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Are user fees bad for equity?
CESifo, Munich, September 27, 2012
(This lecture is based on a joint work with Apostolis Philippopoulos (AUEB and CESifo))
Introduction
 There are publicly provided goods and services where exclusion is not possible (e.g. national
security and defence) or non-desirable from a social point of view (e.g. services provided by
the police and the court system).
 But there are also publicly provided goods and services from which exclusion is possible and
hence user fees, or user prices, could be charged for their use. The most well-known
examples are education, health care, transportation systems, parks, transportation, policing
and protection of neighborhoods, subsidized state theaters etc (see e.g. Atkinson and Stiglitz,
1980, chapter 16).
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 Such goods may have some of the properties of public goods (because of externalities) but,
at the same time, they are also like private goods since the benefit received from them is
mainly personal.
 However, in most countries, user fees are often zero, or partial, with the government
subsidizing the public provision of such goods from other taxes, like income and
consumption taxes.
 For instance, in the euro area, as well as in EU-27, “Payments for non-market output as share
of GDP” are less than 1% as share of GDP. This item represents services provided by the
government “at prices economically non-significant” and includes fees paid by students for
public universities and colleges, tickets for museums, etc.
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 It is interesting to note, however, that in many countries the reliance on user fees has been
increasing over the years. The US is a good example (see Huber and Runkel, 2009, for
details). Also, in Germany, the above ratio is relatively high (2.16%). By contrast, it is less
than 0.5% in France and South European countries.
 The most commonly expressed objection to user fees in the relevant debate is the view that
they are unjust, in the sense that they tend to increase inequality through shifting the burden
of taxation from the rich onto the poor.
 But, is it so? Does really the introduction of user fees worsen inequality? Are there cases, in
which, the conventional belief is not correct? And if yes, under which conditions?
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 To answer the above questions, we are going to study the aggregate and distributional
implications of introducing user fees as an additional public financing policy instrument. We
work in a dynamic general equilibrium setup with Ramsey second-best optimal policy.
 Our main result is that, under certain conditions, the adoption of user fees can, not only
increase aggregate efficiency, but also decrease inequality. Thus, the common view seems to
be, at least in some cases, a fallacy.
 This holds especially when the use of the publicly provided good by one agent creates
external, public-good type benefits for other agents (namely, the publicly provided good is
not private).
 We deliberately work within a simple and recognizable setup. We build on the neoclassical
growth model.
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 Demand for the publicly provided excludable public good, as a negative function of the user
fee, comes from individual agents’ utility maximization problem (see also Ott and
Turnovsky, 2006).
 To study distributional implications, we obviously use a model with heterogeneous agents
and a potential conflict of interests.
 We choose to work with Judd’s (1985) model, in which private agents are divided into two
distinct groups, called capitalists and workers, where workers do not participate in capital
markets.
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 This has been one of the most commonly used models with heterogeneity in the literature on
optimal taxation. (In addition to Judd (1985), see also Lansing (1999), Krusell (2002),
Fowler and Young (2006), Angelopoulos et al. (2011) and many others. As Fowler and
Young (2006) argue, the assumptions of this model are not unreasonable in terms of wealth
concentration.)
 The government is allowed to finance the provision of the excludable public good by a mix
of income taxes, consumption taxes and user fees, all of which are proportional to their own
tax base.
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How we work
 We work in two steps.
(i) We first solve a Ramsey-type second-best optimal policy problem where the
government chooses all the above tax-spending policy instruments optimally.
(ii) In turn, in the second step, we study what happens when the same amount of public
goods, as found in the first step, is financed by income and consumption taxes only, or
by income taxes only, chosen again by the Ramsey government. In other words, in the
second step, we set the total amount of the public good, as well as the amount used by
each group, as in the first step, where these amounts were chosen by utility-maximizing
private agents, and assume away the use of user fees. By working in this way, the two
regimes (with and without user fees) are directly comparable.
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Main results
 When we compare the Ramsey allocations with and without user fees, our main results
are as follows in the long run. (Following most of the related literature on Ramsey
policy, we focus on the long run (see also e.g. Lansing, 1999, Guo and Lansing, 1999,
and Judd, 1999). On the other hand, see e.g. Garcia-Milà et al. (2010), and the
references therein, for redistribution over the transition period in a calibrated model
with heterogeneous agents.)
(i) First, when we compare the economy with income and consumption taxes only to
the economy that also makes use of user fees, the latter is more efficient. Both groups,
capitalists and workers, get better in terms of net income and welfare. This happens
because user fees are less distorting than consumption taxes, which, in turn, are less
distorting than income taxes. Thus, a more efficient economy benefits all groups, which
is as in Judd (1985).
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(ii) Second, under some plausible assumptions specified below, net income inequality
is also reduced by the introduction of user fees when user fees substitute for
consumption taxes. In particular, in this case, although the gross income of capitalists
rises by more than the gross income of workers, at the same time, their tax burden also
gets heavier than that on workers. The tax burden effect more than offsets the gross
income effect, so the introduction of user fees benefits workers more than capitalists in
terms of after-tax income. As a result, the average effective tax rate on the capitalist
rises, while the average effective tax rate on the worker falls, as we switch from an
economy without, to the economy with, user fees. To put it differently, the introduction
of user fees works as a progressive tax that hurts those who have benefited more from
the switch to a more efficient economy and this reduces net income inequality.
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 These results are robust to the presence of external, social effects from the excludable
publicly provided good. Actually, the presence of such effects strengthens our results;
the idea is that, when externalities are also present, the low-income group benefits even
more from the high-income group.
 By the way, we also confirm that the introduction of consumption taxes in a model with
income taxes only, improves efficiency but at the expense of higher inequality
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 The mechanism is as follows. As said, user fees are less distorting than consumption
and income taxes, so that their introduction leads to higher income and higher
consumption for all social groups, especially the rich who own the capital. To the extent
that private agents’ demand for the two goods (the private good and the excludable
public good) follows the standard Samuelson-type condition meaning that the ratio of
marginal utilities equals the ratio of prices, each private agent’s payments for the public
good are proportional to his/her private consumption, where the degree of
proportionality is the weight given by the agent to the public vis-à-vis the private good.
Proportionality arises at least with log-linear and Cobb-Douglas utility functions,
although our results also hold with a CES utility function to the extent that private and
public goods are not perfect substitutes. Since it is the rich who benefit and consume
more after the introduction of user fees, they also pay more and this works like a
progressive tax.
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 Sensitivity analysis shows that the above result holds even if capitalists give a smaller
weight to the public good than workers, to the extent that this is above a critical
minimum value.
 Only when capitalists’ valuation of the public good is too low relative to workers’ the
result does not hold, which means that the introduction of user fees would increase
inequality. This is intuitive given the mechanism discussed above.
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Relevant literature
 Although user prices have always been a debated issue in economic policy circles,
surprisingly the relevant academic literature is relatively (or rather) limited. Exceptions
include, among others, Gertler et al. (1987), Fraser (1996), Ott and Turnovsky (2006),
Swope and Janeba (2006), Fuest and Kolmar (2007) and Huber and Runkel (2009).
 In particular, Gertler et al. (1987) study the impact of the introduction of user fees in the
health care system in Peru. They find that user fees hurt the poor. Fraser (1996) focuses on
the provision of public goods under different public financing schemes including user fees.
Swope and Janeba (2006) analyze how populations with different preferences choose
different public financing schemes. Fuest and Kolmar (2007) focus on the use of user fees
under cross-border externalities. Huber and Runkel (2009) also focus on the use of user fees
under tax competition; they also provide useful empirical evidence for the use of user fees in
the US.
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 We believe that the current work is closer to that of Ott and Turnovsky (2006), who also use
a dynamic general equilibrium model with endogenously determined tax bases. But, when
they study the determination of income taxes and user fees, they focus on the
implementation of the first best; thus, they do not solve for Ramsey policy. More
importantly, they work with a representative agent model so they study efficiency issues
only.
 Thus, to the best of our knowledge, the model I am going to present is the first that evaluates
the aggregate as well as the distributional implications of introducing user fees into a general
equilibrium growth model with income and consumption taxes when all policy instruments
are optimally chosen.
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A model with identical agents
 We start with a situation where agents are identical.
 Although this cannot address issues of equality, it is still useful for addressing efficiency
issues, and more importantly helps us to introduce, later, the more general model, in which
agents differ.
 For simplicity the model is deterministic. Time is discrete and the horizon is infinite.
 The setup is the standard neoclassical growth model extended to include demand for the
publicly provided excludable public good coming from individuals’ utility maximization
problem (see also Ott and Turnovsky, 2006). Subject to this setup, the Ramsey government
maximizes the welfare of the representative individual by choosing once-and-for-all income
taxes, consumption taxes and user fees, as well as the associated amount of the excludable
public good.
15
Households
 There are i  1,2,..., N identical households. Each i maximizes lifetime utility:

  t u (ci , t , li , t , g~i , t )
(1)
t 0
where ci , t is i ’s private consumption, li , t is i ’s work hours, g~i , t is the provision of public
goods and services as perceived by each agent i , and 0    1 is the time preference rate.
We find it convenient to follow e.g. Alesina and Wacziarg (1999) and define g~i , t as:
N
g~i , t  g i , t    g j , t
j i
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(2)
where 0    1 measures the strength of external effects from other agents. If   0 , the
publicly provided good is private. If   1, the good is a pure public good. In this case, we do
not get a well-defined solution. As Ott and Turnovsky (2006) and Hillman (2009) also
explain, user fees can apply to impure public goods, namely, public goods that are rival and
excludable. Here we focus on excludable public goods.
 In our numerical solutions, we will use a simple period utility function of the form:
ui , t  1 log( ci , t )   2 log(1  li , t )   3 log( g~i , t )
where the parameters 1 ,  2 ,  3  0 are preference weights.
 The period budget constraint of each i is:
17
(3)
(1   tc )ci , t  ki , t 1  (1   ) k i , t  pt g i , t  (1   ty )( rt k i , t  wt li , t )
(4)
where k i , t 1 is the end-of-period capital, rt is the return to beginning-of-period capital k i , t , wt
is the return to labor li , t , pt is the user fee paid by household i for the service provided to
him/her, g i , t , 0   tc , ty  1 are consumption and income tax rates respectively, and the
parameter 0    1 is the capital depreciation rate. Profits are omitted since they are zero in
equilibrium.
 Individuals act competitively. The first-order conditions for ci , t , li , t , k i , t 1 , g i , t include the
budget constraint above and:
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 [1    (1   ty1 ) rt 1 ]

c

(1  t ) ci , t
(1   tc1 ) ci , t 1
1
(5a)
1 (1   ty ) wt
2

(1   tc )ci , t
(1  li , t )
(5b)
1 pt
3

(1   tc )ci , t [1   ( N  1)] g i , t
(5c)
where (5a) is a standard Euler equation for capital, (5b) is a standard labor supply condition,
while (5c) is the first-order condition for g i , t . The last condition, (5c), states that the
marginal rate of substitution between the private good and the excludable public good is
equal to the marginal rate of transformation or, equivalently, the ratio of their prices,
(1   tc )
. This condition gives the private agent’s demand for the excludable public good
pt
(recall that N is the number of identical households).
19
Firms
 There are f  1,2,..., N identical firms owned by households. Firms are modeled in the
standard way. Each f maximizes profits:
 f ,t  y f ,t  rt k f ,t  wt l f ,t
(6)
subject to a standard production function, which in our numerical solutions takes the form:
y f ,t  A( k f ,t ) (l f ,t )1
(7)
where k f ,t and l f ,t are capital and labor inputs respectively, while A  0 and 0    1 are
usual technology parameters.
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Firms act competitively. The usual first-order conditions for the two inputs are:
rt 
wt 
 y f ,t
k f ,t
(1   ) y f ,t
l f ,t
so that profits are zero in equilibrium.
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(8a)
(8b)
Government budget constraint
 The period budget constraint of the government is (in aggregate terms):
Gt  N [ ty ( rt ki , t  wt li , t )   tc ci , t  pt g i , t ]
(9)
where G denotes the total provision of the excludable public good.
t
 We use a single income tax, rather than separate taxes on capital income and labour income,
because, if a Ramsey government has access to capital income, labor income and
consumption taxes, it can implement the first-best (see e.g. Lansing, 1999, Coleman, 2000,
and Correia, 2010).
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 We also do not include public debt because as is known, in a Ramsey equilibrium, long-run
debt cannot be pinned down by long-run conditions only (see Chamley, 1986, and Guo and
Lansing, 1999), except if we add some friction.
 Since the presence of public debt is not expected to matter for our main results, we assume it
away to avoid such complexities.
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Decentralized competitive equilibrium (for any feasible policy)
 In the decentralized competitive equilibrium (DCE), households maximize utility, firms
maximize profits, all constraints are satisfied and all markets clear. It is summarized by the
following five equations:
 [1    (1   ty1 ) rt 1 ]

c
(1   t ) ct
(1   tc1 ) ct 1
(10a)
1 (1   ty ) wt
2

(1   tc )ct
(1  lt )
(10b)
ct  k t 1  (1   ) kt  g t  yt
(10c)
g t   ty yt   tc ct  pt g t
(10d)
1
 3 (1   tc )ct
pt 
1 g t [1   ( N  1)]
24
(10e)
where, in the above, we use:
yt  A(kt ) (lt )1
rt 
wt 
 We thus have five equations in

c
t
yt
kt
(1   ) yt
lt
ct , lt , kt 1 , pt t 0
and one of the policy instruments,
, ty , gt t 0 , which adjusts to satisfy the government budget constraint. The DCE is for any

feasible policy. We can now study optimal policy.
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Ramsey policy and allocation
 We focus on Ramsey policy.
 Thus, policy is chosen once-and-for-all.
 The objective of the government is to maximize the representative household’s utility
function subject to the DCE.
 In what follows, we study various cases of this policy problem depending on the public
financing instruments assumed to be available to the government.
 To solve these Ramsey problems, we follow the so-called dual approach.
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Ramsey policy and allocation with user fees
 We start with the relatively general case in which the government chooses all three public
financing instruments (user fees, consumption taxes and income taxes) plus the associated
amount of the public good.
 That is, the government chooses  tc , ty , gt , ct , lt , kt 1 , pt t  0 to maximize the household’s

utility function subject to equations (10a-e).
 Since the resulting equilibrium system cannot be solved analytically, we present numerical
solutions using common parameter values. We report that our results are robust to changes in
parameter values. Following most of the literature on Ramsey policy, we focus on the longrun solution. A numerical long-run solution is presented in Table 1, Regime A. The two
columns of Regime A present the solution with, and without, externalities respectively,
where in the former case we set   0.3 which can be thought as a mild degree of external
effects. Before we comment on results, we solve the same economy without user fees.
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Ramsey policy and allocation without user fees
 We now study the special case in which the government provides the same amount of public
good, as demanded by private agents above, without charging user fees.
 In particular, we set
gt t0
as found above and study two cases: in the first, the
government’s public financing instruments are consumption and income taxes; in the second,
the government can use income taxes only.
(i) In the first special case, the government chooses  tc , ty , ct , lt , kt 1t  0 to maximize the

same objective function as above by taking gt t 0 as given. This is subject to the new

DCE equations which are as in (10a-d) except that now the user fee is set at zero in all
periods.
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(ii) In the second even more special case, the government chooses  ty , ct , lt , kt 1t  0 to

maximize the same objective function by taking gt t 0 as given and subject to the new

DCE equations, which are as in (10a-d) except that now both the user fee and the
consumption tax rate are set at zero in all periods.
 Using the same parameter values as above, numerical solutions for the long run of these two
economies are presented in Table 1, Regimes B and C, respectively. That is, regime B is
without user fees, while Regime C is without user fees and consumption taxes. In each
regime, we again present solutions with, and without, externalities.
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Table 1: Long-run solution with identical agents
Endogenou
s variables
c
l
k
g
y


y
c
p
u
Regime A
Income taxes + consumption
taxes + user fees
  0 .3
0.3305
0.3137
1.7085
0.1102
 0
0.3778
0.3586
1.9530
0.1259
0.5774
0
0.2231
0.3306
-0.6477
0.6600
0
0
1
-0.7656
Regime B
Income taxes + consumption
taxes
  0.3
0.3049
0.2954
1.6090
0.1102
(set)
0.5438
0
0.3615
-0.6562
30
 0
0.2892
0.2954
1.6090
0.1259
(set)
0.5438
0
0.4354
-0.7895
Regime C
Income taxes
  0.3
0.2727
0.2954
1.0571
0.1102
(set)
0.4675
0.2357
-0.6896
 0
0.2494
0.2954
0.9670
0.1259
(set)
0.4527
0.2781
-0.8339
Comparison between regimes A, B and C
 The comparison between regimes A, B and C in Table 1 reveals that the economy with user
fees (Regime A) is the most efficient, both in terms of per capita income and welfare.
 This is because user fees do not distort the decision to work or save, while consumption and
income taxes do (see equations (10a)-(10b)).
 The interesting question here is whether the gain in efficiency happens at the cost of a more
unequal society as is usually believed.
 We address this question in the next section, in which we add two different groups of agents.
 Before we move on to the next section, it is worth pointing out three features of the solution.
31
(i)
First, in all cases in Regimes A and B, the optimal long-run income tax rate is zero.
This is because there are less distorting public financing instruments available to use,
like consumption taxes and user fees.
(ii) Second, in Regime A, where there is a policy choice between consumption taxes
and user fees, both of them should be positive if there are externalities, 0    1; by
contrast, when the publicly provided good is exclusively private,   0 , it is optimal to
use user fees only. The non-zero value of the consumption tax in the case with
externalities is rationalized by the corrective role of tax policy. Namely, the government
uses a relatively distorting policy instrument to correct for a market failure, which here
takes the form of externalities.
(iii) Third, the comparison between regimes B and C confirms the well-recognized
result according to which an economy with income taxes only is less efficient than the
same economy with both income and consumption taxes.
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A model with heterogeneous agents
 We now add heterogeneity among individuals. Following Judd (1985), we assume that
agents differ in capital ownership. In particular, capital is in the hands of a small group
called capitalists, while workers cannot save or borrow. This is the only difference from the
model in the previous section.
33
Capitalists
 There are k  1,2,..., N k identical capitalists. Each k maximizes lifetime utility:

  t u (ck , t , lk , t , g~k , t )
(11)
t 0
where, as in the previous section, we have respectively for the period utility function, the
specification of the excludable public good and the period budget constraint:
u k , t ( ck , t , k k , t , g~k , t )  1 log( ck , t )   2 log(1  lk , t )   3 log( g~k , t )
N
~
g k ,t  g k ,t    g j ,t
(12)
(13)
jk
(1   tc ) ck , t  k k , t 1  (1   ) k k , t  pt g k , t  (1   ty )( rt k k , t  wt lk , t )
34
(14)
The first-order conditions for ck , t , lk , t , k k , t 1 , g k , t include the budget constraint and:
1
(1   tc ) ck , t
 [1    (1   ty1 ) rt 1 ]

(1   tc1 )ck , t 1
(15a)
1 (1   ty ) wt
2

(1   tc ) ck , t
(1  lk , t )
(15b)
1 pt
3

(1   tc ) ck , t [1   ( N k  1)] g k , t  N w g w , t
(15c)
which are like equations (5a-c) above. That is, (15c) is the first-order condition for g k , t and
gives the capitalist’s demand function for the excludable public good (where N k is the
number of identical capitalists and N w  N  N k is the number of identical workers).
35
Workers
 There are w  1,2,..., N w  N  N k identical workers. Each w maximizes lifetime utility:

  t u (cw , t , lw , t , g~w , t )
(16)
t 0
where, as above, we have for each worker, w :
u w , t (cw , t , k w , t , g~w , t )  1 log( cw , t )   2 log(1  lw , t )   3 log( g~w , t )
N
~
g w,t  g w,t    g j ,t
(17)
(18)
jw
(1   tc )cw , t  pt g w , t  (1   ty ) wt lw , t
36
(19)
This is a static problem. The first-order conditions for cw , t , lw , t , g w , t include the budget
constraint and:
1 (1   ty ) wt
2

(1   tc )cw , t
(1  lw , t )
(20a)
1 pt
3

(1   tc )cw , t [1   ( N w  1)] g w , t  N k g k , t
(20b)
which are analogous to (15b-c) above.
37
Firms
 There are f  1,2,..., N k firms owned by capitalists. Thus, each capitalist owns one firm. The
behavior of the firm remains as in the previous section.
Government budget constraint
 The period budget constraint of the government is now (in aggregate terms):
Gt  N k [ ty ( rt k k , t  wt lk , t )   tc ck , t  pt g k , t ]  N w [ ty wt lw , t   tc cw , t  pt g w , t ]
38
(21)
Decentralized competitive equilibrium (for any feasible policy)
 In the decentralized competitive equilibrium (DCE), capitalists and workers maximize
utility, firms maximize profits, all constraints are satisfied and all markets clear. The
market-clearing conditions in the labour and capital markets are respectively:
N f l f ,t  N k l k ,t  N w l w ,t , and N f k f ,t  N k k k ,t . Recall that N f  N k ; namely, the number of
capitalists equals the number of firms.
 It is convenient for what follows to define the population shares of the two groups,
n k  N k / N and n w  N w / N  1  n k . Then, the DCE is summarized by the following nine
equations (quantities are in per capita terms):
39
1
(1   ) ck , t
c
t
 [1    (1   ty1 ) rt 1 ]

(1   tc1 )ck , t 1
(22a)
1 (1   ty ) wt
2

(1   tc ) ck , t
(1  lk , t )
(22b)
1 (1   ty ) wt
2

(1   tc )cw , t
(1  lw , t )
(22c)
{[1   ( N k  1)]cw,t  N k ck ,t }gt pt
(1   )cw,t  k
 (1   ty )wt lw,t (22d)
w
w
w
k
k
n {[1   ( N  1)]ck ,t  N cw,t }  n {[1   ( N  1)]cw,t  N ck ,t }
c
t
n k ck , t  n k [ k k , t 1  (1   ) k k , t ]  n w cw , t  g t  n k y f , t
(22e)
g t   ty n k y f , t   tc ( n k ck , t  n w cw , t )  pt g t
(22f)
3 (1   tc )[nk {[1   ( N w  1)]ck ,t  N wcw,t }  n w{[1   ( N k  1)]cw,t  N k ck ,t }]
pt 
1 gt {[1   ( N k  1)][1   ( N w  1)]   2 N k N w}
(22g)
and, in turn, the amount of the excludable public good used by each type follows from:
40
{[1   ( N w  1)]ck ,t  N wcw,t }
gt
g k ,t  k
[n {[1   ( N w  1)]ck ,t  N wcw,t }  n w{[1   ( N k  1)]cw,t  N k ck ,t }]
(22h)
{[1   ( N k  1)]cw,t  N k ck ,t }
gt
 k
[n {[1   ( N w  1)]ck ,t  N wcw,t }  n w{[1   ( N k  1)]cw,t  N k ck ,t }]
(22i)
g w,t
where, in the above, we use:
nk y f ,t  A(nk kk ,t ) (nk lk ,t  n wlw,t )1
rt 
wt 
y f ,t
kk ,t
(1   )nk y f ,t
(nk lk ,t  n wlw,t )
We thus have nine equations in ck , t , lk , t , kk , t 1 , g k , t , cw, t , lw, t , g w, t , pt t  0 and one of the policy

instruments,  tc , ty , gt t 0 , which adjusts to satisfy the government budget constraint.

41
Ramsey policy and allocation
 The objective of the government is to maximize a weighted average of capitalists’ and
workers’ utility concerning private goods (i.e. consumption and leisure) plus the utility
enjoyed by both groups (i.e. capitalists and workers) from the provision of the public good.
 The maximization is subject to the DCE.
 We will again distinguish optimal policy with and without user fees as we did before.
 We will start by assuming that the government is benevolent, or Benthamite, which means
that in the social welfare function the weights given to the two groups follow their
population fractions.
42
Ramsey policy and allocation with user fees
 The government chooses  ty , tc , g t , ck , t , lk , t , kk , t 1 , g k , t , cw, t , lw, t , g w, t , pt t  0 subject to the DCE

equations above. Thus the objective function of the government is as follows:
1 log(ck ,t )   2 log(1  l k ,t )  (1   )1 log(cw,t )   2 log(1  l w,t )  3 log(g t )



k
k
c
c
k
w













n
k

c


c

W
n
l
n
l


n
k
(
1
)
(
1
)
(
1
)
(
(
)
(
1
)(
1
)
)
k ,t 1 
k ,t 1
t 1
k ,t 1
t
k ,t
t 1
k ,t 1
w,t 1
 1t
   W (1  l )   (1   c )c     W (1  l )   (1   c )c 

2t
1 t
k ,t
2
t
k ,t
3t
1 t
w,t
2
t
w,t


c
c
k
k



t  4t (1   t )c w,t  B(1   t ){[1   ( N  1)]c w,t  N ck ,t }  Wt l w,t 
 

k
w
t 0


Wt (n l k ,t  n l w,t ) c k


k
k
w
1
w







(
n
c
n
c
)
g
A
n
k
n
l
n
l
(
)
(
)


t
k ,t
k ,t
w,t
t
k ,t
w,t
 


1

5t 



c
k
w
w
w
k
k
 B(1   t )n {[1   ( N  1)]ck ,t  N cw,t }  n {[1   ( N  1)]cw,t  N ck ,t }




 6t n k ck ,t  n k k k ,t 1  n k (1   )k k ,t  n w cw,t  g t  A(n k k k ,t ) (n k lk ,t  n wl w,t )1 

43
 We again solve the model numerically.
 We report that our results are robust to changes in parameter values.
 A numerical long-run solution is presented in Table 2, Regime A. The four columns in
Regime A present the solution under four different values of the externality parameter   0 .
In particular, we set   0.4 ,   0.3 ,   0.2 and   0 , where the last case means that the
publicly provided good is private.
 As in the previous section, before we discuss results, we solve for the same economy without
user fees.
44
Ramsey policy and allocation without user fees
 We now solve for the special case in which the government provides the same amount of
public good, as demanded by private agents above, without charging user fees.
 In particular, we set g t , g k , t , g w, t t 0 as found above and study, as we did in the previous

section, two cases:
- in the first, the government’s public financing instruments are consumption and income
taxes;
- in the second, the government can use income taxes only.
45
 In the first special case, the government chooses  ty , tc , ck , t , lk , t , kk , t 1 , cw, t , lw, t t  0 to maximize

the same objective function as above by taking g t , g k , t , g w, t t 0 as given and subject to the

new DCE equations, which are as in (22a-f) except that now the user fee is set at zero in all
periods.
 In the second special case, the government chooses  ty , ck , t , lk , t , kk , t 1 , cw, t , lw, t t  0 to maximize

the same objective function again by taking g t , g k , t , g w, t t 0 as given and subject to the new

DCE equations, which are as in (22a-f) above except that now both user fees and
consumption taxes are set at zero.
 Using the same parameter values as above, numerical solutions for the long run of these two
economies are presented in Table 2, Regimes B and C, respectively.
46
Table 2: Long-run solution with heterogeneous agents
47
Endogenous
variables
ck
  0. 4
0.3661
0.2317
5.6313
0.2005
Regime A
  0.3
  0.2
0.3712 0.3793
0.2325 0.2353
5.6949 5.7978
0.1767 0.1591
 0
0.4348
0.2619
6.5100
0.1449
gw
0.3127
0.3438
0.0671
0.3152
0.3484
0.0796
0.3197
0.3553
0.0905
0.3534
0.4000
0.1178
y
g
1.9032
0.1071
1.9247
0.1087
1.9595
0.1111
2.2001
0.1259
y
c
0
0.2363
0.2749
0.5197
0.6773
0.6300
2.6115
1.1707
0.7673
0
0.2178
0.3350
-0.5360
0
0.1877
0.4296
-0.5570
-0.6997
lk
kk
gk
cw
lw
p
uk
uw
u
yk / yw
ck / cw
uk / u w
0
0
1
-0.6252
  0. 4
0.3466
0.2069
5.3632
0.2005
(set)
0.2914
0.3333
0.0671
(set)
1.8126
0.1071
(set)
0
0.3478
-0.5169
Regime B
  0.3
  0.2
0.3448
0.3421
0.2069
0.2069
5.3632
5.3632
0.1767
0.1591
(set)
(set)
0.2899
0.2876
0.3333
0.3333
0.0796
0.0905
(set)
(set)
1.8126
1.8126
0.1087
0.1111
(set)
(set)
0
0
0.3548
0.3655
-0.5383
-0.5660
 0
0.3255
0.2069
5.3632
0.1449
(set)
0.2736
0.3333
0.1178
(set)
1.8126
0.1259
(set)
0
0.4354
-0.6690
  0. 4
0.3120
0.2069
3.5811
0.2005
(set)
0.2622
0.3333
0.0671
(set)
1.5673
0.1071
(set)
0.2278
-0.5485
-0.7277
-0.8324
-0.6890
-0.7110
-0.7394
-0.8460
-0.7206
-0.6506
-0.6765
-0.7702
-0.6374
-0.6592
-0.6874
-0.7929
-0.6690
2.6234
1.1778
0.7660
2.6369
1.1862
0.7654
2.7038
1.2302
0.7511
2.6622
1.1897
0.7502
2.6699
1.1897
0.7571
2.6816
1.1897
0.7655
2.7580
1.1897
0.7908
2.2822
1.1897
0.7612
48
Regime C
  0.3   0.2
0.3094 0.3055
0.2069 0.2069
3.5515 3.5068
0.1767 0.1591
(set)
(set)
0.2600 0.2568
0.3333 0.3333
0.0796 0.0905
(set)
(set)
1.5626 1.5555
0.1087 0.1111
(set)
(set)
0.2319 0.2381
0.5709 0.6000
0.7436 0.7734
0.6918 0.7214
2.2822 2.2822
1.1897 1.1897
0.7678 0.7758
 0
0.2808
0.2069
3.2232
0.1449
(set)
0.2360
0.3333
0.1178
(set)
1.5090
0.1259
(set)
0.2781
-0.7133
-0.8903
-0.8372
2.2822
1.1897
0.8012
Comparison between regimes A, B and C
 In terms of aggregate efficiency (per capita income and welfare), the results in Table 2 are
similar to those in Table 1. That is, income taxes are the most distorting public financing
instruments.
 Hence, if consumption taxes and/or user fees are also available, the government finds it
optimal to choose   0 in the long run.
y
 Also, consumption taxes are more distorting than user fees. Hence,  falls, and p is
c
positive, as we switch from Regime B (without user fees) to Regime A (with user fees).
Actually, if there is no need to correct for externalities (i.e.   0 ), the government finds it
optimal to choose   0 in Regime A.
c
49
 In terms of equality, comparison of Regimes A and B in Table 2 reveals that the economy
with user fees is not only more efficient, but it is also more just in terms of net income.
 In particular, the net income of capitalists relative to workers,
yk / yw , where
yk  (1   ty )(rkk  wlk )   c ck  pg k and y  (1   ) wl   c  pg , falls as we switch from
y
w
t
c
w
w
w
Regime B (case without user fees) to regime A (case with user fees).
 This happens both when the publicly provided good is private,   0 , and when it creates
public good externalities,   0 .
 To understand the redistribution result when we switch from B to A, recall that, by
definition, changes in net income are driven by changes in gross income and tax payments.
50
 Our solution implies that, as we switch from Regime B to Regime A, the gross income of
both agents rises, since we move to a more efficient economy.
 Actually, the gross income of capitalists rises by more than the gross income of workers, so
it cannot be changes in gross income that drive the redistribution result.
 So this result must be driven by changes in tax payments. This is the case indeed: inspection
of tax payments implies that the amount of taxes paid by each capitalist rises, while the
amount of taxes paid by each worker falls, as we switch from Regime B to Regime A.
 The tax burden effect more than offsets the gross income effect, and it is this that leads to the
net redistribution result. Notice that this is the case both with, and without, externalities,
although redistribution in favor of workers is stronger in the more general case with social
externalities,   0 .
51
 It is useful to examine how these changes in gross income and tax payments translate into
changes in the average effective tax rate, defined as the tax payments as a ratio of gross
income. Irrespectively of the tax rates and user fees chosen by the government, the average
effective tax rates on the capitalist and the worker respectively are:
 (rk  wl )   c  pg
y
c
k
k
rk  wl
k
k
(23a)
k
 wl   c  pg
y
k
c
w
w
wl
w
(23b)
w
52
 Our solutions in Table 2 imply that, as we switch from Regime B to A, the average effective
tax rate rises for the capitalist and falls for the worker. In other words, for the capitalist:
 c  pg
c
c
k
c
k
rk  wl
k
in Regime A is higher than
k
k
rk  wl
k
in Regime B
k
by contrast, for the worker:
 c  pg
c
w
c
c
w
wl
in Regime A is lower than
w
w
wl
in Regime B
w
As a result, inequality falls as we replace consumption taxes with user fees. Actually, our
solutions imply that  c falls by more (strictly speaking, no less) than  c , while pg rises
c
c
k
w
k
by more (strictly speaking, no less) than pg , as we switch from B to A, and that the latter
w
effect dominates.
53
 Depending on the value of the externality parameter,   0 , workers can get better off in
terms of welfare too. This happens when  is sufficiently high.
 Under our baseline parameterization, when   0.3 , u / u rises as we switch from Regime B
k
w
to Regime A, meaning that workers get relatively better off when we introduce user fees
(recall that u , u  0 , so an increase in absolute value denotes lower welfare).
k
w
 In general, as Table 2 shows, the stronger the external social effects from the use of the
public good, the stronger the fall in (both income and welfare) inequality.
 Finally, a switch from Regime C to Regime B worsens inequality. Thus, our results confirm
the common belief that, other things equal, replacing income taxes with consumption taxes is
bad for equality.
54
Interpretation of results and robustness
 Now, we will try first to explain what is special about user fees and then report some
robustness checks.
55
What drives the main result?
 What we have shown so far is that the introduction of user fees to a tax system with income
and consumption taxes, which allows the government to partially replace consumption taxes
with user fees, reduces net income inequality and, depending on the degree of social
externalities, welfare inequality too.
 To understand our results, recall from (23a-b) that the increase in the capitalist’s effective tax
rate, and the simultaneous decrease in the worker’s effective tax rate, as we switch from
Regime B to Regime A, cannot be explained by changes in the denominators (as said,
capitalist’s gross income rises by more) or by changes in consumption taxes in the
numerators (as said, capitalist’s consumption tax payments fall by more).
56
 Hence, the raison d’être for the fall in inequality, as we switch from B to A, must be the
expenditures on the publicly provided private good, pg and pg . Now, focusing for
k
w
simplicity on the simple case without social externalities,   0 , private agents’ first-order
conditions for the publicly provided private good, (15c) and (20b), can be rewritten
respectively as:
pg k 
3
(1   tc )ck
1
(24a)
pg w 
3
(1   tc )cw
1
(24b)
57
so that, for each private agent, expenditures on the publicly provided private good, pg , are
proportional to gross expenditure on private consumption, (1   )c , where the degree of
c
proportionality is the relative weight given to the two goods in the utility function,

.

3
1
 Since  c falls by more than  c , the rise in pg relative to pg is driven by the rise in c
c
c
k
w
k
w
k
relative to c . To put it differently, since capitalists are the winners in terms of gross income
w
and private consumption when we switch from B (without user fees) to A (with user fees),
pg also rises by more than pg and hence inequality falls.
k
w
 It is important to add two clarifications.
58
- First, the implications of the demand conditions, (24a-b), remain the same if we use richer
utility functions. In particular, here we have used a log linear function that is additively
separable in private and public goods.
Our results continue to hold when we use a Cobb-Douglas, or even a CES, utility function
to the extent that the two goods (private and public) are not perfect substitutes. When we
use a Cobb-Douglas function, the ratio of expenditures on each good depends on
preference parameters only, as in (24a-b).
59
When, for instance, we use a CES function, c  (1   ) g


i
i
 (1  l )


t
1
, where i  k , w and

p
0    1, the first-order conditions in (24a-b) become g  
 (1   )(1  
i
1

 c , which
)
 1
c
t
i
again deliver similar results, although now the expenditure shares are not linked linearly.
On the other hand, using more general functions like this narrows the range of parameter
values that can give well-defined solutions. Also notice that in the special case in which
  1 (i.e. the two goods are perfect substitutes), the two optimality conditions of the two
agents are reduced to one so we cannot get a solution without adding more structure. See
also Krusell et al. (1995, section 8) for the implications of different utility functions for
optimal taxation. Thus, what is needed is that private expenditures on the two goods move
together, which is the case in the Samuelson-type first-order conditions (24a-b).
60
- Second, inspection of (24a-b) reveals that the result may not hold if capitalists do not

value enough the public good, namely, if

k ,3
is low enough.
k ,1
In particular, there must be a critical low value for the weight given to the public good
relative to the private good on the part of capitalists that the result does not hold, since in
that case capitalists will be willing to pay relatively little for the provision of the publicly
provided private good.
61
 We report that this is the case indeed when we solve the same model allowing also for
differences in the weights in the utility function between capitalists and workers. In
particular, we enrich (12) and (17) as:
 log(c )   log(1  l )   log( g~ ) and
k ,1
k ,t
2
k ,t
k ,3
k ,t
 log(c )   log(1  l )   log( g~ )
w ,1
w ,t
2
k ,t
w,3
w ,t
respectively, where now    and    .
k ,1
Then, if


k ,3


w,3
k ,3
w,3
is sufficiently low relative to
k ,1
can happen when
while
w ,1


k ,3


w,3
our result does not hold. For instance, this
w ,1
remains as in the baseline parameterization above, namely,
k ,1

1
 ,

3
k ,3
k ,1
rises to around 1 . This is an intuitive result.
2
w ,1
62
Robustness
 We have already studied what happens when the utility function and the weights given by
private agents to the two goods change.
 We also report that our results are robust to a number of changes in parameter values (results
are available upon request). For instance, we have experimented with changes in the weights
given to the two social groups in the government’s objective function. As the weight given to
workers rises, inequality falls, as expected. Nevertheless, for a given weight, benevolent or
partisan, a switch to an economy with user fees reduces inequality, other things equal. The
same applies to changes in the population shares of workers and capitalists; our results do
not change when we experiment with different values of population fractions.
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Concluding remarks and extensions
 We have studied the aggregate and distributional implications of introducing user fees, or
user prices, for a publicly provided excludable public good.
 Using a standard model, we showed that user fees can, under certain conditions, not only
improve aggregate efficiency, but also reduce inequality.
 The current work belongs to the literature on the dilemma between efficiency and equity. In
particular, it belongs to a group of papers that question the validity of some widely perceived
views in public policy regarding this dilemma.
 The paper can be enriched in several ways.
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 For instance, we can also study transition effects. That is, we can study the aggregate and
distributional implications when we depart from the long run of the economy without user
fees and travel towards the long run of the same economy with user fees.
 Besides, we have set aside a lot of interesting issues in public economics (see the discussion
in Hillman, 2009, chapter 3).
- For example, we have not examined the issue of asymmetric information. As is known,
user prices provide useful information about personal benefits from public goods.
- Also, we have focused on the case in which the excludable public good is publicly
provided. There are nevertheless cases where such goods are privately produced (by the
so-called private providers) subject to government regulation.
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Thank you for your attention!
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