# Intertemporal choice

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Intertemporal choice is the process by which people make decisions about what and how much to do at various points in time, when choices at one time influence the possibilities available at other points in time. These choices are influenced by the relative value people assign to two or more payoffs at different points in time. Most choices require decision-makers to trade off costs and benefits at different points in time. These decisions may be about saving, work effort, education, nutrition, exercise, health care and so forth. Greater preference for immediate smaller rewards has been associated with many negative outcomes ranging from lower salary [1] to drug addiction. [2]

## Contents

Since early in the twentieth century, economists have analyzed intertemporal decisions using the discounted utility model, which assumes that people evaluate the pleasures and pains resulting from a decision in much the same way that financial markets evaluate losses and gains, exponentially 'discounting' the value of outcomes according to how delayed they are in time. Discounted utility has been used to describe how people actually make intertemporal choices and it has been used as a tool for public policy. Policy decisions about how much to spend on research and development, health and education all depend on the discount rate used to analyze the decision. [3]

## Portfolio allocation

Intertemporal portfolio choice is the allocation of funds to various assets repeatedly over time, with the amount of investable funds at any future time depending on the portfolio returns at any prior time. Thus the future decisions may depend on the results of current decisions. In general this dependence on prior decisions implies that current decisions must take into account their probabilistic effect on future portfolio constraints. There are some exceptions to this, however: with a logarithmic utility function, or with a HARA utility function and serial independence of returns, it is optimal to act with (rational) myopia, ignoring the effects of current decisions on the future decisions.

## Consumption

The Keynesian consumption function was based on two major hypotheses. Firstly, the marginal propensity to consume lies between 0 and 1. Secondly, the average propensity to consume falls as income rises. Early empirical studies were consistent with these hypotheses. However, after World War II it was observed that saving did not rise as income rose. [ citation needed ] The Keynesian model therefore failed to explain the consumption phenomenon, and thus the theory of intertemporal choice was developed. The analysis of intertemporal choice was introduced by John Rae in 1834 in the "Sociological Theory of Capital". Later, Eugen von Böhm-Bawerk in 1889 and Irving Fisher in 1930 elaborated on the model. A few other models based on intertemporal choice include the life-cycle hypothesis proposed by Franco Modigliani and the permanent income hypothesis proposed by Milton Friedman. The concept of Walrasian equilibrium may also be extended to incorporate intertemporal choice. The Walrasian analysis of such an equilibrium introduces two "new" concepts of prices: futures prices and spot prices.

### Fisher's model of intertemporal consumption

Irving Fisher developed the theory of intertemporal choice in his book Theory of interest (1930). Contrary to Keynes, who related consumption to current income, Fisher's model showed how rational forward looking consumers choose consumption for the present and future to maximize their lifetime satisfaction.

According to Fisher, an individual's impatience depends on four characteristics of his income stream: the size, the time shape, the composition and risk. Besides this, foresight, self-control, habit, expectation of life, and bequest motive (or concern for lives of others) are the five personal factors that determine a person's impatience which in turn determines his time preference. [4]

In order to understand the choice exercised by a consumer across different periods of time we take consumption in one period as a composite commodity. Suppose there is one consumer, ${\displaystyle N}$ commodities, and two periods. Preferences are given by ${\displaystyle U(x_{1},x_{2})}$ where ${\displaystyle x_{t}=(x_{t1},\dots ,x_{tN})}$. Income in period ${\displaystyle t}$ is ${\displaystyle Y_{t}}$. Savings in period 1 is ${\displaystyle S_{1}}$, spending in period ${\displaystyle t}$ is ${\displaystyle C_{t}}$, and ${\displaystyle r}$ is the interest rate. If the person is unable to borrow against future income in the first period, then he is subject to separate budget constraints in each period:

${\displaystyle C_{1}+S_{1}\leq Y_{1},}$    (1)
${\displaystyle C_{2}\leq Y_{2}+S_{1}(1+r).}$     (2)

On the other hand, if such borrowing is possible then the person is subject to a single intertemporal budget constraint:

${\displaystyle C_{1}+{\frac {C_{2}}{1+r}}=Y_{1}+{\frac {Y_{2}}{1+r}}.}$     (3)

The left hand side shows the present value of expenditure and right hand side depicts the present value of income. Multiplying the equation by ${\displaystyle (1+r)}$ would give us the corresponding future values.

Now the consumer has to choose a ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ so as to

Maximize
${\displaystyle U(C_{1},C_{2})}$
subject to
${\displaystyle C_{1}+C_{2}/(1+r)=Y_{1}+Y_{2}/(1+r).}$

A consumer may be a net saver or a net borrower. If he's initially at a level of consumption where he's neither a net borrower nor a net saver, an increase in income may make him a net saver or a net borrower depending on his preferences. An increase in current income or future income will increase current and future consumption (consumption smoothing motives).

Now, consider a scenario where the interest rates are increased. If the consumer is a net saver, he will save more in the current period due to the substitution effect and consume more in the current period due to the income effect. The net effect thus becomes ambiguous. If the consumer is a net borrower, however, he will tend to consume less in the current period due to the substitution effect and income effect thereby reducing his overall current consumption. [5]

### Modigliani's life cycle income hypothesis

The life cycle hypothesis is based on the following model:

${\displaystyle \max U_{t}=\sum _{t}U(C_{t})(1+\delta )^{-t}}$

subject to

${\displaystyle \sum _{t}C_{t}(1+r)^{-t}=\sum _{t}Y_{t}(1+r)^{-t}+W_{0},}$

where

U(Ct) is satisfaction received from consumption in time period t,
Ct is the level of consumption at time t,
Yt is income at time t,
δ is the rate of time preference ( a measure of individual preference between present and future activity),
W0 is the initial level of income producing assets.

Typically, a person's MPC (marginal propensity to consume) is relatively high during young adulthood, decreases during the middle-age years, and increases when the person is near or in retirement. The Life Cycle Hypothesis(LCH) model defines individual behavior as an attempt to smooth out consumption patterns over one's lifetime somewhat independent of current levels of income. This model states that early in one's life consumption expenditure may very well exceed income as the individual may be making major purchases related to buying a new home, starting a family, and beginning a career. At this stage in life the individual will borrow from the future to support these expenditure needs. In mid-life however, these expenditure patterns begin to level off and are supported or perhaps exceeded by increases in income. At this stage the individual repays any past borrowings and begins to save for her or his retirement. Upon retirement, consumption expenditure may begin to decline however income usually declines dramatically. In this stage of life, the individual dis-saves or lives off past savings until death. [6] [7]

### Friedman's permanent income hypothesis

After the Second World War, it was noticed that a model in which current consumption was just a function of current income clearly was too simplistic. It could not explain the fact that the long-run average propensity to consume seemed to be roughly constant despite the marginal propensity to consume being much lower. Thus Milton Friedman's permanent income hypothesis is one of the models which seeks to explain this apparent contradiction.

According to the permanent income hypothesis, permanent consumption, CP, is proportional to permanent income, YP. Permanent income is a subjective notion of likely average future income. Permanent consumption is a similar notion of consumption.

Actual consumption, C, and actual income, Y, consist of these permanent components plus unanticipated transitory components, CT and YT, respectively: [8]

CPt2YPt
Ct = CPt + CTt
Yt = YPt + YTt

## Labor supply

The choice of an individual as to how much labor to currently supply involves a trade-off between current labor and leisure. The amount of labor currently supplied influences not only current consumption opportunities but also future ones, and in particular influences the future choice of when to retire and supply no more labor. Thus the current labor supply choice is an intertemporal choice.

When the laborers face an increase in wage, three things happen: substitution effect, ordinary income effect, and endowment effect. Keep in mind that wage is the price of leisure, since wage is the foregone opportunity cost of consuming leisure. Substitution Effect: As wage goes up, leisure becomes expensive. Therefore, a laborer would consume less leisure and supply more labor. With this, if real income is not changed then there would be an increase in hours worked. Income Effect: As wage goes up, leisure becomes expensive. Therefore, the purchasing power of each dollar would decrease. Since leisure is a normal good, the laborer would buy less leisure. This will create a negative relationship between the number of hours worked and the wage rate. Endowment Effect:As wage goes up, the value of the endowment (= wage times leisure + consumption) goes up. Therefore, income would increase holding labor fixed. Since leisure is a normal good, the laborer would buy more leisure.

## Fixed investment

Fixed investment is the purchasing by firms of newly produced machinery, factories, and the like. The reason for such purchases is to increase the amount of output that can potentially be produced at various times in the future, so this is an intertemporal choice.

## Hyperbolic discounting

The article so far has considered cases where individuals make intertemporal choices by considering the present discounted value of their consumption and income. Every period in the future is exponentially discounted with the same interest rate. A different class of economists, however, argue that individuals are often affected by what is called the temporal myopia. The consumer's typical response to uncertainty in this case is to sharply reduce the importance of the future of their decision making. This effect is called hyperbolic discounting.

Mathematically, it may be represented as follows:

${\displaystyle f_{H}(D)={\frac {1}{1+kD}}\,}$

where

f(D) is the discount factor,
D is the delay in the reward, and
k is a parameter governing the degree of discounting. [9]

When choosing between $100 or$110 a day later, individuals may impatiently choose the immediate $100 rather than wait for tomorrow for an extra$10. Yet, when choosing between $100 in a month or$110 in a month and a day, many of these people will reverse their preferences and now patiently choose to wait the additional day for the extra \$10. [10]

## Related Research Articles

Within economics, the concept of utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness within the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a single consumer's preference ordering over a choice set but is not comparable across consumers. This concept of utility is personal and is based on choice rather than on the pleasure received, and so is more rigorously specified than the original concept but makes it less useful for ethical decisions.

The theory of consumer choice is the branch of microeconomics that relates preferences to consumption expenditures and to consumer demand curves. It analyzes how consumers maximize the desirability of their consumption as measured by their preferences subject to limitations on their expenditures, by maximizing utility subject to a consumer budget constraint.

In economics, time preference is the current relative valuation placed on receiving a good or some cash at an earlier date compared with receiving it at a later date.

In economics, a budget constraint represents all the combinations of goods and services that a consumer may purchase given current prices within his or her given income. Consumer theory uses the concepts of a budget constraint and a preference map as tools to examine the parameters of consumer choices. Both concepts have a ready graphical representation in the two-good case. The consumer can only purchase as much as their income will allow, hence they are constrained by their budget. The equation of a budget constraint is where P_x is the price of good X, and P_y is the price of good Y, and m = income.

Consumption, defined as spending for acquisition of utility, is a major concept in economics and is also studied in many other social sciences. It is seen in contrast to investing, which is spending for acquisition of future income.

In economics, the marginal propensity to consume (MPC) is a metric that quantifies induced consumption, the concept that the increase in personal consumer spending (consumption) occurs with an increase in disposable income. The proportion of disposable income which individuals spend on consumption is known as propensity to consume. MPC is the proportion of additional income that an individual consumes. For example, if a household earns one extra dollar of disposable income, and the marginal propensity to consume is 0.65, then of that dollar, the household will spend 65 cents and save 35 cents. Obviously, the household cannot spend more than the extra dollar.

In economics, the consumption function describes a relationship between consumption and disposable income. The concept is believed to have been introduced into macroeconomics by John Maynard Keynes in 1936, who used it to develop the notion of a government spending multiplier.

In economics, hyperbolic discounting is a time-inconsistent model of delay discounting. It is one of the cornerstones of behavioral economics and its brain-basis is actively being studied by neuroeconomics researchers.

Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stewart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my utility?" It is a type of optimal decision problem. It consists of choosing how much of each available good or service to consume, taking into account a constraint on total spending (income), the prices of the goods and their preferences.

Economic theories of intertemporal consumption seek to explain people's preferences in relation to consumption and saving over the course of their lives. The earliest work on the subject was by Irving Fisher and Roy Harrod, who described 'hump saving', hypothesizing that savings would be highest in the middle years of a person's life as they saved for retirement.

In economics, dynamic inconsistency or time inconsistency is a situation in which a decision-maker's preferences change over time in such a way that a preference can become inconsistent at another point in time. This can be thought of as there being many different "selves" within decision makers, with each "self" representing the decision-maker at a different point in time; the inconsistency occurs when not all preferences are aligned.

In economics, discounted utility is the utility (desirability) of some future event, such as consuming a certain amount of a good, as perceived at the present time as opposed to at the time of its occurrence. It is calculated as the present discounted value of future utility, and for people with time preference for sooner rather than later gratification, it is less than the future utility. The utility of an event x occurring at future time t under utility function u, discounted back to the present using discount factor Is

Average propensity to consume (as well as the marginal propensity to consume) is a concept developed by John Maynard Keynes to analyze the consumption function, which is a formula where total consumption expenditures (C) of a household consist of autonomous consumption (Ca) and income (Y) multiplied by marginal propensity to consume. According to Keynes, the individual´s real income determines saving and consumption decisions.

In economics, the life-cycle hypothesis (LCH) is a model that strives to explain the consumption patterns of individuals.

The permanent income hypothesis (PIH) is a model in the field of economics to explain the formation of consumption patterns. It suggests consumption patterns are formed from future expectations and consumption smoothing. The theory was developed by Milton Friedman and published in his A Theory of Consumption Function, published in 1957 and subsequently formalized by Robert Hall in a rational expectations model. Originally applied to consumption and income, the process of future expectations is thought to influence other phenomena. In its simplest form, the hypothesis states changes in permanent income, rather than changes in temporary income, are what drive changes in consumption.

Consumption smoothing is the economic concept used to express the desire of people to have a stable path of consumption. People desire to translate their consumption from periods of high income to periods of low income to obtain more stability and predictability. There exist many states of the world, which means there are many possible outcomes that can occur throughout an individual's life. Therefore, to reduce the uncertainty that occurs, people choose to give up some consumption today to prevent against an adverse outcome in the future. In order for one to adequately and properly prepare for unforeseen circumstances that can occur in the future, we must start planning today, putting money aside for when these unforeseen circumstances happen.

In economics, the absolute income hypothesis concerns how a consumer divides his disposable income between consumption and saving. It is part of the theory of consumption proposed by economist John Maynard Keynes. The hypothesis was refined extensively during the 1960s and 1970s, notably by American economist James Tobin (1918–2002).

The random walk model of consumption was introduced by economist Robert Hall. This model uses the Euler numerical method to model consumption. He created his consumption theory in response to the Lucas critique. Using Euler equations to model the random walk of consumption has become the dominant approach to modeling consumption.

Precautionary saving is saving that occurs in response to uncertainty regarding future income. The precautionary motive to delay consumption and save in the current period rises due to the lack of completeness of insurance markets. Accordingly, individuals will not be able to insure against some bad state of the economy in the future. They anticipate that if this bad state is realized, they will earn lower income. To avoid adverse effects of future income fluctuations and retain a smooth path of consumption, they set aside a precautionary reserve, called precautionary savings, by consuming less in the current period, and resort to it in case the bad state is realized in the future.

In economics and finance, an intertemporal budget constraint is a constraint faced by a decision maker who is making choices for both the present and the future. The term intertemporal is used to describe any relationship between past, present and future events or conditions. In its general form, the intertemporal budget constraint says that the present value of current and future cash outflows cannot exceed the present value of currently available funds and future cash inflows. Typically this is expressed as

## References

1. Hampton, W. (2018). "Things for Those Who Wait: Predictive Modeling Highlights Importance of Delay Discounting for Income Attainment". Frontiers in Psychology. 9 (1545): 1545. doi:. PMC  . PMID   30233449.
2. MacKillop, J. (2011). "Delayed reward discounting and addictive behavior". Psychopharmacology. 216 (3): 305–321. doi:10.1007/s00213-011-2229-0. PMC  . PMID   21373791.
3. Berns, Gregory S.; Laibson, David; Loewenstein, George (2007). "Intertemporal choice – Toward an Integrative Framework" (PDF). Trends in Cognitive Sciences. 11 (11): 482–8. doi:10.1016/j.tics.2007.08.011. PMID   17980645. S2CID   22282339. Archived from the original on 2016-05-30.CS1 maint: bot: original URL status unknown (link)
4. Thaler, Richard H. (1997). "Irving Fisher: Modern Behavioral Economist" (PDF). The American Economic Review. 87 (2): 439–441. JSTOR   2950963. Archived from the original on 2016-03-04.CS1 maint: bot: original URL status unknown (link)
5. Varian, Hal (2006). Intermediate Micro Economics.
6. Barro, Robert J. (1998). Macroeconomics (5th ed.). Cambridge, Mass.: MIT Press. ISBN   9780262024365.
7. Mankiw, N. Gregory (2008). Principles of Macroeconomics (5th ed.). Cengage Learning. ISBN   9780324589993.
8. "Adaptive expectations: Friedman's permanent income hypothesis". Archived from the original on 2016-03-04. Retrieved 2013-08-22.
9. P. Redden, Joseph. "Hyperbolic Discounting".Cite journal requires |journal= (help)