Utility function

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Utility is a concept in economics used to express the most basic element of well-being for a representative consumer. Utility functions are generally used to show logical inferences about the trade-offs that consumers make and their likely outcome. The concept of using utility functions to represent consumer choice as reflecting likely consumer behavior has long been attacked, partly because of it appears (wrongly) to assume economic actors are rational, and partly because it is tautological.

1 Exposition
2 Indifference Curves
3 Significance of the Utility Function
4 Criticism
5 Notes
6 See Also
7 External Links

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Exposition

Click image to enlarge

Utility is an abstract concept that (conceptually) allows the addition of heterogeneous bundles of goods, such as everything a consumer buys with her annual income. It is actually limited in scope by things the consumer actually consumes, however; yet economists understand that the consumer gets certain advantages by saving also. Consumers are assumed to discount their future utility relative to present utility, which results in the need for an offsetting rate of interest. However, consumers naturally do not spend all their income since they theoretically wish to maintain a constant rate of consumption (See constant relative risk aversion).

In economics, one speaks of "utility" as a state that cannot be measured, but that can be compared; so, for example, in the first chart on the right, the blue line (U1) represents a lower lever level of utility than the red (U2). It is not valid to say U2 represents 1.5 times as much utility, but we can include a very large number of intermediate levels of utility between the two points.^[1]

There are two reasons why utility is nearly always expressed as a function of two goods, despite the fact that we invariably consume so many more than two.

It is graphically impossible to represent a 3-good utility function, because the value of utility itself would require a fourth dimension;
The usual ambition of economists dealing with utility functions is not to actually contrive an actual prediction of a general equilibrium point (which would be impossible), but to demonstrate the logical foundations for certain assumptions.

If we are working with a two-good utility function, we can describe the graph of the function itself as a curved, or sloped, surface (concave downward). Technically, it is possible to have an graphic that could illustrate a 3-good utility function. That could be a Java applet which features an illustration of a two-good utility function, like the one shown on the right, but with a "slider bar" underneath in in which one could vary z. As one slide the cursor from left (z = 0) to right (z >> 0), the surface would move.

Utility functions are usually based on Cobb-Douglas functions, but may be use the Linear expenditure system instead. More fundamentally, they are concave with respect to the x-y plane. This arises from three fundamental assumptions:

preferences are continuous (i.e., the increase in utility is not "all-or-nothing"; the goods x and y are not items like left and right shoes, or disassembled car parts);
preferences are transitive (i.e., given three or more bundles of goods, where b₁ is preferred to b₂, and b₂ is preferred to b₃, then b₁ is preferred to b₃);
preferences are never satiated (no bliss points).

When economists are said to assume that consumers are rational, they are assuming the conditions above. It is impossible to create a utility function under the assumption of intransitive preferences, since that would involve a surface with a region that was downward sloping along a closed loop. That does not mean such a scenario cannot exist, but only that if it accurately describes many different people in any particular society, then that society's economy is highly unstable.

The assumption that consumers do not have lexicographic preferences (all-or-nothing) is not necessarily reasonable at the individual level. In some cases, such as with health insurance and rental apartments, there is a threshold below which one does not, to all intents and purposes, have that particular thing. Slums and predatory health care policies are examples of short-run savings inflicted on poor people, with high long-run costs. "Slums," here, are defined as substandard housing; the rental income from the units probably is less than sufficient to maintain them at levels required by law. In many cases, that's not the case; property owners/renters may simply be in a position to exploit renters. In any event, people live in slums because affordable alternatives are unavailable. However, because they are in a corner solution, they are exposed to other forms of exploitation that are likely to make emergence from poverty much harder. In the case of health insurance, inexpensive individual policies do exist, but they have such onerous terms for paying out claims that they amount to no insurance at all.^[2]

Certainly for lower incomes, preferences do tend to become lexicographic because more and more necessities become unitary; one may substitute cast-off shoes for cheap ones, or various forms of processed starch for actual food, but there are indeed thresholds below which one cannot go when it comes to housing, childcare, medical, utilities, transportation, and so on.

Finally, the assumption of non-satiation is usually interpreted as saying utility functions always have partial derivatives of the same sign (positive) and 2nd derivatives of the same sign (negative).^[3] One could negate this by allowing utility to have increasing returns to scale, but this would violate constant relative risk aversion, which is known to be irrational for other reasons. Allowing satiation would permit people with huge windfalls to become indifferent to their fortunes, which is inherently implausible.

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Indifference Curves

Click image to enlarge

Utility is always described as a function of at least two sources, such as "wages" and "leisure" (from the perspective of the worker). Of course, if you increase wages without reducing leisure, or increase both, then the person obviously has a higher level of utility. But what about when one must trade one for the other?

In the second graph, the economic actor's utility is a function of A and B. The curved red line is the result of a sudden decline in the price of B. When that happened, the "budget line"—the straight dashed lines slicing diagonally across the graph—moved outward, to the right. That diagonal line intersects the A-axis at the point where the consumer spends 100% of her income on A, and the B-axis where she spends 100% of her income on B. So when the price of B fell, the budget line moved outward to intersect with a new, higher, level of utility.

When the consumer had the lower (blue) budget line, she consumed A₁ and B₁. When the price of B fell, her consumption of both increased, to A₂ and B₂. But economists make a distinction between (A₂, B₂) and (A_1.5, B_1.5). While some of the change in consumption (ΔA,ΔB) can be explained by the increased income—i.e., the new, "purple" flashpoint on the red curve above—some of (ΔA,ΔB) is the result of substitution. So, for example, the increased real income caused by a decline in the price of B actually caused the consumption of A to decline in absolute terms. An income effect will always cause both to increase, but a substitution effect will always cause consumption of one to fall relative to the other.

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Significance of the Utility Function

Utility functions were an extremely important feature of neoclassical economics because they formed a basis for estimating how collectives of individuals could reach a finite choice about the amount of what was to be produced, given finite endowments of land, labor and capital. Without utility functions, there were infinitely many possible prices and quantities for goods, since there was no explanation for why a consumer might postpone some consumption (in order to invest it, perhaps) but not all of it; or why the demand for some commodities responded differently to prices than did others (elasticity). Without utility functions, there was no way to evaluate the impact of different taxes on public welfare; there was no way for businesses to decide what an increase in the price of one product might have on the price of a substitute (or complementary) product.^[4]

Two additional features of modern economics depend on the utility function and its applications: econometrics and dynamic general equilibrium (DGE). Econometrics has struggled for years with the thorny and controversial question of measuring inflation and purchasing power parities. Both problems are very similar: how does one weigh inputs? and how does one shift that weight over time (or international boundaries)? Some items in the United Kingdom, for example, are exceptionally inexpensive relative to their equivalent in Italy, and vice versa. Likewise, some items have become cheaper in the last two (or twenty) years, while others have become massively more expensive. And some of those more-expensive items are improved, while others are not. If utility functions were dismissed out of hand as an ideological artifact, one would also have to insist that estimates of inflation were entirely arbitrary, and hence that estimates of GDP growth were, too. Estimates of real exchange rates would be impossible, since foreign exchange rates have little to do with the purchasing power of the currency in its own territory. Hence, estimates of the ability of countries to repay their sovereign debt would be impossible, and international trade in financial services would also be very difficult.

In the case of DGE, this has of course relied intensely on econometric data (mainly, GDP data, interest rates, and inflation) to test models, and it relies on the the Ramsey-Cass-Koopmans system of equations (which incorporates a Cobb-Douglas utility function) to explain how economies respond to technical shocks. At the core of the DGE approach is the assumption that economic actors nearly always operate under unbounded rationality, meaning that they have well-behaved lifetime utility functions which they reliably seek to maximize, and they have unlimited access to credit.

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Criticism

An invalid criticism of the utility function is that it assumes consumers are rational. This poses deeper issues with what "rational" means, with economists assuming it mean that (in the aggregate), consumers have behavior that can be explained by a constrained optimization function. Some consumers might be insanely short-sighted, while others might be given to obsessive hoarding; but these are relatively unusual in the total population at any given time. Moreover, the hoarders and the Madame Bovaries are expected to cancel each other out. In some cases, people violate CRRA, and this may sometimes be exploited by some businesses for profit; but CRRA-violating behavior will likely apply to a limited domain of any one individual's utility function; it will be "punished" as people suffer disappointment from windfalls, or crushing losses.^[5]

A more serious criticism is not that people are economically irrational (sometimes), but that they are incapable of behaving rationally. In order to use a utility function to get a determinate result, one must use it for the longest plausible time horizons. Some economists simplify their math by assuming people live forever, on the grounds that there's little difference between an economy of 160 million immortal adults maximizing their utility over 22-year time horizons, and an economy of 160 million people who turn over completely every 45 years.^[6] Others try to replicate some gradual turnover, mainly because turnover affects worker risk aversion, not consumer choice. However, if we do assume people maximize utility over extremely long time horizons, we need to assume that they are capable of borrowing huge amounts of money early in their working lives, and paying that debt off during their later years, or else borrowing huge amounts of money during economic downturns (as part of a sensible intertemporal substitution of leisure for wages). What we see is that workers are usually devastated and humiliated by unemployment, and usually return to a diminished earning potential.

Another problem is that rational behavior with respect to "indifferent" (i.e., equally valued) bundles of goods is tautological. There is no way, even given unlimited access to consumer purchasing data, to determine how consumer utility is affected by prices or income. The differentiation between substitution effects and income effects is entirely speculative; efforts to infer these effects from income elasticity, price elasticity, or cross-elasticity, always relies on assuming the very things one is attempting to measure.

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Notes

↑ For the original graphic see Manfred Gärtner, eurMacro "Indifference curves," 2002
↑ As an example of predatory health care policies, there is the case of CareCredit, a health-care credit card offered by GE Money (a division of General Electric Corp.). According to MarketWatch,"[New York Attorney General Mario]Cuomo's probe alleges seniors and other patients are misled about financing and often pushed into debt, and that health care providers receive kickbacks based on how much business they generate for the cards." MarketWatch, "NY's Cuomo expands health-care credit card probe" (9 August 2010). Another example, fairly commonplace in 2010, was the $10,000-deductible health care plan (health insurance costs in the USA vary dramatically by county, age, and policy type). Even if one could pay the premiums for such a policy, and was not turned away for pre-existing conditions, availing oneself of care was prohibitively expensive.
↑ Utility functions are always of at least two variables, so each one has at least two 1st (partial) derivatives and four 2nd derivatives. All utility functions I have ever known have 1st derivatives that are positive for all values of x and y, and 2nd derivatives that are positive for cross partial derivatives, but negative for the pure partial derivatives.
↑ Substitute goods are goods that can be replaced, up to a point, by the other. The technical definition is that if a and b are substitute goods, and increase in the price of a will lead to an increase in demand (and therefore, in price) for b.
↑ Mathematically, the domain of a function is the set of values for which it is defined; hence, an irrational person might have a set of values of x and y where her utility function violates the principles of rationality; outside of this limited area on the x-y plane, it is mostly rational. For example, people who go to casinos are not making rational economic choices, although they might possibly be making understandable entertainment choices. However, if a gambler is so addicted that he spends nearly all his income gambling, then he will probably cease to be an economic actor before long.
↑ Most economic models assume that future wealth is discounted at a 7% annual rate; if this is so, then events more than 22 years in the future are inconsequential. The US labor force has about 160 million workers, of whom one sixth are unemployed (U-6), and for whom a working career is about 45 years. Arguably, future wealth is discounted at such a high rate because humans die. If we did live forever, the discount rate would likely be much lower.