834 THE AMERICAN ECONOMIC REVIEW DECEMBER 1997 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 5 -10 5 10 15 20 B.A Simple Model that valu for which the ob For a first look at the data,consider the fol- lowing very simple model of behavior.As. simple class of models such a value of g bes sume that all subjects are s the data.Figure I graphs the ob he ond th e f value is less than or equal to some critical in the range between15 and 20.The bes value,or cutpoint.gbut that they randomly estimate is g=1.at which the deviation rate from thi 'selfish mode selfish h contribute if (n-V)<g overcontribution seems to bee R keep if (n-V)>8 keep or contribute if (n-V)=8. m decis ons to r at andg is one cent.in the 0 =0 in ex in which From our data.we can estimate the maximum likelihood values of(g,q)simply by finding of the public good by no more than one cent
834 THE AMERICAN ECONOMIC REVIEW DECEMBER 1997 u.oo 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 n 1 1 \ \ \ \ \ - 1 1 1 q / / / / / -15 -10 -5 0 5 10 15 20 9 FtGURB 1. CUTPOINT ANALYSIS : FRBQUENCY OF DEVIATIONS FROM THE ^-OPTIMAL DECISION RULE Notes: For each hypothetical warm-glow effect, g, the graph shows the frequency of deviations from the g-optimal decision rule, q. The value g = 1 has the lowest associated q. B. A Simple Model For a first look at the data, consider the following very simple model of behavior. Assume that all subjects are identical and that they contribute if and only if the difference between their token value and the public good value is less than or equal to some critical value, or cutpoint, g, but that they randomly deviate from this decision rule some fraction of the time, q. Call g the warm-glow effect: e.g., if g > 0, then the interpretation is that a subject gains g solely from the act of contribution. Given a fixed value of g, a subject's ^-optimal decision rule is: ( contribute keep keep or contribute iHn-V)<g \f{n-V)>g Despite its simplicity, this class of (g, q) models encompasses a variety of behavior, from completely random decisions {q = 1) to the standard model of completely selfish behavior with no error at all {g = 0, ^ = 0). From our data, we can estimate the maximumlikelihood values of {g, q) simply by finding that value of g for which the observed frequency of deviations from the ^-optimal decision rule is minimized. Within this very simple class of models such a value of g best describes the data. Figure 1 graphs the observed frequency of deviation from the goptimal decision rule, for each integer value of g in the range between —15 and 20. The best estimate is g = I, at which the deviation rate is ^ = 0.11. The standard "selfish" model, g = 0, is nearly as good, with a deviation rate of g = 0.12." The implication of this very simple analysis is that an aggregate warm-glow effect exists, but it is small in magnitude.'' There is overcontribution relative to the selfish theory, but much, if not ail, of this overcontribution seems to be explainable as " Even though the difference in the deviation rate is small, a likelihood ratio test rejects the g = 0 model in favor of the g = I model. The x' statistic is 107.47 with 1 degree of freedom and n = 2,560. " The dollar equivalent of the difference hetween g = 1 and g = 0 is one cent, in the sense thai g = 1 corresponds, in experimental payoffs, to behavior in which a subject is willing to contribute his or her endowment if and only if the value of the endowment exceeds the value of the public good by no more than one cent
VOL 87 NO.5 PALFREY AND PRISBREY:PUBLIC GOODS EXPERIMENTS 83 we examine the nature of the decision rule in detail,giving more consideration to the En=(ra V)-gi-a(N 1)V, errors generating dev od to the role of other factors such as ex where the right-hand side contains all the el. ements of the subject's utility function that de erience and altruism,that are likely to af- termine his or her choicex fect contribution decisions. Accordingly,we estimate a probit model C.The Probit Model The probit model provides a standard way the independent variables in the model.Giver ch don ent trea the public good value.and experience.The the difference (r).which we call structural model underlying this analysis is the diff;and of the ublic c good, that were controlled in the ariables exper,for experience,which takes on a U (xa,x-n) zero for decisions in bloo =yΣ+(8-nx+m period sequence of the same public good value: +aw-1y which takes on values from one to ten. +∑[(g-)x:+w1 D.The Representative Subject Model where V,is the public good value in period. e t in these rep riod and altruism effects a d t be the same across individuals.An observatio riod, is a contribution decision in a single period. player i's ruism term,and odel we assume that for each of subject i's decisions to the warm-glow her subject hen the tw ume that the are independent,identical Normally distrib uted random variables with mean zero and
VOL 87 NO. 5 PALFREY AND PRISBREY: PUBLIC GOODS EXPERIMENTS 835 noise rather than some systematic component of the decision rule. In the next sections, we examine the nature of the decision rule in detail, giving more consideration to the structure of errors generating deviations, to possible heterogeneity across individuals, and to the role of other factors such as experience and altruism, that are likely to affect contribution decisions. C. The Probit Model The probit model provides a standard way to measure the probability of contribution as a function of the different treatment variables, such as the individually assigned token values, the public good value, and experience. The structural model underlying this analysis is the following. We assume that the utility player i gets in period t from contributing ;c,, units of the private good is: = V, X A subject contributes if and r L (Nj*i J where V, is the public good value in period t, gi is player /'s warm-glow term, r,, is player i's token value in period t, Wi, is player I's endowment of tokens in period /, a, is player I's altruism term, and A^ is number of players in /'s group. Finally, in order to estimate the model we assume that for each of subject fs decisions at period t there is a random component, e^,, that is added to the warm-glow term. This error term represents some random added propensity for the subject to either contribute or not contribute. We assume that the e^/s are independent, identical. Normally distributed random variables with mean zero and variance only if where the right-hand side contains all the elements of the subject's utility function that determine his or her choice Xi,. Accordingly, we estimate a probit model, where the probability of contributing a unit of the endowment is given by the cumulative Normal transformation of a linear function of the independent variables in the model. Given our specification of the decision rule of the subject, our independent variables are: • a constant term, which we call constant; • the difference (r, - V^), which we call diff; and • the value of the public good, V. In addition, we include three other variables that were controlled in the experiment: • exper, for experience, which takes on a value of zero for decisions in the first tenperiod sequence with a given public good value, and one for decisions in the second tenperiod sequence of the same public good value; • endow, which takes on a value of zero if the endowment is indivisible and one if it is divisible; and • period, which takes on values from one to ten. D. The Representative Subject Model We present estimates from two probit models which differ only in which independent variables are included. Note that in these representative subject models, the warm-glow and altruism effects are implicitly assumed to be the same across individuals. An observation is a contribution decision in a single period.'" '" We pool observations across all experiments. Decisions in the divisible endowment treatment {endow = 1) are coded as either 0 or 1, depending on whether subjects contributed less than half or more than half their endowment of tokens in a given period, respectively. Similar conclusions obtain when the iwo endowment U'eatment samples are estimated separately. This is addressed in detail in the next section, where some minor differences are also discussed