ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E19-6

Profit Sharing and the Imperfectly Competitive
Product Market: A Conduct Parameter ApproachbyTakanori Adachi

August 2019

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Profit Sharing and the Imperfectly Competitive ProductMarket: A Conduct Parameter Approach∗Takanori Adachi†
August 12, 2019

Abstract
In the framework of Weyl and Fabinger (2013), I analyze how profit sharing between
managers and workers affects the product price and the division of surplus under imperfectcompetition. I discuss a fundamental conflict of interest between workers/consumers andmanagers as to corporate policy and competition policy.
Keywords: Imperfect Competition; Conduct Parameter Approach; Profit Sharing.
JEL classification: D43; J52; L13.

1Introduction

In the standard partial equilibrium analysis, corporate profit is the objective that is maximizedby the choice of the product price. However, corporate profit may be divided by managers
and workers for incentive reasons to weaken internal moral hazard (Shapiro and Stiglitz 1984
and Okuno-Fujiwara 1987). This note takes into account such a scheme of corporate profitsharing (Meade 1972 and Weitzman 1984) in the model of imperfect competition in the productmarket, and analyses the effects of internal bargaining on the product price and the divisionof surplus. In particular, I point out that while an increase in the managers’ bargaining power
is associated with an increase in consumer surplus, consumers/workers can become worse off
because their surplus as workers becomes smaller. This implies a fundamental tension between
consumers/workers and managers as to directions in corporate policy changes. However, itappears that the well-known tension between consumers and producers toward competition∗I am grateful to a Grant-in-Aid for Scientific Research (C) (18K01567) from the Japan Society of thePromotion of Science. All remaining errors 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.jp1policy does not change: consumers/workers support competition policy, whereas managers arealways tempted to be engaged in anti-competitive conduct.

2ModelWe start with pricing by a single-product monopolist. The aggregate market demand is denotedby Q(p), where p ≥ 0 is the market price. This demand is derived from the representative
consumer problem, which we do not spell out for brevity. The monopolist’s profit function iswritten as π = pQ(p) − C(Q), where C(Q) is the industry’s cost of production.
The marginal profit gain by lowering price p at an output level Q is given by m × ∆Q, wherem ≡ p − M C is the markup, whereas, whereas the profit loss is given by −∆p × Q. If the firmchooses to maximize its profit, it equates the marginal profit gain with the marginal profit loss:m × (∆Q) = (−∆p) × Q.
{z}
| {z } |Marginal Gain

Maginal Loss

Here, without loss of generality, we make a unit normalization such that the amount of laborfor producing Q is L = Q, and the unit wage is w≥ 0. In this paper, we do not provide adetailed mechanism that describes the determination of w. In the phrase of Aoki (1988, p.152)
when describing Weitzman’s (1984) “share system,” the monopolist has “the unilateral power
to set the level of employment according to the profit-maximizing motive.” Thus, the marginal
ddcost has two parts: M C(Q) = w + M
C(Q), where M
C(Q) stands for the non-labor marginal
costs.
Now, we introduce imperfect competition in the product market, using Weyl and Fabinger’s(2013) conduct parameter approach. Specifically, symmetrically oligopolistic firms recognizethat their products are not perfectly substitutable as in perfect competition nor they do notincur all marginal profit losses from lowering a price as in monopoly. In other words, theyrecognize that they only incur a 100 × θ percent of the marginal profit losses by lowering
the price, where θ ∈ [0, 1] is the conduct parameter in the product market. Additionally, weintroduce profit sharing in the internal organization of the firm. Specifically, the marginal
profit gains from lowering the price are shared by the manager and the workers, and thus themanager obtains a 100 × λ percent of the marginal gains, where λ ∈ [0, 1] is the manager’s
bargaining power. Now, with these two modifications, the symmetric-firm “syndicate” equates
the marginal profit gains with the marginal profit losses (see Figure 1):(λ · m) × (∆Q) = θ × (−∆p) × Q,{z} |{z}|Marginal Gain

Maginal Loss2

Figure 1: Marginal Profit Gain and Loss from Lowering the Product Price p
or in the form of the Learner formula,p − MCθ=,pελpwhere ε(p) ≡ − Q(p)Q0 (p) is the industry-level price elasticity. The surplus division under theprofit sharing rule is depicted in Figure 2: we define the united surplus (US) by the sumof consumer surplus (CS) and worker surplus (WS), and the manager surplus (MS) by theproducer surplus subtracted by the worker surplus. Social welfare (SW) is defined by: SW =(CS + W S) + M S = U S + M S. Obviously, if λ = 1, the situation is nothing but the ordinary
setting (i.e., the no-profit-sharing economy), where W S = 0 and M S is interpreted as producer
surplus (PS).
Now, it is shown that

∂pQ=−>000∂θ{θ + λ · [1 − mc · Q ]}Q0 + λ{p − M C[Q(p)]}Q00and∂p(p − M C)Q0=−<0∂λ{θ + λ · [1 − mc0 · Q0 ]}Q0 + λ{p − M C[Q(p)]}Q00
where {θ + λ · [1 − mc0 · Q0 ]}Q0 + λ{p − M C[Q(p)]}Q00 < 0 is assumed. Thus, an increase in theworkers’ bargaining power (i.e., a decrease in λ) reduces the consumer surplus.
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Figure 2: Surplus Division under the Profit Sharing Rule

3

A Linear Example

Now, we assume that Q(p) = a − p and M C(Q) = c + dQ, where a > c ≥ 0 and d ≥ 0. Then,the equilibrium product price is obtained asp=λ(c + ad) + aθ,
λ(1 + d) + θ

which is always less than a as long as a > c, andQ=(a − c)λ.
λ(1 + d) + θ

m=(a − c)θ
.
λ(1 + d) + θ

Accordingly, the gross markup is

It is observed that as the workers’ bargaining power increases (a decrease in λ), the aggregate
output decreases:∂Q(a − c)θ
=> 0.
∂λ[λ(1 + d) + θ]2
Now, the manager surplus is4

Surplus
0.20

CS

0.15
0.10

WS

0.05
0.2

0.4

0.6

0.8

1.0

λ

Figure 3: Consumer Surplus and Worker Surplus when θ = 12Surplus
0.4

SW0.3
US0.2

MS

0.1

0.2

0.4

0.6

0.8

1.0λ

Figure 4: United Surplus, Manager Surplus, and Social Welfare when θ = 12MS =λ2 (a − c)2 (2θ + dλ),2[λ(1 + d) + θ]2

whereas the consumer surplus and the worker surplus are
λ2 (a − c)2CS =,2[λ(1 + d) + θ]2andWS =(1 − λ)λ(a − c)2 (2θ + dλ),2[λ(1 + d) + θ]2

respectively.
Now, we further assume that the marginal cost is constant (i.e., d = 0) and a − c = 1. First,
suppose that θ = 12 . Figure 3 depicts how the consumer surplus and the worker surplus changes
as the manager’s bargaining power, λ, increases. While CS increases monotonically because of
the price reduction, WS attains its maximum value at λ = 14 . Thus, a natural question to ask is
how the united surplus changes as λ increases. Figure 4 shows that while the manager surplus
and the social welfare monotonically increase as λ increases, US attains its maximum value at
λ = 12 . In other words, there is a conflict of interest between the workers/consumers and themanagers if λ > 12 : the society as a whole is happier whereas the citizens in the majority areless happier as the managerial authority is strengthened.
Next, we examine the effects of a change in the competitiveness in the product market, θ, by5

Surplus
0.5
0.4
0.3
0.2
WS
0.1
CS
θ
0.2

0.4

0.6

0.8

1.0

Figure 5: Consumer Surplus and Worker Surplus when λ = 140.5
0.4
0.3

SW0.2
US0.1MSθ
0.2

0.4

0.60.81.0

Figure 6: United Surplus and Manager Surplus when λ = 14
supposing that λ = 41 , when WS is maximized for θ = 41 . Naturally, WS attains its maximumvalue at θ = 14 as Figure 5 shows. CS is monotonically decreasing in θ. Figure 6shows that theworkers/consumers are against anti-competitive conduct because US decreases as θ increases.
However, the managers are tempted to increase θ because MS is increasing in θ.
As a summary, when the share economy, where corporate profit sharing is used for incentive reasons, is considered, corporate policy changes favoring management are resisted by themajority of citizens (i.e., workers/consumers) once they break over the critical point (in theabove example, λ = 12 ), competition policy is always welcomed by them, whereas managers arealways tempted to raise θ.

References
Aoki, Masahiko. 1988. Information, Incentives, and Bargaining in the Japanese Economy. Cambridge University Press.
Meade, J. E. 1972. “The Theory of Labour-Managed Firms and of Profit Sharing,” Economic
Journal, 82(325), 402-428.
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Okuno-Fujiwara, Masahiro. 1987. “Monitoring Cost, Agency Relationships, and Equilibrium
Modes of Labor Contracts,” Journal of the Japanese and International Economies, 1(2), 147167.
Shapiro, Carl, and Joseph E. Stiglitz. 1984. “Equilibrium Unemployment as a Worker Discipline
Device,” American Economic Review, 74(3), 433-444.
Weitzman, Martin 1984. The Share Economy. Harvard University Press.
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.

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