# Baker Gibbons Murphy (2002) - Relational Contracts And The Theory Of The Firm

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Has article title | Relational Contracts And The Theory Of The Firm |

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

## Reference(s)

Baker, George, Robert Gibbons and Kevin J. Murphy (2002), "Relational Contracts and the Theory of the Firm", The Quarterly journal of economics, vol. 117, issue 1, page 39 pdf

## Abstract

Relational contracts - informal agreements sustained by the value of future relationships - are prevalent within and between firms. We develop repeatedgame models showing why and how relational contracts within firms (vertical integration) differ from those between (nonintegration). We show that integration affects the parties’ temptations to renege on a given relational contract, and hence affects the best relational contract the parties can sustain. In this sense, the integration decision can be an instrument in the service of the parties’ relationship. Our approach also has implications for joint ventures, alliances, and networks, and for the role of management within and between firms.

## Relational contracting

This is a model of relational contracting, where a relational contract is defined as:

- Informal agreements (this paper)
- Self-enforcing agreements (Telser 1981)
- Implicit contracts (MacLeod and Malcomson 1989)

Like other relational contracting models there are maintained assumptions of:

- Greater surplus from the inside versus the outside option
- Actions are unobservable
- Contracts on inputs or outputs can not be enforced
- Contracts are inherently impossible (due to incompleteness or other reasons) ex ante

The model is novel in that it simultaneously considers relational contracts within and across firms, and allows a comparison of both environments.

## The Model

The basic model is reminiscent of a property rights model with multitasking embedded. Nash Bargaining over surplus is used through-out.

- There is a infinitely repeated game with a discount rate [math]r\;[/math] between and two parties: an upstream ([math]U\;[/math]) supplier and a downstream ([math]D\;[/math]) buyer/user.
- [math]D\;[/math] can take a costly action [math]a\;[/math] (which is a vector to allow multitasking) at cost [math]c(a)\;[/math]
- There is a good produced by [math]U\;[/math] that that has value [math]Q\;[/math] to [math]D\;[/math], and [math]P\;[/math] in an alternative use. Both can take two values [math]H\;[/math] and [math]L\;[/math], such that [math]P_L \lt P_H \lt Q_L \lt Q_H\;[/math].
- The asset in this model could be interpretted as legal title to the good.
- [math]D\;[/math]'s action affect the probability that high values will be realized:

- [math]Pr(Q=Q_H) = q(a) \quad Pr(Q=Q_L) = 1-q(a)\;[/math]

- [math]Pr(P=P_H) = p(a) \quad Pr(P=P_L) = 1-p(a)\;[/math]

For notational simplicity we define:

- [math]\Delta Q = Q_H - Q_L \quad \Delta P = P_H - P_L\;[/math]

It is assumed that:

- [math]c(0) = 0, \; q(0)=0, \; p(0) = 0\;[/math]

So that if no effort is exerted then there is no cost but no chance of a high state.

The efficient use is inside the relationship, so first best effort [math]a^*\;[/math] is given by:

- [math]max_a Q_L + q(a)\Delta Q - c(a)\;[/math]

- [math]\therefore S^* = Q_L + q(a^*)\Delta Q - c(a^*)\;[/math]

Where [math]S\;[/math] is total surplus.

### Spot Outsourcing

The assumptions here are:

- [math]U\;[/math] owns the asset
- There is a one shot transaction
- From Nash bargaining [math]D\;[/math] will pay [math]U\;[/math] the outside option [math]P_j\;[/math] plus half of the surplus [math]Q_i-P_j\;[/math]. Therefore the price is [math]\frac{1}{2}(Q_i + P_j)\;[/math]

The utilities of the two parties are:

- [math]U^{SO} = \frac{1}{2}(Q_L + q(a)\Delta Q) + \frac{1}{2}(P_L + p(a)\Delta P) - c(a)\;[/math]

where [math]a^{SO}\;[/math] solves the maximization problem.

- [math]D^{SO} = \mathbb{E} \left (\frac{1}{2} (Q_i - P_j) | a = a^{SO} \right )\;[/math]

and the total surplus is:

- [math]S^{SO} = D^{SO} + U^{SO} = Q_L + q(a^{SO}) \Delta Q -c(a^{SO})\;[/math]

Note that this is likely to differ from first best. Specifically, if [math]\Delta Q = 0\;[/math] then first best is zero effort, but [math]U\;[/math] will still expend effort to secure more surplus under Nash bargaining.

### Spot Employment

The assumptions here are:

- [math]D\;[/math] owns the asset
- There is a one shot transaction (no relational contract), so [math]D\;[/math] can simply take the output

Given that [math]D\;[/math] can and will just take the output, [math]U\;[/math] will exert no effort, so the value to [math]D\;[/math] will be [math]Q_L\;[/math],which is also the total surplus [math]S^{SE} = Q_L\;[/math].

Note that spot employment dominates spot ownership iff:

- [math]S^{SE} \gt S^{SO} \quad \therefore q(a^{SO}) \Delta Q -c(a^{SO}) \lt 0\;[/math]

### Relational Employment

The assumptions here are:

- [math]D\;[/math] owns the asset
- There is a relational contract between [math]U\;[/math] and [math]D\;[/math] based on the observable but non-contractible output
- The trigger strategy is that if default is observed then spot contracts only will be accepted for all time
- The efficient spot contract will be used as a fall back, and parties can negotiate over asset ownership after default (with side payment [math]\pi\;[/math]).

The relational compensation contract takes the form:

- [math](s,\{ b_{ij} \} ) \; i,j \in \{H,L \} \;[/math]

That is [math]s\;[/math] is a salary and [math]b_ij\;[/math] is a bonus payment paid with [math]Q=Q_i\;[/math] and [math]P=P_j\;[/math]. These bonus may be negative, and both parties are confident that they will be paid.

If [math]U accepts the contract then she chooses \lt math\gt a^{RE}\;[/math] to solve:

- [math]max_a U^{RE} = s + B_{LL}(1-q)(1-p) +B_{HL}q(1-p) +B_{LH}(1-q)p +B_{HH}qp - c(a)\;[/math]

[math]D\;[/math]'s expected payoff is then:

- [math]D^{RE} = \mathbb{E} ( Q_i - s -bij | a= a^{RE}) = Q_L + q(a^{RE}) \Delta Q - (U^{RE} + c(a^{RE}) )\;[/math]

and total surplus is:

- [math]S^{RE} = D^{RE} + U^{RE} Q_L + q(a^{RE}) \Delta Q - c(a^{RE})\;[/math]

This contract is self-enforcing if the present value of a stream of these payoffs is better for both parties than their best default plus the present value of a spot contract.

#### Employment in default

First we examine the case where [math]S^{SE} \gt S^{SO}\;[/math], so that [math]D\;[/math] will be get spot employment contracts forever after. In this case [math]D\;[/math] retains ownership of the asset, and [math]D\;[/math] will honour rather than renege iff:

- [math]-b_{ij} + \frac{1}{r} D^{RE} \ge \frac{1}{r} D^{SE} \quad \forall i,j\;[/math]

And [math]U\;[/math] will honour rather than renege iff:

- [math]b_{ij} + \frac{1}{r} U^{RE} \ge \frac{1}{r} U^{SE} \quad \forall i,j\;[/math]

As the equation for [math]D must hold for all \lt math\gt i,j\;[/math], it must hold for the largest. Likewise as the equation for [math]U\;[/math] must hold for all [math]i,j\;[/math], it must hold for the smallest. Combining these two conditions gives a joint-constraint:

- [math]\max b_{ij} - \min b_{ij} \le \frac{1}{r} (S^{RE}-S^{SE})\;[/math]

#### Outsourcing in default

Now we examine the case where [math]S^{SO} \gt S^{SE}\;[/math], so outsourcing will be used in default, and the asset must be transfer to [math]U\;[/math] for some price [math]\pi\;[/math].

[math]D\;[/math]'s will honour iff:

- [math]-b_{ij} + \frac{1}{r} D^{RE} \ge \frac{1}{r} D^{S0} + \pi \quad \forall i,j\;[/math]

and [math]U\;[/math] will honour iff:

- [math]b_{ij} + \frac{1}{r} U^{RE} \ge \frac{1}{r} U^{SO} - \pi \quad \forall i,j\;[/math]

Again taking the max and the min to get a single joint constraint gives:

[math]\max b_{ij} - \min b_{ij} \le \frac{1}{r} (S^{RE}-S^{SO})\;[/math]

#### A general condition

Given the two conditions from the two default scenarios we can write a single condition that is both necessary and sufficient:

- [math]\max b_{ij} - \min b_{ij} \le \frac{1}{r} (S^{RE}-\max(S^{SO},S^{SE})\;[/math]

### Relational Outsourcing

The assumptions here are:

- [math]U\;[/math] owns the asset
- In this model the value of the good in its alternative usage is key
- Again the trigger strategy is used that calls for spot contracting for all time if default is observed, and again the efficient form of spot contracting is used, with a transfer of the asset if needed.

If U accepts the contract then she chooses [math]a^{RO}\;[/math] to solve:

- [math]max_a U^{RO} = s + B_{LL}(1-q)(1-p) +B_{HL}q(1-p) +B_{LH}(1-q)p +B_{HH}qp - c(a)\;[/math]

[math]D\;[/math]'s expected payoff is then:

- [math]D^{RE} = \mathbb{E} ( Q_i - s -bij | a= a^{RO}) = Q_L + q(a^{RO}) \Delta Q - (U^{RO} + c(a^{RO}) )\;[/math]

Note that [math]U^{RO} = U^{RE}\;[/math] and [math]D^{RO} = D^{RE}\;[/math], but the temptations to renege are different.

If [math]D\;[/math] reneges, then as opposed to receiving [math]Q_i - b_{ij}\;[/math] he renegotiates to buy at the spot price of [math]\frac{1}{2}(Q_i + P_j)\;[/math]. Likewise [math]U\;[/math] is supposed to sell at [math]b_{ij}\;[/math] but instead may demand the spot price.

#### Outsourcing in default

If [math]S^{SO} \gt S^{SE}\;[/math] then outsourcing is efficient in default and is used.

Then [math]D\;[/math] will honour rather than renege iff:

- [math]Q_i - b_{ij} + \frac{1}{r} D^{RO} \ge \frac{1}{2}(Q_i - P_j) + \frac{1}{r}D^{SO}\;[/math]

- [math]\therefore b_{ij} - \frac{1}{2}(Q_i + P_j) \le \frac{1}{r} (D^{RO} - D^{SO})\;[/math]

Likewise [math]U\;[/math] will honour rather than renege iff:

- [math]b_{ij} - \frac{1}{2}(Q_i + P_j) \ge \frac{1}{r} (U^{SO} - U^{RO})\;[/math]

Combining these together using the max and min values as before gives that the relational contract is enforceable if:

- [math]\max(b_{ij} - \frac{1}{2}(Q_i + P_j)) - \min( b_{ij} - \frac{1}{2}(Q_i + P_j) ) \le \frac{1}{r} (S^{RO} - S^{SO})\;[/math]

#### Employment in default

If [math]S^{SE} \gt S^{SO}\;[/math] then employment is efficient in default and is used.

Then [math]D\;[/math] will honour rather than renege iff:

[math]b_{ij} - \frac{1}{2}(Q_i + P_j) \le \frac{1}{r} (D^{RO} - D^{SE}) -\pi\;[/math]

Likewise [math]U\;[/math] will honour rather than renege iff:

[math]b_{ij} - \frac{1}{2}(Q_i + P_j) \ge \frac{1}{r} (U^{SE} - U^{RO}) + \pi\;[/math]

Combining these together using the max and min values as before gives that the relational contract is enforceable if:

[math]\max(b_{ij} - \frac{1}{2}(Q_i + P_j)) - \min( b_{ij} - \frac{1}{2}(Q_i + P_j) ) \le \frac{1}{r} (S^{RO} - S^{SE})\;[/math]

#### A general condition

Given the two conditions from the two default scenarios we can write a single condition that is both necessary and sufficient:

- [math]\max(b_{ij} - \frac{1}{2}(Q_i + P_j)) - \min( b_{ij} - \frac{1}{2}(Q_i + P_j) ) \le \frac{1}{r} (S^{RO} - \max(S^{SE},S^{SO})\;[/math]

### Comparing Relational Contracting

The two (nec. and suff.) constraints for relational contracting are as follows:

**Inside the firm**:

- [math]\max b_{ij} - \min b_{ij} \le \frac{1}{r} (S^{RE}-\max(S^{SO},S^{SE})\;[/math]

**Between firms**:

- [math]\max(b_{ij} - \frac{1}{2}(Q_i + P_j)) - \min( b_{ij} - \frac{1}{2}(Q_i + P_j) ) \le \frac{1}{r} (S^{RO} - \max(S^{SE},S^{SO})\;[/math]

These constraints differ! That is asset ownership affects the temptations to renege on relational contracts!

Proposition: Asset Ownership affects the parties' temptation to renege on a relational contract, and hence affects whether a relational contract is feasible.

A corollary to this finding is also very important. It is that:

Corollary: It is impossible for a firm to mimic the spot-market outcome after it brings the transaction inside the firm because the temptation of reneging is too great.

The payoffs that would match a spot market are:

- [math]s=0, b_ij = - \frac{1}{2}(Q_i + P_j)\;[/math]

Plugging these into the constraint for relational employment:

- [math]\max b_{ij} - \min b_{ij} \le \frac{1}{r} (S^{RE}-\max(S^{SO},S^{SE})\;[/math]

Shows that the condition can not hold - the temptation to renege is too great.

This is as far as the solution can be pushed without functional forms for [math]q(a)\;[/math], [math]p(a)\;[/math] and values for [math]\Delta Q\;[/math], [math]\Delta P\;[/math], and [math]r\;[/math], which is addressed in the paper.