ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E19-9

Robinson meets Weyl and Fabinger: Oligopsonyand exploitation in general equilibrium
byTakanori Adachi

September 2019

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Robinson meets Weyl and Fabinger: Oligopsony andexploitation in general equilibrium∗
Takanori Adachi†
August 29, 2019

Abstract
This paper proposes a general equilibrium model of Robinson’s (1933) oligopsony andexploitation in the labor market coupled with imperfect competition in the product market as well. By using Weyl and Fabinger’s (2013) conduct parameter approach, I examine
how a change in the degree of competition in the labor market or the product market
affects the equilibrium product price and the equilibrium wage.
Keywords: Oligopsony; Exploitation; General Equilibrium; Conduct Parameter Approach.
JEL classification: D43; J42; L13.

1Introduction

It has been increasingly recognized that oligopolistic firms may not only exercise their market
power in the product market but they also take advantage of their dominance in the labormarket as oligopsonists (Azar, Marinescu, and Steinbaum 2017; De Loecker, Eeckhout, andUnger 2018.).1 An unavoidable consequence is that competitive pressure on the price hikeremains weak (controlling for product quality), and what is worse, labor income does not growas much as the national growth rate. This disparity between corporations/business executives
and a majority of citizens may result in their disbelief in free market and free society in general.
∗I have benefited from comments by Takashi Hayashi and Naoki Yoshihara. I am also grateful to a Grant-inAid for Scientific Research (C) (18K01567) from the Japan Society of the Promotion of Science. All remainingerrors are my own.
†School of Economics, Nagoya University, 1 Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan. E-mail:adachi.t@soec.nagoya-u.ac.jp
1See, e.g., The Economist’s special report (November 15, 2018), “(Across the West powerful firms arebecoming even more powerful”). This issue was also extensively discussed at the 2nd and the 3rd hearings on“Competition and Consumer Protection in the 21st Century” organized by the US Federal Trade Commission
in September through December 2018.

1In this paper, I provide a tractable model of the simultaneous determination of the wage andthe product price under imperfectly competitive labor and product markets. To do so, I makeuse of the conduct parameter approach proposed by Weyl and Fabinger (2013). I not only model
the labor market as an imperfectly competitive market, and thereby generalize the concept ofmonopsony (Robinson 1933) to “oligopsony,” I also combine the imperfectly competitive labormarket (L) and the imperfectly competitive product markets (G) in a unified manner.
One of the merits in the conduct parameter approach is that in a symmetric setting, one canfocus on aggregate amount of employment without specifying the market structure in the labormarket such as the number of firms and the mode of competition. In other words, one does
not have to worry about how workers and firms are sorted and matched in the labor market.
This attractive feature of the conduct parameter approach enables one to model oligopsony ina broad manner, including perfect competition as one extreme and monopsony as the otherextreme.
One caveat at this point is that I do not consider search friction as a source of market
imperfection in the labor market. This issue has been extensively studied in the modern
literature on search theory (Pissarides 2000). In this paper, I instead focus on the exerciseof market power by oligopsonic firms in the labor market to extend Robinson’s (1933) theoryof monopsony to “oligopsony” by applying Weyl and Fabinger’s (2013) method of modelingimperfect competition to the labor market.2 Obviously, as an interesting extension, one would
incorporate search friction as another important source of labor market imperfection into themodel below.

2ModelI start with pricing by a coordinated group of (symmetric) single-product firms (i.e., a syndicate). For a while, it is regarded as a monopolist in the product market and a monopsonist in
the labor market until imperfect competition is explicitly introduced by the conduct parameter
approach (see below). This monopolist/monopsonist faces with two imperfectly competitivemarkets: the product market and the labor market. It uses (homogeneous) labor L > 0 as asole input to produce a single product and sells it directly to consumers. As Posner and Weyl(2018, Ch. 5) point out, data may be interpreted as labor input here in the context of IT firms.
Its profit function is written as π = pQ − wL in terms of L, where output Q is produced by
the production function Q = F (L), with the regularity conditions such as F 0 > 0, F 00 < 0,and limL↓0 F 00 = ∞ being satisfied, and p > 0 and w > 0 denote the product price and the(nominal) wage, respectively.
2See, e.g., Manning (2011) and Boeri and van Ours (2013) for existing studies on imperfectly competitive
labor markets in the tradition of Robinson (1933).

2Figure 1: Marginal Profit Gain and Loss from Raising Product Price p: Product Market

2.1

Product Market

To study oligoposony in general equilibrium in a simplified manner, I follow Azar and Vives(2018) and assume that the monopolist choose output, not the product price per se. In otherwords, oligopolistic firms compete in a Cournot manner in the product market. In this sense,
the following analysis is no longer a partial equilibrium analysis because the firms do not takep as given, but recognize that their decision on L affect the product price p. The monopolist’s
profit function is written as π = p(Q)Q − C(Q), where p(Q) is the industry’s inverse demand,and C(Q) = wLe =w
e · F −1 (Q) is the industry’s cost of production, and its optimal production
is summarized in Figure 1. Here, I denote w
e to emphasize that the wage is not a constant, andis an endogenous variable determined in the oligopsonic labor market.
The marginal profit loss by reducing quantity Q at a price level p is given by µ × ∆Q, whereupµ ≡ p − M C is the markup at price p, whereas the marginal profit gain as a result of theincrease in p is given by ∆p × Q. So far, the marginal cost of production is simply a function ofQ: M C(Q). However, it is now observed that the marginal cost is expressed by the terms in the0labor market: M C(Q) = we · (F −1 ) (Q), where F denotes the industry’s production function:
Q = F (L). Now, the monopolist chooses to equate these gains and loss:(∆p) × Q = −µup × (∆Q).
{z}| {z } |Maginal Gain

Marginal Loss3

Now, I introduce imperfect competition in the product market by Weyl and Fabinger’s (2013)
conduct parameter approach. If the product market is imperfectly competitive, firms recognizethat their products are not perfectly substitutable as in perfect competition nor they are able
to seize all marginal gains from raising a price as in monopoly. In other words, they recognizethat they only capture a 100 × θG percent of the marginal gains, where θG ∈ [0, 1] is the conduct
parameter in the product market. Thus, in the imperfectly competitive product market with
θG , the equilibrium output Q solves p(Q) − M C(Q) = −p0 (Q)QθG . This is rewritten in termsof L as:
we= −p0 [F (L)]F (L)θG ,(1)pb(L) − 0F (L)where pb(L) ≡ p[F (L)] (note that p0 [F (L)] is not replaced by pb0 (L)).

2.2

Labor Market

Now, I obtain w
e = H(L; θG ) as a solution of Equation (1) for w.
e This expresses the firm’s
willingness-to-pay or reservation value for the wage in the labor market. It is readily verified∂H00
that ∂H< 0 and ∂θ
G < 0 if the inverse demand is not too convex (that is, p is not too large a∂Lpositive value) so thatwF 000(1 + θG )p0 · F 0 + 0 2 + θG · p00 · F·F| {z } [F ]|{z} | {z }(+)| {z }(−)(+) or (−)(−)

is negative. Workers also recognize that the product price p results from imperfect competitionin the product market. They maximize their utility u(C, L) by choosing their consumption
C = Q and labor supply L, with the budget constraint pb(L)C = wL. Thus, workers choose LwLto maximize u( ρ(L), L), resulting in their labor supply function, LS = LS (w). More precisely,
LS is obtained by solving:∂u[C(L; w), L]w·∂Cdbp
L · dL(L)
1−pb(L)[bp(L)]2!
+∂u[C(L; w), L] = 0,
∂L

where C(L; w) ≡ w · [L/bp(L)]. Thus, I can also define its inverse labor supply function, w =(LS )−1 (L). Figure 2 shows how w
e = H(L; θG ) and w = (LS )−1 (L) look like.
I also introduce imperfect competition in the labor market, using Weyl and Fabinger’s (2013)
conduct parameter approach. If the firm is a monopsonist in the sense of Robinson (1993), thefirm behaves as Stackelberg leader, and chooses the nominal wage w to maximize R[LS (w)] −wLS (w), where R(L) ≡ pb(L)F (L) denotes the firm’s revenue in terms of labor input, and theresulting employment and wage are LM and wM , respectively, as depicted in Figure 2. On
the contrary, if the labor market is perfectly competitive, the equilibrium employment LC is4

Figure 2: Wage Determination under Oligopsonydetermined by H(L; θG ) = [LS ]−1 (L), with the associated wage wC in Figure 2. However, underoligopsony, symmetrically oligopsonic firms recognize that they are not perfectly substitutable
as in perfect competition nor they do not incur all marginal profit losses from raising thewage w, as not all workers are employed in the same firm. Specifically, they recognize thatthey only incur a 100 × θL percent of the marginal profit losses by raising the nominal wage,
where θL ∈ [0, 1] is the conduct parameter in the labor market. Thus, the equilibrium wage isdetermined by the following margin condition:
θL × (∆w) × L = µdown × (∆L),|{z} |{z}Marginal Loss

Maginal Gain

where µdown ≡ H[LS (w); θG ] − w denotes the markdown at employment L = LS (w). In theterminology of Robinson (1933, Ch. 25), exploitation in the labor market is defined by the caseof µdown > 0. More formally, the equilibrium nominal wage w∗ (θL , θG ) is determined by
θL LS (w) = {H[LS (w); θG ] − w}
which implies that∂w∗∂w∗< 0 and< 0,∂θL∂θG
5dLS(w),dw(2)that is, the nominal wage w∗ decreases as the labor market or the product market becomes lesscompetitive (an increase in θL or θG ) , if∂H dLS dLS
d2 L SL·θ +1−− µdown (w, θG ) ·> 0,
∂L dw
dw
dw2
|{z}
(+)

that is, if LS (w) is not too convex.
Now, define the real wage by ŵ ≡ w/bp[LS (w)]. If the labor competition becomes lesscompetitive, it not only has a direct effect in labor market to lower employment and nominalwage as seen a above, it also has an additional effect on the product market: less employment
leads to less output, which raises the product price pb∗ ≡ p[F [LS (w∗ )]]. More formally, the effectof a change in the degree of imperfect competition in the labor market on the equilibrium realwage, ŵ∗ , is described by∂wb∗=∂θL!
S00w∗ · p · F · dL
∂w∗
1dw
−
<0pb∗
(bp∗ )2∂θL|{z}|{z} (−)(+)

0because p < 0. Now, workers as consumers may lose because their labor income, which is equal
to their consumption. As above, the real consumption is defined by bc ≡ [wLS (w)]/p̂[LS (w)].
Then, the utility change by the staggered labor market is captured by∂u dbc∗∂u∂u dLS ∂w∗=[+] L ≷ 0,
∂C |{z}
dw
∂θL
∂θ
|{z}
|∂L{zdw} |{z}(+)

where(+)

(−)

(−)


00
L∗1L∗ · p · F dLSdbc∗∗= ∗ +w ·−> 0.
dwpbpb∗(bp∗ )2dw|{z}|{z}
|{z}
(+)

(+)

(+)

Finally, it is also observed that as concentration rises in the product market, the equilibrium∗wage decreases: ∂∂θwbG < 0.

3

Separable Preferences

To proceed further, I assume additive separability of utility function: u(C, L) = uC (C)−uL (L),where uC and uL are both increasing functions; recall that ∂u/∂L < 0. Notice that the net
surplus for the workers/consumers is u(C ∗ , L∗ ). Under the separability, this net surplus is6

Figure 3: Labor and Product Markets under Imperfect Competition
decomposed into two parts:u(C ∗ , L∗ ) − p∗ C ∗ + w∗ L∗ = [uC (Q∗ ) − p∗ Q∗ ] + [w∗ L∗ − uL (L∗ )],|{z} |{z}Consumer Surplus

Worker Surplus

RL 0e − w]dLe
which I call the united surplus (US). Owner surplus is simply defined by π = 0 [R (L)in terms of employment. Owner surplus can also be defined from a viewpoint of production:
RQe Q.
e Figure 3 is a graphical exposition of OS (owner surplus), WS (workerπ = 0 [p − M C(Q)]dsurplus), and CS (consumer surplus).
Finally, I consider a tractable parameterization for production and utility functions. First,the utility function is given by1εu(C, L) =C 1− ε −ε−1
1 2L + wL ,2

where ε > 1 is the (constant) price elasticity of the demand in the product market, and w ≥ 0 is
the subsistence wage. On the other hand, the production function is given by Q = F (L) = Lα ,7

where α < 1 (decreasing returns). Then, the inverse demand function exhibits the constantα1elasticity: p(Q) = Q− ε , which implies that pb(L) = L− ε . The labor supply function is given byLS (w) =0if w < w

w − w

if w ≥ w,

whereas the firm’s willingness-to-pay for the wage is given by
θG
H(L; θ ) = α 1 −εG

L−α+(1−α)ε,from Equation (1). Equation (2) indicates that the equilibrium nominal wage w∗ (θL , θG ) isobtained by solving:α+(1−α)εθG(w∗ − w)−  .
(1 + θ )w − θ w = α 1 −εL∗LIf w = 0, the equilibrium nominal wage is expressed byLε− α+(2−α)ε1+θw∗ =  G
α 1 − θε.

It is observed that while an increase in α (an improvement of productivity) or in ε (a less degreeof product differentiation) raises w∗ , a higher degree of θG reduces w∗ .

References
Azar, José, Ioana Marinescu, and Marshall I. Steinbaum. 2017. “Labor Market Concentration.”Unpublished.
Azar, José, and Xavier Vives. 2018. “Oligopoly, Macroeconomic Performance, and Competition
Policy.” Unpublished.
Boeri, Tito, and Jan van Ours. 2013. The Economics of Imperfect Labor Markets, 2nd ed.
Princeton University Press.
De Loecker, Jan, Jan Eeckhout, and Gabriel Unger. 2018. “The Rise of Market Power and theMacroeconomic Implications.” Unpublished.
Manning, Alan. 2011. “Imperfect Competition in the Labor Market.” In David Card and Orley
Ashenfelter (eds.), Handbook of Labor Economics, Volume 4, Part B, Elsevier B.V., 973-1041.
8

Pissarides, Christopher A. 2000. Equilibrium Unemployment Theory, 2nd ed. The MIT Press.
Posner, Eric A., and E. Glen Weyl. 2018. Radical Markets: Uprooting Capitalism and Democracy for a Just Society. Princeton University Press.
Robinson, Joan. 1933. The Economics of Imperfect Competition. Macmillan.
Weyl, E. Glen, and Michal Fabinger. 2013. “Pass-Through as an Economic Tool: Principles ofIncidence under Imperfect Competition.” Journal of Political Economy, 121(3), 528-583.

9