The effects of a minimum wage on employment and on poverty have been studied in the literature. This paper characterizes the poverty minimizing minimum wage, and shows how it depends on productivity, inequality and the degree of labor market competitiveness.inequality, labor productivity, market competitiveness, minimum wage, poverty, Food Security and Poverty, International Development, Political Economy, D6, I32, J38, J64,

Goto, Hideaki

English

Research Papers in Economics

 WP  2008-21 
November 2008 
 
  
Working Paper 
 
Department of Applied Economics and Management 
Cornell University, Ithaca, New York  14853-7801  USA 
 
 
 
THE OPTIMAL MINIMUM WAGE 
FOR POVERTY MINIMIZATION   
 
 
Hideaki Goto  
 
It is the Policy of Cornell University actively to support equality of 
educational and employment opportunity.  No person shall be denied 
admission to any educational program or activity or be denied 
employment on the basis of any legally prohibited discrimination 
involving, but not limited to, such factors as race, color, creed, religion, 
national or ethnic origin, sex, age or handicap.  The University is 
committed to the maintenance of affirmative action programs which will 
assure the continuation of such equality of opportunity. The Optimal Minimum Wage for Poverty 
Minimization
*
 
Hideaki Goto
† 
 
October, 2008 
 
 
Contents 
 
1. Introduction 
 
2. The  Model 
 
3.  The Optimal Minimum Wage 
 
4.  The Effects of Competitiveness and Productivity 
 
5.  The Effects of Inequality 
 
6. Conclusion 
 
Abstract 
 
The effects of a minimum wage on employment and on poverty have been 
studied in the literature. This paper characterizes the poverty minimizing 
minimum wage, and shows how it depends on productivity, inequality and the 
degree of labor market competitiveness.  
 
 
Keywords: inequality, labor productivity, market competitiveness, minimum wage, 
poverty.  
 
JEL Classification: D6, I32, J38, J64.  
                                                 
* I am most grateful to Ravi Kanbur and Nancy Chau for many discussions and insightful comments, which 
had a significant influence on the paper.  I would also like to thank Kaushik Basu and Gary Fields for 
helpful comments and suggestions. 
† Department of Applied Economics and Management, Cornell University, Ithaca, NY 14853. Email: 
hg52@cornell.edu. 
 1. Introduction 
There is a large literature on the impact of minimum wages on employment (for a recent 
survey, see Neumark and Wascher 2007). More recently, investigation has begun on the 
impact of a minimum wage on poverty (for example, Fields and Kanbur 2007 for 
competitive labor markets, and Basu, Chau, and Kanbur 2005 for non-competitive labor 
markets). The objective of this paper is to derive the poverty minimizing minimum wage, 
and to show how it depends on productivity, inequality and the degree of labor market 
competitiveness.  
The plan of the paper is as follows. Section 2 presents the basic model. Section 3 derives 
the poverty minimizing minimum wage. Section 4 looks at the partial and the cross effects 
of competitiveness and productivity on the optimal minimum wage. Section 5 does a 
similar exercise for inequality and the minimum wage. Section 6 concludes.  
 
2. The Model 
We will develop a specialized, tractable model that allows us to address the questions 
posed. Let us suppose that individuals are distributed uniformly along a line segment on the 
x-axis,  , as depicted in Figure 1. Firms are located at  . So m  is the 
average distance, or lack of access, to the firms and 
[ mk mk −, +] 0 x =
( ) km ≤  is a parameter describing the 
extent of inequality in terms of access to the labor market. Without loss of generality, 
population size is normalized to unity. Thus the density function of the distribution of 
individuals is given by  ( ) 1 2 f x =/k . If the firms offer some wage rate  , and if an 
individual at 
w
[ ] x mk mk ∈− , + works for a firm, her net income is given by 
() y wx w t x ,=−. The parameter   could be interpreted simply as the cost of mobility 
or, more generally, as transaction costs that are associated with finding and working for a 
firm. We assume that individuals have no earnings opportunity outside the economy. 
Hence given  , the individuals in [
( 0) t ≥
w ] mk w t − ,/ work for the firms, while the individuals in 
 do not. ( wt m k /, + ]
                                        
3 Thus the labor supply function and the inverse labor supply function 
are respectively given by  
         
3In this paper, we focus on the case in which labor productivity is not high enough to ensure full employment. 
2  
1
() [ ( ) ]
2
w
Sw m k
kt
=− −  (1) 
and  
  2( wk t l m k ) t = +− (2) 
 
For the demand side of the labor market, let us suppose that there exist   firms at  n 0 x = , 
where   is treated as a parameter to be varied. All firms have the same revenue function, 
, where   denotes the number of workers employed, and   and 
 are technological parameters describing labor productivity and diminishing marginal 
product, respectively. In what follows, productivity growth is captured by increases in  . 
Given the revenue function and a wage rate  , the firm’s profit function is given by 
n
2 () ii i Rl a a l b l ,= − / 2
i l
i l 0 a >
0 b >
a
w
() ( ) ii lR l a w π =, − . Each firm maximizes profit given the labor supply and the other 
firms’ labor demand. Since the firms’ technology is identical, we restrict ourselves to 
symmetric Nash equilibria in terms of employment.  
The equilibrium employment and wage are calculated as follows. Given (2) and the other 
firms’ labor demand,  , firm  ’s profit function is of the form   i l− i
 
2 ()[ 2 ( ) (
2
ii i i i i i
b
l l a m k t al l kt l l m k t l π −− ;, ,, ,= − − + + − ) ]  
By differentiating π  with respect to  , and then substituting   for  i l ( 1) i nl − i l− , the 
equilibrium labor demand of each firm when there exist n firms in the market is given by  
 
()
2( 1 )
i
am k t
l
bk t n
∗ − −
=
+ +
 (3) 
Thus the equilibrium (total) employment and wage are, respectively,  
 
[( )
2( 1 )
i
na m kt
ln l
bk t n
∗∗ ] − −
==
+ +
 (4) 
and  
 
[2 ( 2 )( )]
2( 1 )
ta k n b k tmk
w
bk t n
∗ + +−
=
++
 (5) 
 
                                                                                                                                                    
wt m k /< + So we suppose that   always holds. 
3 Note that letting n  offers the competitive employment and wage: →∞
4  
 
()
2
c
am k t
l
kt
− −
=  
 
  c wa =  
 
Throughout our analysis, poverty is measured using the poverty measure which has been 
developed by Foster, Greer and Thorbecke (1984):  
 
0 () ( )
z zy
Pg
z
α
α y d y
−
= , ∫  
where   is the (fixed) poverty line and   is the density function of income distribution.  z g α  
is a parameter, increases in which make the measure more sensitive to the gaps between the 
poverty line and income levels below it. We consider  0 α =  and  1 α ≥ .  
By changing the variables, the poverty measure is also expressed as  
 
()
[ ] () ()
w
t
wz w
tt
mk zw t x
P f xd x f xd x
z
α
α −
+ − −
=+ ∫∫  
if ( ) y wm k z ,− >, and  
 
()
[ ] () ()
w
t
w
t
mk
mk
zw t x
Pf x d x
z
α
α
+
− f x d x
− −
=+ ∫∫  
if  . In the case of a uniform distribution,  () ywm k z ,− ≤ () 12 f x =/k
]
 for all 
[ x mk mk ∈− , +, so the above expressions are simplified as:  
1
1
[( ) ] i f ( ) (
2( 1 )
1[ ( ) ]
{[ 1 ( ) ] ( ) } i f 0 ()
2( 1 )
zw
mk y w mk z
kt t
P
zz w t m k w
mk y w mk z
kt z t
α
α
α
α
+
⎧ ++ − , −> ⎪ + ⎪ = ⎨ −− − ⎪ −+ + − < ,
⎪ + ⎩
6 )
( 7 ) − ≤
                                                
 (8) 
 
In this paper, we consider the case where the richest individuals in the economy are not 
poor:  . In this case, by (2) and (8), the poverty measure is further simplified 
as follows:  
() ywm k z ,− >
 
4The numerator of   is the income of the richest individual when the market is competitive.  c l
4   (1 )
2( 1 )
z
P
kt
α α
l
∗ = +−
+
 (9) 
The above expression tells us that, given   and t, the poverty measure solely depends on 
the amount of employment.  
k
 
3. The Optimal Minimum Wage 
Let us restrict attention to minimum wages higher than the market wage. If a minimum 
wage w is set, in any symmetric Nash equilibria, the marginal cost of labor becomes w up 
to  
 
() 1
[( )
2
Sw w
mk
nk n t
] = −−,  
and   for  2 ( 1) ( ) i kt n l m k t ++− () i lS w n ≥/ :
5  
 
for 0 ( ) (10)
()
2( 1 ) ( ) f o r () ( 1 1 )
i
ii
ii
wl S
MC l
kt n l m k t l S w n
≤≤ / ⎧
= ⎨ ++− ≥ / ⎩
w n
 (12) 
 
 
The amount of labor at which the original marginal cost is w is always strictly less than 
() Sw n / . So given any w, the amount of labor a firm could hire at the minimum wage 
always exceeds the amount of labor at which the original marginal cost is equal to the 
minimum wage. Thus the new marginal cost curve jumps at  () i lS w n = /  (Figure 2).  
So we can restrict our attention to the following three cases:   every firm hires the exact 
amount of labor available at the minimum wage,   part of the labor supplied at the 
minimum wage is hired, and   every firm hires more labor than available at the 
minimum wage (though this case never happens as is shown below).  
( ) i
( ) ii
( ) iii
First it can be easily seen that case   occurs if and only if the marginal revenue at  ( ) ii
() Sw n /  is less than w, in which case  
 
[2 ( )]
2
akn b m k t
ww
bk n t
+ −
>≡
+
  
                                                 
5Without government intervention, the marginal revenue and marginal cost functions in symmetric equilibria 
are, respectively,  () ii i MRl a b l =− () 2 ( 1 ) ( ) ii i MCl k tn l m kt = ++− and  for all  .  0 i l ≥
5 holds. Each firm hires () awb −/  so the total employment is () aw n b − / .  
Next, case   occurs if the marginal revenue at  ( ) i () i lS w n = /  is greater than w but the 
marginal cost at  () Sw n /  is (weakly) greater than w. After some calculations, we find that 
this case occurs iff  
  ww w
∗ < ≤,   
where   is the equilibrium wage without government intervention. Since we consider 
minimum wages beyond  , the first inequality always holds. Therefore case (  never 
occurs. The total employment in case (  is 
w
∗
w
∗ ) iii
) i () Sw. It should be noted that the marginal 
revenue at   is   (Figure 3).   ( ) i lS w n =/  w 
() i  ww w
∗ <≤    
Let us start with case  . In this case, if  () i w is set, all the labor supplied at the wage rate is 
hired. By (8), the degree of poverty is given by  
 
1
() [ ( ) ]
2( 1 )
zw
Pw mk
kt
α α t
= ++−
+
,  (13) 
which is clearly decreasing in w. Thus the higher the minimum wage, the lower the degree 
of poverty. Therefore poverty is minimized at ww =  .  
() ii  ww >    
Next, in case ( , since each firm hires   ) ii
 
()
() i
aw S w
lw
bn
−
=< 
total employment is () aw n b −/ . Let us assume that the individuals in [ mk x −,]  are hired, 
where  
 
2( )
()
kn a w
x mk
b
−
= +−  
Then the value of the poverty measure is given by  
 
1 () 1
() [ ] [ ( ) ]
2( 1 ) 2
zz w t x
Pw mk x
kt z k
α
α α
+ − −
=+
+
+ −  (14) 
Differentiating (14) with respect to w gives  
 
() 2(
[
2
Pw nb k n t z wt x
wb b k t z
)
]
α α ∂ + −−
=−
∂
 (15) 
6 Note that (13) and (14) take the same value at ww =  . So the poverty measure is continuous 
for all ww
∗ > . Because the value of (15) evaluated at ww =   is  
 
1
0
2
ww
P
wk t
α
=
∂
| =− <
∂
  
and because the second derivative of (14) with respect to w is positive, the poverty 
minimizing minimum wage is higher than  . In fact, it is high enough for involuntary 
unemployment to occur. The intuition behind this is as follows. First, for minimum wages 
in the range of ( , both employment and wages increase as the minimum wage increases. 
After a critical value, namely  , as the minimum wage increases, the associated level of 
employment begins to fall. While the minimum wage is near  , however, the effect of 
higher minimum wages dominates that of lower employment.  
w 
) i
w 
w 
The optimal minimum wage for poverty reduction can be obtained by equating (15) to 
zero:  
 
1 2
() [ 1 ( ) ]
22
bz knt
nk w w
b knt b knt
α ∗ , =+ − ,
++
  (16) 
where the second term of the right side is positive. The associated poverty level is given by  
 
1 [( ) ( ) ] 2
(( ) ) ( )
22 ( 1 ) 2
bn za mk t z k n t
Pn k w
bk n t k t bk n t
α
α
α
α
α
+ ∗ + −+ +
,= −
++ +
 (17) 
 
In the case of a constant marginal value product of labor ( 0) b = , the optimal minimum 
wage is equal to the competitive wage for any  ,  ,   and  . Otherwise, it depends on 
those variables.  
a n m k
Let  w Δ  denote the difference between the optimal minimum wage and the market wage:  
 
1
22 4[ () ] 2
()() [ 1 ( )
(2) [2 (1 ) ] 2 2
w
kn t a m kt b z k n t
nk wnk w
b knt b kt n b knt b knt
α ∗ −−
Δ≡ , − , = + −
++ + + +
]  
 
First, note that  
  lim 0 w n→∞Δ =  
holds, which means that the optimal minimum wage converges to the competitive wage as 
the number of firms approaches infinity. That in turn means that there is no need for 
government intervention in competitive labor markets.  
7 Second, we have  
 
22 4
0
(2) [2 (1 ) ]
w kn t
a b knt b kt n
∂Δ
= >
∂+ + +
 
So the difference between the poverty minimizing minimum wage and the prevailing 
market wage is greater the higher is labor productivity. Thus there is a greater need for 
government intervention the higher is labor productivity.  
Third, as   increases,  m w Δ  decreases:  
  0
w
m
∂Δ
<
∂
 
Therefore as individuals’ access to the market decreases, the gap between the optimal 
minimum wage and the market wage becomes smaller.  
 
4. The Effects of Competitiveness and Productivity 
Let us begin with the effect of an increase in competitiveness on the optimal minimum 
wage. By differentiating (16) with respect to  , we have   n
 
1 1
2
22 3
2[ ( ) ] 2 2 2 2
[1 ( ) ] ( )
(2) (2) 2 (2) 2
bkt a m k t bktz knt b ktz knt w
n b knt b knt b knt b knt b knt
α
α α
α
− ∗ ∂− −
=− − −
∂+ + + + +
 
The first term of the right side, which is the derivative of   with respect to  , is positive. 
This is clearly seen by Figure 3. With   held constant,   decreases as   increases. 
So   must increase as   increases. Next, (the sum of) the second and third terms, 
which is the derivative of the second term in the right side of (16), are negative. It should 
be remembered that as long as the minimum wage is in the range of 
w  n
w  ( ) Sw n /  n
( ) wn  n
[ w w
∗ ,  ] , the positive 
effect on poverty of increases in the minimum wage is greater than the negative effect of 
decreases in employment. Note that, as we have already seen, the total employment is 
given by () aw n b −/ , the derivative of which with respect to w decreases as   increases. 
In other words, for greater n, employment decreases more rapidly as 
n
w increases. So the 
optimum minimum wage becomes closer to  , which means that the derivative of the 
second term in the right side of (16) with respect to n is negative. Whether or not the 
optimal minimum wage increases as a result of an increase in   is determined by the trade-
w 
n
8 off between those two effects. For example, in the case of  1 α = , the optimal minimum 
wage falls as the degree of competitiveness rises if  
 
{2 [ ( ) ]}
2[ ( ) ]
bz a mk t
n
kt a m k t
− −−
<
−−
 
 
In contrast, the effect of an increase in productivity on the optimal minimum wage is 
determinate:  
  0 w
a
∗ ∂
>
∂
 
The above inequality holds because   always increases with increases in a, and because 
the marginal decrease in employment with an increase in the minimum wage (in the range 
of 
w 
ww ≥  ) is unchanged.  
In addition, we have  
 
2
0 w
an
∗ ∂
>
∂∂
 
We have seen that the optimal minimum wage does not necessarily increase with an 
increase in  . However, the marginal increase in the optimal minimum wage with an 
increase in the number of firms always increases as productivity increases. This is because 
 increases with an increase in productivity while 
n
wn ∂/ ∂  () w w
∗ n ∂ −/ ∂   does not depend on 
labor productivity.  
As for poverty, increases in productivity and competitiveness help decrease the minimized 
poverty:  
 
()
0
2
P n w
ab k n t
α
∗ ∂
= −<
∂+
 
 
 
1
2
() 2
{[ ( ) ] ( ) } 0
(2) 2
P bk w am k tz z
n b knt b knt
α α
∗ ∂
=− − − − + <
∂+ +
n t
z
 
The latter inequality holds because, by assumption, the richest in the economy (in the 
competitive market) is not poor:  () am k t − −> . The above results are intuitive because 
poverty decreases with increases in these variables without government intervention.  
Besides,  
9  
2
2
()
0
(2)
P b w
an b k n t
α
∗ ∂
= −<
∂∂ +
 
also holds. Therefore the higher market competitiveness (labor productivity), the greater 
the marginal decrease in poverty with an increase in productivity (competitiveness). As we 
have already seen,   is greater for greater   but  wn ∂/ ∂  a () w w
∗ n ∂ −/ ∂   does not depend on 
. Since employment increases as   increases, and decreases as the minimum wage 
increases in 
a w 
[ ww w
∗ ∈,  ] , we have the result.  
 
5. The Effects of Inequality 
Finally, let us explore how the optimal minimum wage and the minimized poverty change 
as the degree of inequality changes.  
The derivative of the optimal minimum wage with respect to   is   k
 
1
2
22
{2 [( ) ] } 2
()
(2) (2) 2
bn t am t z b t b z k n t w
k b knt k b knt b knt
α
α
∗ ∂− − −
=−
∂+ + +
 
For example, if the income of the individuals at the average distance from the firm in the 
competitive labor market, a , is less than the poverty line, the optimal minimum wage 
is lower the greater is the degree of inequality.  
m t −
Next, for the sake of simplicity, suppose that the revenue function is linear in terms of 
employment ( ). Then the derivative of the poverty measure with respect to   is given 
by  
0 b = k
  2
() 1
[( ) ]
21
P z w am t
kk t
α
α
∗ ∂
= −−
∂+
,  
where, again,   is the competitive income of the individuals at the average distance 
from the firms. If   satisfies  
am t −
m
  (0 )
(1 )
az
m
tt α
>− >,
+
 
that is, if the extent of poverty is greater than certain degree at the beginning, the 
minimized poverty is lower the higher is inequality. Otherwise, the level of the minimized 
poverty is higher the greater is the degree of inequality.  
 
10 6. Conclusion 
Most of the theoretical literature has conducted comparative statics exercises on the impact 
of minimum wages on employment. This paper derives the poverty minimizing wage and 
conducts comparative statics on that optimal wage with respect to labor market 
competitiveness, productivity and inequality. It is shown that under certain conditions the 
optimal minimum wage falls with the degree of competitiveness, rises with productivity, 
and falls with inequality. Moreover, it is shown that the optimal minimum wage rises more 
with productivity the greater is the degree of competitiveness. These comparative static 
results can guide us in understanding the pattern of minimum wage across societies with 
different degrees of competitiveness, productivity and inequality.  
 
References 
1.  Basu, Arnab K., Nancy H. Chau and Ravi Kanbur (2005), "Turning a Blind Eye: Costly 
Enforcement, Credible Commitment and Minimum Wage Laws", Department of 
Applied Economics and Management, Cornell University.  
2.  Fields, S. Gary, and Ravi Kanbur (2007), "Minimum Wages and Poverty with Income-
Sharing", Journal of Economic Inequality, Vol. 5, pp. 135-147.  
3.  Foster, James, Joel Greer and Erik Thorbecke (1984), "A Class of Decomposable 
Poverty Measures", Econometrica, Vol. 52, pp. 761-66.  
4.  Neumark, David, and William Wascher (2007), "Minimum Wages and Employment", 
IZA Discussion Paper, No. 2570.  
11 Figure 1. "Distance" to the Firms and Individual Incomes 
 
 
Figure 2. Minimum Wages and the Marginal Cost of Labor 
 
 
12 Figure 6.3.    w 
 
 
 
 
 
 
 
 
 
13 

The Optimal Minimum Wage for Poverty Minimization

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