by Filip Palda
In a certain world a person will want to buy more apples if the value of apples (as measured by what he or she would be willing to give up in oranges) is lesser than the actual amount of apples he or she must give up to get an orange. In other words, substitute more apples if your relative valuation is greater than the relative price. To see how this applies to an uncertain world, imagine you are presently uninsured and are once again facing the bankruptcy scenario. Now ask yourself how much your expected utility would change if things were a bit different and you received a dollar less in the good state but a dollar more in the bad state. If you are like most people, then having a dollar less when you are already doing well will not cause much disutility.
But having an extra dollar when you find yourself in a destitute situation could mean a great deal to you. Economists believe this comes about because of diminishing returns to the pleasure we get from spending money. Every dollar spent gives more pleasure but in decreasing amounts. By implication every dollar lost to spending inflicts pain in increasing amounts. If you feel you live in a risky world where the gain of a dollar brings you less pleasure than the pain from losing a dollar then you might want to buy insurance. Economists call you risk-averse. In such a case you might want to trade income from the good state and shift it to the bad state.
Your fear of risk is just one factor determining how eager you are to buy insurance. What also matters is how likely it is that something bad will happen, and the cost of the insurance. Insurance allows you through the premium to shift income from the good state to the bad state.
Because of risk-aversion, that is, diminishing returns to the pleasure from income in any given state, as you buy insurance and shift income to the bad state, the expected value of an extra dollar in that state falls while the expected value of an extra dollar in the good state rises. The ratio of the two represents the extra money you are willing to give up in the good state for money in the bad state. Economists are always considering extras or “margins” because that is where people push things to decide whether to go a little further in their consumption of one good and less of the other. A little less expected utility in any state is called expected “marginal utility”.
The ratio of expected marginal utility from increasing income in the bad state to income in the good state might be five. This means you are willing to give up five dollars in the good state to get one dollar in the bad state, or inversely you are willing to give up a dollar in the good state to get twenty cents in the bad state. This ratio falls as you get more in the bad state and less in the good state. You stop buying insurance when the expected value of extra money to you in both states is equal to how much money you have to give up in the good state to buy a dollar of coverage in the bad state. In more general terms, you keep buying insurance until the rate at which you are willing to trade off income in both states is equal to the rate at which you are able to trade off that income. Once you have attained this equation you can do no better.
Now that is a quite a bit of reasoning to process but we should not be daunted. All we are saying is that people seek a sort of balance in their consumption between possible states that is commensurate with the price of dispersing income between these states and the value they attach to income in these states.
Remember that the price of trading off income in the good and bad states is the odds ratio in a competitive insurance market. Applying the logic of trading off marginal expected utility in different states so that they equal the odds ratio gives a surprisingly clear result. In perfect competition people would choose to be fully insured so as to assure themselves the same level of income in both states.
The mathematical proof can be found on page 161 of Hal Varian’s 1984 textbook but we can see quite easily why it is so. The rate at which you are willing to trade off income in the good state for the bad state is the probability of the bad state multiplied by the added pleasure of extra income in that state, divided by the probability of the good state multiplied by the added pleasure of extra income there (yes, all is meant as you read it). This expresses the relative valuation of an extra dollar in the bad versus the good states. Now suppose income in both states were the same. That means that extra, or “marginal” utility in both states would be the same. So in the ratio these two quantities drop out. All you have left in the ratio of willingness to exchange income in one state for another are the probabilities. And these represent the odds ratio. And there is your result. Consuming equally in both states converts your willingness to trade income in both states into the odds ratio. But the odds ratio is also the market price of shifting income from one state to the other. This proves that under perfectly competitive insurance markets and utility with diminishing returns to income in any given state, the desired policy is full insurance.
The troublesome question of separable utility
THE EMERGENCE OF the odds ratio as the price of insurance under perfect competition and the desire for full insurance under diminishing returns to income in any given state produce the clear and powerful result that people will seek to equalize income across states by purchasing full insurance.
The more precise a theory the more it is open to question. The theory of choice under uncertainty relies critically on the assumption that utility is separable between states. This is what allows us to speak of expected utility. We can separate the utility from income in any given state, weigh it by the probability of that state and add it to all other weighted utilities to arrive at the sum which is called “expected utility”.
The even stronger result of full insurance as the desired objective depends on the assumption of diminishing returns to income. In a way this assumption is the child of the separability of the utility function. In consumer theory, under certainty you can get the result that consumers prefer to evenly balance their consumption of goods, without falling back on the assumption of diminishing marginal utility. You just need to know that the relative desired rates of tradeoff between one good and another will diminish as you consume more of that good. Separable utility forces you to assume diminishing marginal absolute utility to arrive at a scenario where people seek to balance income between states.
Why does any of this matter? Because expected utility is an average this means its mathematical form must be a sum. This means that the income you get in one state should not influence the utility you get from income in another state. Formally, economists call this the separability of utility across states. Being so specific about the form of the utility function troubles economists because it forces them to take a specific stand on the sorts of preferences people have. That opens them to the critique that people do not react to chance in so particular a manner.
To justify their assumptions economists have tried to show that, in a way, they are not assumptions at all but rather features of maximizing utility under uncertainty that emerge from rock-solid “axioms”. The axiomatic approach to expected utility does not start with a specific utility function or any discussion of probabilities. It only makes some very general and reasonable assumptions about peoples’ attitudes to risk. What pops out of these assumptions is a utility function that must be separable across states and that utility in each state must be weighted by some fractional factor that can be interpreted as the probability of that state.
The axiomatic approach may suggest that expected utility maximization is a reasonable pursuit for most people but does not prove it. At worst it may simply prove how far very clever researchers will go to convince themselves that what they are doing makes sense. Most economists do not lose sleep over this issue, but it is a potential weakness of expected utility theory we should keep in tucked in the band of our thinking caps.
The main point to note is that we have shown how to conceive of consumption and the constraints upon it in an uncertain environment. While insurance markets are important and interesting, this analysis also has vast implications for many different forms of choice in uncertain settings. Before getting to the really
meaty topic of stock markets, which are an extension of insurance markets, I want us to take a breather and go on a mental safari into an economic bushland where notions or risk are seldom, but fruitfully applied.
Crime
AN IMPORTANT AND somewhat surprising consequence of studying decisions made in uncertain circumstances arises in the study of crime. A seemingly endless debate rages between those who say that greater punishments deter crime, and those who claim that, instead, greater funding is needed to increase the chance of catching criminals. The economic analysis of uncertainty judges that both sides in the debate have a point. A government that wants to have the most impact on crime needs to find the optimal “mix” of punishment and enforcement. This optimum turns on how people react to the mix.
Consider the white collar crime of tax evasion. The evader faces two possible states of the world: one in which no one detects the evasion and he or she profits; and one in which government discovers the crime, takes back the money owed, and imposes a fine. Every increase in the fine hurts progressively more with every additional dollar of penalty. This is the flip-side of the diminishing returns to getting rich. Fines are increasingly effective.
However, raising the probability of being caught only diminishes a person’s expected utility in constant steps. Going from a 5% to a 6% increase reduces expected utility by one per cent. Going from a 6% to 7% increase reduces expected utility by one per cent, and so on. On the cost side, raising fines is relatively cheap but increasing enforcement is increasingly costly.
The linear effect on crime of the probability of apprehension may be no match for the non-linear power of penalties as deterrents. Being caught is not so dreadful a prospect as being punished. Perhaps this explains why tax authorities can manage to collect vast sums without ever imposing fines on more than a small percentage of the population.
This logic has its limits in modern society. Pushed to the extreme it would prescribe lynching people for parking violations. We are not ready to accept such extreme penalties, perhaps because we fear that we may be mistakenly convicted. Several hundred years ago, punishments for crime were savage, probably because the enforcement “technology” did not exist to make the likelihood of apprehension sufficiently high to act as a deterrent.
While such examples may seem eccentric, they illustrate the principles that underlie a vast subfield of economics that analyzes crime, the underground economy, tax evasion, and criminal law. All are children of the basic economic analysis of choice under uncertainty.
CAPM
A SECOND AND far more widespread application of chance in economics lies in the pricing of stocks. The celebrated Capital Asset Pricing Model or CAPM is the one of the most successful applications of economic concepts to the real world. It tries to do what everyone wants, namely to show you if the market is under or overpricing certain stocks. At best you can use this model to make money. At very least you can use it to manage risk intelligently in your portfolio of investments. Yet despite its origins in the field of economics, the CAPM divides economists and finance academics into mutually disdainful camps.
Economists think of researchers in the finance field as specialized drones looking for patterns in financial data with the aid of models whose conceptual content is slim, and which show little sign of evolution nor scope for the application of economic imagination. Milton Friedman once chided the young Harry Markowitz, a founder of portfolio theory, for being a mathematician rather than an economist. Finance experts think of economists as flabby-thinking generalists who cannot be bothered to probe the details of their asset pricing models. Their lack of intellectual focus prevents economists from really grasping what is going on in finance. Economists simply don’t “get it”. A Linnaeus surveying these squabbles would suggest that finance specialists had broken from the trunk of economics to form a new species, jocus fiscus, or in English “finance jock”. Both camps have a point, but taking sides tends to obscure what asset pricing models are and what they can achieve.
We need to look past the squabbles of either camp and grasp three essential notions. We need to know what is meant by “asset pricing model”, we need an idea of what is important to people in their choice of financial assets, and we need to know how equilibrium comes about in the asset market. I am going to suggest to you a model of asset pricing so ridiculously simple you will wonder what all the fuss is about. Yet as we will see, it contains the major elements of the far more complicated but fundamentally no different models that now dominate finance.
Imagine a company that will close down tomorrow and split between its shareholders a guaranteed payoff. The price a share in this company should sell for is a subject for asset pricing theory. It is pretty clear in this case that a share should sell for exactly the value of the payoff coming to it. If one share is entitled to a hundred dollar payoff then no one would sell it to you for less than that. You could buy it of course, but then your return on the investment would be zero. Return is the net increase in the value of your share relative to what you paid for it, or more simply the percentage increase in share value. There is no net increase over the period of your ownership if you buy it the day before the payout.
In fact it is pretty clear that no matter what stocks you are talking about, an asset with a certain payout tomorrow or some short time thereafter will have zero return for investors. If an asset with certain payout had a positive return this would be an opportunity for massive profit-making by potential investors. If I offer you a share with a guaranteed payout tomorrow of a hundred dollars for the price of fifty you will think I am delusional but will be quite happy to buy as much of this share with whatever money you have in the bank. You will also run to loan-sharks to borrow as much money as you can, even at usurious rates as high as forty-nine percent overnight interest to buy the share, because every dollar you invest will give you a guaranteed rate of return of fifty percent in just one day. Buying low and selling high with a guaranteed return is called “arbitrage”. If others see the opportunity they will bid up the prices of these shares until the return comes back to zero.
Notice what we have just accomplished. We have built an equilibrium model of stock prices based on investor preferences. It may not seem like much of a model but it has some key elements. Preferences are simply that people prefer more to less. The price of the share is equal to its proportional payout. The price adjusts through a process of arbitrage to make the rate of return zero. People care only about share price insofar as it determines the rate of return on a dollar. Put differently, rate of return is the object of focus. Share prices adjust in the background to equilibrate rates of return. This feature of asset pricing models leads to enormous confusion as to why they are called “pricing” models. The only thing that this model of arbitrage determines conclusively is the rate of return. But rate of return by itself is not a sufficient datum for share price. Price is a function as well of the final payout.
Asset pricing models take this payout as given and have nothing to say on how it arises. Payouts are determined by “market fundamentals” such as demand and supply for the firm’s product. Thus a division of labor arose in asset pricing models. Finance jocks would focus on how arbitrage determined rates of return and content themselves with an intellectually narrow activity which could bring in a great deal of money. Economists could be left to go deeper and worry about fundamentals such as what determined the price of a firm’s product, and be original while sewing leather patches over the threadbare fabric of their jacket elbows. So while it makes sense in a remote way to speak of asset pricing models, in reality they are asset return models. The distinction will perhaps become clearer as we spice the pot by making final payouts uncertain.
Suppose a firm’s final payout could be higher than what you paid for the share with a certain probability and lower with another probability. In the first case your return is positive, in the second it is negative. If you are not worried about risk, that is, the pleasure you get from an addition
al dollar is equal to the loss of pleasure you get from having one dollar less, then all you really care about is the average, or more precisely, the expected return. This is pretty much the same scenario as in the certainty case. Prices have to adjust so that expected return goes to zero.
Things start to become interesting if your preferences exhibit diminishing returns to income. Put differently you get less pleasure from an upswing in your revenues than you feel pain from an equivalent loss. In such a case you are said to be shy of risk, or risk averse. The maximum amount a risk averse person would be willing to pay in certain dollars for a share of the stock, and by extension the minimum acceptable return, is not so simple to calculate.
As a utility maximizer you would take certain dollars out of your pocket to pay for a stock with uncertain returns until the loss of utility from a certain dollar just equals the gain in utility from that dollar placed in the risky stock. Put in these terms the arbitrage condition becomes obscured because expected return is no longer the quantity of interest. What matters to people in these circumstances is how returns are dispersed among the different possible states of the world.
A stock which gives the possibility of a very high and a very low return is less desirable than one which gives the possibility of the same average return, but with spreads only half as high and half as low. Both stocks have the same expected return but what matters is how these returns are dispersed. The stock with high dispersion gives you the possibility of lots of money which you value in decreasing increments, and the possibility of very low, negative returns which make you suffer in increasing increments. Thus when we bring expected utility into the picture the individual levels of return in different states take on importance. That was not the case in the certainty and risk-neutral cases where only the average of returns mattered.