Equilibrium selection in an experimental macroeconomy.

Lei, Vivian ; Noussair, Charles N.

1. Introduction

One of the most influential literatures in economics is the theory of growth (for surveys see Azariadis 1993; Barro and Sala-i-Martin 1995; Romer 1996; Sala-i-Martin 2002). The basis of much of the literature is the Ramsey (1928)/Cass (1965)/Koopmans (1965) growth model. In this model the economy is assumed to behave like a benevolent social planner, who chooses capital stock and consumption levels over an infinite time horizon with the goal of maximizing the discounted utility of the consumption stream. The principal result of the model is that consumption and capital stock converge to unique optimal steady state levels that are independent of the initial endowment and the utility function of the social planner.

An implication of the model, therefore, is that different countries would converge toward a common income level even if their initial endowment of capital differed, provided that they have access to the same production technology. Relatively poor countries would exhibit higher growth rates than richer ones. These two predictions are testable with field data. However, field studies have generally failed to support the hypothesis of convergence toward a common income level (see Durlauf and Quah 1999; Temple 1999; and Islam 2003 for surveys). Rather, the data are more consistent with the alternative hypothesis of club convergence (Baumol 1986), which postulates that a small number of steady states exist, and that each country has a tendency to converge toward one of them. (1) Such a framework can explain the observed pattern over time of an increase in income differences between the Organisation for Economic Cooperation and Development countries and the developing world, as well as a decrease in the differences within each of the two groups.

The empirical support for club convergence has encouraged the development of theoretical models with multiple equilibria. While some countries may reach optimal equilibria, unfortunate countries might find themselves in low-income equilibria, which are often labeled as poverty traps. These countries are unable to reach a better equilibrium without coordination. Originally due to Rosenstein-Rodan (1943), the insight that the existence of multiple equilibria might provide an explanation of international income differences has led to a literature that considers a variety of growth models with multiple equilibria. For example, Azariadis and Drazen (1990) construct an overlapping generations model with two stable Pareto-rankable equilibria. In the inferior equilibrium, no agent trades with members of other generations. Murphy, Shleifer, and Vishny (1989) build a model with synergies between industries. Each industry is profitable only if other industries are operating and there are equilibria where all of the industries operate and other, Pareto-dominated equilibria where none operate. Galor and Zeira (1993) and Banerjee et al. (2001) show that inequality and differential access to credit can keep an economy in a Pareto-dominated equilibrium.

Recognizing whether or not an economy has multiple equilibria is important, because policy prescriptions differ depending on whether an economy is in an inferior equilibrium or whether it is in an equilibrium that is unique. Unfortunately, it is generally not possible to identify whether an economy has multiple equilibria (see Cooper 2005 for a discussion of the empirical issues involved). The underlying parametric structure of economies is typically unobservable, and in economies with multiple equilibria, the comparative statics are often ambiguous.

In this paper we take advantage of the fact that experimental methods allow the underlying parameters of the economy to be observed and manipulated, and we construct and study the behavior of dynamic laboratory macroeconomies that are known to have multiple, locally stable, Pareto-rankable stationary steady states. (2) As described in section 3, each steady state corresponds to a stationary competitive equilibrium, and therefore each steady state is a plausible attractor for the economy. The structure of the economies is one for which straightforward application of the Ramsey/Cass/Koopmans optimal growth model, which assumes that a benevolent social planner guides economic activity, makes a prediction that the economy will converge to the optimal of the steady states. These predictions provide null hypotheses about outcomes in our economies. However, another motivation of the paper is exploratory. We look for patterns in the data that might be characteristics of economies with multiple steady states, and that could be helpful in distinguishing between single and multiple steady state economies when the structure is unknown to the observer. While there is no ex ante reason to expect a difference between single and multiple steady state economies, there may be signatures in the economic data that reveal a uniqueness or multiplicity property of the underlying structure. This is potentially important because the right policy to promote growth or efficiency may differ in the two situations.

Two questions are posed with regard to model predictions. The first is whether or not a decentralized dynamic economy with multiple steady states will reach one of the steady states. To facilitate consideration of this question by allowing it to be interpreted within an existing framework, we use an institutional structure employed in Lei and Noussair (2002, hereafter LN), described in section 2, under which economies exhibit convergence to their optimal steady state in cases where the steady state is unique and stable. However, the situation considered here is different in that in economies with multiple steady states, a degree of coordination of actions and expectations is required to reach one of the steady states. We observe that the economy typically does operate at or very close to one of its steady states, and therefore coordination does occur in our dynamic economy.

The second question is whether, given that the economy attains a steady state, there exists any tendency to reach a steady state that is Pareto-dominated. In other words, do the economies fall into their poverty traps? Avoiding or exiting an inferior steady state involves a different and possibly more demanding coordination task than merely converging to some steady state. An ability of our economies to avoid inferior steady states would suggest that such coordination could occur in a natural way, even in economies with a decentralized structure such as ours. On the other hand, if our economies exhibit a tendency to reach inferior steady states, it illustrates that coordination problems are potentially consequential in macroeconomies. Furthermore, a result that the economy reaches inferior steady states is potentially useful for future research because it would create an arena in which different institutions could be introduced into the economy to identify those that might allow an economy to recoordinate on a better steady state. Indeed, we find that the economy often converges to a suboptimal steady state, and will typically do so if the initial endowment of capital is sufficiently low.

The exploratory analysis considers two topics. The first topic is whether an economy with multiple steady states exhibits behavior that is not characteristic of economies with a unique steady state. The existence of such behaviors might provide clues to observers who do not know the underlying parameters of the economy about whether or not the economy has multiple steady states. We study this question by comparing the patterns in our data with those observed by LN, who studied economies with a unique optimal steady state, and we find some suggestive evidence that economies with multiple steady states exhibit larger fluctuations from one period to the next and are more susceptible to severe downturns.

The second topic concerns the behavior of an economy with a similar underlying parametric structure under an idealized institutional arrangement. We consider the outcome when the economy is populated with agents who have incentives to act as benevolent social planners. All members of the economy possess full information about the structure of the economy and have an identical incentive to maximize the overall welfare of the economy. We explore the empirical patterns generated from the decisions of these social planners. We observe that a social planner, who faces no coordination problem, is not susceptible to poverty traps. On the other hand, the absence of trade means that no price information exists, making it difficult for the planner to identify the optimal sequence of consumption and investment.

Of course, the inferences that we make are necessarily valid only for the specific structure of our experimental economy, which, like all economic models, is highly stylized. Economic experiments are subject to the same critique as theoretical models in that, under a narrow interpretation, our results apply only for economies with the precise structure of our experimental environment. However, although theoretical modeling describes the outcomes that are implied from assumptions on the principles of behavior in an economy with a specific structure, the results are used to advance conjectures and to create intuition about a class of related environments, possibly including field economies. Later research may show that a change in assumptions influences the conclusions qualitatively. An analogous argument can be made with experimental research. The extent that any conclusions that we find here carry over to other economies must await further research. However, our environment has no particular features of which we are aware that would render it nongeneric.

2. The Economies

General Structure

The economies we study in our experiment can be approximated by an economy with the following structure. A representative consumer in the economy, who can also be thought of as a benevolent social planner, has a lifetime utility as given in Equation 1,

[[infinity].summation over (t = 0)] [(1 + [rho]).sup -t] U([Csub.t]), (1)

where [rho] is the discount rate, [C.sub.t] is the quantity of consumption at time t, and U([C.sub.t]) is the utility of consumption. The economy faces the resource constraint given in Equation 2,

[C.sub.t] + [K.sub.t] + 1 [less than or greater than] A * F([K.sub.t]) + (1 - [delta])[K.sub.t], (2)

where [delta] is the depreciation rate, [K.sub.t] is the economy's aggregate capital stock at the beginning of period t, and A is an efficiency parameter on the production technology. The value of A depends on the economy's capital stock. There exists a threshold level of capital stock, above which A has the value [bar.A] and below which it has the value < [bar.A]; that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [A.bar] and [bar.A] are low and high production technology efficiency parameters, respectively, and [??] denotes the threshold level of the aggregate capital stock. The threshold can be interpreted as the existence of a positive externality in production, generated by the aggregate quantity of capital stock in the economy. (3)

Decentralization

The economy described in Equations 1-3 can also describe the aggregate structure of a decentralized economy. Suppose that each agent has the utility of consumption given in Equation 4,

[[infinity].summation] t = 0] [(1 + [rho]).sup.-t][u.sup.t]([c.sup.i.sub.t], [m.sup.i.sub.t]), (4)

where [u.sup.i] ([c.sup.i.sub.t], [m.sup.i.sub.t]) is the utility of individual i for his consumption [c.sup.i.sub.t] and his holding of a numeraire good [m.sup.i.sub.t]. Utility functions are quasi-linear so that [u.sup.i] ([c.sup.i.sub.t], [m.sup.i.sub.t]), = [v.sup.i] ([c.sup.i.sub.t] + [m.sup.i.sub.t]. Here m is a good that cannot be produced (it can be thought of as money that can be spent in period t to yield utility). Agents may have a negative holding of m and we assume that [[summation].sub.i][m.sup.i.sub.t] = 0, [for all]t. The functions [v.sup.i] ([c.sup.i.sub.t]) are such that [[summation].sub.i][[dv.sup.i](c.sup.i.sub.t])/ d[c.sup.i.sub.t]].sup.1] = [dU(C.sup.1])/[[dC.sub.t].sup.-1]. That is, the sum of the individual inverse demands in the decentralized economy yields the total market inverse demand for consumption.

Each individual is endowed with a production function, A * [f.sup.i](k.sup.i.sub.t]), which maps his individual capital stock into output, and he faces the resource constraint

[c.sup.i.sub.t] + [k.sup.i.sub.t] + 1 [less than or equal to] A * [f.sup.i]([k.sup.i.sub.t]) + (1 -[delta])[k.sup.i.sub.t] + [d.sup.i.sub.t]. (5)

Here [d.sup.i.sub.t] is individual i's net purchase of output in period t, and A exhibits the production externality described in Equation 3. The threshold is reached when [K.sub.t] + 1 = [[summation].sup.5.sub.1] [k.sup.i.sub.t] + 1 [greater than or less than] [??], so that productivity is a function of the aggregate capital stock holding. After committing to the production activity, the agent can supplement or reduce his output level by exchanging some of his output for [m.sup.i.sub.t]. Because the net quantity individual i purchases is given by [d.sup.i.sub.t], the following constraint has to hold:

[P.sub.t][d.sup.i.sub.t] = [-m.sup.i.sub.t], (6)

where [P.sub.t], is the price of output in the market in terms of the numeraire. The functions [f.sup.i] ([k.sup.i.sub.t]) for each i are chosen so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Parameters of the Experiment

The parameters of the experiment are given in Table 1. The actual production functions used in the experiment were discrete, mapping integers to integers, and were approximations to 7.88 * [K.sup.0.5.sub.t] for [K.sub.t] < 31 and 16.771 * [K.sup.0.5 .sup.t] for [K.sup.t], [greater than] 31. For instance, given 26 units of input, the economy was able to produce 40 units of output. At 31 units, there was a threshold at which production became more efficient. Both segments of the production function, the range [K.sub.t] < 31 and the range [K.sub.t] [greater than] 31, were quasi-concave. The aggregate utility of consumption at time t was approximately U([C.sub.t]) = 400[C.sub.t] - [2([C.sub.t]).sup.2]. It was expressed in terms of an experimental currency, called "francs," which was converted to U.S. dollars at the end of the experiment at a predetermined exchange rate. The parameters A, [delta], and [rho] were always equal to the values specified in Table 1.

The aggregate production capability and the value of consumption of units were divided among the five agents populating the economy. The economy-wide production capability, A * F([K.sub.t]), was allocated among the five agents in the following manner. The marginal product of the first unit of [K.sub.t] was allocated to agent 1, the marginal product of the second unit of [K.sub.t] to agent 2, etc. For instance, the marginal products of the first five units of [K.sub.t] in the economy were 8, 3, 3, 2, and 2 units of output, respectively. Therefore, with the first unit of [k.sup.i.sub.t], agent 1 could produce 8 units of output, agents 2 and 3 could each produce 3 units, and agents 4 and 5 could each produce 2 units. (4) Furthermore, since the aggregate production function consisted of two segments, the individual production schedule contained two parts, one part associated with 7.88 * [K.sup.0.5.sub.t] for [K.sub.t] < 31 and the other part reflecting 16.771 * [K.sup.0.5.sub.t] for [K.sub.t] [greater than] 31. The individual production schedules were private information. In other words, no agent knew any other agent's production capability. A similar rule was applied to allocate the economy-wide valuation for consumption among the five agents. The marginal utility of consumption of agent i was thus a discrete approximation of [v.sup.i]'([c.sup.i.sub.t]) = 396 + 4i - 20[c.sup.i.sub.t].

The experiment had two treatments, the High and Low Endowment treatments. Under High Endowment, the initial level of capital stock [K.sub.0] equaled 35, allocated in equal initial individual endowments of seven units for each of the five members of the economy. Under Low Endowment, each agent received an initial endowment of four units, for an economy-wide total of 20. The economies under the Low Endowment treatment, in which the initial endowment lies below the threshold of 31, may pose a more challenging coordination problem because capital must be accumulated in a range of diminishing returns to production between the initial level of capital stock and the threshold.

The Sessions

Each experimental session consisted of the following sequence of events. Upon arrival at one of the sessions, subjects reviewed a tutorial that lasted approximately 40 minutes on the use of the z-Tree software (Fischbacher 2007), which was used to implement the market for trading output. Afterwards the experimenter handed out and read the instructions for the experiment.

Subjects then participated as agents in an economy where they made a sequence of consumption, investment, purchase, and sale decisions over a series of periods. A practice period, which did not count toward subjects' final earnings, was also implemented to check subjects' understanding of the material in the instructions. All of the materials used in conducting the experiment are available from the authors.

At the beginning of each period, production took place that transformed each agent's current capital stock holding [k.sup.i.sub.t] into output A * [f.sup.i] ([k.sup.i.sub.t]) Agents could then trade output in a market with other members of the economy for a period of two minutes. After the market closed, subjects were required to decide how to allocate their current output between consumption, [c.sup.i.sub.t], and capital stock, [k.sup.i.sub.t] + 1. The period ended after this decision, and subjects were then asked to calculate their period earnings. Each agent's period earnings equaled [u.sup.i]([c.sup.i.sub.t], [m.sup.i.sub.t]) = [v.sup.i]([c.sup.i.sub.t]) + [m.sup.i.sub.t], the utility of consumption and any profit gained from trading on the market.

After each period ended, the experimenter circulated among subjects to record the individual end-of-period capital stock, [k.sup.i.sub.t] + 1, and to calculate the aggregate capital stock, [K.sub.t] + [[summation].sup.5.sub.1] + [k.sup.i.sub.t] + 1. The experimenter announced publicly whether or not aggregate capital stock was above or below the threshold level of 31. (5) The exact value of [K.sub.t+1] was not announced. Along with this piece of public information, the individual end-of-period capital stock, [k.sup.i.sub.t+1], and the production schedule enabled a subject to determine how much output, A * [f.sup.i] ([k.sup.i.sub.t+1]), would be available to him at the beginning of period t + 1.

The market for trading output was computerized and followed continuous double auction rules (Smith 1962). To facilitate trading on the market, each agent was given a loan of 10,000 units of experimental currency at the beginning of each period. The current loan for the period had to be paid back at the end of each period. In other words, the cash balance was always reinitialized to 10,000 at the beginning of each period, but the net change in cash from the beginning to the end of each period counted for or against individual earnings. An agent could make a buy or a sell order at any point in time while the market was operating. Each subject was allowed to buy or sell only one unit at a time. Therefore an offer simply consisted of a price at which the agent submitting the offer would like to purchase or sell. He could also purchase or sell units of output by accepting offers made by other agents. Purchases in the market decreased his cash balance, while sales increased his cash balance.

The Infinite Horizon

A random ending rule was used to induce a decision situation equivalent to an infinite time horizon with discounting. (6) Under the assumption that subjects in the experiment are risk neutral in their final monetary payment, a constant probability of 20% of the horizon ending in each period is equivalent to an infinite horizon in which [rho] = 0.25. A horizon is defined as the time interval that Equations 1-3 describe. The experimenter implemented the ending rule by rolling a 10-sided die at the end of each period to determine if the horizon would end. If the die read number 1 or 2, the horizon ended immediately. Otherwise the horizon continued.

The period of time defined by a horizon is typically distinct from that defined by a session, the length of time a group of subjects interacts in the laboratory. Each session was scheduled for three hours. If the current horizon ended with more than a half hour remaining, a new horizon was started with the initial endowment (20 under Low and 35 under High Endowment) in effect for the treatment. Restarting with the same initial values after an exogenous random ending has no distortionary effect on optimal decisions. If a horizon did not terminate by the end of the third hour, it would be continued on another evening. Subjects were free to return for that session and continue in the same role at the point where they left off. If a subject was unwilling or unable to return, a substitute would be recruited to replace him. The earnings the substitute made would be awarded to the original subject as well as to the substitute himself. This technique, first applied in LN, preserved the incentive for subjects to make optimal decisions, even when they would not be participating when the economy was continued in the future session.

Fifteen sessions were conducted using the procedures described in this section. One of the 15 sessions, MktH1b, was a continuation of a previous session, MktH1a, with the same participants. The session was continued on another evening because the session ended during a horizon. Sessions were conducted at Purdue University and University of Wisconsin-Milwaukee. Subjects were undergraduate students who were recruited from introductory micro- or macroeconomics courses and were inexperienced with any similar experiment. Individual subjects earned an average of $38.80 per session. Information about the number of periods and horizons in each session is contained in Table 2.

3. Models

In section 3.1 we derive the optimal steady state of the economy. These define the values to which the variables in the economy would converge if a benevolent social planner with full information about the structure of the economy made all of the economy's consumption and investment decisions. In section 3.2 we calculate an additional, suboptimal, steady state of the economy. In the decentralized economy, each of the two steady states corresponds to a stationary competitive equilibrium.

Optimality

Suppose the economy behaves as if it is under the direction of a benevolent social planner, who chooses [C.sub.1], ..., [C.sub.[infinity]] to maximize Equation 1 subject to Equations 2, 3, and the constraints that [C.sub.t], [K.sub.t] [greater than or equal to] 0. We also require that [K.sub.t+1] [greater than] (1 - [delta])[K.sub.t] (negative gross investment in a period is not possible). The transversality condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

is assumed to hold (in the experiment the condition is assured by the ending rule we employ). The first-order conditions are

[U.sup.']([C.sub.t]) = [(1 + [rho]).sup.-1] [1 - [delta] + A * [F.sup.']([K.sub.t] + 1)] [U.sup.']([C.sub.t] + 1), [for all] t, (8)

and

[C.sub.t] + [K.sub.t] + 1 = A * F([K.sub.t]) + (1 - [delta])[K.sub. t], [for all]t. (9)

An optimal steady state must satisfy [C.sub.t] = [C.sub.t+1] = [C.sup.*] and [K.sub.t] = [K.sub.t +1] = [K.sup.*], for all t, and the first-order necessary conditions in Equations 8 and 9. This implies that

[C.sup.*] = A * F([K.sup.*]) - [delta][K.sup.*], (10)

and

A * F' ([K.sup.*]) = [delta] + [rho]. (11)

For the parameters given in Table 1, if capital and consumption are constrained to take on only integer quantities, there are three solutions to Equations 10 and 11, two of which are stable. The stable solutions occur at ([K.sup.*], [C.sup.*]) = (45, 68) and ([K.sup.**], [C.sup.**]) = (9, 16). However, numerical simulations show that from any initial level of capital stock [K.sub.0] (including [K.sub.0] = 9), the optimal sequence of consumption and investment decisions converges to ([K.sup.*], [C.sup.*]) = (45, 68). (7) Thus, there is an optimal steady state at ([K.sup.*.sub.spo], [C.sup.*.sub.spo]) = (45, 68). The subscript spo (Social Planner Optimum) indicates the capital stock and consumption levels in the optimal steady state.

Competitive Equilibrium

A rational expectations equilibrium for the decentralized economy described above is a sequence of consumption quantities, capital stock quantities, and prices ([C.sub.t], [K.sub.t], [P.sub.t]) for all periods t, such that each individual is making optimal consumption and investment decisions at each time t given present and future prices, and such that [P.sub.t] clears the market for output. The rational expectations equilibrium has two steady states. That is, there are two profiles ([C.sup.*.sub.t], [K.sup.*.sub.t P*) that are constant over time so that ([C.sup.*.sub.t], [K.sup.*.sub.t], [P.sup.*.sub.t]) = ([C.sup.*], [K.sup.*], [P.sup.*]). Each of these steady states thus can be viewed as describing a stationary competitive equilibrium. These correspond, albeit inexactly, to the values of [C.sub.t] and [K.sub.t] that solve Equations 10 and 11. One of the equilibria is Pareto-optimal and one is Pareto-inferior, with a lower level of consumption and capital stock for each individual and for the economy as a whole. [P.sup.*.sub.opt] = 118 is the price that supports the Pareto-optimal equilibrium, and [P.sup.*.sub.inf]= 334 supports the Pareto-inferior equilibrium. The equilibrium capital stock and consumption levels such that [P.sup.*.sub.opt] clears the market occur at ([K.sup.*.sub.opt], [C.sup.*.sub.opt]) = (45, 70). [P.sup.*.sub.inf] clears the market at ([K.sup.*.sub.inf], [C.sup.*.sub.inf]) = (9, 16). This Pareto-inferior stationary competitive equilibrium can be interpreted as a poverty trap.

At the Pareto-optimal steady state, each agent consumes 14 units per period for an economy-wide total of 70 units of consumption. In equilibrium the capital stock is distributed among the agents in the following manner: [[bar.k].sup.1] = 12, [[bar.k].sup.2] = 9, [[bar.k].sup.3] = 6, [[bar.k].sup.4] = 8, and [[bar.k].sup.5] = 10, where [[bar.k].sup.i] is the equilibrium capital holding of agent i, yielding a total [[summation].sup.5.sub.1][[bar.k].sup.i] = 45. At the Pareto-inferior equilibrium, agents 1-4 each consume 3 units, and agent 5 consumes 4 units per period for a total consumption of 16. The allocation of capital stock in this equilibrium is [[bar.k].sup.1] = 1, [[bar.k].sup.2] = 2, [[bar.k].sup.3] = 1, [[bar.k].sup.4] = 2, and [[bar.k].sup.5] = 3 and yields a total equilibrium capital stock of 9 units. (8)

4. Results

Decentralized Economy

Figure 1 illustrates the movement of capital stock over time in relation to several benchmarks. The optimal steady state level is indicated as [K.sup.*.sub.opt], and the Pareto-inferior steady state is indicated as [K.sup.*.sub.inf]. The threshold level of capital stock, 31, is also shown in the graph as [K.sub.threshold]. [K.sub.gold] is the golden rule level of capital stock, which would result from the decisions of a social planner who treats [rho] as equal to zero. Figure I a shows the data for Low Endowment, and Figure lb contains the data for High Endowment. Figures 2a and 2b illustrate the observed time series of the realized utility of consumption, U([C.sub.t]), in relation to the equilibrium levels. The figures also in essence show the patterns of consumption over time because actual consumption quantities are almost perfectly correlated with the realized utility of consumption. Figures 3a and 3b show the time series of average output prices.

The figures suggest a great deal of stability in the Low Endowment treatment. With the exception of session MktL4, capital stock in the last horizon converges toward the Pareto-inferior equilibrium and shows no tendency to approach any of the other potential attractors shown in the graphs. (9) The time series of the utility of consumption shows a similar strong tendency to converge to the poverty trap level. The price of output in sessions MktL1-3 and MktL6 converges smoothly toward the Pareto-inferior equilibrium in the last horizon. The price remains rather low relative to the poverty trap level in sessions MktL4 and MktL7, and somewhat too high in MktL5. Overall, however, the economy under Low Endowment exhibits a strong tendency to converge to near its poverty trap.

[FIGURE 1a OMITTED]

[FIGURE 1b OMITTED]

A different pattern is observed in the High Endowment data. Session MktH1 has a tendency to converge to near the optimal steady state, and the tendency grows stronger over the three horizons in which the group participates. The other six sessions are characterized by a tendency to converge toward the optimal steady state level, but with the economies experiencing dips below the threshold level of capital at least once, that lowers consumption, capital, and earnings greatly. These episodes usually occur late in the sessions when participants have considerable experience. This occurs, for example, in period 11 in the first horizon of MktH3, period 4 of horizon 5 of MktH4, and period 8 of horizon 6 of MktH6. When below the capital threshold, the economies tend to converge toward the Pareto-inferior steady state. An over-accumulation of capital stock in the following horizon tends to follow an episode of convergence toward the poverty trap.

[FIGURE 2a OMITTED]

To evaluate in a more precise manner whether the economies converge toward equilibrium values, we estimate the model

[Y.sub.jt] = [[beta].sub.1j] 1/t + [[beta].sub.2j] t - 1/t + [[epsilon].sub.jt] (12)

where [Y.sub.jt] is the value of a variable of interest, such as economy-wide consumption, capital stock level, or average transaction price, in period t of a horizon in session j. The [[beta].sub.1j] terms have a natural interpretation as the point of origin of the time series in period 1 of session j, since in that period t = 1 and (t - 1)/t = 0. This means that [[beta].sub.1j] has full weight and [[beta].sub.2j] has zero weight. The [[beta].sub.2j] coefficient can be interpreted as the point to which the time series is converging, since as t [right arrow] [infinity], (t - 1)/t [right arrow] 1, and 1/t converges to zero. If [[beta].sub.2j] is not significantly different from the predictions of a theoretical model, we will say that the time series is converging to the model's prediction for the dependent variable. If [[beta].sub.2j] is closer to the prediction of a model than the corresponding [[beta].sub.1j] term, we will say that the time series is converging toward the prediction of the model. The regression model described above is used to establish Result 1 below.

[FIGURE 2b OMITTED]

RESULT 1. The economies of the Low Endowment treatment converge toward, and in many cases to, the Pareto-inferior steady state.

SUPPORT FOR RESULT 1. The data are reported in Table 3. Each session is estimated separately, and the data from all horizons of a session are used in each estimation. The dependent variables are the level of consumption in the economy, the level of capital stock, and the price of capital. The first pattern that is apparent in each of the equations is that the estimate of [[beta].sub.2j] is closer to the Pareto-inferior equilibrium than the corresponding [[beta].sub.1j] term in a majority of instances. This occurs in only three of seven cases for consumption [C.sub.t], but in six of seven for capital stock [K.sub.t], and five of seven for prices [P.sub.t]. The second pattern is that many of the [[beta].sub.2j] coefficients are not significantly different from the Pareto-inferior equilibrium. This is the case for four of seven cases for consumption [C.sub.t], in four of seven for capital stock [K.sub.t], and five of seven for prices [P.sub.t]. The third pattern is that the direction of convergence is consistent. Consumption and capital stock converge from above, in that [[beta].sub.1j] > [[beta].sub.2j] in six of seven cases for consumption and in six of seven cases for capital stock.

[FIGURE 3a OMITTED]

The economies in the High Endowment treatment tend to converge in the direction of the optimal steady state equilibrium. However, this convergence is not as strong as the tendency of the economies of the Low Endowment treatment to converge toward the poverty trap, and the High Endowment treatment is characterized by occasional episodes of falling below the threshold level of capital, and subsequent declines of capital toward the poverty trap level.

[FIGURE 3b OMITTED]

RESULT 2. Under High Endowment, behavior is less consistent across economies than under Low Endowment. Consistent with convergence to the optimal steady state, the economies typically exhibit increasing consumption and capital stock over time. Consumption tends to converge toward the optimal steady state and capital stock converges toward high levels, which are at times in excess of the optimal steady state level. Six of seven economies fall below the threshold level of capital at least once, and after an economy does so, it usually converges toward the inferior steady state.

SUPPORT FOR RESULT 2. Figures 1-3 illustrate the heterogeneity of capital stock, consumption, and pricing patterns between sessions. To show that, unless they fall below the threshold and dip into the inferior steady state, economies under the High Endowment have a tendency to converge toward the optimal steady state, we adopt the regression model of Equation 12. Table 4 displays the estimates for six of seven sessions of the High Endowment treatment. (10) The estimate of [[beta].sub.2j] is closer to the optimal equilibrium than the corresponding [[beta].sub.1j] estimate in five of six cases for consumption, [C.sub.t], but in only two of six for capital stock, [K.sub.t]. Four of six prices, [P.sub.t], are moving toward optimal equilibrium levels. There are, however, a few cases in which the [[beta].sub.2j] coefficients are not significantly different from the optimal steady state. This occurs in three of six [C.sub.t], one of six [K.sub.t], and one of six [P.sub.t]. The direction of movement is consistent in that consumption and capital stock increase over time so that [[beta].sub.1j] < [[beta].sub.2j] in five of six cases for consumption and in all six cases for capital stock. Nonetheless, the capital stock in six of the seven sessions falls below the threshold in at least one horizon. In five of these six cases, as can be observed in Figures 1-3, the subsequent behavior of the economy within the same horizon is characterized by movement of consumption and capital stock toward the inferior steady state.

There is evidence of more variability in outcomes across economies under the High Endowment treatment than under Low Endowment. In other words, the initial endowment of capital is associated with not only the overall level of income our economies achieve, but also the variance of income. This phenomenon can be seen from Table 5, which shows the normalized variances, [[sigma].sup.2]/[micro], of several of the endogenous variables in the economy. The value of the variable in the last period of each horizon is used as an observation, and the mean and variance of these observations are calculated. Then, for each treatment, the variance is divided by the mean. The normalized variances of consumption, capital stock, prices, and the utility of consumption are all at least 40% higher under High than under Low Endowment. This indicates that, even when the different magnitudes of the variables between treatments are taken into account, more variability exists under High than under Low Endowment. As another test, we calculate the normalized variances of consumption, capital stock, prices, and the utility of consumption within each session (for this calculation we use the value of the variables in the last period of each horizon within each session as an observation). Mann-Whitney tests, employing each session as an observation, indicate that the normalized variances of consumption, utility, and prices are significantly different from each other at the 10% level.

Our study was not designed to generate data for direct comparison to any previous studies. However, a rough comparison of the data obtained here from the High Endowment treatment can be made with the data from economies with a similar institutional structure, but a unique optimal steady state, using the data from the study of LN. Comparison of the two data sets suggests some basic differences between the behavior of economies with single and with multiple steady states. The comparisons are only suggestive since the two data sets were not constructed to be compared to each other and differ from each other in terms of underlying parametric structure, such as the discount rate, the scale of the quantities of consumption and capital stock, as well as the production technology available. We focus on the High Endowment treatment because we believe that the most interesting comparison is between an economy whose parametric structure places it in danger of falling below a threshold and into a poverty trap and one in which no such danger exists. The differences are summarized as Conjecture 1.

CONJECTURE 1. Relative to an economy with a unique steady state, the economies with a poverty trap appear to be characterized by (a) greater heterogeneity in outcomes between economies with identical initial conditions, (b) greater volatility, particularly in consumption, from one period to the next, and (c) more large single-period declines in consumption and capital stock.

BASIS FOR CONJECTURE 1. In support of part (a), we note that in the data from the LN study, all of the economies converge toward the unique optimal steady state. In the High Endowment treatment here, outcomes vary across the economies, even within restarts of the economy with the same agents. To support part (b), we compare the average percentage change in aggregate consumption, capital stock, and utility of consumption, as well as prices, from one period to the next, as described by the expression [absolute value of ([Y.sub.t+1 - [Y.sub.t])/[Y.sub.t]] X 100, between LN and the High Endowment treatment of the current study. These average percentage changes are interpreted as measures of volatility. The data for all periods where t [greater than or equal to] 5 are used in the estimation and are shown in Table 6. The values in the table are those from period 5 and later pooled across all horizons and all sessions of the treatment. The table shows that the volatilities of all variables are greater in the High Endowment economies with multiple steady states than in LN's economics with a unique steady state. Mann-Whitney rank sum tests, taking the average volatility over one horizon as an observation, indicate that only the volatility of consumption in the High Endowment treatment here is significantly different from that in LN (p-value = 0.0311). Finally, to support part (c), we count the number of drops in consumption of more than 50% from one period to the next. There are eight such decreases in consumption in 222 periods of the High Endowment treatment compared to three in 198 periods in the LN study.

The differences between the behavior of aggregate variables in the economies studied here and those that LN investigate are presumably due to the existence of a coordination problem. The heterogeneity in outcomes in the economies with multiple steady states reflects the fact that some groups are more effective in coordinating on Pareto-optima than others. The greater volatility from one period to the next appears to have several different sources. Some is directly associated with surpassing or falling below the threshold, which causes large changes in capital stock and consumption. Some additional volatility appears to be associated with unsuccessful attempts to coordinate, either when individuals invest large amounts in an attempt to surmount the threshold, or consume large amounts, perhaps in response to a failure of one's own earlier investment or when they believe that they are not affecting the probability of crossing the threshold. High consumption in a particular period on the part of too many members of the economy causes capital stock to fall below the threshold and precipitates most of the large sudden decreases in consumption that occur in the multiple steady state economies.

The overall efficiency of the economy U([C.sub.t]), the actual earnings realized divided by the maximum possible earnings that could be realized along the economy's optimal trajectory, is reported in Table 7 for each session. The efficiency attained is considerably less than along the optimal trajectory in both treatments, but lower under Low Endowment, an average of 0.273, than under High Endowment, an average of 0.762. In fact, with p-value = 0.0017, a Mann-Whitney nonparametric test, using each session as an observation, rejects the hypothesis that the overall efficiencies under the two endowment treatments are equal.

Besides the suboptimal level of investment in the economies, there are two other potential contributors to inefficiency in our experiment. We call these production inefficiency and consumption inefficiency. Production inefficiency is the percentage of output that is forgone because of misallocation of capital among producers. The value of the measure is greater than zero if it would be possible for the economy to realize greater output by transferring a unit of capital from one agent to another just before production takes place. Consumption inefficiency, on the other hand, is the percentage of the maximum possible utility of consumption that is foregone because units are consumed by individuals other than those who have the highest marginal utility of consumption. Table 8 reports the production efficiency, defined as 1 minus the production inefficiency, and consumption efficiency, defined as 1 minus the consumption inefficiency, in all sessions under both endowment treatments. The numbers are averaged across periods, and the data are pooled over all of the horizons that make up each session.

The production efficiency averages are nearly identical under Low (0.946) and under High (0.949) Endowment. A Mann-Whitney rank-sum test, using each session as an observation, confirms that production efficiencies are not significantly different from each other under the two endowment treatments (p-value = 0.7483). Consumption efficiency, on the other hand, appears to be higher under Low (0.860) than under High (0.769) Endowment. A Mann-Whitney rank-sum test rejects the hypothesis that consumption efficiencies under the two treatments are the same at the 5% level (p-value = 0.0476). The fact that consumption efficiency is higher under Low than under High Endowment appears to reflect the better convergence toward equilibrium and the more stable prices that accompany this convergence under the Low Endowment treatment. This convergence and the resulting stability of prices allow better consumption decisions (which involve comparing the price of capital to the marginal utility of consumption) than if the price is unstable.

We now consider general patterns in individual level investment decisions. The first hypothesis is that an individual consumes a greater share of his output at the end of the period, the more output he holds. This is not an unambiguous prediction of the theoretical model, but a statement that consumption is a "luxury" good, increasing more rapidly than overall output as output increases. The second hypothesis is that individuals consume less, the higher the price of capital. This is a statement that the demand for the consumption good is downward sloping. We estimate the following model:

[c.sub.it]/([c.sub.it] + [k.sub.i,t+1]) = [[alpha].sub.0] + [[alpha].sub.1] ([c.sub.it] + [k.sub.i,t+1]) + [[alpha].sub.2][P.sub.t]. (13)

The hypotheses of the model are that [[alpha].sub.1] > 0 and that [[alpha].sub.2] < 0. The results of the estimation are shown in Table 9a for Low Endowment and Table 9b for High Endowment. The estimates indicate that the [[alpha].sub.1] coefficient is positive and significant at p < 0.01 for the pooled data from all of the sessions, as well as in a majority (four of seven) of the individual sessions, of the Low Endowment treatment. Here [[alpha].sub.2] is significantly negative in the pooled data from all sessions for each treatment, negative in sign for every session, and significantly negative for two of the seven sessions in each treatment. Thus, there is some support for the idea that consumption and investment decisions respond to income in a simple, predictable manner, particularly in the Low Endowment treatment. There is strong support for the idea that they respond to prices.

Social Planner

Six more sessions were conducted in which groups of subjects were placed in the role of Social Planners. These sessions are indicated as the SP sessions in Table 2 and proceeded in the following manner. After subjects arrived at the laboratory for the experiment, the instructions were distributed to each of the five participants. The instructions explained that the five subjects made up a committee that was given the role of a social planner of an economy. They were required to choose as a group the economy's level of consumption and investment over a sequence of time periods, and their earnings for the experiment were based on the value of the objective function that resulted from their decisions. The experimenter read through the instructions, while subjects were allowed to ask questions. The initial period was for practice and did not count toward subject earnings.

As in the experiment with the decentralized economy described earlier, there were two treatments, differing only in the initial level of capital in the economy. Only one treatment was in effect in a given session. In the Low Endowment treatment of SP, in effect in sessions SpL1-SpL3, the economy began with 20 units of capital. In the High Endowment treatment of SP, in effect in sessions SpH1-SpH3, the initial capital stock level was 35. The values of the other parameters in the Social Planner experiment were also identical to those in Table 1. The production and utility functions were provided as public information to all subjects. Each subject's earnings were proportional to the value of the objective function (1) the group attained over the course of their experimental session, so that the interaction between participants was purely one of common interest. As in the decentralized economies described earlier, the utility and production schedules remained constant throughout the entire session.

A period within a session consisted of the following sequence of events. At the beginning of each period t, production occurred, transforming [K.sub.t] into output according to the production function A * F([K.sub.t]). After production took place, each member of the committee had two minutes to propose an allocation of the output between consumption, [C.sub.t], and investment, [K.sub.t+1]. Their proposals were restricted to those that satisfied (2) with equality for the current level of capital stock, [K.sub.t]. Allocations could be made only in integer quantities. [K.sub.t+1] would be available for production in the subsequent period. After a two-minute time interval, the proposals of four of the five members were collected and delivered to the fifth member, who was designated as the spokesperson for the committee for the current period. The spokesperson was then given one minute to choose [C.sub.t] and [K.sub.t+1] on behalf of the whole group. He was free to adopt one (or none) of the committee members' proposals. After the minute had elapsed, the spokesperson informed the experimenter of the official decision. The experimenter then recorded [C.sub.t], [K.sub.t+1], and the realized utility of consumption U([C.sub.t]) on the blackboard, so that all participants had full information on how [C.sub.t] and [K.sub.t+1] evolved over time. The payoff to each member of the committee in period t was equal to U([C.sub.t]). The role of the spokesperson was rotated among the five members from period to period.

The Social Planner data are reported here as a full information benchmark. The Social Planner is not proposed as a competing institution to the one described in section 2. Rather, the performance of the Social Planner is intended to provide a measure of the complexity of the environment by indicating the level of difficulty of the allocation problem in the absence of a price system to coordinate the allocation. There are four principal features making the decentralized and the Social Planner economies noncomparable. The first is that under the Social Planner system there is a total alignment of objectives of all members of the committee directing the economy. In contrast, in the decentralized economy each agent has his own private payoff and no individual has an incentive to maximize social welfare. The second feature is that the social planner has full information about the structure of the economy, while in the decentralized economy each agent knows only his own utility and production function. The third is that under the Social Planner system, production is restricted to points along the production possibility frontier. In other words, the constraint (2) is always binding. Under the decentralized system, it is possible for the economy to produce inside its frontier if units of capital are not allocated optimally among producers. The fourth difference is that, under the Social Planner system, all units are automatically consumed for their highest possible value, whereas in the decentralized economy, the agent who consumes the last unit may not be the one who has the highest marginal valuation for it.

[FIGURE 4a OMITTED]

The differences listed above present clear advantages for the social planner that place it in a less challenging environment to achieve a high level of welfare than the decentralized economies. Nevertheless, if the decentralized economy could achieve an outcome at least as efficient as the social planner, it would signal that the market system using prices is more than able to compensate for all of the individual incentive misalignments and informational imperfections in the economy. Differences between High and Low Endowment in realized efficiency would give a measure of the relative difficulty of the two optimization problems, even when losses due to coordination failure, misaligned incentives, and production and consumption inefficiency are removed. The outcomes from the Social Planner economies are summarized in Result 3.

[FIGURE 4b OMITTED]

RESULT 3. The Social Planner reaches neither an optimal outcome nor a steady state. There are two principal points to which the economies converge. These are focal rather than equilibrium or optimal outcomes. One attractor is the minimum capital stock level required to exceed the threshold. The other attractor is the golden rule level of capital stock and consumption, an outcome consistent with the Social Planners ignoring the possibility that the economy will terminate. Efficiency is higher when initial endowment is High than when it is Low.

SUPPORT FOR RESULT 3. The evolution of capital stock levels over time in the Social Planner treatment is displayed in Figure 4. We first consider the Low Endowment sessions. The capital stock level in session SpL1 exhibits wide swings early in the horizon, and then stabilizes at 56, the golden rule level of capital stock, which generates the maximum level of consumption of 70 (the level that would be optimal if agents treated the discount rate p as equal to zero). In session SpL2, there is a tendency for capital stock to track (though often to slightly exceed) the threshold capital stock of 31. In the first four horizons of session SpL3, capital stock converges to the threshold value of 31. In horizon 5 the economy overaccumulates its capital stock up to a level of 80.

Under High Endowment, the Social Planner exhibits similar properties. In session Spill, there is a strong tendency for convergence to the threshold capital stock level of 31. In session SpH2, the Social Planner is converging to near the golden rule in the first two horizons. During the last horizon, the economy's capital stock is near the threshold level throughout, although slightly exceeding the minimum needed to remain above the threshold. Session SpH3 shows a similar pattern. Capital stock tends to be slightly above the threshold and is close to the optimal steady state level in horizon 4. However, horizon 5 reverts to a level just above the threshold, indicating that the optimum was not a powerful attractor.

The efficiency in each session of the Social Planner treatment is shown in Table 7. It shows that average efficiency is 72% under Low Endowment and 95.7% under High Endowment. Low Endowment appears to be a more difficult optimization problem, even relative to its own optimal trajectory. This inherent difficulty may be due to the fact that a greater percentage of output must be invested (more consumption as a percentage of output must be foregone) under Low than under High Endowment. If agents are myopic and thus unaware that they benefit from or unwilling to undertake a sacrifice in current consumption, inefficiencies will arise. The difference between the parameters of the two optimization problems is responsible for the nearly 24% difference in efficiency between the Low and High Endowment Social Planner treatments, as the structure and framing is identical in the two treatments. This is almost 50% of the difference in efficiency in the decentralized economies between the High and Low Endowment conditions and thus may account for some of the difference in outcomes between the two treatments of the decentralized economies. In particular, when the optimal trajectory involves accumulating capital, it may be less likely to be attained than when it involves capital depletion.

5. Conclusion

The data in this study support the idea that a decentralized market price system is effective in leading the economy to one of its steady states. LN have already established that in a dynamic experimental macroeconomy in which there is a unique steady state that is optimal, the price system has excellent allocation properties, both intertemporally as well as in allocating production between producers and output among consumers (LN 2002). However, environments with multiple equilibria appear to present a considerable challenge for the price system in terms of leading the economy to a global optimum. The simple price system of the sort we have constructed here gets the economy to a steady state, a local optimum. However, it does not appear very effective at getting an economy out of an inferior steady state and into a better one, or at consistently avoiding an inferior steady state. In our Low Endowment treatment in particular, the Pareto-inferior equilibrium was a very powerful attractor for the economy.

The existence of multiple steady states gives this economy the flavor of a coordination game, although obviously different equilibrium concepts apply than in a normal form game. Experimental evidence indicates that individuals will often select Pareto-dominated equilibria (Cooper et al. 1990; Van Huyck, Battalio, and Beil 1990) in normal form games. The authors of these studies, although they reduced the interaction to a normal form game, motivated their research inquiry in part with coordination issues in macroeconomics. In this paper we consider parallel questions within the context of a laboratory experiment with the actual structure of a canonical macroeconomic model. We obtain similar results to the previous studies of coordination games, in particular that inferior equilibria are frequently attained under certain parametric structures. The results underscore the relevance of the possibility of coordination failure in macroeconomics.

An extensive and well-known literature has been concerned with the factors that influence economic growth. If the world is indeed one in which basic economic principles apply so that economies operate close to equilibrium, then an important empirical issue is to identify the factors that might tend to lead a country to reach a better equilibrium. The experimental data from our decentralized economy under the High Endowment treatment illustrate that the mere heterogeneity of individuals that populate markets can lead an economy to reach different equilibria from identical initial conditions, and the tendency of this population heterogeneity to do so depends on initial conditions. The economies that converge to the poverty trap and those that find their optimum have no difference in policies, institutions, or underlying parametric structure. The differences in outcomes can only be due to heterogeneity in the characteristics of the individual agents and in the endogenous dynamics their interaction creates. Thus, the incomes of the "countries" we have created in the High Endowment treatment differed considerably from each other but not for any structural, parametric, or institutional reason. We also obtain some suggestive evidence that economies with multiple equilibria are more likely than economies with a unique competitive equilibrium to be characterized by instability of the variables in the economy, as well as sudden rapid drops in capital and output.

A challenging institutional design question is posed by the data from this experiment. In this very simple economy, even when the social planner is endowed with the entire structure of the economy and has no incentive problems, the social planner uses simple rules of thumb in choosing consumption and investment sequences. The two outcomes we observed predominantly in our Social Planner data were (1) the golden rule level of consumption and (2) the minimum investment required to realize the threshold externality. Since these outcomes do not generally correspond to optimal behavior, their use may entail substantial opportunity cost. The solution to the selection problem for the decentralized economies may lie in the addition of a missing market, or in a voting institution that facilitates coordination. We believe that the issue of the design of the appropriate institutions to both avoid and escape from a poverty trap is a tantalizing one for future research, and economies of the type constructed here provide a simple arena in which to study this issue.

We are grateful to the College of Letters and Science and Ccnter for Research on International Economics at the University of Wisconsin-Milwaukee for financial support. We thank Colin Camerer, Katarina Kellar, Lutz Hendricks, Nazrul Islam, Tomomi Tanaka, Filip Vesely, and two anonymous referees for helpful comments.

Received December 2004; accepted October 2006.

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(1) A property of the optimal growth model is that it implies that growth rates will converge to zero. This is obviously inconsistent with the empirical evidence. Endogenous growth models (see, e.g., Romer 1986 or Lucas 1988) generate permanent growth.

(2) There has been previous experimental research that is similar to ours in spirit, in the sense that it uses experiments to consider which of multiple equilibria is more likely to be reached. For example, Marimon and Sunder (1993) explore an economy with multiple equilibrium inflationary paths, depending on whether expectations are adaptive or rational. A large literature has investigated normal form games with Pareto-rankable equilibria (see, e.g., Cooper et al. 1990 or Van Huyck, Battalio, and Bell 1990). Plott and George (1992) study one-period markets with multiple competitive equilibria with a focus on the appropriate concept of stability of equilibrium to predict market outcomes. Our paper differs from previous studies in that our focus is mainly on the impact of institutions on equilibrium selection.

(3) Van Boening and Wilcox (1996) study the behavior of double auction markets in environments with non-convex production technologies. They consider an environment in which sellers have a cost structure with avoidable costs. They investigate environments with and without a competitive equilibrium. They find that prices and quantities traded vary erratically in all of their treatments, but particularly when there is no competitive equilibrium. They devise a "bundled unit double auction" to address the problem (Van Boening and Wilcox 2005). An interesting issue for future research would be to consider which types of market mechanisms can successfully and consistently select the optimal from multiple Pareto-ranked competitive equilibria. The literature on combinatorial auctions (see Noussair 2003 for a review) considers the behavior of market mechanisms in environments with non-convex buyer preferences rather than seller costs.

(4) To eliminate additional "nearby" steady states that would arise from the rule used to allocate production capability among agents, we slightly adjusted agent l's production schedule. After the adjustment, agent 1 could produce 28 units of output rather than 29 with 6 units of [k.sup.i.sub.t]. With 8 units of [k.sup.i.sub.t], he could produce 31 units rather than 32. These adjustments ensured that the average marginal products of capital between the 5th and the 7th and between the 7th and the 9th unit of [k.sup.i.sub.t] were equal to 3/2, and the average marginal product of capital between the 9th and the 12th unit of [k.sup.i.sub.t] was 4/3. Since the marginal product of each unit of capital after the 12th unit of [k.sup.i.sub.t] equaled to 1, our adjustments ensured that there was a unique equilibrium capital stock holding, 12, for agent 1 in the region where [K.sub.t] [greater than] 31. The adjustments ensured that there was no unilateral deviation from the steady state in the form of a multi-unit depletion or accumulation of capital stock that was profitable.

(5) Individuals did not know the precise current level of aggregate capital stock. It is possible, but not obvious, that this made the coordination problem more difficult, because individuals might be more willing to increase investment in situations in which they are aware that the current level of economy-wide capital stock is close to the threshold. On the other hand, a lack of information about current capital stock may lead to more investment when the level is considerably below the threshold, if individuals overestimate the investment needed.

(6) Other authors have used a similar rule to create the incentives of infinite horizon models in the laboratory. See for example Camerer and Weigelt (1993) or Noussair and Matheny (2000). Noussair and Matheny explore the implications of the random ending rule in dynamic optimization problems in detail. They compare decisions between a setting with a random ending rule, such as that used here, and a fixed ending rule in which capital could be redeemed for a fixed value at a predetermined terminal period of the decision horizon. They did not find any significant differences between investment and consumption levels in the economies under the two ending rules.

(7) To calculate the optimal sequence of decisions given the initial capital stock, we wrote a computer program to compare the present discounted utility of all possible integer-valued decision sequences. Under our Low Endowment parameters, where [K.sub.0] = 20, the present discounted value of lifetime utility is maximized by increasing capital stock by 11 units in period 1 to 31 units. In the next period, it is optimal to increase capital by five units. The optimal time path of aggregate capital stock under Low Endowment is [K.sub.1] = 31, [K.sub.2] = 36, [K.sub.3] = 39, [K.sub.4] = 42, [K.sub.5] = [K.sub.6] = [K.sub.7] = ... = 45. The optimal sequence of aggregate capital stock holdings under High Endowment where [K.sub.0] = 35 is [K.sub.1] = 39, [K.sub.2] = 42, [K.sub.3] = [K.sub.4] = ... = 45.

(8) Although the total capital stock in the Pareto-optimal equilibrium of the decentralized economy is equal to that in the social planner optimum, the equilibrium output level is two units higher in the competitive equilibrium. The difference is a consequence of the rule we used to allocate the aggregate production capability among the five agents, which in the decentralized economy created a possibility of reallocations of capital between agents that could increase total output.

(9) The power of the Pareto-inferior steady state as an attractor in the Low Endowment treatment is particularly evident in session MktL6. Out of the first eight horizons, capital stock was above the threshold level seven times. Nevertheless, despite this prior experience and the resulting relatively high group earnings, in the last two horizons the economy operated close to the inferior steady state.

(10) Only those horizons that did not experience a plunge to the inferior steady state are included in the regression analysis. As a result, Session MktH3 is excluded in the estimation.

Vivian Lei * and Charles N. Noussair ([dagger])

* Department of Economics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA; E-mail vlei@uwm.edu; corresponding author.

([dagger]) Department of Economics, Faculty of Economics and Business, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands; E-mail C.N.Noussair@uvt.nl.

Table 1. Parameters of the Experimental Economies

U([C.sub.t]): Economy-wide utility function 4000[C.sub.t] -
 2[C.sup.2.sub.t]
F([K.sub.t]): Economy-wide production function [K.sup.0.5.sub.t]
A: Production-efficiency parameter [A.bar] = 7.88; if
 [K.sub.t] < 31
 [bar.A] = 16.771; if
 [K.sub.t] [greater
 than or equal to] 31
K: Threshold level of capital stock 31
[rho]: Discount rate 0.25
[delta]: Depreciation rate 1
[K.sub.0]: Initial capital stock 20 (Low Endowment) or
 35 (High Endowment)

Table 2. The Sessions, Treatment in Effect, Number of Horizons, and
Horizon Lengths

Session Endowment Number of Horizons (Number of Periods within
 Horizon)

MktL1 Low 1(7), 2(6)
MktL2 Low 1(5), 2(7), 3(7)
MktL3 Low 1(3), 2(1), 3(3), 4(1), 5(4), 6(9)
MktL4 Low 1(9), 2(9), 3(5), 4(2)
MktL5 Low 1(1), 2(1), 3(6), 4(1), 5(5), 6(22)
MktL6 Low 1(5), 2(4), 3(1), 4(1), 5(2), 6(1), 7(2), 8(4),
 9(1), 10(9)
MktL7 Low 1(3), 2(2), 3(6), 4(1), 5(3), 6(5), 7(2), 8(18)
MktH1a High 1(19)
MktH1b High 1(7), 2(3), 3(17)
MktH2 High 1(1), 2(6), 3(4), 4(6)
MktH3 High 1(20)
MktH4 High 1(1), 2(1), 3(4), 4(3), 5(4), 6(1), 7(11), 8(6),
 9(6)
MktH5 High 1(7), 2(2), 3(4), 4(6), 5(11), 6(8)
MktH6 High 1(2), 2(2), 3(4), 4(4), 5(1), 6(13), 7(5)
MktH7 High 1(8), 2(4), 3(4), 4(13), 5(6)
SpL1 Low 1(37)
SpL2 Low 1(2), 2(10), 3(3), 4(4), 5(1), 6(5)
SpL3 Low 1(3), 2(2), 3(8), 4(13), 5(4)
SpH1 High 1(1), 2(2), 3(3), 4(19)
SpH2 High 1(8), 2(3), 3(8), 4(4), 5(7)
SpH3 High 1(2), 2(8), 3(5), 4(7), 5(3)

Table 3. Estimates of Model of Convergence, Low Endowment,
All Horizons

 MktL1 MktL2 MktL3

 [[beta]. [[beta]. [[beta]. [[beta]. [[beta]. [[beta].
 sub.1] sub.2] sub.1] sub.2] sub.1] sub.2]

[C.sub.t] 18.7 14.2 17.9 15.0 18.2 11.8
 (4.3) (3.4) (3.5) (2.8) (3.0) (2.9)
[K.sub.t] 19.8 9.8 20.0 12.6 19.9 14.7 ***
 (3.5) (2.7) (2.9) (2.3) (2.0) (2.1)
[P.sub.t] 581.3 336.9 356.6 346.6 393.2 346.0
 (74.6) (62.3) (60.9) (51.6) (52.8) (52.4)

 MktL4 MktL5 MktL6

 [[beta]. [[beta]. [[beta]. [[beta]. [[beta]. [[beta].
 sub.1] sub.2] sub.1] sub.2] sub.1] sub.2]

[C.sub.t] 18.5 9.0 *** 17.5 13.4 9.7 24.2 ***
 (3.0) (2.4) (3.4) (1.8) (2.5) (2.5)
[K.sub.t] 20.2 12.8 ** 19.9 11.0 20.4 33.8 ***
 (2.5) (1.9) (2.0) (1.6) (1.6) (1.7)
[P.sub.t] 232.3 200.6 *** 348.4 367.1 474.6 301.7
 (52.8) (44.3) (60.6) (35.9) (43.1) (45.2)

 MktL7
 Pareto-
 [[beta]. [[beta]. Inferior
 sub.1] sub.2] Equili. N

[C.sub.t] 18.6 12.5 ** 16 174
 (2.3) (1.8)
[K.sub.t] 19.9 7.0 9 223
 (1.8) (1.5)
[P.sub.t] 107.7 225.3*** 334 172
 (39.9) (36.2)

Standard errors are in parentheses. ***, **: significantly different
from Pareto-inferior equilibrium at 1% and 5% levels. respectively.

Table 4. Estimates of Model of Convergence,
High Endowment, All Horizons (a)

 MktH1 MktH2

 [beta. [beta. [beta. [beta.
 sub.1] sub.2] sub.1] sub.2]

[C.sub.t] 53.0 66.2 * 58.8 48.9 ***
 (5.1) (2.0) (5.4) (4.6)
[K.sub.t] 37.0 50.6 ** 33.9 48.7
 (6.2) (2.9) (5.5) (6.4)
[P.sub.t] 134.5 115.2 141.6 160.7 **
 (17.7) (13.6) (18.7) (19.4)

 MktH4 MktH5

 [beta. [beta. [beta. [beta.
 sub.1] sub.2] sub.1] sub.2]

[C.sub.t] 52.4 63.1 45.5 72.5
 (5.4) (4.6) (4.7) (3.9)
[K.sub.t] 33.8 60.6 *** 35.6 83.6 ***
 (4.6) (6.3) (5.4) (5.5)
[P.sub.t] 115.6 153.5 * 184.4 157.6 **
 (18.7) (19.4) (16.1) (17.1)

 MktH6 MktH7

 [beta. [beta. [beta. [beta.
 sub.1] sub.2] sub.1] sub.2]

[C.sub.t] 52.6 61.3 * 38.1 65.0
 (4.3) (5.1) (4.7) (3.9)
[K.sub.t] 35.4 58.1 * 34.0 109.5 ***
 (4.5) (7.3) (5.4) (5.5)
[P.sub.t] 66.0 85.3 ** 68.3 77.7 ***
 (14.7) (17.2) (16.1) (16.7)

 Parcto-
 Optimal
 Equili. N

[C.sub.t] 70 138

[K.sub.t] 45 170

[P.sub.t] 118 138

Standard errors are in parentheses. ***, **, *: significantly
different from Pareto-optimal equilibrium at 1%. 5%, and 10%
levels, respectively.

(a) All horizons in which no plunges toward the Pareto-inferior
equilibrium occur.

Table 5. Normalized Variance in the Last Period, All Horizons

 Low Endowment High Endowment

C 6.00 9.79
U(C) 1502.71 2126.44
K 8.27 12.54
P 44.84 77.09

Table 6. Average Volatility in Percent,
for t [greater than or equal to] 5

 High Endowment LN (2002)

C 18.87 8.90
U(C) 14.97 8.17
K 11.41 9.54
P 15.10 8.55

Table 7. Overall Efficiency in Each Session, All Horizons

 Decentralized Economy Social Planner

Session Low High Low High

1 0.305 0.935 0.935 0.966
2 0.317 0.747 0.592 0.957
3 0.240 0.548 0.634 0.948
4 0.264 0.736
5 0.260 0.827
6 0.256 0.758
7 0.268 0.785
Average 0.273 0.762 0.720 0.957

Table 8. Production and Consumption Efficiency in Each Session,
All Horizons

 Production Efficiency Consumption Efficiency

Session Low High Low High

1 0.949 0.974 0.937 0.928
2 1.000 0.963 0.992 0.759
3 0.961 0.911 0.842 0.548
4 0.867 0.939 0.747 0.715
5 0.961 0.961 0.844 0.809
6 0.929 0.925 0.878 0.673
7 0.956 0.969 0.782 0.753
Average 0.946 0.949 0.860 0.769

Table 9a. Estimates of Individual Behavior, Low Endowment, All Horizons

 MktL1 MktL2 MktL3 MktL4

[[alpha].sub.0] 0.47 *** 3.22 *** 0.85 * 0.57 ***
 -0.09 -0.78 -0.5 -0.11
[[alpha].sub.1] 0.02 * -0.004 0.04 *** -0.0001
 -0.01 -0.007 -0.01 -0.009
[[alpha].sub.2] -0.0001 -0.01 *** -0.002 -0.0004
 -0.0001 (0.00) (0.001) (0.0004)
N 63 95 91 106

 MktL5 MktL6 MktL7 All

[[alpha].sub.0] 0.70 * 0.79 *** 0.36 *** 0.45 ***
 -0.41 -0.11 -0.05 -0.04
[[alpha].sub.1] 0.04 *** -0.0007 0.02 *** 0.02 ***
 -0.01 -0.004 -0.01 -0.003
[[alpha].sub.2] -0.001 -0.001 *** -0.00008 -0.0003 ***
 (0.001) (0.0002) (0.0002) (0.00008)
N 162 127 156 800

Standard errors are in parentheses. ***, **, *: significant at 1%, 5%,
and 10% levels, respectively.

Table 9b. Estimates of Individual Behavior, High Endowment, All Horizons

 MktH1 MktH2 MktH3 MktH4

[[alpha].sub.0] 1.34 *** 0.61 *** 0.41 *** 0.59 ***
 -0.13 -0.11 -0.50 -0.08
[[alpha].sub.1] -0.01 *** 0.004 0.009 ** 0.0003
 0.00 -0.002 -0.005 -0.0003
[[alpha].sub.2] -0.01 *** -0.0008 -0.00008 -0.0002
 0.00 -0.0006 -0.00004 -0.0003
N 230 80 100 159

 MktH5 MktH6 MktH7 All

[[alpha].sub.0] 0.87 *** 0.51 *** 0.55 *** 0.61 ***
 -0.05 -0.07 -0.07 -0.03
[[alpha].sub.1] -0.006 *** 0.004 -0.001 -0.0005
 -0.001 -0.003 -0.002 -0.0008
[[alpha].sub.2] -0.001 *** -0.0004 -0.0006 -0.0005 ***
 -0.0002 -0.0003 (0.0005) (0.0001)
N 184 147 172 1072

Standard errors are in parentheses. ***, **, *: significant at
1%, 5%, and 10% levels, respectively.