Baker Gibbons Murphy (1999) - Informal Authority In Organizations

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© edegan.com, 2016

• This page is referenced in PHDBA602 (Theory of the Firm)

Reference(s)

Baker, G, R Gibbons, and K.J. Murphy (1999), "Informal Authority in Organizations", Journal of Law, Economics & Organization, 15, March pp. 56-73. pdf

Abstract

We assert that decision rights in organizations are not contractible: the boss can always overturn a subordingate's decision, so formal authority resides only at the top. Although decision rights cannot be formally delegated, they might be informally delegated through self-enforcing relational contracts. We examine the feasibility of informal authority in two informational environments. We show that different informations structures priodcute different decusions not only because different information is brought to bear in the decision-making process, but also because different information creates differenty temptations to renege on relational contracts. In addition, we explore the implications of formal delegation achieved through divestitures.

The Basic Model

The is a boss and a subordinate. The boss gets payoffs [math]Y\;[/math], and the subordinate gets payoffs [math]X\;[/math]. Both benefits can take two values:

[math]Y_H \gt 0 \gt Y_L \quad \mbox{and} \quad X_H \gt 0 \gt X_L\;[/math]

The subordinate searches for projects, with the intensity of search affecting the probability of discovering a project that he likes.

[math]a = Pr(X = X_H)\;[/math]

The conditional probability that the boss gets payoff [math]Y_H\;[/math] when an [math]X_H\;[/math] project is found is:

[math]p = Pr(Y = Y_H | X=X_H)\;[/math]

The conditional probability that the boss gets payoff [math]Y_H\;[/math] when an [math]X_L\;[/math] project is found is:

[math]q = Pr(Y = Y_H | X=X_L)\;[/math]

Therefore the joint probabilities are:

[math]Pr(Y = Y_H, X=X_H) = ap\;[/math]

[math]Pr(Y = Y_L, X=X_H) = a(1-p)\;[/math]

[math]Pr(Y = Y_H, X=X_L) = (1-a)q\;[/math]

[math]Pr(Y = Y_L, X=X_L) = (1-a)(1-q)\;[/math]

The timing is as follows:

1. The boss pays the subordinate [math]s\;[/math] (which may be negative)
2. The subordinate searches by choosing [math]a\;[/math], where [math]c(a) = \gamma a^2\;[/math]
3. The subordinate observes the payoffs [math](X,Y)\;[/math], if the payoff is [math]X_L\;[/math] the project is ignored, otherwise the project may be recommended.
4. If the project is recommemded then the boss either implements or rejects the project (perhaps seeing the payoffs).

The next two sections give two simple benchmarks.

Informed Centralization

In this model, the boss is informed of the payoff and then makes the decision. The subordinate knows that the boss will reject decisions with a payoff of [math]Y_L \lt 0\;[/math] and therefore maximizes the expected utility:

[math]\max_a s+ apX_H - c(a)\;[/math]

This solves to:

[math]c'(a^C) = pX_H\;[/math]

The search intensity that maximizes joint welfare conditional on selecting only [math](X_H,Y_H)\;[/math] projects solves:

[math]\max_a ap(X_H + Y_H) -c(a)\;[/math]

Which gives:

[math]c'(a^*) = p(X_H + Y_H)\;[/math]

so [math]a^C\;[/math] is less than efficient. The paper doesn't use the given cost function to make the comparison (any convex cost function would do), but using it gives:

[math]a^C = \frac{1}{\gamma}\cdot pX_H\;[/math]

[math]a^* = \frac{1}{\gamma}\cdot p(X_H + Y_H)\;[/math]

[math]\therefore a^* \gt a^C \; \forall Y_H \gt 0\;[/math]

The expected welfare to the boss under informed centralization is:

[math]a^c p Y_H - s\;[/math]

Total expected welfare is therefore:

[math]V^C = a^C p (X_H + Y_H) - c(a^C)\;[/math]

Contractible Delegation

Now suppose that the boss has contractually delegated the rights to make decisions to the subordinate.

Then the subordinate doesn't worry whether the boss gets [math]Y_H\;[/math] or [math]Y_L\;[/math] and searches to maximize:

[math]\max_a s + a X_H - c(a)\;[/math]

which solves:

[math]c'(a^D) = X_H\;[/math]

Because [math]c''(\cdot) \gt 0\;[/math] and [math]p \lt 1\;[/math], it must be that delegation increases the incentives to search.

Explicitly this can be seen because:

[math]a^D = \frac{1}{\gamma}\cdot X_H \gt a^C = \frac{1}{\gamma}\cdot pX_H\;[/math]

Note that it might increase them too much. The efficient incentives under delegation are given by:

[math]c'(a^*) = p(X_H + X_L) + (1-p)\cdot \max(0,X_H+Y_L)\;[/math]

which can imply either higher or lower incentives.

The expected payoff to the boss under contractible delegation is:

[math]a^D(pY_H + (1-p)Y_L) -s\;[/math]

So total welfare is:

[math]V^D = a^D p(X_H + Y_H) + a^D(1-p)(X_H + Y_L) - c(a^D)\;[/math]

The following points should be made:

• Parties would agree to delegate if [math]V^D \gt V^C\;[/math] and would leave this right with the boss otherwise.
• Ex-ante incentives are stronger under delegation
• Ex-post project choice and its efficacy differ under the two schemes. Which is better depends on the sign of [math]X_H+Y_L\;[/math] - if this is positive then delegation is better, otherwise centralization is better.
• When p is high (interests are aligned) and [math]-Y_L\;[/math] (the cost from delegation) is small, then delegation is more likely.

Models of Informal Authority

In the following models, formal authority can not be delegated within organisations. However, informal authority can be given in a repeated game framework.

In the first model, the boss becomes informed before ratitfying the project, but has a reputation for not interfering to maintain. In the second model the boss is only informed about historic payoffs, and must either rubber stamp or veto the project.

Informal Delegation

In this model the boss "promises" to ratify all of the subordinates decisions and then may renege. The subordinate, in turn, has to believe that the boss will honour the decisions. This is the informal delegation model.

This model would be appropriate when:

• Projects that require careful analysis (increased search) so that the benefits may out weigh the costs of allowing a poor decision through
• The boss may feel regret at allowing a decision yet not overturn it

If the boss promises and the subordinate believes, then the subordinate will choose [math]a^D\;[/math], and the boss will get both [math]Y_H\;[/math] and [math]Y_L\;[/math] projects. We call these [math]Y^D\;[/math] payoffs, and the subordinate gets [math]X^D\;[/math] ([math]=X_H\;[/math]). The joint surplus is therefore [math]V^D\;[/math].

When the boss gets [math]Y_L\;[/math] projects she will be tempted to renege. If she breaks her promise, then subordinate will be playing the informed centralization game from then on, and the payoffs will be [math]X^C\;[/math] ([math]=X_H\;[/math]) and [math]Y^C\;[/math] ([math]=Y_H\;[/math]), for a joint surplus of [math]V^C\;[/math].

Therefore the boss (with a discount rate of [math]r\;[/math]) will honour the promise if:

[math]Y^L + \frac{1}{r} Y^D \gt \frac{1}{r} Y^C \quad \therefore Y^D - Y^C \gt -r Y_L\;[/math]

The subordinate will accept delegation if:

[math]X^D - X^C \gt 0\;[/math]

Together these give the necessary and sufficient condition that for delegation:

[math]V^D - V^C \gt -r Y_L\;[/math]

Of course delegation is efficient if:

[math]V^D - V^C \gt 0\;[/math]

As [math]Y_L\;[/math] is negative, this gives the result that there are times when it would be efficient to delegate authority informally, but that are not feasible because the boss will renege. There are two other important results:

• The attractiveness of delegation over centralization depends solely on the expected surplus
• The feasibility of informal delegation depends on the extreme payoffs - that is holding the expected surplus constant the feasibility is affected by the spread betweem [math]Y_H\;[/math] and [math]Y_L\;[/math].

Informal Authority with an Uniformed Boss

In this model the boss doesn't know the payoffs of this period, only the past payoffs, and must either "Rubber Stamp" all projects or "Veto" all projects.

This model would be appropriate when:

• The subordinate has expertise the boss must rely on
• Small decisions that do not merit monitoring
• Decisions that must be made quickly

The bosses expected benefit from "Rubber Stamping" a project is:

[math]\mathbb{E}(Y|X_H) = pY_H + (1-p)Y_L\;[/math]

The bosses expected benefit from vetoing a project is [math]0\;[/math], and the boss would prefer to veto if:

[math]\mathbb{E}(Y|X_H) \lt 0\;[/math]

Now the suppose that the subordinate is granted informal authority to propose projects that pay [math]Y_H\;[/math] to the boss, and the boss will threaten to retract this authority and either rubber stamp or veto projects is she ever finds that a [math]Y_L\;[/math] project has been proposed. This is the informal authority model. In this model it is the subordinate, not the boss, who is tempted to renege.

Rubber Stamping

When the fall back is rubber stamping we know that:

[math]\mathbb{E}(Y|X_H) \gt 0\;[/math]

The sole question remaining is whether:

[math]V^D \gt V^C \;[/math]

Or whether delegation or centralization is more efficient. When delegation is more efficient that is exactly what happens. When centralization is more efficient, we can determine the knife edge case for when informal authority can occur.

Informal authority can occur if the present value from honouring it exceeds the present value from abusing it. If it is abused, we get the rubber stamp payoffs, which are the same as those to delegation (either [math]Y_L\;[/math] or [math]Y_H\;[/math] projects can be proposed and will be accepted). Formally we get informal authority if:

[math]\frac{1}{r} X^C \gt X_H + \frac{1}{r} X^D \quad \therefore X^C-X^D \gt r X_H\;[/math]

The boss will grant informal authority (over rubber stamping) if:

[math]Y^C - Y^D \gt 0 \;[/math]

Combining terms gives the nec. and suff. conditions for informal authority:

[math]V^C - V^D \gt r X_H\;[/math]

The general result is that holding the expected surplus ([math]V^C - V^D\;[/math]) constant, informal authority becomes less feasible as the temptation to renege ([math]r X_H\;[/math]) increases.

Veto

As with the rubber stamping, the interesting case is when we are trying to achieve centralization through informal authority. Now the conditions are that the subordinate must prefer honouring to abusing:

[math]X^C \gt r X_H\;[/math]

And the boss will grant informal authority iff:

[math]Y^C \gt 0\;[/math]

Together these constraints give:

[math]V^C \gt rX_H\;[/math]

Which does not depend on [math]Y_L\;[/math] at all.

Divestiture as Contractible Delegation

The paper concludes by noting that in a similar vein, firms may divest spin-offs in order to prevent themselves from reneging - that is from retrating informal delegation.