In this paper, we derive conditions under which a minimum-wage law combined with anonymous taxes and transfers and an agent-specific tax-transfer scheme are equivalent policies.

Arantza Gorostiaga

Juan Francisco Rubio-Ramírez

English

Research Papers in Economics

 
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Fiscal Policy and Minimum Wage for 
Redistribution: An Equivalence Result 
 
Arantza Gorostiaga and Juan Francisco Rubio-Ramírez 
 
Working Paper 2005-8 
May 2005  
 
 
Financial support from Fundación Ramón Areces and 9/UPV 00035.321-13511/2001 is acknowledged.  The views expressed here 
are the authors’ and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining 
errors are the authors’ responsibility. 
 
Please address questions regarding content to Arantza Gorostiaga, Universidad del País Vasco, Avenida Lehendakari Aguirre, 
83, 48015 Bilbao (Spain), +34 94 601 38 14, arantza.gorostiaga@ehu.es, or Juan Franciso Rubio-Ramírez, Federal Reserve Bank 
of Atlanta, 1000 Peachtree Street, NE, Atlanta, GA  30309, 404-498-8057, juan.rubio@atl.frb.org. 
 
Federal Reserve Bank of Atlanta working papers, including revised versions, are available on the Atlanta Fed’s Web site at 
www.frbatlanta.org. Click “Publications” and then “Working Papers.” Use the WebScriber Service (at www.frbatlanta.org) to 
receive e-mail notifications about new papers. 
FEDERAL RESERVE BANK of ATLANTA       WORKING PAPER SERIES 
Fiscal Policy and Minimum Wage for  
Redistribution: An Equivalence Result 
 
Arantza Gorostiaga and Juan Francisco Rubio-Ramírez 
 
Working Paper 2005-8 
May 2005 
 
Abstract: In this paper, we derive conditions under which a minimum-wage law combined with anonymous 
taxes and transfers and an agent-specific tax-transfer scheme are equivalent policies. 
 
JEL classification: E62, H21 
 
Key words: redistribution policy, minimum wage, tax-transfer scheme, equivalence result 1
Fiscal Policy and Minimum Wage for Redistribution:
An Equivalence Result
1. Introduction
Many papers in the literature have considered the use of a minimum wage legislation
jointly with a tax-transfer scheme to redistribute income among di⁄erently productive
agents. Moreover, articles such as Allen (1987), Guesnerie and Roberts (1987) and
Gorostiaga and Rubio-Ram￿rez (2004) have shown that minimum wages might be optimal
when combined with linear taxes and transfers in a static general equilibrium model.
In this paper we prove that any equilibrium allocation attained under a minimum
wage law and anonymous taxes and transfers could be implemented through an agent-
speci￿c tax-transfer scheme. We also show that the reverse implication is not always
true. The equivalence depends on imposing some conditions on the initial agent-speci￿c
policy.
2. The basic setup
We consider a static general equilibrium model. The economy is populated by two
types of agent: low and high productivity households. A proportion ￿ of households are
high-skilled (H), and a proportion 1￿￿ are low-skilled (L). This a production economy
where a single consumption good is produced. The resource constraint is:
￿cH + (1 ￿ ￿)cL = y; (1)
where y is the aggregate production, and cH and cL are high-skilled and low-skilled
consumption respectively.
Firms The available constant returns to scale technology can be represented through
the following CES production function:
y = F [￿(1 ￿ ‘H);(1 ￿ ￿)(1 ￿ ‘L)] =
h
￿[￿(1 ￿ ‘H)]
￿ + [(1 ￿ ￿)(1 ￿ ‘L)]
￿
i1=￿
; (2)
where ￿ 2 (0;1), ￿ > 1, and (1 ￿ ‘H) and (1 ￿ ‘L) are high-skilled and low-skilled labor
respectively. Firms are price takers and inverse labor demands are:
!H = ￿
"
￿ +
￿
1 ￿ ￿
￿
￿￿ ￿
1 ￿ ‘L
1 ￿ ‘H
￿￿#1=￿￿1
(3)
!L =
"
1 + ￿
￿
￿
1 ￿ ￿
￿￿ ￿
1 ￿ ‘H
1 ￿ ‘L
￿￿#1=￿￿1
(4)2
Households Households in this economy derive utility from consumption and leisure.
The utility function U(ci;‘i) is assumed to be strictly increasing and concave in both
arguments. Households are endowed with one unit of time which can be devoted to work
or leisure, and face the following budget constraint:
ci = (1 ￿ ￿i)!i(1 ￿ ‘i) + Ti; (5)
where !i is the wage, ￿i is the income tax rate and Ti is a lump-sum transfer.
Government We consider two alternative policy schemes:
Policy scheme A The redistribution policy is implemented through a minimum
wage law that sets a lower bound on low-skilled wages, !min, and an anonymous tax-
transfer scheme f￿;Tg. The following government budget constraint holds:
￿￿!H(1 ￿ ‘H) + (1 ￿ ￿)￿!L(1 ￿ ‘L) = T:
Policy scheme B The redistribution policy is implemented through agent-speci￿c
tax rates f￿H;￿Lg and transfers fTH;TLg. The following government budget constraint
holds:
￿￿H!H(1 ￿ ‘H) + (1 ￿ ￿)￿L!L(1 ￿ ‘L) = ￿TH + (1 ￿ ￿)TL:
3. The equilibrium under alternative policies
Equilibrium under policy scheme A Under policy scheme A wages must be higher
than the legal lower bound, !min. Therefore, if the minimum wage is binding in equilib-
rium, there is an excess labor supply and the demand side determines labor allocations.
Then, households face an additional restriction: there is a maximum number of hours,
SCi, that they can allocate to work at the minimum wage !min. SCi is equal to the
hours of i-skilled labor that the ￿rm demands at the minimum wage in equilibrium.
Therefore, household i￿ s decision problem is:
max
fci;‘ig
U(ci;‘i)
s.t. ci = (1 ￿ ￿)!i(1 ￿ ‘i) + Ti
(1 ￿ ‘i) ￿ SCi
!i, SCi, ￿, T given
The ￿rst order conditions for this problem are the consumer budget constraint (5) and
the following inequalities:
￿Uc(ci;‘i)(1 ￿ ￿)!i + U‘(ci;‘i) + ￿i = 0
￿i
￿
SCi ￿ (1 ￿ ‘i)
￿
= 0 SCi ￿ (1 ￿ ‘i) ￿ 0 ￿i ￿ 03
When the minimum wage is binding, the multiplier ￿i is strictly positive and the i-skilled
labor allocation is demand determined and equal to SCi. We consider economies where
the minimum wage is only binding for low-skilled workers, i.e. high-skilled labor will
never be constrained:
SCH ￿ (1 ￿ ‘H) > 0 ￿L = 0
In this economy, equilibria under policy scheme A are de￿ned as follows:
De￿nition 1: Given a minimum wage !min, allocations fcH;cL;‘H;‘Lg, a tax-transfer
scheme f￿;Tg, wages f!H;!Lg and perceived constraints on the low-skilled labor supply
SCL constitute an equilibrium if the following conditions are satis￿ed:
(i) fcH;‘Hg solves the high-skilled household￿ s decision problem given !H and policies
f￿;Tg. That is,
￿Uc(cH;‘H)(1 ￿ ￿)!H + U‘(cH;‘H) = 0
cH = (1 ￿ ￿)!H(1 ￿ ‘H) + T:
(ii) fcL;‘Lg solves the low-skilled household￿ s decision problem given !L and policies
f￿;Tg and SCL. That is,
￿Uc(cL;‘L)(1 ￿ ￿)!L + U‘(cL;‘L) + ￿L = 0
(1 ￿ ‘L) = SCL ￿L ￿ 0
cL = (1 ￿ ￿)!L(1 ￿ ‘L) + T:
(iii) f‘H;‘Lg maximizes ￿rms￿pro￿ts given !H and !L. That is, f‘H;‘Lg satis￿es (3)
and (4).
(iv) The equilibrium wage !L is equal to or higher than !min. And SCL is the quantity
of low-skilled labor demanded at the minimum wage !min. That is,
!L ￿ !min = F‘L
￿
￿(1 ￿ ‘H);(1 ￿ ￿)SCL
￿
(v) The economy resource constraint (1) holds and the high-skilled labor market clears.
Equilibrium under policy scheme B Under policy scheme B there is no minimum
wage legislation, although every household faces taxes and transfers that depend on
her/his type.
Household i￿ s problem is to maximize utility subject to the budget constraint (5),
taking prices and policies as given. The ￿rst order conditions for this problem are the
consumer budget constraint (5) and the following equation:
￿Uc(ci;‘i)(1 ￿ ￿)!i + U‘(ci;‘i) = 0
Therefore equilibria under policy scheme B are de￿ned as follows:4
De￿nition 2: Allocations fcH;cL;‘H;‘Lg, taxes and transfers f￿H;￿L;TH;TLg and
wages f!H;!Lg constitute an equilibrium if the following conditions are satis￿ed:
(i) fcH;‘Hg solves the high-skilled household￿ s decision problem given !H and policies
f￿H;THg. That is,
￿ Uc(cH;‘H)(1 ￿ ￿H)!H + U‘(cH;‘H) = 0
cH = (1 ￿ ￿H)!H(1 ￿ ‘H) + TH:
(ii) fcL;‘Lg solves the low-skilled household￿ s decision problem given !L, policies f￿L;TLg.
That is,
￿Uc(cL;‘L)(1 ￿ ￿L)!L + U‘(cL;‘L) = 0
cL = (1 ￿ ￿L)!L(1 ￿ ‘L) + TL:
(iii) f‘H;‘Lg maximizes ￿rms￿pro￿ts given !H and !L. That is, f‘H;‘Lg satis￿es (3)
and (4).
(iv) All markets clear.
4. The equivalence result
In this section we show that for any equilibrium allocation under policy scheme A,
a scheme B-type policy exists such that the same allocation is an equilibrium and vice
versa.
Proposition 1: Let f￿;T;!ming be a policy under policy scheme A. Let allocation
fcH;cL;‘H;‘Lg and wages f!H;!Lg constitute an equilibrium under f￿;T;!ming. Con-
sider a scheme B-type policy f￿H;￿L;TH;TLg satisfying:
TH = T; ￿H = ￿; ￿L = 1￿
U‘(cL;‘L)
!minUc(cL;‘L)
; TL = cL￿(1￿￿L)!min(1￿‘L):
Then, fcH;cL;‘H;‘Lg and f!H;!Lg are also an equilibrium for the scheme B type policy
f￿H;￿L;TH;TLg.
Proof. We will prove that allocations fcH;cL;‘H;‘Lg ful￿ll equilibrium conditions
under the agent-speci￿c policy presented in the proposition. On the one hand, when
TH = T and ￿H = ￿, nothing changes in the constraint faced by high-skilled workers.
Thus, the high-skilled household problem￿ s ￿rst order conditions hold . On the other
hand, ￿L is set in order for the marginal rate of substitution between consumption and
leisure to equal the low-skilled after-tax wage and, therefore, make ￿L = 0; and TL is5
set to satisfy the low-skilled budget constraint. Hence, f￿L;TLg are such that fcL;‘Lg
maximizes the low-skilled household￿ s utility given !L. Finally, since allocations are the
same under both policy schemes, the ￿rms￿problem ￿rst conditions and the resource
constraint also hold with an agent-speci￿c tax-transfer scheme f￿H;￿L;TH;TLg.
Proposition 2: Let the scheme B-type policy f￿H;￿L;TH;TLg be such that,
0 > TH ￿ TL = (￿H ￿ ￿L)!L(1 ￿ ‘L):
Let allocations fcH;cL;‘H;‘Lg and wages f!H;!Lg constitute an equilibrium under
f￿H;￿L;TH;TLg. Consider a scheme A-type policy f￿;T;!ming satisfying:
T = TH; ￿ = ￿H; !min = !L:
Then, fcH;cL;‘H;‘Lg and f!H;!Lg are also an equilibrium for the scheme A type policy
f￿;T;!ming.
Proof. As in the previous proof, it is straightforward that fcH;cL;‘H;‘Lg satisfy
the ￿rms￿problem ￿rst conditions and the resource constraint for both policy schemes.
We now prove that allocations fcH;cL;‘H;‘Lg solve households￿problems under the
anonymous policy f￿;T;!ming. Since T = TH and ￿ = ￿H, high-skilled households
solve exactly the same problem under both policies. Thus, the high-skilled household￿ s
problem ￿rst order conditions hold for f￿;Tg. As regards the low-skilled workers, we
have to prove ￿rst that the multiplier ￿L is positive; and then that fcL;‘Lg satis￿es their
budget constraint under the new policy. The multiplier ￿L is such that
￿L = Uc(cL;‘L)(1 ￿ ￿)!L ￿ U‘(cL;‘L)
Since fcL;‘Lg maximizes low-skilled utility for f￿L;TLg, Uc(cL;‘L)(1￿￿)!L￿U‘(cL;‘L) =
0. Hence,
￿L = Uc(cL;‘L)(1 ￿ ￿)!L ￿ Uc(cL;‘L)(1 ￿ ￿)!L = (￿L ￿ ￿)Uc(cL;‘L)!L:
If, as assumed, ￿ = ￿H < ￿L, then ￿L > 0.
Finally, we have to check that the low skilled budget constrained is satis￿ed. Assuming
that TH ￿ TL = (￿H ￿ ￿L)!L(1 ￿ ‘L),
cL = (1 ￿ ￿)!L(1 ￿ ‘L) ￿ T = (1 ￿ ￿H)!L(1 ￿ ‘L) ￿ (￿H ￿ ￿L)!L(1 ￿ ‘L) + TL:
Rearranging terms,
cL = (1 ￿ ￿L)!L(1 ￿ ‘L) + TL
which is the low-skilled budget constraint when taxes and transfers are f￿L;TLg.6
References
Allen, S. (1987): ￿Taxes, Redistribution, and the Minimum Wage: A Theoretical Analy-
sis￿ , The Quarterly Journal of Economics, vol. August, pp 477-489.
Gorostiaga , A. and J.F. Rubio-Ram￿rez (2004): "Optimal Minimum Wage in a Com-
petitive Economy". Universidad del Pa￿s Vasco. Mimeo.
Guesnerie, R and K. Roberts (1987): "Minimum Wage Legislation as a Second Best
Policy", European Economic Review, vol. 31, pp. 490-498.

Fiscal policy and minimum wage for redistribution: an equivalence result

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