Goods-market Desirability of Minimum Wages
Rui Pan∗ and Dao-Zhi Zeng†
Graduate School of Information Sciences
Tohoku University, Sendai, Japan
April 7, 2022

Abstract
This paper develops a general-equilibrium model with heterogeneous ﬁrms anda perfectly competitive labor market to explore the eﬀects of minimum wages on
welfare, inequality, and unemployment. We ﬁnd that a low minimum wage may
increase the social welfare, in which all labor groups are weighted equally. Thisis because minimum wages may reduce the market distortions caused by the weakﬁrm-selection eﬀect in the goods market without minimum wages. Furthermore,
we show that a minimum wage is also desirable for reducing income inequality anddecreasing the unemployment rate.
Keywords: minimum wage, ﬁrm selection, welfare, inequality
JEL Classiﬁcation: H00 J24 J38 J58 J68

1Introduction

This paper aims to examine the eﬀects of minimum wages on welfare, inequality (orantipoverty), and unemployment. Using a general-equilibrium model incorporating heterogeneous ﬁrms and a competitive labor market, we ﬁnd that reducing the market distortions is a new rationale for the desirability of minimum wages. Moreover, a low minimum
wage can reduce inequality and decrease unemployment simultaneously.
Minimum wages are widely used by governments to reduce income inequality. Suchan instrument is also considered to be related to unemployment in the labor market.
However, the policy debate over the desirability of minimum wages is long and remains
intense. In 2014, two open letters both signed by hundreds of economists, including several∗†

E-mail: pan@se.is.tohoku.ac.jp.
E-mail: zeng@se.is.tohoku.ac.jp.

1

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Nobel Prize winners, were sent to US federal policymakers, expressing opposite opinions
on the eﬀect of raising minimum wages.1 Proponents emphasize the antipoverty beneﬁts
as a minimum wage increases the wages of low-skilled workers. Opponents argue thatminimum wages reduce employment, thereby worsening the welfare of workers who lose
their jobs. However, such a negative eﬀect is not supported by recent empirical literature(e.g., Card and Kruger, 1994; Cengiz, Dube, Lindner, and Zipperer, 2019).
For a long time, the desirability of minimum wages was proposed based on two rationales: their eﬃciency in reducing the losses from monopsony power in the labor market(e.g., Ahlfeldt, Roth, and Seidel, 2022) and redistributing the economic output fromhigh-income workers to low-income workers (e.g., Berger, Herkenhoﬀ, and Mongey, 2022).
Two assumptions are crucial for these discussions. First, the labor market is limited to amonopsonistic one. Second, the desirability of minimum wages in redistribution depends
on the relative weights of labor groups in the social welfare. It is not known whether
minimum wages are still desirable without those assumptions.
To ﬁll the theoretic gap, we take another approach by focusing on the goods market.
We use a closed-economy version of the heterogeneous-ﬁrms model of Melitz and Ottaviano (2008). There are two types of workers (high- and low-skilled) and two sectors in
the model. The labor market is perfectly competitive. Two types of workers can work ineither sector. In the numeraire good sector (sector A henceforth), two types of workers
employ themselves to produce homogeneous good A. We assume that high-skilled workers have a higher productivity level in sector A. In the manufacturing sector (sector M
henceforth), ﬁrms produce diﬀerentiated goods under monopolistic competition. Firms
employ high-skilled workers as entry costs and low-skilled workers as the only marginal
input. The minimum wage is binding in sector M for ﬁrms to employ low-skilled workers.
Speciﬁcally, after observing the policy of minimum wages, ﬁrms decide whether to employ
high-skilled workers to enter the goods market. After paying this entry cost, ﬁrms observe
their marginal cost and decide whether to employ low-skilled workers.
To explain our ﬁndings, we highlight two driving forces in the model. The ﬁrst one isa selection eﬀect: an increase in the minimum wage results in a lower cutoﬀ cost level for
producing. Evidence for the selection eﬀect is found in China (Bai, Chatterjee, Krishna,and Ma, 2021). The second eﬀect is a reallocation eﬀect: an increase in minimum wages
enlarges the employment of low-cost ﬁrms and shrinks that of high-cost ﬁrms. Evidence
for the reallocation eﬀect is found in recent literature. (Clemens, Kahn, and Meer, 2021;
Dustmann, Lindner, Schonberg, Umkehrer, and Vom Berge, 2022).
Our ﬁrst important result is that a country can gain from the introduction of minimum
1

See Gerritsen and Jacobs (2020).
The letters are available at www.epi.org/
minimum-wage-statement and http://nebula.wsimg.com/faf44fea2172ad008b46a64835ae2492?
AccessKeyId=D2418B43C2D698C15401&disposition=0&alloworigin=1.

2Electronic copy available at: https://ssrn.com/abstract=4088010

wages. According to Nocco, Ottaviano, and Salto (2014), in the market without minimum
wages, the ﬁrm selection in equilibrium is weaker than that in optimum. Furthermore, lowcost ﬁrms are smaller and high-cost ﬁrms are larger than optimal in the market equilibrium
without minimum wages. Introducing a minimum wage results in a lower cutoﬀ cost levelfor ﬁrms to be active. Compared with the situation without minimum wages, more ﬁrms
are expelled from the market, which reduces the intensity of ﬁrm competition in sector M.
Moreover, the reallocation eﬀect may occur by which (relatively) low-cost ﬁrms produce
more and (relatively) high-cost ﬁrms shrink their production. Thus, minimum wages may
reduce distortions in the goods market and increase the eﬃciency of production, which
ultimately increases welfare. Nevertheless, the minimum wage system is not beneﬁcial to
all workers. We show that only low-skilled workers working in section M are net winners,
while other workers are net losers.
To investigate the eﬃciency of minimum wages in decreasing inequality, we look at two
aspects of income inequality, namely, the inter-group inequality between low-skilled andhigh-skilled workers and the inequality among all individuals. The inter-group inequality
is measured by the ratio of high-skilled workers’ average income to low-skilled workers’
average income, while the Gini coeﬃcient is used to measure the inequality among all individuals. We show that inter-group inequality decreases under minimum wages. The Ginicoeﬃcient decreases with a low minimum wage if the preference intensity for diﬀerentiatedproducts relative to the numeraire is large.
Furthermore, we ﬁnd the positive eﬀects of minimum wages on the employment rateusing an alternative model that considers the costs of work. Workers with positive net
income choose to work. Minimum wages aﬀect employment rate via two channels: theﬁrst is a negative one, increasing the competition tightness of the labor market; the second
is a positive one, encouraging an additional labor supply. We show that a low minimum
wage reduces the unemployment rate.
This paper contributes to three streams of literature. The ﬁrst one clariﬁes a new eﬀect
of minimum wages on welfare from the goods market. Most previous papers examinedthe eﬃciency of minimum wages on welfare by combining the competitive labor markets
with an optimized redistributive tax (Allen, 1987; Guesnerie and Roberts, 1987; Dreze
and Gollier, 1993; Marceau and Boadway, 1994; Boadway and Cuﬀ, 2001; Gerritsen andJacobs, 2020; Lavecchia, 2020). All of these papers show that minimum wages increase
welfare only if they can reduce the labor market distortions caused by redistributive
income taxation. In a setting with competitive labor markets, Lee and Saez (2012) showthat minimum wages increase welfare as long as the welfare weight on low-skilled workers
is greater than the weight on high-skilled workers, the demand elasticity of low-skilledlabor is ﬁnite, the supply elasticity of low-skilled labor is positive, and the rationing of
unemployment is eﬃcient (the lowest-surplus workers are hit by unemployment ﬁrst).
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Simon and Wilson (2021) focus on centralized and decentralized minimum wage policy
setting by extending the Lee and Saez (2012) model to a two-jurisdiction framework withmobile agents, regional governments, and a federal government. They derive suﬃcientconditions for the desirability of a binding minimum wage from the perspective of botha local and central government. Unlike the previous studies focusing on labor market
distortions, we present a mechanism showing that minimum wages are desirable if selecting
ﬁrms brings the goods market closer to its eﬃcient level.
The second stream is the theoretical study of the eﬀects of minimum wages on inequality or antipoverty. Previous empirical studies suggest that minimum wages helped reduce
wage inequality in the United States (DiNardo, Fortin, and Lemieux, 1996; Lee, 1999;
David, Manning, and Smith, 2016; Dube, 2019), Germany (Dustmann, Lindner, Schonberg, Umkehrer, and Vom Berge, 2022), and Brazil (Neumark, Cunningham, and Siga,2006). In contrast, MaCurdy (2015), after examining various mechanisms to understand
who pays for the minimum wages, argues that the introduction of minimum wages is an
ineﬀectual antipoverty policy. Our paper enriches the theoretical study by establishinga general equilibrium including both the labor market and the goods market. We oﬀer
novel analysis of the antipoverty eﬀect by two dimensions: inequalities across groups andamong individuals.
The third stream is the theoretical study of the eﬀect of minimum wages on unemployment, which remains one of the most widely studied and most controversial topics inlabor economics. Card and Kruger (1994) show a positive eﬀect of minimum wages on
employment by examining fast-food employment in New Jersey. More empirical studies
support their work and report a non-negative employment eﬀect of minimum wages (Cardand Krueger, 2000; Zavodny, 2000; Dube, Naidu, and Reich, 2007; Dube, Lester, and Reich, 2010; Allegretto, Dube, and Reich, 2011; Cengiz, Dube, Lindner, and Zipperer, 2019).
Some authors argue that labor market frictions play an important role in the near-zero
employment eﬀect of minimum wages (Rebitzer and Taylor, 1995; Bhaskar, Manning,
and To, 2002; Van Den Berg, 2003; Flinn, 2011; Dube, Lester, and Reich, 2016; Clemens,
and Strain, 2021). In contrast, our paper reveals that the positive employment eﬀect ofminimum wages might result from the fact that more low-skilled workers are encouraged
to search for jobs, which increases the labor supply.
This paper is also related to recent studies on the market distortions of monopolisticcompetition models when consumer preferences are not CES (constant elasticity of substitution). Assuming additively separable preferences with a general subutility function
form, Dhingra and Morrow (2019) consider a one-sector monopolistic competition modelwith heterogeneous ﬁrms. They show that the market equilibrium coincides with the social optimum under the CES demand, but market allocations do not match the optimum
under any non-CES demand. Their work is further extended to a setting with multi4

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ple sectors by Behrens, Mion, Murata, and Suedekum (2020). Our analysis of welfare isconsistent with theirs, although we choose a quasi-linear quadratic utility rather than an
additively separable one.
The rest of the paper is organized as follows. Section 2 presents the basic framework
without minimum wages. In Section 3, we analyze the welfare and Gini coeﬃcient changes
in minimum wages. In Section 4, we take into account costs of work and analyze theimpacts of minimum wages on unemployment rates. Finally, Section 5 concludes.

2

The model

We revise the model of Melitz and Ottaviano (2008) by incorporating minimum wages ina closed economy. There are two sectors: a diﬀerentiated good sector (sector M) and anumeraire good sector (sector A).

2.1

Preference and demand

All individuals share the same preferences given by
Z
2
Z
Z
η
γ
c 2
c
c
c
(q ) di −qi ,U = q0 + α
qi di −
2 i∈Ω i2i∈Ωi∈Ωα > 0, β > 0, γ > 0,(1)

where q0c and qic represent the individual consumption levels of the numeraire good andvariety i ∈ Ω, respectively. Parameter α represents the preference intensity for the differentiated products relative to the numeraire, γ is the degree of love for variety, andη describes substitutability between the diﬀerentiated varieties. In particular, varieties
are perfect substitutes and consumers only care about their consumption level over all
varieties when γ is close to 0.
The budget constraint is written as:
Z
I + q̄0 =qic pi di + q0c ,i∈Ωwhere I is the individual’s net income, q̄0 is the initial endowment. We assume that theinitial endowment is large enough, (e.g., q̄0 > 1), that the demand of each consumer forthe numeraire good is always positive (q0c > 0).
From the FOC of the utility maximization problem, the inverse demand for eachvariety i is given byZ
pi = α − γqic − ηQc ,Qc =qic di,(2)i∈Ωwhenever qic > 0. Let Ω∗ ⊂ Ω be the subset of varieties that are consumed (i.e., qic > 0)
and N = |Ω∗ |. Equation (2) can be inverted to yield the linear market demand system
5Electronic copy available at: https://ssrn.com/abstract=4088010

for these varieties:qi ≡ Lqic =αLL
ηN Lp̄
− pi +,
γ + ηNγγ + ηN γ

Letpmax ≡

1
p̄ =
N

Z
i∈Ω∗

pi di.

(3)αγ + ηN p̄
.
γ + ηN

This represents the price by which the demand for a variety (3) is driven to 0. Note that(2) implies pmax ≤ α.

2.2

Labor supply and production

The labor market has two types of workers: (1 − λ)L high-skilled workers and λL lowskilled workers. The costs to individuals of work are 0, thus all individuals will choose
to work.2 In sector A, workers are employed by themselves. One unit of low-skilled labor
can produce one unit of the numeraire good and the wages are wAl = 1. Meanwhile, one
unit of high-skilled labor can produce a > 1 units of the numeraire good and the wages
are wAh = a.
In sector M, there is a continuum of monopolistically competitive ﬁrms. Following
Melitz and Ottaviano (2008), ﬁrms are ex-ante identical. After making the irreversible
investment of one unit of high-skilled labor as the sunk cost of entry into the market, aﬁrm observes its marginal unskilled-labor requirement cl , which is randomly drawn froma cumulative distribution G(cl ).3 Each ﬁrm then decides whether to produce or exit.
We also follow the previous literature by assuming a Pareto distribution of cl suchthat
 c κl
G(cl ) =,cl ∈ (0, cM ], κ ≥ 1,
(4)
cM
where κ is the shape parameter describing the degree of ﬁrm heterogeneity and cM > 0
represents the upper bound of cl . Furthermore, we impose some technical assumptions:γa
L > 2(κ + 1)(κ + 2) 2 ,cM(5)

α > 2cM ,(6)

α2 (κ + 1),
4(κ + 2)4cM κ 
cM 
4cM 
cM 
1−
<λ<1−
1−
.
α(κ + 1)αaα(κ + 1)
α

η>

2

(7)
(8)

Section 4 relaxes this assumption using an alternative model with working cost.
Since workers are homogeneous in Melitz and Ottavianao (2008), that model does not distinguishthe labor inputs of marginal costs and entry costs.
3

6

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Inequality (5) requires a large total population. Inequality (6) requires that the preference intensity of diﬀerentiated goods is large enough. By (7), the substitutability between
varieties is large enough. Melitz and Ottaviano (2008) explain this condition in terms of
α in their footnote 7. The inequalities of (8) imply that the amounts of low-skilled laborand high-skilled labor are not extremely large or small. Given a > 1, it is easy to verify
that its left-hand side is smaller than its right-hand side.
Let wM l be the wage rate of low-skilled labor working in sector M. Without a minimum
wage system, wM l = wAl = 1 holds because of the free mobility of low-skilled workers
between sectors M and A. However, wM I = w if a minimum wage policy of w is imposed.
In contrast, wAl = 1 remains true because sector A is an informal labor market sector inwhich workers employ themselves.
The marginal cost of a ﬁrm with cl is given as c(cl ) = cl wM l . The gross proﬁt of aﬁrm in sector M is π(cl ) = [p(cl ) − cl wM l ]q(cl ). The FOC to maximize the proﬁt gives thefollowing relationship between the optimal price p(cl ) and the output level q(cl ):q(cl ) = (L/γ)[p(cl ) − cl wM l ].
Using (2), the optimal price and the optimal gross proﬁt arep(cl ) =clD + clwM l
2and π(cl ) =L
2[clD − cl ]2 wM
l,
4γ

(9)

respectively, where clD is the cutoﬀ marginal labor requirement at which the ﬁrm is
indiﬀerent between remaining in the industry and exiting. The expression of clD is givenbyαγ + ηN p̄
p(clD ) = c(clD ) = clD wM l = pmax =.
(10)ηN + γThe above equations give the mass of active ﬁrms
γ α − clD wM l.
(11)
η clD wM l − p̄
Rc
Before the entry, the expected proﬁt of a ﬁrm is 0 lD π(cl )dG(cl ) − wM h , where wM h is
the wage rate of high-skilled labor in sector M. We have wM h = wAh = a because of thefree mobility of high-skilled workers between two sectors. The free entry condition gives
Z
Z clD
L clD
[c(clD ) − c(cl )]2 dG(cl ) = a.
(12)π(cl )dG(cl ) =
4γ
0
0
N=

2.3

Market equilibrium without minimum wage system

In an economy without a minimum wage requirement, we have wM l = 1. The marginal
cost and price of each ﬁrm are given byc(cl ) = cl ,p(cl ) =clD + cl.
2

7

Electronic copy available at: https://ssrn.com/abstract=4088010(13)

Combining equations (3), (10), (12), and (13), the equilibrium cutoﬀ clD and the massof ﬁrms N are determined. The cutoﬀ marginal labor requirement is given by
c0lD ≡1h 2(κ + 1)(κ + 2)γa(c )κ i κ+2
M
.
L

(14)

Assumption (5) gives
c0lD < cM(15)

immediately.
According to (11), the mass of surviving ﬁrms is
N0 =
2(κ + 1)γ  α
−
1
,
η
c0lD

where the positiveness of N 0 (i.e., α > c0lD ) is ensured by (6) and (15). The low-skilledlabor demand of ﬁrm cl is q(cl )cl . Therefore, the amount of low-skilled labor in sector M,
L0M l , is given by
Z c 0
lD
N0
κ(c0lD )2 LN 0
κ(α − c0lD )c0lD L
0LM l =q(cl )cl dG(cl ) ==G(clD ) 02(κ + 1)(κ + 2)γ(κ + 2)η
<λL,

(16)

where inequality (16) is ensured by (5)—(8).4 Thus, the low-skilled labor supply is large
enough to meet the demand in sector M. Meanwhile, the low-skilled workers who are not
employed by ﬁrms in sector M choose to employ themselves in sector A. The technology
of producing the numeraire good is constant returns to scale, which generates suﬃcientdemand for low-skilled labor in sector A. Thus, the amount of low-skilled workers in sector
0
A is LAl= λL − L0M l .
Each ﬁrm employs one high-skilled labor as the entry cost. Therefore, the demand forhigh-skilled workers in sector M equals the mass of entrants given by
L0M h =

 0
N02(κ + 1)γ(cM )κ
(α − c0lD ) c0lD L
0
=
α
−
c
c
=
lD
lD
κ+2
G(clD )a(κ + 2)ηηc0lD<(1 − λ)L,
where the inequality is ensured by (6), (7), and (8) for reasons similar to those given
in footnote 4. Thus, the high-skilled labor supply is large enough to meet the demandin sector M. Note that the remaining skilled labor works in sector A, whose amount is
L0Ah = (1 − λ)L − L0M h .
Inequalities (6) and (15) give (α − c0lD )c0lD < (α − cM )cM . From (7) and (8), we have L0M l <L[4κ/(κ + 1)](1 − cM /α)cM /α < λL.
4

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3

Minimum wage system

Now we assume that the government imposes a minimum wage rate w > 1,5 which isbinding in sector M. In sector A, individuals are employed by themselves, thus minimum
wages are not guaranteed.6
Following Lee and Saez (2012) and Lavecchia (2020), we mostly focus on minimum
wages just above the market equilibrium wages earned by low-skilled workers. Speciﬁcally,
we assume   κ+2
cM κ
.
(17)
w ≤ min a, 0clD
The ﬁrst part implies that wages of high-skilled workers are not directly related to theminimum wage rate. Accordingly, the high-skilled workers’ wages remain a. The second
part is imposed to ensure the existence of active ﬁrms. In fact, pmax ≤ α holds from (2).
No ﬁrm is active if the labor costs are too high to allow for positive proﬁts in sector M.
Given wM l = w, after substituting G(cl ) using (4), π(cl ) using (9), and c(clD ) using(10) into equation (12), the cutoﬀ marginal labor requirement level clD is derived as2αc1lD ≡ c0lD w− κ+2 < c0lD < cM <,(18)2where inequalities are obtained from w > 1 and the results of Section 2.3.
The averages of ﬁrm-level performance measures arep̄ =2κ + 1 1c w,
2κ + 2 lD

c̄ =

κ 1
c w,
κ + 1 lD

q̄ =

L 1 1
c w.
2γ κ + 1 lD

(19)

The mass of surviving ﬁrms is2(κ + 1)γ  α
N =
−1 ,
η
c1lD w
1(20)

where the positiveness of N 1 is ensured by (17) and (18).

3.1

Labor market

The demand for low-skilled workers employed in sector M is
L1M l =N1
G(c1lD )Z c1lDq(cl )cl dG(cl ) =0

κL
c1lD (α − c1lD w)(κ + 2)η

5(21)

For convenience, with slight abuse of notation, the system without a minimum wage is also treatedas a special case of w = 1 later.
6
Sector M can be regarded as the formal labor market sector, and sector A is the informal labormarket sector in the two-sector models of Welch (1974); Gramlich, Flanagan, and Wachter (1976); andMincer (1976).

9Electronic copy available at: https://ssrn.com/abstract=4088010

<L0M l < λL,
where the ﬁrst inequality is from w > 1 and (18) directly, and the second inequality isfrom (16). The amount of low-skilled labor in sector A is L1Al = λL − L1M l . The demandfor high-skilled workers in sector M is given byN1
c1lD w  c1lD w
α2
L1M h ==
1−
L < (1 − λ)L,G(c1lD )a(κ + 2)ηα
αwhere the inequality is ensured by (6), (7), (8), and (17). The amount of high-skilled labor
in sector A is L1Ah = (1 − λ)L − L1M h . The average income of all groups of individuals is
given by1L1
I¯ = L1Al + wL1M l + a(L1Ah + L1M h ) = I0 + (w − 1) M l ,(22)
LL
where I0 ≡ 1 + (1 − λ)(a − 1) represents the average income without minimum wages,
L1M l /L is the share of workers covered by minimum wages, and (w − 1) is the additional
increase in income of a low-skilled worker working in sector M.

3.2

WelfareIn the heterogeneous-ﬁrms model considered here, we ﬁrst check the eﬀect of a minimum
wage system on entering ﬁrms. A higher minimum wage increases the marginal costs, so
ﬁrms must be more eﬃcient to survive. The result below follows from (18) directly.
Lemma 1. (Selection eﬀect): An increase in minimum wages strengthens the selectioneﬀect monotonically: dc1lD /dw < 0.
Bai et al. (2021) also predict the selection eﬀect of minimum wages by considering
heterogeneous ﬁrms under perfect competition in a Heckscher-Ohlin setting. Firms aremonopolistic competition in this paper. Our ﬁnding is consistent with and complementary
to theirs.
This selection eﬀect on high-cost ﬁrms decreases the mass of active ﬁrms and reduces
the intensity of competition in sector M. This results in a larger amount of production bymore productive ﬁrms and a smaller amount of production by less productive ﬁrms.
Lemma 2. (Reallocation eﬀect): An increase in minimum wages enlarges the employmentof low-cost ﬁrms (cl < c1lD κ/(κ +2)) and shrinks that of high-cost ﬁrms (cl > c1lD κ/(κ +2)).
Proof: Let ι(cl ) = q(cl )cl be the unskilled-labor input of ﬁrm with cost cl . The result
follows immediately from the derivative below.
dι(cl )
Lcl c1lD κ=− cl .
dw
2γ κ + 2
10

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□This reallocation eﬀect is supported by recent empirical evidence in Germany (Dustmann et al., 2022). Berger et al. (2022) also predict the reallocation eﬀect of minimum
wages by considering a general-equilibrium model with an oligopsony labor market. Although the labor market is perfectly competitive in this paper, our ﬁnding is consistent
with and complementary to theirs.
These two lemmas are informative if we compare them with the results in Noccoet al. (2014), which shows that the selection eﬀect is weaker in equilibrium than in
optimum. Furthermore, low-cost ﬁrms are smaller and high-cost ﬁrms larger in the market
equilibrium without minimum wages than in the optimal allocation. In contrast, we showthat the new selection eﬀect in Lemma 1 and the reallocation eﬀect in Lemma 2 generated
by minimum wages may reduce the bias between the market equilibrium and the social
optimum, which inspires us to investigate the welfare change due to the minimum wages.
Welfare can be evaluated using the indirect utility function associated with equation
(1):
γ −1
1 N1 21η+ 1(α − p̄)2 +σ ,(23)
W = V̄ = q̄0 + I¯ +
2
N
2 γ p
where
σp2 =

1
N1

Z

2

κ (wc1lD )(pi − p̄) di =.
4(κ + 1)2 (κ + 2)
i∈Ω
2

(24)

Using (18), (19), (23), and (24), we rewrite the expression for welfare as
W = q̄0 + I¯ + CS,(25)

where CS is the consumer surplus
α − wc1lD
CS =
2η



κ+1
α − wc1lD
κ+2


.

(26)

Equations (23) and (25) indicate that minimum wages aﬀect the social welfare via twochannels: the ﬁrst one is changing the average income directly; the second one is changing
consumer surplus through price and the mass of varieties.
Thus the derivative of W with respect to w is divided into two parts as follows.
dW dI¯ dCS
=
+
dw dw
dw

(27)

i
κ
κ
c0 κw− κ+2 h0
κ+2 ) + (2 + κ)c0 w κ+2
= lD
(2
+
κw)(α
−
2c
w
lD
lD
η(κ + 2)22iκc0lD κw− κ+2 h0
κ+2 − (3 + 2κ)α .
+
2(1
+
κ)c
w
lD
2η(κ + 2)2
κ+4

11Electronic copy available at: https://ssrn.com/abstract=4088010

(27a)
(27b)

Assumptions (6) and (17) give the positiveness of (27a) and the negativeness of (27b).
Speciﬁcally, the minimum wage raises the wages of low-skilled workers, which links theminimum wage to a gain in social welfare. At the same time, the higher wages reduce
consumer surplus by the decrease in the mass of varieties and the increase in the priceof diﬀerentiated goods. The total eﬀect on welfare becomes ambiguous. However, we areable to show that the positive eﬀect dominates the negative one when the minimum wage
is small.
Proposition 1. A low minimum wage increases the social welfare.
Proof: The derivative of (27) at w = 1 is
dW
dI¯
dCS
c0 κ(α − 2c0lD )=+= lD
dw w=1 dw w=1
dw w=12η(κ + 2)2

(28)

>0,
where the inequality follows from (6) and (15).

□The inequality of α > 2c0lD is crucial for Proposition 1. Noting that c0lD of (14)decreases with L and increases with a, cM , and γ, we know that the positive eﬀect ofa low minimum wage is more likely in a larger country with a smaller productivity gap
between workers and across ﬁrms, a lower degree of love for variety, and a higher demandintensity for the diﬀerentiated goods.
The technical assumptions of (5) to (8) are imposed to derive Proposition 1. These
conditions are not related to the suﬃcient conditions of Lee and Saez (2012) derived
from a competitive labor market, where they conclude that for the minimum wage tobe desirable, the government needs to redistribute the weights in the social welfare fromhigh-skilled workers toward low-skilled workers. In our paper, the weights of all workers
are equal in the social welfare function; in other word, there is no redistribution eﬀect.
Ahlfeldt et al. (2022) also predict a welfare gain from a low minimum wage using aquantitative spatial model with heterogeneous ﬁrms and a monopsonistic labor market.
Their welfare gain directly depends on the monopsony power of the labor market, namely,
the eﬃciency in reducing the losses from the monopsony power. In contrast, the labormarket is perfectly competitive, and therefore, there is no monopsony power in the current
paper.
Without a minimum wage, the market is distorted and biased from the social optimumin the Melitz-Ottaviano Model (Nocco et al., 2014). The market provides too much entry
if the demand parameter α is high. The introduction of a low binding minimum wage
generates higher costs of employing low-skilled labor in sector M, which increases theselection eﬀect that expels low-productivity ﬁrms from the market. This stronger selection
12

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eﬀect reduces the distortion of the market and increases welfare. Bagwell and Lee (2020)
also ﬁnd that starting at the market equilibrium, a new entrant generates a negative
externality, which decreases welfare if α is higher than a critical level based on the modelof Melitz and Ottaviano (2008). Similar facts are found in models based on additively
separable preferences. Dhingra and Morrow (2019) consider a one-sector monopolisticcompetition model with heterogeneous ﬁrms and show that the market equilibrium isthe social optimum under the CES demand while market allocations do not maximize
individual welfare under any non-CES demand. Furthermore, Behrens et al. (2020)
extend Dhingra and Morrow’(2019) analysis to a setting with multiple sectors and showthat intersectoral distortions exist with regard to labor allocation and ﬁrm entry evenin the CES form. Our ﬁnding is related and complementary to theirs. We show that
imposing minimum wages can be an eﬃcient policy to reduce the aforementioned market
distortions.
We have three groups receiving diﬀerent impacts from the minimum wage policy:
low-skilled workers working in sector M, low-skilled workers working in sector A, andhigh-skilled workers. Having clariﬁed the overall impact in Proposition 1, we now explore
another important question: which group gains and which loses as a result of the minimum
wages?
Among all workers, it is clear that LM l low-skilled workers working in sector M beneﬁt
from the minimum wage directly; their nominal wages increase from 1 to w. However,
the amount of LM l decreases with w.
In the goods market, the increase in wage costs has two eﬀects. The ﬁrst is an increasein prices. From (19), the average prices of diﬀerentiated goods increase with minimum
wages, which is passed to the consumers directly. The second eﬀect is the selection ofﬁrms. Low-productivity ﬁrms are expelled from the market under minimum wages. It iseasy to verify that the mass of surviving ﬁrms (as well as the mass of varieties) decreases
with minimum wages from equation (20). However, the expected proﬁts of ﬁrms arestill zero under the free-entry condition, which implies that the reduction in varieties isultimately passed on to the consumers which decreases the consumer surplus. The quasilinear utility function assumed in this paper indicates that all individuals share the samelevel of consumption of diﬀerentiated goods. Therefore, the reduction in consumer surplus
is identical and given by (27b) for all individuals.
The derivative of utility of low-skilled workers in sector M, WM l , with respect to w at
w = 1 is given bydWM lc0lD κ
=1 −[(3 + 2κ)α − 2(1 + κ)c0lD ]2dw w=12η(κ + 2)
>0,

(29)
(30)

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and inequality (30) is given in Appendix A.
The derivative of utility of low-skilled workers in sector A and high-skilled workers,
WAl and Wh with respect to w, is given by
dWAl
dWh
dCS
=
=
< 0.
dw
dw
dw
The above equations indicate that LM l low-skilled workers are net winners under alow minimum wage. Meanwhile, all high-skilled workers and other low-skilled workers are
net losers.
We use a numerical simulation to show the features of the eﬀects of a general w on
the utility of each group. The parameters are set as
α = 4, η = 4, γ = 2, a = 2, L = 400, κ = 2, cM = 1, λ = 0.7, q0 = 1.1.
It is easily veriﬁed that these parameters satisfy assumptions (5), (6), (7), and (8). 7
Meanwhile, (cM /c0lD )(κ+2)/κ ≈ 2.04124 > 2 = a holds. According to (17), the upperbound of w is 2.
Figure 1 plots the average income and consumer surplus of all individuals. The solid
curve is the consumer surplus of (26), whose monotonicity shows that raising the minimum
wage is harmful to consumers. The dashed curve is the average income of (22), whosemonotonicity shows that raising the minimum wage improves the average income of all
individuals.
1.50

W
3.835

1.45

Income

3.830

1.40

3.825
1.35

3.820

CS

1.30

3.815

1.25
1.2

1.4

1.6

1.8

2.0

1.2

Figure 1: Consumer surplus and average income

1.4

1.6

1.8

2.0

w

Figure 2: Welfare

Figure 2 shows how the minimum wage changes the social welfare. We observe a bell
shape. Although a low minimum wage improves the society, a high minimum wage is
not desirable. This nonmonotonicity provides an explanation for the long debate over theminimum wage policy introduced in footnote 1.
7

Some empirical literature estimates that the average κ is close to 2. See footnote 6 of Melitz andOttaviano (2008). Meanwhile, in the CES framework, κ is supposed to be larger: κ > 4.

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Utility
4.6

4.4

Wh

4.2

4.0

WMl

3.8

3.6

WAl
3.4
1.2

1.4

1.6

1.8

2.0

w

Figure 3: Utility of each group of workers
Figure 3 provides three utility curves: one for high-skilled workers, one for low-skilledworkers in sector M, and one for low-skilled workers in sector A. They are the dotted(WM l ), dashed (WAl ), and solid (Wh ), respectively. All of them are monotonic. The risein WM l dominates the fall in WAl and Wh when w is small, but it is dominated when w
is large. The interaction among groups results in the nonmonotonic property observed in
Figure 2.

3.3

Income distribution and inequality

Evidently, minimum wages have an eﬀect on wage inequality. Three groups of workers
have incomes, a, w, and 1. Following Kohl (2020), we ﬁrst look at the inequality between
high- and low-skilled workers by calculating the ratio of their average income. The incomeof high-skilled workers is independent of minimum wages and equal to a. The income of
low-skilled workers employed by themselves in sector A is still equal to 1, while sector
M’s workers beneﬁt from a minimum wage system and get w. The inter-group inequality
is given byaaλL
InterIneq =
=
.
(31)
1
1
wLM l /Ll + LAl /Ll
λL + (w − 1)L1M l
The derivative of this inter-group inequality with respect to w is given by
dInterIneq
aλL2 c0lD κw− κ+2=−dw
η[λL + (w − 1)L1M l ]2 (κ + 2)2
h
i
κ
κ
0
0
κ+2
κ+2
× (2 + κw)(α − 2clD w ) + (2 + κ)clD w
κ+4

(32)

Assumptions (6) and (17) give the negativeness of (32). Therefore, an increase in theminimum wage decreases the inter-group inequality monotonically.
One more inequality source is the inequality within low-skilled workers, which evidently
increases with w. To see the whole eﬀect, we plot the Lorenz curves in Figure 4. The
dotted curve represents the inequality without minimum wages and the solid line shows
the inequality with minimum wages.
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1

1 − I0I−λ¯λ
I0λL−LM l
¯
IL

(0,0)

λL−LM lL

λ1

Figure 4: Lorenz curve
The Gini coeﬃcient is the ratio of the area that lies between the 45-degree line andthe Lorenz curves. With a minimum wage, the Gini coeﬃcient is given by
Gini =2λ − 1 −L1M l (I¯ − I0 ) (1 − λ)I0 − λ
+
¯
IL
I¯
4

η(2 + κ)[I0 − (1 + I0 )λ]w κ+2
=2λ − 1 +
2
4
0
0
κ+2
clD κ(w − 1) αw− clD w + ηI0 (κ + 2)w κ+22κ(c0lD )2 κ2 (w − 1) α − c0lD w κ+2
h


i,
−
2
4
0
0
κ+2
κ+2
η(κ + 2) clD κ(w − 1) αw− clD w + ηI0 (κ + 2)w
where the last equality is from (18), (21), and (22).
Without the minimum wage requirement, the Gini coeﬃcient is simply
Gini

=
w=1

λ(I0 − 1).
I0

Taking the derivative of Gini with respect to w at w = 1 gives2

dGini
c0 κ2 (α − c0lD )= lD
×(α1 − α),dw w=1(2 + κ)2 I0 η 2
|
{z
}
+

where

(2 + κ)η
α1 ≡ c0lD +c0lD κ



λ
λ−1+
I0


.

Equation (33) immediately leads to the following result.
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Proposition 2. A low minimum wage reduces the Gini coeﬃcient if α > α1 .
Note that the value of α1 depends on the share of low-skilled workers λ and thesubstitutability between the diﬀerentiated varieties η. α1 is large if λ and η are large.
Propositions 1 and 2 tell us that the minimum wage policy may be desirable in terms ofboth welfare and inequality when α is large.
An intuitive explanation of Proposition 2 is obtained if we notice that there are somelow-skilled workers self-employed in the numeraire sector and not covered by the minimum
wage system. Recall that α represents the preference intensity for diﬀerentiated products
relative to the numeraire. A large α indicates a large demand for diﬀerentiated products, which leads to a large share of low-skilled labor working in sector M who beneﬁts
from the minimum wages. Therefore, the average income of low-skilled workers becomeslarge, which narrows the inter-group inequality. Furthermore, the eﬀects on inter-groupinequality dominate and the overall inequality decreases.
Gini
0.168

0.166

0.164

0.162

1.2

1.4

1.6

1.8

2.0

w

Figure 5: Gini coeﬃcient
Figure 5 shows how the minimum wage changes the Gini coeﬃcient, using the same
parameters as in Section 3.2. Noting that α1 ≈ 3.42549 < 4 = α holds, the Gini coeﬃcientdecreases in the beginning. We observe a nonmonotonic shape again. A low minimum
wage decreases the inequality, while a high minimum wage is not desirable. This is anatural result because the inter-group inequality decreases and the inequality within the
low-skilled workers increases with w.

4

Working costs and unemployment

To investigate the employment eﬀect of minimum wages, we borrow the idea of working
cost in Lee and Saez (2012) and Simon and Wilson (2021). Note that the employment ofskilled workers is not related to the minimum wages in our model, we simply assume that
only low-skilled workers face cost θ of work, which is randomly drawn from a distribution
H(θ) on interval [0, θ̄], where θ̄ > 1. The low-skilled workers pay these costs only if they

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work.8 There are three low-skilled labor supply options: (a) not working and earning zero,(b) working in sector A with wages 1, and (c) working in sector M with wages w. We
assume that ﬁrms can not observe the individual cost θ of work. Workers with positive
net income choose to work.
To ensure a suﬃcient supply of unskilled labor, we assume that θ̄ is not too large.
Speciﬁcally, the inequality
1
λ>× (LHS of (8))
(34)
H(1)
is assumed to replace the ﬁrst inequality of assumption (8). Meanwhile, the demand sideof labor does not change since the wages are the same as in the case of no working costs.
Finally, in this alternative model, we add one more assumptionw ≤ θ̄

(35)

in addition to (17).

4.1

Unemployment rate without minimum wages

Without the minimum wage policy, wages are 1 for all low-skilled workers and a for all
high-skilled workers. All low-skilled workers whose costs are smaller than 1 choose towork. Therefore, the labor supplies of two types of workers are
Lh = (1 − λ)L.

Ll = λLH(1),Inequality Ll > L0M l is ensured by (5)—(7), and (34).9 The technology of producing thenumeraire good is constant returns to scale, whose market results in an inﬁnity demand
for two types of labor in sector A. Therefore, all workers who choose to work can get jobs.
The unemployment rate in the economy can be easily calculated as follows.
µ0 = λ(1 − H(1)).

4.2

Unemployment rate with minimum wages

Under a minimum wage policy, the wages for low-skilled workers in sector M become w.
The wages remain 1 for low-skilled workers in sector A and a for all high-skilled workers.
All low-skilled workers whose costs θ are smaller than w will search for jobs in sector
M. Meanwhile, the low-skilled labor demand in sector M is the L1M l of (21). Firms do
8

Therefore, these costs are diﬀerent from the participation cost of Lavecchia (2020), which needs tobe paid as long as workers start searching jobs.
9
Inequalities (6) and (15) give (α − c0lD )c0lD < (α − cM )cM . From (7) and (34), we have L0M l <L[4κ/(κ + 1)](1 − cM /α)cM /α < H(1)λL.

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not know the working costs θ of each individual. Therefore, all low-skilled workers with
θ < w share an identical possibility to get a job in sector M. The probability of a successful
match is given by
L1M l
< 1,
(36)
e=
λLH(w)
where the inequality comes from the fact of L1M l (< L0M l < Ll ) < λLH(w). Meanwhile,H(w) increases and L1M l decreases with w, which indicates that e decreases with w. Notethat e also represents the tightness of the labor market in sector M .10 Thus, raising w
reduces the tightness and increases the diﬃculty for unskilled workers to get jobs.
Low-skilled workers who failed to match with a ﬁrm in sector M choose to work in
sector A if their costs θ are smaller than 1. Thus, the amount of low-skilled labor in sector
A is
Lwc
(37)
Al = (1 − e)H(1)λL.
All high-skilled workers choose working. Therefore, the high-skilled labor supply is Lh =(1 − λ)L. The unemployment rate in the economy is easily calculated asµ1 = µ0 − |{z}e × λ[H(w) − H(1)] .
|{z}tightness(38)

supplyEquations (36) and (38) indicate that the unemployment rate is jointly aﬀected by two
eﬀects of minimum wages: the ﬁrst one is increasing the competition tightness of thelabor market (lower e) through H(w) and L1M l ; the second one is encouraging additional
labor supply through H(w).
Equation (38) implies that µ1 < µ0 as long as w > 1. Moreover, the derivative of µ1
with respect to w is


dµ1
de
dH(w)
− eλ
,
(39)
= λ[H(w) − H(1)] −
dw
dw
dw
|
{z
} | {z }
+

+

where

dµ1
dH(w)
= −eλ
< 0.
dw w=1
dw w=1(40)

Thus, a low minimum wage decreases the unemployment rate from (40), although therelationship between µ1 and w is ambiguous for a general w according to (39).
In summary, we have the following proposition:
Proposition 3. A minimum wage policy generates a lower unemployment rate.
10

The tightness is deﬁned as the ratio of the number of vacancies to the number of job seekers in
Lavecchia (2020). The number of job vacancies equals to the total low-skilled labor demand in thispaper.

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When a minimum wage is introduced, more low-skilled workers are encouraged tosearch for jobs in sector M, which increases the low-skilled labor supply. The low-skilledworkers with θ ∈ (1, w) become new entrants in the job market. They face a possibility
e of being employed in sector M. Meanwhile, all low-skilled workers with costs θ ≤ 1 canstill be employed either by ﬁrms in sector M or by themselves in sector A, which is thesame as the case without minimum wages. Therefore, the new job searchers (θ ∈ (1, w))
who are successfully employed in sector M become additional employees, contributing to
the reduction in the unemployment rate.
The theoretical result of Proposition 3 is supported by a series of empirical studies.
Non-negative or positive employment eﬀects of minimum wages are reported by Cardand Krueger (1994,2000); Zavodny (2000); Dube, Naidu, and Reich (2007); Dube, Lester,and Reich (2010); Allegretto, Dube, and Reich (2011); and Cengiz, Dube, Lindner, andZipperer (2019).
We conduct numerical examples by assuming H(w) = θ/θ̄ and using following parameters:
α = 2, η = 5, γ = 0.001, a = 10, L = 100000, κ = 2, cM = 0.01, λ = 0.1, θ̄ = 5.
It is easily veriﬁed that these parameters satisfy assumptions (5), (6), (7), (8), and (34).
The upper bound of w in this model is 5 because θ̄ = 5 < (cM /c0lD )(κ+2)/κ ≈ 6.45497 <
10 = a holds. Figure 6 shows how the unemployment rate changes with minimum wages.
We observe a nonmonotonic shape again. A low minimum wage is desirable to reduce theunemployment rate, while a large one is not. This nonmonotonicity gives an explanation
for the empirical evidence of both negative (e.g., Clemens and Wither, 2019) and nonnegative employment eﬀects of minimum wages.
μ10.079820.079800.079780.07976

0.07974

0.07972

2

3

4

5

w

Figure 6: Unemployment rate

4.3

Robustness

We now examine the robustness of the welfare and inequality results in Section 3 byconsidering the working costs. For convenience, we simply assume that the working costis randomly drawn from a uniform distribution H(θ) = θ/θ̄.
20Electronic copy available at: https://ssrn.com/abstract=4088010

4.3.1

WelfareGiven minimum wages w, the average net income of each group is
Z 1
Z w
1
1
¯
¯
IAl = 1 −
θdH(θ), IM l = w −θdH(θ),
H(1) 0H(w) 0I¯h = a, I¯u = 0.
Using the uniform distribution of θ, the average net income of all groups of individuals is
given by1 ¯ wc ¯
I¯ =IAl LAl + IM l L1M l + Lh I¯hL
1 wc=(L + wL1M l + 2aLh ),
2L Al



where Lwc
Al is the amount of low-skilled labor in sector A from (37). Similarly, welfare isgiven by (25). The derivative of W with respect to w is given bydW dI¯ dCS
=
+
dw dw
dw



2κ+6
κ
c0lD κw− κ+220
κ+2× (2 + κ)α + (2 + κw )(α − 2clD w )=2η(κ + 2)22κc0lD κw− κ+20
κ+2+× 2clD (1 + κ)w− (3 + 2κ)α .
2η(κ + 2)2

We then obtain the eﬀect of minimum wages at w = 1:dI¯dCS
c0 κ(α − 2c0lD )
dW
=
+
= lD
,
dw w=1 dw w=1
dw w=12η(κ + 2)2
which is the same as equation (28). Thus, the results of Proposition 1 are robust evenwhen working costs are included. These results also extend the suﬃcient condition for thedesirability of minimum wages given by Lee and Saez (2012). Their eﬃcient rationing is
not required here. In fact, all low-skilled workers with working cost θ < w have the same
possibility of being employed in sector M.
Furthermore, Figure 7 shows how the minimum wage changes the welfare in the caseof working costs, using the same parameters as in Section 3.2 and θ̄ = 1.3. It is easily
veriﬁed that these parameters also satisfy assumption (34). Noting that θ̄ = 1.3 < a =2 < (cM /c0lD )(κ+2)/κ ≈ 2.04124 holds, the upper bound of w is 1.3 according to (35).
Figure 7 has the same bell shape as Figure 2. a low minimum wage increases welfare, anda high minimum wage decreases welfare.

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W
3.404

3.402

3.400

3.398

1.05

1.10

1.15

1.20

1.25

1.30

w

Figure 7: Welfare
4.3.2

Income inequality

The inter-group inequality is given byInterIneq =aIM l L1M l /Ll + IAl Lwc
Al /Ll=2aθ̄λL.
1
λL + θ̄(w − w1 )LMl

The above equation leads to a counterpart of (32):2aθ̄2 λL2 c0lD κw− κ+2
dInterIneq=−dw
η[λwL + θ̄(w2 − 1)L1M l ]2 (κ + 2)2hiκ× (2 + κw2 )(α − 2c0lD w κ+2 ) + (2 + κ)α2(41)

Assumptions (6) and (17) give the negativeness of (41). Thus, an increase in minimum
wages decreases the inter-group inequality monotonically again.
All high-skilled workers are homogeneous since they share the same wage rate a andwork without costs. However, low-skilled workers are heterogeneous along two dimensions:working place (sector M or A) and working cost (θ). Let ξ(I) denote the bottom percentilesof the population whose net income is I. Therefore, ξ(I) = 0 implies that I is the lowest
income and ξ(I) = 1 means that I is the highest income.
We obtain11
[0, µm ],if I = 0,




λ


µm + I,
if 0 < I ≤ 1,θ̄ξ(I) =(42)
λ L1M l I − 1


µ
+
+
,
if
1
<
I
≤
w,
m


L w
θ̄



(λ, 1],
if I = a.

11

See Appendix B for the proof.

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The reverse function of (42) is the income schedule of all individuals:0,
if 0 ≤ ξ ≤ µm ,




λ
θ̄


 (ξ − µm ),if µm < ξ < µm + ,θ̄I(ξ) = λ Lw(λ + µ θ̄)
λ
Lw
m


+ 1 ξ, if µm + < ξ ≤ λ,
1−


LM l
L1M l θ̄
θ̄




a,
if λ < ξ ≤ 1,
which is plotted in Figure 8(a).
Thus, the Lorenz curve is


0,



Z


1 ξ θ̄(ξ − µm )dξ,I¯ µm λλZ
 1 n µm + θ̄ θ̄
(x − µm )dx
Q(ξ) = I¯
λ
µmZ


h
ξ


Lw(λ + µm θ̄)
Lw i o+1−+ 1 ξ dξ ,LM l
L1M l θ̄
µm + λ

θ̄



2
1


a(ξ
−
λ)
1
(w
−
1)L
λ
Ml


+
,
+
¯
wL
2I
I¯
θ̄

if 0 ≤ ξ ≤ µm ,λif µm < ξ < µm + ,θ̄

if µm +λ< ξ ≤ λ.
θ̄if λ < ξ ≤ 1,

which is illustrated in Figure 8(b).
Income
1
a
w
1

µmµ0

µmµ0

µm + λθ̄ λ 1 ξ(a): Income schedulesµm + λθ̄ λ 1 ξ(b): Lorenz curve

Figure 8: Income schedule and Lorenz curve
The dotted line indicates the inequality without minimum wages, and the solid line
shows the inequality with minimum wages. The corresponding Gini coeﬃcient can then
23

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Gini

0.593154

0.593152

0.593150

0.593148

0.593146

1.1

1.2

1.3

1.4

1.5

1.6

w

Figure 9: Gini coeﬃcientbe calculated as

Z 1
Gini =1 − 2Q(ξ)dξ
02[aθ̄wL(1 − λ)2 − λ(w − 1)L1M l ]=2λ − 1 + 1
θ̄LM l (w2 − 1) + wL[2aθ̄(1 − λ) + λ]2[λ2 w2 L2 + θ̄2 (w3 − 3w + 2)(L1M l )2 ].
−
3θ̄wL θ̄L1M l (w2 − 1) + wL[2aθ̄(1 − λ) + λ](43)

Substituting L1M l of (21) into (43) and taking its derivative with respect to w at w = 1,
we obtain
2L1M ldGini=−[6aθ̄2 (1 − λ)2 + 6aθ̄λ(1 − λ) + λ2 ] < 0.
dw w=13L[2aθ̄(1 − λ) + λ]2
Thus, a low minimum wage decreases inequality again. Compared with Proposition 2, inthe model of working costs, the antipoverty eﬀects of a low minimum wage always happen
without any additional condition regarding α.
We conduct a simulation to see how the minimum wage changes the Gini coeﬃcientin the case of working costs, using the following parameters:
α = 2, η = 5, γ = 0.001, a = 2, L = 100000, κ = 2, cM = 0.01, λ = 0.99, θ̄ = 1.6.
It is easily veriﬁed that these parameters satisfy assumptions (5), (6), (7), (8), and (34).
Noting that θ̄ = 1.6 < a = 2 < (cM /c0lD )(κ+2)/κ ≈ 4.56435 holds, the upper bound of w is
1.6. The result is plotted in Figure 9. We observe that a low minimum wage decreases theinequality, but a high minimum wage may not be desirable. The nonmonotonic property
observed in Figure 5 is robust when working costs are incorporated.

5

Conclusion

We analyze the eﬃciency of a minimum wage policy in a closed economy with heterogeneous workers using a general-equilibrium framework. We show that a country can
24Electronic copy available at: https://ssrn.com/abstract=4088010

gain from a minimum wage policy from the viewpoint of the goods market. Under themarket equilibrium, the ﬁrm selection is weaker than that in the social optimum. The
new selection eﬀect under minimum wages is more eﬃcient. The market distortions are
reduced; therefore, welfare is increased. Moreover, we investigate the desirability of minimum wages in terms of inequality. The Gini coeﬃcient decreases if the demand intensity
for the manufactured varieties is large. Finally, we discuss the eﬀects of minimum wages
on unemployment rates using an alternative model with working costs. It is shown that aminimum wage reduces the unemployment rate by encouraging more low-skilled workers
to search for jobs. The desirability in terms of welfare and inequality remains true whenworking costs are included in the model.

ACKNOWLEDGEMENTS
Zeng acknowledges the ﬁnancial support from JSPS KAKENHI of Japan (Grant numbers
20H01485).

DISCLOSURE
The authors have no ﬁnancial arrangements that might give rise to conﬂicts of interest
with respect to the research reported in this paper. The views expressed here are solely
those of the authors and do not reﬂect those of any institution with which they areaﬃliated.

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AppendicesA. Proof of (30)
The inequality(κ + 1)(κ + 2) 2
α − (3 + 2κ)αc0lD + 2(1 + κ)(c0lD )22κ
h
3 + 2κ i2
8 + 11κ + 4κ2= 2(1 + κ) c0lD −α + α24(1 + κ)8κ(1 + κ)>0leads toα2 (κ + 1)(κ + 2)
> c0lD [(3 + 2κ)α − 2(1 + κ)c0lD ].
2κ

(A.1)

Then (29) gives
dWM l
c0lD κ
=1−[(3 + 2κ)α − 2(1 + κ)c0lD ]2dw w=12η(κ + 2)>1−2κc0lD[(3 + 2κ)α − 2(1 + κ)c0lD ]α2 (κ + 1)(κ + 2)

> 0,
where the ﬁrst inequality is from (7) and the second inequality is from (A.1).

B. The derivation of (42)First, noting that only unemployed workers have income 0, we have ξ(0) = [0, µm ], whereµm is the unemployment rate.
Second, we consider the situation of I ∈ (0, 1]. The working costs of low-skilled workers
are randomly drawn from a uniform distribution, which implies that the net income of
low-skilled workers within a sector is also drawn from a uniform distribution. In sector
M, the net income range of low-skilled workers is (0, w]. Accordingly, the number ofworkers whose net income is smaller than I is L1M l I/w in sector M. Similarly, the number
wc
of workers whose net income is smaller than I is Lwc
Al I in sector A, where LAl is givenby (37). Combining the working labor in two sectors and the unemployed labor, ξ(I) is
given byξ(I) = µm +λL1M l I  λ L1M l +−
I = µm + I,
wL
wL
θ̄
θ̄

if I ∈ (0, 1].

29Electronic copy available at: https://ssrn.com/abstract=4088010

Third, we consider the situation of I ∈ (1, w]. In sector M, the number of workers
whose net income is smaller than I is LM l I/w in sector M. Meanwhile, in sector A, theincome of each low-skilled worker is smaller than I. Combining the working labor in two
sectors and the unemployed labor, ξ(I) is given by
L1M l I  λ L1M l 
λ L1 I − 1= µm + + M l,
ξ(I) = µm +
+
−
wL
wL
L w
θ̄
θ̄

if 1 < I ≤ w.

Finally, all high-skilled workers have the same wages, a, which implies that ξ(I) = (λ, 1]
if I = a.
Combining all situations above, function ξ(I) is given byξ(I) =

[0, µm ],µ + λ I,

if I = 0,

m

θ̄
L1
λ


,
µm + θ̄ + LM l I−1

w





(λ, 1],

if 0 < I ≤ 1,
if 1 < I ≤ w,
if I = a.

30

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