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CESifo, a Munich-based, globe-spanning economic research and policy advice institution
Venice Summer Institute 2014
Venice Summer Institute
July 2014
REFORMING THE PUBLIC SECTOR
Organiser: Apostolis Philippopoulos
Workshop to be held on 25 – 26 July 2014 on the island of San Servolo in the Bay of Venice, Italy
INCENTIVES TO WORK AND PRODUCTION
EFFICIENCY IN THE PUBLIC SECTOR
George Economides, Apostolis Philippopoulos and Petros Varthalitis
CESifo GmbH • Poschingerstr. 5 • 81679 Munich, Germany
Tel.: +49 (0) 89 92 24 - 1410 • Fax: +49 (0) 89 92 24 - 1409
E-Mail: [email protected] • www.cesifo.org/venice
Incentives to work and productive efficiency
in the public sector
George Economides ∗, Apostolis Philippopoulos* and Petros Varthalitis #
July 16, 2014
Abstract: This paper develops a general equilibrium growth model with three distinct
social groups, capitalists, private workers and public employees. After solving for the
status quo equilibrium, which can mimic the advantages of employment in the public
sector in the EU, the paper looks for policy reforms that can improve work incentives
and productive efficiency in the public sector. We focus on reforms that establish
parity between employment in the public and the private sector.
Acknowledgements: We thank Christos Kotsogiannis, Dimitris Papageorgiou, Hyun
Park and Vanghelis Vassilatos for discussions. This research is co-financed by the
European Union (European Social Fund - ESF) and Greek national funds through the
Operational Program “Education and Lifelong Learning” of the National Strategic
Reference Framework (NSRF) - Research Funding Program ARISTEIA II_Public
Sector Reform_5328; we are grateful for their support. Any views and errors are ours.
Corresponding author: George Economides, School of Economic Sciences, Athens
University of Economics and Business, 76 Patission street, Athens 10434, Greece.
Tel: +30-210-8203729. Email: [email protected]
∗
#
Athens University of Economics and Business, and CESifo.
Athens University of Economics and Business.
1
1.
Introduction
The 2008-2009 financial and economic crisis, and the subsequent sharp rise in
sovereign debts in most countries, are forcing governments to reform their public
sectors. A particularly debated reform is to establish parity between work conditions at
the private and the public sectors. For instance, according to the lead article in The
Economist, January 8th 2011, “The struggle with public-sector unions should be about
productivity and parity, not just spending cuts”. It is thus useful to study the potential
efficiency gains, but also the distributional conflict, arising from such reforms.
It is widely recognized that employment in the public sector differs from
employment in the private sector. Two distinct differences are in job security and
benefits: in most countries, public employees enjoy a high degree of job security and
receive a wide range of “family-friendly” benefits on top of their salaries that are
unrelated to work performance. 1 There is a lot of anecdotal evidence of these
differences (see e.g. Alesina, 1999, Alesina and Giavazzi, 2006, and The Economist,
January 8th 2011), while examples of “advantages of government employment” in
various countries can be found in a plethora of web links.
In this paper, we first construct a neoclassical general equilibrium model that
incorporates the above characteristics. The model is solved numerically using
common parameter values and European data. In turn, departing from this status quo
solution, we study the quantitative implications of a number of reforms that aim to
strengthen the incentive to work, and hence to improve productive efficiency, in the
public sector. We focus on reforms that establish parity between employment in the
public and the private sector. In particular, in light of the above empirical evidence,
we introduce job insecurity and change the mix between wages and transfers in favor
of the former similarly to the situation in the private sector. We study both aggregate
and distributional implications of these reforms.
Our main results are as follows. Introducing job insecurity like in the private
sector, and/or cutting wages in the public sector, does not help the aggregate economy
substantially. Especially, with cuts in public wages, the work incentives of public
1
These benefits come in the form of medicare, pensions, paid personal days, sick days and holidays,
student loan repayments, child care subsidies, etc. Advantages of employment in the public sector also
come in the form of reduced working hours and a more relaxed work environment. In addition, wage
differentials between the public and the private sector have been reduced in most industrial countries.
2
employees deteriorate further. What does help the aggregate economy is a policy
reform that equalizes the ratio of non-labor transfers to the wage rate in the two
sectors and, at the same time, uses the efficiency savings - being enjoyed by the switch
to a more efficient economy - to cut income taxes. The importance of the ratio of nonlabor transfers to the wage rate should not come as a surprise: Fang and Rogerson
(2011) have shown the importance of this ratio for hours of work.
The problem, as it usually happens, is that this policy reform is not a free
lunch: although it is good for the general interest (private and public output, public
sector efficiency and per capita welfare, they all rise), public employees are hurt
relative to the status quo. Their net income and consumption level fall. These
distributional effects can possibly provide an explanation why public sector unions are
negative to such reforms.
We therefore combine the above reform with a policy that keeps the wage rate
in the public sector above a minimum value. We show that it is possible to find a
wage rate in the public sector that, in combination with the above explained reform
(namely, the equalization of the ratio of non-labor transfers to the wage rate in the two
sectors), allows a rise in the net income and consumption levels of capitalists and
private workers without hurting the net income and consumption level of public
employees relative to the status quo. The latter get, of course, worse off in terms of
utility but this happens simply because they have to work harder than in the status
quo. On the negative side, this new mix of reforms comes at the cost of smaller gains
in aggregate efficiency.
The bottom line is that, although there are options, at the end of the day, there
is a value judgment to be made. We cannot find Pareto efficient reforms. As it
happens in most cases, there is a tradeoff between efficiency and distribution.
The rest of the paper is organized as follows. Section 2 describes informally
the economic environment. Section 3 solves for the status quo economy. Reforms are
studied in section 4. Section 5 closes the paper. An Appendix includes technical
details.
3
2.
Informal description of the economic environment
We study a closed economy populated by three distinct types of households, called
capitalists, private workers and public employees (see also e.g. Ardagna, 2007, and
Economides et al., 2013). Time is discrete and infinite. The model is deterministic.
2.1
Population shares and households’ roles
The population size at time t , N t , is exogenous. Among N t , there is a pool of
identical capitalists or entrepreneurs k = 1,2,..., N tk who own the firms, hold capital and
keep profits; a pool of identical private workers w = 1,2,..., N tw who work in the private
sector; and a pool of identical public sector employees b = 1,2,..., N tb who work in the
public sector. Thus, N tk + N tw + N tb = N t at each t . There are also f = 1,2,..., N t f
identical private firms owned by capitalists. The number of capitalists equals the
number of private firms or each firm is run by a capitalist. For simplicity, the fractions
of the three agents in total population are exogenously set and remain constant over
time. We also rule out occupational choice and mobility across groups. 2
2.2
Production of private and public goods
Private goods are produced by private firms. These firms choose capital, supplied by
capitalists, and labour, supplied by both capitalists and private workers. They also
make use of public infrastructure. The public good is produced by the state or the
government. The government purchases part of the private good produced and hires
public employees to produce a public good. 3 The public good provides utilityenhancing services to households, as well as productivity-enhancing services to
private firms. The private good is converted into the public good by a production
function so that each is expressed in the same units. In order to finance its various
types of public spending, the government levies distorting taxes and issues bonds.
2.3
How private and public workers differ and how we model job insecurity
2
See e.g. Acemoglu (2009, chapter 23) for occupational choice although in a model with capitalists and
workers only.
4
Public and private employees can differ in wages earned, non-labor benefits received
and, perhaps more importantly, job security. Regarding the degree of job security, in
the status quo economy, we assume that only private workers face job insecurity.
Then, in the reformed economy, one possible reform is to allow for job insecurity in
the public sector too.
To model job insecurity, we use the employment lotteries model of Hansen
(1985) and Rogerson (1988) and also used by e.g. Hansen and Prescott (1995). Thus,
agents face an exogenous employment lottery that determines whether or not they are
employed. Although this is a rather stylized model of unemployment, it keeps the
algebra simple and allows minimal deviations from both the associated marketclearing paradigm and the related literature on public employment.
In particular, following the lotteries literature, we assume that there is a standin representative private worker w = 1,2,..., N tw who faces an exogenous probability
0 ≤ ϕ ≤ 1 of retaining his/her job and an exogenous probability 0 ≤ 1 − ϕ ≤ 1 of losing
his/her job in each time period. Following the same literature, we also assume that the
worker can insure himself/herself through (private and public) unemployment
insurance. 4 The worker makes his/her decisions (including employment contracts)
prior to the lottery draw. 5
In turn, one of the policy reforms we consider below is to introduce a similar
type of job insecurity for public employees. In other words, departing form the status
quo case in which there is job insecurity in the private sector only, we will see what
happens when, not only the stand-in representative private worker, but also the standin representative public employee expects, or fears, to be assigned to work by a
similar random lottery.
3
This should be contrasted to most of the literature where it is typically assumed that the government
transforms units of private goods into units of public goods at a one-to-one rate.
4
There is also government insurance via a transfer program. However, unlike private contracts that are
privately optimally chosen, the government forces agents to be part of its transfer program by financing
its spending by compulsory taxation. See also the discussion in Hassler et al. (2005) for the coexistence
of both private and public unemployment insurance in modern societies. Note that we could use private
savings instead of private insurance; the main features would emerge but the model is simpler with
private insurance than with savings.
5
See Hassler et al. (2005) for a case in which private decisions are made after current uncertainty is
realized.
5
3.
The model (status quo economy)
This section formalizes the above story starting with the case in which only private
workers face job insecurity.
3.1
Capitalists’ problem
The lifetime utility of each capitalist k = 1,2,..., N tk is:
∞
∑ β u (c , e , Y
t
t =0
k
t
k
t
t
g
)
(1)
where ctk and etk are each k ’s consumption and work effort respectively, Yt g is per
household public goods and services (i.e. Yt g ≡
Yt g
) and 0 < β < 1 is a time preference
Nt
parameter. In what follows, it is more intuitive to assume that that those employed
work a fixed shift length of h > 0 hours so that the disutility from working a shift of
length h with a degree of effort et is measured by het . Thus, hours of leisure, for
those currently employed, is 1 − het where 1 is the total time endowment. For our
numerical solutions below, we will use a period utility function of the additive form:
u (ctk , etk =
; Yt g ) µ1 log ctk + µ2 log(1 − hetk ) + (1 − µ1 − µ2 ) log Yt g
(2)
where 0 < µ1 , µ2 < 1 are standard preference parameters.
Each k enters period t with predetermined holdings of physical capital and
government bonds, ktk and btk , whose gross returns are rt and it respectively. The
within-period budget constraint of each k is:
(1 + τ tc )ctk + ktk+1 − (1 − δ )ktk + btk+1 − btk = (1 − τ tk )(rt ktk + π tk ) + (1 − τ tl ) wtk etk h + it btk + Gttr ,k
6
(3)
where 0 < δ < 1 is the capital depreciation rate, π tk is dividends received from private
firms, wtk is the wage rate earned by capitalists, Gt tr , k is government transfers to each
k and 0 < τ tk ,τ tl ,τ tc < 1 , are tax rates on capital income, labor income and private
consumption respectively.
Each k chooses {ctk , ktk+1 , btk+1 , etk }t∞= 0 acting competitively. The first-order
conditions include the budget constraint in (3) and:
µ1 (1 − τ tl ) wtk
µ2
=
(1 − hetk )
(1 + τ tc )ctk
(4a)
(1 + τ tc+1 )ctk+1
= β [1 − δ + (1 − τ tk+1 )rt +1 ]
(1 + τ tc )ctk
(4b)
(1 + τ tc+1 )ctk+1
= β (1 + it +1 )
(1 + τ tc )ctk
(4c)
where (4a) is a labor supply condition, while (4b) and (4c) are standard Euler
equations for capital and government bonds respectively.
3.2
Private workers’ problem
The lifetime utility of each private worker, w = 1,2,..., N tw , is:
∞
∑ β u (c
t
w
t
, etw , Yt g )
(5)
t =0
where ctw and etw are w ’s consumption and work effort respectively. For our
numerical solutions below, we will again use:
u (ctw , etw=
; Yt g ) µ1 log ctw + µ2 log(1 − hetw ) + (1 − µ1 − µ2 ) log Yt g
(6)
As said above, private workers face employment uncertainty as in the lotteries
model. If the worker is employed, which happens with probability 0 < ϕ ≤ 1 , the
budget constraint is:
7
(1 + τ tc )ctw,e + µtw ytw =
(1 − τ tl ) wtwetw h + Gttr , w
(7a)
while, if the worker loses his/her job, which happens with probability 0 ≤ 1 − ϕ < 1 , the
budget constraint is:
(1 + τ tc )ctw,u + µtw ytw = ytw + Gt tr ,u
(7b)
where ctw, e and ctw,u are consumption when the worker is employed and unemployed
respectively, wtw is the wage rate earned by private workers when employed, Gt tr , w is
government transfers to each private worker when employed, Gt tr ,u is government
transfers to each agent when unemployed, ytw is the worker’s private unemployment
insurance whose price is denoted as µtw .
We solve the problem working as in e.g. Rogerson (1988) and Hansen and
Prescott (1995). Thus, the private worker maximizes expected utility:
ϕ  µ1 log ctw,e + µ2 log(1 − hetw ) + (1 − µ1 − µ2 ) log Yt g  + (1 − ϕ )  µ1 log ctw,u + (1 − µ1 − µ2 ) log Yt g 
(8)
subject to the single ex ante budget constraint:
ϕ (1 + τ tc )ctw,e + (1 − ϕ )(1 + τ tc )ctw,u + µtw ytw = ϕ (1 − τ tl ) wtwetw h + ϕ Gttr ,w + (1 − ϕ ) ytw + (1 − ϕ )Gttr ,u
(9)
Each w chooses {ctw, e , ctw,u , etw , ytw }t∞=0 acting competitively. The first-order
conditions include the budget constraint in (9) and:
w,e
c=
ctw,u ≡ ctw
t
(10a)
µ1 (1 − τ tl ) wtw
µ2
=
(1 − hetw ) (1 + τ tc )ctw,e
(10b)
µtw = 1 − ϕ
(10c)
8
where (10a) implies that private insurance allows consumption smoothing across
different employment states, (10b) is a standard labor supply condition and (10c)
equates costs and benefits of private insurance.
If we combine these first-order conditions, (10b) can be rewritten as:
µ1 (1 − τ tl ) wtw
µ2
=
(1 − hetw ) ϕ (1 − τ tl ) wtw hetw + ϕ Gttr , w + (1 − ϕ )Gttr ,u
(11)
Gttr , w
Gttr ,u
which implies that work effort decreases with
,
and ϕ . That is,
(1 − τ tl ) wtw (1 − τ tl ) wtw
work effort, or hours of work, fall when transfers relative to net-of-tax labor income
rise or when job security rises. Keep in mind that these are direct effects holding
everything else constant. The overall or general equilibrium effects associated with
changes in tax rates, etc, are studied below. All this is conceptually similar to Fang
and Rogerson (2011), although these authors focus on the effects of labor tax rate on
work hours and also abstain from job insecurity issues.
3.3
Public employees’ problem
Public employees are modeled similarly to private employees with the exception that
(in the status quo economy) they do not face job insecurity. Thus, the lifetime utility
of each b = 1,2,..., N tb is:
∞
∑ β u (c , e , Y
t
b
t
b
t
t
g
)
(12)
t =0
where ctb and etb are b ’s consumption and work effort respectively. For our numerical
solutions below, we will again use:
u (ctb , etb =
; Yt g ) µ1 log ctb + µ2 log(1 − hetb ) + (1 − µ1 − µ2 ) log Yt g
subject to the budget constraint:
9
(13)
(1 + τ tc )ctb =
(1 − τ tl ) wtg etb h + Gttr ,b
(14)
where wtg is the wage rate in the public sector and Gt tr ,b is government transfers to
each public employee.
Each b chooses {ctb , e , ctb ,u , etb , ytb }t∞=0 acting competitively. The first-order
conditions include the budget constraint in (14) and the condition for work effort:
µ1 (1 − τ tl ) wtg
µ2
=
(1 − hetb ) (1 + τ tc )ctb ,e
(15)
which, using the budget constraint, can be rewritten as:
µ1 (1 − τ tl ) wtg
µ2
=
(1 − hetb ) (1 − τ tl ) wtg hetb + Gttr ,b
(16)
It is useful for what follows, to compare (16) to (11). Inspection of these two
equations implies that work effort in the private sector and work effort in the public
sector can differ because of differences in: (i) the degree of job security, ϕ < 1 ; (ii) the
wage rate wtw ≠ wtg ; (iii) the ratio of non-labor benefits to wages,
Gttr , w Gttr ,b
≠ g . One of
wtw
wt
them, or a combination of them, is enough to cause differences between work effort in
the private sector and work effort in the public sector.
3.4
Private firms and production of the private good
In each period, each private firm f = 1,2,..., N t f chooses capital, kt f , and the two types
of labor services, etf ,k and etf , w , to maximize period-by-period profits:
π t f =ytf − rt kt f − wtwetf , w − wtk etf ,k
(17)
where, for our numerical solutions below, we use a production function of the form:
10
1−α1 −α 2
f α1
=
yt A(kt ) ( A et
f
k
f ,k
+ A et
w
 Yt g 
)  f 
 Nt 
f ,w α2
(18)
where A, Ak , Aw > 0 and 0 < α1 , α 2 < 1 are technology parameters. The part of the
production function related to the labor input is modeled as in Hornstein et al. (2005)
while we also allow the public good to provide production services as in e.g. Barro
(1990). None of these assumptions is important to our main results.
The firm’s first-order conditions are simply:
rt = α1
ytf
kt f
(19a)
ytf
( Ak etf ,k + Awetf , w )
(19b)
ytf
w = α2 A
( Ak etf ,k + Awetf , w )
(19c)
wtw = α 2 Aw
k
t
k
so that profits are:
π t f =(1 − α1 − α 2 ) ytf
3.5
(19d)
Government budget constraint and policy instruments
The period budget constraint of the government is (aggregate quantities are denoted by
capital-letters):
Gtg + Gtw + Gttr , k + Gttr , w + Gttr ,b + Gttr ,u + (1 + it ) Bt = Bt +1 + Tt
(20a)
where Gtg is total public spending on goods and services purchased from the private
sector, Gtw is the total public wage bill, Gttr , k , Gttr , w , Gttr ,k and Gttr ,u are respectively
transfers to capitalists, private workers, public employees and the unemployed people,
Bt is the beginning-of-period total stock of government bonds and Tt denotes total tax
revenues. Total tax revenues are:
11
Tt ≡ τ tc ( Ntk ctk + Ntwctw + Ntb ctb ) + τ tk Ntk (rt ktk + π tk ) + τ tl ( Ntk wtk etk h + ϕ Ntw wtwetw h + Ntb wtg etb h)
(20b)
Therefore, as in e.g. Alesina et al. (2002), we include the three main types of
government spending (purchases of goods and services from the private sector, public
wages, and transfers to individuals). We also include the three main types of taxes
(taxes on consumption, capital income and labor income).
Inspection of (20a-b) implies that, in each time period, there are eleven policy
instruments (Gtg , Gtw , Gttr ,k , Gttr , w , Gttr ,b , Gttr ,u ,τ tc ,τ tk ,τ tl , Bt +1 , N tb ) out of which one has to
adjust residually to satisfy the government budget constraint. Following most of the
related literature, we will assume that, along the transition path, the adjusting
instrument is the end-of-period public debt, Bt +1 , so that the rest can be set
exogenously by the government. At steady state, we will instead set the debt-to-output
ratio as in the data and allow the labor tax rate to be the residually determined
instrument (see also e.g. Mendoza and Tesar, 2005, in a model for the European
Union).
For convenience, concerning public spending instruments, we will work in
terms of their GDP shares. Thus, we define stg ≡
sttr , w ≡
Gttr , w
G tr ,b
G tr ,u
, sttr ,b ≡ t , and sttr ,u ≡ t
Yt
Yt
Yt
G tr , k
Gtg
Gw
, stw ≡ t , sttr , k ≡ t ,
Yt
Yt
Yt
, where Yt denotes total output (see
Appendix 1 for details). Similarly, concerning the number of public employees, we
will work in terms of their population share, vtb ≡
N tb
. It is convenient for what
Nt
follows to define also the population share of capitalists, vtk ≡
N tw
population share of workers follows residually, v ≡
= 1 − vtk − vtb .
Nt
w
t
3.6
State enterprise and production of the public good
12
N tk
. Hence the
Nt
Following most of the related literature, 6 we assume that total public goods and
services, Yt g , are produced using goods and services purchased from the private
sector, Gtg , and total public employment, Ltg . In particular, following e.g. Linnemann
(2009) and Economides et al. (2013), we use a Cobb-Douglas production function of
the form:
Yt g = A(Gtg )θ ( Ltg )1−θ
(21)
where 0 ≤ θ ≤ 1 is a technology parameter. Notice that our modeling can nest most of
the specifications used in the literature. Also notice that both private and public good
production face the same TFP; this is because we do not want our results to be driven
by exogenous factors. The total cost of public good, Gtg + wtg Ltg , is financed by the
government through taxes and bonds (see the government budget constraint above).
3.7
Decentralized competitive equilibrium (DCE) of the status quo economy
Combining the above, we now solve for a DCE. This is for any feasible policy. In this
DCE: (i) all types of households maximize utility acting competitively; (ii) all firms in
the private sector maximize profits acting competitively; (iii) the insurance company
maximizes profits acting competitively (see Appendix 2 for details); (iv) all
constraints are satisfied and (v) all markets clear (see Appendix 3 for details). When
{stg , stw , sttr ,k , sttr ,w , sttr ,b , sttr ,u ,τ tc ,τ tk ,τ tl , vtb }t∞=0 are set by the government, we end up with a dynamic
system
of
16
equations
in
16
endogenous
variables,
{ctk , ctw , ctb , ktk+1 , btk+1 , etk , etw , etb , ytf , ytg , rt , it , wtw , wtk , wtg , π tk }t∞=
0 . This equilibrium system is presented in
Appendix 4. It is solved numerically first for its steady state and then for its transition
path when linearized around the steady state.
3.8
Numerical solution of the status quo economy
In this subsection, we provide a numerical solution of the above economy. We start
with parameterization.
6
See Economides et al. (2013) for details.
13
3.8.1 Parameter values and policy instruments
Table 1 reports the baseline parameter values for technology and preference, as well as
the values of the exogenously set policy instruments, used to solve the model
economy developed above. The time unit is meant to be a year. Regarding parameters
for technology and preference, we use relatively standard values used by the business
cycle literature. Regarding fiscal policy variables, we use data averages of the
Eurozone over 2000-2012.
Table 1 around here
Baseline parameterization
Let us briefly discuss the parameter and policy values summarized in Table 1.
In the private sector production function, the Cobb-Douglas exponents of labour and
capital are set at 0.64 and 0.33 respectively, while the exponent of public capital is set
at 0.03, which is public infrastructure investment as share of output in the data. The
TFP parameter, A , is normalized at 1. The time preference rate, β , is set at 0.99. The
weight given to public goods and services in the utility function is set at 0.1, as is
usually the case in similar studies. The other preference parameters related to private
consumption and leisure, µ1 and µ2 , are set at 0.3 and 0.6 respectively; these
parameter values imply hours of work within usual ranges. The capital depreciation
rate, δ , is set at 0.05. In the public sector production function, the share of public
employment vis-à-vis the share of goods purchased from the private sector, 1 − θ , is
set at 0.67. This value is the sample average of payments to public wages, as a share
of total public payments to inputs used in the production of public goods (see also e.g.
Linnemann, 2009, and Economides et al., 2013). We report that our qualitative results
are robust to changes in all parameter values, including the (relatively unknown) value
of 1 − θ .
Regarding policy variables, the share of public employees in total population,
ν b , is set at 0.2150, which is the average value in the data. The share of capitalists,
defined as those self-employed, is set at 0.1480 again as in the data. The data values of
the output share of public spending on public wage payments, stw , on goods and
services purchased from the private sector, stg , on total transfers, sttr , and on
14
unemployment benefits, sttr ,u , are respectively 0.1051, 0.1020, 0.1970 and 0.0110.
The effective tax rates on consumption, capital income and labor income, τ tc , τ tk and
τ tl , are respectively 0.1938, 0.2903 and 0.3780 in the data, while, at steady state, the
public debt to GDP ratio is set at 0.9.
Regarding the (relatively unknown in the data) allocation of total government
transfers ( sttr ) among the three social groups ( sttr ,k , sttr , w and sttr ,b ), we set the share of
transfers going to entrepreneurs equal to their fraction in population, while we
calibrate the share of transfers going to public employees vis-à-vis to private workers
so as public employees receive the same net income, and hence enjoy the same level
of consumption, as private workers. Thus, our calibration forces the status quo long
run solution to equalize workers’ net income and consumption in the two sectors. To
put it differently, in terms of net income and consumption, workers are indifferent
between working in the private or the public sector. This is similar to Acemoglu and
Verdier (2000). Notice that this can happen when public employees receive a larger
share of government transfers than their fraction in population (see below).
3.8.2 Steady state solution of the status quo economy
Given the parameter and policy values in Table 1, the steady state solution of the
status quo economy is reported in the first column of Table 2. The solution is well
defined and does relatively well at mimicking the GDP ratios of key macroeconomic
aggregates like consumption and capital. The resulting wage rate of the public sector
relative to the wage rate in the private sector is also close to its value in the data. All
these values are produced endogenously. We also report that, when linearized around
its steady state, the above economy is saddle-path stable.
In this solution (first column, Table 2), although - by construction - private
workers and public employees enjoy the same net income, and the same level of
consumption, this is achieved by a different mix of wages and benefits. In particular,
private workers work harder and enjoy a higher labour income than public employees.
On the other hand, public employees enjoy a relatively large share of non-labour
transfers, which compensates for a lower labour income and hence allows them to
enjoy the same net income and the same level of consumption. As a result, in terms of
utility, public employees are better off than private workers, while it is the capitalists
15
that enjoy the highest utility level. We feel this is consistent with the common belief
that, in most European countries as well as in the US, public employees find it
beneficial to exchange lower wages with higher job security and higher benefits.
Table 2 around here
Steady state solutions
It is also worth noticing, in the status quo solution in Table 2, that productive
efficiency in the public sector is lower than in the private sector, where productive
efficiency is typically measured as an output-to-input ratio (see Appendix 5 for
details). 7
4.
Policy reforms
In this section, departing from the steady state solution of the status quo economy, we
study the implications of various exogenous changes in fiscal policy. Motivated by the
discussion in the Introduction, the aim is to study the effects of policy changes aiming
at parity. As we saw above, work effort in the public sector and work effort in the
private sector can differ because of differences in the degree of job security, the wage
rate earned in the two sectors and the amount of income that is being received as a
transfer relative to labor income. Hence, we now study what happens when private
workers and public employees are at par regarding the degree of job security, the
manner the wage rate is determined and the ratio of non-labor transfers to the wage
rate.
4.1
Reforms studied
The reforms studied are listed in Table 3. To understand the logic of our results, and
following usual practice, we start by experimenting with one reform at a time and only
in turn study reform mixes.
7
See Afonso et al. (2005 and 2006) and Angelopoulos et al. (2008) for computations of public sector
efficiency in various countries and various policy areas. See Pestieau (2007) for a critical review and
methodology issues.
16
Table 3 around here
Description of policy reforms
In Reform no. 1, we add job insecurity in the public sector like in the private
sector. Thus, now public employees know that can keep their job with a nonzero
probability 0 < q < 1 only, like their counterparts do in the private sector. In particular,
we set q= ϕ= 0.9 , meaning that both private workers and public employees fear that
there is a probability 10% of losing their jobs. Since we do not want the ex ante
employed public employees to receive higher per capita wages, we also cut the
exogenously set output share of public wages, stw , by 10%.
Reform no. 2 assumes that the wage rate in the public sector is set in a manner
similar to that in the private sector. Specifically, we assume that the wage rate in the
(1 − θ )Yt g
public sector equals the marginal product of labor in this sector, w =
.
Ltg
g
t
Algebraically,
this adds an extra equation to the equilibrium system. We thus also
have an extra endogenous variable and, given the nature of the extra equation, we find
it natural to choose the share of public spending going to public wages, stw , to play the
role of the extra endogenous variable.
Reform no. 3 assumes that the ratio of non-labor transfers to the wage rate is
equal in the two sectors,
Gttr , w Gttr ,b
≠ g . Again this adds an extra equation to the system
wtw
wt
and we now choose the share of transfers allocated to public employees, sttr ,b , to be the
extra endogenous variable.
Reform no. 4 is a mix of reforms that simply combines reforms 1, 2 and 3
together.
Reform no. 5 is like no. 4 with the exception that now the wage rate is
proportional to the wage rate in the private sector, where the degree of proportionality
is discussed below. That is, instead of assuming wtg =
(1 − θ )Yt g
as in Reform no. 4,
Ltg
we now assume wtg = ρ wtp , where the degree of proportionality, 0 < ρ ≤ 1 , is
specified below. The logic of this reform is also explained below.
17
4.2
Adding job insecurity in the public sector and the new equilibrium system
In terms of modeling, the main new ingredient is to add job insecurity in the public
sector. We thus now assume that both private workers and public employees face job
uncertainty and make their decisions prior to the lottery draw. In what follows, we
only model what changes relative to the status quo model in the previous section.
Public sector employees, b = 1,2,..., N tb , are now modeled similarly to private
workers. Thus, if we denote by 0 < q ≤ 1 the probability that a public employee keeps
his/her job, while 0 < 1 − q ≤ 1 is the probability of losing his/her job, then with
probability 0 < q ≤ 1 , the budget constraint is:
(1 + τ tc )ctb ,e + µtb ytb =
(1 − τ tl ) wtg etb h + Gttr ,b
(22a)
while, with probability 0 < 1 − q ≤ 1 , the budget constraint is:
(1 + τ tc )ctb ,u + µtb ytb = ytb + Gt tr ,u
(22b)
where, as above, ctb , e and ctb ,u are consumption when the public employee is employed
and unemployed respectively, Gt tr ,b is government transfers to each public sector
employee when employed whereas Gt tr ,u is government transfers to each public sector
employee when unemployed, ytb is a private unemployment insurance whose price is
µtb .
Thus, the ex ante budget constraint is:
q(1 + τ tc )ctb ,e + (1 − q)(1 + τ tc )ctb ,u + µtb ytb = q(1 − τ tl ) wtg etb h + qGttr ,b + (1 − q) ytb + (1 − q)Gttr ,u (23)
Each b chooses {ctb , e , ctb ,u , etb , ytb }t∞=0 acting competitively. The first-order
conditions include the budget constraint in (23) and:
ctb , e = ctb ,u = ctb
(24a)
18
µ1 (1 − τ tl ) wtg
µ2
=
(1 − hetb ) (1 + τ tc )ctb ,e
(24b)
µtb = 1 − q
(24c)
If we combine these first-order conditions, the labor supply condition (24b)
can be rewritten as:
µ1 (1 − τ tl ) wtg
µ2
=
(1 − hetb ) q(1 − τ tl ) wtg hetb + qGttr ,b + (1 − q)Gttr ,u
(25)
which is like (11) above for the private worker and can give (16) as a special case.
The new DCE system, as well as the associated new market-clearing
conditions, is presented in the Appendix (see Appendices 3 and 4).
4.3
Numerical solution of the steady state of the reformed economy
In this subsection, we provide numerical steady state solutions under the reforms
listed in Table 3. The parameterization is as in Table 1 except that now
q= ϕ= 0.9 < 1 . The solution, under each type of reform, is reported in Table 2. Notice
that some ratios remain unchanged across regimes; this happens because of the
functional forms used in the numerical solution but it is not important to the main
results.
We start with Reform no. 1 and compare it to the status quo. On the positive
side, Reform no. 1 improves work incentives in the public sector and enhances private
output. On the negative side, public output and per capita welfare all fall. But all these
effects are marginal in magnitude so we claim that the introduction of job insecurity in
the public sector does not appear by itself to generate substantial social benefits.
In Reform no. 2, work incentives in the public sector clearly deteriorate and
this is damaging for public sector output. This happens mainly because the resulting
fall in wages in the public sector further damages the work incentive of public
employees. Public sector efficiency rises but this happens only because public wages
have fallen substantially. Thus, as with job insecurity, the attempt to link public wages
to productivity in the public sector cannot by itself lead to social benefits.
19
Reform no. 3 is different. Now, work incentives in the public sector clearly
improve and, at same time, private output, public output and per capita utility all rise.
The same applies to public sector efficiency which rises substantially. Notice that a
larger tax base allows a cut in the labour tax rate that further stimulates the aggregate
economy. On the negative side, public employees get worse off; public wages and
transfers allocated to public employees both fall, so their net income and consumption
fall. Reform no. 4 (which is a mix of Reforms 1, 2 and 3) is like Reform no. 3, both
qualitatively and quantitatively, indicating that the main results are driven by Reform
no. 3.
Taking stock of the results so far, the key reform is no. 3. It is good for the
general interest to equalize the ratio of non-labor transfers to the wage rate in the two
sectors,
Gttr , w Gttr ,b
≠ g , and, at the same time, to use the efficiency savings - enjoyed by
wtw
wt
the switch to a more efficient economy - to cut labor taxes. The latter creates a second
round of social benefits. The problem, as it happens in most cases, is that this reform
is not a free lunch: although per capita private and public output, public sector
efficiency and per capita welfare all rise, public employees get impoverished by a fall
in their net income and get worse off relative to the status quo.
Is it possible to find a mix of policy reforms that are not only socially
beneficial, but also do not hurt the net income and the consumption level of public
employees? Since the above reforms were accompanied by a substantial fall in public
wages that impoverished public employees, we now search for a mix that combines
the key reform (Reform no. 3) with a policy that keeps the wage rate in the public
sector above a minimum value. This is why, in Reform no. 5, we replace the condition
wtg =
(1 − θ )Yt g
with the condition wtg = ρ wtp , where 0 < ρ ≤ 1 . As said above, in all
g
Lt
other aspects, Reform no. 5 is like Reform no. 4. In particular, we choose/calibrate
0 < ρ ≤ 1 so as public employees maintain the same net income and the same
consumption level as in the status quo economy. Our solutions show that, under our
baseline parameterization in Table 1, this happens when ρ = 0.975 . Then, the results
reported in the last column of Table 2 imply that, in this case, there is an increase in
net income and consumption levels of capitalists and private workers without hurting
20
the net income and consumption level of public employees (the latter are of course
worse off in terms of utility but this happens only because they have to work harder
than in the status quo). Also, private output, public output as well as per capita welfare
all rise relative to the status quo, although now this rise is smaller than in Reform no.
3. On the negative side, since public wages are relatively high, public sector efficiency
falls relative to the status quo. In sum, we cannot avoid the classical tradeoff between
efficiency and equity.
We report that for values of ρ above 0.975, public employees get better off
but at the cost of the general interest, while the opposite holds for value of values of
ρ below 0.975. There is thus a genuine tradeoff and a social value judgment to be
made.
5.
Concluding remarks
We searched for policy reforms that can strengthen the incentive to work, and hence
improve public sector efficiency. Using a general equilibrium setup with
heterogeneous agents and a potential conflict of interests, we showed that the adoption
of specific reforms, the main feature of which was the establishment of parity between
employment in the public sector and employment in the private sector, can improve
both aggregate and public sector efficiency but, in most cases, at the cost of making
the existing public employees worse off.
21
References
Acemoglu D. (2009): Introduction to Modern Economic Growth. Princeton University
Press, Princeton.
Acemoglu D. and T. Verdier (2000): The choice between market failures and
corruption, American Economic Review, 90, 194-211.
Afonso A., L. Schuknecht and V. Tanzi (2005): Public sector efficiency: An
international comparison, Public Choice, 123, 321-347.
Afonso A., L. Schuknecht and V. Tanzi (2006): Public sector efficiency: Evidence for
new EU member states and emerging markets, ECB Working Paper, no. 581.
Alesina A. (1999): Too large and too small governments, in Economic Policy and
Equity, edited by V. Tanzi, K. Chu and S. Gupta, International Monetary Fund,
Washington, D.C.
Alesina A. and F. Giavazzi (2006): The Future of Europe: Reform or Decline. MIT
Press, Cambridge, Mass.
Angelopoulos K., A. Philippopoulos and M. Tsionas (2008): Does public sector
efficiency matter? Revisiting the relation between fiscal size and economic growth in
a world sample, Public Choice, 137, 245-278.
Ardagna S. (2007): Fiscal policy in unionized labor markets, Journal of Economic
Dynamics and Control, 31, 1498-1534.
Atkinson A. and J. Stiglitz (1980): Lectures on Public Economics. McGraw Hill,
London.
Economides G., D. Papageorgiou, A. Philippopoulos and V. Vassilatos (2013):
Smaller public sectors in the euro area: aggregate and distributional implications,
CESifo Economic Studies, 59, 536-558.
Fang L. and R. Rogerson (2011): Product market regulation and market work: A
benchmark analysis, American Economic Journal: Macroeconomics, 3, 163-188.
Hansen G. D. (1985): Indivisible labor and the business cycle, Journal of Monetary
Economics, 16, 309-327.
Hassler J., P. Krusell, K. Storesletten and F. Zilibotti (2005): The dynamics of
government, Journal of Monetary Economics, 52, 1331-1358.
Hassler J., K. Storesletten and F. Zilibotti (2007): Democratic public good provision,
Journal of Economic Theory, 133, 127-151.
22
Hillman A. (2009): Public Finance and Public Policy. Cambridge University Press,
second edition, Cambridge.
Hornstein A., P. Krusell and G. Violante (2005): The effects of technical change on
labor market inequalities, in Handbook of Economic Growth, volume 1B, edited by P.
Aghion and S. Durlauf, North-Holland, Amsterdam.
Judd K. (1985): Redistributive taxation in a simple perfect foresight model, Journal of
Public Economics, 28, 59-83.
King R. and S. Rebelo (1999): Resuscitating real business cycles, in Handbook of
Macroeconomics, volume 1B, edited by J. Taylor and M. Woodford, North-Holland,
Amsterdam.
Linnemann L. (2009): Macroeconomic effects of shocks to public employment,
Journal of Macroeconomics, 31, 252-267.
Mendoza E. and L. Tesar (2005): Why hasn’t tax competition triggered a race to the
bottom? Some quantitative lessons from the EU, Journal of Monetary Economics, 52,
163-204.
Pappa E. (2009): The effects of fiscal shocks on employment and the real wage,
International Economic Review, 50, 217-244.
Pestieau P. (2007): Assessing the performance of the public sector, CREPP Working
Paper, no. 2007/03, University of Liege.
Rogerson R. (1988): Indivisible labor, lotteries and equilibrium, Journal of Monetary
Economics, 21, 3-16.
23
APPENDIX
Appendix 1: Public spending categories expressed as shares of output
Gtg
(a) G = s N yt so that Gt =
= stg vtk ytf .
Nt
g
t
g
t
k
t
g
f
Gtw = stw N tk ytf
(b)
so
Gt w =
that
Gtw
= stw vtk ytf .
Nt
g g
w k f
, Gtw s=
=
Gtw w=
wtg N tb qetb h=
wtg vtb qetb h or wtg =
t Lt
t vt yt
(c) Gttr , k = sttr , k N tk ytf , so that Gt tr , k =
tr , w
t
(d) G
=s
tr , w
t
k
t
tr , w
t
f
N yt , so that G
(e) Gttr ,b = sttr ,b N tk ytf so that Gttr ,b =
In
turn,
since
stwvtk ytf
.
vtb qetb h
Gttr , k
= sttr , k ytf .
k
Nt
k
Gttr , w
tr , w vt
≡
=
st
yf .
w
w t
ϕ Nt
ϕ vt
Gttr ,b tr ,b vtk f
= st
y where sttr ,k + sttr , w + sttr ,b ≡ sttr , and
b
b t
qN t
qvt
where sttr is the share of transfers in the data.
tr , u
t
(f) G
= s N y so that Gt
tr , u
t
k f
t t
tr , u
Gttr ,u
vtk
tr , u
=
= st
yf .
w
b
w
b t
(1 − φ ) N t + (1 − q) N t
(1 − φ )vt + (1 − q)vt
Appendix 2: The insurance company
The insurance company maximizes period-by-period profits by choosing a supply of
unemployment insurance, denoted as I . Thus, it maximizes µtw I − (1 − ϕ ) I . Since
µtw= (1 − ϕ ) from the worker’s optimization problem, the insurance company is at its
break-even point.
Appendix 3: Market-clearing conditions
The equations below are the market-clearing conditions in the general case in which
there is job uncertainty in both the private and the public sector. The market-clearing
conditions of the status quo model, in which there is job uncertainty in the private
sector only, follows as a special case if set qt = 1 .
The market-clearing conditions in the labor markets are:
24
N t f etf ,k= N tk etf ,k= N tk etk h ⇒ etf ,k= etk h
N t f etf , w= N tk etf , w= ϕ N twetw h ⇒ etf , w= ϕ
ν tw w
e h
ν tk t
Ltg = qN tb etb h
The market-clearing conditions in capital, bond and dividend markets are respectively:
N t f kt f = N tk ktk ⇒ kt f = ktk
N tk btk = Bt
N t f π t f= N tk π tk ⇒ π t f= π tk
Finally, the market-clearing condition in the goods market (this is also the economy’s
resource constraint) is:
N tk ctk + N tw ctw + N tb ctb + N tk (ktk+1 − (1 − δ )ktk ) + Gtg = N t f ytf = N tk ytf
Appendix 4: Decentralized Competitive Equilibrium (I need to rewrite some
equations using the log linear utility function)
The system below summarizes the DCE in the general case in which there is job
uncertainty in both the private and the public sector. The DCE of the status quo
model, in which there is job uncertainty in the private sector only, follows as a special
case if set qt = 1 .
Capitalists:
1/ ξ
 µ1 (1 − τ tl ) wtk 
e h=
c
k σ 
 µ2 (1 + τ t )(ct ) 
(A.4a)
(1 + τ tc+1 )(ctk+1 )σ = β (1 + τ tc )(ctk )σ [1 − δ + (1 − τ tk+1 )rt +1 ]
(A.4b)
β (1 + τ tc )(ctk )σ (1 + it +1 )
(1 + τ tc+1 )(ctk+1 )σ =
(A.4c)
k
t
Private workers:
25
(1 + τ tc )ctw= ϕt (1 − τ tl ) wtwetw h + ϕt Gttr , w + (1 − ϕt )Gttr ,u
(A.4d)
1/ ξ




µ1 (1 − τ tl ) wtw


w
et h = 
σ 
l
w w
tr , w
tr ,u
 µ (1 + τ c )  ϕt (1 − τ t ) wt et h + ϕt Gt + (1 − ϕt )Gt  
 
t 
 2
(1 + τ tc )

 
where we use Gttr , w = sttr , w
(A.4e)
vtk
vtk
tr ,u
tr ,u
f
and
=
y
G
s
ytf
t
t
ϕt vtw t
(1 − ϕt )vtw + (1 − qt )vtb
Public employees:
(1 + τ tc )ctb = qt (1 − τ tl ) wtg etb h + qt Gttr ,b + (1 − qt )Gttr ,u
(A.4f)
1/ ξ




µ1 (1 − τ tl ) wtg


b
et h = 
σ

l
g b
tr ,b
tr ,u
 µ (1 + τ c )  qt (1 − τ t ) wt et h + qt Gt + (1 − qt )Gt  
 
2
t 
(1 + τ tc )

 

where we use Gttr ,b = sttr ,b
(A.4g)
vtk f
vtk
tr ,u
tr ,u
and
y
=
G
s
yf .
t
t
b t
w
b t
qt vt
(1 − ϕt )vt + (1 − qt )vt
Other equations:
=
vtk ytf A(vtk ktk )α1 ( Ak vtk etk h + Awϕt vtwetw h)α 2 (Yt g )
1−α1 −α 2
ytf
rt = α1 k
kt
(A.4h)
(A.4i)
vtk ytf
w = α2 A
w w
( Ak vtk etk h + Awϕν
t t et h)
(A.4j)
vtk ytf
w = α2 A
w w
( Ak vtk etk h + Awϕν
t t et h)
(A.4k)
w
t
k
t
w
k
( stg + stw + sttr ,k + sttr , w + sttr ,b + sttr ,u )vtk ytf + (1 + it )bt = bt +1 +
+τ tc (vtk ctk + vtwctw + vtb ctb ) + τ tk vtk (rt ktk + π tk ) + τ tl (vtk wtk etk h + ϕt vtw wtwetw h + qt vtb wtg etb h)
(A.4l)
Yt g ≡ ytg =
A( stg vtk ytf )θ (qt vtb etb h)1−θ
(A.4m)
vtk ctk + vtw ctw + vtb ctb + vtk (ktk+1 − (1 − δ )ktk ) + stg vtk ytf = vtk ytf
(A.4n)
26
stwvtk ytf
w = b b
vt qt et h
(A.4o)
π tk =(1 − α1 − α 2 ) ytf
(A.4p)
g
t
We thus have a system of 16 equations in 16 endogenous variables.
Exogenous stochastic variables
The two employment probabilities follow:
ϕ
1− ρϕ
(ϕt −1 ) ϕ eε t
1− ρ q
(qt −1 ) q eε t
ϕt = (ϕ0 )
qt = (q0 )
ρ
ρ
q
where the shocks capture employment volatility.
Appendix 5: Measures of productive efficiency
Public sector efficiency is defined as public output as a share of public sector
expenditure that the government allocates to achieve this particular output. Thus,
PubSEt ≡
Yt g
ytg
=
Gtg + Gtw stg vtk ytf + stwvtk ytf
where Yt g is “imputed” from the model solution.
Similarly, we measure private sector efficiency. Thus,
Pr ivSEt ≡
ytf
vtw w
rt k + w [e + ϕt k et ]
vt
k
t
p
t
k
t
27
Table 1: Baseline parameterization
Parameters
and policy
instruments
Description
Value
α1
α2
Share of capital in private production
0.33
Share of labor in private production
0.64
1−θ
Share of public employment in public production
0.67
δ
Capital depreciation rate
0.05
β
Rate of time preference
0.99
µ1
Preference parameter on private consumption in utility
0.3
µ2
Preference parameter on leisure in utility
0.6
sw
Public wage payments as share of GDP (data)
0.1051
sg
Public purchases as share of GDP (data)
0.1020
s tr
Public transfers as share of GDP (data)
0.1970
s tr ,k
Public transfers as share of GDP to capitalists (set)
0.0292
s tr , w
Public transfers as share of GDP to private workers (calibrated)
0.0731
s tr ,b
Public transfers as share of GDP to public sector employees (calibrated)
0.0948
s tr ,u
Public transfers as share of GDP to unemployed persons (data)
0.0110
τc
Tax rate on consumption (data)
0.1938
τk
Tax rate on capital income (data)
0.2903
τl
Tax rate on labor (data)
0.3780
B /Y
Public debt as a share of GDP (data)
0.9
vk
Capitalists as share of population (data)
0.1480
vw
Workers as share of population (data)
0.6370
vb
Public employees as share of population (data)
0.2150
1−ϕ
Probability of unemployment for private workers (set)
0.10
1− q
A
Probability of unemployment for public sector employees (set)
0 and 0.10
Long-run TFP (set)
1
28
Table 2: Steady state solutions
Variable
Status Quo
Reform 1
Reform 2
Reform 3
Reform 4
Reform 5
ck
0.3134
0.3165
0.3275
0.3559
0.3609
0.3210
ek
0.1454
0.1470
0.1547
0.1651
0.1672
0.1467
kk
9.7242
9.7563
9.7639
10.1206
10.1607
9.9057
cw
0.2504
0.2527
0.2637
0.2892
0.2933
0.2562
ew
0.3174
0.3190
0.3193
0.3215
0.3231
0.3190
cb
0.2504
0.2452
0.2061
0.1553
0.1447
0.2504
eb
0.1855
0.1913
0.1438
0.3044
0.3196
0.3189
wg / w p
0.8380
0.8172
0.6215
0.5238
0.4908
0.977
y
0.3693
0.3705
0.3708
0.3844
0.3859
0.3762
yg
0.0391
0.0373
0.0330
0.0553
0.0533
0.0528
c/ y
k/y
b/ y
0.7032
0.7032
0.7032
0.7032
0.7032
0.7032
3.8968
3.8968
3.8968
3.8968
3.8968
3.8968
0.9
0.9
0.9
0.9
0.9
0.9
τ
sw
s tr ,b
uk
0.2462
0.2354
0.1972
0.1363
0.1203
0.2368
0.1051
0.0946
0.0597
0.1051
0.0925
0.1886
0.0948
0.0948
0.0948
0.0129
0.0121
0.0241
-0.7664
-0.7695
-0.7768
-0.7077
-0.7087
-0.7303
uw
-0.9457
-0.9491
-0.9486
-0.8712
-0.8720
-0.9102
ub
u
-0.8626
-0.8653
-0.9079
-1.0661
-1.0811
-0.9170
-0.9013
-0.9045
-0.9144
-0.8889
-0.8928
-0.8850
PubSE
0.5116
0.5117
0.5511
0.6943
0.7098
0.4825
PrivSE
1.0309
1.0309
1.0309
1.0309
1.0309
1.0309
PubSE /
PrivSE
0.4963
0.4963
0.5345
0.6735
0.6885
0.4680
l
29
Table 3: Description of policy reforms
Reform
Description
1
decrease in q and
sw
(by 10%)
2
wg = marginal product of Lg
3
G tr ,b / wg = G tr , w / wp
4
G tr ,b / wg = G tr , w / wp
wg = marginal product of Lg
decrease in q
G tr ,b / wg = G tr , w / wp
5
wg = ρ wp
decrease in q
30