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Reforming the public sector:
Public providers, or private providers, of public goods?
CESifo, Munich, September 26, 2012
(This lecture is based on a joint work with Apostolis Philippopoulos (AUEB and CESifo), and Vanghelis Vassilatos (AUEB))
Motivation
 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; for instance, 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.
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 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
the 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 classic textbook, Atkinson and Stiglitz (1980, p. 482) emphasize that
“the two are often confused, though both logically and in practice they are distinct”. 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). Here the focus is on the way of
provision.
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 In policy, there is an ongoing debate on the role of the state and, in particular, the idea of
opening up public services to new providers; an example is the recent debate in the UK.
 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 public-sector providers”
(The Guardian, 10 February 2011, p. 15).
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Basic Questions
 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?
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The aim of the lecture
 In the present lecture, we will try to answer the above questions. To do so, we will study the
difference between public production and public finance in a dynamic general equilibrium
setup.
 We will study 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.
 We will first model the case of public production (for similar models see e.g. Finn, 1998,
Cavallo, 2005, Ardagna, 2007, and Pappa, 2009).
 There are two distinct groups of households: those that work in the private sector and those
that are employed in the public sector.
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 The latter (called public employees), together with goods purchased from the private sector,
are used as inputs in the government production function. Solving the model numerically
when the values of fiscal policy instruments are in line with the UK averages over 19902008, we will specify, among other variables, the time-path of public goods as induced by
the existing fiscal policy mix.
 In turn, 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 (for a similar model see Turnovsky and Pintea, 2006).
 We also study what would change if, again other things equal, the same amount of public
goods continues to be produced by the public sector but now public enterprises minimize
their costs like their private counterparts do in the case of public finance.
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 The latter is 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.
 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. Before listing the main results below, it is worth reporting that these results are
robust to a number of changes, including the endogenous reallocation of employees across
sectors, the specific type of the status quo we depart from, the productivity of public
employees and the specification of the production function used by the public sector for the
production of public goods.
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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)
fall when they turn from public employees into employees at cost-minimizing private
providers. Since private providers find it optimal to pay lower wages and hence produce the
public good at a lower cost, the switch allows the government to make efficiency savings.
 Second, the effect of this switch on per capita output and welfare (i.e. on aggregate
efficiency) depends crucially on the way the government uses its efficiency savings. When
the efficiency savings achieved by the government - through the use of private providers are used to cut distorting income taxes, then per capita output and welfare also rise.
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 Third, for a large range of parameter values, when it is public providers/enterprises that
choose inputs in a cost-minimizing way, the numerical solution is very similar to that under
the status quo case where the associated variables are exogenously set at their data averages.
Thus, one could argue 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.
 Fourth, since the above policy allows aggregate efficiency gains, but only at the cost of
making those that used to be public employees worse off, there is need to search for Paretoimproving changes. In such a search, we show that everybody can become better off relative
to the status quo if this policy is supplemented by redistributive government transfers that
compensate the ex public employees, and/or a voluntary reallocation of employees across
sectors. Actually, the latter, namely a voluntary reallocation of employees from the
production of the public to the production of the private good, is particularly beneficial
because it also boosts the supply side of the economy. By contrast, the former, namely
redistributive government transfers, works through the demand side of the economy.
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 Therefore, although we are aware that one should treat quantitative results with caution, our
policy message is as follows:
If the government wishes to increase the aggregate pie and also make everybody better off, it
should adopt a mix of reforms that:
(i) assigns the production of public goods to cost-minimizing private providers
(ii) uses its efficiency savings, achieved through the use of private providers, to cut distorting
income taxes
(iii) adds a mechanism (redistributive government transfers and, in particular, a reallocation
of employees) to reduce the rise in inequality caused by switching to private providers.
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Clarifications
 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.
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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 fraction 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 was 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 given and looks at the efficiency of the tax system. Here,
similarly, we look at the efficiency of the system of public goods provision.
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An economy with public production of public goods (status quo)
 We will 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 produced 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.
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 To finance total public spending, including the cost of the public good, the government
levies distorting taxes and issues bonds.
 Notice that irrespectively of the producer, we assume that public goods are provided freely
without user charges. See also Atkinson and Stiglitz (1980, chapter 16).
 For simplicity, the model is deterministic. Time is discrete and infinite.
 As said above, the status quo model, presented in this section, is similar to that used by most
of the related literature (see Finn, 1998, Cavallo, 2005, Ardagna, 2007, Pappa, 2009,
Linnemann, 2009, Forni et al., 2009, and Fernández-de-Córdoba et al., 2010).
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Population composition and agents’ economic roles
 The population size at time t , Nt , is exogenous. Among Nt , there are p  1,2,..., Ntp identical
households that work in the private sector and b  1,2,..., Ntb identical households that work
in the public sector, where N tp  N tb  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, N t f  N tp , or equivalently each
household employed in the private sector owns one private firm. This population
composition is not important to our results and also allows us to avoid scale effects in
Ntb
, is exogenously set by
equilibrium. The fraction of public employees in population, v 
Nt
b
t
the government (see below for an endogenous determination of vtb ).
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 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. 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).
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Households working in the private sector
 The lifetime utility of each household working in the private sector, p  1,2,..., Ntp , is:

t
p
p
g
  u(ct , et , Yt )
(1)
t 0
where ctp and etp are p ’s consumption and work hours respectively; Yt g is per capita public
Yt g
goods and services; and 0    1 is a time preference parameter. Notice that Yt 
,
Nt
g
where Yt g is total public goods and services.
 The period utility function is (see also e.g. Christiano and Eichenbaum, 1992):
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(etp )1
u(c , e ,Yt )  log(c  Yt )  
1 
p
t
p
t
g
p
t
g
(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  d tp  (1   tk )(rt ktp   tp )  (1   tl ) wtp etp  t btp  Gt tr , p
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(3a)
where itp is savings in the form of physical capital, d tp is savings in the form of government
bonds,  tp is dividends received from private firms, wtp is the wage rate in the private sector,
Gt tr , 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, 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:
ktp1  (1   ) ktp  itp 
btp1  btp  d tp 

k
 t
2  Yt
p,k
 p ,b  btp 
 
2  Yt 
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p



2
(3b)
2
(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. Notice that Yt 
Yt
, where Yt is total output in the economy.
Nt
 Regarding the 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 root problems in
the transition path and also get a solution for the portfolio share of each agent in the long run
(see below for details). None of our main results depend on these transaction costs.
 Each p chooses c , k , b , e
p
t
p
p
t 1
t 1
t
p


t 0
taking factor prices, economy-wide quantities and policy
variables as given.
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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,..., Ntb , is:

t
b
b
g
  u(ct , et ,Yt )
t 0
(etb )1
u(c , e , Yt )  log(c  Yt )  
1 
b
t
b
t
g
b
t
g
(4)
(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  Gt tr , b
(6a)
where wtg is the wage rate in the public sector and Gt tr ,b is government transfers to each b .
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 The laws of motion of physical capital and government bonds for each b are:
ktb1  (1   )ktb  itb 
 b , k  ktb 
 
2  Yt 
  btb 
b
b
b
 
bt 1  bt  d t 
2  Yt 
b,b
2
(6b)
2
(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.
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Private firms producing the private good
 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 technology parameters. Notice that we could assume that
public goods provide productivity-enhancing services in addition to utility-enhancing ones
(see e.g. Ardagna, 2007). We report that our main results do not change.
 In each period, each f chooses kt f and etf taking factor prices as given.
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Public sector
 We now present the public sector. We start with the government budget constraint and then
specify the production function of public goods and services.
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 the total public wage bill; Gttr , p and Gttr , b are respectively transfers to private and
public employees; Bt is the beginning-of-period total stock of government bonds; and Tt
denotes total tax revenues defined as:
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Tt   tc ( N tp ctp  N tb ctb )   tk [ N tp (rt ktp   tp )  N tb rt ktb ]   tl ( N tp wtp etp  N tb wtg etb )
(9b)
 Thus, as in Alesina et al. (2002), we include the three main types of government spending.
We also include the three main types of taxes (taxes on consumption, capital income and
labor income).Inspection of (9a-b) implies that, in each time period, there are nine policy
instruments (Gtg , Gtw , Gttr , p , Gttr , b , tc , tk , tl , Bt 1 , N tb ) out of which one needs to adjust to satisfy
the government budget constraint. Following most of the related literature, we start by
assuming that the adjusting instrument is the end-of-period total public debt, Bt 1 , so that the
other eight policy instruments can be set exogenously by the government. For convenience,
concerning spending policy instruments, we will work in terms of their GDP shares,
Gtg w Gtw tr , p Gttr , p tr , b Gttr , b
, st 
, st 
, st 
, where Yt denotes total output.
s 
Yt
Yt
Yt
Yt
g
t
 Similarly, concerning the number of public employees, we will work in terms of their
N tb
population share, v 
. The processes of exogenous variables are defined below.
Nt
b
t
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Public sector production function
 Following most of the related literature, we start by assuming that total public goods and
services, Yt g , are produced using goods purchased from the private sector, Gtg , and public
employment, Ltg (where, in equilibrium, Ltg  N tb etb ). In particular, following Linnemann
(2009), we start by using a CRS Cobb-Douglas production function of the form:
Yt g  A(Gtg ) ( Ltg )1
(10)
where 0    1 is a technology parameter.
 We wish to emphasize four things in (10).
- First, as you will see, later we generalize the public production function (10) and also
report what happens when this function is the same as that of private firms’ in (8).
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- Second, our modeling in (10) nests most specifications used before. For instance,
Ardanga (2007) assumes that the sole input is public employment. At the other extreme,
the business cycle and endogenous growth literatures assume that there is a one-to-one
relationship between the amount of public goods and goods purchased from the private
sector. 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; to the extent that public goods used for public investment
are also purchased from the private sector, adding public capital as an input in (10) does
not affect our main results.
- Third, 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.
- Fourth, in our numerical solutions below, we experiment with various values of the
relatively unknown parameter, 0    1.
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Decentralized competitive equilibrium (DCE) with public production
 Combining the above, we solve for a DCE in which:
(i) all households maximize utility acting competitively
(ii) all firms in the private sector maximize profits acting competitively
(iii) all markets clear (see Appendix C for market-clearing conditions)
(iv) all constraints are satisfied.
 The DCE is summarized by the following eleven equilibrium conditions, where
Gtg
Yt g
Gttr , p
Gttr , b w Gtw wtg Lgt wtgvtbetb
tr , p
tr , b
g
g p f
 p f :
, Gt 
 st vt yt , Gt  p , Gt  b , st  
Yt  N t yt , Yt 
Nt
Nt
Nt
Nt
Yt
Yt
vt yt
p
f
g
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 (etp ) (1   tc )(ctp   Yt g )  (1   tl ) wtp
(11a)

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 ) 




(11c)
2
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
1    (1   t 1 )rt 1   ( p y f ) 2 
1
t 1 t 1


(1   tc )(ctb  Yt g )
(1   tc1 )(ctb1  Yt g1 )






(11f)
29

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)
 tb b  p 1
yt  A( kt  p kt ) (et )
t
(11h)
Yt g  A( stg tp ytf ) ( tb etb )1
(11i)
f
p
( stw  stg  sttr , p  sttr ,b ) tp ytf  (1  t )( tp btp   tbbtb )   tp1btp1   tb1btb1 
+

 p ,k
 t ctp  ktp1  (1   ) ktp 
2

p
c
t
( tp ctp   tb ctb )   tk rt ( tp ktp   tb ktb )  tl ( wtp tp etp  wtg tb etb )
(11j)
2
2
 ktp   p ,b  btp  
 p f  

 
2   tp ytf  
  t yt 

2
2

 b ,k  ktb   b ,b  btb   g p f
b
b
b
p f
  ct  kt 1  (1   ) kt 
 p f  
 p f    st  t yt   t yt
2   t yt 
2   t yt  


b
t
30
(11k)
where, in the above equations, we use the factor returns:
rt 
ytf vtp
v k v k
p p
t t
b b
t t
(12a)
(1   ) ytf
w 
etp
(12b)
stwvtp ytf
w  b b
vt et
(12c)
p
t
g
t
Equations (12a-b) follow from the optimality conditions of the private firm and the related
market-clearing conditions, while equation (12c) follows from the policy rule
Gtw wtg Ltg wtg Ntbetb wtg vtbetb
s 

 p f  p f .
Yt
Yt
Nt yt
vt yt
w
t
31
 We therefore have eleven equations, (11a-k), in eleven endogenous variables,
c
p
t
, 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 , vtb t 0 .

 Equations (11a-c) and (11e-g) are the optimality conditions of private and public employees
respectively, with respect to labor, savings in capital and savings in bonds. Equations (11d),
(11j) and (11k) are the three linearly independent budget constraints (private employees’, the
government’s and the economy’s resource constraint). Equations (11h) and (11i) are the
production functions for the private and the public good. These equilibrium equations, (11ak), are log-linearized around their long-run solution. The model is solved numerically.
Notice 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). This is our “status quo” model.
32
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 (see also Turnovsky and Pintea, 2006). Thus, now the
government is not involved in any production itself.
33
Population composition and agents’ economic roles
 To make the comparison meaningful, we allow for private providers keeping the rest of the
model unchanged. Thus, as above, the number of private firms producing the private good,
f , equals the number of households working in these firms, p . This number remains as
before.
 Analogously, we assume that the number of private providers producing the public good
ordered by the government, denoted as g , equals the number of households working in these
firms, b . Again, this number remains as before.
 In other words, the allocation of employees/households to sectors, as well as the total
population, remains as before. We report that our qualitative results do not depend on these
scaling assumptions, while some generalizations will be presented below.
34
 What changes, relative to the status quo model, is the introduction of private firms producing
the public good, the so-called private providers, indexed by g  1,2,..., N tg , 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 numerical 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 also finances the total
cost of private providers, N tg [rt g ktg  wtg etg ] , where rt g and wtg are the rental costs of capital
and labor paid by private providers, and ktg and etg are the capital and labor inputs used by
each private provider.
35
 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.
 Notice that the total cost, N tg [ rt g ktg  wtg etg ] , replaces spending on public wages, Gtw , and
goods purchased from the private sector, Gtg , which were among the government spending
items in the status quo economy.
 In what follows, we present what changes relative to the status quo economy.
36
Private firms producing a given amount of the public good (private providers)
 In each period, each private provider of public goods, g  1,2,..., N tg chooses ktg and etg 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):

 Yt g
rt k  w e  t  b  ytg 

 Nt
g
g
t
g g
t t
(13)
where rt g , wtg and Yt g have been defined above; t is a multiplier measuring the marginal
cost of production; 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
37
(14)
 Each g chooses ktg and etg taking factor prices and economy-wide quantities as given. The
first-order conditions, working as in Mas-Colell et al. (1995, pp. 139-143), are:
rt  t
g
ytg
(15a)
ktg
(1   ) ytg
w  t
etg
(15b)
Yt g
g 
g 1 

A
(
k
)
(
e
) 0
t
t
b
Nt
(15c)
g
t
38
 It is useful to point out three things.
- First, the determination of wtg is different from the status quo economy. In particular,
while it was determined by the policy rule for the share of the public wage bill in the
status quo economy (see equation (12c) above), it is now market-determined as shown by
equation (15b).
- Second, the solution to the cost-minimization problem above implies that the profits of
private providers,  tg , are zero (recall that private firms producing the private good also
make zero profits).
- Third, now both types of firms, f and g , participate in the factor markets (see also the
market-clearing conditions below).
39
Government budget constraint
- The budget constraint of the government changes from (9a) to:
N tg [rt g ktg  wtg etg ]  Gttr , p  Gttr , b  (1  t ) Bt  Bt 1  Tt
(16a)
where tax revenues change from (9b) to:
Tt   tc ( N tp ctp  N tb ctb )   tk [ N tp (rt ktp   tp )  N tb (rt g ktg   tb ]   tl ( N tp wtp etp  N tb wtg etb ) (16b)
where the first term on the left-hand side of (16a) is the total cost of public goods produced
by private firms and all other variables are as defined above.
 In each period, there are seven policy instruments (Gttr , p , Gttr , b , tc , tk , tl , Bt 1 , vtb ) or
equivalently in ratios ( sttr , p , sttr , b , tc , tk , tl , Bt 1 , vtb ) . As in the status quo economy, we will start
by assuming that the residually determined policy instrument is the end-of-period public
debt, Bt 1 .
40
Decentralized competitive equilibrium (DCE) with cost-minimizing private providers
 Combining the above, we solve for a DCE in which:
(i) all households maximize utility acting competitively
(ii) all private firms that produce the private good maximize profits, and all private firms that
produce the public good minimize costs, acting competitively
(iii) all markets clear (see Appendix F for the new market-clearing conditions)
(iv) all constraints are satisfied.
 The new DCE is summarized by the following eleven equilibrium conditions:
41
 (etp ) (1   tc )(ctp   Yt g )  (1   tl ) wtp
(17a)

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
 (etb ) (1   tc )(ctb   Yt g )  (1   tl ) wtg
(17d)
(17e)

ktb1 
k
g
b,k
1    (1   t 1 )rt 1   ( p y f ) 2 
1
t 1 t 1


(1   tc )(ctb  Yt g )
(1   tc1 )(ctb1  Yt g1 )






(17f)
42

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   tb btb )   tp1btp1   tb1btb1 
+

 p ,k
p
p
p

 t ct  kt 1  (1   ) kt 
2

p
c
t
( tp ctp   tb ctb )   tk ( rt p tp ktp  rt g tb ktb )  tl ( wtp tp etp  wtg tb etb )
(17j)
2
2
 ktp   p ,b  btp  
 p f  
 p f  
y

2
 t t 
  t yt  

 b,k
  ctb  ktb1  (1   ) ktb 
2

b
t
2
2
 ktb   b ,b  btb  
p f
 p f  
 p f     t yt
y
y
2


 t t 
 t t  
where, in the above equations, we use the factor returns:
43
(17k)
rt   t
p
rt  t
g
ytf
(18a)
kt p
ytg
(18b)
ktg
(1   ) ytf
w 
etp
p
t
(18c)
(1   ) ytg
w  t
etg
g
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 , vtb ) 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).
44
 The new equilibrium conditions (17a-k) are similar to those in (11a-k) except that now:
(i) public wages, wtg , are determined in a cost-minimizing way
(ii) all producers have the same production function
(iii) the government finances the cost of private providers, while it spent on stw and stg in the
status quo economy
(iv) the market-clearing conditions differ from the status quo economy
(v) we do not have spending on private goods purchased by the government in the
economy’s resource constraint.
 These equilibrium equations, (17a-k), are log-linearized around their long-run solution. The
model is solved numerically in the next section.
45
Numerical solutions and comparison of the two model economies
 We solve numerically the two model economies presented above and then compare them.
How we work
 We work in two steps.
- We first solve numerically the status quo economy using conventional parameter values
and fiscal data from the UK economy. The numerical 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.
46
- 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 which 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 .

- We will then 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. 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.
- Thus, we will first evaluate various policy regimes based on a comparison of long-run
equilibria. Transitional dynamics, as well as lifetime welfare gains from moving from one
regime to another, are discussed later.
47
Parameter values and policy instruments
 Table 1 reports the baseline parameter values for technology and preference, as well as the
values of the exogenous policy instruments, used to solve the status quo model economy.
The time unit is meant to be a year.
 Regarding parameters for technology and preference, we use conventional values used by the
business cycle literature (see e.g. Malley et al., 2009, and Angelopoulos et al., 2011a, who
have calibrated aggregate DSGE models to the UK economy). When we have no a priori
information about a parameter value, or when different authors use different values, we
experiment with a range of values. Regarding fiscal data, 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. 1970-2008 or 1996-2008.
48
Table 1: Baseline parameterization
Parameters
and policy
instruments
Description
Value
Share of capital in private production
0.399
Share of public employment in public production
0.493
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 (data)
0.1090
sg
Public purchases as share of GDP (data)
0.1119
s tr
Public transfers as share of GDP (data)
0.2199
c
Tax rate on consumption (data)
0.1852

1
k




Parameters
and policy
instruments
Description
Value
k
l
Tax rate on capital income (data)
0.3875
Tax rate on labor income (data)
0.2685
vb
A
Public employees as share of population (data)
0.1904
Long-run TFP
1


 p ,k
 p ,b
 b,k
 b ,b
Autoregressive parameter of TFP
0.9
Standard deviation of TFP
0.01
Transaction cost incurred by private agents in capital
market
Transaction cost incurred by private agents in bond
market
Transaction cost incurred by public employees in
capital market
Transaction cost incurred by public employees in
bond market
0.002
 Let us discuss, briefly, the values summarized in Table 1.
49
0.002
0.002
0.002
 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.
 The other preference parameters related to hours of work,  and  , are set at 5 and 1
respectively; these parameter values imply hours of work within usual ranges.
 The capital depreciation rate,  k , is set at 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 all these parameter values.
50
 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).
 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, stw and sttr , are
respectively 0.109 and 0.2199, again as in the data.
 We assume that transfers are allocated to private and public employees according to their
shares in population, sttr , p  vtp sttr  (1  vtb ) sttr and sttr ,b  vtb sttr .
 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.
51
 The effective tax rates on consumption, capital and labor,  tc ,  tk and  tl , are respectively
0.1852, 0.3875 and 0.2685 over 1990-2008.
 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.
Notice that:
 Without transaction costs, that is   0 , the long-run system would be “under-identified” in
the sense that there would be nine equations and eleven variables.
52
 This happens because, in the long run, if   0 , the two agents’ Euler conditions for capital
are reduced to one equation only. The same applies to the two Euler conditions for bonds.
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. 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 extraneous assumption is usual in the literature (see Mendoza and Tesar, 1998, in a
two-country model).
53
Long-run solution when the consumption tax rate is the adjusting instrument
 We stabilize the public debt-to-output ratio at 80% and allow the consumption tax rate,  c , to
adjust to satisfy the government budget constraint in the long run.
 Using the parameterization in Table 1, the long-run solutions of the status quo economy and
the reformed economy are reported respectively in columns 1 and 2 in Table 2. These longrun solutions follow from solving the systems (11a-k) and (17a-k) respectively when
variables do not change.
 Recall that, in the reformed economy, the same amount of public goods, as found in the
status quo economy, 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 cost-minimizing private providers/firms in the reformed economy, while the
superscript p denotes those households that work in private firms producing the private
good.
54
Table 2
Long-run solution when the consumption tax rate is the residual policy instrument
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
yg
c/ y
k/y
b/ y
c
sw
sg
s t, p
s t ,b
total cost of
public good (GDP share)
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
-1.0345
-1.1656
-1.0595
0.4853
0.4118
0.3611
0.3438
0.8100
0.6879
0.0711
0.6851
3.6282
0.8000
0.1634
0.1090
0.1119
0.8096* str
0.1904* str
0.2209
-0.8253
-1.4933
-0.9525
0.5992
0.2555
0.3605
0.2499
0.3001
0.6767
0.0711
0.7887
3.7411
0.8000
-0.0675
0.0294
0.8096* str
0.1904* str
0.0489
-1.0295
-1.1651
-1.0553
0.4889
0.4127
0.3624
0.3447
0.8052
0.6902
0.0711
0.6872
3.6282
0.8000
0.1511
0.1083
0.1113
0.8096* str
0.1904* str
0.2196
55
Discussion of the status quo solution
 Before we compare the two regimes, we point out that the long-run solution of our status quo
economy in column 1 of Table 2 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, w g / w p , is found to be 0.81 in column 1 of Table 2, which is close to that
in the actual data over the 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.
 Notice that in the long run of the status quo economy, since w g  w p , public employees are
worse off than private employees, u b  u p (see column 1 in Table 2).
 Possibly, one could question whether this is a reasonable departure point in the sense that, in
a long-run equilibrium, private agents should be indifferent between being of b or p type.
56
 We have experimented with this case by allowing the fraction of public employees, or the
allocation of total government transfers to the two types of households, to be endogenous so
as u p  u b in the long run of the status quo economy. None of our qualitative results is
affected by this. Hence, since the aim of the paper is to study the implications of policy
reforms, rather than to specify how private workers and public employees differ in the status
quo economy, we proceed with the status quo solution as reported in column 1.
 We can now 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).
57
Distributional implications of switching to private providers
 In the long run, the ratio of public to private wages, w g / w p , falls from 0.81 in column 1 to
only 0.30 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 fall in the long-run
utility of b households, u b , as we switch from the status quo to the reformed economy.
 By contrast, the long-run utility of p households, u p , rises in column 2 . This is thanks to
higher consumption, c p , enjoyed by p households under private provision (see below for
details). Notice that, in this particular experiment, the beneficial welfare effects on p
households dominate the adverse effects on b households, so that per capita long-run utility,
denoted as u , rises under private provision in column 2.
58
Macroeconomic implications of switching to private providers
 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 Gt g releases ceteris paribus resources for private consumption and
investment. 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.
59
 The above are direct effects that work through resource reallocation. 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 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 0.0294 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
the consumption tax rate, these efficiency savings 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 in Table 2.
60
 The combination of direct-resource effects and indirect-public financing effects shapes, in
turn, the value of per capita output, y . In the numerical experiment reported in Table 2, y
slightly falls as we switch to the reformed economy.
 This seemingly paradoxical result arises simply because we have assumed that it is the
consumption tax rate that adjusts to close the government budget. As said above, in this
baseline case, efficiency savings allow the government to afford lower consumption taxes.
But the resulting rise in the consumption of p households is not strong enough to offset the
adverse effects coming from the fall in consumption of b households and less public
spending. At the same time, on the supply side, capital is also used for the private production
of the public good which is not marketed, while the reduction in consumption taxes cannot
boost the production side of the economy.
 Hence, combining adverse demand effects and trivial supply effects, y falls as we switch to
private providers.
61
Long-run solution when the labor tax rate is the adjusting instrument
 We now study a more interesting way of public financing.
 In particular, we stabilize the public debt-to-output ratio at 80% and allow the labor tax rate,
 l , to adjust to satisfy the government budget constraint in the long run.
 Using again the parameterization in Table 1, the long-run solutions of the status quo
economy and the reformed economy are reported respectively in columns 1 and 2 in Table 3.
These long-run solutions follow from solving the systems (11a-k) and (17a-k) respectively
when variables do not change.
 Recall again that, in the reformed economy, the same amount of public goods, as found in
the status quo economy, is supplied by cost-minimizing private providers.
62
Table 3
Long-run solution when the labor tax rate is the residual policy instrument
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
yg
c/ y
k/y
b/ y
l
sw
sg
s t, p
s t ,b
vb
total cost of
public good (GDP share)
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
4
Cost-minimizing private providers
plus endogenous
redistributive transfers
5
Cost-minimizing private
providers plus endogenous
allocation of employees
-1.0280
-1.1632
-1.0537
0.4918
0.4158
0.3648
0.3480
0.8085
0.6949
0.0719
0.6868
3.6282
0.8000
0.2371
0.1090
0.1119
0.8096* str
0.1904* str
0.1904
0.2209
-0.7900
-1.6245
-0.9489
0.6453
0.2289
0.3811
0.2690
0.2554
0.7153
0.0719
0.7913
3.7162
0.8000
-0.0576
0.0255
0.8096* str
0.1904* str
0.1904
0.0424
-1.0282
-1.1635
-1.0540
0.4917
0.4156
0.3648
0.3479
0.8083
0.6949
0.0719
0.6867
3.6282
0.8000
0.2373
0.1090
0.1120
0.8096* str
0.1904* str
0.1904
0.2210
-0.8682
-0.8682
-0.8682
0.6119
0.4665
0.3943
0.2200
0.4268
0.7401
0.0719
0.7894
3.7680
0.8000
-0.0381
0.0337
0.5302* str
0.4698* str
0.1904
0.0560
-0.8035
-0.8035
-0.8035
0.6254
0.6254
0.3718
0.3718
1
0.7978
0.0719
0.7839
3.9211
0.8000
0.0086
0.0542
0.9173* str
0.0827* str
0.0827
0.0902
63
 In Table 3, 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 in Table 3).
 Since labor taxes are particularly distorting (see also e.g. Angelopoulos et al., 2011a, for the
UK economy), their reduction not only strongly stimulates c p , e p and in turn u (per capita
welfare, u , increases from -1.0537 in column 1 to -0.9489 in column 2 in Table 3), 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 3).
 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.
64
Summary
 A switch from the status quo economy to a reformed economy, where ceteris paribus the
same amount of public goods is produced by cost-minimizing private providers, increases
the welfare of private employees but makes public employees worse off.
 The effect on per capita welfare and output is ambiguous depending on the choice of the
adjusting public finance instrument.
 When the efficiency savings, enjoyed from a more efficient way of delivering the public
good, are used to cut distorting income (labor) taxes, per capita welfare and output can both
rise.
 Keep in mind that these are steady state comparisons; transition results, when we depart from
the status quo economy and travel to a reformed economy with private providers over time,
are presented below.
65
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, input decisions were exogenously set as in the data, while, in
the latter, the same decisions were made by cost-minimizing private providers.
 One is wondering what would happen when we compare the cases in which, not only private
providers, but also public providers, choose their inputs in a cost-minimizing way, 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, 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.
66
 In particular, as we did in section with private providers, we assume that a single public
provider 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 before, the path Yt g t 0


is exogenously set.
 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.
Cost-minimizing public enterprise
 The economy is basically as in status quo but now, in addition, in each period, the public
enterprise chooses its two inputs, Gtg and Ltg , or equivalently their output shares, stg and stw ,
to minimize its costs. The cost-minimization problem of the public enterprise is:
67
Gtg  wtg Ltg   t [Yt g  A(Gtg ) (Ltg )1 ]
(19)
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 first-order conditions combined imply:

stg

 stg (1   )  stw
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.
68
Decentralized competitive equilibrium (DCE) with cost-minimizing public provider
 In the new DCE, we have twelve equations, namely, the eleven equations of the status quo
economy,
c
p
t
(11a-k),
plus
equation
(20),
in
twelve
endogenous
variables,
, 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 , vtb t  0 , and the path of

Y 
g 
t 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.
69
Long-run solutions
 Long-run solutions of the new model economy, under the three different ways of public
financing, are reported in Tables 2 and 3 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.
70
Table 2
Long-run solution when the consumption tax rate is the residual policy instrument
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
yg
c/ y
k/y
b/ y
c
sw
sg
s t, p
s t ,b
total cost of
public good (GDP share)
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
-1.0345
-1.1656
-1.0595
0.4853
0.4118
0.3611
0.3438
0.8100
0.6879
0.0711
0.6851
3.6282
0.8000
0.1634
0.1090
0.1119
0.8096* str
0.1904* str
0.2209
-0.8253
-1.4933
-0.9525
0.5992
0.2555
0.3605
0.2499
0.3001
0.6767
0.0711
0.7887
3.7411
0.8000
-0.0675
0.0294
0.8096* str
0.1904* str
0.0489
-1.0295
-1.1651
-1.0553
0.4889
0.4127
0.3624
0.3447
0.8052
0.6902
0.0711
0.6872
3.6282
0.8000
0.1511
0.1083
0.1113
0.8096* str
0.1904* str
0.2196
71
Table 3
Long-run solution when the labor tax rate is the residual policy instrument
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
yg
c/ y
k/y
b/ y
l
sw
sg
s t, p
s t ,b
vb
total cost of
public good (GDP share)
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
4
Cost-minimizing private providers
plus endogenous
redistributive transfers
5
Cost-minimizing private
providers plus endogenous
allocation of employees
-1.0280
-1.1632
-1.0537
0.4918
0.4158
0.3648
0.3480
0.8085
0.6949
0.0719
0.6868
3.6282
0.8000
0.2371
0.1090
0.1119
0.8096* str
0.1904* str
0.1904
0.2209
-0.7900
-1.6245
-0.9489
0.6453
0.2289
0.3811
0.2690
0.2554
0.7153
0.0719
0.7913
3.7162
0.8000
-0.0576
0.0255
0.8096* str
0.1904* str
0.1904
0.0424
-1.0282
-1.1635
-1.0540
0.4917
0.4156
0.3648
0.3479
0.8083
0.6949
0.0719
0.6867
3.6282
0.8000
0.2373
0.1090
0.1120
0.8096* str
0.1904* str
0.1904
0.2210
-0.8682
-0.8682
-0.8682
0.6119
0.4665
0.3943
0.2200
0.4268
0.7401
0.0719
0.7894
3.7680
0.8000
-0.0381
0.0337
0.5302* str
0.4698* str
0.1904
0.0560
-0.8035
-0.8035
-0.8035
0.6254
0.6254
0.3718
0.3718
1
0.7978
0.0719
0.7839
3.9211
0.8000
0.0086
0.0542
0.9173* str
0.0827* str
0.0827
0.0902
72
Summary
 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 cost-minimizing private
providers is superior to public production, even when public providers act as costminimizers. Thus, one could argue 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.
 Below we generalize the public production function and explain why cost-minimizing
private providers do better than cost-minimizing public providers even when they both
choose inputs optimally.
73
Searching for a Pareto-improving mix of reforms
 As said, although per capita welfare can increase when we move from the status quo
economy to an economy with private providers of public goods, public employees become
worse off once they become 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 a switch to private provision/public finance - by introducing a supplementary
reform that improves the welfare of both types of agents relative to the status quo economy.
 Now, we are going to study two such reforms.
74
(i) First, a government transfer scheme that compensates the losers from a switch to
private provision/public finance.
(ii) Second, a reallocation of employees between the two sectors (i.e. the one producing
the private and the one producing the public good).
 We find it natural to report results only for those cases in which private provision/public
finance increases the aggregate pie (as measured by per capita output) relative to the status
quo economy. As shown above, and in particular in Table 3, this happens when the
efficiency savings from private provision/public finance are used to cut distorting income
labor taxes.
75
Endogenizing government transfers and a new DCE
 We search for a government transfer scheme that, in combination with private
provision/public finance of public goods and use of 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 fractions in the data, 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.
 Algebraically, the DCE consists of equations (17a-k) plus a new equation that equates longrun utility across the two agents, u b  u p , where the associated new endogenous variable is
the share of government transfers between the two agents, x b , where s tr , b  x b s tr ,
s tr , p  (1  x b ) s tr and s tr is the value of total transfers in the data as share of output.
Computationally, we simply search for a value of x b so as u b  u p . The long-run solution of
this economy is reported in column 4 in Table 3.
76
Table 3
Long-run solution when the labor tax rate is the residual policy instrument
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
yg
c/ y
k/y
b/ y
l
sw
sg
s t, p
s t ,b
vb
total cost of
public good (GDP share)
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
4
Cost-minimizing private providers
plus endogenous
redistributive transfers
5
Cost-minimizing private
providers plus endogenous
allocation of employees
-1.0280
-1.1632
-1.0537
0.4918
0.4158
0.3648
0.3480
0.8085
0.6949
0.0719
0.6868
3.6282
0.8000
0.2371
0.1090
0.1119
0.8096* str
0.1904* str
0.1904
0.2209
-0.7900
-1.6245
-0.9489
0.6453
0.2289
0.3811
0.2690
0.2554
0.7153
0.0719
0.7913
3.7162
0.8000
-0.0576
0.0255
0.8096* str
0.1904* str
0.1904
0.0424
-1.0282
-1.1635
-1.0540
0.4917
0.4156
0.3648
0.3479
0.8083
0.6949
0.0719
0.6867
3.6282
0.8000
0.2373
0.1090
0.1120
0.8096* str
0.1904* str
0.1904
0.2210
-0.8682
-0.8682
-0.8682
0.6119
0.4665
0.3943
0.2200
0.4268
0.7401
0.0719
0.7894
3.7680
0.8000
-0.0381
0.0337
0.5302* str
0.4698* str
0.1904
0.0560
-0.8035
-0.8035
-0.8035
0.6254
0.6254
0.3718
0.3718
1
0.7978
0.0719
0.7839
3.9211
0.8000
0.0086
0.0542
0.9173* str
0.0827* str
0.0827
0.0902
77
 Three results should be underlined.
(i) First, when we compare this reformed economy (in column 4) to the status quo economy
(in column 1), there are welfare gains for both types of agents.
(ii) Second, 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.
(iii) Third, per capita output ( y ) increases relatively to columns 1 (status quo) and 2
(private providers without redistribution) thanks to stronger demand effects.
78
Endogenizing the allocation of employees and a new DCE
 We now allow the fraction of households employed in the production of public goods to be
endogenous in the reformed economy.
 In particular, instead of assuming that this fraction, v b , is set by the government, we
b
endogenize v so as both agents are equally off in the long run of the reformed economy.
This is as if households “vote with the feet” choosing their profession so as to be indifferent
between being p or b type.
 Acemoglu and Verdier (2000) also allow the population fraction to adjust so as to ensure that
agents are willing to become public employees rather than entrepreneurs.
79
 On the other hand, although this is a popular way of endogenizing the allocation of workers,
or firms, across sectors, we realize it is a simplistic one. For instance, one could introduce
labour flows between the two sectors using a search model (see e.g. Quadrini and Trigari,
2008, and Brückner and Pappa, 2010).
 Algebraically, the DCE consists of equations (17a-k) plus a new equation that equates longrun utility across the two agents, u b  u p , where the associated new endogenous variable is
the long-run population fraction, v b . Computationally, we simply search for a value of v b so
as w g  w p which in turn implies u b  u p .
 The long-run solution of this economy is reported in column 5 in Table 3.
80
Table 3
Long-run solution when the labor tax rate is the residual policy instrument
Variable
up
ub
u
cp
cb
ep
eb
wg / w p
y
yg
c/ y
k/y
b/ y
l
sw
sg
s t, p
s t ,b
vb
total cost of
public good (GDP share)
1
Status quo
economy
2
Cost-minimizing
private providers
3
Cost-minimizing
public providers
4
Cost-minimizing private providers
plus endogenous
redistributive transfers
5
Cost-minimizing private
providers plus endogenous
allocation of employees
-1.0280
-1.1632
-1.0537
0.4918
0.4158
0.3648
0.3480
0.8085
0.6949
0.0719
0.6868
3.6282
0.8000
0.2371
0.1090
0.1119
0.8096* str
0.1904* str
0.1904
0.2209
-0.7900
-1.6245
-0.9489
0.6453
0.2289
0.3811
0.2690
0.2554
0.7153
0.0719
0.7913
3.7162
0.8000
-0.0576
0.0255
0.8096* str
0.1904* str
0.1904
0.0424
-1.0282
-1.1635
-1.0540
0.4917
0.4156
0.3648
0.3479
0.8083
0.6949
0.0719
0.6867
3.6282
0.8000
0.2373
0.1090
0.1120
0.8096* str
0.1904* str
0.1904
0.2210
-0.8682
-0.8682
-0.8682
0.6119
0.4665
0.3943
0.2200
0.4268
0.7401
0.0719
0.7894
3.7680
0.8000
-0.0381
0.0337
0.5302* str
0.4698* str
0.1904
0.0560
-0.8035
-0.8035
-0.8035
0.6254
0.6254
0.3718
0.3718
1
0.7978
0.0719
0.7839
3.9211
0.8000
0.0086
0.0542
0.9173* str
0.0827* str
0.0827
0.0902
81
 Qualitatively, we get the same results as in the previous subsection where the endogenous
variable was government transfers. Nevertheless, three extra results should be underlined:
(i) First, wages are equalized via labor mobility.
(ii) Second, there is a fall in the fraction of households used in the production of the public
good: v b falls from 0.1904 in the data (see columns 1-4) to 0.0827 where agents vote with
the feet (see column 5). This reallocation of workers is voluntary since the fall in wages -
suffered once public employees become employees in private providers of public goods (see
the fall in w g / w p in column 2) - makes employment in the sector producing the private
good a more attractive choice.
(iii) Third, the increase in per capita output ( y ) is more pronounced in column 5 than in
column 4. This is because the reallocation of workers has, not only demand effects as in
the previous subsection, but also supply-side effects.
82
Summary
 Since a switch to private providers, in combination with a cut in income (labor) taxes,
increases per capita output and welfare but at the cost of making those that used to be public
employees worse off, we need to search for a Pareto-improving mix of reforms.
 In this section, we showed that redistributive government transfers that compensate those
that used to be public employees, and/or a voluntary reallocation of employees from the
production of the public good to the production of the private good, can complement the
switch to private providers and the cut in income (labor) taxes and hence provide a mix of
reforms that is Pareto improving. A reallocation of employees to the production of the
private good is particularly productive.
83
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.
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. This is under
all types of reform and all methods of public financing studied.
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 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.
 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 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.
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 This percentage is denoted as z in the Tables, 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).
Results for lifetime utility
 Results are reported in Table 4. 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, it is the case, in which a reform increases the aggregate pie (per capita output)
relative to the status quo economy in the long run.
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Table 4
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 the status
quo economy to
cost-minimizing
private providers
3
From the status quo
economy to costminimizing public
providers
4
From the status quo economy
to cost-minimizing private
providers plus endogenous
redistributive transfers
5
From the status quo economy to
cost-minimizing private providers
plus endogenous allocation of
employees
Up
-102.8009
-80.2693
-102.8184
-88.3975
-81.9682
z
-
0.2564
-0.0002
0.1572
0.2350
Ub
-116.3176
-160.1591
-116.3541
-85.9722
-80.9733
z
-
-0.3611
-0.0004
0.3606
0.4313
U
-105.3745
-95.4803
-105.3956
-87.9357
-81.8859
z
-
0.1388
-0.0002
0.1959
0.2512
87
 In Table 4, U p and U b denote respectively the discounted lifetime utility of the p household
and the b household, while U is the weighted 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), to cost-minimizing private providers in combination with
transfers that compensate those suffered from the reform (column 4) and to cost-minimizing
private providers in combination with a voluntary reallocation of employees between the two
sectors (column 5).
 In each case of regime switch (columns 2-5), we also report the associated value of the
welfare measure, z , as defined above.
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 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 the government also adjusts transfers to compensate the losers or
when there is a reallocation of employees, both groups of agents get better off relative to the
status quo (see columns 4 and 5).
 Finally, again as in the long run, public providers cannot beat private providers even when
both are assumed to minimize their costs.
89
Summary
 When the criterion is lifetime utility, the results are qualitatively the same as those derived
by the study of the long run.
 Namely, there are Pareto benefits from a mix of reforms that combines:
(i) a transition to cost-minimizing private providers which allows the government to make
efficiency savings
(ii) a reduction in income (labor) taxes made affordable by efficiency savings
(iii) a mechanism to reduce the rise in inequality caused by (i), like redistributive
government transfers and, in particular, a reallocation of employees to the production of the
private good.
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Robustness and some extensions
 We finally check the sensitivity of our results to a number of changes.
 We first check robustness to changes in the parameter values used and, in particular, the
value of the relatively unknown parameter, 1   , measuring the productivity of public
employees in the public sector production function (see (10) above).
 Second, we generalize the public sector production function, (10).
 Third, we study what happens when in the status quo, from which we depart to study various
reforms, agents are indifferent between being private or public employees.
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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.
 The main results remain unchanged. When 1   falls from 0.493 to 0.3, the only difference is
that, in the latter case, public employees become worse off as we move from the status quo
to cost-minimizing public providers, while recall that their utility did not change much in
Tables 2-3 above. This is intuitive: when their productivity is low, public employees suffer
under cost-minimizing public providers as they do under cost-minimizing private providers.
This is also the case when we compute lifetime utility.
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 When 1   rises from 0.493 to 0.7, the opposite happens: the public wage bill, stw , and hence
the welfare of public employees, rise as we move from the status quo to cost-minimizing
public providers.
 This might look paradoxical but it happens simply because of the optimality condition (20):
since 1    0.7 is relatively high, the cost-minimizing public enterprise finds it optimal to
choose a relatively high stw (or a relatively low stg ) which in turn makes public employees
better off in column 3. This is also the case when we compute lifetime utility.
 Nevertheless, the key result of the paper holds even in this case: the mix of reforms that can
make both groups better off relative to status quo is the one that includes cost-minimizing
private providers with a reduction in income labor taxes.
 We finally report that our results are robust to changes in other parameter values.
93
Generalizing the production function of the public sector
 In equation (10), the public enterprise used goods purchased from the private sector, Gtg , and
public employment, Ltg , to produce the public good.
 While, as said, this is as in most of the related literature, we now generalize this production
function. Our aim is to work with an encompassing production structure that generates more
special cases (including the one where the production functions of private and public firms
are identical) and also helps us to understand the working of the model.
We assume that equation (10) changes to:
Y  A(G ) ( K ) ( L )
g
t
g
1
g
t
t
where 0   ,  1 are technology parameters.
1
2
94
2
g
t
11  2
(21)
 Thus, we now assume that the public enterprise purchases goods from the private sector, G ,
g
t
and also participates in the factor markets hiring capital and labour inputs, K and L , like
g
t
g
t
private firms do. Notice that:
(i) when   0 , we go back to equation (10) and section 2
2
(ii) when   0 , the production function is the same as that of the private sector.
1
Also notice that, in order to make the production function comparable to that in the private
sector, we assume that K is borrowed directly from the capital market. As said above, the
g
t
qualitative role of public investment, when public investment goods are purchased from the
private sector, is similar to that of G .
g
t
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 In this new model specification, the solution with cost-minimizing private providers remains
as in section 3 above, while the solutions for the status quo economy and the economy with
cost-minimizing public providers change a bit.
 However, the main results do not change.
 What changes is that in the special case in which   0 , cost-minimizing public providers
1
become equivalent to cost-minimizing private providers. Thus, they become equally
superior to the status quo. This is not surprising.
 When there is no need to use (intermediate) goods purchased from the private sector, and
they both act optimally resorting directly to competitive factor markets, public and private
providers do exactly the same job.
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 But, in the “real world”, where governments do purchase (intermediate) goods produced by
the private sector in order to use them as inputs in the production of public goods and
services, i.e. 1  0 , cost-minimizing public providers do worse than cost-minimizing private
providers.
 Therefore, what is crucial is the assumption that the government purchases some goods
from the private sector and uses them as inputs in the production of public goods. Recall
that in most of the business cycle literature 1  1 (and  2  1  1   2  0 ).
97
Allowing agents be indifferent between being private or public employees
 In the analysis above, we started with a long-run status quo where agents differed. In
particular, our solutions implied that public employees were worse off than private
employees in the long run of the status quo economy. As said, one might like to look at
policy reforms when we depart from an equilibrium in which agents are indifferent if they
work in the private or public sector. We have therefore redone all the above experiments
when, in the long run status quo, agents are equally off and this is achieved via endogenous
government redistributive transfers or via an endogenous reallocation of households between
the two sectors. We report that all qualitative results remain unchanged.
Summary
 Our main results are robust to changes in parameter values as well as to a number of changes
in the model.
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Conclusions
 Here we 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 then showed that one can design redistributive schemes, and/or
allow for a reallocation of workers to the private sector, that allow everybody, including ex
public employees, to benefit from such a switch.
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 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 with transfers that
compensate those previously employed by the state and/or a reallocation of workers, can
benefit everybody.
 Our work can be enriched in several ways. For instance, we could allow for imperfect
competition on the part of private providers of public goods, which is a usual social fear. We
could allow for unemployment. We could also allow for substitutability between public
employment and goods purchased from the private sector in the production of public goods.
Finally, we could introduce various politico-economy issues, like extra benefits on the part
of public employees coming from rent seeking. We leave these extensions for future work.
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Thank you for your attention!
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