the share of the surplus going to the worker is lower than its bargaining power. For every unit that the firm gives up in favour of the worker, the pair lose a fraction τ_t to the government. So they economise on their tax payments by agreeing an lower wage.

Notice that, from equation 18, there is only one capital level that maximizes the surplus of the match and hence the wages. Given the capital-skill complementarity, the optimal level of capital is increasing with ability, provided the price of the good is not decreasing in ability, which is guaranteed in the numerical exercise. This is the key mechanism that insures that, even the ability was not observable, we could design a separating equilibrium.

If firms commit to supplying a capital stock of the high type in every period, the low ability workers would not want to pretend to be high ability workers. Even if they would have higher job-finding rate, they would be paired with too much capital for the duration of the match, which would imply lower surplus and lower wages.⁵

2.5 Wholesale firms

The representative wholesale firm buys intermediate inputs in a competitive market, pro- duces a final good and sells it at price ˜p^y_t. The objective is to choose the inputs to maximize profits given by

maxxt

[˜p^y_tF (x_t) −X

i

˜

p^x,i_t xⁱ_t], (24)

where bold denotes a vector, i.e. x_t denotes a vector with all four intermediate inputs. The solution is given by the first-order conditions:

˜

p^y_tF_x⁰i = ˜p^x,i_t , ∀i. (25)

5For details see Fern´andez-Blanco and Gomes (2013).

2.6 Retails firms

There are a continuum of retailers facing monopolistic competition. Each firm n buys the intermediate good, y_n,t and sells it as a differentiated good. Each firm faces a sequence of downward slopping demand curves:

yn,t+s= pⁿ_t+s p_t+s

−ξ

Yt+s, s = 0, 1, ... (26)

where Y_t is the aggregate demand of differentiated final goods and ξ is the elasticity of substitution between them. The real marginal cost is

mc_t= ˜p^y_t + ϕ^c_t, (27)

where ϕ^c_t is a cost-push shock. I follow the Calvo price setting model. In each quarter, a share θ of firms does not reset their price. All firms re-optimizing at date t solve an identical problem given by

max

p^n,∗_t

E_t ( _∞

X

s=0

θ^sβ_t,t+s p^n,∗_t

p_t+s − mc_t+s



y_n,t+s|t )

s.t.

y_n,t+s|t = p^n,∗_t p_t+s

−ξ

Y_t+s.

The optimal pricing decision and the law of motion for the price level are given by

E_t

∞

X

s=0

(θ)^sβ_t,t+sY_t+sp^ξ_t+s( p^∗_t

p_t+s − ξ

ξ − 1mc_t+s) = 0. (28)

p^1−ξ_t = θp^1−ξ_t−1 + (1 − θ)p^∗1−ξ_t . (29)

2.7 Government

I assume the government needs to produce a minimum number of services, gt, that is stochas- tic. To produce these services the government has to hire different types of workers. I consider the public sector wages are exogenous policy variables determined one period in advance when vacancies are posted. Given a wage schedule, the government chooses the vacancies of each type of worker to minimize the total cost of providing the government

services. The total costs, in real terms encompasses the wage bill and the recruitment costs.

min

v_t^g,i

X

i

ωⁱκ^g,iv_t^g,i+ E_tβ_t,t+1[X

i

ωⁱw_t+1^g,i p_t+1l^g,i_t+1] s.t.

gt+1= g(l^g_t+1)

l^g,i_t+1= (1 − λ^j,i)l_t^g,i+ q_t^g,iv_t^g,i, ∀i,

where g(l^g_t) is the production function of government services that uses the four types of workers, l^g_t. Given the level of public wages and a market tightness, the government has to guarantee that it post enough vacancies to maintain an employment level capable of providing the government services. The first-order conditions are

ωⁱκ^g,i

q^g,i_t + E_tβ_t,t+1[ωⁱw^g,i_t+1

p_t+1] = ζ_tE_tg_i,t+1⁰ , ∀i, (30) where ζ_t is the real multiplier of the constraint on government services and g_i,t⁰ is the partial derivative of the government services with respect to government employment of type i. This problem incorporates the two opposite forces that are important to understand the role of public sector wages. When wages of one type go down, the government would save on the wage bill if it hired more workers. But at the same time it might find it more expensive to recruit them. The overall effect depends on the tightness of the market.

The government budget constraint in real terms is given by:

τ_t X

j

X

i

ωⁱl^j,i_t w_t^j,i pt

+ r_t pt

K_t

!

=X

i

ωⁱl_t^g,iw_t^g,i pt

+X

i

ωⁱv_t^g,iκ^g,i+X

i

χ^bωⁱuⁱ_t+ T_t+ ¯g^int, (31)

where T_tare lump-sum transfers and ¯g^intare exogenous purchases of intermediate goods. The costs of recruiting are external, meaning that the cost comes out of the budget constraint.

Throughout the paper I consider two cases: one where any adjustment of the government budget is guaranteed by changes in lump-sum transfers and one where the distortionary income tax rate adjusts to balance the budget.

2.8 Central bank

Finally, the central bank sets the following nominal interest rate i_t : 1 + i_t= ρ^m(1 + i_t−1) + (1 − ρ^m)(1

β + φ(π_t− 1)), (32)

where πt= _p^pt

t−1 is the inflation rate, φ is the response of the target interest rate to changes in inflation and ρ^m is the degree of persistence of the interest rate.

2.9 Market clearing

The market clearing conditions in the intermediate and final goods’ markets are:

xⁱ_t= ωl^p,i_t fⁱ(at, zⁱ, k_tⁱ), ∀i, (33) Yt = F (xt) = ct+ ¯g^int+ Kt+1− (1 − δ)Kt+X

i

X

j

ωⁱv_t^j,iκ^j,i. (34) In this economy the measure of GDP as in the national accounts would be GDPt= F (xt) + P

iωⁱl^g,i_t w^g,i_t . The market clearing in the capital market implies that all capital is rented to the intermediate good producers:

K_t=X

i

ωⁱkⁱ_tl^p,i_t . (35)

As bonds are in zero-net supply, the market clearing for bonds is

B_t = 0. (36)

2.10 Business cycle

I assume three main sources of fluctuations: technology shocks, cost-push shocks and gov- ernment employment shocks.

a_t = (1 − ρ^a)¯a + ρ^aa_t−1+ ε^a_t, (37) ϕ^c_t = ρ^cϕ^c_t−1+ ε^c_t, (38) g_t+1 = (1 − ρ^g)¯g + ρ^gg_t+ ε^g_t, (39) where ε^a_t, ε^c_t and ε^g_t are iid innovations with standard deviations σ^a, σ^c and σ^g. ¯a and ¯g are

I also need to characterize the business cycle policy for public sector wages. I assume that, over the business cycle, the government adjusts the wages of all types of workers proportionally. I consider that the government sets the nominal wage as:

log(w^g,i_t+1

p_t ) = log( ¯w^g) + ι[log(w_t^p

p_t) − log( ¯w^p)] + ϕ^w_t, (40) where w_t^p ≡ ^w_l^ptp^lpt

t and w_t^g ≡ ^w_l^gtg^lgt

t represent the average nominal wage in the private and public sectors. The parameter ι measures the cyclicality of wages. If ι = 0 public sector wages are acyclical. If ι > 0 they are procyclical. If ι < 0 they are countercyclical. ϕ^w_t is an autocorrelated public sector wage shock given by:

ϕ^w_t = ρ^wϕ^w_t−1+ ε^w_t. (41)

I propose and evaluate an alternative simple rule that generates procyclical public sector wages. The government sets the growth rate of public sector wages for the subsequent period Ξ_t+1, such that an aggregate target for the average wage is met:

Rule : Ξ_t+1w^g_tl^g_t

l_t^g = Υ × w^p_tl^p_t

l^p_t . (42)

The idea is that government aims to maintain the same proportion of the average nominal public sector wages relative to the average nominal private sector wages as in steady-state, given by Υ ≡ ^w_w^¯_¯^gp. Alternatively we could interpret the rule as the government maintaining the public wage bill relative to private wage bill proportional to the ratio of public to private sector workers as in Figure 1b in the introduction. Notice that this rule is equivalent to equation (40), when ι = 1, but without wage shocks. In other words, by explicitly assuming this rule, the government eliminates uncertainty around the public sector wage.

3 Calibration

To solve the model, I consider the following functional forms for the matching functions, production functions and preferences.

m^j,i_t = ζ^j,i(u^j,i_t )^ηj(v_t^j,i)^1−ηj, ∀i, j, u(ct) = c^1−σ_t

1 − σ, ν(u) = χ^uu,

f (at, zⁱ, kⁱ) = atzⁱ(kⁱ)^α ∀i F (x_t) =

Ψ((x^¯h_t)^%+ (x^h_t)^%)^ς% + (1 − Ψ)((x^µ_t^¯)^%+ (x^µ_t)^%)^ς%¹_ς g(l^g_t+1) = 

Φ((ω^¯hz^h¯l^g,¯_t+1^h)^%+ (ω^hz^hl^g,h_t+1)^%)^ς% + (1 − Φ)((ω^µ¯z^µ¯l^g,¯_t+1^µ)^%+ (ω^µz^µl^g,µ_t+1)^%)^ς%¹_ς I assume a CRRA utility function with a coefficient of risk aversion σ and linear util- ity of unemployment. For the matching function, the matching elasticity with respect to unemployment, η^j, can be different across sectors but not across types, while the matching efficiency, ζ^j,i, differs across sectors and education but not by ability. For the production function of individual firms, I assume an elasticity of output with respect to the capital per worker of α. The final output is produced by two nested CES functions. Both the skill and unskilled inputs are an aggregation of the low and high ability, with the parameter % determining the elasticity of substitution between types. The final good is then produced by a CES of the skilled and unskilled intermediate inputs with a parameter ς. Ψ governs the importance of the skilled input in production. Finally, the production function of the government has the same elasticity of substitutions between high and low ability workers (%) and between skilled and unskilled inputs (ς) as in the private sector.

The model is calibrated to match the UK economy at a quarterly frequency, drawing largely from the Labour Force Survey (LFS) microdata for the period of 1996-2010. The education attainment of the labour force has improved significantly over the past two decades, as documented in Gomes (2012). I take an average of the period 1996-2010 that places the share of university graduates at 32 percent of the population. I consider that the high and low ability workers have the same mass, so ω^¯h = ω^h = 0.16 and ω^µ¯ = ω^µ = 0.34.

The contribution of skilled workers to the provision of government services, Φ, and their steady-state level ¯g are such that the government hires 37.3 percent of university graduates and 16.7 percent of workers without university degree. These numbers, taken from the LFS, reflect the fact that government hire predominantly skilled workers. Following Gomes (2012), I construct flows data to calibrate the separation rates. I assume that the separation rates are equal for workers of different abilities, but are different by education and sector. The numbers are λ^p,h = 0.012, λ^p,µ = 0.0173, λ^g,h = 0.004, λ^p,µ = 0.006. The private sector has two to three times more separations than the public sector. The unskilled workers are more likely to lose their jobs than skilled workers.

To calibrate the public sector wage premium for skilled workers, I run quantile regressions of the log of net wages of college graduates on a dummy for the public sector. I control for:

sex, industry and occupation dummies, status in previous quarter, tenure, age and its square, marital status, time and region dummies, average hours worked and its square. The sample from 1996 to 2006 includes around 300 thousand observations. I take the coefficients of the public sector dummy of the 25 and 75 percentiles as the premium of the low and high- ability skilled workers. I repeat the regressions for non-college graduates. The public sector wages of the four types are set such that: ^wg,¯h

w^p,¯h = 1.016, ^w_w^g,hp,h = 1.039, ^w_w^{g, ¯}p, ¯^µµ = 1.037 and

w^g,µ

w^p,µ = 1.071. These numbers are consistent with several studies using micro data from the United Kingdom, such as Disney and Gosling (1998) or Postel-Vinay and Turon (2007), that document a wage compression in the public sector within and across education groups.

The matching elasticities with respect to unemployment are set to η^p = 0.4 and η^g = 0.15, estimated by Gomes (2014). The United Kingdom has a unique source on recruitment costs by sector. Every year, the Chartered Institute of Personal Development performs a recruitment practice survey covering 800 organizations from manufacturing, private sector services and public sector services (CIPD (2009)). The costs of recruiting a worker, which encompass advertising and agency costs, are approximately £13000 for a skilled worker in the private sector and £8000 for the public sector, corresponding to 26 and 16 weeks of the UK median income. For a low skilled worker the costs are £3500 and £2000 for private and public sector respectively. The costs of posting vacancies are set to target these numbers (κ^p,h = 1.35, κ^g,h = 0.90, κ^p,µ = 0.14, κ^g,µ = 0.13). The CIPD data also reports vacancy duration. It takes 14.5 weeks to hire a skilled worker in the private sector and 16 weeks in the public sector. For unskilled workers, it takes 5.5 weeks in the private sector and 9.1 weeks in the public. The matching elasticities are set to match these moments (ζ^g,h = 0.73, ζ^p,h = 0.56, ζ^g,µ= 0.99 and ζ^p,µ= 0.98).

The parameter of the private production function Ψ is set to 0.408 to target a college premium of 40 percent, found by regressing the log net wages on a dummy for college education, on average hours and its square. I normalize z^h = z^µ¯ = 1. I link the productivity differences within skilled and unskilled workers to a measure of within group wage dispersion.

I run mincer regression of log net wages on several controls, and retrieve the 25-75 percentile difference of the wage residuals. The difference is 0.461 for skilled and 0.416 for unskilled.

It is a strong assumption to consider that all the wage dispersion is due to productivity differences. Other factors, namely, search frictions might also contribute. Abowd, Kramarz, and Margolis (1999) find that search frictions can explain 7-25 percent of the French inter- industry differential. Tjaden and Wellschmied (2014) find that 13.7 percent of overall wage inequality is due to the presence of search frictions. I assume that 20 percent of the wage

¯h µ

between high and low ability of 0.368 for the skilled and 0.332 for the unskilled.⁶

To accurately predict the welfare and budgetary effects of public sector pay, we have to distinguish the value of unemployment due to home production and due to unemployment benefits. Salom¨aki and Munzi (1999) find that the net replacement rate is 61 percent for low educated workers and 49 percent for high education workers in the United Kingdom. I set χ^b = 0.21 such that the replacement rate of the low ability unskilled worker is 60 percent of the net wage. It implies the replacement rate is between 40 and 45 percent for the high- ability unskilled and low-ability skilled workers, and 30 percent for the high ability skilled workers. I calibrate the utility value of an unemployment (χ^u = 0.37) and the bargaining power of workers (b = 0.35) to target an average unemployment rate of 6 percent and of 7.3 percent for the unskilled, the average values extracted from the LFS (the unemployment rate of the skilled workers is 3 percent). The joint value of unemployment encompassing both utility and unemployment benefits varies from 53 of the net wage for the high-ability skilled worker to 99 percent for the low-ability unskilled worker. On average is between 70-75 percent, near the point suggested by Hall and Milgrom (2008).

Regarding the technology parameters, the elasticity of output with respect to capital α is set to 0.34 to target a labour share of 61.8 percent, the average of the UK economy between 1996-2010. The parameter determining the elasticity of substitution between skilled and unskilled input, ς is set to 0.401, value estimated by Krusell et al. (2000). I assume that the workers of the same skill group are close to perfect substitutes with % = 0.95.

The rest of the parameters are consensual: β is set to 0.99, σ to 2 and the depreciation rate δ to 0.02. I use textbook values [Gal´ı (2008)] for the nominal frictions. The elasticity of substitution between different varieties, ξ, is set to 6, implying a markup of 20 percent. The share of firms not allowed to reset prices, θ, is ²₃. The central bank responds to inflation with φ = 1.5 and with an inertia of ρ^m = 0.8. For the government, I set the income tax equal to 0.2 and the purchase of intermediate inputs such that the total government consumption is 20 percent of GDP , the average from 1996 to 2010 of the UK economy (¯g^int= 0.034). The lump-sum transfers balance the budget in steady-state.

We now turn to the business cycle policies. I assume in the baseline case that the gov- ernment keeps the tax rate constant over the business cycle, and budgetary imbalance is neutralized by lump-sum taxes. The objective is to examine how the volatility of unem- ployment depends on the cyclicality of government wages, ι. To estimate it I regress the

6In Appendix I show the results assuming all wage dispersion is due to productivity differences and one

log of the average public sector wage on the lagged log of average private sector wage. The quarterly data series, from 1970 to 1996 are taken from OECD and are previously detrended using a linear, quadratic and cubic trend. I found a cyclicality of ι = 0.58, implying that public sector wages are less procyclical than private sector wages. I use the residual of the regression to estimate the stochastic components: ρ^w = 0.83 and σ^w = 0.026.

I consider three sources of fluctuations: technology shocks, government employment shocks and cost-push shocks. I estimate the technology process, using OECD detrended quarterly productivity index, resulting in ρ^a = 0.74 and σ^a = 0.010. I follow a similar pro- cedure for the government employment. I found a much higher persistence (ρ^g = 0.96) and a standard deviation σ^g = 0.0005. The volatility of the cost-push shock is such that the standard deviation of inflation is 0.005, average for the period of 1996-2010. The autocorre- lation of the cost-push shock is set to 0.7, implying that the autocorrelation of inflation of zero close to the negative correlation of -0.05 observed in the United Kingdom.

An important element for the business cycle quantitative exercise is the distribution of sector preference Γ. A higher dispersion, implies lower volatility of search in response of shocks that affect asymmetrically the two sectors. I assume a uniform distribution with parameters [ν₁, ν₂].⁷ Given that the search pattern of the unemployed is unobservable, there is no obvious data source to use. I exploit data from Google Trends as a proxy. Google Trends provides indexes of keyword searches reflecting how many times people have “Googled” a specific word or combination of words relative to overall traffic. These indexes are available at a weekly frequency since 2004.⁸ I retrieve the index of keyword searches of “jobs” and another one which include several keyword searches related to the public sector like: “government jobs”, “council jobs”, “nhs jobs” or “army jobs”. The average ratio of the two indexes is 0.14 and the quarterly standard deviation is 0.037. The constructed series has a correlation of 0.9 with the aggregate public sector wage premium, for the common sample between 2004 and 2010. I calibrate the two parameters of the distribution, ν₁ and ν₂ to match these two moments.⁹

7In Appendix I report all the results when Γ follows a logistic distribution,

8Researchers have used these data to forecast: financial markets, labour market, housing market, au- tomobile industry, inflation expectations or private consumption. See the review in Gomes and Taamouti (2014).

9Further details of the calibration and data sources can by found in Appendix.

4 Effect of public sector wages in steady-state

I assume, in the steady-state analysis, that the income tax adjusts to balance the budget.

I start by examining the effects of changing the public sector wages of unskilled and skilled workers separately, shown in Figure 2.

As the government reduces the unskilled wages (top panel), it shifts the composition of public employment from skilled to unskilled workers. Lowering wages have two opposite ef- fects: a wage bill effect and a recruitment effect. As workers become cheaper, the government wants to employ more to save on the wage bill. However, offering lower wages makes the public sector less attractive, fewer unemployed will search in the public sector, making the recruitment more costly. When the government reduces the unskilled wages, the first effect dominates, because there are still unemployed queueing in the public sector. To maintain the same level of services, the government hires more workers but reduces spending on the total wage bill plus recruitment costs.

When the government reduces the wages of unskilled workers, it faces a constraint: the wage has to satisfy the participation constraint of the unemployed. For the baseline calibra- tion, the government cannot cut the wages by more than 7 percent, or the low-ability unskill workers would prefer staying unemployed.

Still, the consequence in the labour market are dramatic. With a 7 percent wage cut, the unemployment rate of the unskilled falls from 7.3 percent to 5.5 percent, keeping the unemployment rate of the skilled constant. Lowering wages shifts the search to the private sector, firms post more vacancies, thus reducing unemployment.

Perhaps surprising, the elasticities of private sector wages with respect to the public sector unskilled wages are negative. Wage cuts in the public sector raise the wages in the private sector. Two effects explain the negative elasticity. First, by lowering unemployment and raising total production and private consumption they entail a wealth effect. As the marginal utility decreases, the utility value of unemployment increases, putting pressure on the wage bargaining. Second, as the government saves on the wage bill, it cuts income taxes and hence the distortions on the wage bargaining and the capital accumulation.

The bottom panel of Figure 2 shows the consequences of reducing skilled wages. First, it shifts the composition of public employment to unskilled workers. In the case of skilled wage cuts, the recruitment effect dominates the wage bill effect. By offering too low wages, only a few devoted skilled unemployed look for public sector jobs. The government faces

Figure 2: Steady-state effects of public sector wages adjustments Unskilled public sector wages only

0.8 0.9 1 1.1

2 6 10

Unemployment Rate

Unskilled public wages relative to baseline

%

All workers Skilled workers Unskilled workers

Min: 0.93

0.8 0.9 1 1.1

10 20 30 40

Public sector employment

Unskilled public wages relative to baseline

%

All workers Skilled workers Unskilled workers

0.8 0.9 1 1.1

15 35 55

Public employment: share of high ability

Unskilled public wages relative to baseline

%

Skilled workers Unskilled workers

0.8 0.9 1 1.1

−0.4 0 0.4 0.8

Elasticity of private sector wages

Unskilled public wages relative to baseline Skilled (high ability)

Skilled (low ability) Unskilled (high ability) Unskilled (low ability)

0.8 0.9 1 1.1

0 8 16 24

Share of unemployed searching in public sector

Unskilled public wages relative to baseline

%

Skilled (high ability) Skilled (low ability) Unskilled (high ability) Unskilled (low ability)

0.8 0.9 1 1.1

15 16 17 18

Wage bill plus recruitment costs

Unskilled public wages relative to baseline

% of GDP

Skilled public sector wages only

0.8 0.9 1 1.1

2 6 10

Unemployment Rate

Skilled public wages relative to baseline

%

All workers Skilled workers Unskilled workers

0.8 0.9 1 1.1

10 20 30 40

Public sector employment

Skilled public wages relative to baseline

%

All workers Skilled workers Unskilled workers

0.8 0.9 1 1.1

15 35 55

Public employment: share of high ability

Skilled public wages relative to baseline

%

Skilled workers Unskilled workers

0.8 0.9 1 1.1

−0.4 0 0.4 0.8

Elasticity of private sector wages

Skilled public wages relative to baseline Skilled (high ability)

Skilled (low ability) Unskilled (high ability) Unskilled (low ability)

0.8 0.9 1 1.1

0 8 16 24

Share of unemployed searching in public sector

Skilled public wages relative to baseline

%

Skilled (high ability) Skilled (low ability) Unskilled (high ability) Unskilled (low ability)

0.8 0.9 1 1.1

15 16 17 18

Wage bill plus recruitment costs

Skilled public wages relative to baseline

% of GDP