Dynamic decisions in a laboratory setting.

Olson, Mark A.

I. Introduction

Many economic models involve a trade-off between current and future rewards. For example, in neoclassical one-sector growth models there is a trade-off between immediate consumption and investment for future consumption. In sequential job search models of labor economics, workers must decide between continuing their job search at a cost or accepting the best wage offered so far. In oligopoly models of tacit collusion, firms decide whether or not to defect from collusive behavior, and the defection leads to short-term benefit but future punishment. These models are all dynamic in the sense that an action at one stage influences the available actions or rewards at a future stage.

Many such examples fall into the class of dynamic programming problems and for large classes of such problems, techniques for finding optima are well understood. Less well understood, however, are the decisions which individuals actually make in such dynamic settings. This is an empirical issue which lends itself to the methodology of laboratory investigation. There have already been a number of experiments in which subjects face a sequence of related decisions. Examples include the asset market studies of Camerer and Weigelt [3] and Smith, Suchanek, and Williams [13]. The predictions of dynamic game theory have been tested extensively, for example, in the contexts of the centipede game by McKelvey and Palfrey [9], of the repeated prisoner's dilemma by Selten and Stoecker [11] and Andreoni and Miller [1] and of bargaining by Ochs and Roth [10]. Dynamic job search has been studied by Cox and Oaxaca [4] and two armed bandits have been studied by Banks, Olson, and Porter [2].

In the market and game theory studies the decisions made by subjects are complicated by the strategic aspects of the games; subjects' decisions must take into account the strategies they expect the other players to use. In asset markets, subjects seem to have difficulty making decisions and speculative bubbles frequently result. As noted by McCabe, Rassenti, and Smith [8], "The robust tendency of laboratory stock markets to produce bubbles is attributable to the myopic trading behavior of subjects. In effect, subjects fail to act according to the backward induction principle." The game-theoretic equilibrium refinement of subgame perfection, which depends on backward induction, fails to describe the observed outcomes in the game experiments cited above. The experiments in job search, in which the wage offers are stochastic, find that early termination of the search is observed (compared to a rational risk-neutral agent) though the data is consistent with the ability to solve complex optimization problems, if the presence of risk-aversion is postulated. In the two-armed bandit study, in which exogenous uncertainty is present, consistent deviations from the "maximizing" strategy, described by the authors as experimentation and hedging, are observed.

In this paper we report data from an experimental study in which subjects are given a monetary incentive to solve a simple, non-stochastic, non-strategic, multi-stage, dynamic decision problem. The structure of the problem is motivated by the consumption vs. investment tradeoff of the one-sector growth problem. We structure the problem in very simple terms; subjects are asked to make ten decisions at discrete time intervals. There is no strategic uncertainty, no risk, no exogenous uncertainty, and no complex computation. These complicating features are removed from the decision problem. Decisions do not have to take into account beliefs about other players, there is no incentive for risk averse subjects to smooth out their payoffs, and no probability calculations need to be made. The optimum can be found using a simple rule. Under investigation is the empirical validity of a basic assumption of a variety of models such as those mentioned in the first paragraph. If subjects can successfully find the optimum for our decision problem, we need to consider the complicating elements as the source of the difficulty subjects have in the experiments cited above. If they cannot find the optimum, then one source of the difficulty may be purely in the sequential structure of the decision problems.

The remainder of the paper is divided into sections as follows. In section II we discuss the structure of the decision problem and describe the experimental design. In section III we analyze the data. In section IV we give our concluding thoughts.

II. The Experiment

In this section we describe the dynamic decision problem presented to the subjects.(1) For purposes of exposition we will describe the problem in the traditional terms of consumption and capital stock.(2) Subjects were faced with a discrete integer-valued approximation to the following dynamic decision problem:

[Mathematical Expression Omitted] (1)

subject to:

[C.sub.t + 1] + [K.sub.t + 1] = f([K.sub.t]) + 0.5[K.sub.t] (2)

[K.sub.t + 1], [C.sub.t + 1] [greater than or equal to] 0 (3)

and

[Mathematical Expression Omitted] (4)

with t = 1, ..., 10 and [K.sub.1] given.

[U.sub.t] is utility in each "round" t derived from consumption [C.sub.t], [K.sub.t] is the capital stock at time t, Q([K.sub.11]) is the utility from the remaining capital stock after time 10, f([K.sub.t]) is the production function and the existing capital stock depreciates by 50% each round. Consuming too much at any time drives the future capital stock down too much, and hence decreases future production and consumption. Investing too much at any time decreases utility from current consumption by too much. The optimal decision consists of a constant level of investment each round (except for the last round in some treatments). The optimal level of investment is independent of the utility function but does depend on the initial endowment.

The experiments, which were computerized, took place at the CREED laboratory of the University of Amsterdam. The 48 participants were undergraduates at the university. Participants were not allowed to communicate with each other during the experiment. Each participant played 4 periods, numbered 0-3, each consisting of 10 rounds, for a total of 40 decisions. Each subject faced the same decision problem for all four periods. A participant's possibilities for investment in a given round depended on decisions he/she made in earlier rounds of the same period. The data from period 0 were discarded as they were for practice only and did not affect participants' final earnings. Participants took between 60 and 90 minutes to complete the experiment. Final earnings of subjects varied from 20 to 40 Dfl. (12-20 $U.S.). 8 Dfl. is a good hourly net wage for a college student in Amsterdam.

Participants were given a table of production possibilities, Table XI in the appendix, that was a discrete approximation off ([K.sub.t]) + 0.5[K.sub.t]. The table lists, for any integer level of the capital stock in round t, all feasible combinations of consumption in round t and capital stock in round t + 1. Each participant was also given an initial endowment of capital stock (either 4 or 7) and a table (see tables XII and XIV in the appendix) of total utility associated with consumption in round t and corresponding marginal utilities, M[U.sub.t], in terms of "tokens," an experimental currency. The production function and the utility from consumption remained the same each round. Each subject's marginal utility was a discrete approximation of either

[Mathematical Expression Omitted], (5)

or

[Mathematical Expression Omitted]. (6)

Participants received "tokens" based on their consumption level. For example, if in a particular round, a subject with marginal utility [Mathematical Expression Omitted] consumed two units of C, the subject received 400/4 = 100 tokens for the first unit and 400/5 = 80 tokens for the second unit, or a total of 180 tokens for the round. These tokens were converted to Dutch guilders at the end of the experiment at a conversion rate known in advance to the subjects. Participants were asked at each round t to give their desired level of consumption and capital stock for round t + 1 and then to verify their choices. They then observed their amount consumed, their remaining capital stock, and their payoff from consumption before the program moved on to the next round. Subjects never observed the terms consumption (C) and capital stock (K), but saw the neutral terms X and A instead. Subjects did not observe any of the decisions made by any other subjects.

[TABULAR DATA FOR TABLE I OMITTED]

Subjects consumed all of the capital stock in their inventory after round 10 of each period. The marginal utility of consumption of the capital stock corresponding to the two utility functions was 90 - 5[K.sub.11] and 400/[K.sub.11] for M[U.sup.1] and M[U.sup.2] respectively (see Tables XIII and XV in the appendix). Thus, in round 10, equating marginal utilities of investment and consumption implied choosing a level of capital stock 3 units greater than round 10 consumption. Table I contains the optimal decisions in the four treatment cells of the experiment.

The total sample was divided into 4 groups, each corresponding to one treatment, with 12 subjects in each group. Throughout the remainder of the paper initial endowments of 4 and 7 are labelled as Low and High respectively. The marginal utilities in equations (5) and (6) are labelled as Lin and Con respectively, due to the linearity or convexity of the marginal utility. Thus, treatment Lin/High, refers to the condition in which the linear utility function and the high endowment of 7 are in effect.

In Table I, in the column labeled Optimum, the optimal solution to the dynamic optimization problem faced by the subjects is given. The optimum is independent of the utility function of the agent but does depend on the initial level of capital stock. If the initial endowment is 4 units of K, the optimal strategy is to maintain 4 units of K and consume 3 units of C in rounds 1-9 and then to maintain 5 units of K and consume 2 units of C in the last round (10). If the initial level of K is 7 units, the optimal strategy is to maintain 6 units of K and consume 3 units of C in each round. The optimal decision is straightforward, in that it always involves a constant level of capital stock and consumption in all rounds for endowment level 7, and in the first nine rounds for endowment level 4. It should be kept in mind, however, that while the decisions given in table I are optimal, different choices of K can yield close-to-optimal payments. For example, in Lin/High, a constant investment level of 8, consumption of 1 in round 1 and consumption of 3 in each of the remaining rounds, provides the subject with 99.7% of the total possible payment. Therefore, in our analysis of the data we emphasize the percentage of the maximum possible payment actually obtained by the subjects.

The formal analysis in the next section concentrates on the effect of different utility functions, of different endowments, and of repetition on the decisions of subjects. It can be seen that the optimal solution is independent of the utility function, but does depend on the initial endowment. The optimal solution does not depend on experience or repetition in solving the problem, but previous experiments lead us to believe that experience may be an important explanatory variable.

[TABULAR DATA FOR TABLE II OMITTED]

III. Results

The three variables we analyse are the two choice variables of the subjects, consumption and investment, as well as the efficiency of the outcome of their decisions. The term efficiency(3) denotes the percentage of the maximum possible money income obtainable that was actually obtained by the subject and is a way of measuring subjects' performance on a decision problem. In our analysis of efficiency we focus on: 1) whether subjects make money payoff maximizing (optimal) choices; 2) the amount of variation within treatments; 3) whether and how much outcomes differ by treatment; and 4) whether subjects improve their performance with experience.

We begin the analysis by giving some simple statistics on the decisions made by subjects. The mean, median, and standard deviation of the decisions (for rounds 1 to 9), made by subjects and the resulting efficiencies attained are given in Table II.(4)

The complete data for period 3, when subjects had the most experience, are given in Tables VII-X. Figure 1 shows the distribution of observed efficiency and illustrates the level of heterogeneity of subject performance.

The overall mean values of [K.sub.t+1] and [C.sub.t] show a small but consistent difference from the optimal levels. There is some tendency to overinvest and underconsume, especially in treatment [TABULAR DATA FOR TABLE III OMITTED] Con/Low, reducing efficiency below the optimal level of 1. There is substantial variation in the decisions made by different subjects within each treatment. None of the subjects used the optimal decision in all three periods, though 25 of the 48 subjects made over 94% of the total possible payment. Six subjects employed the optimal decision in at least one of the three periods.

The effect of treatment and experience on efficiency are explored by estimation of the following GLS error components model:

ln ((1 - [eff.sub.ip])/[eff.sub.ip] + 0.01) = const + [a.sub.2] x P2 + [a.sub.3] x P3 + [b.sub.1] x U1 + [b.sub.2] x E7 + [b.sub.3] x UE + [[Alpha].sub.i] + [[Epsilon].sub.ip]. (7)

P2 and P3 are dummy variables which equal 1 in periods 2 and 3 respectively (and 0 otherwise). U1 is a dummy variable which equals 1 if utility function 1 is in effect, E7 is a dummy variable which equals 1 if an endowment of 7 is in effect, and UE is an interaction term for utility and endowment. [[Alpha].sub.i] is the random effect of subject i. [eff.sub.ip] is the efficiency attained by subject i in period p. [[Epsilon].sub.ip], is the error term, which is assumed to be normally distributed with mean 0 and variance [Mathematical Expression Omitted]. Since all subjects are drawn from the same population, we assume [[Alpha].sub.i] to be normally distributed with mean 0 and variance [Mathematical Expression Omitted]. The transformation of efficiency was used because of the non-normality of the distribution of efficiency.

Table III gives the results of the error components model. The column labelled "Effect" gives the estimates of the effect of the variable on efficiency. The Lagrange Multiplier Test rejects the null hypothesis that individual error components do not exist; the Chi-squared statistic with 1 degree of freedom is 75.6108 with a P-value of less than 0.0001, strong evidence of heterogeneity among subjects. The constant coefficient, which equals the estimated efficiency attained by group 1 in period 1, is significantly less than 1 (the maximum possible). None of the treatments show a significant effect, indicating that the different decision problems were of roughly equal difficulty. The coefficients of the dummy variables P2 and P3 are both significant at the p [less than] 0.01 level, and the effect of P3 is slightly greater than that of P2; indicating that efficiency is increasing with experience but at a decreasing rate.

Our results on efficiency can be summarized as follows. Subjects exhibit heterogeneous behavior. [TABULAR DATA FOR TABLE IV OMITTED] [TABULAR DATA FOR TABLE V OMITTED] They are generally able to get higher payoffs as they gain experience with the decision process. The observed levels of efficiency are less than the optimum. The performance of subjects does not vary significantly across treatments.

We now turn to the actual choices made by subjects. Since the efficiencies are less than 1, it may be the case that consumption behavior differs across treatments. To test whether the choices differ by treatment, we construct the following MANOVA model:

[Mathematical Expression Omitted]. (8)

Each observation of [C.sub.ip] is a 10 by 1 vector indicating the observed level of consumption in each round t(t = 1, ..., 10) by subject i in period p. [Mathematical Expression Omitted] is a 10 by 1 vector denoting the optimal level of consumption by subject i in each round of period p. The analysis treats the data from each period as one observation on ten dependent variables, one for each round.

The results of the estimation are given in Tables IV and V. The last column in table IV gives the results of the decision on whether to accept or reject the null hypothesis (at the 5% level) that the independent variable has no effect on the vector of dependent variables. The F-ratios in Table IV indicate that the utility function and the period dummy variables have no significant [TABULAR DATA FOR TABLE VI OMITTED] effect on the deviation from the optimum, but that the level of the initial endowment does have an effect. Table V gives the results of F-tests from the MANOVA which isolate the effect of each of the individual rounds on the level of deviation from optimal consumption. Of the ten rounds in a period, the level of initial endowment has a significant effect on the deviation from optimal consumption only in rounds 9 and 10.

The pattern of observed consumption and investment relative to the optimum over rounds is given in Table VI for each of the four treatments. Several observations can be made from the table. The consumption of C is generally below the optimal level for rounds 1-8. In rounds 9 and 10 the pattern of consumption differs depending on the initial endowment, as mentioned previously. In Lin/Low and Con/Low, there is a sharp increase in consumption in round 10 relative to the optimum? For Lin/High and Con/High, there is a sharp decrease in consumption in round 10.

The deviations of investment from the optimum vary across treatments more than those of consumption. In Lin/High and Con/Low, the treatments in which earnings were the lowest, subjects accumulate too much K early in the period and then deplete it late in the period. In Lin/Low and Con/High the investment level is closer to the optimum, suggesting an interaction between utility function and initial endowment. In all four treatments the level of capital stock was greater in round 5 than in round 10, revealing a tendency to reduce capital stock at the end of the period in all treatments.

Though subjects may not choose the optimal consumption/investment path it is possible that they tend to choose the optimal path beginning in round t + 1, conditional on their choice in round t. If this is the case, the extent of deviations from the unconditional optimum may provide a misleading picture of the ability of subjects to successfully solve the decision problem. Figure 2 displays the percentage of choices that were conditionally optimal, as well as the percentage of the time that choices reflected overconsumption or underconsumption, in each round of period 3. At least 40% of the subjects' choices differed from the conditional optima in every round. Deviations are more likely to be in the direction of lower than of higher consumption for the first seven rounds. The incidence of optimal choices decreases toward the end of the period, despite the fact that the required backward induction becomes shorter.

The decision problem is simplest in round 10. No backward induction need take place; there is only one decision, after which the period ends. As can be deduced from Tables VII-X. of [TABULAR DATA FOR TABLE VII OMITTED] the 48 participants who participated in the study, only 14 made the optimal decision in round 10 of period 3 given their inventory of capital at the end of round 9. Most subjects do not equate marginal utilities of consumption and investment. In none of the four treatments did more than 4 subjects make the optimal choice given their level of capital stock in round 10. Roughly one-half of those subjects who did not optimize chose too much consumption, the other half too little consumption. In addition, 5 subjects actually consumed 0 units in round 10, even though a positive consumption level was optimal.(6)

The next paragraphs summarize the individual level data within each treatment.

Lin/Low. The data for period 3 are given in Table VII. In this treatment there was a tendency to make choices close to the optimum. Subjects 3, 4, and 7 made the optimal decision and subjects [TABULAR DATA FOR TABLE VIII OMITTED] [TABULAR DATA FOR TABLE IX OMITTED] 1 and 37 held a constant level of capital stock of 4, with a payoff very slightly below the optimum. Three of the subjects ran down their capital stock completely and two subjects, 27 and 41, built up a very high level of K.

Lin/High. The data are in Table VIII. Although there was a larger variation in choices in Lin/High than in Lin/Low, there was less variance in the earnings of subjects with the higher endowment. The higher endowment appears to prevent subjects from completely running down their capital stock before round 10, which is very costly. However, no subject made the optimal decision in Lin/High. Unlike Lin/Low, in which five of the subjects maintained a constant level of capital stock, only one subject in Lin/High chose the same level of capital or consumption for at least nine rounds. There seemed to be little recognition that the problem was identical in each round.

Con/Low. The data from Con/Low are much different from the data from Lin/Low, though the optima are the same. A higher level of capital stock was held under Con/Low. Three subjects, numbers 8, 23, and 24, held a roughly constant level of capital stock until the end of the period [TABULAR DATA FOR TABLE X OMITTED] and increased it sharply in rounds 9 or 10. Subjects 2, 25, 31, 39 and 47 built up very high levels of capital stock. All but one of the subjects had at least as much K at the end of the period as at the beginning. The high level of capital stock in Con/Low may indicate that subjects focus largely on the total values of capital and consumption, rather on the marginal values, since in Con/Low, the first units of K held in round 11 have a very high value.

Con/High. The data, given in Table X below, indicate a strong contrast between Lin/High and Con/High. Under Con/High subjects had a tendency to maintain constant levels of consumption and investment and no subject depleted her capital stock completely. Subject 36 used the optimal decision rule and 5 of the 12 subjects maintained a constant capital stock for at least 8 rounds.

IV. Conclusion

We find that subjects' decisions differ from the optimal decision for the four particular decision problems we studied. The removal of all strategic uncertainty, risk, exogenous uncertainty, and complex computations is not sufficient to ensure that subjects choose an optimal decision. The problem we studied had a solution which was obvious if backward reasoning was used to determine the answer: equate marginal utilities of consumption and investment in round 10, and then use a constant level of investment to reach it. However, it seems that many subjects analyse each round separately and myopically, changing their choices from round to round. It appears that dynamic decision problems are difficult to solve. Even after repetition of the same simple problem, it continued to pose a challenge for many of the subjects.(7)

About half of the subjects seem to have developed a reasonably successful strategy by period 3, in that they achieve at least 94% of the maximum possible earnings in period 3. This proportion is quite stable across the four different treatments; 6/12 in Lin/Low, 5/12 in Lin/High and 7/12 in Con/Low and Con/High. The four decision problems seemed comparable in terms of the level of difficulty. The earnings of subjects were not affected by the treatment variables, the utility function or the endowment treatment in effect.

The level of consumption was also not different across treatments, although we detected differences in behavior in the last two rounds of the period depending on the initial endowment. The levels of capital stock chosen were affected by the treatment. The differences in the level of investment between treatments appear to reflect an interaction between the utility function and the initial endowment. In each of the treatments, however, subjects tended to reduce their capital stock toward the end of the period, whereas the optimum specifies a constant level of capital. Over time subjects' earnings increased. As has been widely observed in other experimental studies, the feedback received in early periods improves the ability of subjects to make decisions in later periods.

It might be argued that the monetary amounts we offer are too small to induce the optimizing behavior that would occur under stronger incentives. This argument fails to explain the fact that in one of our treatments, Lin/Low, a substantial fraction of the subjects made the optimal choices in period 3 while subjects did not optimize in Con/Low, in which the problem was exactly the same except for a monotonic transformation of the utility function.

Appendix: Instructions and Tables

The following pages contain the instructions which subjects read at the beginning of the experimental sessions and the tables which they had available to them. Table XI is the Production Schedule, Tables XII and XIII display the token values for [C.sub.t] and [K.sub.11] for utility function LIN, and Tables XIV and XV indicate the token values for [C.sub.t] and [K.sub.11] under utility function CON.

Instructions

This experiment is part of a study of decision making. Various research foundations have provided funds for this research. The instructions are simple and, if you follow the instructions carefully you can generally expect to make a substantial amount of money, which will be paid to you IN CASH at the end of the experiment.

One important rule of this experiment is that once we begin, no one is allowed to talk or communicate in any way to anyone else. Anyone that does talk or communicate to someone else will lose their right to payment.

I. What determines how much you will be paid?

A. The amount of your payment depends partly on your decisions and partly on chance.

B. The payoffs in the experiment are not necessarily fair, and we cannot guarantee that you will earn any specified amount.

C. However, if you are careful you can generally expect to make a substantial amount of money.

D. During the experiment payoffs will be given in "tokens" or "game points". The tokens will be exchanged for guilders at the end of the experiment. Each ......... tokens are worth 1 guilder.

II. How does the experiment work?

A. The experiment consists of a series of games (or periods).

B. Each period will consist of rounds. In each round you will make decisions about how much of two goods, A and X, to produce.

C. At the start of each period you will be given ......... units of good A.

D. You will then be asked to make A and X by choosing quantities from the Production Schedule given to you.

E. You will then be awarded a fixed number of tokens based on how much X that you choose. You can see how many tokens you get for each unit of X on the sheet entitled "Token Value for X." You can receive tokens in every round. In each round the tokens that you receive are added to your total.

F. You will then be asked to make new A and X from the A you created in the last round. You will be awarded new tokens based on your new X. The new tokens will be added to your previous total.

G. Notice that if you make too much X and too little A in the early rounds, you may not have enough A remaining to make as much X as you would like in the later rounds.

H. There will be ......... periods played for money, each of which will consist of ......... rounds.

I. There will be ......... practice periods; after which there will be ......... periods played for money. The practice period will also consist of ......... rounds.

J. After the last period you will be paid in guilders at the rate of ......... tokens to 1 Dfl.

Before we begin some practice games the INPUT SCREEN will be explained.

The INPUT SCREEN allows you to input your choice. It also

* Shows you the current period.

* Shows you the choices of A and X you have made.

* Allows you to see outcomes of all past games.

It is divided into 2 parts:

The HISTORY Window:

* Shows the outcome of the last game and all past games by pressing the PageUp and PageDown keys.

The INPUT Window:

To input your choice move the cursor to the Input A or Input X position by using the left and right arrow keys. Then type in your choice and press [Enter]. After you enter your choices for both A and X you can complete your choice by pressing the [F10] key; you will be asked to confirm it by pressing the Y (for YES that is the right choice) or by pressing the N (for NO that is not the right choice). Pressing the N key will allow you to change your choice.

The Page Up and Page Down keys will allow you to see the outcomes of the past periods.

You have three sheets in front of you: the Production Schedule and 2 Redemption Value sheets.

The Production Schedule:

This sheet indicates the amount of X and A which you can make from a given amount of A. This sheet is to be used in all of the periods and rounds. The first column indicates the amount of A you currently have. Columns 1-8 show the possible combinations of X and A which you can make. For example, if you have 3 units of A, you can make either:

4 units of X and 1 unit of A, or 3 units of X and 2 units of A, or 2 units of X and 3 units of A, or 1 unit of X and 4 units of A, or 0 units of X and 5 units of A.

The Redemption Value Sheets

One sheet is for X and the other sheet is for A.

[TABULAR DATA FOR TABLE XI OMITTED]

Table XII. Redemption Value Sheet for X: Linear Marginal Utility

Unit X Unit Value X Total Value

(1) 70 70
(2) 65 135
(3) 60 195
(4) 55 250
(5) 50 300
(6) 45 345
(7) 40 385
(8) 35 420
(9) 30 450
(10) 25 475
(11) 20 495
(12) 15 510
(13) 10 520
(14) 5 525
Table XIII. Redemption Value Sheet for A: Linear Marginal Utility

Unit A Unit Value A Total Value

(1) 85 85
(2) 80 165
(3) 75 240
(4) 70 310
(5) 65 375
(6) 60 435
(7) 55 490
(8) 50 540
(9) 45 585
(10) 40 625
(11) 35 660
(12) 30 690
(13) 25 715
(14) 20 735

The sheet for X is to be used in every round and the sheet for A is to be used only in the last round of each period.

The first column contains the number of units that you made in the round. The last column, entitled Total Value, contains the TOTAL number of tokens you receive from those units. The second column, entitled Unit Value, contains the additional number of tokens that you receive from the last unit you made. For example, in row 5, the number in the Unit Value column gives the additional number of tokens you receive from making 5 units instead of making 4 units.

The redemption value sheet for A contains the number of tokens you will receive for the number of units of A you have in the last round. It is used just like the redemption value sheet for X.

Table XIV. Redemption Value Sheet for X: Convex Marginal Utility

Unit X Unit Value X Total Value

(1) 100 100
(2) 80 180
(3) 67 247
(4) 57 304
(5) 50 354
(6) 44 398
(7) 40 438
(8) 36 474
(9) 33 507
(10) 31 538
(11) 29 567
(12) 26 593
(13) 25 618
(14) 24 642
Table XV. Redemption Value Sheet for A: Convex Marginal Utility

Unit A Unit Value A Total Value

(1) 400 400
(2) 200 600
(3) 133 733
(4) 100 833
(5) 80 913
(6) 67 980
(7) 57 1037
(8) 50 1087
(9) 44 1131
(10) 40 1171
(11) 36 1207
(12) 33 1240
(13) 31 1271
(14) 29 1300

1. For more detail on the design of the experiment see Van Marrewijk, Noussair, and Olson. [7].

2. In the experiment itself, the unbiased variable names A and X were used. Unbiasedness refers to the property that the variable names themselves do not suggest any particular behavior to subjects. For a discussion of the concept of unbiasedness see Davis and Holt [5].

3. This is a widely used measure in experimental research [5].

4. The data reported here for [K.sub.t+1] are from rounds t = 1, ..., 9. The 10th round choice must be considered separately from the first 9 rounds because under the endowment of 4, round 10 has a different optimal level of consumption and investment than the other rounds. The efficiency data are reported with each period as the unit of observation.

5. However, since the optimal level of consumption is 3 in round 9 and 2 in round 10, the increase in consumption relative to the optimum in Lin/Low is actually a decrease in an absolute sense.

6. Subject 26 chose [C.sub.10] = 0 and [K.sub.11] = 4, which was one of two optimal decisions given that [K.sub.10] = 2. For two subjects [K.sub.10] equalled 0, so they had only one feasible choice of consumption and investment in round 10, which was 0 for both variables. In the numbers reported in this paragraph, they are not considered to have made an optimal decision in round 10, and are not counted among the 5 subjects who consumed 0 in round 10 when a positive consumption level was optimal given their capital stock at the beginning of round 10.

7. Recent work by Fehr and Zyck [6] on addiction also indicates that dynamic decision problems are very challenging for subjects. In their experiment, subjects face a decision problem in which they have an incentive to maximize the utility of consumption of an "addictive" commodity over 30 rounds. They have a fixed income each period but can save and borrow against income in future rounds in perfect capital markets. Consuming more at any point in time lowers future consumption by reducing future income but it also lowers the marginal utility of consumption in future rounds, in the same way as the building up of a tolerance to an addictive substance. The optimal level of consumption is increasing in each round. Theirs is a somewhat more complicated problem than ours because ours involves a constant optimal level of consumption (except in the last round of some treatments) and ours consists of only 10 rounds. Fehr and Zych find that over-consumption relative to the optimum occurs consistently. Relative to the conditional optimum, overconsumption occurs until the final two rounds of the 30 round decision problem. The fact that we did not observe a strong tendency toward excess consumption may be due to the fact that their optimal consumption is increasing from round to round whereas ours is constant.

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