Moral Hazard and Bargaining Power∗ Dominique Demougin and Carsten Helm† School of Business and Economics Humboldt University Berlin

January, 2004

Abstract We introduce bargaining power in a moral hazard framework where parties are risk-neutral and the agent is financially constrained. We show that the same contract emerges if the concept of bargaining power is analyzed in either of the following three frameworks; a standard P-A framework by varying the agent’s outside opportunity, in an alternating offer game and in a generalized Nash bargaining game.

JEL Classification: D2; D8; L14 Keywords: Principal-agent model; bargaining power; moral hazard.

∗

We wish to thank Oliver Fabel, Eberhard Fees, Roland Strausz, Anja Sch¨ ottner and Veikko

Thiele for valuable comments. † Address: Spandauer Str. 1, D-10178 Berlin, Germany, Tel: +49 30 2093-1592, Fax: +49 30 2093-1343, e-mails: [email protected] and [email protected].

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1

Introduction

A standard assumption in the Principal-Agent model is that the principal can present a contract as a ‘take it or leave it offer’ to the agent. However, in most real world relationships both parties will hold some bargaining power. Furthermore, in a recent contribution Pitchford (1998) pointed out that for a large class of cases where the agent has limited liability the distribution of bargaining power between principal and agent affects the joint surplus generated by the contract. This may have important policy implications, e.g. for the design of labor market institutions, which play an essential role in determining the distribution of bargaining power between employers and employees. To model bargaining power, Pitchford (1998) varies the agent’s reservation utility in a standard Principal-Agent framework with ‘take it or leave it offers’. A different approach is adopted by Balkenborg (2001), who uses the Nash bargaining solution to analyze a similar moral hazard problem. Alternatively, one could think of representing bargaining power in a moral hazard framework as an alternating offer game. For the case of risk-neutral parties and a financially constrained agent we show that these three different ways of analyzing bargaining power are equivalent. In particular, the same set of contract arises from varying the agent’s reservation utility in the standard P-A model, the discount factor in an alternating offer game a` la Rubinstein (1982), or the bargaining power coefficient in a Nash bargaining game. In the following, we analyze these three approaches in turn.

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The P-A model with varying outside options

We consider a moral hazard environment with one principal and one agent, both of which are risk-neutral. The value of output for the principal is v(e), where e ∈ R+ is the agent’s effort associated with costs c(e). We impose standard requirements, assuming that v(e) is increasing and concave with v
(0) = ∞, while c(e) is increasing and convex. Moral hazard results because none of the above variables are assumed verifiable. Instead, the principal and the agent observe a contractible binary signal s ∈ {0, 1}, where s = 1 is a favorable signal (see Milgrom 1981).1 We denote with p(e) the 1

This is a generalized version of the problem analyzed by Pitchford (1998). Note that the

assumption s ∈ {0, 1} is without loss of generality, as in the risk-neutral agency problem all relevant information from a mechanism design point of view can be summarized by a binary statistic (see,

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probability of observing the favorable signal given the agent’s effort and assume p
(e) > 0, p

(e) < 0.2 Due to the informational assumptions contracts will also be binary. We denote with F the fixed payment and with b the bonus which the agent receives when s = 1. Finally, we assume that the agent is financially constrained since otherwise the first-best is always obtainable, as is well known. For parsimony, we simply require that F, F + b ≥ 0. In the standard P-A model, the principal makes a take-or-leave-it offer that leads to expected profits π(¯ u) ≡ max

{F,b,e}

v(e) − [F + bp(e)] subject to

(1)

bp
(e) = c
(e)

(2)

F ≥0

(3)

F + bp(e) − c(e) ≥ u ¯,

(4)

where (2) is the incentive compatibility condition, (3) the liability limit and (4) the constraint on the agent’s utility.3 Substituting b from (2) yields the the expected bonus B(e) = c
(e)p(e)/p
(e), which we assume to be convex. This assumption ensures that the first-order condition of the Lagrangian is sufficient. Keep in mind that B(e) includes the contract’s optimal adjustment to induce effort e. Accordingly, the concavity of the agent’s optimization problem, which takes b as a constant, and the convexity of B(e) are not in contradiction.4 Upon varying the agent’s reservation utility, we get the following result.5 Proposition 1 In the P-A model, the principal’s profits are decreasing concave in the agent’s utility u. For low values of u, the optimal contract has F = 0, the agent extracts rent and effort is constant at the second best e∗∗ . For high values of u, the ¯ + c(e∗ ) − B(e∗ ). For optimal contract implements first-best effort e∗ and has F = u intermediate values of u, F = 0, and effort is increasing in u. e.g., Kim 1997). 2 These conditions guarantee that the agent’s problem is well behaved. They are equivalent to considering binary signals satisfying MLRC and CDFC within the class of differentiable signals with constant support. 3 Note that we dropped the condition F + b ≥ 0 as b ≥ 0 by (2). 4 For example, suppose c(e) = 0.5e2 and p(e) = eθ with θ ∈ [0, 1], then B(e) is convex (see Demougin and Fluet 2001 for an extensive discussion of the example). 5 For similar results see Pitchford (1998) or Demougin and Fluet (2001) and, for an adverse selection context, Demougin and Garvie (1991) and Inderst (2002).

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Proof. The Lagrangian of the principal’s optimization problem is L(e, F, λ, µ) = v(e) − F − B(e) + λF + µ [F + B(e) − c(e) − u ¯] .

(5)

From the first-order conditions w.r.t. e and F , v
(e) − B
(e) + µ(B
(e) − c
(e)) = 0

(6)

−1 + λ + µ = 0,

(7)

there are three possible cases, which we examine successively. When λ = 1, µ = 0, fixed payments F must be 0 by complementary slackness and the standard second¯ best effort, e∗∗ , obtains from (6). Obviously, this can only arise if B(e∗∗ ) ≥ c(e∗∗ )+ u so that the agent extracts a rent. If the inequality is strict, small variations in u ¯ leave e∗∗ , π ∗∗ and u∗∗ unaffected. As u ¯ continues to increase, the constraint on the agent’s utility must become binding at some point, in which case λ, µ > 0. Thus, fixed payments remain at 0 and effort follows from the binding constraint on the agent’s utility. Implicitly differentiating w.r.t. u ¯ yields eu¯ =

B
(e)

1 > 0, − c
(e)

(8)

where the sign of B
(e) − c
(e) follows straightforwardly from the definition of B(e) and the curvature assumptions. In addition, from the envelope theorem we have ¯ the πu¯ = −µ < 0. Moreover, µu¯ > 0 which implies that for intermediate values of u Pareto frontier is decreasing and concave.6 That process can only go on until µ = 1 as then λ = 0. At this point, effort attains the social optimum as can be seen from (6). Finally, F follows from the binding participation constraint.7

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We illustrate the proposition in Figure 1 which depicts the constrained Pareto frontier for the standard P-A model. When u ¯ < u∗∗ , the principal offers the agent a ¯. Obviously, raising contract yielding utility u∗∗ , i.e. the agent extracts a rent u∗∗ − u u ¯ reduces the agent’s rent until it falls to zero. Accordingly, the dotted line above 6

To verify the claim, totally differentiating (6) and rearranging yields µu¯ =

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(1 − µ)B  (e) − v  (e) + µc (e) > 0. [B  (e) − c (e)]2

This third case can never arise in the framework analyzed by Pitchford due to the binary nature

of v(e) in his model.

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π πi

I u

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . ∗∗

... ... .. .. ... ... .. II .... .. ... u∗

III umax

u

Figure 1: The Pareto frontier region I does not really belong to the Pareto frontier. When u ¯ ≥ u∗∗ , increases in the agent’s reservation utility must be compensated by either raising b or F . Obviously, increasing b to keep the rent to zero is initially advantageous as it also raises effort above the second best (region II). However, once first best effort is attained any further increase in u ¯ is best compensated by lump sum transfers (region III).

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Bargaining game with alternating offers

We now consider the case where the principal and the agent bargain over the contract elements F, b, and e. We model this as an alternating offer game. In round 1, the principal offers the agent a contract Cp = {F, b, e, u}, where F, b are the contract elements that induce effort e and provide the agent with utility u. Consistent with the agent’s financial constraint, we further require F, F + b ≥ 0. For any given u, the principal will obviously choose Cp so as to maximize her own profits. Of course, this is just equivalent to solving problem (1) to (4) and yields a solution along the constrained Pareto frontier in Figure 1. If the agent accepts the offer, the contract is implemented, i.e. the agent undertakes effort, the verifiable signal is realized and payments are made. If the agent declines, the game proceeds to a second round where now the agent proposes a contract Ca = {F, b, e, π}. Just as before F and b are required to be feasible contract elements that credibly implement e and leave the principal with

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profit π. It is easily verified that varying π, the agent’s resulting optimization problems yield exactly the same constrained Pareto frontier as in Figure 1. If the principal accepts, the contract is implemented. If she rejects, the game continues in the same manner with alternating offers. Parties are assumed to be impatient. We denote with δp ∈ [0, 1] and δa ∈ [0, 1] the respective discount factor of the principal and agent. For the moment, we assume that there exists a unique subgame perfect equilibrium and denote it with Cp∗ = {Fp , bp , ep , up } and Ca∗ = {Fa , ba , ea , πa }. We know from Rubinstein (1982) that this leads to no delay. To determine the equilibrium, suppose the game were to attain period 3. The principal would make the offer and guarantee herself π(up ). Going one period back, the agent will match the principal’s present value of her period 3 profit, i.e. offer πa = δp π(up ). Again going back one period, the principal will offer up = max{u∗∗ , δa u(πa )} as we know from proposition 1 that it can never be optimal to offer less than u∗∗ (see Figure 1). Upon substitution, we find up = max{u∗∗ , δa u(δp π(up ))}.

(9)

To prove existence of a unique subgame perfect equilibrium, consider the function δa u(δp π(up )). Its slope is 0≤

∂δa u(δp π(up )) = δa u
(·)δp π
(up ) ≤ 1. ∂u

(10)

To verify the inequality, observe that from up ≤ u(δp π(up )) and the concavity of the Pareto frontier, we obtain u
π
≤ 1. Moreover, u
and π
are both negative and δp , δa ≤ 1. Figure 2 illustrates the result and allows a characterization of the fixed point. The figure maps the minimal acceptable offer which the principal must make in the first round if the agent can guarantee himself u in the third round. Note that in period 3 the agent’s utility will always be between u∗∗ and umax . Starting with the latter point, observe that point C always lies below the diagonal since umax ≥ δa u(δp π(umax )). Hence, if δa u(δp π(u∗∗ )) lies above the diagonal (like point A), then we have an interior fixed point like up . Otherwise, δa u(δp π(u)) is like BC and the fixed point is u∗∗ . Note that variations in the discount factors shift the curves and, therefore, the fixed point provided that δa u(δp π(up ) lies on the diagonal. Otherwise small variations in the discount factors will leave the fixed point at u∗∗ and, therefore, have no effect on the solution.

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δa u(δp π(u))

u∗∗

.. A.. .. .. .. .. .. .. .............. .. ... .. . .. B.... .. .. . .∗∗ .. u p u

..C .. .. .. .. .. .. .. .. .. .. .. .. .

umax

u

Figure 2: Fix points in alternating offer game Finally, it is straightforward to show that any point along the constrained Pareto frontier can be represented by different profiles of the discount factors. For example, suppose δp = 0, i.e. the principal is completely impatient. Letting δa vary between 0 and 1 completely describes the Pareto frontier. Indeed, with δa = 0 the agent receives u∗∗ , while with δa = 1, the agent receives u(0) = umax . Thus by continuity as δa varies between 0 and 1, the agent’s utility must take all the values between u∗∗ and umax . We summarize the foregoing results, which imply that we can represent variations in the bargaining power equivalently through changes in discount factors or by changes in u ¯. Proposition 2 In the alternating offer game for each δa , δp there is a unique subgame perfect equilibrium in contracts. Equilibrium profit for the principal and utility for the agent are on the same constrained Pareto frontier as in the P-A model. Moreover, varying δa , δp yields every point along the constrained Pareto frontier and any such point is associated with the same contract elements F, b, e.

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The Nash bargaining solution

Binmore, Rubinstein, and Wolinsky (1986) have shown that the standard bargaining process with alternating offers can be approximated by the Nash bargaining solution. We extend their result to the current moral hazard set up. To do so, we initially hold bargaining power α constant and maximize the Nash bargaining product (F + bp(e) − c(e))α (v(e) − F − bp(e))1−α ,

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(11)

with respect to contracts that are ex-post incentive compatible and satisfy the agent’s financial constraint.8 With α = 0, the Nash bargaining product equals the principal’s profit, and with α = 1 the agent’s utility. Obviously, for these extreme cases the solution corresponds to the boundary points of the constrained Pareto frontier (see Figure 1). For α ∈ (0, 1), the corresponding Lagrangian becomes L(e, F, ξ) = α ln [F + B(e) − c(e)] + (1 − α) ln [v(e) − F − B(e)] + ξF, with first-order conditions α (B
(e) − c
(e)) (1 − α)(v
(e) − B
(e)) + = 0 F + B(e) − c(e) v(e) − F − B(e) (1 − α) α − + ξ = 0. F + B(e) − c(e) v(e) − F − B(e)

(12) (13)

When ξ = 0, the first best solution obtains since from substituting (13) into (12) we get

α(B
(e) − c
(e)) α(v
(e) − B
(e)) + = 0, F + B(e) − c(e) F + B(e) − c(e)

(14)

implying v
(e) = c
(e). Furthermore, (13) can be solved for α=

B(e∗ ) − c(e∗ ) F + , v(e∗ ) − c(e∗ ) v(e∗ ) − c(e∗ )

(15)

where the second term on the r.h.s defines a critical level of bargaining power αc . For α ≥ αc the first best effort obtains and any increase in bargaining power results in a larger F . Moreover, from (15) as α approaches 1 so that the agent has the entire bargaining power, he extracts all the profit and attains utility umax . When ξ > 0, complementary slackness implies F = 0. Implicitly differentiating of (12) then yields de/dα > 0. Finally observe that as α approaches 0 the standard second best obtains. We get the following result. Proposition 3 In the Nash bargaining solution, variations in the bargaining power completely map the constrained Pareto frontier that characterizes the alternating offer game and the P-A model. Moreover, points along the Pareto frontier are associated with the same contract elements F, b, e in all three models. 8

Note that participation is guaranteed by construction of the Nash-bargaining solution, except

for the extreme cases of α = 0 and α = 1, where the respective participation constraints have to be added to the problem.

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In conclusion, as the agent’s bargaining power goes from α = 0 to α = αc effort increases from the second best to the first best level and F remains at 0. As α increases further, effort stays at the first best level and F adjusts. Finally, observe that even though it is desirable from a welfare point of view to attain first best effort, the principal will not willingly relinquish bargaining power as it lowers her profit.

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Conclusion

In this note we have analyzed three approaches to account for bargaining power in a moral hazard framework, each of them leading to the same set of contracts. Nevertheless, their usefulness will vary depending on the particular problem under consideration. For example, solving the alternating offer game may be quite cumbersome. Similarly, measuring changes in bargaining power by u ¯ is unsatisfactory if one wants to understand the impact of bargaining power on equilibrium utility. Moreover, for sufficiently low levels of the agent’s bargaining power, increasing it marginally does affect the equilibrium in the Nash bargaining game, but not in the P-A model and in the alternating offer game (see Figures 1 and 2). This is in clear contrast to the standard bargaining problem without moral hazard – such as dividing a cake.

References Balkenborg, D. (2001). How liable should a lender be? The case of judgementproof firms and environmental risk: Comment. American Economic Review 91 (3), 731–738. Binmore, K., A. Rubinstein, and A. Wolinsky (1986). The Nash bargaining solution in economic modelling. RAND Journal of Economics 17 (2), 176–188. Demougin, D. and C. Fluet (2001). Monitoring versus incentives. European Economic Review 45, 1741–1764. Demougin, D. and D. Garvie (1991). Contractual design with correlated information under limited liability. RAND Journal of Economics 22, 477–489. Inderst, R. (2002). Contract design and bargaining power. Economics Letters 74, 171–176.

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Kim, S. (1997). Limited liability and bonus contracts. Journal of Economics and Management Strategy 6, 899–913. Milgrom, P. (1981). Good news and bad news: Representation theorems and applications. Bell Journal of Economics 12, 380–391. Pitchford, R. (1998). Moral hazard and limited liability: The real effects of contract bargaining. Economics Letters 61 (2), 251–259. Rubinstein, A. (1982). Perfect equilibrium in a bargaining model. Econometrica 50, 97–109.

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