Set-up of the model and the social optimum

Let V(x) be the brewery's gross profit from the sale of beer, where x is its initial investment (i.e., how much beer it brews), and V′ > 0 and V(x)″ < 0. Storage and sale of the beer, however, requires the delivery of ice at a cost C, which is not known with certainty at the time the brewery chooses x. The sequence of decisions is therefore crucial: first, the brewery must choose x, then the cost of supplying ice is realized, and finally, the decision of whether or not to supply the ice is made. If the ice is delivered, the transaction goes ahead as planned, yielding a joint return of V(x) − C − x, but if it is not, the initial investment, x, is lost.

As usual, the social optimum is derived in reverse sequence of time. Thus, once C has been realized, it is optimal for delivery of the ice to occur if C ≤ V(x) for any value of the sunk investment x. In contrast, if it turns out that C > V(x), then the ice should not be supplied, even though this means the loss of the brewery's initial investment in beer. Now move back to the time when the brewery has to make its choice of x. From a social perspective, it should choose x to maximize the expected value of the beer, subject to uncertainty regarding the delivery of ice. Specifically, because there are some states of the world where the realized cost of ice will make delivery inefficient, the brewer should invest less than if delivery were certain. In other words, the brewer should hedge against the possibility of non-delivery so as to mitigate the amount of the loss in that state.

The actual performance and investment decisions will be made by the relevant decision-makers based on the organizational form governing the ‘transaction’ between the brewer and ice supplier. The next three sections derive the resulting equilibria: first, under a market (spot) transaction, then under a contractual arrangement, and finally under vertical integration.

Market transaction

When the brewery and ice company interact in a market setting, they engage in a spot transaction for the sale of ice after the cost of ice production is realized. I assume that they complete a transaction whenever it is jointly profitable given the realized cost, C, and the brewer's prior investment, x; that is, whenever V(x) ≥ C. In other words, I assume Coasian bargaining takes place at this point (Coase, Reference Coase1960).Footnote ⁹ I further assume that the price of the ice is determined by generalized Nash bargaining. The resulting price is therefore

(1)\begin{equation} P = {\alpha}V(x) + (1 - {\alpha})C,\end{equation}

where α represents the strength of the ice company's bargaining power.Footnote ¹⁰ The key thing to note here is that the price is increasing in x as long as α > 0. Thus, the more that the brewery has invested in reliance on the promised delivery of ice, the higher will be the price that the latter can extract at the time of the transaction. This represents the source of the holdup problem that plagues spot transactions in the presence of prior transaction-specific investments. In particular, because the brewery rationally anticipates the outcome of its subsequent bargaining with the ice company, it will underinvest in beer compared to the efficient level in order to lessen its vulnerability to the holdup problem. This is the primary disadvantage of market exchange in this context.

Court-enforced contract

As an alternative to engaging in spot transactions, the brewery and ice company may choose to enter into a legally enforceable contract that commits the ice company to deliver ice at a pre-determined price over some prescribed time period. The advantage of this arrangement from the brewery's point of view is that it avoids the need for the parties to negotiate a price for each transaction after C has been realized. As a result, it eliminates the holdup problem. In addition, it creates an obligation on the part of the ice company to perform, which the brewery can enforce by legal means. In other words, in those states where the ice company chooses not to perform, the brewery can seek damages in court. (I ignore here the risk of bankruptcy on the part of the input supplier in high-cost states.) Let R be the pre-determined contract price, payable on performance,Footnote ¹¹ and let D be the damages in the event of breach as set by the court.

The timing of events in this case is as follows. First, the brewery and the ice company sign the contract and specify the contract price, R;Footnote ¹² then the brewery chooses x; and finally, the ice company realizes the cost of performance, C, and decides whether or not to perform. In the event that it chooses not to perform, the parties go to court, which orders the ice company to pay damages of D to the brewery.

Considering first the ice company's performance decision, once C is realized, it will deliver the ice if R − C ≥ − D, or if

(2)\begin{equation} C \le R + D,\end{equation}

and breach if the reverse is true. Recall that the condition for efficient performance, given x, is C ≤ V(x). Thus, the measure of damages for breach that induces the ice company to perform efficiently is D = V(x) − R, which is the ‘expectation damage’ measure (Shavell, Reference Shavell2004: 343–344). Under this measure, the ice company is required to pay the brewery its surplus from the transaction, conditional on its actual level of investment. Expectation damages induce efficient performance because they force the ice company to internalize the full cost of breach to the brewery, given its prior investment of x. (Note that such a measure assumes the court is able, after the fact, to observe and condition the amount of damages on the brewery's actual loss of income as a result of the breach; namely, V(x)).

Given the above damage measure, the brewery anticipates that the breach decision will be made efficiently by the ice company for any amount of x that it invests. In other words, the transaction will occur efficiently, as was the case under the spot transaction. The problem, however, is that the brewery now has an incentive to overinvest in beer production. The reason is that the expectation damage measure fully insures it against non-delivery of ice and hence creates a moral hazard problem (Shavell, Reference Shavell1980).Footnote ¹³ The result here is therefore the opposite of the outcome under market organization, which resulted in underinvestment by the brewery.

Vertical integration

The third alternative to the market or contract is for brewery and ice company to vertically integrate; that is, to organize their transaction within a single firm. For concreteness, I assume that the brewery takes control of the ice company, though the reverse is also possible (see, for example, Grossman and Hart, Reference Grossman and Hart1986). Thus, I will treat decisions concerning the production and delivery of ice as being made by ‘employees’ of the firm in return for a wage, while the ‘owner’ (the residual claimant) decides how much to invest in brewing the beer. I will model the relationship between the residual claimant and the workers as a principal-agent problem.

The distinguishing feature of this type of organizational form is that it outlives the individual transaction. As a consequence, the employees must be paid in both the delivery and non-delivery states, though I will allow the wages to differ in those states. Specifically, suppose that in the states where the (single) worker is asked to produce and ‘deliver’ ice, he is paid a wage of w, but when he is not asked to produce due to high costs, he is paid a wage of w ₀, where w ₀ is defined to be the minimum wage that prevents the worker from quitting (hence, w ≥ w ₀). In this specification, the cost of delivery, C, is interpreted to be the worker's realized, and privately known, cost of performance.

The owner of the integrated firm chooses the maximum cost under which it will produce. Let Ĉ be this threshold cost level. Thus, the firm will produce when C ≤ Ĉ and not produce when C > Ĉ. However, because the owner cannot observe the realization of C, he must create an incentive contract for the employee to produce only in those states where C ≤ Ĉ. In other words, the owner must specify a contract consisting of the wages, w and w ₀, and the threshold cost, Ĉ, to maximize profits, subject to the following constraints:

(3)\begin{equation} w - C \ge w_0 ,\quad C \le \hat C,\vspace*{-18pt}\end{equation}

(4)\begin{equation} w - C < w_0 ,\quad C > \hat C.\end{equation}

Constraint (3) says that the worker, after privately observing the cost of production, prefers to produce in the ‘low-cost’ state, while (4) says that the worker prefers not to produce in the high-cost state but instead to receive the reservation wage. Figure 1 depicts the ranges for C graphically. Note that, given w ₀, which is determined by the next-best option of workers, the choice of w uniquely determines Ĉ (and vice versa). Thus, only one of these variables needs to be independently chosen.

Figure 1.

Ranges for production and non-production under vertical integration.

The problem for the owner/brewer is therefore to choose x and w to maximize profit, subject to constraints (3) and (4). It turns out that, in the optimal solution, the firm will underproduce given x – that is, it will take delivery of ice too infrequently from a social point of view. Intuitively, because the brewer/owner cannot observe the worker's cost of delivery, he must pay the worker a high enough wage to produce in the desired range (i.e., when C ≤ Ĉ). Thus, by reducing the range of production slightly, he can lower the wage he has to pay in production states. (In Figure 1, a reduction in Ĉ coincides with a lower w.) At the optimum, the marginal reduction in the wage bill just equals the lost profit from production. Given the underproduction, however, the firm invests efficiently in x because the owner fully internalizes the cost and benefit of the investment.

Comparison of the various forms

The conclusions from the preceding sections are summarized in Table 1. Note that all of the organizational forms involve some inefficiency: the spot market and contract both result in efficient production (delivery of ice) once the cost of performing is realized, but the brewery invests inefficiently under both. In particular, it underinvests under market exchange in a strategic effort to mitigate the holdup problem, and it overinvests under the long-term contract due to the moral hazard problem created by the expectation damage measure. In contrast, when the brewery and ice company merge, the integrated firm invests efficiently given the anticipated level of production, but it underproduces as a result of the agency costs associated with the workers’ private knowledge of the cost of ice production.

Table 1.

Outcomes under the various organizational forms

Since none of the organizational forms achieves the first-best outcome, the choice among them depends on which involves the least distortion. In the context of the current model, this will depend on three factors: (1) the importance of the brewery's up-front investment; (2) the severity of the holdup problem that may arise after beer production is complete and delivery costs are realized (which depends on the relative ex post bargaining strengths of the parties), and (3) the degree of ex ante uncertainty over the ice company's costs of supplying the essential input (which determines how likely breach is.) To illustrate the nature of the choice, I first consider some special cases and then develop a numerical example.

Suppose first that investment by the brewery in anticipation of performance is not important (i.e., V is fixed). In other words, the asset specificity problem is not severe in the sense that the brewery could easily salvage the initial investment if delivery of the ice turns out to be infeasible or undesirable. In that case, the risk of a holdup by the ice company is small or non-existent, and so both the market and contract yield the efficient (first-best) outcome because they both result in efficient production once costs are realized. In other words, when asset specificity is not a severe problem, arm's length transactions yield the maximum gains from trade. In contrast, the firm results in underproduction because of the agency costs. Thus, when up-front investments are not important, we would expect spot market transactions and contracts to dominate vertical integration.

Alternatively, suppose that the up-front investments are important but production costs are known, which makes performance perfectly predictable. In this case, only the holdup problem remains, so the spot market will result in too little up-front investment. In this case, the transaction is best governed within the confines of a firm or by a contract. In the context of the simple example, both organizational forms will be efficient because of the hypothesized absence of cost uncertainty, which, recall, was the source of both the agency costs under vertical integration and moral hazard under contracting. In reality, of course, the choice will depend on other factors that affect the relative costs of the two forms, which will likely depend on both the scale and expected duration of the relationship. In the brewery context, for example, one might expect small breweries to enter into contracts with ice companies, as was the case in Goebel, and large breweries to build and operate their own ice houses.

In the most general case where production costs are uncertain and up-front investments are important, none of the forms yields the efficient outcome. Thus, all we can say is that, as the likelihood of breach increases, the level of investment under the market more closely tracks the efficient level of investment as compared to the contract. This is true because the level of investment under the contract is not sensitive to the probability of breach, owing to the moral hazard problem arising from the breach remedy, and thus diverges further from the efficient level as that probability increases. In contrast, the level of investment under the market, though inefficiently low, at least declines with the probability of breach. Consequently, as the probability of breach increases, the market and firm will tend to dominate the contract, with the choice between the former two options depending on the relative costs of the holdup and agency problems. Again, one would expect that for more enduring relationships, the firm would be the preferred choice.

To gain further insights, we consider a simple example. Let the brewer's gross profit be V(x) = θx ^1/3, and assume that C is uniformly distributed on [0,4]. A comparison of the various organizational forms involves computing the net value of production under each form while varying the production parameter θ, and the bargaining power parameter α. Figure 2 provides a stylized representation of the optimal organizational form for different combinations of θ and α.Footnote ¹⁴ In the figure, the vertical axis, represented by α, reflects the severity of the holdup problem, with higher α corresponding to greater bargaining power on the part of the ice company, while the horizontal axis, represented by θ, reflects the importance of the brewery's up-front investment, with higher θ corresponding to an increased probability of performance, given the (fixed) distribution of costs.

Figure 2.

Regions where different organizational forms dominate.

Note first that the market dominates for all θ when α is low (near zero). This is true because the brewery expects to receive a large share of the gains from any ex post bargaining under a spot transaction and thus internalizes most of the gains from trade. In other words, the holdup problem is not severe. (In the extreme case where α = 0, the holdup problem is absent and the outcome is first-best under market exchange.) As α rises, however, the holdup problem worsens, and eventually the firm or contract dominates the market, with the choice depending on the value of θ. Specifically, when θ is high, meaning that the likelihood (and value) of production is high, the contract dominates the firm. In this case, the risk of breach is low, and so the moral hazard problem associated with contract breach is small compared to the agency costs under a firm. In other words, the value of the contract in allowing the parties to pre-commit to a price, coupled with the high probability of performance, makes it the preferred form. In contrast, when θ is low, and consequently the likelihood of breach is high, the opposite is true, and the firm dominates the contract.