Plate xi in print

The Contingency

the contract as a state-contingent claim, and the dream of complete markets in which every future could be insured against, were it only nameable.

A claim can divide the harvest; it cannot promise more grain than the harvest supplies.

— the argument of this plate, after Arrow (1964)

An unconditional promise carries one instruction into several possible futures. A contingent promise gives those futures different instructions. Pay if the shipment arrives; replace the crop if the specified loss occurs; deliver a larger quantity when the agreed measure of demand rises. The condition is part of what is bought. A claim that pays in a prosperous year and a claim that pays in a disastrous one can have the same face amount while meeting very different needs. The idea of a complete system of claims extends this observation to its limit. Imagine that every relevant future can be distinguished, and that people can exchange promises appropriate to each. The question becomes what such promises could accomplish, and which parts of the imagined completeness reside in the world rather than in the words describing it.

Arrow's analysis of securities supplies a precise benchmark. It considers finitely many states and a pure exchange economy, with available resources specified in each state. Contingent claims can allocate those resources before uncertainty resolves. Financial claims followed by commodity trading after the state becomes known can also support the allocation under the model's conditions.1 The benchmark separates a useful question from a much larger fantasy. Who bears a loss can be changed by an agreement. Whether there is a loss to bear depends on what happens to resources. A promise written before the harvest can determine which household receives the grain that survives. Its wording cannot alone replace grain that was never grown. Completeness concerns the available arrangements for claims across specified states; abundance is another matter.

1. K. J. Arrow, “The Role of Securities in the Optimal Allocation of Risk-bearing”, Review of Economic Studies 31 (1964), §§I–III, pp. 91–94. This is a finite-state exchange benchmark with feasibility and preference assumptions, not a claim that all uncertainty disappears. Its securities construction retains later commodity markets.

Take two households and two equally likely harvest states. In the first, one household has eighty units and the other twenty; in the second, their positions reverse. If each transfers thirty to the other when it is the prosperous household, both receive fifty in either state. Total resources remain one hundred throughout. Each household's expected resources were fifty before the agreement and remain fifty afterward. What changes is the distribution of its own possible consumption: eighty or twenty becomes fifty for certain. A household that prefers a stable fifty to this equal-probability fluctuation gains from the arrangement without a larger expected harvest. The counterparties can insure each other because their initial exposures differ. Their agreement changes who receives what the harvest supplies.2 If both instead have twenty in the same bad state, no transfer between them can give both fifty there.

2. The example assumes enforceable transfers, costless contracting, equal state probabilities, and no change in production incentives. In state one, 80 − 30 = 20 + 30 = 50; state two reverses the transfer. Initial expected consumption is (80 + 20)/2 = 50. The joint bad-state control has only forty units available. Preferences determine whether smoothing is desirable.
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Even naming the states does not establish that the available instruments can separate them. Imagine three possible outcomes, but only a bond paying one unit in every outcome and a claim paying one unit in the first alone. A portfolio holding quantities a and b of these instruments pays a + b in the first outcome, a in the second, and a in the third. However the quantities are chosen, the last two payments are identical. A household seeking protection only in the third outcome cannot obtain it from this pair. Add a claim paying only in the second, and the three outcomes can be distinguished algebraically: the third-state claim is one bond less one first-state claim and one second-state claim. This replication assumes the household may take those negative positions and can meet the resulting obligations. Trading restrictions can prevent a possible formula from becoming an available transaction.3

3. The payoff vectors are (1,1,1) and (1,0,0). Their portfolios have form (a + b,a,a), so cannot produce (0,0,1). With (0,1,0) also available, (1,1,1) − (1,0,0) − (0,1,0) = (0,0,1). This is an illustrative linear replication, with divisible positions, permitted short sales, and reliable settlement.

A price for a contingent claim is also more than a forecast written in currency. In an illustrative market with no discounting, let claims paying one unit in the first and second of two exhaustive states cost 0.6 and 0.4. Buying both costs one and pays one whichever state occurs, consistent with a one-unit sure payment costing one. This arithmetic does not establish that the first state is sixty percent likely. The prices can reflect how valuable resources are when each payment arrives as well as beliefs about which state will occur. Even equal probabilities need not make protection against hunger as desirable as another unit during abundance. To infer probabilities from prices requires further assumptions about valuation and the pricing system. The contract records a priced claim, while the probability is one ingredient in deciding what that claim is worth.4

4. The two claim prices are stipulated, not estimated. Their sum is checked against the stated price of a sure unit under costless replication and reliable payment. For example, with equal probabilities and valuation weights 1.2 and 0.8, weighted prices are 0.5 × 1.2 = 0.6 and 0.5 × 0.8 = 0.4. The weights illustrate why probabilities alone do not determine prices.

The recovered inquiries in The Unforeseen and The Gap bring the benchmark back to the work of contracting. The parties must specify the event, observe or establish its occurrence, and arrange dependable performance. These are separate tasks. Rainfall can be named yet measured at a station that poorly represents a particular farm. Damage can be visible to the owner yet difficult for another party to verify. A payment can be perfectly described yet promised by someone unable to make it when the whole region suffers. Calling every future by a name addresses only the first difficulty. The design of evidence, reserves, settlement, and remedies determines which named promises can actually be used. A more detailed agreement may improve that design, but additional sentences do not automatically supply missing resources or reliable information.5

5. These are analytical distinctions developed from the preserved writing on incomplete contracts. The rainfall, verification, and common-shock examples are illustrative. They distinguish defining a condition, proving it, and funding the promised consequence; no claim is made that an unobservable or disputed event can never be described in an agreement.

The contingency is thus an achievement of differentiation. It allows the same parties to prepare different performances for different events, and lets resources flow toward the circumstances in which they are most useful. Its limit is equally concrete. A list of futures is not yet a set of feasible, affordable, enforceable exchanges, and even a complete set of claims must divide the resources that each future permits. The unfinished work lies in the connections between the sentence and the event, the event and the evidence, the evidence and the transfer. The final plate, The Code, examines an attempt to tighten the last connection: once a specified condition is reported, the machine carries out the promised step. What has been made automatic, and what remains outside its reach?