by Filip Palda
Conclusion
TIME IN ECONOMICS has two faces. The first face, glanced upon by Milton Friedman and Franco Modigliani was that if you expected a fixed stream of income over your life then you could reduce all your decisions about how much to consume in any give year to one variable: the present discounted value of the sum of your income stream. The Friedman-Modigliani synthesis produced some insights on the effect of government stimulus packages. But it ignored that people’s income streams are not fixed in a present value sense. That was the second face of time.
By investing in new technology or education you can raise the present value of your lifetime income. The effects of investment may be cumulative. This cumulative feature of investment presents people with a dilemma. You can save heavily today and massively increase your future income but this reduces your present enjoyment. How should you choose the best from an infinite number of investment paths to maximize the happiness derived from consumption over your lifetime? This turns out to be a very difficult problem with no pat answer. Economists applying the techniques of dynamic optimization to the consumer problem had some success but also got it into their heads to see if government could not apply similar techniques to guide the economy. The problem with government’s use of these techniques was that individuals could anticipate what government was doing and subvert the optimal program. The result was a government policy that was “time-inconsistent”.
There are many applications of the theory of time in economics I have not talked about, such as the Swan-Solow model of economic growth, and the Lucas-Romer approach to economic development. Surveying the field of dynamic macroeconomics was not my goal. I have discussed the essentials of time with an eye to understanding the manner in which people decide to consume and produce. With these insights in your pocket you can easily move to a mastery of the entire field of time in economics.
References
Barro, Robert J. 1974. “Are Government Bonds Net Wealth?” Journal of Political Economy, volume 82: 1095-1117.
Becker, Gary, and Kevin M. Murphy. 1988. “A Theory of Rational Addiction.” The Journal of Political Economy, volume 96: 675-700.
Blanchard, Olivier Jean and Stanley Fischer. 1989. Lectures on Macroeconomics. The MIT Press.
Chang, Alfa. 1992. Dynamic Optimization. McGraw-Hill.
Dreyfuss, Stuart. 2002. “Richard Bellman on the Birth of Dynamic Programming.” Operations Research, volume 50: 48-51.
Hall, Robert E. 1978. “Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence.” Journal of Political Economy, volume 86: 971-987.
Hendry, David F. 1980. “Econometrics—Alchemy or Science?” Economica, volume 47: 387-406.
Keynes, John Maynard. 1939. “Reviewed Works.” The Economic Journal, volume 49: 558-577.
Kydland, Finn, and Edward C. Prescott. 1977. “Rules Rather than Discretion: The Inconsistency of Optimal Plans.” The Journal of Political Economy, volume 85: 473-492.
Kydland, Finn, and Edward C. Prescott. 1982. “Time to Build and Aggregate Fluctuations.” Econometrica, volume 50: 1345-1370.
Leeson, Robert. 2000. The Eclipse of Keynesianism: The Political Economy of the Chicago Counter-Revolution. Palgrave.
Lucas, Robert E., Jr. 1976. “Econometric Policy Evaluation: A Critique.” Carnegie-Rochester Conference Series on Public Policy, volume 1: 19-46.
Sargent, Thomas J. 1987. Dynamic Macroeconomic Theory. Harvard University Press.
CHANCE 4
CHANCE OFTEN STRIKES PEOPLE AS the sole domain of statistics. Is it not the statistician’s first duty to calculate how likely some uncertain event is, such as a coin flip coming up heads? The “probability calculus” that statisticians developed to aid them in this pursuit can be quite daunting to learn but life as we know it in the modern world could not proceed without it. The entire life insurance business depends on the mathematics developed by statisticians to predict life expectancy. Yet knowing how likely an event is does not tell you how to react in the face of uncertainty. If you know that your life will end with a certain probability at a certain age and someone tries to sell you life insurance, you have to ask if, given the probabilities, the price of the insurance is reasonable. Thus probability by itself is no guide to action. People must weigh the probabilities they observe along with other factors to decide how to face an uncertain future. Understanding how they decide has become largely the domain of economists.
People of course had been shielding themselves from the vagaries of chance long before the economist came on the scene to explain to them how they are thinking. The extreme conservatism of most cultures throughout history shows how daunting a force chance has been in limiting human endeavors. Hamlet summarized this attitude to risk in extreme form when he ruminated “that the dread of something after death, the undiscovered country, from whose bourn no traveller returns, puzzles the will, and makes us rather bear those ills we have, than fly to others that we know not of.” Economists only really started having something to say when risk became a marketable commodity in the 19th century. This happened with the development of insurance markets. They had even more to say with the development of stock markets in the 20th century.
The study of choice under uncertainty is not just a meaningful way of understanding how insurance and stock markets work, but may also cast light on seemingly unrelated social changes. By allowing people to shield themselves from risks, modern financial innovations have liberated the individual from the terrors of uncertainty. Without these terrors, human creativity and self-expression have increased exponentially. We become acutely aware of this liberation when we compare developed economies to economies where insurance and stock markets are poorly developed. The extremely conservative social attitudes that prevail there may reflect an economic necessity to tone down individualism and the concomitant risky exploration of possibilities this entails. Without insurance markets such risks may be deemed excessive. The message for those wishing to foster economic development is that understanding the relation between insurance markets and social attitudes may be the key to helping some societies break out of excessively conservative, and possibly limiting outlooks.
Thus we see that in the case of chance, as in so many other economic fields, the seemingly dry topic of optimization in the face of some constraint is actually a key to understanding profound changes in society. However, before we can understand these social consequences, we need to understand what exactly is being optimized in an uncertain world and what the relevant constraint is. Let us start with the perhaps simpler notion of a constraint, then move on to analyzing objectives under uncertainty.
The constraint
A CONSTRAINT IN economic markets shows consumers the possibilities for purchase that they face. What the consumer garners from his or her budget constraint is how much income is available for him or her to spend and the price of various goods on which it can be spent. This may seem quite straightforward when we are considering how many apples and oranges we can buy. If an orange costs twice as much as an apple then your budget constraint enables to you to trade-off two apples in exchange for one orange. But how does any of this translate to an uncertain world?
Economists ask us to think of uncertainty as splitting the world into a variety of possible future states. Apples and oranges will still taste the same in all states and our preferences will be constant. The only factor that distinguishes states is how likely they are to be realized. There may be an extremely unlikely state of the world in which you lose all your money and a likely state in which your money is intact and ready to be spent on apples and oranges. The issue of trade now becomes one not of how many oranges and apples to consume, but of how much income to give up in the good state in exchange for income in the bad state. Did you catch that? Few people do the first hundred-or-so times around, but the idea is really quite familiar to us in our everyday lives.
Suppose you have $100,000 in a bank that might or might not go bankrupt next year.
This fear has robbed the ordinary person of sleep since banks for the masses came into prominence in the 18th century. In a way, you are looking at a store shelf upon which sit two possible products. In the “state of the world” in which the bank goes under you come out of the “store” with your shopping cart empty. If the bank survives you leave with $100,000 in your cart. Those are quite extreme alternatives. It would be nice if you could trade off money in the good state (no bankruptcy) to “buy” extra money in the bad state (bankruptcy) just as in a normal store you can give up a few apples to buy more oranges.
This sort of trade-off exists in the real world. It is called insurance. Suppose that if you pay $1000 now for full insurance, then in the bad state of the world you get $100,000 of insurance claim less your $1000 premium. And in the good state you just keep your $100,000 but are out of pocket for the $1000 premium. You have “bought” $100,000 of income in the bad state with $1000 from the good and bad states. So you end up with $99,000 of net income no matter what state occurs through the purchase of full insurance. Insurance here helps you “span” future uncertain states of the world so that you find yourself at the average of states rather than at their extremes. Surprisingly the price of trading income between states is not the premium, though it is related to the premium. The price is how much of one dollar in one state must be given up to get a dollar in another state. In this example you have traded off income in the good state for income in the bad state at a rate of one dollar for ninety-nine. That is the relative price of trading income between states. You give up a thousand dollars in the good state and get a hundred thousand in the bad state, less the thousand you paid as a premium, for a payoff of ninety-nine thousand. Hence the cost is one cent in the good state to buy ninety-nine cents in the bad state.
Let us store this piece of information and go see now how much a firm will charge you in a competitive insurance market. Bear with me because the result is quite surprising and of crucial importance to understanding what happens not just in insurance markets but also in stock markets.
The odds ratio
I DO NOT really like to keep people in suspense so allow me to blurt out what the price will be. Competitive insurance markets bring the price of a dollar in the bad state (in terms of money in the good state) down to the ratio of the probability of the bad state divided by the probability of the good state, the “odds ratio”.
Economists arrive at this result by conceiving perfect competition as a condition of the market in which the expected profits of insurance firms are zero or close to it. Firms are held to be effectively neutral with respect to risk not because their owners are fearless in the face of the vagaries of chance but because, in a very precise manner, chance poses no risk for firms. Firms attain this condition by pooling thousands of individuals into their clientele. While one individual’s fate may be highly uncertain, the fates of the many assume a near certain character.
To grasp the absence of risk to firms managing risk, think of playing a slot machine where in one game out of ten you will double your dollar. You might play ten times and win nothing. But if you played ten thousand times your winnings would come close to being a thousand dollars. Repeated play is similar to the law of large numbers by which extreme runs of good and bad luck cancel each other out, and all that is left are the average or “central” tendencies of the game.
Insurance companies, especially those selling life insurance, employ statisticians to calculate these probabilities for the human “slot machine”. The statistics tell them that people of a certain age share a common chance of dying. What this means, strangely enough, is that the best price an insurance firm can offer you depends not on your own personal risk of dying, but on the group of people, or “pool” into which the company puts you. There are two types of risk in this pool. One is called idiosyncratic, and the other is called systemic or “aggregate” risk.
Idiosyncratic risk is the particular deviation from the norm an individual brings with him into the insurance pool. It should have nothing to do with what the insurance company charges him or her. Pooling large numbers of people together and adding in the positive elements of idiosyncratic chance cancels the negative elements. An efficient insurance firm “diversifies away” all idiosyncratic risks to offer its clients cheaper insurance than the competitor. The competitor must follow suit. The result is that no one charges you for any genetic quirks you bring to the pool.
After diversifying away idiosyncratic risk, what an insurance company is left with is that an almost certain fraction of an age cohort, representing the common element of chance, will die in some future year. These fractions determine the price of insurance, due to the economic condition that perfect competition drives expected profits to zero. If the chance of death is one percent and the payout is one hundred thousand dollars, then what premium needs to be charged to assure zero expected profits? The answer is one thousand dollars. The firm pockets a thousand with certainty, but has an expected payout to make of a hundred thousand dollars times one percent, which is one thousand. From the point of view of the insurance consumer he or she has to give up a certain dollar in both states to get a hundred dollars in the bad state and zero dollars in the good state. Meaning that the tradeoff rate between money in the good and bad state is one in ninety-nine. But one in ninety-nine is exactly the odds ratio of the bad and good states happening. In a perfectly competitive insurance market the rate of tradeoff between good and bad states is the odds ratio. That is the relative price of spanning income states.
The first thing that strikes one about the relative price of spreading income between possible future states of the world is that here is the quest of an actuary pumped high on amphetamines. What did Caesar know about possible states when he crossed the Rubicon and exclaimed “let the dice fly high”? We have come a long way since the civil wars of the Roman Republic. We now know how to calculate the risks of letting the dice fly in most of the important situations that concern ordinary people, such as life, fire, and deposit insurance. This knowledge helps drive down insurance to its lowest possible price, as companies use this knowledge to outcompete each other. The perfect competition result of pricing insurance according to odds ratios leads to the highest possible level of wellbeing for people concerned about risk.
The third thing that strikes one is that demand and supply do not have much to do with the price. Most people think that supply and demand should interact so that high demand leads to high price. Not so in the insurance world. Only the supply side seems to matter. The implications of this weird aspect of the insurance market will rebound on our analysis of the stock market, where a similar insight holds.
Happiness in an uncertain world
WHAT HE HAVE achieved so far in our analysis of choice in uncertain circumstances is to establish that perfect competition creates insurance markets where the cost of shifting income between possible states is equal to the odds ratio of these states. How much income should the consumer buy in the bad state? The answer depends on his or her fear of the bad state coupled with the likelihood of this state. Economists believe that the individual comes up with a likely number for his or her satisfaction (or “expected utility”) if insured, and compares it to his or her utility if not insured. If the first number is greater, then you buy insurance.
Now here comes the tricky part. What “happiness number” do you attach to your choice of not buying insurance? You don’t know which of two possible ways the world will go. In the good state you would have much disposable income, in the bad state little. How do you attach one number then to happiness, or “utility”, arising from a situation in which two possible outcomes are possible? Put differently, since income in different states cannot be consumed simultaneously the way goods in the here and now can be consumed, we have to ask in what sense income levels in each state can be added up to produce a single number that tells you on average how well off you will be in the future. Economists call that number “expected utility”. It is the ce
nterpiece of choice under uncertainty because it enables an individual to compare his or her level of satisfaction with and without insurance. This, however, is not a simple average where you add the utility of riches in the good state and deprivation in the bad state and divide by two. Rather, it is a weighted average where you give a higher score to states that are more probable. Then you multiply each state by its score and add the two. The scores in question are the probabilities that one or the other state will be realized.
If your bank has only a one in a hundred chance of going under you would attach a score of only one in a hundred to the utility of the destitute state, and a score of ninety-nine out of a hundred to the utility of the prosperous state. Economists call the sum of these two weighted scores “expected utility”. You would make the same sort of calculation if you bought partial insurance, applying the one in a hundred chance to your level of compensation in case of accident, less the premium you pay, and the ninety-nine in a hundred chance in case of no accident, to your full wealth less the insurance premium. If expected utility is greater in the insured than in the uninsured state, then you buy insurance.
Risk aversion
SO FAR, ALL we have done is identify the factors that are important in the individual’s choices and which frame his or her decision-making process in terms of these factors. This does not tell us whether he or she will buy insurance or not. Nor does it even begin to show why chance occupies such an important place in economics.