Public, or private, providers of public goods?

A dynamic general equilibrium study

November 19, 2013

George Economides*, Apostolis Philippopoulos* and Vanghelis Vassilatos#

Abstract: We study the difference between public production and public finance of public goods in a dynamic general equilibrium framework. Under public production, public goods are produced by the government itself using public employees and goods purchased from the private sector. Under public finance, the same amount of public goods is produced by cost-minimizing private providers with the government financing their costs. When the model is solved numerically using fiscal data from the UK economy, we find that a switch from public production to public finance can have substantial aggregate and distributional implications. Also, public providers cannot beat private providers in terms of efficiency. The main policy message is that the following mix of reforms can be Pareto improving: (i) a transition to cost-minimizing private providers that allows the government to achieve efficiency savings (ii) a reduction in distorting income taxes made affordable by these efficiency savings (iii) a mechanism to compensate the ex public employees. Keywords: Public goods, growth, welfare. JEL classification: H4, D9, D6. Acknowledgements: We thank Konstantinos Angelopoulos, Fabrice Collard, Saqib Jafarey, Jim Malley, Dimitris Papageorgiou, Hyun Park and Heraklis Polemarchakis for discussions and comments. We also thank seminar participants at Bilgi University and the Athens University of Economics and Business. Any errors are ours. *Athens University of Economics and Business, and CESifo # Athens University of Economics and Business Corresponding author: Apostolis Philippopoulos, Department of Economics, Athens University of Economics and Business, 76 Patission street, Athens 10434, Greece. Tel: +30-210-8203357. Email: aphil@aueb.gr

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1. Introduction

Concerning the provision of public goods, 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 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 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 (see e.g. Atkinson and Stiglitz, 1980, and Hillman, 2009).1 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, heated debate in the UK.2

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 policy 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 the difference between public production and

public finance in a dynamic general equilibrium model. We fill this gap by 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.

1 As Atkinson and Stiglitz (1980, p. 482) emphasize, “the two are often confused, though both logically and in practice they are distinct”. Note that 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, and Hillman, 2009). Here the focus is on the way of provision or production. 2 Opening up new areas for private providers of public goods is a key part of the policy of the current UK government. In particular, 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). At the same time, the British Deputy PM, Mr 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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We first model the case of public production by following the related literature.3

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. Solving the model numerically when the values of fiscal policy instruments are in

line with the UK averages over 1990-2008, we 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 was 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 (see e.g. 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 (this 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 our 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,

and the specification of the production function used by the public sector for the production

of public goods.

We study both the long run and the transition, where the latter means that we depart

from the status quo long run solution and travel to a new reformed long run as defined

above. There are five 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

3 See e.g. Finn (1998), Cavallo (2005), Ardagna (2007), Pappa (2009), Linnemann (2009), Forni et al. (2009),

Fernández-de-Córdoba et al. (2010) and Economides et al. (2012). See below for further details.

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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 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.

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 (of switching to private providers and cutting income

taxes) 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 Pareto-improving 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.

Fifth, it should be noted that what is crucial to the key result (namely, that private

producers do better, at least in terms of aggregate efficiency, than public producers, and this

happens even if they both act as cost minimizers) is the assumption that public sector

production implies an extra step of intermediation relative to the case in which it is private

producers that produce the public good. The fact that the presence, or absence, of

intermediate goods plays an important role should not come as a surprise. Jones (2011) has

shown that incorporating intermediate goods can have a first-order impact (either positive or

negative) on factor productivity and multipliers. Here, a similar mechanism drives our result

with our model implying that public production is less efficient. Note that this result can be

reversed if one is willing to assume that TFP in the public sector is sufficiently higher than in

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the private sector. Also note that all this happens irrespectively of political economy stories

which are expected to strengthen the argument for private providers.

Although we are aware that one should treat quantitative results with caution, and

that efficiency savings are not the only driver for switching to private providers (see the

discussion in the closing section below), our normative 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 the efficiency savings or fiscal space, created by the

use of private providers, to cut distorting income taxes (iii) adds a mechanism (like

redistributive government transfers and, in particular, a reallocation of employees) to reduce

the rise in inequality caused by switching to private providers.

Before we start, there are three features of our model that should be noted. First, here

we focus on polar cases. 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, here 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

private providers with the government just financing their costs. Third, irrespectively of who

produces the public good, we assume that this good is provided freely without user charges.

Thus, in all cases studied, the cost is covered by the government or the tax-payer.4

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

4 Ellingsen and Paltseva (2012) study access pricing and the efficient provision of excludable public goods. Economides and Philippopoulos (2012) study the implications of introducing user prices of excludable public goods into a dynamic general equilibrium model. For a review, see Hillman (2009, chapter 3).

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8 presents robustness checks. Section 9 studies what drives our main result. Section 10

closes the paper.

2. An economy with public production of public goods (status quo)

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 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. To finance total public spending, including

the cost of the public good, the government levies distorting taxes and issues bonds. 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, Fernández-de-Córdoba et al., 2010, and Economides et

al. 2012).

2.1 Population composition and agents’ economic roles

The population size at time t , tN , is exogenous. Among tN , there are 1,2,..., ptp N

identical households that work in the private sector and 1,2,..., btb N identical households

that work in the public sector, where p bt t tN N N . There are also 1,2,..., f

tf N identical

private firms. Each household employed in the private sector owns one private firm,

p ft tN N . This composition is not important to our results and allows us to avoid scale

effects in equilibrium. The fraction of public employees in population, b

b tt

t

N

N , is set by the

government (see below for endogenous determination of bt ).

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

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save in capital and bonds subject to transaction costs.5 Thus, the key difference between

public and private employees is that they earn different wages.6

2.2 Households working in the private sector

The lifetime utility of each household working in the private sector, 1, 2,..., ptp N , is:

0

( , , )t p p gt t t

t

u c e Y

(1)

where ptc and p

te are p ’s consumption and work hours respectively; gtY is per capita public

goods and services;7 and 0 1 is a time preference parameter.

The period utility function is (see also e.g. Christiano and Eichenbaum, 1992):

1( )( , , ) log( )

1

pp p g p g t

t t t t t

eu c u Y c Y

(2)

where , , 0 are preference parameters. Thus, p gt tc Y 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, ptk and p

tb , whose gross returns are tr and t respectively. The

within-period budget constraint of each p is:

,(1 ) (1 )( ) (1 )c p p p k p p l p p p tr pt t t t t t t t t t t t t tc i d r k w e b G (3a)

5 By allowing both groups of households to participate in asset markets, we enrich the baseline model 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). 6 Public and private employees can differ in many other dimensions, like job security and non-monetary privileges. Here we focus on differences in wages only. Adding these extra dimensions is expected to strengthen our main results.

7 g

g tt

t

YY

N , where g

tY is total public goods and services.

7

where pti is savings in the form of physical capital; p

td is savings in the form of government

bonds; pt is dividends received from private firms;8 p

tw is the wage rate in the private

sector; ,tr ptG is government transfers to each p ; and 0 k

t , lt , 1c

t 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:

2,

1 (1 )2

pp kp p p t

t t tt

kk k i

Y

(3b)

2,

1 2

pp bp p p t

t t tt

bb b d

Y

(3c)

where 0 1 is the capital depreciation rate; , ,, 0p k p b capture the transaction costs

paid by each p associated with participation in the capital and bond market respectively;

and tY denotes per capita output.9 Regarding the transaction costs, , ,, 0p k p b , similar

quadratic cost functions have been used by e.g. Persson and Tabellini (1992), Benigno

(2009) and Angelopoulos et al. (2011). Technically, these transaction costs allow us to avoid

unit root problems 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 main results depend on the

presence of transaction costs.

Each p chooses 1 1 0{ , , , }p p p pt t t t tc k b e

taking factor prices, economy-wide quantities

and policy variables as given. The first-order conditions are presented 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, 1, 2,..., btb N , is:

8 We assume that only private employees own the private firms and receive dividends from them (see (3a) and (6a) below). This is unimportant because private firms producing the private good make zero profits in equilibrium. Our main results do not depend on this assumption.

9 tt

t

YY

N , where tY is total output in the economy.

8

0

( , , )t b b gt t t

t

u c e Y

(4)

where

1( )( , , ) log( )

1

bb b g b g tt t t t t

eu c u Y c Y

(5)

The within-period budget constraint of each b is:

,(1 ) (1 ) (1 )c b b b k b l g b b tr bt t t t t t t t t t t t tc i d r k w e b G (6a)

where gtw is the wage rate in the public sector; and ,tr b

tG is government transfers to each b .

The laws of motion of physical capital and government bonds for each b are:

2,

1 (1 )2

bb kb b b tt t t

t

kk k i

Y

(6b)

2,

1 2

bb bb b b tt t t

t

bb b d

Y

(6c)

where , ,, 0b k b b capture the transaction costs paid by each b associated with

participation in the capital and bond market respectively.

Each b chooses 1 1 0{ , , , }b b b bt t t t tc k b e

taking factor prices, economy-wide quantities and

policy variables as given. The first-order conditions are presented in Appendix A.

2.4 Private firms producing the private good

In each period, each private firm 1,2,..., ftf N chooses capital and labor inputs, f

tk and

fte , to maximize profits:

f f f p ft t t t t ty r k w e (7)

9

where output is produced by a CRS Cobb-Douglas function:

1( ) ( )f f ft t ty A k e (8)

where 0A and 0 1 are technology parameters. Note that we could assume that

public goods also provide productivity-enhancing services, in addition to utility-enhancing

ones; we report that our main results do not change.

Each f chooses ftk and f

te taking factor prices as given. The simple first-order

conditions of this static problem are in Appendix B.

2.5 Government budget constraint

The within-period budget constraint of the government is (quantities are in aggregate terms):

, ,1(1 )g w tr p tr b

t t t t t t t tG G G G B B T (9a)

where gtG is total public spending on goods and services purchased from the private sector;

wtG is the total public wage bill; ,tr p

tG and ,tr btG are respectively transfers to private and

public employees;10 tB is the beginning-of-period total stock of one-period maturity

government bonds; and tT is total tax revenues defined as:

( )c p p b bt t t t t tT N c N c [ ( ) ]k p p p b b

t t t t t t t tN r k N r k ( )l p p p b g bt t t t t t tN w e N w e (9b)

That is, we include the three main types of government spending (purchases of goods and

services from the private sector, public wages, and transfers to individuals) typically

included in related studies, as well as the three main types of taxes (on consumption, capital

income and labor income) in the data.

Inspection of (9a-b) implies that, in each period, there are nine policy instruments

( , ,, , , ,g w tr p tr bt t t tG G G G c

t , kt , ,l

t 1,b

t tB N ) out of which one follows residually to satisfy the

government budget constraint. As in most of the related literature, we assume that, along the

10 ,

,tr p

tr p tt p

t

GG

N and

,,

tr btr b tt b

t

GG

N .

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transition path, the adjusting policy instrument is the end-of-period public debt, 1tB , so that

the other eight policy instruments can be set exogenously. For convenience, concerning

spending policy instruments, we will work in terms of their GDP shares, g

g tt

t

Gs

Y ,

ww tt

t

Gs

Y ,

,,

tr ptr p tt

t

Gs

Y ,

,,

tr btr b tt

t

Gs

Y , where tY denotes total output (defined below).

Similarly, concerning the number of public employees, we will work in terms of their

population share, b

b tt

t

N

N . The processes of exogenous variables are defined below.

2.6 Public sector production function

Following most of the related literature, we start by assuming that total public goods and

services, gtY , are produced using goods purchased from the private sector, g

tG , and public

employment, gtL (where, in equilibrium, g b b

t t tL N e ). In particular, following Linnemann

(2009), we start by using a CRS Cobb-Douglas production function of the form:

1( ) ( )g g gt t tY A G L (10)

where 0 1 is a technology parameter.

It is important to emphasize four things in (10). First, in section 9 below, we

generalize the public production function (10) and report what happens when this function is

the same as that of private firms’ in (8). Second, our modeling in (10) can nest most

specifications used so far.11 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 (see section 8).

11 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.

11

2.7 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), and

(iv) all constraints are satisfied. The DCE is summarized by the following eleven equilibrium

conditions:12

( ) (1 )( ) (1 )p c p g l pt t t t t te c Y w (11a)

, 11 1 2

1 1

1 1 1

1 (1 )( )1

(1 )( ) (1 )( )

pk p k tt t p f

t tc p g c p gt t t t t t

kr

y

c Y c Y

(11b)

, 11 2

1 1

1 1 1

1( )1

(1 )( ) (1 )( )

pp b t

t p ft t

c p g c p gt t t t t t

b

y

c Y c Y

(11c)

2 2, ,

1 1(1 ) (1 )2 2

p pp k p bc p p p p pt tt t t t t tp f p f

t t t t

k bc k k b b

y y

,(1 ) (1 )k p l p p p tr p ft t t t t t t t t tr k w e b s y (11d)

( ) (1 )( ) (1 )b c b g l gt t t t t te c Y w (11e)

, 11 1 2

1 1

1 1 1

1 (1 )( )1

(1 )( ) (1 )( )

bk b k tt t p f

t tc b g c b gt t t t t t

kr

y

c Y c Y

(11f)

, 11 2

1 1

1 1 1

1( )1

(1 )( ) (1 )( )

bb b t

t p ft t

c b g c b gt t t t t t

b

y

c Y c Y

(11g)

1( ) ( )b

f p b ptt t t tp

t

y A k k e

(11h)

1( ) ( )g g p f b bt t t t t tY A s y e (11i)

12 p ft t tY N y ,

gg t

tt

YY

N ,

gg g p ftt t t t

t

GG s y

N ,

,,

tr ptr p tt p

t

GG

N ,

,,

tr btr b tt b

t

GG

N ,

w g g g b bw t t t t t tt p f

t t t t

G w L w es

Y Y y

.

12

, ,1 1 1 1( ) (1 )( )w g tr p tr b p f p p b b p p b b

t t t t t t t t t t t t t t ts s s s y b b b b

+ ( )c p p b bt t t t tc c ( )k p p b b

t t t t t tr k k ( )l p p p g b bt t t t t t tw e w e (11j)

2 2, ,

1 (1 )2 2

p pp k p bp p p p t t

t t t t p f p ft t t t

k bc k k

y y

2 2, ,

1 (1 )2 2

b bb k b bb b b b g p f p ft tt t t t t t t t tp f p f

t t t t

k bc k k s y y

y y

(11k)

where, in the above equations, we use the factor returns:13

f pt t

t p p b bt t t t

yr

k k

(12a)

(1 ) fp t

t pt

yw

e

(12b)

w p fg t t tt b b

t t

s yw

e

(12c)

We therefore have eleven equations, (11a-k), in eleven endogenous variables,

1 1 1 1 0{ , , , , , , , , , , }p b p b p b p b f gt t t t t t t t t t t tc c k k b b e e y Y

. This is for any feasible policy, where the latter is

summarized by the paths of the exogenous policy instruments, , ,0{ , , , , , , , }g w tr p tr b c k l b

t t t t t t t t ts s s s .

For simplicity, we will assume that all exogenous policy instruments are constant and set at

their data average values (see below).

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, (11a-k), are log-linearized around their long-run

solution. The model is solved numerically in section 4.14 This is our “status quo” model.

13 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 w g g g b b g b b

w t t t t t t t t tt p f p f

t t t t t t

G w L w N e w es

Y Y N y y

.

13

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 (see also Turnovsky and Pintea, 2006). Thus,

now the government is not involved in any production itself.

3.1 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 in section 2. We

report that our qualitative results do not depend on these scaling assumptions, while some

generalizations are reported in section 8 below.

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 1,2,..., gtg N , and

the new role of the government. Regarding private providers, each g produces a given

amount of the public good ordered by the government, /g bt tY N , by choosing capital and

labor inputs in a cost-minimizing way, where the path 0{ }gt tY

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, 0{ }gt tY

, or equivalently the per capita amount of public goods, 0{ }gt tY

, 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,

[ ]g g g g gt t t t tN r k w e , where g

tr and gtw are the rental costs of capital and labor paid by private

14 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).

14

providers, and gtk and g

te are the capital and labor inputs used by each private provider.15

Notice that the total cost, [ ]g g g g gt t t t tN r k w e , replaces spending on public wages, w

tG , 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 (details on the

problems of the two types of households and those firms producing the private good are in

Appendix D; except from notational differences, these problems are as in section 2).

3.2 Private firms producing a given amount of the public good (private providers)

In each period, each private provider of public goods, 1, 2,..., gtg N , chooses g

tk and gte 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):

gg g g g gt

t t t t t tbt

Yr k w e y

N

(13)

where gtr , g

tw and gtY have been defined above; t is a multiplier measuring the marginal

cost of production; and gty is each private provider’s output which is produced by using the

same production function as in (8), namely:

1( ) ( )g g gt t ty A k e (14)

Each g chooses gtk and g

te taking factor prices and economy-wide quantities as

given. The first-order conditions are:

gg t

t t gt

yr

k

(15a)

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.

15

(1 ) gg tt t g

t

yw

e

(15b)

1( ) ( ) 0g

g gtt tb

t

YA k e

N (15c)

where Appendix E provides details working as in Mas-Colell et al. (1995, pp. 139-143).

It is useful to point out three things. First, the determination of gtw 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, the solution to the cost-minimization problem above implies that the

profits of private providers, gt , 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).

3.3 Government budget constraint

The budget constraint of the government changes from (9a) to:

, ,1[ ] (1 )b g g g g tr p tr b

t t t t t t t t t t tN r k w e G G B B T (16a)

where tax revenues change from (9b) to:

( )c p p b bt t t t t tT N c N c [ ( ) ( )]k p p p p b g b b

t t t t t t t t tN r k N r k ( )l p p p b g bt t t t t t tN w e N w e (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 ( , ,, ,tr p tr bt tG G c

t , , ,k lt t 1,

bt tB N ) or

equivalently in ratios ( , ,, ,tr p tr bt ts s c

tkt 1, ,l b

t t tB ). As in section 2, we will start by assuming

that the residually determined policy instrument is the end-of-period public debt, 1tB .

16

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

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) and (iv) all

constraints are satisfied. The new DCE is summarized by the following eleven equilibrium

conditions:

( ) (1 )( ) (1 )p c p g l pt t t t t te c Y w (17a)

, 11 1 2

1 1

1 1 1

1 (1 )( )1

(1 )( ) (1 )( )

pk p p k tt t p f

t tc p g c p gt t t t t t

kr

y

c Y c Y

(17b)

, 11 2

1 1

1 1 1

1( )1

(1 )( ) (1 )( )

pp b t

t p ft t

c p g c p gt t t t t t

b

y

c Y c Y

(17c)

2 2, ,

1 1(1 ) (1 )2 2

p pp k p bc p p p p pt tt t t t t tp f p f

t t t t

k bc k k b b

y y

,(1 ) (1 )k p p l p p p tr p ft t t t t t t t t tr k w e b s y (17d)

( ) (1 )( ) (1 )b c b g l gt t t t t te c Y w (17e)

, 11 1 2

1 1

1 1 1

1 (1 )( )1

(1 )( ) (1 )( )

bk g b k tt t p f

t tc b g c b gt t t t t t

kr

y

c Y c Y

(17f)

, 11 2

1 1

1 1 1

1( )1

(1 )( ) (1 )( )

bb b t

t p ft t

c b g c b gt t t t t t

b

y

c Y c Y

(17g)

1( ) ( )f p pt t ty A k e (17h)

1( ) ( )g b b bt t t tY A k e (17i)

, ,1 1 1 1( ) ( ) (1 )( )b g b g b tr p tr b p f p p b b p p b b

t t t t t t t t t t t t t t t t t tr k w e s s y b b b b

17

+ ( )c p p b bt t t t tc c ( )k p p p g b b

t t t t t t tr k r k ( )l p p p g b bt t t t t t tw e w e (17j)

2 2, ,

1 (1 )2 2

p pp k p bp p p p t t

t t t t p f p ft t t t

k bc k k

y y

2 2, ,

1 (1 )2 2

b bb k b bb b b b p ft tt t t t t tp f p f

t t t t

k bc k k y

y y

(17k)

where, in the above equations, we use the factor returns (see Appendix G for details):

fp t

t pt

yr

k

(18a)

gg t

t t b bt t

Yr

k

(18b)

(1 ) fp t

t pt

yw

e

(18c)

(1 ) gg tt t b b

t t

Yw

e

(18d)

Therefore, in this new system, we have eleven equations, (17a-k), in eleven

endogenous variables, 1 1 1 1 0{ , , , , , , , , , , }p b p b p b p b ft t t t t t t t t t t tc c k k b b e e y

. This is for any feasible

policy, where the latter is summarized by the paths of the exogenous policy instruments,

, ,{ , ,tr p tr bt ts s c

tkt 0, }l b

t t t , and the path of the per capita amount of public goods, 0{ }g

t tY , 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 new equilibrium conditions (17a-k) are similar to those in (11a-k) except that

now: (i) public wages, gtw , 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 wts and g

ts in section 2 (iv) we allow for separate returns to capital in the

two sectors (v) the market-clearing conditions differ from section 2 (vi) we do not have

spending on private goods purchased by the government in the economy’s resource

constraint, as we had in section 2.

18

These 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 solve numerically the two model economies in sections 2 and 3 and then compare them.

4.1 How we work

We work in two steps. We first solve numerically the model in section 2 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,

0{ }gt tY

, 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, 0{ }gt tY

.

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 below for details). 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 traveling from one regime to another over time, are discussed

in section 7.

4.2 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 in

section 2. 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. Angelopoulos et al., 2012, for a DSGE model

with tax reforms calibrated 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,

19

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.

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.

(2012). 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 of work, and , are set at 5 and 1 respectively; these

parameter values imply hours of work within usual ranges. The capital depreciation rate, ,

is set at 0.05. The transaction cost parameter associated with participation in asset markets is

set at , , , , 0.002p k p b b k b b across both agents and both assets. Our results are

robust to changes in all these parameter values.

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 also experiment with other values of 1

(see section 8 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, wts and tr

ts , 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,

, (1 )tr p p tr b trt t t t ts s s and ,tr b b tr

t t ts s (see below for other cases considered). The output

share of public spending on goods and services purchased from the private sector, gts , 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, ct , k

t and lt , are respectively 0.1852, 0.3875 and 0.2685

over 1990-2008; the data are taken from Angelopoulos et al. (2012), who have followed the

methodology of Conesa et al. (2007) in constructing effective tax rates for the UK economy.

20

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 the consumption tax rate is the adjusting 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 solving the

systems (11a-k) and (17a-k) respectively when variables do not change.16 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 (see below for other public financing

cases).

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 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.

Table 2 around here

(Long-run solution when the consumption tax rate is the residual policy instrument)

4.3.1 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

16 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. 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 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 extraneous assumption is usual in the literature (see Mendoza and Tesar, 1998, in a two-country model).

21

private wage ratio, pg ww / , 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 g pw w , public

employees are worse off than private employees, b pu u (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.

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 b pu u in the long run of the status quo economy. None of our qualitative results is

affected by this (see section 8 below for details). 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).

4.3.2 Distributional implications of switching to private providers

In the long run, the ratio of public to private wages, /g pw w , 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, bc , and the willingness to work, be , of b households in column 2. Despite the

increase in leisure time, (1 )be , the fall in consumption, bc , leads to a fall in the long-run

utility of b households, bu , as we switch from the status quo to the reformed economy. By

contrast, the long-run utility of p households, pu , rises in column 2 . This is thanks to

higher consumption, pc , enjoyed by p households under private provision (see below for

details). Notice that, in this particular experiment, the beneficial welfare effects on p

22

households dominate the adverse effects on b households, so that per capita long-run utility,

denoted as u ,17 rises under private provision in column 2.

4.3.3 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 gtG 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 bc (caused by the fall in /g pw w )

allows an increase in pc , as discussed above.

The above are direct effects that work through resource reallocation. But there are

also indirect effects that work through public financing. The fall in gw under private

providers leads to a fall in the total cost of public good production as share of output, ws .

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.

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 ( y falls from 0.6879 in

column 1 to 0.6767 in column 2 of Table 2). 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

17 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 p p b bu v u v u .

23

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. To confirm all this, we next study a more interesting public financing case.

4.4 Long-run solution when the labor tax rate is the adjusting instrument

We now study a more interesting way of public financing. In Table 3, the residually

determined long-run policy instrument is the labor tax rate (always, the long-run public debt-

to-output ratio is set at 80%).

Table 3 around here (Long-run solution when the labor tax rate is the residual policy instrument)

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

Angelopoulos et al., 2012, for the UK), their reduction not only strongly stimulates pc , pu

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 (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 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

24

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.

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, 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. In

particular, as we did in section 3 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, 0{ }gt tY

, as offered by the status quo economy. Thus, as in section 3, the path

0{ }gt tY

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 enterprise

The economy is as in section 2 but now, in addition, in each period, the public enterprise

chooses its two inputs, gtG and g

tL , or equivalently their output shares, gts and w

ts , to

minimize its costs. The cost-minimization problem of the public enterprise is:

1[ ( ) ( ) ]g g g g g gt t t t t t tG w L Y A G L (19)

where gtw denotes the new wage rate received by public employees; t is a multiplier

measuring the marginal cost of producing the public good; and gtY is the total amount of

25

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:

1

gtwt

s

s

(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, namely, the eleven equations of the status quo

economy, (11a-k), plus equation (20), in twelve endogenous variables,

1 1 1 1 0{ , , , , , , , , , , , }p b p b p b p b f g wt t t t t t t t t t t t tc c k k b b e e y s s

. This is for any feasible policy, as

summarized by , ,{ , ,tr p tr bt ts s c

tkt 0, }l b

t t t , and the path of 0{ }g

t tY , 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 two 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. See section 8 below for richer numerical experiments.

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 cost-

minimizing private providers is superior to public production, even when public providers

act as cost-minimizers. 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. Section 9 below generalizes the public production function and explains

26

why cost-minimizing private providers do better then cost-minimizing public providers even

when they both choose inputs optimally.

6. 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. In this section, we study two such reforms. First, a government transfer

scheme that compensates the losers from a switch to private provision/public finance.

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 in section 4 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 (results for the other case, where the residual public finance

instrument is consumption taxes, are available upon request).

6.1 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

27

allocation of transfers that makes both agents equally off in the long run of the reformed

economy modeled in section 3.18

Algebraically, the DCE consists of equations (17a-k) plus a new equation that

equates long-run utility across the two agents, b pu u , where the associated new

endogenous variable is the share of government transfers between the two agents, bx , where

,tr b b trs x s , , (1 )tr p b trs x s and trs is the value of total transfers in the data as share of

output. Computationally, we simply search for a value of bx so as b pu u .

The long-run solution of this economy is reported in column 4 in Table 3. Three

results should be underlined. 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.

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. Third, per capita output ( y ) increases relatively

to columns 1 (status quo) and 2 (private providers without redistribution) thanks to stronger

demand effects.

6.2 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,

bt , is set by the government, we endogenize b

t so as both agents are equally off in the long

run of the reformed economy modeled in section 3. This is as if households “vote with the

feet” choosing their profession so as to be indifferent between being p or b type. 19

Algebraically, the DCE consists of equations (17a-k) plus a new equation that

equates long-run utility across the two agents, b pu u , where the associated new

endogenous variable is the long-run population fraction, b . Computationally, we simply

search for a value of b so as g pw w which in turn implies b pu u .

18 See e.g. Park and Philippopoulos (2003) for other redistributive transfer schemes. 19 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. 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). See also the last section.

28

The long-run solution of this economy is reported in column 5 in Table 3.

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:

First, wages are equalized via labor mobility. Second, there is a fall in the fraction of

households used in the production of the public good: 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 /g pw w in column 2) - makes

employment in the sector producing the private good a more attractive choice. 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.

6.3 Summary of this section

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.

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), Mendoza and Tesar (1998) and

Angelopoulos et al. (2012). We first check, using our baseline parameterization, that when

29

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 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 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).

7.2 Results for lifetime utility

Results are reported in Table 4. Again, to save on space, 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 4 around here (Lifetime utility under regime switches)

In Table 4, pU and bU 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.

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. Results are available upon request.

30

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.

7.3 Summary of this section

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.

8. Robustness checks

Our results are robust to a number of changes in parameter values and the model

specification. Here, to save on space, we present two robustness checks only. We first check

robustness to changes in 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 check what happens when in the status quo, from which we depart to study

various reforms, agents are indifferent between being private or public employees.

Numerical solutions and the associated new tables are available upon request.

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 1 0.493 in the baseline parameterization.

31

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.

When 1 rises from 0.493 to 0.7, the opposite happens: the public wage bill, wts ,

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 (or 0.3 is relatively

low), the cost-minimizing public enterprise finds it optimal to choose a relatively high wts (or

a relatively low gts ) 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.

8.2 Departing from a status quo where agents are indifferent if they are b or p type

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.

9. What drives the main result?

We now investigate what drives the main result of the paper, namely, that private producers

do better, at least in terms of aggregate efficiency, than public producers, and this happens

even if they both act as cost minimizers. Recall that here we have deliberately abstained

32

from any political-economy issues in the case of public production so that, any differences in

efficiency, are driven by technological reasons only.

Inspection of the two production functions, under public production in section 2 and

under private production in section 3, reveals that, other things equal, the two equations

differ because a part of private output is used as an intermediate good in the production of

the public good, when the latter is produced by the public enterprise in section 2. The fact

that it is the presence of the intermediate good that drives the main result can be confirmed

numerically in a number of ways. It is convenient to assume that, in the case of public

production, equation (10) generalizes to:

1 2 1 21( ) ( ) ( )g g g gt t t tY A G K L (21)

where 1 20 , 1 are technology parameters. Thus, we now assume that the public

enterprise purchases goods from the private sector, gtG , and also participates in the factor

markets directly by hiring capital and labour inputs, gtK and g

tL , as private firms do. Notice

that if 2 0 , we go back to equation (10) in section 2, while if 1 0 , the production

function becomes the same as that of the private provider in equation (14) in section 3. 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 are in Appendix H.21

We report that, to the extent that 10 1 , the main results do not change. Only if

1 0 , cost-minimizing public providers become equivalent to cost-minimizing private

providers (and both become equally superior to the status quo). Thus, only if there is no need

to use any intermediate goods purchased from the private sector, and they both act optimally,

public and private providers do exactly the same job. By contrast, when 1 0 , cost-

minimizing public providers do worse than cost-minimizing private providers. Therefore,

what is crucial to our numerical results is the assumption that the public producer purchases

some goods from the private sector and uses them as intermediate goods in the production of

21 Note that, in order to make the production function comparable to that in the private sector, we assume that

gtK is borrowed directly from the capital market (see Appendix H for details). As said above, the qualitative

role of public investment, when public investment goods are purchased from the private sector, is similar to that of g

tG . Recall also that, in most of the business cycle literature, 1 1 (and 2 1 21 0 ).

33

the public good or, strictly speaking, that, public sector production implies an extra step of

intermediation relative to the case in which it is private producers that produce the public

good.

The fact that the presence, or absence, of intermediate goods plays a crucial role in

aggregate efficiency should not come as a surprise. Jones (2011) has shown that

incorporating intermediate goods can have a large first-order impact on factor productivity,

multipliers and, ultimately, output determination. Here we get a similar story. Combining a

neoclassical model with a standard treatment of intermediate goods, used as an input by the

public enterprise, provides one way of explaining why private providers can do better, and

this is without resorting to political economy issues which are expected to make our result

even stronger.

To show the above formally, working as in Jones (2011), let us use a more abstract

model. Say that when a public good, gtY , is produced directly by private providers, we have:

(1 )( ) ( )gt t tY A K L (22)

where tK and tL denote the inputs used by the private producer.

On the other hand, when the same good is produced by the public enterprise, we

have:

1 1( ) [( ) ( ) ]g g g gt t t tY A G K L

where gtK and g

tL denote the inputs used by the public enterprise, the intermediate good is a

fraction of private output, g g pt t tG s Y , and private output is produced by

(1 )( ) ( )p p pt t tY A K L . Thus, in the case of public production, we have:

1 1 1 1( ) [( ) ( ) ] [( ) ( ) ]g g p g p gt t t t t tY A s K K L L (23)

Inspection of (22) and (23) reveals that they can differ in three ways when 0 . In

particular, they can differ because: (i) 1A A ; (ii) ( ) 1gts ; (iii) depending on how we

34

define the mean, tL can differ from 1( ) ( )p gt tL L and similarly for the capital input.22 Now,

what do these differences mean in terms of our model? Regarding (i), recall that here we

have simply set 1A , because we do not want our results to be driven by assumptions about

TFP parameter values. Regarding (ii), since 0 ( ) 1gts , log( ) 0g

ts , which makes public

production in (23) less efficient, other things equal. Regarding (iii), we have not made any

assumptions about the type of the mean (after all, this argument is used only to understand

how the model works, because, in our solutions, all these inputs are endogenously

determined). Nevertheless, since there is no heterogeneity in labor skills, we find it more

intuitive to think in terms of an arithmetic mean, which means a higher degree of

substitutability, or a lower degree of complementarity, than a geometric mean, which would

again make public production in (23) less efficient, other things equal. Combining (i)-(iii),

our solutions imply that output is smaller under public producers, except when 0 .

10. Conclusions

This paper has 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

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 can happen 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.

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.

22 Only if we use a geometric mean, namely, 1( ) ( ) ( )p g

t t tL L L , type (iii) difference disappears. If, on the

other hand, we use an arithmetic mean, namely, ( ) (1 )p gt t tL L L , concavity implies

log( ) log[ (1 ) ] log( ) (1 ) log( )p g p gt t t t tL L L L L , which favors private producers. Similar arguments

hold for the capital input. See Jones (2011) for the importance of the choice of the mean.

35

Our work can be extended in several ways. We can extend the model to address some

usually expressed social fears from a switch to private providers of public goods (e.g. an

increase in market power, a fall in the quality of goods provided and a rise in

unemployment). Finally, we could study richer production functions allowing, for instance,

for substitutability between public employment and goods purchased from the private sector

in the production of public goods. We leave these extensions for future work.

36

APPENDIX

Appendix A: First-order conditions of household ,h p b in section 2

The first-order conditions include the budget constraints and:

( ) (1 )( ) (1 )h c h g l pt t t t t te c Y w (A.1)

, 11 1 2

1

1 1 1

1 (1 )( )1

(1 )( ) (1 )( )

hk h k tt t

tc h g c h gt t t t t t

kr

Y

c Y c Y

(A.2)

, 11 2

1

1 1 1

1( )1

(1 )( ) (1 )( )

hh b t

tt

c h g c h gt t t t t t

b

Y

c Y c Y

(A.3)

Appendix B: First-order conditions of firm f in section 2

ft

t ft

yr

k

(B.1)

(1 ) fp t

t ft

yw

e

(B.2)

so that profits are zero.

Appendix C: Market-clearing conditions in section 2

In the labor market:

f f p pt t t tN e N e (C.1a)

g b bt t tL N e (C.1b)

In the capital market:

f f p p b bt t t t t tN k N k N k (C.2)

In the dividend market:

0f f p pt t t tN N (C.3)

In the bond market:

p p b bt t t t tB N b N b (C.4)

In the goods market (economy’s resource constraint):

37

p p b b p p b b g f ft t t t t t t t t t tN c N c N i N i G N y (C.5)

where we also set f pt tN N .

Appendix D: Households and firms in section 3

The problem of households 1,2,..., ptp N , who work at private firms producing the private

good, as well as the problem of households 1,2,..., btb N , who work at private firms

producing the public good ordered by the government, remain as in section 2 (see equations

(1)-(3) and (4)-(6) respectively). The only difference is that each p rents capital to private

firms producing the private good earning a return, ptr , and getting dividends, p

t , while each

b rents capital to private firms producing the public good earning a return, gtr , and getting

dividends, bt (see Appendix F below for details on the capital market). Thus, their budget

constraints are:

,(1 ) (1 )( ) (1 )c p p p k p p p l p p p tr pt t t t t t t t t t t t t tc i d r k w e b G (D.1)

,(1 ) (1 )( ) (1 )c b b b k g b b l g b b tr bt t t t t t t t t t t t t tc i d r k w e b G (D.2)

The problem of private firms producing the private good, 1, 2,..., ftf N , also remains as in

section 2 (see equations (7)-(8)). The only difference is that the rental cost of capital for

these firms is now denoted as ptr . The problem of private firms/providers producing the

public good, 1, 2,..., gtg N , is in the main text.

Appendix E: Cost minimization of private provider g in section 3

We follow Mas-Colell et al. (1995, pp. 139-143). The first-order conditions imply:

gg t

t t gt

yr

k

(E.1a)

(1 ) gg tt t g

t

yw

e

(E1b)

1

1

( ) ( ) ( ) ( )

(1 )

g g g g g g g g g g b g g g g bt t t t t t t t t t t t t t t t

t g g gt t t

r w r k w e r k w e N r k w e

A y Y Y

(E.1c)

where (E.1c) follows if we use (E.1a-b) to get expressions for gtk and g

te respectively, and

use them back in the production function, 1( ) ( )g g

g g g t tt t t b b

t t

Y YA k e y

N

.

38

In turn, we use (E.1c) to substitute out the multiplier, t , in (E.1a) and (E.1b):

1 1

1

g g gg t t tt

y r wk

A

(E.2a)

1

g g gg t t tt

Y r we

A

(E.2b)

so that the total cost of each firm can be written as:

11( ) ( ) 1

1

g g gg g g g t t t

t t t t

y r wr k w e

A

11( ) ( ) 1

1

g g gt t t

bt

Y r w

N A

(E.3)

Notice that profits are zero (thanks to CRS). To show this, consider profits:

11( ) ( ) 1

1

g g gg g g g g g t t tt t t t t t

y r wy r k w e y

A

(E.4)

so that (thanks to linearity) the first-order condition is:

11( ) ( ) 11

1

g gt tr w

A

(E.5)

but, if this condition holds, total profits are zero in each period.

Appendix F: Market-clearing conditions in section 3

In the labor market:

f f p pt t t tN e N e (F.1a)

g g b bt t t tN e N e (F.1b)

In the capital market:

0f f p pt t t tN k N k (F.2a)

0g g b bt t t tN k N k (F.2b)

In the dividend market:

f f p pt t t tN N (F.3a)

g g b bt t t tN N (F.3b)

In the bond market:

p p b bt t t t tB N b N b (F.4)

39

In the goods market (economy’s resource constraint):

p p b b p p b b f ft t t t t t t t t tN c N c N i N i N y (F.5)

where we also set f pt tN N and g b

t tN N . Also recall that the privately produced public

good is provided without charge as in section 2.

Appendix G: Factor returns in section 3

fp t

t pt

yr

k

(G.1)

g g g gg t t t t t

t t t t tg b b b b b bt t t t t t t t

y Y Y N Yr

k k N k N N k

(G.2)

(1 ) fp t

t pt

yw

e

(G.3)

(1 ) (1 ) (1 ) (1 )g g g gg t t t t tt t t t tb b b b b b b

t t t t t t t t

y Y Y N Yw

e e N e N N e

(G.4)

Appendix H: The model in subsection 8.2

We first solve for the status quo economy. The problems of private firms and households

working in private firms are as in section 2. What changes is the production function of the

public sector (see equation (21)), the budget constraint of households working in the public

sector, the government budget constraint and the market-clearing conditions.

In particular, the budget constraint of households working in the public sector becomes:

,(1 ) (1 )( ) (1 )c b b b k g b b l g b b tr bt t t t t t t t t t t t t tc i d r k w e b G (H.1)

The government budget constraint becomes:

, ,1(1 )g k w tr p tr b

t t t t t t t t tG G G G G B B T (H.2)

where k g g

kt t tt

t t

G r Ks

Y Y is government expenditure on capital renting expressed as share of

output. This spending share is another policy instrument. Recall from section 2, that the

other spending instruments are g

gtt

t

Gs

Y and

w g gwt t tt

t t

G w Ls

Y Y .

In the capital market, the market-clearing conditions are:

f f p pt t t tN k N k (H.3a)

40

g b bt t tK N k (H.3b)

Then, the DCE equations are as in (11a-k) and (12a-c) above, except that now:

(i) Equation (11h) becomes:

1( ) ( )f p pt t ty A k e (H.4a)

(ii) Equation (11i) becomes:

1 2 1 21( ) ( ) ( )g g p f b b b bt t t t t t t tY A s y k e (H.4b)

(iii) Equation (11j) becomes:

, ,1 1 1 1( ) (1 )( )w k g tr p tr b p f p p b b p p b b

t t t t t t t t t t t t t t t ts s s s s y b b b b

+ ( )c p p b bt t t t tc c ( )k p p b b

t t t t t tr k k ( )l p p p g b bt t t t t t tw e w e (H.4c)

(iv) In the factor returns, (12a-c), we add a fourth equation:

k p fg t t t

t b bt t

s yr

k

(H.4d)

This gives the DCE of the status quo economy given the exogenously set policy instruments,

which are now , ,0{ , , , , , , , , }g k w tr p tr b c k l b

t t t t t t t t t ts s s s s .

We now add a cost-minimizing public provider. The problem of the latter changes from (19)

to:

1 2 1 21[ ( ) ( ) ( ) ]g g g g g g g g gt t t t t t t t t tG w L r K Y A G K L (H.5)

where the first-order conditions for the three inputs give:

1g

k t tt p f

t t

Ys

y

(H.6a)

1 2(1 ) gw t tt p f

t t

Ys

y

(H.6b)

2g

k t tt p f

t t

Ys

y

(H.6c)

We thus have the eleven equations of the status quo plus the above three optimality

conditions. This means that we have fourteen equations in fourteen variables,

1 1 1 1 0{ , , , , , , , , , , , , , }p b p b p b p b f g w kt t t t t t t t t t t t t t tc c k k b b e e y s s s

. This is for any feasible policy, as

summarized by , ,{ , ,tr p tr bt ts s c

tkt 0, }l b

t t t , and the path of 0{ }g

t tY , which is exogenously set

as found in the status quo economy. Compare this system to that in subsection 5.2.

41

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

ws Public wage payments as share of GDP (data) 0.1090

gs Public purchases as share of GDP (data) 0.1119

trs Public transfers as share of GDP (data) 0.2199

c Tax rate on consumption (data) 0.1852

k Tax rate on capital income (data) 0.3875

l Tax rate on labor income (data) 0.2685

bv Public employees as share of population (data) 0.1904

A Long-run TFP 1

a Autoregressive parameter of TFP 0.9

a Standard deviation of TFP 0.01

kp, Transaction cost incurred by private agents in capital market 0.002

bp, Transaction cost incurred by private agents in bond market 0.002

kb, Transaction cost incurred by public employees in capital market 0.002

bb, Transaction cost incurred by public employees in bond market 0.002

42

Table 2 Long-run solution when the consumption tax rate is the residual policy instrument

Variable

1 Status quo economy

2 Cost-minimizing private providers

3 Cost-minimizing public providers

pu -1.0345 -0.8253 -1.0295

bu -1.1656 -1.4933 -1.1651

u -1.0595 -0.9525 -1.0553 pc 0.4853 0.5992 0.4889

bc 0.4118 0.2555 0.4127

pe 0.3611 0.3605 0.3624

be 0.3438 0.2499 0.3447

pg ww / 0.8100 0.3001 0.8052

y 0.6879 0.6767 0.6902 gy 0.0711 0.0711 0.0711

/c y 0.6851 0.7887 0.6872

/k y 3.6282 3.7411 3.6282

yb / 0.8000 0.8000 0.8000

c 0.1634 -0.0675 0.1511

ws 0.1090 0.0294 0.1083

gs 0.1119 - 0.1113

pts , 0.8096* trs 0.8096* trs 0.8096* trs bts , 0.1904* trs 0.1904* trs 0.1904* trs

total cost of public good (GDP share)

0.2209

0.0489

0.2196

Notes: (i) We use the baseline parameterization in Table 1. (ii) p p b bu v u v u (the same formula is used for all per capita quantities).

43

Table 3 Long-run solution when the labor tax rate is the residual policy instrument

Variable

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

pu -1.0280 -0.7900 -1.0282 -0.8682 -0.8035

bu -1.1632 -1.6245 -1.1635 -0.8682 -0.8035

u -1.0537 -0.9489 -1.0540 -0.8682 -0.8035 pc 0.4918 0.6453 0.4917 0.6119 0.6254

bc 0.4158 0.2289 0.4156 0.4665 0.6254

pe 0.3648 0.3811 0.3648 0.3943 0.3718

be 0.3480 0.2690 0.3479 0.2200 0.3718

pg ww / 0.8085 0.2554 0.8083 0.4268 1

y 0.6949 0.7153 0.6949 0.7401 0.7978 gy 0.0719 0.0719 0.0719 0.0719 0.0719

/c y 0.6868 0.7913 0.6867 0.7894 0.7839

/k y 3.6282 3.7162 3.6282 3.7680 3.9211

yb / 0.8000 0.8000 0.8000 0.8000 0.8000

l 0.2371 -0.0576 0.2373 -0.0381 0.0086

ws 0.1090 0.0255 0.1090 0.0337 0.0542

gs 0.1119 - 0.1120 - -

pts , 0.8096* trs 0.8096* trs 0.8096* trs 0.5302* trs 0.9173* trs bts , 0.1904* trs 0.1904* trs 0.1904* trs 0.4698* trs 0.0827* trs

bv 0.1904 0.1904 0.1904 0.1904 0.0827

total cost of public good (GDP share)

0.2209

0.0424

0.2210

0.0560

0.0902

Notes: See notes of Table 2.

44

Table 4 Lifetime discounted 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 cost-minimizing 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

pU

-102.8009

-80.2693

-102.8184

-88.3975

-81.9682

z

-

0.2564

-0.0002

0.1572

0.2350

bU

-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

Notes: (i) See notes of Table 2. (ii) For ,h p b , 0

( , , )h t h h gt t t

t

U u c e Y

. (iii) p p b bU v U v U . (iv) z is

the constant private consumption supplement which makes i jU U where i j denotes regimes 1-5.

45

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