The Effect of Temporal Constraints on the Value of ...

Journal of Behavioral Decision Making, Voi 6, 77-94(1993)

The Effect of Temporal Constraints on the Value of Money and Other Commodities ANDRES RAINERI and HOWARD RACHLIN State University of New York at Stony Brook, USA ^

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ABSTRACT I 1 ,|.

The value of hypothetical rewards of various delays and durations, consisting of money or other commodities, such as automobile use and vacations, was measuredby the psychophysical up-down staircase method using an amount of unrestricted immediate money as the titrated variable. A model of temporal discounting based on implied constraints on anticipated consumption rate described the data. Further explicit constraints on consumption produced predictable deviations from the model. The concept that consumption is normally constrained together with time discounting may also explain the interaction between amount and delay of monetary reward in determining value and may provide the basis for a theory of addiction based on learning to increase consumption rate. KEY WORDS

Value Delay Duration Discounting Choice Humans

The meaning of consumption depends on your point of view. For an economist, consumption occurs when a consumer buys a consumable commodity. Psychologists studying animal choice distinguish between an animal's obtaining food and eating it (Herrnstein, 1970); consumption is said to occur at the point when the food is eaten. A biologist might not consider consumption to have occurred until the food is digested or converted to energy. In the chain of events leading from an organism's choice of a given commodity to its ultimate exhaustion, the maximum rate of consumption at each level is constrained by the next. For instance, in the long run, food cannot be eaten faster than it can be digested. The present experiments are based on a theory of choice as it is affected by expected consumption. In these experiments people chose between a certain amount of money (hypothetically) available immediately and unrestricted in how it might be spent and a greater amount of money (or a commodity) with various implicit or explicit temporal constraints on how it might be obtained or spent. Although the packages of goods a person expects to buy with different amounts of money may differ, and although in practice money must be spent in chunks, we assume that in prospect, as people consider between alternatives, the package of goods they anticipate buying with a given number of dollars is the same for each dollar. Thus, dollars for a person, like seconds of access to a hopper of mixed grain for a pigeon, are assumed to be perfectly substitutable for and commensurate with each other. Ten dollars are meaningfully ten times the amount (if not the value) of one dollar. Given that dollars are perfectly substitutable for one another (like Coke and Pepsi) they must be expected to be consumed in series. At one extreme, highly substitutable commodities must be consumed in series. For instance, 0894~3257/93/020077-18$14.00 © 1993 by John Wiley & Sons, Ltd.

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Received 6 February 1991 Revised21 August 1992

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marginal addition of a can of Coke to ten cans of Pepsi would extend the expected time to consume the total. As degree of substitutability decreases toward zero, a given pair of commodities may be consumed in parallel (i.e. together) rather than in series. As substitutability becomes negative (the goods are complements), each good gains in value by parallel consumption with the other. At the extreme of complementarity the package of commodities must be consumed in parallel; left shoes and right shoes or bicycle frames and bicycle wheels are the standard examples. As a person contemplates a choice between a smaller and larger amount of money, he or she must consider that it will take longer to spend the larger amount. At the extreme, this implied constraint is obvious. The movie Brewster's Millions and its several remakes and variations (a man inherits an immense sum of money on condition that he spend a smaller but still very large sum in a limited time) vividly illustrates this restriction. In the context of the above considerations the fundamental economic law of diminishing marginal value may be viewed as a consequence of implied temporal constraints. The law of diminishing marginal value says that as the amount of a commodity increases, its value increases at a diminishing rate; thus the increase in value from $1 to $2 is greater than that from SlOl to $102. If consumption rate is limited by the biology and physiology of the organism (the size of the mouth, the digestive organs, and so forth) then consumption of the last (marginal) unit of a large amount of a commodity will be delayed longer than consumption of the last unit of a small amount. For example, an additional apple will be consumed sooner if you already have only one apple than if you already have ten apples. The additional apple has to wait on line, so to speak, to be consumed; the longer the line, the longer the wait. Commodities for which a person (or other animal) has to wait longer are valued less; they are discounted by delay. Thus, as the amount of a commodity increases uniformly, its value increases at ever-diminishing increments. If marginal units are discounted by their delay, the more they are delayed, the less they are worth. Hence, the greater the amount of a commodity, the less the value of its marginal unit. The I02nd dollar is worth less than the 2nd dollar, not (or not only) because 102 is a higher number than 2 but because the 102nd dollar must be spent later and is therefore more discounted by time than the 2nd dollar. (See, Rachlin, 1992, for a more detailed argument.) Note that this biological conception of the origin of diminishing marginal value differs from the standard economic conception (Newman, 1965). According to the standard economic conception, the 2nd dollar is worth more than the I02nd dollar because the 2nd dollar will be used to buy necessities (like food) while the 102nd dollar will be used to buy luxuries (like a movie) which are intrinsically less valuable than food. But one might then ask, if food is so valuable why not spend all $102 immediately on food? The answer to that question could well be given in terms of constraints on food consumption, at the level of eating, digestion, or energy conversion. Thus, with either biological or economic conceptions, consumption constraints may be conceived as underlying diminishing marginal value. The larger question whether a//diminishing marginal utility may be due to consumption constraints is far beyond the scope of the present research. However, it should be noted that it is not a trivial exercise to create an example where diminishing marginal value is unconfounded with temporal constraints on consumption. For example, most people's preference for tourist over more expensive first-class airplane seats may seem to be a case of non-temporal marginality of first-class amenities. But (unlike an 11th apple) first-class amenities would not generally be bought without the fundamental good, the transportation; such amenities are therefore not strictly marginal. On the other hand, upgrading to faster transportation (from regular first-class to Concorde, for instance) is marginal (like the 11th apple) because it presumably provides more time at one's destination. But, of course, 'more time' is marginal in a strictly temporal way as the present model supposes.

Value of Money

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Time Exhibit I. Uniform consumption at rate, R, units per day beginning 7, days from the present moment (0), and ending Ti days from the present moment. The striped bar represents a day Tdays from now For the sake of simplicity, we assume that an amount of money to be received after a delay, r,, is expected to be spent at a constant rate, R, until it is used up at time T2. This uniform-spending assumption is illustrated in Exhibit 1. A second assumption is that the present value (the value at T-Q) of each unit of money spent {Rts,T in Exhibit I) is discounted by the expected delay, T, to the point in time when that unit would be spent. The particular form of the discount function is irrelevant to the present theory. Moreover, the present experiments are not precise enough to test between specific forms. However, any delay discount function must have the following properties: ' i (1) When delay {T) is zero, discounted value (v) should equal undiscounted value {V)\ (2) As delay increases, discounted value should decrease monotonically; (3) As delay becomes infinitely large, discounted value should approach zero. The following delay discount function (from Loewenstein and Prelec, in press) satisfies these requirements: - -•:' I (1) {) where v is present (discounted) value, V is undiscounted value, T is expected delay, and k and b are positive constants. When k = b, equation (1) becomes the simple hyperbola: (2) Several researchers of reward delay have suggested a general form identical or similar to equation (1) (Logue, 1988; Mazur, 1987; Prelec, 1989; Rachlin, 1989), which corresponds to Baum's (1974) generalization of Herrnstein's (1970) matching equation. However, the simpler form of equation (2) has empirical support in studies of hungry pigeons choosing between various amounts, delays, and probabilities of food (Mazur, 1987), in studies of quite satiated humans choosing among various amounts, delays, and probabilities of small monetary rewards (Ainslie and Haendel, 1983; Rachlin et ai, 1987), and in studies of humans choosing between various amounts, delays, and probabilities

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of hypothetical, large monetary rewards (Ainsiie and Haendel, 1983; Rachlin ei al., 1991). Equation (2) will suffice also to describe the present data. Prelec (1989) has shown that equation (1) becomes an exponential function as the constant k of equation (1) approaches zero; •-'•

V=VQ-'^

(3)

Exponential time discount functions are inconsistent with the nearly universal finding with human and nonhuman subjects of preference reversal over time (Ainsiie, 1974; Logue, 1988; Thaler, 1981). For instance, a child may prefer two candy bars 10 days hence to one candy bar 9 days hence. But subtracting 9 days from both delays (assuming nine days pass but the calendar time of candy-bar availability remains fixed), the child may reverse and prefer one candy bar today to two tomorrow. Such a reversal is predicted by many delay discount functions, including equation (2), but is not possible with the constant-interest-rate exponential discount function (used by banks to calculate compound interest) and considered by economists to be 'rational' (Strotz, 1955). Relative to the exponential function upon which standard compound interest is based, hyperbolic functions imply diminishing interest rates over time. Equation (2) is not the only non-exponential discount function that has been proposed. Loewenstein (1987), Prelec (1989), Stevenson (1986) and others have proposed more or less different functions. As indicated previously, it is not the object of the present series of experiments to test between the various proposed functions. Many non-exponential discount functions account for reversals in intertemporal choice such as those described above (Prelec, 1989). Alternatively, reversals may arise from the action of a cognitive bias or heuristic upon a fundamentally rational (i.e. exponential) discount function. (See Loewenstein, 1987, for a discussion of cognitive mechanisms in intertemporal choice.) However, it may be expected that as collections of heuristics and biases become more and more unwieldy, their effects may be efficiently summarized by some such account as equation (2). Given the above assumptions, the present value of a monetary reward is the sum of the hyperbolically discounted values of each of its units. Integrating equation (2) between Tj and T2:

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Equation (4) predicts diminishing marginal value (as other time discount functions would, given implied temporal constraints on consumption). With delay to reward (T,) and consumption rate (R) constant, the time at which the reward is completely consumed (Tj) must (according to Exhibit 1) be directly proportional to the amount of reward (A). Since equation (4) postulates a logarithmic (diminishing or concave downward) relationship between value (v) and T2, the same relationship must also apply between value and amount — hence fixed marginal additions to amount must produce diminishing marginal additions to value. Temporal consumption constraints imply not only diminishing marginal value of delayed rewards (r, > 0 ) and nondelayed rewards (T, = 0), they also, with an additional assumption, account for the interaction often found between amount and delay of reward (Benzion et ai, 1989; Loewenstein and Prelec, in press). One can intuit that a million dollars delayed by a month is worth almost as much as a million dollars now but a dollar delayed by a month is worth a small fraction of a dollar now. Generally, large amounts are proportionally less discounted by a given delay than are smaller amounts. Note that with constant rate of consumption, the amount {A) of a commodity consumed would be proportional to the time taken to consume it {T2-T^ = AIR). Replacing 7", by the symbol D, for delay, T2 = D + {AIR). Suppose (as is typical in studies of delay discounting and as will be the case in Experiment 1)

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that the amount (AQ) of an immediate reward (DQ = 0) is adjusted so the immediate reward is equal in value to a delayed reward of greater amount (A). Then, assuming immediate and delayed rewards are each consumed at a constant rate, according to equation (4): VQ=V

(5) where v is the value of a delayed reward and vo is the value of an equivalent immediate reward. To predict the immediate amount (AQ) that would balance a larger delayed amount, the present mode! must make some assumption about consumption rates (RQ and R). If we were to assume RQ = R, then, according to equation (5), AQ = A/(\ +kD) and there would be no interaction. Loewenstein and Prelec (in press) show that in order to obtain an amount-delay interaction the relation between value and amount must be subproportional. (Value as a function of amount must be negatively accelerated, concave downwards, on log-log co-ordinates.) One reasonable assumption about consumption rate that would produce subproportionahty in equation (5) is that anticipated consumption rate is a negatively accelerated function of amount. Since value is already a logarithmic function of amount in equation (5), any further deceleration of the function (through a relation between anticipated consumption rate and amount) creates subproportionality. The assumption that anticipated consumption rate is a decelerating function of amount of reward is intuitively reasonable. It says we expect to spend large amounts of additional money at a higher rate than small amounts (perhaps because of different composition of the package of goods we expect to purchase with the high amount), but not proportionally faster. For instance, we would expect to spend a million-dollar prize at a higher rate (dollars per day) than a thousand-dollar prize but not a thousand times as fast. As a first, quite arbitrary, approximation we assume: (6) Combining equations (5) and (6):

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(7)

Assuming, as is typical in studies of human time discounting, that immediate equivalent amount (^o) is a measure of the discounted value of a delayed reward (A, D), the basic requirements for a discount function are satisfied by equation (7). First, if delay is zero, AQ equals A; second, as delay increases indefinitely, AQ monotonically approaches zero. Experiment 1, which balances values of immediate rewards and delayed rewards of various amounts, further tests the assumptions underlying equation (7). '

GENERAL METHOD

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Unless otherwise indicated, the procedure in these experiments was a variation of the up-down staircase (titration) method of psychophysics (Stevens, !975). Subjects chose between two hypothetical rewards, each stated on an index card. The two index cards were placed on a table by the experimenter who sat across from the subject. Subjects indicated their preference by pointing to one of the cards. One reward, the titrated reward, was always a given amount of money available immediately with no explicit constraints on how it might be spent. While the other reward (the other card) remained in front of the subject, the experimenter, starting with either a very low or very high titrated reward.

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increased or decreased its amount by presenting cards in sequence until the preference of the subject switched to or from the titrated reward. After two successive choices, representing a switch in preference, the experimenter changed the other (non-titrated) reward and began again. Half of the subjects were first exposed to the ascending and then to the descending sequence of titrated rewards, half, the reverse. The data were determined as follows. First, for each titration, the arithmetic mean was calculated of the amounts (stated on the cards) just before and just after the switch; then, for each subject, the arithmetic mean was calculated of the ascending and descending titrations; then, for each condition, the median was determined across subjects. These data serve as the standard against which the values of various amounts of money or other commodities constrained by delay or duration were measured. The data represent the amount of money, available immediately and unconstrained in use, between which and a given comparison reward subjects were indifferent. The delay, the amount, the time available for consumption, and the kind of comparison reward (money or other commodities) differed in the various experiments. Following is an example of the instructions read to the subjects (in the first experiment): Imagine that you have won $100 in a lottery. However, the $100 will be given to you some time in the future (to be indicated in the card at your left). Now, imagine that the experimenter gives you the choice between waiting for the $100 or exchanging it for a lesser amount of money that you would receive right now (to be indicated in the card at your right). Please show the experimenter by pointing with your hand which card you prefer. Thanks for your co-operation.

EXPERIMENT 1

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The purpose of this experiment was to investigate the relationship between the amount of a reward and the degree of delay discounting.

Method Subjects One hundred and twenty students enrolled in an undergraduate psychology course at the State University of New York at Stony Brook served as subjects. Their participation in this experiment was a course requirement. Subjects were unsystematically assigned to one of three groups, with 40 subjects in each group. Each group was used to obtain a delay discount function for a different amount ofmoney ($100, $10,000,51 million).

Procedure Cards were presented in pairs to all subjects. One card (the comparison reward card) stated an amount of money to be obtained after a delay (1 month, 6 months, 1 year, 5 years, 10 years, 25 years, and 50 years). The amount stated on the card was different for each group ($100, $10,000, and $1 million). The other card (the titration card) stated an amount of money to be paid right now. For each group these amounts were a fraction of the delayed reward, in the following proportions: 1, 0.99, 0.98, 0.96, 0.94, 0.92, 0.90, 0.85, 0.80, 0.75, 0.70, 0.65, 0.60, 0.55, 0.50, 0.45, 0.40, 0.35, 0.30, 0.25, 0.20, 0.15, O.IO, 0.08, 0.06, 0.04, 0.02, 0.01, 0.005, 0.001, 0.0005, and 0.0001. The $1 million group also had additional titration cards of $50 and $10. Half of the subjects in each group were presented with the titrated amounts in ascending order and the delayed rewards in descending

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order, the other half with the reverse. Thus, the method of this experiment differed from the general method in that order was not counterbalanced within subjects but it was between subjects. Results and discussion The ratio between the equivalent immediate amount (^o) 3 " ^ the delayed amount (A) was calculated for each reward amount and delay and is shown in Exhibit 2.

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Exhibit 2. Fractions of reward value (A(/A) due to delay discounting with various amounts and delays of reward

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Delay 1 month 6 months 1 year 5 years 10 years 25 years 50 years ' '

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SIOO

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0.90 (0.95) 0.70 (0.77) 0.58 (0.66) 0.30 (0.26) 0.20(0.15) 0.10(0.06) O.OI (0.03)

0.99 (0.97) 0.85 (0.85) 0.75 (0.76) 0.50 (0.44) 0.28(0.31) 0.10(0.17) 0.04 (0.09)

0.99 (0.98) 0.96 (0.89) 0.90 (0.82) 0.55 (0.58) 0.33 (0.46) 0.09(0.31) 0.001 (0.23)

Thenumbersin parentheses arepredicted by equation (7), t = 0.001.

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Looking across the rows of the exhibit is it clear that the discount caused by a given delay was greater for lower than higher amounts except at the longest delays of 25 and 50 years. Judging from the subjects' spontaneous informal comments, the larger delays were beyond their subjective time horizons. The longer delays undoubtedly interacted with subjective probability of receipt of the reward. Even were they to live long enough to receive the million dollars 25 or 50 years hence. Stony Brook undergraduates did not feel they would live long enough to spend it nor could they perceive much of a use for it if they did live that long. Expected inflation may have also distorted the results at long delays although, as we shall see in a later experiment, our subjects did not seem to adjust for inflation. Because of these issues, in subsequent experiments, all with this same subject population, delay of reward was not varied beyond 10 years. For delays between 1 month and 10 years, however, there is a clear upward trend to the obtained fractions across rows of Exhibit 2. For instance, $1 million had to be delayed by a year to be discounted by the same fraction (0.90) as $100 was in a month. A mixed-design MANOVA on the individualsubject data {AQ/A) including the longer delays reveals a significant interaction between delay and amount flI2,702] = 3.86,/J < 0.001. Distributions of individual subjects' data are shown in Exhibit 3. These distributions are typical of those obtained in this series of experiments; they are skewed at high and low delays (degree of skewedness highest at the extremes) and symmetrical in the middle. The skewedness is caused by the implicit floor and ceiling on choices; a delayed reward can neither be worth less than nothing nor more than its nominal (undelayed) value). Exhibit 4 shows the median obtained amounts as points along with the functions predicted by equation (7) (solid lines) for delays from 1 month to 10 years. (It is not possible to solve equation (7) for A(,. Instead, a value (v) was first calculated for the right side of equation (7) and then AQ was varied until the value of the left side (v^,) equalled that of the right.) In fitting equation (7) to the data we used a constant, /: =0.001, obtained in previous experiments with human subjects (Rachlin and Raineri, in press). Thus, there were no free parameters. The data are predicted reasonably well by equation (7) except at delays of 25 and 50 years, which, as noted above, are apparently

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$10,000

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DISCOUNT A/Ao Exhibit 3. Distribution of individual-subject ratios of present value to delayed amounts of SlOO, $10,000, and $1 million with various delays well beyond our subjects' time horizons. Including the 25- and 50-year points, r^ for log-transformed data = 0.936; slope between predicted and obtained values = 0.931. Not including the 25- and 50-year points, r^ = 0.999 and slope = 0.968. Using the same set of calculations by which equation (7) was obtained from equation (2), the following exponential discount function may be derived from equation (3): = V

In equation (8), as D->oo, ^^-^O and if Z) = O,AQ = A; thus equation (8) satisfies the basic requirements of a discount function. However, the exponential function does not predict the main result of this experiment — the amount-delay interaction. In fact, the small interaction predicted by equation (8) is in the opposite direction to the obtained one. For instance, at a 1-year delay, equation (8) (best fitting k = 0.0003) predicts amount ratios (AQ/A) of 0.90, 0.90, and 0.89 compared to obtained ratios of 0.58, 0.75, and 0.90 for the $100, $10,000, and $1 million rewards. As with a previous test of exponential versus hyperbolic functions on these sorts of data (Rachlin et ai, 1991), exponential functions do a relatively bad job. "•; . " '

EXPERIMENT 2 In this experiment amount of reward was varied indirectly by varying duration of consumption. Subjects were instructed to Imagine receiving a certain fixed amount each day for a given time period. Within that period money could be saved from day to day but at the end of the period, subjects were told, any excess money would be taken back as would all items bought but not consumed; that is, the money would have to be spent within the period. This mode of reward would impose a more or less rectangular pattern of consumption as assumed in Exhibit 1. Two per-diem amounts

A. Raineri and H. Rachlin

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Value 0

PRESENT VALUE (Ao)

$1,000,000 $1,000,000 $100,000 $10,000

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Exhibit 4. Median amounts of money available immediately equivalent in value to $1 million (diamonds), $10,000 (circles), and $100 (squares) available with delays of 1 month, 6 months, 1 year, and 5 years, All scales in this and subsequent exhibits are logarithmic. The solid lines are predicted by equation (7) were used: $50/day, which we believed could be consumed fairly easily by our subjects, and $500/day, which we believed could be consumed only with difficulty by our subjects.

Method Subjects Forty students enrolled in an undergraduate psychology course at the State University of New York at Stony Brook served as subjects. Their participation in this experiment was a course requirement.All subjects participated in both conditions ($50/day and $500/day). Procedure Again, subjects chose between two cards. One card stated an amount of money ($50 or $500) to be obtained for sure every day for a duration (1 week, I month, or 1 year). This daily amount could be accumulated within the stated duration but had to be consumed by the time the duration had expired. Any remaining money or items bought but not consumed would have to be returned.

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The period of receipt of money was to begin after a delay (immediately, 1 week, 1 month, 1 year, and 10 years). A total of six different discount functions were obtained for each subject (two amountsper-day by three durations). For example, one of the comparison reward cards read: — — — — —

$50 every day for a period of 1 week to be consumed within that week to be obtained 10 years from today for a total of $350

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Subjects were asked to choose between each of these comparison reward cards and a titration reward card which stated an amount of money to be paid right now and consumed at any rate. These amounts were fractions of the total amount to be received within the specified period in the same proportion as in Experiment 1. The order of presentation of amount, duration, and delay was counterbalanced between subjects. Results and discussion

Exhibit 5 shows the results of the $50/day and $500/day groups. The leftmost points (at zero delay on the abscissa) represent immediate amounts of money unrestricted in use judged equal in value to $50/day or $500/day, undelayed, for durations of one week (squares), one month (circles), and one year (diamonds). The squares, circles, and diamonds stemming to the right and downward represent delay discount functions for each reward duration. *S" PRESENT VALUE (Ao)

PRESENT VALUE (Ao)

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Exhibit 5. Median equivalent amounts of (a) $50 per day and (b) $500/day for various durations (one week, squares; one months, circles; one year, diamonds) available immediately (leftmost symbols) or after various delays (one week; one month; one year; 10 years). The solid line represents predictions of equation (7) based on equivalent total amounts of money to be spent without constraints

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Exhibit 5(b) with $500/day shows the same pattern of attenuation of value by constraint as does Exhibit 5(a) except that the degree of attenuation is greater with the greater imposed rate of spending. For instance, $500/day for 7 days is nominally worth $3,500 but actually worth $1,500, or 42.9% of its nominal value (as shown by the distance of the leftmost square from the line in Exhibit 5(b). But $50/day for 7 days, actually worth $350, is 85.7% of its corresponding nominal value (as shown by the distance of the leftmost square from the line in Exhibit 5(a). While greater durations and greater imposed rates of spending had greater attenuating effects on value, delay as such had no consistent effect. When the predicted values are normalized to the zero delay values (the lines of Exhibits 5(a) and 5(b) brought down so as to run through the leftmost points) the delayed values are predicted fairly well (r^ for log transformed data = 0.97; slope between predicted and obtained non-zero delayed points = 1.06). The lines above the points represent the predictions of equation (7) based on unrestricted use of the total amount of money obtained. For instance, for $50/day with a duration of one year, A was assumed to be $50/day x 365 days or $18,250. Since that amount is undelayed (Z) = 0), equation (7) predicts y4o = /4 =$18,250. Note that the actually obtained AQ (the leftmost diamond in Exhibit 5(a)) is considerably below this predicted value.The difference must be due to the constraint on spending the $18,250. In other words, $50/day for a year with any remainder to be returned is worth much less than $ 18,250 all delivered immediately and to be spent as one chooses. As the duration decreases from a year to a month to a week, the attenuation of value by the constraint diminishes. Thus, S50/day for a week is worth only slightly less than $350 immediately — as one would expect.

EXPERIMENT 3 As in Experiment 2, the comparison rewards for each subject varied along three dimensions: amount, duration, and delay. However, instead of a stated fixed amount per day to be spent over a given time period, subjects were asked to imagine receiving a single total amount (A) at 7", which would nevertheless have to be spent within a fixed duration (7^, - 7,). It would be possible in this experiment for subjects to imagine spending the entire amount on the first day, whereas in Experiment 2, where the money was hypothetically doled out every day, such a pattern would not have been possible. In other words, in this experiment a rectangular spending pattern (constant rate, R) was imposed less strictly than it was in Experiment 2. •

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Subjects Forty undergraduate students at the State University of New York at Stony Brook were paid $5 for their participation in this experiment. Participation was voluntary and was announced through flyers on local bulletin boards. Procedure Again, subjects chose between two cards. One card stated an amount of money ($1,000, $10,000, or $100,000) to be obtained for sure, after a delay (immediately, I week, 1 month, 1 year, and 10 years), and to be consumed within a period of time (1 week, 1 month, and 1 year). For example, one of the comparison cards read: — $100,000 — 10 years from today — You must use this money within 1 week

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PRESENT VALUE (Ao) $100,000 F

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1 month

1 year

DURATION (days) Exhibit 6. Median values of lump sum amounts of $1,000 (squares), $10,000 (filled circles), and $100,000 (triangles) to be spent within durations of one week, one month, and one year. The vertical lines running through the filled circles are interquartile ranges Subjects were asked to choose between each of these comparison-reward cards and a titration card which stated an amount of money to be paid right now. These amounts were fractions of the delayed amount in the same proportion as in Experiment 1. The order of presentation of nominal amount, duration, and delay was counterbalanced between subjects. Results and discussion Exhibit 6 shows results for only those conditions with zero delay for the three amounts ($100,000, $10,000, and $1,000). The only restriction on these comparison rewards relative to the titrated values was the duration, the time period, over which they had to be consumed. For each amount, as duration increases, equivalent amount approaches the nominal value (dotted line). Since it is easier to spend $1,000 than $100,000 in a given time period, the lower set of points approaches its asymptote faster than does the middle or upper set. Thus, the smallest amount, $1,000, to be spent and consumed within a year, was worth its full value to all subjects; they presumably anticipate consuming it well within a year. Obtained and predicted values of various nominal delays and durations were calculated as in Experiment 2. Exhibit 7 shows the data and discount functions predicted, as in Experiment 2, for

A. Raineri and H. Rachlin

Value of Money

89

PRESENT VALUE (Ao)

$10,000 -I

$5,000

-

. $1,000

-

$600

1 year

J 0

1 week 1 month '

l year

10 years

DELAY (days)

Exhibit 7. The three leftmost symbols represent the corresponding filled circles of Exhibit 6. The filled circles to the right represent these amount/durations delayed by a week, a month, a year, and 10 years The thin solid Imes represent predictions of equation (4)

the 510,000 group. To show the effect of delay alone on discounting, the solid lines in this figure are normalized at the zero-delay points. (The actual deviations of these lines from their corresponding pomts are shown in Exhibit 6 by the distances between the filled circles and the $10,000 dotted horizontal line.) For log-transformed non-zero delay data of Exhibit 6, r^ = 0.9S, slope between predicted and obtained points = 0.95. Fits for the $ 1,000 and $ 100,000 amounts were comparable.

EXPERIMENT 4 The generally good fit of the discount functions of Exhibit 7 as well as the consistency of the parameters from experiment to experiment tend to support (but, of course, do not prove) the validity of our various assumptions and methods. However, money may be thought to be a peculiar commodity in Its susceptibility to quantitative description. Experiment 4 was undertaken to discover the degree to which the present analysis applies to other commodities.

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Method Subjects Sixty students enrolled in an undergraduate psychology course at the State University of New York at Stony Brook served as subjects. Their participation was a course requirement. Subjects were randomly assigned to one of two groups of thirty. For one group the comparison reward was a vacation, for the other group the comparison reward was a car.

Procedure Again, cards were presented in pairs to all subjects. The comparison reward card stated a prize, an all-expenses-pa id vacation trip for one group, an economy car for the other group. These cards stated that the prize could be used for a specified period of time: for example, 'You have won a one-week (1-year. 10-year)... vacation trip . . . ' The cards also stated that subjects would be allowed to use the prizes after a period of time ('immediately, I week, 1 month, 1 year, and 10 years from today'). One of these cards for the 'car' group read: You have won one-week use of an economy car (Toyota Tercel, VW Rabbit, or equivalent). You can start to make use of this prize one year from today or any time thereafter. The titration card stated an amount of money to be paid right now ($1, $5; SIO; S20; S40; S60; $80; $100; $150; $200; $250; $300; $350; $400; $450; S500; $550; $600; $650; $700; $750; $800; $850; $900; $1,000; $1,100; $1,200; $1,300; $1,400; $1,500; $1,700; $1,900; $2,200; $2,500; $2,800; $3,100; $3,500; $4,000; $5,000; $6,000; $7,000; $8,000; S9,000; $10,000; $12,000; $14,000; $16,000; $18,000; $20,000; $25,000; $30,000; $35,000; $40,000; $50,000; $60,000; $70,000; $80,000; $90,000; $100,000; S120,000; $ 150,000; $200,000; $250,000; $300,000; $400,000; $500,000; $750,000; $ I million; $2 milHon; $3 million; $4 million; $5 million; S7 million; $9 milJion; $12 million; $15 million; $18 million; or $25 million). Half of the subjects in each group were presented with the comparison rewards in ascending order of delay and duration and descending order of titrated amounts. The other half were presented with the comparison rewards in descending order of delay and duration and ascending order of titrated amounts. A subject was considered to have switched to the initially rejected alternative after the first time that that alternative was chosen.

Results and discussion Since we have no way of knowing R in dollars per day for commodities, we estimated R for each commodity duration as follows. First, using the left side of equation (7), we calculated the value of the average obtained dollar amount equivalent to an undelayed commodity. Then we set that equal to the value of an undelayed commodity (equation (5), zero-delay). Thus, VQ (money) = VQ (commodity)

4J^\n{\+k4J^ = R\n(\+kA/R) Assuming linear consumption, duration [dur] = A/R\ V ] 4 o l n ( l + i V ^ ) = /?ln(l+fc[dur])

-

(9)

With /t = 0.001, and using obtained zero-delay AQ, equation (9) may be solved for R. Exhibit 8 shows these derived consumption rates for zero-delay cars and vacations of various durations. Once consumption rate at zero delay is known, it is possible to calculate the value (v) of delayed

Value of Money

A. Raineri and H Rachlin

91

Exhibit 8. Derived consumption rates (R) Duration Car 1 week 1 year 10 years

R ($/day) Vacation

36 16 7

625 40 40

commodity rewards using equation (5). Setting that calculated value equal to the value of an undelayed monetary reward (obtained by equation (7)), the equivalent amount (AQ) of an undelayed monetary reward may be estimated. Exhibit 9 shows the obtained (points) and predicted (lines) values of delayed cars and vacations of various durations. Exhibit 8 indicates that as duration increased, derived consumption rate diminished sharply for both car and vacation, but much more steeply for the vacation. Perhaps the reason for the decrease in overall rate of consumption is the diminished use these college students would make of a car or vacation over long durations. This would be especially true of the vacation. A one-week vacation might be completely used but a 1- or 10-year vacation would disrupt these students' lives and leave them with no additional assets at the end. Hence, they would not be able to use the longer-duration commodities at the maximum rate for the entire duration. Although the obtained fits of predicted to actual points in Exhibit 9 are reasonably good (for

Equivalent Amount (Ao)

Equivalent Amount (Ao)

$100,000

$10,000

910,000 $1,000 $1,000 $100

$100 DURATION

$10

• 1 waak 9 1 ynr ^ W yatra

DURATION • 1 waak • 1 ywr ^ 10 y«ir«

$10

1 waah

DELAY (days)

1 month

1 yaar

10 yawa

DELAY (days)

Exhibit 9. Median immediate amounts of money equal in value to (a) an economy car and (b) a vacation of 1 week (square). 1 year (circles), and 10 years (diamonds). The solid lines are predicted values based on the derived consumption rates of Exhibit 8

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car, f2 = 0.972; s]ope = 0.9I3; for vacation, ^2 = 0.948, slope = 0.840), all points fall below predicted equivalent amounts. Exhibit 10. Fractions of reward value (equivalent amount of delayed reward divided by equivalent amount of undelayed reward) for one year's use of $500/day, $50/day, an economy car, and a vacation Delay 0 1 week 1 month 1 year 10 years

$500/day

$50/day

1.00 1.00 0.90 0,64 0.48

1.00 1.00 0.79 0.71 0.21

Vacation Car (/? = $40/day) (.R = $16/day) 1.00 0.92 0.65 0.42 0.19

1.00 0.75 0.63 0.34 0.10

Exhibit 10 compares normalized discount functions of Experiment 4 for 1-year durations of commodities (car, vacation) at various delays with discount functions of Experiment 2 for 1-year duration of cash amounts ($500/day, $50/day) with the same delays. The derived consumption rates for the one-year duration, zero-delay commodities (from Exhibit 8) are shown in parentheses. Note that, as with amounts in Experiment 1, the lower the consumption rate (stated or derived from the zero-delay data), the steeper the discounting. Since the duration was the same (one year) for all the items in Exhibit 10, amount was proportional to rate {R). The commodity data are thus consistent with the money data in showing an interaction between amount and delay.

CONCLUSIONS These four experiments demonstrate that the value of money and other commodities as measured by the choices of undergraduate subjects among hypothetical rewards depends on their anticipated rate and duration of consumption. The fundamental constraint on consumption is its limited rate. Because the commodities that money buys can only be consumed at a limited rate, marginal increments of those commodities as well as that of money itself are discounted in value by delay. For example, a steak dinner for most of us is more valuable when we have not eaten for a day than when we have just had a heavy meal. The reason for this, according to the present conception, is that right after a heavy meal our digestive systems are still handling the meal. We cannot process the steak right after the meal with the same efficiency as after a day's fasting; in terms of the present conception, consumption rate is slower {R is smaller) right after a meal. Furthermore, R for the steak should be lower, the more substitutable the meal just eaten is for the steak. Many delay discount functions could account for diminishing marginal value. The particular discount function applied here, a hyperbolic function originally used to explain choices of hungry pigeons among various amounts and delays of food reward (equation (2)), provided a good fit to the data in Experiment 1, where delayed rewards could be consumed freely once they were obtained. In the other experiments, deviation from predicted values depended on further consumption constraints; the more constrained, the more deviation. Together with the assumption that anticipated consumption rate is a negatively accelerated function of amount, the hyperbolic function accounts for the interaction between amount and delay of money as these variables affect value; large amounts of money are discounted relatively less by delay than small amounts. (For instance, the present value of a million dollars delayed by a month is almost a million dollars, but the present value of a dollar delayed by a month is much, much less than

A. Raineri and H. Rachlin

\ '' '

"

^

Value of Money

a dollar.) The hyperbolic delay discount function also accounts for reversals of preference over time (failures of'self-control'), reversals that appear irrational in the light of the commonly used exponential discount function (continuous compounding of interest) of economics (Rachlin and Rained, 1991) as well as the enhanced value of gambles such as those at casinos and racetracks of zero or negative expected value (Rachlin, 1990). The fact that consumption rate is the fundamental constraint on value suggests a mechanism that might contribute to addiction (Rachlin, 1992). Certain commodities have to be learned to be consumed. Alcohol and cigarettes are notoriously difficult for neophytes to consume, but, with practice, both taste and physiology alter; only experienced drinkers and smokers can consume two quarts of whiskey or two packs of cigarettes a day. The same pattern may well hold true for other addictive substances. Equation (4) implies that value increases logarithmically with increases in amount of a commodity but directly (i.e. much more steeply) with increases in consumption rate. Thus, a person may increase the value of a substance, without obtaining more of it, by learning to consume it faster. Once value is increased (relative to other available cotnmodities), people will spend more of their resources on that substance. An addictive cycle would then consist of alternate increases in value: by increasing consumption rate (direct value increase), then by increasing amount obtained (logarithmic value increase), then by further increases in consumption rate, then by further increases in amount obtained, and so forth, until the absolute limits of consumption rate were at last reached. The model suggests that substances that can be learned to be consumed unusually rapidly will tend to be addictive substances and people who are unusually capable of learning to consume substances rapidly (for instance, to absorb alcohol or nicotine rapidly into their bloodstreams) are most prone to become addicts. Questions naturally arise about the degree to which the current methodology (undergraduate subjects choosing among hypothetical rewards) applies to everyday life. From a behavioral viewpoint (our viewpoint), experimental instructions are discriminative stimuli. An experimenter who asks a subject to imagine some state of affairs is asking the subject to behave as he or she would behave if that state of affairs existed. Whether or not our subjects did indeed follow our instructions may be judged in terms of the internal consistency of the results, their plausibility, their applicability to everyday life, and their compatibility with (1) other laboratory studies with humans choosing among real but non-meaningful rewards and (2) other laboratory studies with non-humans choosing among real and meaningful rewards. The present studies qualify on all of these grounds, but we especially stress the quantitative consistency (specific functional relationships and parameters carried over from experiment to experiment), unusual in human decision research. The present experiments further extend the range of applicability of Mazur's discount function (equation (2)), which is in turn based on Herrnstein's (1970) matching law, which in its turn is explicable in terms of maximization of utility (Rachlin, 1989). Thus, despite apparent inconsistencies and irrationalities, the subjects of these experiments may be conceived as choosing within the imposed constraints so as to maximize subjective expected utility. It is a fundamental assumption of behavioral theory that an animal cannot but choose what it values most at the time of choice. The question we ask (which we believe should be asked in all studies of choice) is not. Are subjects maximizing utility? or Are subjects rational? but. What is the utility function (or discount function) that best accounts for their behavior? (Becker, 1976). The specific constraints and parameters of these experiments may be expected to apply only to the population tested. The subjects were all undergraduates, almost all around 20 years old, with limited experience in earning and spending money and limited social responsibilities. Other groups, specifically older subjects supporting families, perhaps anticipating paying for a child's education, might well be expected to operate under different constraints. However, the fundamental patterns observed here should still apply.

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ACKNOWLEDGEMENTS This research was done with the aid of a grant from the National Institute of Mental Health. Experiment 4 was designed in collaboration with Walter Mischel. We thank David Cross, Walter Mischel, and George Loewenstein for many helpful criticisms, comments, and suggestions. Correspondence should be addressed to Howard Rachlin. REFERENCES Ainslie, G., 'Impulse control in pigeons'. Journal of the Experimental Analysis of Behavior, 21 (1974), 485-9. Ainslie, G. and Haendel, V., T h e motives of the will', in Gottheil, E., Druley, K. A., Skoloda, T. E. and Waxman, H. M. (eds), Eiiologic Aspects of Alcohol and Drug Abuse, Springfield, ILL: Charles C Thomas 1983,119^0. Becker, G. S.. The Economic Approach to Human Behavior, Chicago: University of Chicago Press, 1976. Benzion, U., Rapoport, A. and Yagil, J., 'Discount rates inferred from decisions: an experimental study' Management Science, 35 (1989), 270-84. Herrnstein, R. J., 'On the law of effect'. Journal of the Experimental Analysis of Behavior, 13(1970), 243-66. Logue, A. W.. "Research on self-control: an integrating framework', The Behavioral and Brain Sciences 11 (1988), 665-79. Loewenstein, G., 'Anticipation and the valuation of delayed consumption'. Economic Journal, 97 (1987), 666-84. Loewenstein, G, and Frelec, D., 'Anomalies in intertemporal choice: evidence and an interpretation'. Quarterly Journal of Economics, in press. Mazur, J. E., 'An adjusting procedure for studying delayed reinforcement', in Commons, M., Mazur, J., Nevin, J. A. and Rachlin, H. (eds). Quantitative Analysis of Behavior: Volume 5, The Effect of Delay and of Intervening Events on Reinforcement Value, Hillsdale, NJ: Lawrence Erlbaum Associates, 1987. Newman, P., The Theory of Exchange, Englewood Cliffs, NJ: Prentice-Hall, 1965. Preiec, D., 'Decreasing impatience: definition and consequences'. Working paper, Russell Sage Foundation New York. 1989. Rachlin, H., Judgment. Decision And Choice, New York: W. H. Freeman, 1989. Rachlin, H.. 'Why do people gamble and keep gambling despite heavy losses?' Psychological Science, I (1990), 294-7. Rachlin, H., 'Diminishing marginal value as delay discounting', Journal of the Experimental Analysis of Behavior 57 (1992), 407-16. ^ ./ . Rachlin, H.,Castrogiovanni, A. and Cross, D.,'Probability and delay in commitment', yourna/o/Me£jcperj>nen/a/ Analysis of Behavior, 48 (1987), 347-54. Rachlin, H., Raineri, A. and Cross, D., 'Subjective probability and delay'. Journal of the Experimental Analysis of Behavior, 55 (1991), 233-44. Rachlin, H. and Raineri. A., 'An objective view of irrationality and impulsiveness', in Loewenstein, G. F. and Elster, J. (eds). Perspectives On Intertemporal Choice, New York: Russell Sage Foundation, in press. Stevens, S. S., Psychophysics, New York: John Wiley, 1975. Stevenson, M. K., 'A discounting model for decisions with delayed positive or negative outcomes'. Journal of Experimental Psychology: General, 115(1986), 131-54. Strotz, R. H., 'Myopia and inconsistency in dynamic utility maximization'. Review of Economic Studies 23 (1955), 165-80. Thaler, R., 'Some empirical evidence on dynamic inconsistency'. Economics Letters, 8 (1981), 201-7.

A uthors' biographies: Andres Raineri has just received his PhD in Psychology from SUNY, Stony Brook and currently resides in New York City. Howard Rachlin is a Professor of Psychology at SUNY, Stony Brook. He is associate editor of Philosophy And Behavior and author of a forthcoming book. Behavior and Mind: The Two Psychologies, to be published by Oxford University Press.

Authors'address: Andres Raineri and Howard Rachlin, Department of Psychology, SUNY at Stony Brook, Stony Brook NY 11794-2500, USA.