Distributing scarce jobs and output: Experimental
evidence on the dynamic effects of rationing
Guidon Fenig∗
Luba Petersen†
University of British Columbia
Simon Fraser University
October 20, 2015
Abstract
How does the allocation of scarce jobs and production influence their supply? We
present the results of a macroeconomics laboratory experiment that investigates
the effects of alternative rationing schemes on economic stability. Participants play
the role of consumer-workers who interact in labor and output markets. All output, which yields a reward to participants, must be produced through costly labor.
Automated firms hire workers to produce output so long as there is sufficient demand for all production. In every period either output or labor hours are rationed.
Random queue, equitable, and priority (i.e., property rights) rationing schemes are
compared. Production volatility is the lowest under a priority rationing rule and
is significantly higher under a scheme that allocates the scarce resource through
a random queue. Production converges toward the steady state under a priority
rule, but can diverge to significantly low levels under a random queue or equitable
rule where there is the opportunity for and perception of free-riding. At the individual level, rationing in the output market leads consumer-workers to supply less
labor in subsequent periods. A model of myopic decision-making is developed to
rationalize the results.
JEL classifications: C92, E13, H31, H4, E62
Keywords: Rationing · allocation rules · unemployment · experimental
macroeconomics · laboratory experiment · general equilibrium
∗
Vancouver School of Economics, University of British Columbia, 997-1873 East Mall, Vancouver,
BC, V6T 1Z1, Canada, gfenig@mail.ubc.ca.
†
Corresponding author. Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6,
Canada, lubap@sfu.ca. We thank the Sury Initiative for Global Finance and International Risk Management and the Social Sciences and Humanities Research Council of Canada for generous financial
support and Camila Cordoba for excellent research assistance.
1
1
Introduction
In a world that faces rationing due to the constant flux between periods of excess supply
of labor and periods of increased demand for output, does it matter how scarce jobs
and goods are allocated? Can certain allocation schemes bring about greater welfare
and economic stability? Representative agent macroeconomic frameworks have little to
say on these questions. The representative household optimally demands the amount of
output associated with its labor supply, resulting in no need for rationing.
But markets do not always clear and rationing of scarce goods or jobs frequently
occurs. Rationing has been approached using many different allocation schemes. To
deal with a very low supply of organ donations, the Israeli government implemented a
policy in 2008 to give priority on organ waiting lists to those willing to sign an organ
donation card. By 2011, the policy had led to a dramatic increase in the number of
deceased and living donors relative to previous years (Lavee et al. 2013). Food rationing
occurred throughout North America and Europe during the two World Wars. Rationing
was undertaken in such a way that every person would receive an equal portion of
food. Victory or war gardens were planted at private residences and in public parks in
many countries to alleviate demand on rationed food supplies. The gardens provided
households an opportunity to supplement their weekly rations with their private food
production. Those who put forth more effort tending to their gardens were able to eat
more, leading to millions of tons of household food production. By contrast, evidence
suggests that voluntary rationing of food during World War I was ineffective at ensuring
equitable allocations: “While many better-educated and more affluent Americans did
observe wheatless and meatless days, immigrants and those in the working class . .
. increased their food intake; beef consumption . . . actually went up during the
war” (Bentley (1998)). Other rationing system have been associated with panic and
instability. Before the advent of deposit insurance in the United States in 1933, banks
allowed depositors to withdraw their money on a first-come, first-served basis until they
ran out of funds. Depositors’ expectations that others would withdraw their deposits
1
would caused panic and a run on the bank, leading to a fragile banking system. Large
price cuts on Black Friday in the U.S. or on Boxing Day in Canada often result in
consumers’ waiting long hours in line and in buying frenzies.
The rationing of labor hours has been employed in dynamic macroeconomic models
to generate cyclical fluctuations of involuntary unemployment observed in the United
States and Europe (Michaillat (2012)). Labor rationing has also been modeled in partial equilibrium settings as the result of efficiency wages (Stiglitz 1976; Solow 1980;
Shapiro and Stiglitz 1984), gift exchange (Akerlof 1984), and search costs and turnover
costs (Salop 1979; Akerlof 1984), or in general equilibrium environments as a consequence of matching frictions in labor and product markets (Michaillat and Saez (2015)).
Unemployment risk associated with labor rationing can lead to precautionary saving,
endogenous underemployment, and potentially deep recessions (Ravn and Sterk 2013;
Kreamer 2014).
We design and implement a laboratory experiment to develop a better understanding
of how a macroeconomy evolves under alternative rationing mechanisms. In a laboratory
environment, we are able to study the implications of allocation schemes on individual
decision-making and aggregate macroeconomic outcomes with significant control over the
implementation of the rationing scheme and without considerable external consequences.
Moreover, participants’ heterogeneous preferences and reactions to rationing provide us
a richer understanding of the implications of alternative allocation schemes. While a
number of empirical and experimental studies have explored the effects of rationing in
single markets (e.g., common-pool resources, public goods), none have investigated how
the nature of labor and output rationing influences household decisions. These decisions
critically determine the extent of rationing and are essential to our understanding of
how distribution schemes influence the willingness of agents to continue to supply scarce
labor hours or demand output when aggregate demand is low.
This paper explores the effects of rationing within a macroeconomic setting where
households supply costly labor that produces utility-yielding output. In our framework,
rationing occurs either when households are unwilling to purchase all the output they
2
wish to produce, or when they prefer to consume more than they are willing to work to
produce. Rationing occurs as a consequence of aggregate household decisions, an inability for prices to adjust fully, and a lack of inventories. An optimal consumption–leisure
tradeoff condition shows that as individuals’ expected consumption falls, their willingness
to supply labor also decreases, and vice versa. We investigate whether alternative allocation schemes lead to a greater reaction to and incidence of rationing, increased spillover
effects into other markets, and overall greater volatility in production. Decision-making
and aggregate outcomes are compared under three non-manipulable rationing schemes:
a random queue where the rationed market is distributed on a first-come, first-served
basis, an equitable allocation scheme, and a priority scheme where those willing to buy
what they produce (or produce what they demand) receive priority for scarce labor hours
(output).
Our contribution is to provide causal evidence of the implications of rationing rules
on the availability of scarce labor opportunities and output. First, unlike the predictions
of standard equilibrium models, we observe rationing of jobs and output in all of our
sessions. We also observe that the mechanism by which the short side of the market is
rationed does matter for welfare and macroeconomic stability. Participants are willing
to supply high levels of costly labor if they are given priority to purchase the output they
produce. Under an equitable allocation scheme, the willingness to supply labor decreases
in the presence of rationing, and output volatility is considerably higher. Occasionally—
and sometimes permanently—aggregate labor supply will collapse to low levels due to
output rationing under random and equitable distribution schemes. Allocating scarce
output and jobs according to a priority scheme, by contrast, results in significantly
more stable production by reducing subjects’ reaction to rationing. We develop a model
of myopic decision-making in which agents focus only on maximizing current utility
and pessimistically expect others in the market to possess extremely high demands.
Consistent with our experimental findings, the model predicts suboptimal equilibria
under equitable and random allocation schemes that involve high aggregate demand for
output and low labor supply.
3
Our findings also provide important political economy insights into redistribution
policies. In an environment where individuals are equally skilled, redistributive policies
that generate highly inequitable or equitable outcomes and that fail to appropriately
compensate individuals for their costly labor can decrease willingness to work and lead
to periods of economic turmoil. Conversely, minimal redistribution and property rights
to the fruits of one’s labor creates a sufficient incentive to consistently supply labor and
can foster greater macroeconomic stability.
2
Rationing in theory and experiments
Our experimental analysis is most directly inspired by substantial theoretical work that
introduces non-market clearing and quantity rationing to general equilibrium settings.
The disequilibrium approach was born out of the earliest work by Patinkin (1956), in
which involuntary unemployment occurred because of constraints on how much could
be sold. His work spans nearly two decades and aims to understand the necessary and
sufficient conditions by which general equilibrium environments can persistently exist
out of equilibrium.1 We develop an environment and set of rationing schemes most
closely related to Svensson (1980). Svensson, building on earlier work by Gale (1979)
and Futia (1975), develops the notion of stochastic rationing, whereby a consumer must
submit demands to the market before it is known whether there will be rationing or
not. The extent of trade is random due to the stochastic rationing mechanism. Such
rationing is in contrast to the framework of Drèze (1975) in which consumers, facing no
uncertainty, simultaneously take into account the extent of rationing and their budget
constraints when forming their demands.2 More recently, Michaillat and Saez (2015),
develop a tractable equilibrium model of macroeconomic rationing that sidesteps the
1
For an excellent survey of the macroeconomic disequilibrium theory literature, see Drazen (1980).
An alternative approach to rationing was developed by Clower (1965), Barro and Grossman (1971),
and Benassy (1975, 1977), whereby the consumer maximizes demand for each good separately subject
to her budget constraint. In forming the demand for a good, the agent disregards quantity rationing for
the particular good, but optimizes as though all other demands for goods in her consumption set have
faced rationing. Gale (1979) and Svensson (1980) explore the existence of disequilibria under stochastic
manipulable and non-manipulable schemes.
2
4
disequilibrium approach by employing a matching function that governs the probability
of trade and imposes a cost of matching on buyers. Like the Barro-Grossman framework,
Michaillat and Saez’s model is able to capture the spillover of demand shocks to labor
markets.
While the disequilibrium literature has addressed the implications of manipulable
versus non-manipulable rationing rules, little attention has been paid to the behavioral
responses associated with alternative allocation schemes. The laboratory experiments
we discuss in this paper provide the first causal evidence of the effects of different distribution rules on decision-making in a macroeconomic setting.
We begin by developing a laboratory production economy in which participants playing the role of worker-consumers supply the necessary labor to produce the output
they later purchase and consume. Such experimental environments have been used to
study the effects of money supply and monetary policy (Lian and Plott (1998); BoschDomènech and Silvestre (1997); Petersen (2015)), exogenous shocks (Noussair et al.
(2014, 2015)), and asset price stabilization policies (Fenig et al. (2015)). While these
environments all experience some degree of rationing, there has yet to be a comprehensive analysis of how the nature of rationing influences aggregate outcomes. Our
experiment directly builds on Fenig et al. (2015), by systematically investigating how
rationing schemes influence decision-making and economic dynamics in a laboratory
macroeconomy.
In a related paper, Lefebvre (2013) designs a common-pool resource game to compare
four rationing rules the ability of four rationing rules–proportional, constrained-equalawards, constrained-equal-losses, and no-allocation rules—in their ability to coordinate
agents to optimal levels of self-insurance, efficiency, and reliability. Under the Nash
equilibrium predictions, the rationing rule should not influence aggregate usage of the
common resource or self-insurance. Lefebvre finds that no-allocation and constrainedequal-awards rules lead to more efficient coordination. Welfare gains are, however,
highest under the constrained-equal-awards rule. On the other hand, proportional and
constrained-equal-losses rules were shown to be easily manipulable and to lead to sub5
optimal investment in alternative safe resources. Lefebvre argues that the success of
the constrained-equal-awards rule can be attributed to its ability to fully allocate the
resource and to the fact that it reduces the strategic interaction among agents. While
the experiment yields valuable insights into the effects of alternative rationing schemes,
the environment studied does not allow for the endogenous creation of scarce resources
or for rationing to influence the production of resources.
As this paper will demonstrate, rationing does have important effects on the supply
of scarce resources. In a partial equilibrium laboratory experiment to study the effects
of allocation rules on organ donation, Kessler and Roth (2012) observe that priority on
waiting lists for registered donors leads to significantly more donations than does a firstcome, first-served scheme. Buckley et al. (2012) investigate the willingness to pay for
private health insurance under different public sector health-care allocation rules. They
observe that the willingness to pay for private insurance is significantly higher when
public health care is allocated randomly than when it is allocated on a needs or severity
basis. In both these environments, the implementation of specific rationing schemes
effectively reduces the excess demand for scarce resources.
3
Experimental design and implementation
The experimental design and implementation extend the baseline macroeconomy developed in Fenig et al. (2015) by considering alternative rationing schemes. The environment we consider is a dynamic general equilibrium economy with nominal rigidities and
monopolistic competition. We provide a summary a fully derived model and parameterization in Appendix A.
3.1
Experimental economy
Groups of nine participants were assigned the roles of households and were tasked with
making decisions about how much to work and consume over a number of temporally
linked periods. In each period, they gained points by buying (and automatically con6
suming) units of the output good, ct , at a price of Pt , and lost points by selling labor
hours, ht , to automated firms in exchange for an hourly wage, Wt . Points in any given
period were awarded according to the following formula:
− 0.4h2.5
P oints = 1.51c0.66
t .
t
Participants automatically borrowed and saved in the form of one-period bonds, Bt ,
at the prevailing interest rate, it . They received a one-time endowment of 10 units of
lab money to make purchases within the sequence. Additional lab money was earned
through supplying labor and earning interest on savings. Each participant also received
an equal share of the firms’ profits, Πt . Thus, all participant faced a per-period budget
constraint given by Pt ct + Bt = Wt ht + Bt−1 (1 + it−1 ) + Πt . Nominal interest rates were
set by an automated central bank and adjusted automatically in response to changes in
current inflation. Specifically, the central bank set the nominal interest rate according
to the following Taylor rule:
(1 + it ) = (1 + ρ)(1 + it−1 )(1 + πt )1.5
0.5
,
where ρ = 0.0363 is the natural nominal interest rate.
At the beginning of each period, participants were asked to submit the maximum
number of hours they would be willing to work (up to a maximum of 10 hours) and
the maximum units of output they would be willing to purchase (up to a maximum of
100 units). Participants were allowed to submit fractions of labor supply and output
demand.
Automated monopolistically competitive firms produced output using labor as their
sole input: firms were able to produce 10 units of output with each hour of labor hired.
After all participants submitted their output demands and labor supplies, an aggregate
P
P
supply of labor (HtS = nj=1 hSj,t ) and demand for output (CtD = nj=1 cD
j,t ) were calculated and used to determine the aggregate level of labor demand and production. If
there was more labor supplied than necessary to produce the total amount of output
7
demanded, only the necessary amount of labor would be hired and hours would be rationed. On the other hand, if there was insufficient labor to produce the total amount
of output demanded, all workers would be hired to work their desired labor and output
would be rationed. Thus, output was made to order; no output was produced that was
not demanded and sold. We parameterized the firms’ probability of being unable to update their prices to 1 − ω = 0.1. Such nominal rigidities prevented firms from adjusting
prices sufficiently in response to aggregate demand.
Wages, prices, and the central bank’s nominal interest rate evolved based on aggregate outcomes. Specifically, prices were determined by the evolution of inflation:
Πt = 1 + 0.0016(cmed
− cSS ) + 0.0744(hmed
− hSS ).
t
t
The nominal wage and the output price were then calculated using median realized labor
supply and output consumed as
Pt = Pt−1 Πt , and
Wt = Pt−1 Πt hmed
t
1.5
cmed
t
0.33
.
Importantly, we assumed that the firms’ pricing rule did not take into consideration the
extent of rationing. Wages and prices were unable to adjust fully to accommodate excess
aggregate labor supply or output demand. This was an important design decision that
increased the occurrence of rationing when aggregate behavior was inconsistent with the
predictions of a rational utility-maximizing framework.
To induce exponential discounting within an infinite horizon environment, we generated indefinite length sequences that ended randomly with a probability of 3.5%. This
implied an average of 28 periods. To make this salient to subjects, in each period we
drew a marble from a bag containing 193 blue marbles and 7 green marbles. If a green
marble was drawn, the sequence ended and a new one began.3
3
Stationary repetition allows us to control for learning and is especially important in macroeconomic
experiments. In our environment, subjects carry cash balances and debt from one period into the next.
8
When a sequence ended, participants would have either a positive or a negative cash
balance in their bank account. If they had a positive balance, the participants would
be required to buy up output and would be credited the points received for that final
consumption. On the other hand, if the participants had a negative balance, they would
be required to work the necessary hours to pay off their debt, and points would be
deducted accordingly. To make discounting salient, we provided participants with a
hypothetical adjusted score assuming that the previous period was the last period of
a sequence. Finally, to be consistent with most macroeconomic models that assume
that agents know the steady state values, at the beginning of the experiment we showed
subjects the steady state values for labor and consumption.
Participants had extensive information at their disposal to make decisions. First,
the interactive computer interface enabled subjects to experiment with different combinations of labor and output decisions for both themselves and the average person in
the economy, in order to derive predictions about their own potential points and bank
account balances as well as aggregate wages, prices, and interest rates. We believe this
dramatically facilitated learning of what would otherwise be a relatively complicated
payoff function. Second, participants had access to all historical information up to the
current period for a given sequence. They could toggle between personal history and
market history to receive detailed information about past outcomes. Finally, we informed
all participants what the steady state values of labor and consumption were and told
them that if everyone in their group were to play such values for an extended amount
of time, wages and prices would stop adjusting and the interest rate would converge to
its steady state level. We provided such detailed information becuase the model is derived under the assumption that agents have full information about the data-generating
process and the steady state values of the economy.
At the beginning of an experiment, it is not unreasonable for subjects to experiment with their decisions
or make decision errors that will influence their bank account balances. Bank account balances, however,
have important implications for optimal consumption and labor decision-making, and errors during
learning can potentially bias subjects’ behavior.
9
3.2
Testable hypotheses under the assumption of homogeneous
utility maximization
We now outline our testable hypotheses formed under the assumption that participants
behave consistently with the predictions for a representative utility-maximizing household with rational expectations.
Hypothesis 1a. Household-consumers will individually supply hi,t = hSS = 2.24
hours of work and demand ci,t = cSS = 22.4 units of output.
In the steady state, individuals consume 22.24 units of output and work 2.24 hours.
This is the equilibrium solution to the Appendix A model.
Hypothesis 1b. The average labor supply will be Ht /n = hSS = 2.24 and the
average output demand will be Ct /n = cSS = 22.4 units.
If Hypothesis 1a holds then Hypothesis 1b will hold also. This is a weaker hypothesis
that tests whether the economy converges on average to the steady state even if some
individuals deviate from equilibrium consumption and labor.
Hypothesis 2. There will not be rationing in labor or consumption.
This is a consequence of Hypothesis 1a. If consumption demand and labor supply are
symmetric among individuals, then there will not be rationing.
Hypothesis 3. The allocation scheme will not affect participants’ behavior.
If Hypothesis 1a holds then Hypothesis 3 will also hold. Labor supply and output
demand decisions should not be influenced by the different allocation rules.
3.3
Rationing rules
Note that in the above model, agents are assumed to optimize their labor and output
decisions identically and have no reason to form expectations about future rationing.
10
Thus, as Hypothesis 3 states, the equilibrium predictions should be unaffected by the
choice of a specific rationing rule.
In our environment, n households simultaneously submitted their desired labor supply and output demand. Given aggregate output demand (CtD ) and labor supply (HtS ),
individual actual consumption (ci,t ) and labor (hi,t ) in the experiment were allocated
according to one of three scenarios at any point in time:
S
1. If CtD = HtS , neither output nor labor was rationed, ci,t = cD
i,t and hi,t = hi,t .
2. If CtD > HtS , subjects obtain the hours of work they requested, hi,t = hSi,t , and
R
output was rationed, ci,t = min θci,t
, cD
i,t .
3. If CtD < HtS , subjects obtain the output they requested, ci,t = cD
i,t , and labor was
R
rationed, hi,t = min θhi,t
, hSi,t .
Here, θcR (θhR ) is the individual consumption (hours) when output (labor) is rationed
according to a specific rule R. Note that households never obtained more than their
desired consumption and labor.
A household that supplied the equilibrium level of labor but faced rationing in terms
of output would experience an increase in its money balance. As noted by Barro and
Grossman (1971), the household’s best response to frustrated demand is to increase its
output demand and/or decrease its labor supply over the following periods. All else
equal, both will generate further excess demand. Likewise, following involuntary unemployment or underemployment relative to consumption, the household’s best response
is to smooth its consumption over the horizon by increasing its labor supply and/or reducing its output demand. Thus, involuntary unemployment can further increase excess
labor supply in the future. This leads us to our fourth testable hypothesis:
Hypothesis 4. Excess output demands and labor supplies are persistent.
In this paper, we explicitly test whether different rationing rules influence economic
stability and welfare. We focus on rationing rules for which realized outcomes for an
11
individual i are a function of her own effective supplies and demands, as well as the
aggregate effective supply and demand on the market. The rules are similar in that
no individual is forced to trade more than she likes and only the market with excess
supply/demand is rationed. Moreover, all the rationing rules are efficient: the aggregate
output produced is consumed by all agents and the total hours of hired work are allocated among them. Below we describe in detail the rationing rules we considered in our
experiments.
1. Random Queue (Random): In each period households were assigned a position
in a queue. Households at the front of the queue had priority for the scarce hours or
output, and positions were randomized in each period. When there was excess output
demand, expected consumption was given by
(
(
)
)
1
X
1
1
S D
S
D
E(ci,t ) =
min ZHt , ci,t + min max ZHt −
[cj,t ]q , 0 , ci,t +
|n
{z
} n
q=1
|
{z
}
First Position
Second Position
(
(
)
)
n−1
X
1
. . . + min max ZHtS −
[cj,t ]q , 0 , cD
i,t ,
n
q=1
|
{z
}
Last Position
while in instances of excess labor supply, expected labor was
(
(
)
)
D
1
D
X
Ct
C
1
1
t
min
, hSi,t + min max
−
[hj,t ]q , 0 , hSi,t +
E(hi,t ) =
n
Z
n
Z
q=1
|
{z
} |
{z
}
First Position
Second Position
(
(
)
)
n−1
CtD X
1
S
−
[hj,t ]q , 0 , hi,t ,
. . . + min max
n
Z
q=1
|
{z
}
Last Position
where [cj,t ]q and [hj,t ]q denote consumption and hours of work of agent j where j 6= i,
respectively, and q ∈ {1, n} is the position in the queue.
2. Equitable Rule (Equitable): All households equally shared the rationed hours
o
n S
D
or output up to their desired demand. Households obtained ci = min ZH
,
c
when
i
n
n D
o
there was excess output demand, and hi = min C n/Z , hSi when there was excess labor
12
supply. Any undesired hours of work or units of output were allocated in equal shares
among those with excess demands for labor hours or output.
3.
Priority Rule (Priority): Households were given priority to purchase the
output they personally produced: ci = min{ZhSi , cD
i }. Similarly, if labor hours were
rationed, participants were given priority to work the hours associated with their purs
chased output, hi = min{cD
i /Z, hi }. Any undesired hours of work or units of output were
randomly allocated among those with unsatisfied demands for labor hours or output.
Table 1 presents an example of how resources are allocated in a four-household economy under each of the rationing rules. In the example, output is rationed due to excess
demand. Columns 2 and 3 display the desired labor and consumption. In column 4, the
assigned hours are shown; they are the same as the desired hours. Finally, since total
output produced is lower than the desired output, columns 5, 6, and 7 show how output
is allocated under each rule.
To gain some intuition about the relative effects of the different rationing schemes
on allocations, consider the following example of excess output demand.4 Suppose eight
participants are demanding output and supplying labor in a manner consistent with the
model predictions (hsi = 2.24 and csi = 22.4), while the ninth participant supplies the
same amount of labor but demands more output (csi = 22.4+j, where j > 0). Aggregate
labor supply is H = 201.6 and output demand is C = 201.6 + j.
If rationing is conducted according to the Priority Rule, one person’s demanding an
excessive amount of output has no effect on the allocations for the other eight participants. These participants will receive their requested labor hours and units of output,
while the excess demander receives her requested labor hours and is rationed on output,
where her realized consumption is ci = 22.4. Increasing j has no additional effect on
any participant’s final allocations.
Under the Equitable Rule, the scarce output will be distributed equally among all
4
The intuition for excess labor supply follows a similar thought experiment.
13
participants up to their desired demand. Given that the excess demander also supplies
hsi = 2.24, the output allocations will be identical to that observed under a Priority Rule,
ci = 22.4. In this case, output rationing should have no effect on other participants’
decisions. If, however, output rationing were due to a single participant supplying less
labor than would be predicted by the optimizing model (hsi = 2.24 − k and csi = 22.4),
aggregate labor supply and output demand would be H = 20.16−k and C = 201.6−10k.
Each participant would receive ci = 22.4 − (10/9)k, which is less than her original
demand. As k grows large, the impact of one person’s reduction in labor supply on
others’ output allocations grows large. Furthermore, all participants will spend less
than they desired, resulting in an increase in cash balances. In the following period,
participants will best respond to this excess cash balances by either lowering their labor
supply or increasing their output demands—both of which may generate even more
rationing of output.
Under a Random Queue Rule, the opportunity for the excess demander to influence
others’ allocations of output will depend on her position in the queue. Unless she is at the
end of the queue, at least one other participant will experience a reduction in her output
allocation. As the amount by which the participant overdemands output, j, grows large,
an increasing number of participants will be unable to receive their desired output. The
alternative scenario, in which a single participant undersupplies labor, resulting in excess
demand, will have similar effects. For the participants who experience output rationing,
their best response in the next period will be to undersupply labor or increase their
demand for output.
Thus, for the same amount of excess output demand generated by a reduction in labor supply by k = 1, leading to a 10-unit reduction of total production, cEquitable
= 21.28
i
for all participants while cRandom
= 22.4 for all participants except the last person in
i
the queue, who receives cRandom
= 12.4. The extent to which labor supplies will dei
crease in the next period depends on how much participants smooth their unanticipated
increases in their cash balances. The increases in individual cash balances due to underconsumption are relatively modest, as they are spread equally among all participants
14
under the Equitable Rule. By contrast, the increase in individual cash balances are quite
large and isolated to a single participant under the Random Queue Rule. In general,
the reduction in the following period’s aggregate labor supply will be larger under a
Random Queue Rule.5 This leads us to our fifth testable hypothesis:
Hypothesis 5. The size of the adjustment in labor supply in response to past output rationing varies by rule, as follows: Random Queue > Equitable > Priority.
3.4
Experimental procedures
The experiment was conducted at the CRABE Laboratory at Simon Fraser University.
Subjects were undergraduate participants recruited from a wide variety of disciplines.
We conducted six sessions of the Random and Priority treatments and seven of the
Equitable treatment. Each session had eight or nine inexperienced participants and
consisted of only one treatment. At the beginning of each session we conducted a 35minute instruction phase that involved a discussion of the game, the rationing rule and
four periods of guided practice through the visual interface.6 Payoffs, including a $7
show-up fee, ranged from $10 to $38.7
4
Aggregate findings
In this section, we summarize our findings across treatments. Our analysis includes
decisions made by all participants over all periods of play. The data from all sequences
are treated as one time series, unless otherwise noted.
5
Note that nominal wages and prices are more likely to respond minimally to rationing under an
Equitable Rule, as the median participant will also be changing her output demands and/or labor
supplies. By contrast, under a Random Queue Rule, the median participant in the initial stages of
rationing will be unaffected by small amounts of labor shading. Under a Priority Rule, wages and prices
are unaffected by a single participant deviating from the representative agent prediction, regardless of
the size of the deviation.
6
Screenshots of the computer interface can be found in Figures 10-12 in Appendix D.
7
The instructions can be found in Appendix E and Appendix F.
15
4.1
Decisions, production, and rationing
Cumulative distributions of median and individual labor supply and median output
demand decisions are presented in Figures 1 and 2 for each treatment, respectively. The
dashed vertical line is the steady state predicted individual labor supply of 2.24 hours
and output demand of 22.4 units. Histograms of labor supply, output demand, and
realized consumption are provided in Figure 3.
Mean labor supply under the Priority treatment is 2.76 hours (SD = 1.83), while it is
modestly higher in the Random treatment, with participants supplying an average of 2.91
hours (SD = 2.17). By contrast, mean labor supply is lower in the Equitable treatment,
with 2.58 hours (SD = 1.93). Signed-rank tests reject the null hypothesis that the sessionlevel mean labor supply is equal to the equilibrium prediction in the Priority and Random
treatments (p = 0.046 and p = 0.028, respectively), but detect no significant differences
from equilibrium behavior in the Equitable treatment (p = 0.3980). Two-sided Wilcoxon
rank-sum tests are unable to reject the null hypothesis that session-level mean labor
supplies are identical across treatments (p > 0.31 for each pairwise comparison). Labor
supply in the Priority treatment is heterogeneous but largely centered around the steady
state. By contrast, labor supply in the Random treatment exhibit greater heterogeneity
and a distribution closer to bipolar. Participants facing rationing according to a random
queue have a tendency either to work very little or to work a lot. In the Equitable
treatment, we observe considerably lower labor supplies across the entire distribution,
with a large mass of decisions on hours below the steady state.
Output demand differs more clearly across treatments. We observe the highest average demands under the Priority treatment (mean = 48.13, SD = 9.12), followed by
the Equitable treatment (mean = 42.66, SD = 7.31) and the Random treatment (mean
= 38.97, SD = 4.12). Median demands follow a similar order. Mean output demands
in all treatments are significantly above the equilibrium prediction (p < 0.028 in all
cases). While the session-level mean output demands are not significantly different between the Random and Equitable treatments (p = 0.317) or the Equitable and Priority
16
treatments (p = 0.253), the differences are significant between the Random and Priority
(p = 0.055). Output demands in the Random treatment stochastically dominate at first
order the output demands in the Priority treatment. Output demands are also considerably lower in the Equitable treatment than in the Priority treatment for most of the
distribution. From the histograms of the distribution of output demands, we see that
approximately 8% of Random decisions, 12% of Equitable decisions, and 14% of Priority
decisions are for the maximum allowed (100 units).8
The differences in labor supply and output demand do not translate into significantly
different levels of mean production across treatments. Table 2 presents the sessionlevel summary statistics, with two-sided Wilcoxon rank-sum tests provided to denote
statistical differences between treatments. Mean total output produced is lowest in
the Equitable treatment at 219.09 units (SD = 54.51), and is relatively higher in the
Random with 238.44 units (SD = 45.11) and Priority with 241.68 (SD = 43.44). While
mean production is above the steady state prediction in all treatments, the differences
are statistically significant only in the Random and Priority treatments. Moreover,
the treatment differences in mean production at the session level are not statistically
significant, with p > 0.39 in all pairwise comparisons.
Rationing occurs in all periods of play and we confidently reject Hypothesis 2. We
observe high frequencies of output rationing in all treatments, occurring on average
between 80 and 87% of the time. Rationing of labor hours occurs minimally in five
of six sessions of the Random treatment (the exception is in Random2, in which labor
rationing never occurs), in four of seven Equitable sessions (Equitable1, Equitable3,
Equitable5, and Equitable7), and in only two of six sessions of the Priority treatment
(Priority4 and Priority6).
Rationing of both output and labor is highly persistent over time. Figure 4 plots,
for all periods and sessions, the relationship between the quantity of lagged and current
output and labor rationed. The green 45-degree line denotes observations in which the
8
The frequency of subjects submitting the maximum levels of consumption and labor was extremely
low. Only 0.6% (23/3928) of submitted decisions in the Random treatment, 0.1% (6/4723) in the
S
Equitable treatment, and 0.3% (14/4177) in the Priority treatment were for cD
i,t = 100 and li,t = 10.
17
aggregate quantity of output or labor rationed remains constant across two consecutive
periods. Observations above (below) the diagonal line denote instances of rationing
increasing (decreasing) in the following period. The solid red line denotes a local polynomial smoothed line. The vast majority of periods in which rationing occurs is followed
by further rationing of the same market. More than 91% of output rationing and 61%
of labor rationing are followed by further rationing in the following period. However,
the quantity of output rationed increases in the following period roughly half the time
across all treatments, while the quantity of labor rationed increases between 28% and
41% of the time.9
Observation 1. Output demands are significantly higher than the steady state
equilibrium in all treatments. As output is typically the rationed market, labor supplies
are the key driver of production in most sessions. Mean labor supply and realized production is on average higher than the steady state equilibrium in all treatments, and
significantly higher in the Priority and Random treatments. Hypothesis 1a is rejected
completely in the Priority and Random treatments, and for output demand decisions in
the Equitable treatment.
Observation 2. Both output and labor rationing is persistent. While labor rationing tends to subside in the following period in the Equitable and Priority treatments,
it worsens on average in the Random treatment. Output rationing worsens in half of
the following periods consistently in all treatments.
Production volatility is influenced by the form of rationing. The lowest levels of
volatility are observed in the Priority treatment (mean = 0.20, SD = 0.02), whereas
volatility increases in the Equitable treatment (mean = 0.25, SD = 0.10, p = 0.333) and
9
We also compute a session-level measure of the likelihood of worsening output and labor rationing
given past rationing. Wilcoxon signed-rank tests reject the null hypothesis that output rationing remains
constant in favor of increased output rationing in all treatments (p < 0.027). By contrast, only in the
Random treatment does output rationing significantly lead to increased output rationing in the following
period (p = 0.06).
18
significantly increases in the Random treatment (mean = 0.25, SD = 0.03, p = 0.037).
In terms of average points earned, all subjects earn significantly less than the steady
state equilibrium prediction of 8.625. Equitable subjects earn the highest with 5.04
points, followed by Priority subjects at 4.26 points, and Random subjects at 1.01 points.
There is no statistical difference between earnings in the Equitable and Priority treatments. Participants in the Random treatment, however, receive significantly less output
and earn significantly fewer points on average than both Equitable and Priority participants. Figure 5 presents the wealth distribution for each treatment. The solid black line
is a reference line of perfect equality among subjects. As expected, the highest levels of
equality are observed in the Equitable treatment, where output is equally distributed up
to individual demands. The fact that the Equitable treatment exhibits some inequality
is due to heterogeneity in participants’ preferences for labor and output, as well as to
fluctuating decisions and outcomes over time. Inequality worsens under the Priority
rationing rule. When participants are largely responsible for their points, as they are in
the Priority treatment, the heterogeneity in labor supply decisions leads to significantly
different levels of output and points received. Finally, the greatest inequality is observed
when output is allocated according to a random queue. In the Random treatment, 50%
of the participants receive, on average, less than 10% of the points earned. The inequality in the Random treatment is driven by consumers demanding the highest levels of
output each period in the hopes of being at the front of the queue. Such impulses can
leave little output remaining for other participants later in the queue, especially in later
periods, when labor supply and output demand fall significantly.
4.2
Convergence
As in most dynamic experimental environments, the main macroeconomic variables in
our economies do not immediately reach their steady state values. However, one would
expect that after some learning, subjects would become familiarized with the environment and their choices would gradually converge to the equilibrium predictions. In this
section we analyze whether median labor supply and output demand converged to the
19
equilibrium predictions, and if so how fast the process was. We first contrast behavior
across treatments graphically. Figures 6.a and 6.b show box-plots of the average median
labor supply and output demand for each sequence, while Figure 6.c and 6.d depict
the aggregate outcomes. Figure 7 contrasts the time series of average labor supply and
output demand across treatments. The horizontal red lines in these figures represent
the corresponding steady state levels of labor and output.
Median and aggregate labor supplies appear to be converging toward the steady
state after many stationary repetitions. In the Random treatment, there is considerable heterogeneity in both median and aggregate labor supplies, where some sessions
experience very high levels of labor supply. By contrast, in the Equitable treatment,
labor supplies in some sessions become quite low after a few repetitions. In terms of
output demand, there is little convergence, either at the median or aggregate level, to
the steady state. Over time, we see that the differences in output demand become quite
stark under the Random and Priority rules, with average demand often 10 units higher
under the Priority rule.
We next identify the period of convergence following Bao et al. (2013). In each
session, we calculate the absolute deviation of median labor supply and output demand
from the steady state. We then claim that convergence occurs in the first period in which
the absolute deviation from the steady state is less than 1 in the case of labor supply
and less than 10 for output demand, and this is preserved until the end of the session.10
The second column of Table 3 shows the number of periods before convergence for the
median labor supply. It takes only 22 periods on average for labor supply to converge
to the steady state in the Priority treatment, whereas in the Random and Equitable
treatments it takes almost three times as many periods (64 and 62, respectively). There
is not much difference across treatments in terms of the number of periods it takes
for output demand to converge (74, 64, and 74 periods on average for the Random,
Equitable, and Priority treatments, respectively).
− hss > 1/ : (cmed
− css > 10), where hmed
and cmed
are median labor
If at period t, hmed
t
t
t
t
supply and median output demand at period t, respectively,
as a converging
period
med we still
count it
only if convergence is restored in the following period, ht+1 − hss < 1/ : (cmed
t+1 − css < 10).
10
20
Finally, we formally test convergence, following the regression model of Noussair et al.
(1995). They were the first to propose this econometric procedure to study convergence
of experimental panel data. The regression model for each treatment is the following:
S
1X
t−1
yst =
αs Ds + β
+ εst ,
t s=1
t
where yit is the dependent variable (in this case, median labor supply/output demand),
s = 1, 2, ..., S is the session, Ds is a dummy variable for each of the sessions within
a treatment, and εst is an error term. The αs coefficients capture the initial value of
the variable of interest at the beginning of the sessions, and β represents the value of
the variable y to which each of the treatments converge. Table 3 shows the generalized
least squares (GLS) estimates of αs and β. For median labor supply, β̂ is not significantly different than the steady state value in the Random and Priority treatments;
thus there is evidence of asymptotic convergence. However, median labor supply in the
Equitable treatment converges to a value that is significantly lower than the predicted
one. Median output demand converges to values above the theoretical predictions in all
the treatments.
5
The effects of output rationing on decisions
We now utilize our rich panel-level data with subject-level observations collected every
period to gain insight into the effects of rationing schemes on individual labor and
output decisions. Our main estimating equation is motivated by the intra-temporal
optimization equation of households, which suggests that increases in real wages and
output demand (labor supply) is associated with an increase in labor supply (output
demand). We further consider the effects of entering bank account balances, as well as
the effects of experiencing output rationing and the quantity of rationing incurred in
the previous period on current decisions. Our focus on output rationing stems from the
earlier observation that the vast majority of instances of rationing are of output.
21
A series of pre-estimation diagnostic tests are conducted to determine appropriate
estimation strategies. We use our first specification in Table 5 as our baseline testing
specification and apply the recommended estimation strategy to all other specifications.
First, a Hausman test rejects the null hypothesis that the preferred model is one with
random effects in favour of the alternative of fixed effects (p = 0.000). A further test
for random effects (Breusch-Pagan Lagrange multiplier test) is unable to reject the null
hypothesis that the variance across subjects is zero (p = 1.000). That is, we are advised
to use ordinary least squares (OLS) rather than assume random effects. A Pesaran crosssectional dependence test rejects the null hypothesis that residuals are not correlated
across subjects (p = 0.000), implying that our standard errors should be corrected for
cross-sectional dependence. A modified Wald test identifies group heteroskedasticity
within a fixed effects regression specification (p = 0.000). Finally, we test for serial
correlation between variables and reject the null hypothesis of no serial correlation (p =
0.000). The results of the diagnostic tests motivate us to consider a fixed effect panel
regression in which we employ robust standard errors to correct for heteroskedasticity
and autocorrelation. Given our limited number of sessions per treatment, we do not
cluster our standard errors.
We conduct numerous regressions to understand how rationing influences decisions
within a treatment. The treatment-specific results are presented in Tables 5, with labor
supply decisions presented in Panel A and output demand decisions in Panel B. All specifications include controls for real wages, past and current decisions, and end-of-period
bank account balances from the previous period, and they vary by the modelling of output rationing. Specification (1) includes the dummy variable OutputRationedi,t−1 , which
takes the value of 1 if a subject received less output than she demanded in the previous
period. Specification (2) includes the variable QuantityOutputRationedi,t−1 , which is a
continuous variable measuring the difference between what a subject demanded in output and what she received in the previous period. Specification (3) instead considers an
alternative measure of output rationing, AltQuantityOutputRationedi,t−1 , which is measured as the difference between the amount of output a participant was willing to produce
22
D
S
) − ci,t−1 ).
, Ct−1
and the potentially rationed amount she received, min(0, min(10Ni,t−1
This alternative measure allows us to account for rationing that an individual subject
incurred because another participant was allocated her production. This alternative rationing notion is absent from the Priority Rule specifications, as under a Priority Rule
all participants are able to receive the output they were hired to produce. To identify
differentiated reactions of labor supply and output demand to rationing across treatments, we conduct a further set of regressions that pools data from all three treatments
in which our various rationing measures interact with treatment dummies. The results
can be found in Table 6.
Labor Supply Response to Output Rationing. As seen in Table 5, evaluating each treatment independently, we find that experiencing any output rationing in the
previous period leads subjects to significantly reduce their current labor supply, from 0.2
hours in the Equitable treatment to 0.43 hours in the Random treatment. Comparing
the treatments in a pooled regression with treatment interactions, we observe that the
negative labor supply response to past output rationing is significantly more pronounced
in the Equitable and Random treatments. Controlling for other determinants of labor
supply, we find that output-rationed Equitable and Random participants will work 0.23
and 0.38 hours less, respectively, than their Priority counterparts. Similar results are
observed in Specification (2) when we instead consider the effects of increasing the degree of rationing. Participants in the Random and Equitable significantly decrease their
labor supply by 0.008 and 0.009 hours for every unit of output rationed. By contrast,
the average Priority participant adjusts her labor supply downward by only 0.003 hours
for each unit of output rationed and this reaction is not significantly different from zero.
Comparing across treatments, we again observe a significantly larger response to output rationing in the Equitable and Random treatments than in the Priority treatment.
Compared to rationed participants in the Priority treatment, rationed Equitable workers
supply 0.23 fewer hours while rationed Random participants supply 0.38 fewer hours.
These differences are statistically significant at the 1% level.
23
Labor supplies in the Random and Equitable treatments also respond adversely to
increases in the quantity of output rationed. Random and Equitable participants significantly reduce their labor supply by 0.08 and 0.09 hours, respectively, for every 10 units
they were unable to purchase in the previous period. In contrast, Priority participants
reduce their labor supply by 0.03 hours for every 10 units rationed, but this reaction is
not statistically significant. Compared to their counterparts in the Priority treatment,
Equitable and Random participants have a significantly more adverse reaction to output rationing. We observe an even more pronounced response when we instead consider
the AltQuantityOutputRationedi,t−1 measure of output rationing. Ten units of output
rationing leads the average Random and Equitable participants to significantly reduce
their labor supply by 0.16 and 0.29 hours, respectively. Compared to their Random
counterparts, Equitable participants’ labor supplies are significantly more sensitive to
being rationed output they personally produced.
Output Demand Response to Rationing. Results for output demand decisions
are presented in Panel B of Table 5. Simply being unable to satiate last period’s demands leads Random participants to significantly increase their demands by 7.2 units,
but leads to small and insignificantly lower demands in the Equitable treatment and
small and insignificantly higher demands in the Priority treatment. Rationed Random
participants demand significantly more than their Priority and Equitable counterparts,
but the difference between the latter two treatments is not significant.
The treatment-specific reactions to facing output rationing is presented in each column (1) of Panel B. We observe Priority participants increase their output demand on
average by 14.3 units in response to rationing. The reaction is significantly muted in
the Equitable treatment, with consumption demand increasing by only 10 units. Compared to reactions under the Priority treatment, reactions in the Random treatment
are also smaller on average but the difference is not statistically significant. In column
(6) of the pooled regressions, we observe that compared to participants in the Random
treatment, Equitable participants are not significantly more or less reactive to output
rationing. In terms of quantity rationing, the greater the extent of rationing, the more a
24
participant will demand in the following periods. Relative to Priority participants, the
increase in consumer demand is significantly and quantitatively larger among Equitable
and Random participants (with no significant differences between the latter two).
Each column (2) in Panel B of Table 5 presents output demand responses to the
quantity of output rationed. Rationed participants in all treatments increase their output
demands in response to greater rationing in the previous period. While demands increase
from an additional 0.134 units per unit previously rationed in the Priority treatment
up to 0.317 units in the Equitable treatment, the differences are not significant across
treatments. When we instead consider the AltQuantityOutputRationedi,t−1 measure of
output rationing in column (3), we observe significant increases in output demand only
in the Random treatment as the number of units of output rationed increases. Equitable
demands are largely unresponsive to this form of output rationing.
The above results suggest that rationing schemes do have important effects on labor
and consumption decisions. Random and Equitable labor supply decisions are considerably more reactive to rationing. In these treatments, rationing may be a consequence
of others’ free-riding. By contrast, in the Priority treatment, rationing can occur only
if a participant wants to consume more than she has produced and there is insufficient
excess supply to draw on. Conversely, consumption decisions are less reactive to the
quantity of output rationing in the Random and Equitable treatments. Participants in
the Random treatment who demand excessive quantities of output face a greater probability of receiving the units (and a higher consumption bill) than those in the Priority
treatment, resulting in more cautious decisions. Moreover, asking for relatively more
output in the Random treatment drains the available pool for others. In the Equitable
treatment, demanding higher levels of output is less likely to influence overall output
received since everyone receives an equal share of the production up to their personal demands. Taken together, these results suggest that in response to past output rationing,
a priority rationing scheme provides the greatest stability in labor hours supplied at the
cost of increased demand for output. However, because one subjects’ excess demands
do not influence others’ allocations, the Priority allocation scheme ensures the greatest
25
stability in aggregate labor supply and production.
6
A model of myopic decision-making
The underworking pattern observed in some of the sessions when the Random Queue
rule and the Equitable rule were implemented and the persistently high levels of output demand can be explained by assuming that agents form their decisions myopically,
without regard to their budget constraints and future utility maximization. Under these
rationing schemes, agents will find it optimal to undersupply labor.
Suppose that there are n agents in the economy. The labor supply and output
demand of agent i ∈ {1, ..., n} is denoted by hSi ∈ {0, 10} and cD
i ∈ {0, 100}, respectively,
whereas hours worked and output purchased are denoted by hi and ci , respectively.
Agent i chooses nSi and cD
i to maximize:
max U (ci , hi )
S
cD
i ,hi
(1)
subject to
R
D
S
S
ci = min cD
if C D > ZH S
i , θci (C−i , H ) , hi = hi
S R D S ci = cD
,
h
=
min
hi , θhi (C , H−i )
if C D < ZH S ,
i
i
where U (ci , hi ) =
1
1−σ
c1−σ
i
−
1
1+η
h1+η
, and θcR (θhR ) is the individual consumption
i
(hours) when output (labor) is rationed according to a rule, R, which is either Random, Equitable, or Priority. Aggregate output demand and labor supply are denoted
P
P
S
by C D ≡ ni=1 cD
≡ ni=1 hSi , respectively. We can compare the symmetric
i and H
equilibrium for labor supply under the different rationing rules.11
11
We rule out of the analysis the excess labor supply case. Whenever there is excess labor supply,
agents would find it profitable to deviate by cutting their hours of work, up to the point at which there
would be excess output demand. This is consistent with the data; in the Random treatment there was
no excess labor supply in any of the sessions, while in the Equitable treatment only 12% of the periods
26
Random Queue: Under a random queue rationing scheme, the probability of obtaining output depends on the quantity of excess output demanded. Increasing one’s
own labor supply raises the probability of consuming, but it is costly in terms of utility.
As long as aggregate labor supply is positive, at least the first individual in the queue
will be able to purchase a positive amount of output. To simplify the analysis, suppose
D
D
that agents demand the same amount of output, cD
1 = c2 = . . . = cn = c̄. Agent i
chooses how many hours to work to maximize her expected utility:
max
s
hi
n
h
i o
n
h
i o
n
h
i o
i
1h Random
Random
Random
U min cD
, hsi + U min cD
, hsi + . . . + U min cD
, hsi ,
i , θci
i , θci
i , θci
1
2
n
n
Random S
=
max
ZH
−
(q
−
1)c̄,
0
, and q ∈ {1, . . . , n} is the position of
where θci
q
individual i in the queue. Positions are randomly assigned, and there is a
1
n
probability
of obtaining each one of the spots.
The maximization problem can be solved numerically. There are multiple Nash
equilibria for labor supply. The equilibrium depends on output demand. Assuming that
Random
= 0.471. For c̄ < 48, the equilibrium
there are nine agents, if c̄ ≥ 48 then hSi
Random
range is hSi
= [0.471, 0.65].
Equitable Rule: Under an equitable allocation scheme, each agent receives an equal
ZH S
,
n
Equitable
share of the total production, θci
=
up to her specified demand. The first
order condition with respect to hSi from equation (1) is
1−σ
η
Z
(H S )−σ = hSi .
n
In a symmetric equilibrium, H S = nhsi . Thus, each agent will choose to work
Equitable
hSi
1
=
n
1
η+σ
1−σ
Z η+σ .
With n = 9, the Nash equilibrium labor supply is 0.699.
saw excess labor supply (see Table 2).
27
Priority Rule: Under a priority rationing scheme, ci depends on hSi . Specifically,
P riority
θic
= ZhSi . The first order condition with respect to hSi from equation (1) is given
by the following equation:
Z 1−σ hSi
−σ
= hSi
η
.
Thus, the Nash equilibrium labor supply is
hSi
P riority
1−σ
= Z η+σ .
Under the parameterization of the experimental environment, the Nash equilibrium labor
supply is 2.32.12
There are two important conclusions from the above analysis. First, unlike under the
Equitable rule and the Random rule, under the Priority rule equilibrium labor supply
does not depend on the number of agents. Under the Equitable rule and the Random
rule, as the number of agents increases, the optimal labor supply decreases. The subjects’
best response when aggregate labor increases is to cut their own labor supply; that
is, labor supply is a strategic substitute. Second, optimal labor supply under myopic
decision-making is highest under the Priority rule, followed by the Equitable rule, and
lowest under a Random rule.
Our individual-level findings support many of the predictions of the myopic model.
In response to past rationing, participants significantly reduce their labor supply and increase their output demands. Compared to their Priority rule counterparts, participants
faced with Random Queue and Equitable rationing schemes reduce their labor supplies
by significantly more. Moreover, participants in the Random rule respond more aversely
to past output rationing than those facing an Equitable allocation scheme. While average labor supply, measured at the session level, is not significantly different across
12
Given that this is the equilibrium solution for the static model, labor supply is higher than in
the steady state. However, for the dynamic case, the Nash equilibrium labor supply and the implied
consumption do not maximize individuals’ lifetime utility because they imply a positive level of indebtedness.
28
treatments, the myopic model does predict extreme outcomes in our data. We observe
that the minimum aggregate labor supply observed at the session level is significantly
lower in the Random Queue treatment (mean = 11.77, SD = 4.14) than in the Priority
rule (mean = 15.05, SD = 3.48, p = 0.078). Aggregate labor supplies also reach very low
levels in the Equitable treatment (mean = 10.49, SD = 6.46), but are not significantly
different from that observed in the Random Queue (p = 0.668). While the average
minimum aggregate labor supply is lower under an Equitable rule than under a Priority
rule, the difference is not statistically significant (p=0.153).
7
Discussion
In this paper we present the first experimental evidence that the nature of rationing has
important implications for macroeconomic stability. Equilibrium models with representative agents and market clearing abstract away from these important issues and consequently miss out on important, realistic dynamics driven by market spillovers. However,
in our experimental economies populated with heterogeneous participants, distribution
schemes are shown to play an important role in fostering economic stability. Our findings
suggest that convergence to the steady state equilibrium depends significantly on the
presence of property rights and on the opportunity for free-riding. Under random queue
and equitable allocation rules, consumer-workers are not obliged to supply costly labor
in order to purchase output and can instead rely to some extent on others to do the
work for them. Such behavior results in greater incentives to free-ride on others’ costly
labor. In turn, hard-working individuals may respond to others’ excess consumption by
reducing their labor supply, which leads the aggregate production to fall to extremely
low levels. The presence of property rights in a priority system allows individuals to
confidently supply labor with the expectation that they will be able to purchase at least
what they have produced. Likewise, these individuals can purchase output with minimal
uncertainty about their employment opportunities. In this case, aggregate production is
significantly less volatile, and it quickly and consistently converges to the steady state.
29
We must be cautious not to argue that the undersupplying of labor in our environments is driven entirely by free-riding. If an individual faces rationing of output and
cannot spend her money balances sufficiently, she should optimally reduce her labor supply in an effort to smooth her leisure. Such reductions in labor supply lead to further
output rationing and the potential for a downward spiral in production. Similarly, pessimistic employment expectations should motivate optimizing individuals to reduce their
consumption demands and can result in self-fulfilling recessions. It is not unreasonable
for some participants to perceive the consumption-smoothing behavior of other market participants as free-riding and reciprocate by cutting their own labor supplies. We
emphasize that such low-production outcomes are more likely to occur under rationing
rules where there are minimal or nonexistent property rights to workers’ production.
Rationing schemes have important implications for inequality. Inequality is significantly greater when jobs and output are distributed on a first-come, first-served basis,
as in the Random treatment. A tendency to work many hours or buy up as much output
as possible when given an opportunity to do so leads to less work and output for others
at the end of the queue. Considerable inequality exists when participants are allocated
output based on their willingness to work. Heterogeneity in participants’ willingness to
work and save results in a wide range of payoffs in the Priority treatment. As expected,
equality is the highest (though still not perfect) when policies are in place to provide
equally up to individual demands, as is the case in the Equitable treatment. This equality, however, comes at the cost of increased instability as a result of rationing and may
lead to periods of significantly lower individual and aggregate welfare.
30
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33
Tables and figures
Table 1: Rationing rules example
Subject
hSi
cD
i
hi
1
2
3
4
2
5
1
4
100
80
50
30
Aggregate
12
260
ci
I
Random
UT
PriorityII
2
5
1
4
100
20
0
0
30
30
30
30
20+x
50+y
10+z
30
12
120
120
120
(I) Here it is assumed that subject i ∈ 1, 2, 3, 4 has the ith
position in the queue.
(II) x + y + z = 10.
Table 2: Session-level statistics on production, rationing and welfareI
Treatment
Sessions
Statistic
Total Output
Produced
Steady State Equilibrium
Freq. Excess
Output
Labor SupplyII Volatility
Avg.
Points
200.16
0
0
8.625
mean
238.44**
0.20**
0.25**
1.01**
SD
45.11
0.25
0.03
3.38
mean
219.09
0.19**
0.25**
5.04**
SD
54.51
0.24
0.10
1.88
mean
241.68**
0.13*
0.20**
4.26**
SD
43.44
0.22
0.02
1.58
Random vs. Equitable
p-value
0.391
0.943
0.253
0.015
Random vs. Priority
p-value
0.749
0.333
0.037
0.025
Equitable vs. Priority
p-value
0.567
0.389
0.317
0.475
Random
6
Equitable
7
Priority
6
(I) Summary statistics for the following session-level results from all periods of play are presented: total
output produced, frequency of excess labor supply, output volatility, and the average points earned in a
period by subjects for consumption and labor decisions. All variables are adjusted to account for only
submitted decisions in a given round.
(II) All sessions in all treatments exhibit rationing of either output or labor. The asterisks in this column
indicate whether session-level rationing is significantly different from zero.
34
Table 3: Number of periods before convergenceI
No. of Periods Before Convergence
Total No.
Sessions
Med. Labor
Supply
Med. Output
Demand
of
Periods
Random1
Random2
Random3
Random4
Random5
Random6
76
Never
66
1
Never
80
74
Never
Never
62
65
63
79
90
79
69
73
84
Equitable1
Equitable2
Equitable3
Equitable4
Equitable5
Equitable6
Equitable7
Never
38
49
Never
41
Never
61
84
Never
Never
Never
Never
40
26
97
75
70
72
81
79
74
Priority1
Priority2
Priority3
Priority4
Priority5
Priority6
6
9
34
10
3
Never
Never
Never
Never
66
65
Never
77
78
84
100
70
71
(I) In each period t, we compute cmed
− css and hmed
− hss .
t
t
We then
occurs
in the first period in
convergence
claim that
this is pre− hss < 1), and
which cmed
− css < 10 (hmed
t
t
med
c
served
until
the
end
of
the
session.
If
at
period
t
− css >
t
med
period
10 (ht
− hss > 1), we still count it as a converging
if convergence
is restored in the following period, cmed
t+1 − css <
med
10 (ht+1 − hss < 1).
Table 4: Convergence model estimatesI
Treatment
Dependent
Variable
α̂1
α̂2
α̂3
α̂4
α̂5
α̂6
med Nt
2.29
(0.76)
2.75
(0.82)
3.86
(0.94)
1.71
(0.48)
3.61
(1.91)
med Ct
22.41
(9.11)
44.38
(9.00)
27.21
(13.04)
18.77
(8.62)
med Nt
1.73
(0.47)
2.12
(0.58)
3.43
(1.46)
med Ct
17.31
(11.15)
43.12
(10.87)
med Nt
2.87
(0.50)
med Ct
41.73
(12.33)
Random
Equitable
Priority
β̂
Model
Prediction
ρ
2.90
(0.71)
2.34
(0.10)
2.24
0.75
45.56
(9.88)
28.88
(11.20)
33.73
(0.81)
22.37
0.81
2.22
(1.12)
4.49
(0.82)
2.59
(0.59)
3.79
(0.97)
2.13
(0.10)
2.24
0.66
41.75
(11.05)
61.28
(16.90)
41.93
(12.39)
33.24
(7.45)
33.11
(9.61)
32.48
(0.83)
22.37
0.43
2.36
(0.84)
1.58
(0.78)
3.30
(0.58)
4.34
(0.47)
3.60
(1.12)
2.43
(0.07)
2.24
0.47
76.14
(25.85)
23.33
(12.69)
38.83
(15.49)
56.14
(12.95)
37.36
(14.63)
43.03
(1.33)
22.37
0.55
α̂7
(I) Standard errors are in parentheses. The α̂s coefficients capture the initial values for each session; while the β̂ coefficient
captures the value of the variable labor supply (output demand) to which each treatment converges. Following Noussair, Plott,
and Riezman (1995) the standard errors are corrected for heterosedasticity and first order autocorrelation (where ρ is the
correlation parameter).
35
Table 5: Labor supply and output demand decisions by treatmentI
Panel A
Dep.Var. Labor Supply
Wt /Pt
S
Ni,t−1
Bt−1
D
Ci,t
OutputRationedt−1
QuantityOutputRationedt−1
Alt.QuantityOutputRationedt−1
α
N
F
Panel B
Dep.Var. Output Demand
Wt /Pt
D
Ci,t−1
Bt−1
S
Ni,t
OutputRationedt−1
QuantityOutputRationedt−1
Alt.QuantityOutputRationedt−1
α
N
F
Random Queue Rule
(1)
(2)
(3)
0.039***
0.040***
0.041***
(0.01)
(0.01)
(0.01)
0.362***
0.363***
0.387***
(0.04)
(0.04)
(0.04)
-0.000
-0.000
-0.000
(0.00)
(0.00)
(0.00)
-0.004
-0.004
-0.005
(0.00)
(0.00)
(0.00)
-0.432***
(0.08)
-0.008***
(0.00)
-0.016***
(0.00)
1.829***
1.735***
1.694***
(0.17)
(0.16)
(0.17)
3881
3881
3881
61.99
58.45
52.04
Random Queue Rule
(1)
(2)
(3)
0.197**
0.208**
0.145*
(0.08)
(0.08)
(0.08)
0.391***
0.341***
0.402***
(0.03)
(0.04)
(0.03)
0.001
0.001
0.001
(0.00)
(0.00)
(0.00)
-0.673
-0.656
-0.904
(0.63)
(0.64)
(0.61)
7.187***
(1.57)
0.186***
(0.04)
0.199***
(0.07)
21.967***
23.859***
24.143***
(2.18)
(2.12)
(2.09)
3881
3881
3881
49.92
49.55
36.37
(1)
0.068***
(0.01)
0.302***
(0.03)
-0.000
(0.00)
-0.006
(0.00)
-0.200**
(0.09)
Equitable Rule
(2)
(3)
0.066***
0.068***
(0.01)
(0.01)
0.287***
0.341***
(0.03)
(0.04)
-0.000
-0.000
(0.00)
(0.00)
-0.002
-0.006
(0.01)
(0.00)
Priority
(1)
0.063***
(0.01)
0.197***
(0.04)
-0.000**
(0.00)
0.001
(0.00)
-0.204**
(0.09)
-0.009***
(0.00)
1.497***
(0.25)
4668
47.84
(1)
0.097
(0.12)
0.457***
(0.05)
0.000
(0.00)
-0.915
(0.89)
-1.724
(1.18)
1.486***
(0.25)
4668
92.59
-0.003
(0.00)
-0.029***
(0.01)
1.367***
(0.26)
4668
43.68
Equitable Rule
(2)
(3)
0.331***
0.125
(0.11)
(0.12)
0.222***
0.446***
(0.04)
(0.05)
0.000
0.000
(0.00)
(0.00)
-0.297
-0.868
(0.90)
(0.90)
1.543***
(0.16)
4133
27.90
1.475***
(0.16)
4133
29.32
Priority
(1)
0.303***
(0.10)
0.486***
(0.03)
0.002***
(0.00)
0.189
(0.81)
0.035
(1.31)
Rule
(2)
0.340***
(0.10)
0.391***
(0.04)
0.002***
(0.00)
0.383
(0.84)
0.317***
(0.06)
26.708***
(2.82)
4668
27.05
25.822***
(2.82)
4668
32.72
Rule
(2)
0.063***
(0.01)
0.196***
(0.04)
-0.000***
(0.00)
0.002
(0.00)
0.134***
(0.04)
-0.080
(0.11)
26.014***
(2.88)
4668
25.48
21.418***
(2.74)
4133
55.51
22.118***
(2.54)
4133
60.06
D and N S refer to current period output demands and labor
(I) This table presents results from a series of fixed effect panel regressions. Ci,t
i,t
supplies. Banki,t−1 refers to the end-of-period bank account balance in period t − 1. OutputRationedi,t−1 takes the value of 1 if, in the
previous period, participant i received less output than she demanded, and 0 otherwise. QuantityOutputRationedi,t−1 measures the amount
by which the participant was rationed on output in the previous period. AltQuantityOutputRationedi,t−1 measures the difference between
the amount of output a participant was willing to produce and the potentially rationed amount received. Robust standard errors are employed.
*p < 0.10, **p < 0.05, and ***p < 0.01.
36
Table 6: Treatments effects of rationing schemes on labor supply and output demand decisionsI
Wt /Pt
S
Ni,t−1
(1)
0.057***
(0.01)
0.294***
(0.02)
Labor Supply Decisions
(2)
(3)
(4)
0.057***
0.056***
0.056***
(0.01)
(0.01)
(0.01)
0.294***
0.289***
0.289***
(0.02)
(0.02)
(0.02)
(5)
0.054***
(0.01)
0.369***
(0.03)
S
Ni,t
Bt−1
D
Ci,t
-0.000**
(0.00)
-0.003
(0.00)
-0.000**
(0.00)
-0.003
(0.00)
-0.000**
(0.00)
-0.002
(0.00)
-0.000**
(0.00)
-0.002
(0.00)
-0.000*
(0.00)
-0.006*
(0.00)
D
Ci,t−1
OutputRationedt−1
OutputRationedt−1 × EQ
OutputRationedt−1 × RQ
-0.059
(0.08)
-0.227**
(0.10)
-0.379***
(0.12)
OutputRationedt−1 × P R
-0.438***
(0.08)
0.153
(0.11)
37
(5)
0.133*
(0.07)
-0.463
(0.45)
0.001**
(0.00)
-0.463
(0.45)
0.001**
(0.00)
-0.225
(0.46)
0.001**
(0.00)
-0.225
(0.46)
0.001**
(0.00)
-0.885
(0.54)
0.001
(0.00)
0.447***
(0.02)
1.098
(1.45)
-1.735
(1.79)
5.747***
(1.91)
0.447***
(0.02)
6.845***
(1.45)
-7.482***
(1.69)
0.333***
(0.03)
0.333***
(0.03)
0.425***
(0.03)
0.186***
(0.04)
0.028
(0.06)
0.012
(0.04)
0.198***
(0.04)
0.016
(0.06)
-5.747***
(1.91)
0.000
(0.00)
-0.010***
(0.00)
-0.008***
(0.00)
QuantityOutputRationedt−1 × EQ
QuantityOutputRationedt−1 × RQ
QuantityOutputRationedt−1 × P R
-0.008***
(0.00)
-0.002
(0.00)
0.008***
(0.00)
Alt.QuantityOutputRationedt−1
Alt.QuantityOutputRationedt−1 × RQ
Treatments
N
F
Output Demand Decisions
(2)
(3)
(4)
0.197***
0.276***
0.276***
(0.06)
(0.05)
(0.05)
0.379***
(0.12)
QuantityOutputRationedt−1
α
(1)
0.197***
(0.06)
1.607***
(0.12)
All
12682
81.51
1.607***
(0.12)
All
12682
81.51
1.560***
(0.11)
All
12682
120.8
1.560***
(0.11)
All
12682
120.8
-0.012
(0.04)
-0.034***
(0.01)
0.019**
(0.01)
1.516***
(0.16)
EQ, RQ
8549
71.56
23.326***
(1.56)
All
12682
83.16
23.326***
(1.56)
All
12682
83.16
23.961***
(1.50)
All
12682
96.13
23.961***
(1.50)
All
12682
96.13
-0.061
(0.11)
0.254**
(0.13)
25.241***
(1.85)
EQ, RQ
8549
47.20
D and N S refer to current period output demands and labor supplies.Bank
(I) This table presents results from a series of fixed effect panel regressions. Ci,t
i,t−1 refers
i,t
to the end-of-period bank account balance in period t − 1. OutputRationedi,t−1 takes the value of 1 if, in the previous period, participant i received less output
than she demanded, and 0 otherwise. QuantityOutputRationedi,t−1 measures the amount by which the participant was rationed on output in the previous period.
AltQuantityOutputRationedi,t−1 measures the difference between the amount of output a participant was willing to produce and the potentially rationed amount
received In specification (5), PR is omitted from the regression as all observations of Alt.QuantityRationed are equal to zero in that treatment. Robust standard errors
are employed. *p < 0.10, **p < 0.05, and ***p < 0.01.
Figure 1: Cumulative Distribution Functions of Median Labor Supply and Output Demand
Random
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Priority
Equitable
Random
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Steady State
1
2
3
4
5
6
7
8
9
10
Equitable
Priority
Steady State
0
Hours of Work
10 20 30 40 50 60 70 80 90 100
Units of Output
(a) CDF Median Labor Supply
(b) CDF Median Output Demand
Note: These figures display median labor supply and median output demand starting from period 1.
To calculate the medians we excluded observations for which subjects did not submit their decisions
on time (when this was the case the decisions were recorded as zero units of labor requested and zero
units of output requested).
Figure 2: Cumulative Distribution Functions of Individual Labor Supply and Output
Demand
Random
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Priority
Equitable
Random
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Steady State
1
2
3
4
5
6
7
8
9
10
Hours of Work
Equitable
Priority
Steady State
0
10 20 30 40 50 60 70 80 90 100
Units of Output
(a) CDF Individual Labor Supply
(b) CDF Individual Output Demand
Note: These figures display labor supply and median output demand for each subject/period starting
from period 1. We excluded observations for which subjects did not submit their decisions on time
(when this was the case the decisions were recorded as zero units of labor requested and zero units of
output requested).
38
Figure 3: Relative Frequency (First Row: Labor Supply, Second Row: Output Demand,
Third Row: Consumption)
Equitable
Random
0.35
0.35
Steady State
0.3
Priority
0.35
Steady State
0.3
0.25
0.25
0.2
0.2
0.2
0.15
0.15
0.15
0.25
0.1
0.1
0.1
0.05
0.05
0.05
0
Steady State
0.3
0
0
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Labor Supply
Labor Supply
Equitable
Labor Supply
Priority
Random
0.2
0.2
0.2
Steady State
Steady State
Steady State
0.15
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0
0
0
20
40
60
0
80 100
0
Output Demand
0.4
Steady State
40
60
80 100
0
Output Demand
Equitable
Random
0.4
20
0.4
Steady State
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
20
40
60
80 100
Consumption
20
40
60
80 100
Output Demand
Priority
Steady State
0
0
20
40
60
80 100
Consumption
0
20
40
60
80 100
Consumption
Note: These figures display data for each subject/period starting from period 1. We excluded
observations for which subjects did not submit their decisions on time (when this was the case the
decisions were recorded as zero units of labor requested and zero units of output requested).
39
Figure 4: Persistence of Rationing
(a)
Priority
200
400
Equitable
0
Excess output demand in current period
600
Random
0
200
400
600 0
200
400
600 0
200
400
600
Excess output demand in previous period
(b)
Equitable
Priority
30
20
10
0
Excess labor supplied in current period
40
Random
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
Excess labor supplied in previous period
The green 45-degree line denotes observations in which the aggregate quantity of output or labor
rationed remains constant across two periods. Observations above (below) the diagonal line denote
instances of rationing increasing (decreasing) in the next period. The solid red line denotes a local
polynomial smoothed line.
40
Figure 5: Lorenz Curve
Random
Priority
Equitable
Line of Equality
100
Percentage of Points
50
0
−50
−100
−150
−200
−250
−300
−350
0
10
20
30
40
50
60
70
80
90 100
Percentage of Subjects
These are cumulative frequency curves showing the distribution of subjects against their average
points (not including the bank account).
41
Units of Output
60
Hours of Work
2
3
4
80
5
Figure 6: Labor Supply and Output Demand
0
20
1
40
Steady State
1
2
3
4
5
1
Random
2
3
4
5
1
2
Equitable
3
4
5
Steady State
1
Priority
2
3
4
5
1
Random
3
4
5
1
2
Equitable
3
4
5
Priority
(b) Average of Median Output Demand
300
20
Hours of Work
30
Units of Output
400
500
40
600
50
700
(a) Average of Median Labor Supply
2
Steady State
200
10
Steady State
1
2
3
4
Random
5
1
2
3
4
Equitable
5
1
2
3
4
5
1
Priority
2
3
4
Random
(c) Aggregate Labor Supply (Per-Period Average)
5
1
2
3
4
Equitable
5
1
2
3
4
5
Priority
(d) Aggregate Output Demand (Per-Period
Average)
Notes: The x-axis displays i ∈ {1, ..., 5}; i = 1 represents the range from the first period 1 to
T otalP eriod × 0.20 of each session, similarly i = 2 represents the range from the first period
T otalP eriod × 0.20 + 1 to T otalP eriod × 0.40. The bottom and top of the box are the first and
third quartiles; the band inside the box is the second quartile (the median). The upper (lower)
limit of the whisker is the highest (lowest) value within the 1.5 interquartile range of the upper
(lower) quartile. The red dashed line is the steady state value predicted by the theoretical model.
The dots are the outliers. To be consistent with the sessions in which there were fewer than nine
participants, we adjusted the aggregate labor supply (output demand): Adj.AggregateLabor =
AggregateLaborSupply × 9/N (Adj.OutputDemand = AggregateOutputDemand × 9/N ).
42
Figure 7: Average Labor Supply and Output Demand Over Time
Random
Priority
Equitable
Random
Priority
Equitable
50
3.4
3.2
45
3
40
2.8
35
2.6
30
2.4
Steady State
2.2
25
Steady State
2
1
2
3
4
20
5
Period
1
2
3
4
Period
(a) Average Labor Supply
(b) Average Output Demand
Notes: The x-axis displays i ∈ {1, .., 5}; i = 1 represents the range from the first period 1
to T otalP eriod × 0.20 of each session, and i = 2 represents the range from the first period
T otalP eriod × 0.20 + 1 to T otalP eriod × 0.40.
43
5
Appendix A: Theoretical model
The framework that we use to implement our experimental design is based on a representative agent dynamic general equilibrium (DGE) model with sticky prices and monopolistic competition. In order to have a constant fundamental value for the asset that
we introduce, we assume that the economy is subject to no shocks and therefore our
model is not stochastic. However, the framework can be easily extended to include various shocks, in which case it will be a dynamic stochastic general equilibrium (DSGE)
model. The choice of this particular framework is motivated by the fact that DGE (and
DSGE) models are widely used for monetary policy analysis and forecasting among central banks. In this model households optimally choose their consumption of final goods,
labor supply, and savings. Final goods are produced by monoplistically competitive
firms that use labor as their only input. Firms set their prices based on the staggered
pricing mechanism à la Calvo (1983). Finally, the central bank sets the nominal interest
rate in response to fluctuations in inflation. We begin with a description of the model
and provide a characterization of the behavior of households and their optimal decisions.
Then we describe the production and price-setting decisions of firms, and finally we show
how the central bank conducts monetary policy.
Households
Households maximize the present discounted value of their utility associated with consumption and labor as follows:
Ut =
∞
X
β
c1−σ
h1+η
t+j
t+j
−
1−σ 1+η
j
j=0
where
Z
ct =
0
1
θ−1
θ
cit di
!
,
(2)
θ
θ−1
.
They obtain utility from the immediate consumption of a bundle of differentiated varieties, each variety denoted by cit , and disutility from working ht hours. The coefficient
44
of relative risk aversion is represented by σ, the elasticity of labor supply by 1/η, and
the elasticity of substitution between different varieties by θ.
Equation (3) is the household’s budget constraint that equates expenditures to income:
Pt ct + Bt = (1 + it−1 )Bt−1 + Wt ht + Tt .
(3)
Households may purchase a consumption good, ct , at a price, Pt ; save or borrow through
a risk-free nominal bond, Bt , and earn interest-rate income or pay interest on debt,
(1 + it−1 )Bt−1 , on bond holdings; earn wage income from working, Wt ht ; and receive a
transfer, Tt , from the monopolistic firms. The representative household maximizes its
utility stream (2) by making optimal choices on ct , ht , and Bt subject to the budget
constraint (3).
From the household’s first order conditions the following equations are derived:
hηt
Wt
, and
−σ =
Pt
ct
"
β
ct+1
ct
−σ
(4)
#
(1 + it )
= 1.
(1 + πt+1 )
(5)
Equation (4) describes the labor–leisure intratemporal trade-off taking the real wage
as given. Equation (5) represents the intertemporal tradeoff between current and future
consumption in terms of the risk-free bond. Real interest can be defined using the Fisher
equation:
1 + rt =
1 + it
(1 + πt+1 )
.
Firms
Firms possess a linear production function and operate in a monopolistically competitive
environment. They sell differentiated goods, Yi , using labor as the sole input in the
production process:
Yit = Zhit .
45
(6)
Here, hit is the number of hours of work hired by the firms and Z is a productivity
parameter. Firms must decide what price to set for the output. In each period, only a
fraction 1 − ω of the firms are allowed to adjust their prices (Calvo (1983) mechanism).
The prices set by the firms determine the demand for each variety:
1
Yit =
I
Pit
Pt
−θ
Yt ,
where I is the number of firms in the economy. Pt is the aggregate price index and is
defined as
("
1
Pt ≡ I θ−1
I
X
1
#) 1−θ
(Pit )1−θ
.
i=1
The Calvo assumption about price stickiness can be also written as
Pt1−θ = (1 − ω) (Pto )1−θ + ω (Pt−1 )1−θ .
(7)
One important feature about the firms is that output is “made-to-order”, which implies
that all output that is produced has to be consumed. In other words, there are no
inventories and this is why excess resources have to be rationed.
Monetary Policy
The central bank sets the nominal interest rate on bonds according to the following
Taylor rule:
(1 + it )
=
(1 + ρ)
1 + it−1
1+ρ
γ
(1 + πt )δ
1−γ
,
where ρ is the natural nominal interest rate. Also important to notice is that when γ > 0,
the central bank exhibits interest-rate smoothing behavior. Our decision to incorporate
interest-rate smoothing was motivated by a desire to provide stability in policy.
46
Market Clearing
In order to close the model, we need to impose market clearing conditions on asset
markets. Note that in a DGE, the net supply of bonds is zero and the fixed supply of
the shares of the risky asset is normalized to one:
Bt = 0, and Xt = 1.
We also impose that total demand for output is financed by income from output production:
C t = Yt .
Steady State Equilibrium
To derive the steady state equilibrium, one can start by solving the cost minimization
problem for firm i (equation (6)):
min
hit
Wt
Pt
hit − ϕt (Yit − Zhit )
where ϕt is the firm’s real marginal cost. From the first order condition, the following
equation is obtained:
mct ≡ ϕt =
Wt 1
.
Pt Z
It can be shown that the ratio of the price set by the firms when they are able to update
it, Pto , relative to the aggregate price index, Pt , is
P∞
i i 1−σ
i=1 ω β ct+i mct+i
Pto
θ
=
Pt
θ − 1 P∞
i=1
ω i β i c1−σ
t+i
47
Pt+i
Pt
Pt+i
Pt
θ
θ−1
≡
θ S1t
,
θ − 1 S2t
where
S1t = c1−σ
mct + βωEt S1t+1 , and
t
+ βωEt S2t+1 .
S2t = c1−σ
t
With no shocks in the economy, S1t = S1t+1 and S2t = S2t+1 . This implies that
Pto
θ
=
mct .
Pt
θ−1
(8)
Thus, in the steady state,
W
θ−1
=
Z.
P
θ
Using the market clearing conditions and equation (4), the steady state values for labor
and consumption are obtained:
SS
h
cSS
1
θ − 1 1−σ η+σ
=
Z
, and
θ
1
θ − 1 1−σ η+σ
= Z
Z
.
θ
From Equation (5) the steady state for the interest rate can be obtained:
iSS =
1
− 1.
β
Derivation of the inflation equation
Combining equations (4), (7), and (8), the following equation for inflation is obtained:
1
πt = 1 − (1 − ω)
ω 1−θ
θ
θ−1
48
hηt cσ
t
Z
1
1−θ 1−θ
− 1.
(9)
Linearizing (9), by using a first order Taylor approximation around the steady state,
linearized inflation is obtained:
πt = 1 + γ c (cmed
− cSS ) + γ h (hmed
− hSS ),
t
t
where
σ−1 SS η
1
1
σθ(ω − 1)ω 1−θ Ψ,
h
cSS
(1 − θ) Z
σ SS η−1
1
1
=
cSS
h
ηθ(ω − 1)ω 1−θ Ψ,
(1 − θ) Z
γc =
γh
and

Ψ = 1 − (1 − ω)
c
SS σ
SS η
h
θ
(θ − 1) Z
1
!1−θ −(1+ 1−θ
)

49
σ SS η !−θ
cSS
h
θ
.
(θ − 1) Z
Appendix B: Parametrization
We choose parameter values for our environment based on two considerations. First,
we aimed to be close to U.S. quarterly data. Second, our aim was ensure a sufficiently
interior steady state and steep expected payoff hills to clearly observe actively chosen
deviations from equilibrium behavior (these parameters are presented in Table 7). We
set the discount factor (framed in our environment as the probability of continuation of
the sequence) β equal to 0.965. This implies that a particular sequence of periods would
last for an average of 28 periods. The parameter of risk aversion, σ, is calibrated to be
0.33, while the labor supply parameter, η, is set to 1.5. The elasticity of substitution
between varieties, θ, is 15, implying a markup of 7% over marginal cost. The Calvo
parameter, ω, is 0.9, implying that 10% of firms have the ability to update their prices
each period. The interest-rate smoothing parameter used by the central bank is 0.5,
while the Taylor rule parameter, indicating how responsive the nominal interest rate is
to inflation, is δ = 1.5. The per-period dividend paid on assets is 0.035, which means
that the asset is worth 1 after an average of 28 periods. We set a fixed fundamental
value to minimize subjects’ confusion. Each firm produces Z = 10 units of output with
1 unit of labor. In the steady state, the selected calibration implies steady state levels of
individual consumption and labor of 22.4 and 2.24 units, respectively. The steady state
real wage is set to 9.35, and the steady state nominal rate of return is 0.036.
50
Table 7: Parameters and Steady State Values
Parameter
Parameter Description
Z
Productivity level
1−ω
Fraction of firms updating
δ
Inflation target of the central bank
γ
Interest smoothing parameter
θ
Measure of substitutability
β
Rate of discounting
ρ
Natural nominal rate of return
1/σ
Elasticity of intertemporal substitution
1/η
Frisch labor supply elasticity
∗
µ
Steady state markup (θ/(θ − 1))
C∗
Steady state consumption
∗
N
Steady state labor
∗
W
Steady state nominal wage
P∗
Steady state output price
51
Value
10
0.1
1.5
0.5
15
0.965
0.0363
3.03
0.67
1.07
22.37
2.237
10
1.07
Appendix C: Aggregate Variables
Figure 8: Median Labor Supply per Session
Equitable 1
Random 1
Priority 1
10
10
10
8
8
8
6
6
6
4
4
4
2
2
0
0
20
40
0
60
2
0
20
Period
40
0
60
10
8
8
8
6
6
6
4
4
4
2
2
40
60
0
80
2
0
20
Period
40
0
60
8
8
8
6
6
6
4
4
4
2
2
0
60
2
0
20
Period
40
0
60
8
8
8
6
6
6
4
4
4
2
2
40
0
60
20
40
0
60
0
Equitable 5
Random 5
10
8
8
8
6
6
6
4
4
4
2
2
2
0
0
60
0
20
Period
40
60
0
80
20
Equitable 6
10
8
8
8
6
6
6
4
4
4
2
2
2
0
0
60
80
0
20
40
60
0
0
40
20
40
Period
Period
Equitable 7
10
8
6
4
2
0
0
20
40
60
Period
Med. Labor Supply
1st-3rd quartiles Labor Supply
1
100
60
Priority 6
10
Period
80
Period
10
40
0
Period
Random 6
60
Priority 5
10
20
40
Period
10
0
20
Period
40
80
2
0
Period
20
60
Priority 4
10
0
40
Period
10
20
20
Equitable 4
Random 4
0
0
Period
10
0
60
Priority 3
10
40
40
Period
10
20
20
Equitable 3
Random 3
0
0
Period
10
0
60
Priority 2
10
20
40
Equitable 2
Random 2
0
20
Period
10
0
0
Period
Begin Sequence
Steady State
60
Figure 9: Median Output Demand per Session
Equitable 1
Random 1
Priority 1
100
100
100
80
80
80
60
60
60
40
40
40
20
20
0
0
20
40
0
60
20
0
20
Period
40
0
60
100
80
80
80
60
60
60
40
40
40
20
20
40
60
0
80
20
0
20
Period
40
0
60
80
80
80
60
60
60
40
40
40
20
20
0
60
20
0
20
Period
40
0
60
80
80
80
60
60
60
40
40
40
20
20
20
0
0
40
60
0
20
Period
40
0
60
0
80
80
80
60
60
60
40
40
40
20
20
20
0
0
40
60
0
20
40
60
0
80
20
Equitable 6
100
80
80
80
60
60
60
40
40
40
20
20
60
80
0
40
20
40
60
0
0
20
40
Period
Period
Equitable 7
100
80
60
40
20
0
20
40
60
Period
Med. Labor Supply
60
20
0
Period
0
100
Priority 6
100
40
80
Period
100
20
0
Period
Random 6
60
Priority 5
100
0
40
Equitable 5
100
Period
80
Period
100
0
20
Period
Random 5
60
Priority 4
100
20
40
Period
100
0
20
Equitable 4
Random 4
20
0
Period
100
0
60
Priority 3
100
40
40
Period
100
20
20
Equitable 3
Random 3
0
0
Period
100
0
60
Priority 2
100
20
40
Equitable 2
Random 2
0
20
Period
100
0
0
Period
1st-3rd quartiles Labor Supply
1
Begin Sequence
Steady State
60
Appendix D: Computer Interfaces
Figure 10: Main screen
54
Figure 11: Personal history screen
55
Figure 12: Market history screen
56
Appendix E: Instructions
The instructions distributed to subjects in all the treatments (Random, Equitable, and
Priority) are reproduced on the following pages. Subjects received identical instructions
with the exception of the tables in Appendix F.
57
INTRODUCTION
You are participating in an economics experiment at the University of British Columbia. The purpose of this
experiment is to analyze decision making in experimental markets. If you read these instructions carefully and
make appropriate decisions, you may earn a considerable amount of money. At the end of the experiment all
the money you earned will be immediately paid out in cash.
Each participant is paid 5 CAD for attending. During the experiment your income will not be calculated in
dollars, but in points. All points earned throughout this game will be converted into CAD by applying the
exchange rates found on the whiteboard.
During the experiment you are not allowed to communicate with any other participant. If you have any
questions, the experimenter(s) will be glad to answer them. If you do not follow these instructions you will be
excluded from the experiment and deprived of all payments aside from the minimum payment of 5 CAD for
attending.
You will play the role of a household over a sequence of several periods (trading days). You will be interacting
with other human consumers. There will be also computerized firms and a central bank operating in this
experimental economy.
In this experiment, you will have the opportunity to work and purchase output in two markets. All transactions
in all markets will be conducted using laboratory money.
OVERVIEW
The objective of each player is to make as many points as possible. You will receive points for purchasing more
units of output in your bank account. You will lose points by working. You may borrow and save at the current
interest rate.
LABOR & OUTPUT MARKETS
At the top of the screen you’ll see a graph representing the different combinations of output (x-axis) and labor
(y-axis) you can choose. Each of the different combinations defines:


A current hourly wage
A current price for a single unit of output
This information will be located on the right hand side of the graph. Notice that these 2 pieces of information
are only potential outcomes. The actual outcomes will be computed based on everyone’s actual choices.
You may agree to trade none, some or all of your labor hours to firms in exchange for potential wage. You will
input the very maximum you would like to work. You may end up working less than your desired amount, but
you will never work more than that. You are able to work a maximum of 10 hours per period and may also
work fractions of an hour, up to 1 decimal place. eg. 4.3 or 7.2 hours. Each worker is able to produce 10 units
per hour and this will never change. Wage income will be deposited from your bank account.
You may also choose to purchase output. You will input the very maximum you would like to purchase. You
may end up purchasing less than your desired amount. Spending on output will be debited from your bank
account. You will also receive a dividend from firms that will also help you to pay for the varieties you will
purchase. This is an equal share of the positive or negative profits the firms earned in the current period.
To better understand how your labor and consumption decisions translate into points and how the balance on
your bank account changes, you will have the opportunity to move the red dot to your preferred point on the
payoff space. Notice that as you increase the amount of labor, you will lose points at an increasing rate. As you
increase the amount of output, you will gain points at a decreasing rate.
Actual wage, output price and the interest rate will be computed based on your choices and everyone else’s
choices. That’s why you will be able to move around 2 different dots, the red one that represents your own
decisions and the green one that characterizes the average of everyone else’s choices. This way you will
visualize different predictions on wages and prices for different combinations of aggregate consumption and
aggregate labor.
** You will have an initial balance of 10 experimental units of money on your bank account. Whenever your
bank account is negative, ie. you spent more than you earned, you will owe the bank the remainder PLUS
interest in the next period. So long as you pay the interest on your debt, you may continue to borrow. Any
money owing at the end of the experiment will be repaid through points. In particular, you will lose:
(
you will gain:
) . Similarly, If your bank account has a positive balance at the end of the experiment
(
)
.
**If your bank account is positive, you will receive interest on the saving in your bank account. This will be
credited to your account in the next period.
After all subjects submit their labor, consumption, and investment decisions, firms will decide how many
hours to hire. Wage and output price will be computed. There will be no unsold output. If the total number of
labor hours supplied in the economy is in excess of what is necessary to satisfy consumers’ output demands,
firms will hire fewer hours and you may find yourself working a fraction of the hours you requested. Similarly,
if the worker supplied hours is insufficient to cover consumer demand, you may find yourself able to purchase
only a fraction of the output you requested.
As you purchase more units, you will gain more points but at a decreasing rate. As you work more hours, you
will lose more points at an increasing rate. You do NOT obtain points from your holdings of cash.
Worker Points = (Points Gained from Consuming – Points Lost from Working)
The interest rate at which you spend or save will depend on inflation. Particularly, for every 1% that prices
increase from yesterday, the automated central bank will increase the borrowing and saving rate by more
than 1%. Over the long run, the central bank will aim to keep the interest rate around 3.5%, but it will
fluctuate as inflation on output occurs. Lower interest rates make it cheaper to borrow but more challenging
to accumulate savings, and vice versa.
Notice that interest rate might also be negative. In that case you will lose money by saving and gain money by
borrowing.
Each sequence will have a random number of periods determined by a continuation rate of 0.965. That is,
there is a 3.5% chance of a period ending at any period. To make the termination rule as transparent as
possible, the experimenter will carry a bag containing 200 marbles, 193 of them are blue and only 7 of them
are green. Each period a marble will be drawn. If a blue marble is drawn the sequence will end, otherwise the
sequence will continue. You will play multiple sequences. On average you will play 28 periods in each
sequence.
Screens
Throughout the experiment you will have a chance to flip back and forth between 4 different screens:
1) Action Screen. - This is the main screen. This screen is divided in two:
a) On the left hand side of the screen you’ll find a graph that represents all possible combinations of labor
and output. On the graph you’ll see two different dots. The red one represents your own choices. By
moving around the red dot you will be able to visualize the points you might earn by selecting different
combinations of labor and output.
The green dot denotes the average values of output and labor of the rest of the participants. By
moving around both dots you’ll have a better sense on how your choices as well as everyone else’s
decisions affect the potential wage and output price of the economy. Your predicted banking account
balance (without interest rate) will be also displayed.
Notice that by positioning the dots together you will be assuming that everyone else’s choices are the
same as yours.
b) On the right hand side of the screen (SUBMIT YOUR DECISIONS) you will have to enter your final
choices on output and labor. Immediately after everyone submits their decisions, the total amount of
output and labor will be computed.
2) Personal History.- You will find a summary of your previous decisions on consumption, labor, as well as the
points you earned and your bank account balance.
3) Market History. - On this screen you will be able to observe information on interest rates and inflation rate
from previous periods. Information on total output and labor is also included.
Some useful Information
(
)
(
)
Appendix F: Specific Examples of the Rationing Rules
In addition to the instructions in Appendix E we showed subjects specific examples of
how output and labor were allocated when there was rationing. The first table shows
an example of rationing under excess output demand and the second table shows an
example of rationing under excess labor supply for the Random, Equitable, and Priority
treatments, respectively.
62