A Test of Diminishing Marginal Value

INTRODUCTION

The assumption that having more of a good will lead an individual to place a lower value on an additional unit of that good, which we call diminishing marginal value, is a pervasive component of economists' beliefs about human behaviour. Robert Frank (2000), summarizing this intuition, writes that ‘the more one consumes of something, the less one is willing to pay to obtain more of it. Even a hungry person would be willing to pay less for a second sandwich than for the first.’¹

This concept was highly influential in the development of economic thought. Initiated in the 1870s, the ‘Marginalist Revolution’ represented such a severe break from the prevailing approaches that many contemporaries reasoned that the new paradigm was ‘quackery’ because it lacked a direct connection between utility and demand. Combining the critical assumption of diminishing marginal value with equivalent marginal utilities (per dollar spent) across goods, H. H. Gossen (1854) initially made this connection.² In 1871 Jevons independently developed his own results linking demand and utility, making use of diminishing marginal utility (see Jevons 1965). Other figures in the Marginalist School also appealed frequently to diminishing utility. Walras wrote: ‘The want which we have for things, or utility which things have for us, diminishes gradually as consumption increases’ (see Walras, 1954); and Jevons wrote: ‘Each increment of food is less necessary, or possesses less utility, than the previous one.’

This presumption of diminishing marginal value was based on its plausibility and intuition, not on specific empirical evidence. Our goal is to fill this gap. A great many other aspects of neoclassical behaviour have been shown to fail when examined ‘under the microscope’ of experimental economics. Our goal is to subject diminishing marginal value to this same scrutiny.

The task requires us to be more precise about what is being measured. Early writers wrote in terms of diminishing utility. In this paper we look at diminishing marginal value (DMV) where value is measured in terms of either money or another good that must be exchanged for the good in question. Diminishing marginal value must be measured in one of these units. Note that the link between preferences and demand has been considerably refined since the nineteenth century; we focus here directly on the idea expressed by Jevons and Walras, which formed the germ of early economic progress.

To test for DMV, we conducted three types of experimental treatments. The first set of treatments uses a standard willingness-to-accept format and examines whether the compensation an individual requires to give up a unit of some good decreases when the individual has more of that good. This is a money-based measure of diminishing marginal value and it has the most general applicability. For example, public finance economists would frequently be interested in knowing whether additional units of a publicly provided good become less valuable as more of the good is provided. Since public provision of such goods typically require a ‘general sacrifice’ that might best be measured in money terms, a money-based measure of diminishing marginal value (for the good in question) is most informative for making the necessary inference.

In a second set of treatments we examined subjects' willingness to trade one unit of an endowed good for one unit of another good. We gave our experimental subjects different amounts of the endowed good. Diminishing marginal value means that more individuals are willing to make a trade when they have more of the endowed good. This is a goods-based measure of DMV. This formulation comes closer, we feel, to the kind of exchange situation that the early writers envisioned. Furthermore, it enables us, with some additional experiments, to trace out a demand curve, thereby roughly following the early linkage.

The properties of the utility function are most interesting for public goods since these are goods for which demand curves are usually missing. Although a true policy-relevant experiment is essentially impossible to conduct, we designed a third set of treatments to test whether DMV is exhibited in collective choice. In these treatments, subjects were endowed with one good and asked whether they were willing to trade it for one unit of another good; however, the trade would take place only if a majority of subjects agreed to it, and all subjects would be required to trade their good in that case. We tested for DMV by changing the endowment of the initial good for all subjects and examining whether more individuals were willing to vote to make the trade when they had more of the initial good.

The evidence we found for DMV is strong. In the money-based treatments, which elicited a continuous measure of value, all possible comparisons show a decrease in the value of the items as the endowment of the good in question increased. We found consonant results in the exchange treatments: increases in the endowment significantly increased subjects' willingness to trade part of it away (i.e. the trading rate) in all of our treatments, in both individual choice and collective choice situations.

Previous research has largely been content to test whether estimated demand curves are consistent with utility maximization. These studies, which are legion, frequently imply an underlying utility function that would indeed exhibit DMV.³ What is at stake in our research is an explicit test of the ‘utility-based’ behaviour, which may then be connected with the more common ‘demand-based’ picture of behaviour.

In Section II below we report a set of demand curve experiments which we then link to the DMV results using a specific utility function. If we allow for a reference-point effect, these experiments show demand curves that are highly consistent with the DMV behaviour we previously uncovered.

Although DMV is highly intuitive, it may be worthwhile to consider the basis of this intuition. A sufficient condition for DMV is that utility be concave in the good in question and that an increase in the endowment of the good not reduce the marginal utility of money ‘too much.’⁴ The first of these properties is the historical neoclassical idea. The second property means that having more of the good in question does not so greatly reduce utility from other goods that money itself becomes less valuable.

DMV may also be understood by framing the issue in terms of the quasi-concavity of preferences, hence without explicitly invoking concavity. If preferences are quasi-concave, then either {good A valued in terms of good B} or {good B valued in terms of good A} must exhibit DMV. It is difficult to see why one of these values should exhibit DMV and not the other. Therefore, DMV might better be thought of as following from quasi-concavity when the goods in question are such that preferences for them can be expected to exhibit similar properties.

I. EXPERIMENTAL DESIGN AND RESULTS

Our treatments fall into three categories, depending on whether value is measured in terms of money or another good and, for the goods-based experiments, on whether there is individual or collective choice. Within each category, we statistically test for diminishing marginal value, using subjects who are randomly allocated into one treatment cell.

Treatments are summarized in Appendix A. Except for treatment A, our subject pool consists of non-student subjects operating in non-classroom settings. The subject pool consists of ‘middle America’, although we cannot claim to have a representative sample of the US population.

II. UTILITY AND DEMAND

Diminishing marginal value played a key historical role in connecting demand and utility. In this section we use our experimental sports card results to show the empirical link. This exercise requires (i) a specific utility function, (ii) the DMV treatments reported above and (iii) a further set of treatments with different trading prices, which allows us to observe demand. We first describe the demand treatments.

Linking utility and demand

To connect the demand and DMV results we use a common utility function, shown in (1). Our functional form is similar to the minimum-distance utility specification in animal commodity–choice behaviour (e.g. Kagel et al. 1995), but we substitute reference points for their ‘minimum needs’:

(1)

where α, A₀ and B₀ are parameters and A and B are the ‘consumption’ of the two sports cards. Note that we expect α>0.5 since the data show a slight preference for good a over good b.

To convert trading decisions to ‘consumption’, we calculated items per 100 subjects. This is easiest to understand in terms of the budget constraints for each particular treatment, which are shown in Table 11. In treatment J, 100 subjects begin the treatment with 3a each; thus, the endowment is A=300. One unit of a can be traded for one unit of b. Thus, the budget constraint is A+B300. If the trading rate were 75% (75% of subjects opt to trade an a for a b) then those 100 subjects would end up with 75b, leaving them with 225a. The final consumption levels would be A=225 and B=75.

Table 11.
BUDGET CONSTRAINTS AND REFERENCE POINTS

Treatment	Budget constraint	A0	B0
H	a+b=100	100β	100γ
I	a+b=100	100γ	100β
J	a+b=300, a200	300β	300γ
K	a+2b=300, a100	300β	300γ
L	a+b=300, b200	300γ	300β
M	2a+b=300, b100	300γ	300β

An appropriate scale is also needed for A₀ and B₀. The reference-point effect described above requires us to recognize that this scale may be specific to each treatment, dependent on both the endowment and the framing of the decision problem. We set the baseline for the endowed good at βY, where Y is income (Table 11), and the baseline for the non-endowed good at γY. A reference-point effect implies β>γ; a finding of β=γ implies no reference-point effect. This format is arbitrary but useful. Our main goal here is to construct a model that provides a transparent role for reference-point effects yet is also well behaved as a model of choice and demand in all other regards. Values for A₀ and B₀ based on this approach are shown in Table 11.

Maximizing utility subject to the budget constraint yields demand for A given by

(2)

Results We used data from experiments H–M to estimate (2) using maximum likelihood methods. Estimated parameters and approximate standard errors are α=0.56 (0.04), β=0.60 (0.05) and γ=0.12 (0.04). Actual and predicted values for choice variables are shown in Table 12. These results show a high degree conformity with observed choices, although we recognize that such an assessment is necessarily qualitative.

Table 12.
PREDICTED v. ACTUAL CHOICES

Treatment	Predictions	Actuala
	A* (from (2))	A (Tables 5, 6, 9, 10)
H	76	77.5
I	28	27.5
J	227	222.5
K	207	205
L	84	87.5
M	50	42.5

• a All numbers normalize the number of subjects to 100.

The hypothesis β=γ is rejected at above the 99% level. Since in our experiments, being endowed with a good makes the good roughly five times more valuable than not being so endowed.

Previous research has largely been content to find well behaved demand curves, while the underlying behaviours such as DMV have been largely unexplored. Our calculations yield a joint model of DMV and demand, provided we make allowance for the reference-point effect.⁹

III. CONCLUDING COMMENTS

DMV is an historically important idea that helped lead the early neoclassicists to adopt a utility-based model of behaviour, although it is not the link between utility and demand in a theoretical sense, and early writers sometimes conflated it with other relevant concepts. For modern economists, this property remains a widely assumed behavioural feature, often the core of our intuition—indeed, it is often difficult to imagine behaviour without DMV. Yet, the empirical evidence for it has remained unexplored and untested. This gap has clearly needed redressing. Given the rising evidence of anomalies in neoclassical preferences, it is desirable to look for empirical support for even the most basic postulates of demand theory.

In our tests, the range of endowments being valued is small. Such small ranges provide powerful tests of our hypothesis. They do not, however, allow us to predict behaviour under larger endowments. We also have looked at only three types of good, while we expect the effect of the endowment on marginal value to be different for different goods. Economists will be most interested next, we feel, in knowing which kinds of goods are likely to have strong DMV and which are likely to have constant marginal value or even increasing marginal value. We leave these questions for a subsequent paper.