Contingent Consumption

Since we now know all about the standard theory of consumer choice, let's try to use what we know to understand choice under uncertainty. The first question to ask is what is the basic "thing" that is being chosen?

The consumer is presumably concerned with the probability distribution of getting different consumption bundles of goods. A probability distribution consists of a list of different outcomes—in this case, consumption bundles—and the probability associated with each outcome. When a consumer decides how much automobile insurance to buy or how much to

invest in the stock market, he is in effect deciding on a pattern of probability distribution across different amounts of consumption.

For example, suppose that you have $100 now and that you are contemplating Jouying lottery ticket number 13. If number 13 is drawn in the lottery, the holder will be paid $200. This ticket costs, say, $5. The two outcomes that are of interest are the event that the ticket is drawn and the event that it isn't.

Your original endowment of wealth—the amount that you would have if you did not purchase the lottery ticket—is $100 if 13 is drawn, and $100 if it isn't drawn. But if you buy the lottery ticket for $5, you will have a wealth distribution consisting of $295 if the ticket is a winner, and $95 if it is not a winner. The original endowment of probabilities of wealth in different circumstances has been changed by the purchase of the lottery ticket. Let us examine this point in more detail.

In this discussion we'll restrict ourselves to examining monetary gambles for convenience of exposition. Of course, it is not money alone that matters; it is the consumption that money can buy that is the ultimate "good" being chosen. The same principles apply to gambles over goods, but restricting ourselves to monetary outcomes makes things simpler. Second, we will restrict ourselves to very simple situations where there are only a few possible outcomes. Again, this is only for reasons of simplicity.

Above we described the case of gambling in a lottery; here we'll consider the case of insurance. Suppose that an individual initially has $35,000 worth of assets, but there is a possibility that he may lose $10,000. For example, his car may be stolen, or a storm may damage his house. Suppose that the probability of this event happening is p = .01. Then the probability distribution the person is facing is a 1 percent probability of having $25,000 of assets, and a 99 percent probability of having $35,000.

Insurance offers a way to change this probability distribution. Suppose that there is an insurance contract that will pay the person $100 if the loss occurs in exchange for a $1 premium. Of course the premium must be paid whether or not the loss occurs. If the person decides to purchase $10,000 dollars of insurance, it will cost him $100. In this case he will have a 1 percent chance of having $34,900 ($35,000 of other assets — $10,000 loss -f $10,000 payment from the insurance payment - $100 insurance premium) and a 99 percent chance of having $34,900 ($35,000 of assets - $100 insurance premium). Thus the consumer ends up with the same wealth no matter what happens. He is now fully insured against loss.

In general, if this person purchases K dollars of insurance and has to pay a premium 7K, then he will face the gamble:1

1 The Greek letter 7, gamma, is pronounced "gam-ma."

and probability .99 of getting $35,000 - 7K.

What kind of insurance will this person choose? Well, that depends on his preferences. He might be very conservative and choose to purchase a lot of insurance, or he might like to take risks and not purchase any insurance at all. People have different preferences over probability distributions in the same way that they have different preferences over the consumption of ordinary goods.

In fact, one very fruitful way to look at decision making under uncertainty is just to think of the money available under different circumstances as different goods. A thousand dollars after a large loss has occurred may mean a very different thing from a thousand dollars when it hasn't. Of course, we don't have to apply this idea just to money: an ice cream cone if it happens to be hot and sunny tomorrow is a very different good from an ice cream cone if it is rainy and cold. In general, consumption goods will be of different value to a person depending upon the circumstances under which they become available.

Let us think of the different outcomes of some random event as being different states of nature. In the insurance example given above there were two states of nature: the loss occurs or it doesn't. But in general there could be many different states of nature. We can then think of a contingent consumption plan as being a specification of what will be consumed in each different state of nature—each different outcome of the random process. Contingent means depending on something not yet certain, so a contingent consumption plan means a plan that depends on the outcome of some event. In the case of insurance purchases, the contingent consumption was described by the terms of the insurance contract: how much money you would have if a loss occurred and how much you would have if it didn't. In the case of the rainy and sunny days, the contingent consumption would just be the plan of what would be consumed given the various outcomes of the weather.

People have preferences over different plans of consumption, just like they have preferences over actual consumption. It certainly might make you feel better now to know that you are fully insured. People make choices that reflect their preferences over consumption in different circumstances, and we can use the theory of choice that we have developed to analyze those choices.

If we think about a contingent consumption plan as being just an ordinary consumption bundle, we are right back in the framework described in the previous chapters. We can think of preferences as being defined over different consumption plans, with the "terms of trade" being given by the budget constraint. We can then model the consumer as choosing the best consumption plan he or she can afford, just as we have done all along.

Let's describe the insurance purchase in terms of the indifference-curve analysis we've been using. The two states of nature are the event that the loss occurs and the event that it doesn't. The contingent consumptions are the values of how much money you would have in each circumstance. We can plot this on a graph as in Figure 12.1.

Insurance. The budget line associated with the purchase of insurance. The insurance premium y allows us to give up some consumption in the good outcome (Cg) in order to have more consumption to the bad outcome (C&).

Your endowment of contingent consumption is $25,000 in the "bad" state^—if the loss occurs—and $35,000 in the "good" state—if it doesn't occur. Insurance offers you a way to move away from this endowment point. If you purchase K dollars' worth of insurance, you give up 7K dollars of consumption possibilities in the good state in exchange for K — 7K dollars of consumption possibilities in the bad state. Thus the consumption you lose in the good state, divided by the extra consumption you gain in the bad state, is

A Cb K-yK 1-7' This is the slope of the budget line through your endowment. It is just as if the price of consumption in the good state is 1 — 7 and the price in the bad state is 7.

We can draw in the indifference curves that a person might have for contingent consumption. Here again it is very natural for indifference curves to have a convex shape: this means that the person would rather have a constant amount of consumption in each state than a large amount in one state and a low amount in the other.

Given the indifference curves for consumption in each state of nature, we can look at the choice of how much insurance to purchase. As usual, this will be characterized by a tangency condition: the marginal rate of substitution between consumption in each state of nature should be equal to the price at which you can trade off consumption in those states.

Of course, once we have a model of optimal choicc, wc can apply all of the machinery developed in early chapters to its analysis. We can examine how the demand for insurance changes as the price of insurance changes, as the wealth of the consumer changes, and so on. The theory of consumer behavior is perfectly adequate to model behavior under uncertainty as well as certainty.

EXAMPLE: Catastrophe Bonds

We have seen that insurance is a way to transfer wealth from good states of nature to bad states of nature. Of course there are two sides to these transactions: those who buy insurance and those who sell it. Here we focus on the sell side of insurance.

The sell side of the insurance market is divided into a retail component, which deals directly with end buyers, and a wholesale component, in which insurers sell risks to other parties. The wholesale part of the market is known as the reinsurance market.

Typically, the reinsurance market has relied on large investors such as pension funds to provide financial backing for risks. However, some reinsurers rely on large individual investors. Lloyd's of London, one of the most famous reinsurance consortia, generally uses private investors.

Recently, the reinsurance industry has been experimenting with catastrophe bonds, which, according to some, are a more flexible way to provide reinsurance. These bonds, generally sold to large institutions, have typically been tied to natural disasters, like earthquakes or hurricanes.

A financial intermediary, such as a reinsurance company or an investment bank, issues a bond tied to a particular insurable event, such as an earthquake involving, say, at least $500 million in insurance claims. If there is no earthquake, investors are paid a generous interest rate. But if the earthquake occurs and the clairiis exceed the amount specified in the bond, investors sacrifice their principal and interest.

Catastrophe bonds have some attractive features. They can spread risks widely and can be subdivided indefinitely, allowing each investor to bear only a small part of the risk. The money backing up the insurance is paid in advance, so there is no default risk to the insured.

Prom the economist's point of view, "cat bonds" are a form of state contingent security, that is, a security that pays off if and only if some particular event occurs. This concept was first introduced by Nobel laureate Kenneth J. Arrow in a paper published in 1952 and was long thought to be of only theoretical interest. But it turned out that all sorts of options and other derivatives could be best understood using contingent securities. Now Wall Street rocket scientists draw on this 50-year-old work when creating exotic new derivatives such as catastrophe bonds.

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