In a model where trade unions dominate the labor market, a relationship is derived between the rate of unemployment and the provision of a public input in the production. This relationship implies that for conventional rates of unemployment, the public input will be overprovided compared to the first-best level.Public Inputs, Trade Unions.

Diego Martinez

Tomas Sjögren

English

Research Papers in Economics

 
 
    Working papers series 
   Department of Economics 
 
 
 
WP ECON 09.13 
 
 
Can Labor Market Imperfections Cause 
Overprovision of Public Inputs? 
 
 
 
Diego Martínez (U. Pablo de Olavide) 
 
Tomas Sjögren (Umeå University) 
 
 
 
 
 
 
JEL Classification numbers: H41, J51 
 
Keywords: Public Inputs, Trade Unions.  
 
 
 
 
http://www.upo.es/econ 
 
Can Labor Market Imperfections Cause Overprovision of
Public Inputs?￿
Diego Martinez
Department of Economics, University Pablo de Olavide
Tomas Sj￿gren
Department of Economics, Ume￿ University
November 2009
Abstract
In a model where trade unions dominate the labor market, a relationship is derived
between the rate of unemployment and the provision of a public input in the production.
This relationship implies that for conventional rates of unemployment, the public input
will be overprovided compared to the ￿rst-best level.
Keywords: Public Inputs, Trade Unions.
JEL Classi￿cation: H41, J51.
￿The authors would like to thank Thomas Aronsson for helpful comments and suggestions. The authors
would also like to thank the Spanish Ministry of Science, Junta de Andalucia, the Bank of Sweden Tercentenary
Foundation (Stiftelsen Riksbankens Jubileumsfond), the Swedish Council for Working Life and Social Research
(FAS) and the National Tax Board (Skatteverket) for research grants. Corresponding author: Tomas Sj￿gren,
Department of Economics, Ume￿ University, SE - 901 87 Ume￿, Sweden. Phone: +46 90 786 99 94. Fax: +46
90 77 23 02. Email: tomas.sjogren@econ.umu.se
1 
 
 
 
 
http://www.upo.es/econ 
 
1 Introduction
An important question in public economics is whether the provision of a public good in a
second-best economy will exceed, or fall short of, the quantity provided in a ￿rst-best setting?
In the literature, the main focus has been to analyze how the use of distortionary taxes,1 or
the presence of labor market distortions,2 may in￿ uence the provision of a public good.
These issues have received far less attention in the context of public inputs in the produc-
tion.3 In particular, no previous study has made an attempt to relate the quantity provided of
a public input to the rate of unemployment. This is somewhat surprising since the provision of
a public input will be in￿ uenced by the presence of unemployment.4 The argument is that if
the public input is complementary with labor in the production, then the government will have
an incentive to provide more of the public input. The question is then whether this incentive
will be su¢ ciently strong to cause overprovision compared to the ￿rst-best quantity?
The purpose of this paper is to analyze this question. Since we want to focus on the
connection between labor market imperfections and public input provision, we do not include
other types of imperfections, such as distortionary taxes, in the model. The labor market
imperfection is assumed to arise because of trade union wage setting and the main result is
that when agents are risk-neutral and the production is of Cobb-Douglas type, then the public
input will be overprovided in relation the ￿rst-best quantity as long as the rate of unemployment
does not exceed approximately 60%.
In section 2, we present the basic model while the main result is derived in Section 3. The
paper is concluded in Section 4.
2 The Basic Model
Consider an economy made up of ￿rms, consumers, trade unions and a government. The ￿rms
are identical and their number is normalized to one. The production function is of Cobb-Douglas
1See, for example, Pigou (1947), Wilson (1991), Chang (2000), Gaube (2000, 2007).
2See, for example, Bovenberg and van der Ploeg (1996), and Aronsson and Sj￿gren (2003).
3Exceptions are Martinez and Sanchez (2009a,b) who analyze the e⁄ects of distortionary taxes on the pro-
vision of a public input.
4See Aronsson and Wehke (2008) who characterize the optimal provision of public inputs in an economy
where the labor market is dominated by trade unions, and Aronsson and Koskela (2008) who characterize
public input provision in the presence of unemployment and oursourcing.
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type
F (N;K;G) = N
￿ ￿ K
1￿￿ ￿ G
￿ (1)
where ￿;￿ 2 (0;1), N is labor, K a ￿xed factor and G a public input. This formulation of the
technology means that the public input is factor augmenting. In the following, the ￿xed factor
will be normalized to one, in which case we can write the pro￿t (i.e. the return to the ￿xed
factor) as
￿ = F (N;G) ￿ w ￿ N (2)
where w is the wage rate. The ￿rst-order condition for pro￿t maximization implies w =
FN (N;G), which implicitly de￿nes the following labor demand function
N (w;G) = ￿
1
1￿￿ ￿ G
￿
1￿￿ ￿ w
￿ 1
1￿￿ (3)
Substituting the labor demand function into equation (2) de￿nes the pro￿t function ￿ (w;G).
Turning to the consumption side of the economy, all consumers have identical preferences for
consumption, c, and these preferences are described by a utility function u(c) which is increas-
ing in c. There are three types of consumers: a ￿rm-owner, employed workers and unemployed
workers, and they are distinguished by the superindices "f", "e" and "u", respectively. Be-
ginning with the ￿rm-owner, he is endowed with the ￿xed factor. The ￿rm-owner does not
work and he receives the pro￿t income in return for providing the ￿xed factor to the ￿rm. The
￿rm-owner￿ s budget constraint is given by cf = ￿ ￿ T f where T f is a lump-sum tax. Turning
to the labor force, it is made up of M workers out of whom N are employed and M ￿ N are
unemployed. An unemployed worker receives a net of tax unemployment bene￿t, b, from the
government meaning that his utility is given by uu = u(b). As for an employed worker, he faces
the budget constraint ce = w ￿ T e, where T e is a lump-sum tax.
All workers are assumed to belong to a trade union which dominates the labor market.5
The trade union has a utilitarian objective function
V = N ￿ u(w ￿ T
e) + [M ￿ N] ￿ u(b) (4)
where M ￿ N is the number of unemployed workers. The trade union chooses w so as to
maximize equation (4) subject to the restriction N = N (w;G). The ￿rst-order condition
becomes
Nw (w;G) ￿ [u(w ￿ T
e) ￿ u(b)] + N (w;G) ￿ u
0 (w ￿ T
e) = 0 (5)
5Since the number of ￿rms is normalized to one, so is also the number of trade unions.
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Equation (5) implicitly determines the wage as a function6 w = w(b;T e). If this wage exceeds
the market-clearing wage, there will be unemployment in the equilibrium.
3 Optimal Policy
The government maximizes a utilitarian welfare function
W = N ￿ u
e + [M ￿ N] ￿ u
u + u
f (6)
and the budget constraint is given by
N ￿ T
e + T
f ￿ G ￿ [M ￿ N] ￿ b = 0 (7)
The policy instruments are b, G, T e and T f, and the Lagrangian corresponding to the govern-
ment￿ s problem can be written as
L = N ￿ u(w ￿ T
e) + [M ￿ N] ￿ u(b) + u
￿
￿ ￿ T
f￿
+ ￿ ￿
￿
N ￿ T
e + T
f ￿ G ￿ (M ￿ N) ￿ b
￿
(8)
where ￿ is a Lagrange multiplier.
Let us ￿rst consider the optimal policy chosen in a ￿rst-best setting. Within the context of
this model, this corresponds to the policy chosen when there is no unemployment. Substituting
N = M into equation (8) and maximizing w.r.t. to the policy instruments, it is straightforward
to show that the solution to this problem produces the standard condition for the optimal
provision of the public input
FG (M;G
￿) = 1 (9)
where G￿ denotes the ￿rst-best quantity.
Next, we turn to the outcome when there is unemployment in equilibrium. Di⁄erentiating
the Lagrangian w.r.t. b, G, T e and T f, and combining the ￿rst-order conditions for T f and G
while using that w = FN, it is straightforward to derive the following condition for the optimal
provision of the public input
0 = FG (N (w;G
￿);G
￿) ￿ 1
+ NG (w;G
￿) ￿
￿
u(w ￿ T e) ￿ u(b)
￿
+ (T
e + b)
￿
(10)
6Note that the functional form of N (w;G) in equation (3) implies that w will be independent of G.
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Let G￿ denote the quantity of the public input chosen in the presence of unemployment. Equa-
tion (10) shows that two incentives in￿ uence the provision of the public input in this case; a
￿ pure￿motive captured by the term FG ￿ 1 and an employment motive re￿ ected by the terms
in the second row of equation (10). Beginning with the ￿ pure￿motive, observe ￿rst that since
N and G are complements in production, it follows that FG (N;G) < FG (M;G) for any level
of employment satisfying N < M. This means that the ￿ pure￿motive for providing the pub-
lic input is reduced in the presence of unemployment. The employment motive, on the other
hand, will induce the government to provide more of the public input as long as NG > 0. Each
worker who goes from the state of unemployment to employment will give rise to two welfare
e⁄ects. First, there will be a direct and positive utility e⁄ect equal to ue ￿ uu > 0 and second,
each worker who leaves unemployment will give rise to a net tax revenue improvement equal to
T e + b > 0.
The discussion above indicates that the presence of unemployment will give rise to two
con￿ icting e⁄ects regarding the provision of the public input. On one hand, the ￿ pure￿motive
for providing G will be weakened whereas the employment motive will provide an incentive to
increase the provision of G. The question is then if the employment motive is su¢ ciently strong
to cause overprovision of the public input in the sense that G￿ > G￿? Without making further
assumptions we cannot answer this question. Let us, therefore, assume that the individuals
are risk-neutral, meaning that we can write the utility function u(c) = c. In this situation,
equation (10) can be simpli￿ed to7
FG (N;G
￿) ￿ 1 + NG (w;G
￿) ￿ w = 0 (11)
where N is determined by equation (3). To be able to compare G￿ with G￿, let us add and
subtract FG (M;G￿) in equation (11) and use that FG (M;G￿) ￿ 1 = 0. This produces
FG (M;G
￿) = FG (N;G
￿) + NG (w;G
￿) ￿ w (12)
Next, substituting
FG (N;G) = ￿ ￿ N
￿ ￿ G
￿￿1 (13)
NG (w;G) =
￿
1 ￿ ￿
￿
N
G
(14)
w = FN (N;G) = ￿ ￿ N
￿￿1 ￿ G
￿ (15)
7Here we use that risk neutrality will imply ￿ = 1.
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into equation (12), we can derive the following relationship
￿
G￿
G￿
￿1￿￿
=
1
1 ￿ ￿
￿
￿
N
M
￿￿
(16)
If there is overprovision of the public input in the presence of unemployment, then G￿=G￿ > 1.
As such, equation (16) implies that employment rates associated with overprovision will satisfy
the following inequality
N
M
> h(￿) (17)
where
h(￿) = (1 ￿ ￿)
1
￿ (18)
The function h(￿) is decreasing in ￿ and the start- and the endpoints of this function are given
by
lim
￿!0 h(￿) = e
￿1 ￿ 0:3679 (19)
lim
￿!1 h(￿) = 0 (20)
Three observations can be made from this analysis. First, since h(0) ￿ 0:3679 and since
h(￿) is decreasing in ￿, the public input will always be overprovided as long as the rate
of employment exceeds 36:79% (or equivalently, as long as the rate of unemployment does not
exceed approximately 63%). Second, the larger ￿ is, the smaller will be the rates of employment
associated with underprovision of the public input. Third, if ￿ = 1 then the public input will
always be overprovided in the presence of unemployment.
The analysis above can be summarized in the following proposition;
Proposition: Consider an economy where monopoly union wage setting creates unemployment,
individuals are risk-neutral and the production technology is of Cobb-Douglas type. If the level
of employment in this setting satis￿es the inequality N > M ￿ (1 ￿ ￿)
1
￿, then the public input
will be overprovided compared to the ￿rst-best quantity.
4 Concluding Remarks
In this paper, we show that in the presence of unemployment, the government has an employ-
ment motive to overprovide a public input in the production. An avenue for future research
would be to analyze to what degree this result will hold in the presence of distortionary taxes.
It would also be interesting to study whether the result derived in the paper is re￿ ected in
empirical data.
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References
[1] Aronsson, T. and Sj￿gren, T. (2003) Income Taxation, Commodity Taxation and Provision
of Public Goods under Labor Market Distortions. FinanzArchiv 59 (3), 347-70.
[2] Aronsson, T. and Wehke, S. (2008) Public Goods, Unemployment and Policy Coordination.
Regional Science and Urban Economics 38 (3), 285-98.
[3] Aronsson, T. and Koskela, E. (2008) Optimal Redistributive Taxation and Provision of
Public Input Goods in an Economy with Outsourcing and Unemployment. CESifo Working
Paper No. 2481.
[4] Bovenberg, A. L. and van der Ploeg, F. (1996) Optimal Taxation, Public Goods and
Environmental Policy with Involuntary Unemployment. Journal of Public Economics 62
(1-2), 59-83.
[5] Chang, M. (2000) Rules and Levels in the Provision of Public Goods: The Role of Com-
plementarities between the Public Good and Taxed Commodites. International Tax and
Public Finance 7, 83-91.
[6] Gaube, T. (2000) When Do Distortionary Taxes Reduce the Optimal Public Spending on
Productive Activities. Journal of Public Economics 76, 151-180.
[7] Gaube, T. (2007) A Note on the Link Between Expenditures and Distortionary Taxation.
Economics Bulletin 8 (9), 1-10.
[8] Martinez, D. and Sanchez, A. J. (2009) A Diagrammatic Approach to the Optimal Level
of Public Inputs. Social Choice and Welfare, forthcoming.
[9] Martinez, D. and Sanchez, A. J. (2009) Returns to Scale and the Optimal Level of
Productivity-Enhancing Public Expenditures: A Numerical Approach. Mimeo University
Pablo de Olavide.
[10] Pigou, A. C. (1947) A Study in Public Finance. Macmillan. London.
[11] Wilson, J. D. (1991) Optimal Public Good Provision with Limited Lump-Sum Taxation.
American Economic Review 81 (1), 153-166.
7

Can Labor Market Imperfections Cause Overprovision of Public Inputs?

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Can Labor Market Imperfections Cause Overprovision of Public Inputs?

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