Monopsonistic wage discrimination and employment effect under conditions of constant labor supply elasticit

Shieh, Yeung-Nan

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San Jose State UniversitySJSU ScholarWorksFaculty Publications Economics1-1-2006Monopsonistic Wage Discrimination andEmployment Effect under Conditions of ConstantLabor Supply ElasticityYeung-Nan ShiehSan Jose State University, yeung-nan.shieh@sjsu.eduFollow this and additional works at: https://scholarworks.sjsu.edu/econ_pubPart of the Economics CommonsThis Article is brought to you for free and open access by the Economics at SJSU ScholarWorks. It has been accepted for inclusion in FacultyPublications by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu.Recommended CitationYeung-Nan Shieh. "Monopsonistic Wage Discrimination and Employment Effect under Conditions of Constant Labor SupplyElasticity" Economics Bulletin (2006): 1-4.Monopsonistic wage discrimination and employment effect under conditions of constant labor supply elasticity Yeung-Nan Shieh Department of Economics, San Jose State University Abstract This note exploits an alternative but simple way to examine the employment effect of wage discrimination when the constant elasticity labor supply curves are strictly concave. The Bernoulli inequality applied in this paper allows us to show that wage discrimination increases total employment in a relative simple way, without resorting to complicated manipulations as were used by Formby, Layson and Smith (1983) and Shieh (2001). I am grateful to one excellent anonymous referee for very valuable comments and suggestions. Responsibility for any remaining errors is strictly mine. Citation: Shieh, Yeung-Nan, (2006) "Monopsonistic wage discrimination and employment effect under conditions of constant labor supply elasticity." Economics Bulletin, Vol. 4, No. 32 pp. 1-4 Submitted: September 1, 2006. Accepted: October 3, 2006. URL: http://economicsbulletin.vanderbilt.edu/2006/volume4/EB-06D40013A.pdf                                                                                                                                                                   1. Introduction Formby, Layson and Smith (henceforth FLS) in their well-known 1983 paper utilized Lagrangean techniques to examine the output effect of monopolistic third degree price discrimination under constant demand elasticity conditions.  They also examined the employment effect of monopsonistic third degree wage discrimination under constant labor supply elasticity conditions.  Recently, in this journal, Aguirre (2006) provided a much simple approach, the Bernoulli inequality, to show FLS’s result that monopolistic price discrimination increases total output under constant demand elasticity if the demand curves are strictly concave.  However, Aguirre didn’t consider the impact of wage discrimination on total employment.  It would be interesting and important to apply the Bernoulli inequality to investigate the employment effect of wage discrimination.  The purpose of this paper is to fill this gap.  We will set up a monopsony model with nlabor markets and examine the effect of wage discrimination on total employment by using the Bernoulli inequality.  It will be shown that monopsonistic wage discrimination increases total employment if the labor supply curves belong to the class of strictly concave and constant elasticity. 2. Analysis Consider a monopsonistic firm which uses a single input, labor, to produce a product that was sold in a perfectly competitive market, e.g., FLS (1982, 1983) and Sandmo (1994). Assume also that the firm hires workers from all n segregated markets. The labor supply function in market i (i = 1, 2, 3,..., n) has constant elasticity and is given by Li(wi) = aiwi e i where wi = wage rate, ei = constant elasticity, ai > 0 and ei > 1. If the third degree wage discrimination is allowed, the firm’s problem in market i is: max πi = pq – wiLi where q = f(L) = ΣLi, p is determined at the perfectly competitive product market n ΣLi = L (total employment) and Σ ≡ Σ . For simplicity, we drop the index throughout the i=1 paper. It should be noted that the perfectly competitive output market and the fixedcoefficient production function assumptions are not too restrictive. Ekelund, Higgins and Smithson pointed out that marginal revenue product is independent of monopsony’s ability to discriminate, (1981, 665).  FLS maintained that “the slope of MRP curve is directly pertinent only to the magnitude of the employment change but not to thedirection of that change.” (1982, footnote 1, 551).  Shieh (1995) showed Ekelund, Higgins and Smithson’s (1981) and FLS’s (1982) assertion mathematically. Following Boal and Ransom (1997, p. 87), via the first order condition, we obtain thePigou index in market i is inversely proportional to ei, i.e., [(p – wi*)/wi*] = (1/ei), i = 1, 2, 3, …, n (1) Solving (1), we obtain the optimal wage in market i wi* = p/[1 + (1/ei)], i = 1, 2, 3, …, n (2) 1                                                                                                                                                                                                                                                             Thus, the monopsonistic firm will offer a higher wage in the market with the higherelasticity of labor supply.  The employment in market i will be Li* = ai[pei/(1 + ei)]e i , i = 1, 2, 3, …, n (3) and total employment under wage discrimination would be L* = ΣLi* = Σ ai[pei/(1 + ei)]e i  (4) Under simple monopsony, the firm’s problem is: max π = pL – wL. Via the first order condition, the Pigou index is [(p – wo)/wo] = [1/e(wo)] (5) where e(wo) = the elasticity of the aggregate supply of labor at wo . Since L = Σ  Li, the weighted average of the elasticities of individual markets, e(wo), can be written as: e(wo) = [Σ Li’(wo)][wo/ΣLi(wo)] = Σeiai(wo)e i /Σai(wo)e i  = Σωi(wo)ei  (6) where Li’(wo) is dL(w)/dw at w = wo and ωi(wo) = Li(wo)/ ΣLi(wo) is the share of market i in total employment at the optimal uniform wage (wo). For convenience and without loss of generality, following FLS (1983), we assume that the units of employment (L) are defined so that wo = 1 and p = 1+ (1/e). With this convention, (6) can be rewritten as: e(1) = Σeiai/Σai  (7) Substituting (7) into p = 1 + [1/e(1)], we obtain p = Σai(1 + ei)/ Σaiei  (8) Further, we can obtain employment in market i, Li o = ai and total employment Lo = ΣLi o = Σai. The effect of wage discrimination on total employment can be obtained by comparing L* and Lo .  If L* > Lo, we can conclude that wage discrimination increases total employment.  Following Aguirre (2006) and Galera and Zaratiegui (2006), we will use the Bernoulli inequality to show that L* > Lo . According to the Bernoulli inequality, “ if– 1 < x ≠ 0 and a >1 are real values, then (1 + x)a > 1 + ax”, Mitrinovic (1970, p. 34). Let (1 + x) = wi* = [pei/(1 + ei)] and a = ei > 1. The Bernoulli inequality implies that [pei/(1 + ei)]e i  > 1 + ei{[pei/(1 + ei)] – 1} (9) Given (8) we have [pei/(1 + ei)] – 1 = (eiΣai -Σaiei)/(1+ei)Σaiei  (10) 2                                                                                                                                                                                                                                     Substituting (10) into (9), we obtain [pei/(1 + ei)]e i > 1 + ei[(eiΣai -Σaiei)/(1+ei)Σaiei] (11) Multiplying both side of (11) by ai and then summing i from 1 to n, we obtain L* =ΣLi* = Σai[pei/(1 + ei)]e i  > Lo + Σaiei[(eiΣai -Σaiei)/(1+ei)Σaiei] (12) where Lo = Σai . It is clear that L* > Lo if the last term in (12) is non-negative, i.e., Σ{[aiei/(1 + ei)](eiΣai -Σaiei)}/Σaiei] = (1/Σaiei){Σ[aiei 2/(1 + ei)]Σai - Σ[aiei/(1 + ei)] Σaiei)} > 0 (13) Since (Σaiei) > 0, we have to check whether A = {Σ[aiei 2/(1 + ei)]Σai - Σ[aiei/(1 + ei)] Σaiei)}, i = 1, 2, 3,…, n is non-negative or not.    By using the mathematical induction method, we first obtain  A(n = 2) = [1/(1 + e1)(1 + e2)]a1a2(e1 – e2)2 > 0 (14) Next, we have A(n + 1) = A(n) + an+1{Σ[ai(ei – en+1)2/(1 + en+1)(1 + ei)]} (15) Since the last term in (15) is positive, it is easy to see that if A(n) > 0 then A(n + 1) > 0. This shows that the sign of A is positive for any n ≥ 2. Thus L* > Lo, i.e., wagediscrimination increases total employment if the labor supply curves belong to the class of strictly concave and constant labor supply elasticity.  This result is consistent with FLS (1983, p. 896 and p. 898) and Shieh (2001, p. 186). 3. Concluding remarks We have attempted to exploit an alternative but much simple way to examine the effect of wage discrimination on total employment.  Following Aguirre (2006) and Galera and Zaratiegui (2006), we utilize the Bernoulli inequality to show that the wage discrimination always increases total employment if the labor supply curves are strictly concave and have constant supply elasticity. We obtain this result with a much simpleway without resorting to complicated manipulations as were used by FLS (1983) and Shieh (2001). 3                       ReferencesAguirre, I. (2006) “Monopolistic price discrimination and output effect under conditionsof constant elasticity demand” Economics Bulletin 4, 23, 1-6. Boal, W. M. and Ransom, M. R. (1997) “Monopsony in the labor market” Journal of Economic Literature 35, 86-112. Ekelund, R. B., Jr., Higgins, R. S. and Smithson, C. W. (1981) “Can discrimination increase employment? a neoclassical perspective” Southern Economic Journal 47, 664-73. Formby, J. P., Layson, S. and Smith, W. J. (1982) “Discriminatory changes in employment: comment” Southern Economic Journal 48, 550-555. Formby, J. P., Layson, S. and Smith, W. J. (1983) “Price discrimination, adjusted concavity, and output change under conditions of constant elasticity” Economic Journal 93, 892-899. Galera, F. and Zaratiegui, J. M. (2006) “Welfare and output in third-degree price discrimination: a note” International Journal of Industry Organization 24, 605-611. Mitrinovic, D. S. (1970) Analytical Inequalities. Springer – Verlag: New York. Sandmo, A. (1994) “Monopsonistic wage discrimination, incentives and efficiency” Labour Economics 1, 151-170. Shieh, Y. N. (1995) “A note on employment under wage discrimination” Rivista Internazionale di Scienze Economiche e Commerciali 42, 383-390. Shieh, Y. N. (2001) “Employment effects of third degree wage discrimination” International Review of Economics and Business 48, 179-188. 4 

Monopsonistic Wage Discrimination and Employment Effect under Conditions of Constant Labor Supply Elasticity

Yeung-nan Shieh

CiteSeerX

San Jose State University

San Jose State University
SJSU ScholarWorks
Faculty Publications Economics
1-1-2006
Monopsonistic Wage Discrimination and
Employment Effect under Conditions of Constant
Labor Supply Elasticity
Yeung-Nan Shieh
San Jose State University, yeung-nan.shieh@sjsu.edu
Follow this and additional works at: https://scholarworks.sjsu.edu/econ_pub
Part of the Economics Commons
This Article is brought to you for free and open access by the Economics at SJSU ScholarWorks. It has been accepted for inclusion in Faculty
Publications by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu.
Recommended Citation
Yeung-Nan Shieh. "Monopsonistic Wage Discrimination and Employment Effect under Conditions of Constant Labor Supply
Elasticity" Economics Bulletin (2006): 1-4.
Monopsonistic wage discrimination and employment effect 
under conditions of constant labor supply elasticity 
Yeung-Nan Shieh 
Department of Economics, San Jose State University 
Abstract 
This note exploits an alternative but simple way to examine the employment effect of wage 
discrimination when the constant elasticity labor supply curves are strictly concave. The 
Bernoulli inequality applied in this paper allows us to show that wage discrimination 
increases total employment in a relative simple way, without resorting to complicated 
manipulations as were used by Formby, Layson and Smith (1983) and Shieh (2001). 
I am grateful to one excellent anonymous referee for very valuable comments and suggestions. Responsibility for any remaining 
errors is strictly mine. 
Citation: Shieh, Yeung-Nan, (2006) "Monopsonistic wage discrimination and employment effect under conditions of constant 
labor supply elasticity." Economics Bulletin, Vol. 4, No. 32 pp. 1-4 
Submitted: September 1, 2006. Accepted: October 3, 2006. 
URL: http://economicsbulletin.vanderbilt.edu/2006/volume4/EB-06D40013A.pdf 
      
 
 
 
 
 
  
 
 
     
 
 
 
 
 
                                                                 
                                                                        
1. Introduction 
Formby, Layson and Smith (henceforth FLS) in their well-known 1983 paper utilized 
Lagrangean techniques to examine the output effect of monopolistic third degree price 
discrimination under constant demand elasticity conditions.  They also examined the 
employment effect of monopsonistic third degree wage discrimination under constant 
labor supply elasticity conditions.  Recently, in this journal, Aguirre (2006) provided a 
much simple approach, the Bernoulli inequality, to show FLS’s result that monopolistic 
price discrimination increases total output under constant demand elasticity if the demand 
curves are strictly concave.  However, Aguirre didn’t consider the impact of wage 
discrimination on total employment.  It would be interesting and important to apply the 
Bernoulli inequality to investigate the employment effect of wage discrimination.  
The purpose of this paper is to fill this gap.  We will set up a monopsony model with n
labor markets and examine the effect of wage discrimination on total employment by 
using the Bernoulli inequality.  It will be shown that monopsonistic wage discrimination 
increases total employment if the labor supply curves belong to the class of strictly 
concave and constant elasticity. 
2. Analysis 
Consider a monopsonistic firm which uses a single input, labor, to produce a product that 
was sold in a perfectly competitive market, e.g., FLS (1982, 1983) and Sandmo (1994). 
Assume also that the firm hires workers from all n segregated markets. The labor supply 
function in market i (i = 1, 2, 3,..., n) has constant elasticity and is given by Li(wi) = 
aiwi e i where wi = wage rate, ei = constant elasticity, ai > 0 and ei > 1. If the third degree 
wage discrimination is allowed, the firm’s problem in market i is: max πi = pq – wiLi 
where q = f(L) = ΣLi, p is determined at the perfectly competitive product market 
n 
ΣLi = L (total employment) and Σ ≡ Σ . For simplicity, we drop the index throughout the 
i=1 
paper. It should be noted that the perfectly competitive output market and the fixed
coefficient production function assumptions are not too restrictive. Ekelund, Higgins and 
Smithson pointed out that marginal revenue product is independent of monopsony’s 
ability to discriminate, (1981, 665).  FLS maintained that “the slope of MRP curve is 
directly pertinent only to the magnitude of the employment change but not to the
direction of that change.” (1982, footnote 1, 551).  Shieh (1995) showed Ekelund, 
Higgins and Smithson’s (1981) and FLS’s (1982) assertion mathematically. 
Following Boal and Ransom (1997, p. 87), via the first order condition, we obtain the
Pigou index in market i is inversely proportional to ei, i.e., 
[(p – wi*)/wi*] = (1/ei), i = 1, 2, 3, …, n (1) 
Solving (1), we obtain the optimal wage in market i 
wi* = p/[1 + (1/ei)], i = 1, 2, 3, …, n (2) 
1
 
 
 
 
 
 
 
    
 
 
 
 
 
 
 
 
    
 
 
 
 
 
 
                                                                    
                                                                                          
                                                                  
Thus, the monopsonistic firm will offer a higher wage in the market with the higher
elasticity of labor supply.  The employment in market i will be
 Li* = ai[pei/(1 + ei)]e i , i = 1, 2, 3, …, n (3) 
and total employment under wage discrimination would be 
L* = ΣLi* = Σ ai[pei/(1 + ei)]e i  (4) 
Under simple monopsony, the firm’s problem is: max π = pL – wL. Via the first order 
condition, the Pigou index is 
[(p – wo)/wo] = [1/e(wo)] (5) 
where e(wo) = the elasticity of the aggregate supply of labor at wo . Since L = Σ  Li, the 
weighted average of the elasticities of individual markets, e(wo), can be written as: 
e(wo) = [Σ Li’(wo)][wo/ΣLi(wo)] = Σeiai(wo)e i /Σai(wo)e i  = Σωi(wo)ei  (6) 
where Li’(wo) is dL(w)/dw at w = wo and ωi(wo) = Li(wo)/ ΣLi(wo) is the share of market i 
in total employment at the optimal uniform wage (wo). For convenience and without loss 
of generality, following FLS (1983), we assume that the units of employment (L) are 
defined so that wo = 1 and p = 1+ (1/e). With this convention, (6) can be rewritten as: 
e(1) = Σeiai/Σai  (7) 
Substituting (7) into p = 1 + [1/e(1)], we obtain 
p = Σai(1 + ei)/ Σaiei  (8) 
Further, we can obtain employment in market i, Li o = ai and total employment Lo = ΣLi o = 
Σai. 
The effect of wage discrimination on total employment can be obtained by comparing 
L* and Lo .  If L* > Lo, we can conclude that wage discrimination increases total 
employment.  Following Aguirre (2006) and Galera and Zaratiegui (2006), we will use 
the Bernoulli inequality to show that L* > Lo . According to the Bernoulli inequality, “ if
– 1 < x ≠ 0 and a >1 are real values, then (1 + x)a > 1 + ax”, Mitrinovic (1970, p. 34). Let 
(1 + x) = wi* = [pei/(1 + ei)] and a = ei > 1. The Bernoulli inequality implies that 
[pei/(1 + ei)]e i  > 1 + ei{[pei/(1 + ei)] – 1} (9) 
Given (8) we have 
[pei/(1 + ei)] – 1 = (eiΣai -Σaiei)/(1+ei)Σaiei  (10) 
2
 
  
 
 
 
  
 
         
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                      
                     
 
                            
                                                     
                                    
Substituting (10) into (9), we obtain 
[pei/(1 + ei)]e i > 1 + ei[(eiΣai -Σaiei)/(1+ei)Σaiei] (11) 
Multiplying both side of (11) by ai and then summing i from 1 to n, we obtain 
L* =ΣLi* = Σai[pei/(1 + ei)]e i  > Lo + Σaiei[(eiΣai -Σaiei)/(1+ei)Σaiei] (12) 
where Lo = Σai . It is clear that L* > Lo if the last term in (12) is non-negative, i.e., 
Σ{[aiei/(1 + ei)](eiΣai -Σaiei)}/Σaiei] = (1/Σaiei){Σ[aiei 2/(1 + ei)]Σai
 - Σ[aiei/(1 + ei)] Σaiei)} > 0 (13) 
Since (Σaiei) > 0, we have to check whether A = {Σ[aiei 2/(1 + ei)]Σai - Σ[aiei/(1 + ei)] 
Σaiei)}, i = 1, 2, 3,…, n is non-negative or not. 
   By using the mathematical induction method, we first obtain  
A(n = 2) = [1/(1 + e1)(1 + e2)]a1a2(e1 – e2)2 > 0 (14) 
Next, we have 
A(n + 1) = A(n) + an+1{Σ[ai(ei – en+1)2/(1 + en+1)(1 + ei)]} (15) 
Since the last term in (15) is positive, it is easy to see that if A(n) > 0 then A(n + 1) > 0. 
This shows that the sign of A is positive for any n ≥ 2. Thus L* > Lo, i.e., wage
discrimination increases total employment if the labor supply curves belong to the class 
of strictly concave and constant labor supply elasticity.  This result is consistent with FLS 
(1983, p. 896 and p. 898) and Shieh (2001, p. 186). 
3. Concluding remarks 
We have attempted to exploit an alternative but much simple way to examine the effect 
of wage discrimination on total employment.  Following Aguirre (2006) and Galera and 
Zaratiegui (2006), we utilize the Bernoulli inequality to show that the wage 
discrimination always increases total employment if the labor supply curves are strictly 
concave and have constant supply elasticity. We obtain this result with a much simple
way without resorting to complicated manipulations as were used by FLS (1983) and 
Shieh (2001). 
3
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
References
Aguirre, I. (2006) “Monopolistic price discrimination and output effect under conditions
of constant elasticity demand” Economics Bulletin 4, 23, 1-6. 
Boal, W. M. and Ransom, M. R. (1997) “Monopsony in the labor market” Journal of 
Economic Literature 35, 86-112. 
Ekelund, R. B., Jr., Higgins, R. S. and Smithson, C. W. (1981) “Can discrimination 
increase employment? a neoclassical perspective” Southern Economic Journal 47, 
664-73. 
Formby, J. P., Layson, S. and Smith, W. J. (1982) “Discriminatory changes in 
employment: comment” Southern Economic Journal 48, 550-555. 
Formby, J. P., Layson, S. and Smith, W. J. (1983) “Price discrimination, adjusted 
concavity, and output change under conditions of constant elasticity” Economic 
Journal 93, 892-899. 
Galera, F. and Zaratiegui, J. M. (2006) “Welfare and output in third-degree price 
discrimination: a note” International Journal of Industry Organization 24, 605-611. 
Mitrinovic, D. S. (1970) Analytical Inequalities. Springer – Verlag: New York. 
Sandmo, A. (1994) “Monopsonistic wage discrimination, incentives and efficiency” 
Labour Economics 1, 151-170. 
Shieh, Y. N. (1995) “A note on employment under wage discrimination” Rivista 
Internazionale di Scienze Economiche e Commerciali 42, 383-390. 
Shieh, Y. N. (2001) “Employment effects of third degree wage discrimination” 
International Review of Economics and Business 48, 179-188. 
4
 


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