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Public providers, or private providers,
of public goods?
A general equilibrium study
Apostolis Philippopoulos, George
Economides and Vangelis Vassilatos
Public providers, or private providers, of public goods?
A general equilibrium study
September 1, 2011
George Economides*, Apostolis Philippopoulos, and Vangelis Vassilatos*
Abstract: This paper studies the difference between public production and public finance of
public goods in a dynamic general equilibrium model. By public production, the public good
is produced by the government itself using public employees and goods purchased from the
private sector as inputs. By public finance, the public good is produced by cost-minimizing
private providers with the government financing their costs. When the model is calibrated to
match fiscal data from the UK economy, we find that a switch from public production to
public finance has substantial aggregate and distributional implications. Public providers
cannot beat private providers in terms of aggregate efficiency. The key policy message is
that there are Pareto benefits from a mix of reforms that combines: (i) a transition to costminimizing private providers that allows the government to achieve efficiency savings (ii)
redistributive transfers that compensate those previously working as public employees (ii) a
reduction in distorting income taxes made affordable by efficiency savings.
Keywords: Public goods, growth, welfare.
JEL classification: H4, D9, D6.
Acknowledgements: We thank Kostas Angelopoulos, Fabrice Collard, Saqib Jafarey, Jim
Malley, Dimitris Papageorgiou, Evi Pappa, Hyun Park and Heraklis Polemarchakis for
discussions and comments. We also thank seminar participants at Bilgi University, Istanbul.
Any remaining errors are ours.
*


Athens University of Economics and Business.
Athens University of Economics and Business, and CESifo.
Corresponding author: Apostolis Philippopoulos, Department of Economics, Athens University of
Economics and Business, 76 Patission street, Athens 10434, Greece. Tel: 0030-210-8203357, fax: 0030-2108203301, email: [email protected]
1.
Introduction
Concerning the provision of public goods and services, an important distinction is between
public production and public finance. In the former case, the goods are produced by the
government itself. In particular, the government hires public employees and purchases final
goods from the private sector to produce public goods and services. In the latter case, public
goods and services are produced by private firms, the so-called private providers, with the
government financing the cost of production of an agreed-upon quantity. Examples of public
goods and services that can belong to either category include hospitals, television and radio,
schools, prisons, environmental protection, most services provided by local authorities, etc.
The issue of public goods provision has attracted increasing interest in both academic
and policy circles. In academia, production and finance are two distinct ways of public goods
provision. For instance, in their early classic textbook, Atkinson and Stiglitz (1980, p. 482)
emphasize that “the two are often confused, though both logically and in practice they are
distinct”.1 In policy, there is a big debate nowadays on the role of the state and, in particular,
the idea of opening up public services to new providers.
For instance, in the UK, reforms are designed to encourage “any qualified provider of
public goods” (The Observer, 22.05.2011, p. 7) and “across much of the public sector, from
health and education to local authorities and prisoner rehabilitation, the provision of public
services is increasingly being contracted out to private suppliers” (The Economist, January
22nd, 2011, p. 41). But, at the same time, the British Deputy PM, Nick Clegg, questions
private sector involvement saying that the real issue is about “diversifying providers” and
that this does not extend to a belief “that private providers are inherently better than publicsector providers” (The Guardian, 10 February 2011, p. 15).
What are the implications of switching from public production to public finance? Can
public providers beat private providers? Is this switch good for the general interest and bad
for public employees? If yes, is there a mix of reforms that can be good for both private and
public employees?
The present paper tries to answer the above questions. To the best of our knowledge,
so far there has not been an attempt to study, and quantify, the differences between public
production and public finance in a dynamic general equilibrium setup. We fill this gap by
1
There is a rich taxonomy of public goods and services depending on the way of provision, financing and
distribution. See e.g. Cullis and Jones (1998, chapter 5).
1
studying issues of both efficiency and redistribution, where efficiency has to do with per
capita output and welfare, while redistribution refers to differences in income and welfare
between private and public employees. When one studies reforms, efficiency gains need to
be traded-off against distributional implications.
We build upon the neoclassical growth model. We first model the case of public
production. Following a growing macroeconomic literature (see e.g. Ardagna, 2007), there
are two distinct groups of households: those that work in the private sector and those that are
employed in the public sector. The latter (called public employees), together with goods
purchased from the private sector, are used as inputs in the government production function.
Calibrating the model to match the tax-spending data of the UK economy over 1990-2010,
we solve it and specify, among other variables, the time-path of public goods as induced by
the existing tax-spending policy mix. Then, using this “status quo” solution as a point of
departure, we study what would change if, other things equal, the same time-path of public
goods were produced by private firms, the so-called private providers. These firms produce
the amount of public goods ordered by the government by solving a cost-minimization
problem with the government financing their total cost. We also study the case in which the
same amount of public goods continues to be produced by the public sector but now, again
other things equal, public firms minimize their costs like their private counterparts do in the
case of public finance.2 These three model economies (namely, the status quo one, the one
with cost-minimizing private providers and the one with cost-minimizing public providers)
are directly comparable. It is worth emphasizing that, for comparability reasons, we assume
that, not only the amount of public goods produced, but also the number of households
employed in the production of public goods, remains the same across all regimes.
There are four main results. First, a switch from the status quo economy to an
economy with cost-minimizing private providers increases the welfare of private employees
but makes public employees worse off; the latter happens because the wages of those
involved in the production of public goods falls when it is private providers that supply these
goods. The same switch allows the government to make efficiency savings. Second, the
effect of this switch on per capita output and per capita welfare depends crucially on the
method of public financing. When the efficiency savings achieved by the government through the use of private providers - are used to cut distorting income taxes, then per capita
2
This is similar to what Atkinson and Stiglitz (1980, chapter 15.3) call public production efficiency in the
sense that the state enterprise chooses its optimal mix of inputs.
2
output and welfare also rise. Third, when we assume that it is public providers/enterprises
that choose inputs in a cost-minimizing way, the solution is very similar to the status quo
case in which the associated variables are exogenously set at their data averages. This seems
to imply that in the UK, over 1990-2018, the public sector has exhausted its role, at least in
terms of aggregate efficiency, as a provider of public goods and services. Fourth, since,
depending also on the public financing method, there can be aggregate efficiency gains from
switching to private providers, we show that the government can design a redistributive
transfer scheme that makes everybody better off relative to the status quo, including those
previously employed in the public sector.
Although we are aware that one should treat quantitative results with caution, our key
policy implication is robust and, we believe, intuitive: If the government wants to increase
the aggregate pie and also make everybody better off, then it should adopt a mix of reforms
that: (i) assigns the production of public goods and services to cost-minimizing private
providers (ii) redistributes transfers to compensate those previously employed as public
employees (iii) uses the efficiency savings, achieved through the use of private providers, to
cut distorting income taxes.
We wish to clarify two things at the outset. First, we focus on polar cases. For
instance, in the status quo economy, we assume that there is public production only. But we
are aware that actually some public services have been contracted out to private suppliers
already. At the other end, in the reformed economy, we assume that there are private
providers only with the government financing their costs. But we are aware that some public
production is always desirable (e.g. police and courts). In any case, our main results are not
expected to be affected by the presence, or not, of such public goods; one could take them as
given, and then compare public production versus public finance of the remaining public
goods. Second, we do not take a stance on the socially optimal amount of public goods. We
just take the size/mix of public spending, the share of public employees in population, and
the tax rates, as in the data, and compute the induced amount of public goods by using a
relatively standard general equilibrium model. In turn, we ask what would have happened in
the case in which the same amount of public goods, socially optimal or not, were supplied by
cost-minimizing firms with the government just financing their costs. This is consistent with
the Mirrlees Review in the UK (Mirrlees et al 2010, 2011) that also takes public spending as
3
given and looks at the efficiency of the tax system. Here, similarly, we look at the efficiency
of the system of public goods provision.3
The rest of the paper is organized as follows. Section 2 models the status quo case of
public production. Section 3 models the case of private providers. Their long-run comparison
is in Section 4. Section 5 asks whether public providers can beat private providers. Section 6
looks for Pareto improving policy packages. Section 7 studies transitional dynamics. Section
8 reports on robustness. Section 9 closes the paper.
2.
An economy with public production of public goods
We add public employees, used as an input in the production of public goods and services, to
the baseline neoclassical growth model. Consider a two-sector general equilibrium model in
which private firms choose capital and labor supplied by private employees to produce a
private good, while the government purchases part of the private good and hires public
employees to produce a public good. The latter provides utility-enhancing services to all
households. The private good is converted into the public good by a production function so
that each can be expressed in the same units. To finance total public spending, including the
cost of the public good, the government levies distorting taxes and issues bonds. Thus,
irrespectively of the producer, we assume that public goods are provided freely without user
charges.4 For simplicity, the model is deterministic. Time is discrete and infinite.
Our model in this section is similar to that used by most of the related
macroeconomic literature (in particular, Ardagna, 2007), in the sense that the roles of private
and public employees are distinct, there is no labor mobility between the private and public
sector, and economic policy is exogenous.5
3
For an early review of models on the optimum provision of public goods, see Atkinson and Stiglitz (1980,
chapters 15 and 16). See e.g. Angelopoulos et al (2011a, b) for exogenous policy reforms in the UK economy
based on computable general equilibrium models.
4
See also e.g. Atkinson and Stiglitz (1980, chapter 16) for public goods provided without charge to all
members of society. On the other hand, in chapter 15, the same authors study the pricing of publicly provided
(private) goods.
5
Fernández-de-Córdoba et al (2010) provide a survey of this literature. On the other hand, see Quadrini and
Trigari (2008) for job search and matching mechanisms.
4
2.1 Population composition and agents’ economic roles
Total population is N t at time t . Among N t , there are p  1, 2,..., N tp identical households
that work in the private sector and b  1, 2,..., N tb identical households that work in the public
sector, where N tb  N tp  N t . There are also f  1, 2,..., N t f identical private firms. The
number of private firms equals the number of households that work in the private sector,
Nt f  N tp , or equivalently each household employed in the private sector owns one private
firm. This population composition allows us to avoid scale or size effects in equilibrium. The
population sizes, N t and N tb , or equivalently the share of public employees in population,
N tb
, are exogenous (defined below).
Nt
 tb 
Thus, there are four agents in the economy: households that work in the private sector
(private employees), households that work in the public sector (public employees), private
firms that produce the private good and are owned by private employees, and a consolidated
public sector that also produces the public good. All households consume, work, and can
save in capital and bonds subject to transaction costs.
Note that, by allowing both groups of households to participate in asset markets, we
enrich the related literature in which either public employees do not save (see e.g. Ardagna,
2007), or there is a representative household that allocates its work time between working in
the private and the public sector (see e.g. Finn, 1998, Cavallo, 2005, Pappa, 2009,
Linnemann, 2009, Forni et al, 2009, and Fernández-de-Córdoba et al, 2010).
Also note that, since the focus of the paper is on the implications of policy reforms,
we construct the baseline model, and choose its parameterization, so as private and public
employees differ only in the way their wages are formed rather than in other ways like
participation, or not, in asset markets.
2.2 Households working in the private sector
The lifetime utility of each household working in the private sector, p  1, 2,..., N tp , is:

  u (c
t
t 0
p
t
, etp , Yt g )
(1)
5
where ctp and etp are p ’s consumption and work hours respectively; Yt g is per capita public
goods and services;6 and 0    1 is a time preference parameter.
The period utility function is (see also e.g. Christiano and Eichenbaum, 1992):
u (ctp , utp , Yt g )  log(ctp  Yt g )  
(etp )1
1 
(2)
where  ,  ,   0 are preference parameters. Thus, ctp   Yt g is composite consumption,
where public goods and services influence private utility through the parameter  .
Each household p enters period t with predetermined holdings of physical capital
and government bonds, ktp and btp , whose gross returns are rt and t respectively. The
within-period budget constraint of each p is:
(1   tc )ctp  itp  dtp  (1   tk )(rt ktp   tp )  (1   tl ) wtp etp  t btp  Gttr , p
(3a)
where itp is savings in the form of physical capital; dtp is savings in the form of government
bonds;  tp is dividends received from private firms;7 wtp is the wage rate in the private
sector; Gttr , p is government transfers to each p ; and 0   tk ,  tl ,  tc  1 are tax rates on capital
income, labor income and private consumption respectively. Regarding notation, note that
economy-wide quantities, which are treated as given by private agents, are denoted by
capital-letters.
The laws of motion of physical capital and government bonds for each p are:
k
6
p
t 1
 (1   )kt  it 
Thus, Yt g 
p
p
 p , k  kt p 
2
 
2  Yt 
(3b)
Yt g
, where Yt g is total public goods and services (see below).
Nt
7
We assume that only private employees receive dividends from private firms (see (3a) and (6a) below). This
is unimportant because, for simplicity, there are no profits in equilibrium. We report that our main results do
not depend on having profits or not.
6
p
t 1
b
 bt  dt 
p
p
 p ,b  btp 
2
 
2  Yt 
(3c)
where 0    1 is the capital depreciation rate;  p ,k ,  p ,b  0 capture the transaction costs
paid by each p associated with participation in the capital and bond market respectively;
and Yt denotes per capita output.8 Regarding transaction costs,  p , k ,  p ,b  0 , similar
quadratic cost functions have been used by e.g. Persson and Tabellini (1992), Benigno
(2009) and Angelopoulos et al (2011b). The usefulness of such transaction costs is that they
allow us to avoid unit roots in the transition path and get a solution for the portfolio share of
each agent in the long run (see below for details). None of our qualitative results depend on
these transaction costs.
Each p chooses {ctp , ktp1 , btp1 , etp }t0 taking factor prices, economy-wide quantities
and policy variables as given. The first-order conditions are written in Appendix A.
2.3 Households working in the public sector (public employees)
Public employees are modeled similarly to private employees. Thus, the lifetime utility of
each household working in the public sector, b  1, 2,..., N tb , is:

  u (c , e , Y
t
t 0
b
t
b
t
t
g
)
(4)
where
u (ctb , utb , Yt g )  log(ctb  Yt g )  
(etb )1
1 
(5)
The within-period budget constraint of each b is:
(1   tc )ctb  itb  dtb  (1   tk )rt ktb  (1   tl ) wtg etb  t btb  Gttr ,b
8
Thus, Yt 
Yt
, where Yt is total output in the economy (see below).
Nt
7
(6a)
where wtg is the wage rate in the public sector; and Gttr ,b is government transfers to each b .
The laws of motion of physical capital and government bonds for each b are:
b
t 1
 (1   )k  i 
b
t 1
b d 
k
b
b
t
b
t
b
t
b
t
 b ,k  ktb 
2
 
2  Yt 
 b ,b  btb 
(6b)
2
 
2  Yt 
(6c)
where  b ,k ,  b ,b  0 capture the transaction costs paid by each b associated with
participation in the capital and bond market respectively.
Each b chooses {ctb , ktb1 , btb1 , etb }t0 taking factor prices, economy-wide quantities and
policy variables as given. The first-order conditions are as in Appendix A if we replace the
superscript p with the superscript b .
2.4 Firms in the private sector
In each period, each private firm f  1, 2,..., N t f chooses capital and labor inputs, kt f and
etf , to maximize profits:
 t f  ytf  rt kt f  wtp etf
(7)
where output is produced by a CRS Cobb-Douglas function:
ytf  A(kt f ) (etf )1
(8)
where A  0 and 0    1 are parameters.9
Each f chooses kt f and etf taking factor prices as given. The standard first-order
conditions of this static problem are written in Appendix B.
9
We could assume that public goods provide productivity-enhancing services in addition to utility-enhancing
ones (see e.g. Ardagna, 2007, in a similar model). We report that our main results do not change.
8
2.5 Government budget constraint
The within-period budget constraint of the government is (quantities are in aggregate terms):
Gtg  Gtw  Gttr , p  Gttr ,b  (1  t ) Bt  Bt 1  Tt
(9a)
where Gtg is total public spending on goods and services purchased from the private sector;
Gtw is total public wage payments; Gttr , p and Gttr ,b are respectively transfers to all private
and all public employees;10 Bt is the beginning-of-period total stock of government bonds;
and Tt denotes total tax revenues, where:
Tt   tc ( N tp ctp  N tb ctb )   tk rt ( N tp ktp  N tb ktb ) +  tk N tp tp  tl ( wtp N tp etp  wtg N tb etb )
(9b)
In other words, we include the same types of government spending as in e.g. Alesina
et al (2002), namely, purchases of goods and services from the private sector, the public
wage bill and transfers to individuals. We also include the three main types of taxes, namely,
taxes on consumption, capital income and labor income.
Inspection of (9a-b) implies that, in each period, there are nine policy instruments
( Gtg , Gtw , Gttr , p , Gttr ,b ,  tc ,  tk ,  tl , Bt 1 , N tb ) out of which only eight can be set independently,
with the ninth following residually to satisfy the government budget constraint. As in most of
the related literature, we will start by assuming that the adjusting policy instrument is the
end-of-period public debt, Bt 1 , so that the other eight policy instruments can be set
exogenously (their processes are defined below). For convenience, concerning spending
Gtg
Gtw
w
policy instruments, we will work in terms of their GDP shares, s 
, st 
,
Yt
Yt
g
t
sttr , p 
Gttr , p tr ,b Gttr ,b
, st 
, where Yt denotes total output ( Yt  N t f ytf  N tp ytf in equilibrium).
Yt
Yt
Similarly, concerning the number of public employees, we will work in terms of their
population share,  tb 
10
Thus, Gttr , p 
N tb
(its process is defined below).
Nt
Gttr , p
Gttr ,b
tr , b
and
.
G

t
N tp
N tb
9
2.6 Public sector production function
Following the related literature, we assume that total public goods and services, Yt g , are
produced using goods purchased from the private sector, Gtg , and public employment, Ltg
( Lgt  N tb etb in equilibrium). In particular, following Linnemann (2009), we use a CRS CobbDouglas production function of the form:
Yt g  A(Gtg ) ( Ltg )1
(10)
where 0    1 is a parameter.
Notice three things in (10). First, our specification can nest several specifications
used by the literature. For instance, Ardanga (2007) assumes that the only input is public
employment. Cavallo (2005) and Linnemann (2009) use the same inputs as in (10). Pappa
(2009) assumes that the inputs are public employment and public capital, where the latter
changes over time via public investment (we have experimented with adding public capital
as an input in (10) and the main results do not change). Second, the TFP in (10) is assumed
to be the same as in the private sector (see (8) above); this is because we do not want our
results to be driven by exogenous factors.11 Third, in our numerical solutions below, we will
experiment with different values of the relatively unknown parameter, 0    1 .
2.7 Decentralized competitive equilibrium (DCE) with public production
Combining the above, we solve for a DCE in which (i) all households maximize utility, (ii)
all firms in the private sector maximize profits, (iii) all markets clear (see Appendix C for
market-clearing conditions in labor, capital, bond and goods markets) and (iv) all constraints
are satisfied. The DCE consists of the following eleven equilibrium conditions:12
 (etp ) (1   tc )(ctp   Yt g )  (1   tl ) wtp
(11a)
11
See e.g. Angelopoulos et al (2008) for efficiency in the public sector and its effect on economic growth.
Yg
G tr , p
G tr ,b
Gg
12
Note that Yt g  t , Gtg  t  stg tp ytf , Gttr , p  t p , Gttr ,b  t b ,
Nt
Nt
Nt
Nt
stw 
Gtw wtg Lgt
w g N b eb w g b eb

 t p t f t  t p t f t , Yt  N tp ytf .
Yt
Yt
N t yt
 t yt
10

ktp1
k
p ,k
r




1

(1

)

t 1 t 1

( tp1 ytf1 ) 2
1



(1   tc )(ctp  Yt g )
(1   tc1 )(ctp1  Yt g1 )









(11b)


btp1
p ,b


1


t 1

p
f
2 
( t 1 yt 1 ) 
1

c
p
g
c
p
(1   t )(ct  Yt )
 (1   t 1 )(ct 1  Yt g1 ) 




2
(11c)
2
 p , k  kt p 
 p ,b  btp 
p
p
b
b
(1   )c  k  (1   ) kt 






 
t 1
t
2   tp ytf 
2   tp ytf 
c
t
p
t
p
t 1
p
 (1   tk )rt ktp  (1   tl ) wtp etp  t btp  sttr , p ytf
(11d)
 (etb ) (1   tc )(ctb   Yt g )  (1   tl ) wtg
(11e)

ktb1 
k
b,k
r




1

(1

)

t 1 t 1

( tp1 ytf1 ) 2 
1


(1   tc )(ctb  Yt g )
(1   tc1 )(ctb1  Yt g1 )






(11f)

btb1
b ,b


1


t 1

( tp1 ytf1 )2
1



(1   tc )(ctb  Yt g )
 (1   tc1 )(ctb1  Yt g1 )


(11g)
ytf  A( ktp 






 tb b  p 1
k ) (et )
 tp t
(11h)
Yt g  A( stg tp ytf ) ( tb etb )1
(11i)
( stw  stg  sttr , p  sttr ,b ) tp ytf  (1   t )( tp btp   tbbtb )   tp1btp1   tb1btb1 
+  tc ( tp ctp   tb ctb )   tk rt ( tp ktp   tb ktb )  tl ( wtp tp etp  wtg tb etb )

 p ,k
 t ctp  ktp1  (1   ) ktp 
2

p

 b ,k
b
b
b
  ct  kt 1  (1   ) kt 
2

b
t
(11j)
2
2
 ktp   p ,b  btp  
 p f  

 
2   tp ytf  
  t yt 

2
2
 ktb   b ,b  btb   g p f
p f
 p f  
 p f    st  t yt   t yt
2   t yt  
  t yt 

11
(11k)
where, in the above equations, we use the factor returns:13
rt 
 ytf tp
 tp ktp  tb ktb
(12a)
(1   ) ytf
etp
(12b)
wtp 
stw tp ytf
w  b b
 t et
g
t
(12c)
We therefore have eleven equations, (11a-k), in eleven endogenous variables,
{ctp , ctb , ktp1, ktb1, btp1, btb1, etp , etb , t , ytf ,Yt g }t0 . This is for any feasible policy, where the latter is
summarized
by
the
paths
of
the
exogenous
policy
instruments,
{stg , stw , sttr , p , sttr ,b , tc , tk , tl , tb }t0 . For simplicity, we will assume that all exogenous policy
instruments are constant and set at their data average values (see below).
The equilibrium equations, (11a-k), are log-linearized around their long-run solution.
This model serves as a benchmark and is solved numerically in section 4.14 It is our “status
quo” model.
3.
The same economy with private providers of public goods
We now study what changes when, other things equal, the same amount of public goods, as
implied by the above solution, is produced by private firms, the so-called private providers,
in each time period. These private providers choose capital and labor inputs to produce the
amount of public goods ordered by the government by solving a cost minimization problem
with the government financing their total cost. Thus, now the government is not involved in
any production itself.
13
Equations (12a-b) follow from the optimality conditions of the private firm and the related market-clearing
G w w g Lg w g N b eb w g b eb
conditions, while equation (12c) follows from the policy rule stw  t  t t  t p t f t  t p t f t .
Yt
Yt
N t yt
 t yt
12
3.1. Population composition and agents’ economic roles
As above, the number of private firms producing the private good equals the number of
households working in these firms. This is indexed as p  1, 2,..., N tp , as before. We
analogously assume that the number of private firms producing the public good ordered by
the government equals the number of households working in these firms. Again, this is
indexed as b  1, 2,..., N tb , as before. In other words, the allocation of employees/households
to sectors, as well as the total population, remains as above.
Thus, the optimization problem of p  1, 2,..., N tp households, which work at the
private firms producing the private good, as well as the optimization problem of
b  1, 2,..., N tb households, which work at the private firms producing the public good
ordered by the government, remain as before (see equations (1)-(3) and (4)-(6) respectively).
The only difference, at household level, is notational: each p household rents capital to
private firms producing the private good earning a capital return denoted as rt p , while each
b household rents capital to private firms producing the public good earning a capital return
denoted as rt g .15 The problem of private firms f  1, 2,..., N p producing the private good
also remains as above (see equations (7)-(8)). The only difference, at firm level, is again
notational: the rental cost of capital for these firms is rt p .
What changes, relative to the model in section 2, is the introduction of private firms
producing the public good, the so-called private providers, indexed by g  1, 2,..., N tb , and
the new role of the government. Regarding private providers, each g produces a given
amount of the public good ordered by the government, Yt g / N tb , by choosing capital and
labor inputs in a cost-minimizing way, where the path {Yt g }t0 is exogenously set as found by
the solution of (11a-k) in the previous regime. In other words, the total amount of public
goods, {Yt g }t0 , or equivalently the per capita amount of public goods, {Yt g }t0 , is treated as
an exogenous variable in this new regime. Regarding the government, it makes lump-sum
transfers as before and finances the total cost of private providers, N tb [rt g ktg  wtg etg ] ; the
14
Note that the equilibrium equations are in terms of individual variables directly (i.e. private and public
employees) without using any aggregation results. See the related discussion in Garcia-Milà et al (2010).
15
We have experimented with various specifications of this regime. The one we use here, and in particular the
assumption that households b rent capital to firms g , while households p rent capital to firms f , instead of
assuming a single capital market in which both types of households meet both types of firms, allows us to get a
well-defined saddlepath that meets the Blanchard-Kahn criterion. Details are available upon request.
13
latter replaces spending on public wages, Gtw , and goods purchased from the private sector,
Gtg , which were among the government spending items in section 2.
In what follows, we present what changes relative to section 2.
3.2 Private firms producing a given amount of the public good (private providers)
In each period, each private provider g  1, 2,..., N tb chooses capital and labor inputs, denoted
as ktg and etg respectively, to minimize its costs. The cost-minimization problem is (as said,
economy-wide quantities, denoted by capital letters, are taken as given by private agents):
Y g

rt g ktg  wtg etg  t  t b  ytg 
 Nt

(13)
where rt g and wtg are respectively the rental cost of capital and the wage rate paid by private
providers (see below for details), t is a multiplier measuring the marginal cost of
production, Yt g is the total amount of public goods which is exogenously given by the
previous problem, and ytg is each private provider’s output which is produced by using the
same production function as in (8), namely:
ytg  A(ktg ) (etg )1
(14)
Each g chooses ktg and etg taking factor prices and economy-wide quantities as
given. The first-order conditions are:
rt  t
g
 ytg
(15a)
ktg
(1   ) ytg
etg
(15b)
Yt g
 A(ktg ) (etg )1  0
N tb
(15c)
wtg  t
14
where Appendix D provides details based on Mas-Colell et al (1995, pp. 139-143).
It is useful to point out two things. First, the determination of wtg is different from
section 2. In particular, while it was determined by the policy rule for the share of the public
wage bill in section 2 (see equation (12c) above), it is now market determined as shown by
equation (15b). Second, now both groups of firms, f and g , participate in the factor
markets (see also the market-clearing conditions below).
3.3 Government budget constraint
The within-period budget constraint of the government changes from (9a) to:
N tb [rt g ktg  wtg etg ]  Gttr , p  Gttr ,b  (1  t ) Bt  Bt 1  Tt
(16)
where the first term on the left-hand side is the total cost of public goods produced by private
firms and the other variables are as defined in (9a-b) above.
In each period, there are seven policy instruments ( Gttr , p , Gttr ,b ,  tc ,  tk , tl , Bt 1 , N tb ) or
equivalently in ratios ( sttr , p , sttr ,b ,  tc  tk  tl , Bt 1 , tb ). As in section 2, we will start by assuming
that the residually determined policy instrument is the end-of-period public debt, Bt 1 .
Compare the vector of policy instruments to that in section 2, where Gtg and Gtw , or
equivalently their output shares, stg and stw , were also among the policy instruments.
3.4 Decentralized competitive equilibrium (DCE) with cost-minimizing private providers
Combining the above, we solve for a DCE in which (i) all households maximize utility, (ii)
all private firms that produce the private good maximize profits and all private firms that
produce the public good minimize costs, (iii) all markets clear (see Appendix E for the new
market-clearing conditions) and (iv) all constraints are satisfied. The new DCE consists of
the following new eleven equilibrium conditions:
 (etp ) (1   tc )(ctp   Yt g )  (1   tl ) wtp
(17a)
15

ktp1
k
p
p ,k
r




1

(1

)

t 1 t 1

( tp1 ytf1 ) 2
1



(1   tc )(ctp  Yt g )
(1   tc1 )(ctp1  Yt g1 )









(17b)


btp1
p ,b


1


t 1

p
f
2 
( t 1 yt 1 ) 
1

c
p
g
c
p
(1   t )(ct  Yt )
 (1   t 1 )(ct 1  Yt g1 ) 




2
(17c)
2
 p , k  kt p 
 p ,b  btp 
p
p
b
b
(1   )c  k  (1   ) kt 






 
t 1
t
2   tp ytf 
2   tp ytf 
c
t
p
t
p
t 1
p
 (1   tk ) rt p ktp  (1   tl ) wtp etp  t btp  sttr , p ytf
(17d)
 (etb ) (1   tc )(ctb   Yt g )  (1   tl ) wtg
(17e)

ktb1 
k
g
b,k
r




1

(1

)

t 1 t 1

( tp1 ytf1 ) 2 
1


(1   tc )(ctb  Yt g )
(1   tc1 )(ctb1  Yt g1 )






(17f)

btb1
b ,b


1


t 1

( tp1 ytf1 )2
1



(1   tc )(ctb  Yt g )
 (1   tc1 )(ctb1  Yt g1 )


(17g)






ytf  A( ktp ) (etp )1
(17h)
Yt g   tb A( ktb ) (etb )1
(17i)
 tb (rt g ktb  wtg etb )  ( sttr , p  sttr ,b ) tp ytf  (1  t )( tp btp   tbbtb )   tp1btp1   tb1btb1 
+  tc ( tp ctp   tb ctb )   tk ( rt p tp ktp  rt g tb ktb )  tl ( wtp tp etp  wtg tb etb )

 p ,k
 t ctp  ktp1  (1   ) ktp 
2

p
(17j)
2
2
 ktp   p ,b  btp  
 p f  

 
2   tp ytf  
  t yt 

2
2

 b , k  ktb   b ,b  btb  
b
b
b
p f
  ct  kt 1  (1   ) kt 
 p f  
 p f     t yt
2   t yt 
2   t yt  


b
t
where, in the above equations, we use the factor returns (see Appendix F for details):
16
(17k)
rt 
p
 ytf
(18a)
kt p
 Yt g
rt  t b b
kt  t
g
wtp 
(18b)
(1   ) ytf
etp
(18c)
(1   )Yt g
etb tb
wtg  t
(18d)
Therefore, in this new system, we have eleven equations, (17a-k), in eleven
endogenous variables, {ctp , ctb , ktp1 , ktb1 , t , btp1 , btb1 , etp , etb , t , ytf }t0 . This is for any feasible
policy, where the latter is summarized by the paths of the exogenous policy instruments,
{sttr , p , sttr ,b ,  tc  tk  tl , tb }t0 , and the path of the per capita amount of public goods, {Yt g }t0 ,
which is exogenously set as in the previous, status quo, regime. We will again assume that
all exogenous policy instruments are constant and set at their data average values (see
below).
The equilibrium equations, (17a-k), are log-linearized around their long-run solution.
The model is solved numerically in the next section.
4. Numerical solutions and comparison of the two model economies
We now solve and compare the two model economies developed in sections 2 and 3.
4.1 How we work to solve the models
We work in two steps. We first solve the model in section 2, when this model is calibrated to
match some stylized facts of the UK economy, in particular the tax-spending policy mix over
1990-2010. This solution will give us, among other endogenous variables, the path of the per
capita amount of public goods, {Yt g }t0 , induced by the existing UK tax-spending policy
mix. In turn, this status quo economy will be used as a point of reference for evaluating
various policy reforms. For instance, in this section, we solve the model economy in section
3, where it is cost-minimizing private providers, rather than the government itself, that
produce the same path of per capita public goods, {Yt g }t0 .
17
We will compare the status quo economy to the reformed economy both in the long
run and in the transition path. The way we work follows most of the literature on policy
reforms.16 Thus, we will first evaluate various policy regimes based on a comparison of
long-run equilibria (this is in sections 4-6). Transitional dynamics, as well as lifetime welfare
gains from moving from one regime to another, are discussed in section 7.
4.2 Parameters and policy instruments
Table 1 reports the baseline parameter values for technology and preference, as well as the
values of exogenous policy instruments, used to solve the status quo model economy in
section 2. The time unit is meant to be a year.
Our parameterization is standard with most parameter values for technology and
preference being borrowed from Angelopoulos et al (2011a), who have recently calibrated an
aggregate DSGE model to annual data for the UK economy. When we have no a priori
information about a technology or preference parameter value, or when different authors use
different values, we will consider a range of values. In general, we can report that all main
results are robust to changes in the parameter values.
Public spending and tax rate values are those of sample averages of the UK economy
over 1990-2008. The data are obtained from OECD, Economic Outlook, no. 88. We report
that our main results do not change when we consider alternative time periods, e.g. 19702008 or 1996-2008.
Table 1 around here
(Baseline parameterization)
Let us discuss, briefly, the values summarized in Table 1. The labour share in the
private production function, 1   , is set at 0.601, which is the value in Angelopoulos et al
(2011a). The scale parameter in the technology function, A , is set at 1. The time preference
rate is set at 0.99. The weight given to public goods and services in composite consumption,
 , is set at 0.1, as is usually the case in similar studies. The other preference parameters
related to hours worked,  and  , are set at 5 and 1 respectively; these parameter values
jointly imply hours worked within usual ranges. The capital depreciation rate is set at
16
See e.g. Lucas (1990), Cooley and Hansen (1992) and Mendoza and Tesar (1998). Recall that Lucas (1990)
compared the macroeconomic allocation implied by the existing US tax mix to that under optimal Ramsey
policy according to which the capital tax rate is set to be zero.
18
  0.05 . The transaction cost parameter associated with participation in asset markets is set
at  p ,k   p ,b   b ,k   b ,b  0.002 across both agents and both assets. Our results are robust
to changes in these parameter values (see below).
In the baseline calibration, the productivity of public employment, vis-à-vis the
productivity of goods purchased from the private sector, in the public sector production
function, 1   , is set at 0.493. This value is the sample average of public wage payments, as
share of total public spending on inputs used in the production of public goods (see also e.g.
Linnemann, 2009, for similar practice). But we will also experiment with other values of
1   (see below).
Public employees as a share of total population,  b , are set at 0.1904 as in the data.
Public spending on wage payments and transfers, as shares of output, are respectively
stw  0.109 and sttr  0.2199 in the data. We assume that transfers are allocated to private
and public employees according to their shares in population, sttr , p   tp sttr  (1  tb ) sttr and
sttr ,b   tb sttr (see below for other cases considered). The output share of public spending on
goods and services purchased from the private sector, stg , is then calculated residually from
total public spending minus spending on public wage payments, transfers and interest
payments; this is found to be 0.1119. The effective tax rates on consumption, capital and
labor are respectively  c  0.1852 ,  k  0.3875 and  l  0.2685 over 1990-2008; the data
are taken from Angelopoulos et al (2011a), who have followed the methodology of Conesa
et al (2007) in constructing effective tax rates for the UK economy.
We can now present numerical solutions. As said, we start with a comparison of
long-run equilibria. We report that, using the parameterization of Table 1, all regimes studied
feature local determinacy.
4.3 Long-run solution when public debt is the adjusting public finance instrument
Using the parameterization in Table 1, the long-run solutions of the status quo economy
presented in section 2 and the reformed economy presented in section 3 are reported
respectively in columns 1 and 2 in Table 2. These long-run solutions follow from the
systems (11a-k) and (17a-k) respectively when variables do not change.17
Without transaction costs,   0 , the long-run system would be “under-identified” in the sense that there
would be nine equations and eleven variables. This happens because, in the long run, if   0 , the two agents’
(i.e. private and public workers’) Euler conditions for capital (see equations (11b) and (11f) written in the long
17
19
Recall that, in the reformed economy in section 3, the same amount of public goods,
as found in section 2, is supplied by cost-minimizing private providers. Also recall that the
superscript b denotes those households that are involved in the production of the public
good, either as public employees in the status quo economy, or as workers at the costminimizing private providers/firms in the reformed economy, while the superscript p
denotes those households that work in private firms producing the private good.
Table 2 around here
(Long-run solution when public debt is the residual policy instrument)
Before we compare the two regimes, we report that the long-run solution of our status
quo economy in column 1 can mimic rather well some key macroeconomic averages in the
actual data in the UK. For instance, our long-run solution for the public wage to private wage
ratio is found to be w g / w p  0.8109 in column 1 of Table 2, which is close to that in the
actual data over our sample period, which is 0.8884. We also report that our long-run output
shares of consumption, capital, etc, are close to their average values in the data.
We now proceed to compare the status quo economy to the reformed economy. We
start with distributional implications and then discuss macroeconomic or aggregate
implications (we do so only for presentational convenience because distribution and
efficiency are obviously interrelated).
4.3.1 Distributional implications
In the long run, the ratio of public to private wages, w g / w p , falls from 0.8109 in column 1
to only 0.3499 in column 2 of Table 2. Lower labor income explains, in turn, the fall in
consumption, c b , and the willingness to work, eb , of b households in column 2. Despite the
increase in leisure time, (1  eb ) , the fall in consumption, c b , leads to a clear fall in the longrun utility of b households, u b , as we switch from the status quo to the reformed economy.
run) are reduced to one equation only. The same applies to the two Euler conditions for bonds (equations (11c)
and (11g), written in the long run, are also reduced to one equation only). Thus, the model could pin down the
total long-run stocks of capital and bonds but not their allocation to the two types of agents. The same feature
characterizes the system in (17a-k). The presence of transaction costs,   0 , help us to circumvent this
problem. Alternatively, we could use an ad hoc rule for the allocation of the total long-run stocks of assets to
each agent (our main results do not change). In any case, as is known, with perfect capital markets and
common discount factors, the allocation of the aggregate stock of capital and bonds to different types of
individual investors cannot be pinned down by the equilibrium conditions. This is why resorting to some
20
By contrast, the long-run utility of p households, u p , rises in column 2 . This is thanks to
higher consumption, c p , and more leisure time, (1  e p ) , enjoyed by p households under
private provision (see below for further details).
Notice that, in this particular experiment, the adverse welfare effects on b
households dominate the beneficial effects on p households, so that per capita long-run
utility, denoted as u ,18 falls under private provision in column 2 (as we shall see below, this
aggregate depends heavily on the way the government uses its efficiency savings enjoyed by
private provision).
4.3.2 Macroeconomic implications
Per capita private consumption and per capita capital, both as levels and as shares of output,
rise in column 2 relative to column 1. This happens because the switch to private provision
in column 2 releases resources for private use. In particular, the comparison of the resource
constraints (11k) and (17k) implies that, in the latter, the elimination of Gtg releases ceteris
paribus resources for private use (private consumption and capital accumulation). This is
like a traditional wealth effect in the sense that, given output, government spending on goods
and services works as a resource drain. This partly explains the rise in per capita
consumption and capital. The rise in per capita consumption also explains how the reduction
in c b (caused by the fall in w g / w p ) allows an increase in c p , as discussed above.
The above are direct effects that work through reallocation of resources. But there are
also indirect effects that work through public financing. The fall in w g under private
providers leads to a fall in the total labor cost of public good production as share of output,
s w . The latter falls from 0.1090 in the data (see column 1 in Table 2) to only 0.0338 in the
reformed economy (see column 2 in Table 2). Since this cost is always financed by the
government, irrespectively of who is the provider, a more efficient way of delivering the
public good in column 2 allows the government to make efficiency savings. In the baseline
public financing case studied so far, where the residual policy instrument is - by assumption
- the end-of-period stock of public debt, these efficiency savings allow the government to
afford a much larger debt burden through the long-run government budget constraint.
extraneous assumption, in order to get a portfolio share for each individual in models with different agents and
perfect capital markets, is usual in the literature (see e.g. Mendoza and Tesar, 1998, in a two-country model).
18
Per capita values are defined as the weighted average of p households and b households, where the
weights are their shares in population. For instance, per capita utility is u  v u  v u .
p
21
p
b
b
Indeed, as reported in Table 2a, the endogenously determined output share of public debt,
b / y , rises from 212.71% of GDP in column 1 to a very high number in column 2.
It is the combination of direct-resource effects and indirect-public financing effects
that explains the value of per capita output, y . In the numerical experiment reported in
Table 2, y falls as we switch to the reformed economy ( y falls from 0.6838 in column 1 to
0.6418 in column 2 of Table 2). This seemingly paradoxical result arises simply because we
have assumed that it is public debt that adjusts to close the government budget. As said
above, in this baseline case, efficiency savings only allow the government to afford the
financing of a much higher debt burden. Thus, although we move to a more efficient way of
delivering the public good, we do not use - by assumption - the resources saved in a way that
benefits the economy. At the same time, the decrease in public spending creates an adverse
demand effect on output. The combination of those two effects, namely, the trivial use of
government efficiency savings and the adverse effect on the demand side, explains the drop
in the level of y , even if we have switched to a more efficient way of delivering the public
good in column 2. To confirm this, we next study two more interesting public financing
cases.
4.4 Long-run solutions when distorting taxes are the adjusting public finance instruments
We now study two more interesting ways of public financing. In Table 3, the residually
determined long-run policy instrument is the consumption tax rate, while, in Table 4, the
residually determined long-run policy instrument is the labor tax rate. In both cases, the
long-run public debt-to-output ratio is exogenously set at its average value in the data, 80%.
Table 3 around here
(Long-run solution when the consumption tax rate is the residual policy instrument)
As shown in columns 1 and 2 of Table 3, efficiency savings from private provision
allow the government to afford a much lower consumption tax rate (actually, in our
experiment,  c turns from a tax in column 1 to a small subsidy in column 2). Lower
consumption taxes stimulate the consumption and welfare of p households (compare the
values of c p and u p in column 2 of Table 3 to those in column 2 of Table 2). The rise in u p
is now high enough to lead to higher per capita welfare (per capita welfare, u , increases
from -1.0595 in column 1 to -0.9525 in column 2 in Table 3, while it decreased from -1.0677
22
in column 1 to -1.0774 in column 2 in Table 2). In other words, with consumption taxes as
the adjusting instrument, per capita welfare rises when we switch to private provision.
Nevertheless, as it was the case in Table 2, per capita output falls again when we switch to
private provision. In particular, y falls from 0.6879 in column 1 to 0.6767 in column 2 in
Table 3 (this fall is smaller than that in Table 2, where y fell from 0.6838 in column 1 to
0.6418 in column 2). In other words, although the rise in c p is much higher than it was under
debt financing in Table 2, this rise is still not strong enough to offset the adverse demand
effect on output coming from the fall in consumption of b households and less public
spending. This is not surprising: consumption taxes are not very distorting in this class of
models so, loosely speaking, their reduction cannot work as an engine of growth.
Table 4 around here
(Long-run solution when the labor tax rate is the residual policy instrument)
In Table 4, the residual public finance policy instrument is the labor tax rate in the
long run. Efficiency savings from private provision allow the government to afford a much
lower labor tax rate (actually, in our experiment,  l turns from a tax in column 1 to a small
subsidy in column 2). Since labor taxes are distorting (see also e.g. Angelopoulos et al,
2011a, for the UK economy), their reduction not only strongly stimulates c p , u p and in turn
u (per capita welfare, u , increases from -1.0537 in column 1 to -0.9489 in column 2 in
Table 4), but it also stimulates long-run per capita output ( y rises from 0.6949 in column 1
to 0.7153 in column 2 in Table 4). In other words, via the public financing channel, we now
have substantial supply-side benefits, which more than offset the adverse demand effects on
output coming from a smaller public sector. Thus, private provision now leads to a larger
national pie and higher per capita welfare (as we show below, a larger pie can allow the
government to afford Pareto-improving redistributive policies).
4.5 Summary of this section
A switch from the status quo economy to a reformed economy, where the same amount of
public goods is produced by cost-minimizing private providers, increases the welfare of
private employees but makes public employees clearly worse off. The effect on per capita
welfare and per capita output is ambiguous depending crucially on the adjusting public
finance instrument. Only when the efficiency savings, coming from a more efficient way of
23
delivering the public good, are used to cut distorting income (labor) taxes, per capita welfare
and per capita output can both rise. Recall that these are long-run results. Transition results,
when we depart from the status quo economy and switch to private providers over time, are
presented below.
5.
Can cost-minimizing public providers beat cost-minimizing private providers?
One could argue that so far we have been “unfair” to the public sector. In particular, we have
compared the status quo economy to an economy with private providers, where, in the
former, public production decisions were exogenously set as in the data, while, in the latter,
public production decisions were made by cost-minimizing private providers. Although, as
said in the beginning of section 4 above, comparisons of this type are common in the
literature on policy reforms, one is wondering what would happen when we compare the
cases in which, not only private providers, but also public providers minimize their costs,
always with the general taxpayer (i.e. the government) financing these costs. We turn to this
question now.
Although there are several ways of modeling the behavior of public
providers/enterprises, here we choose a simple way that also makes the solution of this new
regime directly comparable to the solutions of the two other regimes studied above. In
particular, like we did in section 3 with private providers, we assume that there is a single
public provider who, in each period, chooses its inputs in a cost-minimizing way so as to
produce the same amount of public goods, {Yt g }t0 , as offered by the status quo economy.
Thus, as in section 3, the path {Yt g }t0 is exogenously set. As said above, our modeling of
public providers is not different from Atkinson and Stiglitz (1980, chapter 15.3), where the
government tells state enterprises to choose their mix of inputs so as to minimize their costs.
5.1 Cost-minimizing public provider
The economy is as in section 2 but now, in addition, in each period, the public provider
chooses its two inputs, Gtg and Ltg , or equivalently their output shares, stg and stw , to
minimize its costs. The cost-minimization problem is:
Gtg  wtg Ltg   t [Yt g  At (Gtg ) ( Ltg )1 ]
(19)
24
where wtg denotes the new wage rate received by public employees,  t is a multiplier
measuring the marginal cost of producing the public good and Yt g is the total amount of
public goods which is exogenously set as found by the solution of the status quo model in
section 2.
It is straightforward to show that the three first-order conditions combined imply:
stg


w
st 1  
(20)
which says that the ratio of public spending on the two inputs should be equal to the ratio of
their productivities.
5.2 Decentralized competitive equilibrium (DCE) with cost-minimizing public provider
In the new DCE, we have twelve equations, the eleven equations of the status quo economy,
(11a-k),
plus
equation
(20),
in
twelve
endogenous
variables,
{ctp , ctb , ktp1 , ktb1 , btp1 , btb1 , etp , etb , t , ytf , stg , stw }t 0 . This is for any feasible policy, as
summarized by {sttr , p , sttr ,b ,  tc  tk  tl , tb }t0 , and the path of {Yt g }t0 , which is exogenously set
as found in the status quo economy. These new equilibrium equations are log-linearized
around their long-run solution.
5.3 Long-run solutions
Long-run solutions of the new model economy, under the three different ways of public
financing, are reported in Tables 2, 3 and 4 respectively, column 3. We again use the
baseline parameterization in Table 1. Inspection of the results reveals that any differences
between the status quo economy in column 1 and the economy in column 3, where the public
provider acts optimally, are minor.
5.4 Summary of this section
When public providers choose their inputs in a cost-minimizing way, the results are very
similar to those under the status quo regime, at least when we use the baseline
parameterization. This implies that contracting out the production of public goods to costminimizing private providers is superior to public production, even when public providers
25
act as cost minimizers. It also seems to imply that in the UK, over 1990-2008, the public
sector has exhausted its role, at least in terms of aggregate efficiency, as a provider of public
goods and services.
6.
Searching for Pareto improving reforms
As we have seen, although per capita welfare can increase when we move from the status
quo economy to an economy with public finance only, public employees clearly become
worse off by becoming employees at cost-minimizing private providers. This means that
such reforms, although good for the general interest, are unlikely to be implemented,
especially, when public sector employees, or their trade unions, have a strong influence in
blocking reforms.
The question is whether the society can take advantage of the aggregate efficiency
gains, generated by private provision/public finance, and design a transfer scheme that
improves the welfare of both types of agents, namely both private and public employees,
relative to the status quo economy.
We find it natural to report results only for those cases in which private
provision/public finance increases the aggregate pie (per capita output) relative to the status
quo economy. As we showed in section 4, and in particular in Table 4, this happens when the
efficiency savings from private provision/public finance are used to cut distorting labor
taxes. (Results for the other cases, where the residual public finance instrument is public debt
or consumption taxes, are available upon request.)
6.1 Endogenizing transfers and a new DCE
We search for a government transfer scheme that, in combination with private
provision/public finance of public goods and labor taxes as the residual public finance
instrument, makes everybody equally well off in the long run. In particular, instead of
assuming that government transfers are exogenously allocated to the two groups according to
their population shares, we now endogenize this scheme by solving for an allocation of
transfers that makes both agents equally off in the long run of the reformed economy
modeled in section 3.19
19
See e.g. Park and Philippopoulos (2003) for other redistributive transfer mechanisms in a dynamic general
equilibrium model.
26
Algebraically, the new DCE consists of equations (17a-k) plus a new equation that
equates long-run utility across the two agents, u b  u p , while the associated new endogenous
variable is the long-run share of government transfers, xb , where s tr ,b  x b s tr and
s tr , p  (1  x b ) s tr . Numerically, we compute xb so as u b  u p . Results for the long run of this
economy are reported in column 4 in Table 4. As can be seen, when we compare this new
economy (in column 4) to the status quo economy (in column 1), there is room for
substantial welfare gains, now for both types of agents. Notice that, although private
employees are worse off in column 4 than in column 2, which was the case with private
provision without redistribution of transfers, they are still better off than in the status quo
economy in column 1.
6.2 Summary of this section
A switch to private provision, in combination with redistributive transfers and use of the
efficiency savings from private provision to cut labor taxes, is Pareto improving relative to
the status quo economy.
7. Transition and discounted lifetime utility
The above results compared long-run equilibria with and without reforms. We now study
lifetime utility between pre- and post reform steady states when we depart from initial
conditions corresponding to the pre-reform, status quo, economy.
7.1 How we work
We work as in e.g. Lucas (1990), Cooley and Hansen (1992) and Mendoza and Tesar (1998).
We first check, using our baseline parameterization, that when log-linearized around its
steady state solution, each model economy studied so far is saddle-path stable.20 This is
under all types of reform and all methods of public financing studied. Then, setting, as initial
conditions for the state variables, the steady state solution of the status quo economy, we
compute the equilibrium transition path of each reformed economy and calculate the
20
Without asset transaction costs there are unit roots, at least in some regimes. Although there are papers that
work with unit roots (see e.g. Schmitt-Grohé and Uribe, 2004, p. 219), we prefer to avoid this feature since it
implies that we may not converge to the long-run around which we have approximated. We also report that
when we make the model stochastic by adding shocks to e.g. policy instruments and TFP, the impulse response
functions give intuitive results.
27
associated discounted lifetime utilities of the two types of households. We also calculate the
permanent supplement to private consumption, expressed as a constant percentage, which
would leave the household indifferent between two regimes. This percentage is denoted as
z , where a positive (resp. negative) value of z will mean that discounted lifetime utility is
higher under the reformed economy (resp. the status quo economy).
7.2 Results for lifetime utility
Results are reported in Table 5. Again, we report results only for the case in which the
efficiency savings from a reform are used to cut a distorting income (labor) tax rate. Recall
that, only in this case, a reform increases the aggregate pie (per capita output) relative to the
status quo economy in the long run.
Table 5 around here
(Lifetime utility under regime switches)
In Table 5, U p and U b denote respectively the discounted lifetime utility of the p
household and the b household, while U is the per capita value. Column 1 describes the
case in which we remain forever in the long-run of the status quo economy, while in the
other columns we study what happens over time when we switch from the status quo
economy to cost-minimizing private providers (column 2), to cost-minimizing public
providers (column 3), and finally to cost-minimizing private providers in combination with
transfers that compensate those suffered from the reform (column 4). In each case of regime
switch (columns 2-4), we also report the associated value of the welfare measure, z , as
defined above.
As can be seen, the transition results are qualitatively the same as the long-run
results. Namely, in all cases studied, the transition from the status quo economy to an
economy with private providers is good for private employees and the aggregate economy,
but this is clearly at the loss of those employed in the public sector. On the other hand, when
we also adjust transfers to compensate the losers, both groups of agents get better off relative
to the status quo (see column 4). Finally, again as in the long run, public providers cannot
beat private providers even when are both assumed to minimize their costs.
28
7.3 Summary of this section
When the criterion is lifetime utility, there seems to be substantial Pareto benefits from a mix
of reforms that combines: (i) a transition to cost-minimizing private providers that implies
efficiency savings for the government (ii) redistributive transfers that compensate those
previously working as public employees (ii) a reduction in labor taxes made affordable by
efficiency savings.
8. Robustness
We finally check the sensitivity of our results to changes in the parameter values used. In
particular, we focus on the value of the relatively unknown parameter, 1   , measuring the
productivity of public employees in the public sector production function (see equation (10)
above).
8.1 Various ad hoc values of the productivity of public employees
Keeping everything else as in the baseline parameterization of Table 1, we now arbitrarily
set a low productivity of public employees, say 1    0.3 , and a high productivity, say
1    0.7 . Recall that in the baseline parameterization so far, the calibrated value of 1  
was 0.493. Results for these two new cases are reported in Tables 6-9 and 10-13
respectively. Thus, these new tables are as Tables 2-5 with the only difference being that
1    0.3 in Tables 6-9 and 1    0.7 in Tables 10-13.
Tables 6-9 around here
(This is like Tables 2-5 with 1    0.3 )
Tables 10-13 around here
(This is like Tables 2-5 with 1    0.7 )
The main message is that all key results remain unchanged. Let us first briefly look at
the case with low productivity, 1    0.3 . Comparison of Tables 2-4 to Tables 6-8 (longrun results) reveals that the only difference is that, in the latter, public employees become
worse off as we move from column 1 to column 3, while recall that their utility did not
change much in Tables 2-4 above. This is intuitive: when their productivity is low, public
employees suffer under cost-minimizing public providers as they do under cost-minimizing
29
private providers. This is also the case when we compute lifetime utility. Namely,
comparison of Table 5 to Table 9 reveals that lifetime utility of public employees is lower in
Table 9 than in Table 5 in all regimes with cost-minimizing producers; for the symmetrically
opposite reason, public employees can gain more in Table 9 than in Table 5, when we
combine cost-minimization with redistributive transfers.
Let us also look at the case with high productivity of public employees, 1    0.7 .
Comparison of Tables 2-4 to Tables 10-12 (long-run results) reveals that now the opposite
happens: the public wage bill, stw , and hence the welfare of public employees, rise as we
move from column 1 to column 3 in Tables 10-12. This might look paradoxical but it
happens simply because of the optimality condition (20): since 1    0.7 is relatively high
(or   0.3 is relatively low), it is optimal to choose a relatively high stw (or a relatively low
stg ) which in turn makes public employees better off in column 3 in Tables 10-12. This also
explains the results for lifetime utility in Table 13. Nevertheless, Table 13 also implies that
the key result of the paper holds even in this case: the reform that makes both groups better
off relative to status quo is the one that combines cost-minimizing private providers with
redistributive transfers and a reduction in labor taxes.
8. 2 Summary of this section
The key results are robust to a wide range of the parameter value measuring the productivity
of public employees. We also report that our results are robust to changes in the other
parameter values (results are available upon request).
9.
Conclusions
This paper studied a much debated reform of the state - the idea of opening up public
services to new providers - in a dynamic general equilibrium setup. We showed that
substantial aggregate gains are possible if the society switches to private provision/public
finance of public goods and if the government uses the resulting efficiency savings to reduce
distorting income taxes. It is remarkable that this happens even when the amount of public
goods produced, and the number of households employed in the production of public goods,
remain the same as in the status quo economy. We also showed that one can design
30
redistributive schemes that allow everybody, including public employees, to benefit from
such a switch.
Our results are another example of the importance of social contracts (see also the
discussion in Garcia-Milà et al, 2010). In our model, social contracts that terminate the
monopoly of the public sector as a producer of public goods, in combination of transfers that
compensate those previously employed by the state, can benefit everybody.
Our work can be extended in several ways. For instance, we could include
uncertainty coming from shocks to e.g. technology and policy instruments. We could also
use a richer production function for the public good allowing for substitutability between
public employment and goods purchased from the private sector. Also, we could introduce
various politico-economy issues, like extra benefits on the part of public employees coming
from rent seeking. We leave such extensions for future work.
31
APPENDIX
Appendix A: First-order conditions of household p in section 2
The first-order conditions include the budget constraints and:
 (etp ) (1   tc )(ctp   Yt g )  (1   tl ) wtp
(A.1)

ktp1
k
p,k




r
1

(1

)

t 1 t 1

(Yt 1 ) 2
1



(1   tc )(ctp  Yt g )
(1   tc1 )(ctp1  Yt g1 )









p


p ,b bt 1


1


t 1

2 
(Yt 1 ) 
1

c
p
g
c
p
(1   t )(ct  Yt )
 (1   t 1 )(ct 1  Yt g1 ) 




(A.2)
(A.3)
Appendix B: First-order conditions of private firm f in section 2
rt 
 ytf
wtp 
(B.1)
kt f
(1   ) ytf
etf
(B.2)
so that profits are zero.
Appendix C: Market-clearing conditions in section 2
In the labor market:
N t f etf  N tp etp
(C.1a)
Lgt  N tb etb
(C.1b)
In the capital market:
N t f kt f  N tp ktp  N tb ktb
(C.2)
In the bond market:
Bt  N tp btp  N tbbtb
(C.3)
In the goods market (economy’s resource constraint):
N tp ctp  N tb ctb  N tp itp  N tb itb  Gtg  N t f ytf
(C.4)
where we set Nt f  N tp .
32
Appendix D: Cost minimization of private provider g in section 3
Here we follow Mas-Colell et al (1995, pp. 139-143). The first-order conditions imply:
rt g  t
 ytg
(D.1a)
ktg
(1   ) ytg
w  t
etg
g
t
t 
(D.1b)
(rt g ) ( wtg )1
rt g ktg  wtg etg (rt g ktg  wtg etg ) N tb (rt g ktg  wtg etg ) tb



A  (1   )1
ytg
Yt g
Yt g
(D.1c)
where (D.1c) follows if we use (D.1a)-(D.1b) to get expressions for ktg and etg respectively,
g 
t
g 1
t
and use them back in the production function, A(k ) (e )
Yt g Yt g
y  b  b .
Nt  t
g
t
In turn, we use (D.1c) to substitute out the multiplier, t , in (D.1a) and (D.1b):
 1
yg  rg 
k  t  t 
A  
g
t
1
 wtg 


 1 

Y g  r g   wg 
e  t  t   t 
A     1 
(D.2a)

g
t
(D.2b)
so that the total cost of each firm can be written as:
y g (r g ) ( wtg )1
rt k  w e  t t
A
g
g
t
g g
t t
 1      1 

 
 
    1    
Yt g (rt g ) ( wtg )1

N tb A
 1      1 

 
 
    1    
(D.3)
Notice that profits are zero (thanks to CRS). To show this, consider profits:
ytg (rt g ) ( wtg )1
y  rt k  w e  y 
A
g
t
g
g
t
g g
t t
g
t
 1      1 

 
 
    1    
(D.4)
so that (thanks to linearity) the first-order condition is:
(rt g ) ( wtg )1
1
A
 1      1 

 
 
    1    
(D.5)
but, if this condition holds, total profits are zero in each period.
Appendix E: Market-clearing conditions in section 3
In the labor market:
33
N t f etf  N tp etp
(E.1a)
Ntg etg  N tb etb
(E.1b)
In the capital market:
N t f k t f  N t p kt p
(E.2a)
N tg ktg  N tb ktb
(E.2b)
In the bond market:
Bt  N tp btp  N tbbtb
(E.3)
In the goods market (economy’s resource constraint):
N tp ctp  N tb ctb  N tp itp  N tb itb  N t f ytf
(E.4)
where we set N t f  N tp and N tg  N tb . Also recall that the privately produced public good is
provided without charge as in section 2.
Appendix F: Factor returns in section 3
rt p 
 ytf
rt  t
g
wtp 
(F.1)
kt p
 ytg
ktg
 Yt g
 Yt g N t
 Yt g
 t b b  t b b  t b b
kt N t
kt N t N t
kt  t
(1   ) ytf
etp
wtg  t
(F.2)
(F.3)
(1   ) ytg
(1   )Yt g
(1   )Yt g N t
(1   )Yt g






t
t
t
etb
etb N tb
etb N tb N t
etb tb
34
(F.4)
Table 1
Baseline parameterization
Parameters
and policy
instruments
Description
Value

Share of capital in private production
0.399
1
Share of public employment in public production
0.493*
k
Capital depreciation rate
0.05

Rate of time preference
0.99

Public consumption weight in utility
0.1

Preference parameter on work hours in utility
5

Elasticity of work hours in utility
1
sw
Public wage payments as share of GDP
0.1090
sg
Public purchases as share of GDP
0.1119
s tr
Public transfers as share of GDP
0.2199
c
Tax rate on consumption
0.1852
k
Tax rate on capital income
0.3875
l
Tax rate on labor income
0.2685
vb
Public employees as share of population
0.1904
A
Long-run TFP
1
a
Autoregressive parameter of TFP
0.9
a
Standard deviation of TFP
0.01
 p,k
Transaction cost incurred by private agents in capital market
0.002
 p,b
Transaction cost incurred by private agents in bond market
0.002
 b,k
Transaction cost incurred by public employees in capital market
0.002
 b,b
Transaction cost incurred by public employees in bond market
0.002
Notes: * We also experiment with
1    0.3 (see Tables 6-9) and 1    0.7 (see Tables 10-13).
35
Table 2
Long-run solution when public debt is the residual policy instrument
1
Status quo
economy
2
Cost-minimizing
Private providers
3
Cost-minimizing
public providers
-1.0432
-0.9794
-1.0432
-1.1720
-1.4938
-1.1726
-1.0677
0.4792
-1.0774
0.4959
-1.0678
0.4792
0.4075
0.2503
0.4071
0.3590
0.3419
0.3590
0.3414
0.2338
0.3413
0.8109
0.3499
0.8099
0.6838
0.6418
0.6837
yg
c/ y
0.0706
0.0706
0.0706
0.6808
0.6999
0.6808
k/y
3.6282
3.7690
3.6282
b/ y
2.1271
7.4738
2.1316
sw
sg
s t, p
s t ,b
0.1090
0.0338
0.1088
0.1119
-
0.1119
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
0.8096* s
tr
0.8096* s
tr
0.8096* s
tr
0.1904* s
tr
0.1904* s
tr
0.1904* s
tr
Notes: (i) We use the baseline parameterization in Table 1. (ii) u  v u  v u
for all per capita quantities).
p
36
p
b
b
(the same formula is used
Table 3
Long-run solution when the consumption tax rate is the residual policy instrument
1
Status quo
economy
2
Cost-minimizing
Private providers
3
Cost-minimizing
public providers
-1.0345
-0.8253
-1.0295
-1.1656
-1.4933
-1.1651
-1.0595
0.4853
-0.9525
0.5992
-1.0553
0.4889
0.4118
0.2555
0.4127
0.3611
0.3605
0.3624
0.3438
0.2499
0.3447
0.8100
0.3001
0.8052
0.6879
0.0711
0.6767
0.0711
0.6902
0.0711
0.6851
0.7887
0.6872
3.6282
3.7411
3.6282
0.8000
0.8000
0.8000
c
0.1634
-0.0675
0.1511
sw
sg
s t, p
s t ,b
0.1090
0.0294
0.1083
0.1119
-
0.1113
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
g
y
c/ y
k/y
b/ y
0.8096* s
tr
0.8096* s
tr
0.8096* s
tr
0.1904* s
tr
0.1904* s
tr
0.1904* s
tr
Notes: See notes of Table 2.
37
Table 4
Long-run solution when the labor tax rate is the residual policy instrument
1
Status quo
economy
2
Cost-minimizing
Private providers
3
Cost-minimizing
public providers
-1.0280
-0.7900
-1.0282
4
Cost-minimizing private
providers
plus
redistributive transfers
-0.8682
-1.1632
-1.6245
-1.1635
-0.8682
-1.0537
0.4918
-0.9489
0.6453
-1.0540
0.4917
-0.8682
0.6119
0.4158
0.2289
0.4156
0.4665
0.3648
0.3811
0.3648
0.3943
0.3480
0.2690
0.3479
0.2200
0.8085
0.2554
0.8083
0.4268
0.6949
0.0719
0.7153
0.0719
0.6949
0.0719
0.7401
0.0719
0.6868
0.7913
0.6867
0.7894
3.6282
3.7162
3.6282
3.7680
0.8000
0.8000
0.8000
0.8000
l
0.2371
-0.0576
0.2373
-0.0381
sw
sg
s t, p
s t ,b
0.1090
0.0255
0.1090
0.0337
0.1119
-
0.1120
-
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
g
y
c/ y
k/y
b/ y
0.8096* s
tr
0.8096* s
tr
0.8096* s
tr
0.5302* s
tr
0.1904* s
Notes: See notes of Table 2.
tr
0.1904* s
tr
0.1904* s
tr
0.4698* s
tr
38
Table 5
Lifetime utility under regime switches,
with the labor tax rate as the residual policy instrument in the long run
1
Status quo
economy
Up
-
z
U
-102.8009
b
z
U
z
-116.3176
-105.3745
-
2
From status quo
economy to
cost-minimizing
private providers
-80.2693
0.2564
-160.1591
-0.3611
-95.4803
0.1388
3
From status quo
economy to costminimizing
public providers
-102.8184
-0.0002
-116.3541
-0.0004
-105.3956
-0.0002

4
From status quo economy
to cost-minimizing private
providers plus
redistributive transfers
-88.3975
0.1572
-85.9722
0.3606
-87.9357
0.1959
Notes: (i) See notes of Table 2. (ii) For h  p, b , U h    t u (cth , eth , Yt g ) . (iii) U  v U  v U . (iv) z is
p
t 0
the constant private consumption supplement which makes U
39
p
Ub.
p
b
b
Table 6
Long-run solution when public debt is the residual policy instrument
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
1
Status quo
economy
2
Cost-minimizing
Private providers
3
Cost-minimizing
public providers
-1.0428
-0.9798
-1.0275
-1.1715
-1.4839
-1.4115
-1.0673
0.4791
-1.0758
0.4956
-1.1006
0.4836
0.4074
0.2540
0.2890
0.3589
0.3420
0.3555
0.3413
0.2381
0.2794
0.8109
0.3617
0.4744
0.6836
0.6419
0.6772
yg
c/ y
0.0728
0.0728
0.0728
0.6808
0.7004
0.6594
k/y
3.6282
3.7801
3.6282
b/ y
2.1271
7.4257
3.3678
sw
sg
s t, p
s t ,b
0.1090
0.0356
0.0527
0.1119
-
0.1230
0.8096* s
tr
0.8096* s
tr
0.8096* s
tr
0.1904* s
tr
0.1904* s
tr
0.1904* s
tr
Notes: (i) The parameterization is as Table 1 except that now 1  
formula is used for all per capita quantities).
40
 0.3 . (ii) u  v p u p  v b u b (the same
Table 7
Long-run solution when the consumption tax rate is the residual instrument
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
-1.0342
-0.8274
-0.9892
-1.1652
-1.4819
-1.4057
-1.0591
0.4852
-0.9521
0.5976
-1.0685
0.5091
0.4117
0.2597
0.2938
0.3611
0.3605
0.3624
0.3437
0.2541
0.2868
0.8100
0.3111
0.4616
0.6877
0.0733
0.6766
0.0733
0.6903
0.0733
0.6851
0.7882
0.6781
3.6282
3.7512
3.6282
0.8000
0.8000
0.8000
c
0.1634
-0.0653
0.1055
sw
sg
s t, p
s t ,b
0.1090
0.0310
0.0516
0.1119
-
0.1205
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
g
y
c/ y
k/y
b/ y
0.8096* s
tr
0.8096* s
tr
0.8096* s
tr
0.1904* s
Notes: See notes of Table 6.
tr
0.1904* s
tr
0.1904* s
tr
41
Table 8
Long-run solution when the labor tax rate is the residual instrument
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
-1.0276
-0.7922
-0.9765
4
Cost-minimizing
private providers
plus
redistributive transfers
-0.8697
-1.1627
-1.6094
-1.4251
-0.8697
-1.0534
0.4917
-0.9478
0.6433
-1.0619
0.5219
-0.8697
0.6101
0.4157
0.2336
0.2914
0.4679
0.3648
0.3808
0.3689
0.3938
0.3479
0.2731
0.2947
0.2244
0.8086
0.2656
0.4509
0.4386
0.6948
0.0741
0.7148
0.0741
0.7028
0.0741
0.7391
0.0741
0.6868
0.7908
0.6802
0.7888
3.6282
3.7254
3.6282
3.7784
0.8000
0.8000
0.8000
0.8000
l
0.2371
-0.0541
0.1816
-0.0342
sw
sg
s t, p
s t ,b
0.1090
0.0269
0.0509
0.0353
0.1119
-
0.1188
-
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
g
y
c/ y
k/y
b/ y
0.8096* s
tr
0.8096* s
tr
0.8096* s
tr
0.5353* s
tr
0.1904* s
Notes: See notes of Table 6.
tr
0.1904* s
tr
0.1904* s
tr
0.4648* s
tr
42
Table 9
Lifetime utility under regime switches,
with the labor tax rate as the residual policy instrument in the long run
1
Status quo
economy
2
From status quo
economy to
cost-minimizing
private providers
3
From status quo
economy to costminimizing public
providers
Up
-102.7638
z
-
-80.4913
0.2532
-158.6424
-0.3515
-95.3712
0.1381
-97.7318
0.0524
-142.6349
-0.2359
-106.2813
-0.0025
U
b
z
U
z
-116.2714
-105.3357
-

4
From status quo
economy to costminimizing private
providers plus
redistributive transfers
-88.5456
0.1551
-86.1088
0.3583
-88.0816
0.1938
Notes: (i) See notes of Table 6. (ii) For h  p, b , U h    t u (cth , eth , Yt g ) . (iii) U  v U  v U . (iv) z is
p
t 0
the constant private consumption supplement which makes U
43
p
Ub.
p
b
b
Table 10
Long-run solution when public debt is the residual instrument
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
-1.0436
-0.9791
-1.0675
-1.1724
-1.5041
-0.8698
-1.0681
0.4793
-1.0791
0.4963
-1.0299
0.4723
0.4076
0.2466
0.5978
0.3591
0.3419
0.3643
0.3415
0.2292
0.3830
0.8108
0.3378
1.3267
0.6839
0.6417
0.6939
yg
c/ y
0.0683
0.0683
0.0683
0.6808
0.6993
0.7151
k/y
3.6282
3.7576
3.6282
b/ y
2.1272
7.5226
-0.0184
sw
sg
s t, p
s t ,b
0.1090
0.0320
0.1971
0.1119
-
0.0845
0.8096* s
tr
0.8096* s
tr
0.8096* s
tr
0.1904* s
tr
0.1904* s
tr
0.1904* s
tr
Notes: (i) The parameterization is as Table 1 except that now 1  
formula is used for all per capita quantities).
44
 0.7 . (ii) u  v p u p  v b u b (the same
Table 11
Long-run solution when the consumption tax rate is the residual instrument
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
-1.0349
-0.8231
-1.0777
-1.1661
-1.5056
-0.8659
-1.0599
0.4854
-0.9531
0.6009
-1.0374
0.4657
0.4119
0.2510
0.5998
0.3612
0.3606
0.3623
0.3439
0.2451
0.3827
0.8100
0.2884
1.3560
0.6880
0.0687
0.6769
0.0687
0.6902
0.0687
0.6851
0.7893
0.7118
3.6282
3.7304
3.6282
0.8000
0.8000
0.8000
c
0.1634
-0.0699
0.2081
sw
sg
s t, p
s t ,b
0.1090
0.0277
0.2024
0.1119
-
0.0868
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
g
y
c/ y
k/y
b/ y
0.8096* s
tr
0.8096* s
tr
0.8096* s
tr
0.1904* s
Notes: See notes of Table 10.
tr
0.1904* s
tr
0.1904* s
tr
45
Table 12
Long-run solution when the labor tax rate is the residual instrument
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
-1.0284
-0.7876
-1.0937
4
Cost-minimizing private
providers
plus
redistributive transfers
-0.8666
-1.1637
-1.6407
-0.8565
-0.8666
-1.0541
0.4919
-0.9501
0.6474
-1.0485
0.4556
-0.8666
0.6138
0.4159
0.2239
0.6054
0.4650
0.3649
0.3813
0.3593
0.3948
0.3481
0.2644
0.3827
0.2150
0.8085
0.2446
1.4100
0.4141
0.6951
0.0695
0.7157
0.0695
0.6843
0.0695
0.7411
0.0695
0.6868
0.7919
0.7074
0.7900
3.6282
3.7065
3.6282
3.7568
0.8000
0.8000
0.8000
0.8000
l
0.2371
-0.0613
0.3036
-0.0424
sw
sg
s t, p
s t ,b
0.1090
0.0240
0.2123
0.0319
0.1119
-
0.0910
-
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
g
y
c/ y
k/y
b/ y
0.8096* s
tr
0.8096* s
tr
0.8096* s
tr
0.5248* s
tr
0.1904* s
Notes: See notes of Table 10.
tr
0.1904* s
tr
0.1904* s
tr
0.4752* s
tr
46
Table 13
Lifetime utility under regime switches,
with the labor tax rate as the residual policy instrument in the long run
1
Status quo
economy
2
From status quo
economy to
cost-minimizing
private providers
3
From status quo
economy to costminimizing public
providers
Up
-102.8394
z
-
-80.0365
0.2597
-161.8003
-0.3712
-95.6043
0.1396
-109.2500
-0.0630
-85.5912
0.3664
-104.7453
0.0188
U
b
z
U
z
-116.3656
-105.4148
-

4
From status quo
economy to costminimizing private
providers plus
redistributive transfers
-88.2394
0.1594
-85.8381
0.3630
-87.7822
0.1982
Notes: (i) See notes of Table 10. (ii) For h  p, b , U h    t u (cth , eth , Yt g ) . (iii) U  v U  v U . (iv) z
p
t 0
is the constant private consumption supplement which makes U
47
p
Ub.
p
b
b
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