10ISSN 1183-1057

SIMON FRASER UNIVERSITY

Department of Economics

Working Papers

15-02

“Distributing scarce jobs and output: Experimental

evidence on the effects of rationing”

Guidon Fenig & Luba

Petersen

March 2015

Economics

Distributing scarce jobs and output: Experimental

evidence on the effects of rationing

Guidon Fenig∗

University of British Columbia

Luba Petersen†

Simon Fraser University

March 10, 2015

Abstract

How does the allocation of scarce jobs and production influence their supply? Wepresent the results of a macroeconomics laboratory experiment that investigatesthe effects of alternative rationing schemes on economic stability. Participantsplay the role of consumer-workers who interact in labor and output markets. Alloutput, which yields a reward to participants, must be produced through costlylabor. Automated firms hire workers to produce output so long as there is suf-ficient demand for all production. Thus, either labor hours or output units arerationed. Random queue, equitable, and priority (i.e., property rights) schemes arecompared. Production volatility is the lowest under a priority rationing rule andis significantly higher under a scheme that allocates the scarce resource through arandom queue. Production converges toward the high steady state under a priorityrule, but can diverge to significantly low levels under a random queue or equitablerule where there is the opportunity for and perception of free-riding. At the indi-vidual level, rationing in the output market leads consumer-workers to supply lesslabor in subsequent periods. A model of myopic decision making is developed torationalize the results.

JEL classifications: C92, E13, H31, H4, E62Keywords: Rationing, allocation rules, unemployment, experimental macroe-conomics, 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 Man-agement and the Social Sciences and Humanities Research Council of Canada for generous financialsupport.

1

1 Introduction

In a world that faces rationing due to the constant flux between periods of excess supply

of labor and demand for output, does it matter how scarce jobs and goods are allocated?

Can certain allocation schemes bring about greater welfare and economic stability? Rep-

resentative 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 alternative mechanisms to ration scarce re-

sources has had varying degrees of success. To deal with a very low supply of organ

donations, the Israeli government implemented a policy in 2008 to give priority on or-

gan 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 Amer-

ica 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 during the First

and Second World War to alleviate demand on rationed food supplies. The gardens

provided households an opportunity to supplement their weekly rations with their own

produce. 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... in-

creased 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.

Expectations that others would withdraw their deposits would cause depositors to panic

1

and run on the bank, leading to a very fragile banking system. Large price cuts on

Black Friday in the U.S. or Boxing Day in Canada often result in consumers’ waiting

long hours in line and 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 par-

tial equilibrium settings as the result of efficiency wages (Stiglitz, 1976; Solow, 1980;

Shapiro and Stiglitz, 1984), gift-exchange (Akerlof, 1984), search costs and turnover

costs (Salop, 1979; Akerlof 1984) or in general equilibrium environments as a conse-

quence 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 understand-

ing of how macroeconomic dynamics evolve 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 on the implementation of the rationing scheme and without considerable ex-

ternal consequences. Moreover, participants’ heterogeneous preferences and reactions to

rationing provide us a more natural and richer understanding of the implications of alter-

native 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,

2

rationing occurs when either households are unwilling to purchase all the output they

wish to produce, or they are prefer to consume more than they are willing to work to

produce. Rationing occurs as a consequence of aggregate household decisions, an inabil-

ity for prices to adjust fully, and a lack of inventories. An optimal consumption–leisure

tradeoff condition shows that as individuals’ expected consumption falls, their willing-

ness 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-manipuable schemes: a

random queue where the rationed market is distributed on a first-come, first-serve basis,

an equitable allocation scheme, and a priority scheme where those willing to buy what

they produce (or produce what they demand) receive priority on scarce labor hours

(output).

Our contribution is to provide the first causal evidence of the implications of ra-

tioning rules on the availability of scarce labor opportunities and output. We observe

rationing of jobs and output in all of our experiments and 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-though sometimes permanently-aggregate labor sup-

ply will collapse to low levels due to output rationing. Without any guaranteed rights

to the output that they produce, participants who face output rationing according to a

random queue experience vastly higher inequality in terms of welfare and are generally

unwilling to supply high levels of labor. These rationed individuals more aggressively

decrease their labor supply in response to rationing, leading to significantly greater out-

put volatility than under a random queue. We attribute the adverse reaction to output

rationing under the random queue and equitable allocation schemes to a combination

of myopic spending and perceived free-riding of others. We develop a model of myopic

3

decision-making where agents focus only on maximizing current utility and pessimisti-

cally 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.

2 Rationing in Theory and Experiments

Our experimental analysis is most directly inspired by substantial theoretical work that

introduced 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 spawned nearly two decades and aimed to understand the necessary and

sufficient conditions by which general equilibrium environments can persistently exist

out of equilibrium.1 While many different notions of rationing have been considered, we

develop an environment and set of rationing schemes most closely related to Svensson

(1980). Svensson, building off of earlier work by Gale (1979) and Futia (1975), devel-

oped 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 Dreze (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 equilib-

rium model of macroeconomic rationing that side-steps the disequilibrium approach by

employing a matching function that governs the probability of trade and impose a cost

1For an excellent survey of the macroeconomic disequilibrium theory literature, see Drazen (1980).2An 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 subjectto her budget constraint. In forming the demand for a good, the agent disregards quantity rationing forthe particular good, but optimize as though all other demands for goods in their consumption set havefaced rationing. Gale (1979) and Svensson (1980) explore the existence of disequilibria under stochasticmanipulable and non-manipulable schemes.

4

of matching on buyers. As in 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 dis-

tribution rules on decision making in a macroeconomic setting.

We begin by developing a laboratory production economy where 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); Bosch-Domenech

and Silvestre (1997); Petersen (2015)), exogenous shocks (Noussair et al. (2014, 2015)),

and asset price stabilization policies (Fenig, Mileva, and Petersen (2014)). 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. (2014), by systematically investigating how rationing

schemes influence decision-making and economic dynamics.

Two experimental papers are closely related to ours in terms of themes and environ-

ment. Lefebvre (2013) designed a common-pool resource game to compare four rationing

rules in their ability to coordinate agents to optimal levels of self-insurance, efficiency,

and reliability: proportional, constrained equal awards, constrained equal losses, and

no-allocation rules. Under the Nash equilibrium predictions, the rationing rule should

not influence aggregate usage of the common resource or self-insurance. Lefebvre found

that no-allocation and constrained equal awards rules lead to more efficient coordina-

tion. 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 suboptimal investment in alternative safe resources. Lefebvre

argued that the success of the constrained equal awards rule can be attributed to its

ability to fully allocate the resource and the fact that it reduces the strategic interaction

5

among agents. While the experiment yields valuable insights into the effects of alter-

native rationing schemes, the environment studied does not allow for the endogenous

creation of scarce resources or for rationing to influence the production of resources.

Huber et al. (2013) design and conduct a dynamic general equilibrium experiment

to study alternative approaches to financing public goods. Subjects play the role of

producer-consumers who individually allocate their private goods between consumption

and investment in production. A depreciating public good is financed either by anony-

mous voluntary contributions or taxation. The authors observe that the sustainability of

the public good is considerably higher when financing comes from taxation. By contrast,

financing through anonymous voluntary contributions leads to a rapid decline in provi-

sion. In a democracy treatment, subjects are willing to vote for taxation to ensure public

provision. Their environment is similar to ours in that subjects’ consumption-production

tradeoff decisions have important consequences for the stock of scarce resources. More-

over, their findings yield important insight into what should be expected under equitable

distribution schemes.

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 a first-come,

first-serve scheme. Buckley, Cuff, Hurley, McLeod, Nuscheler, and Cameron (2012)

investigated the willingness to pay for private health insurance under different public

sector health-care allocation rules. They observed that the willingness to pay for private

insurance is significantly higher when the 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.

6

3 Experimental Design and Implementation

The experimental design and implementation extend the baseline macroeconomy devel-

oped in Fenig et al. (2014) by considering alternative rationing schemes. The environ-

ment we consider is a dynamic general equilibrium economy with nominal rigidities and

monopolistic competition. We provide a summary a fully derived model and parameter-

ization in the Appendix A.

3.1 Experimental Economy

Participants were assigned the role of households that made decisions about how much

to work and consume over a number of temporally linked periods. Each period, they

gained points by buying (and automatically consuming) 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:

Points = 1.51c0.66t − 0.4h2.5

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. They also received an equal share of

the firms’ profits, Πt. Thus, each participant faced a per-period budget constraint given

by: Ptct + Bt = Wtht + 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 + ρ

)0.5 [(1 + πt)

1.5]0.5

,

7

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).

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 supply of labor (HSt =

∑nj=1 h

Sj,t) and demand for output (CD

t =∑n

j=1 cDj,t)

were calculated and used to determine the level of labor demand and production. If there

was more labor supply than necessary to produce the total amount of output 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. Firms faced a constant probability 1 − ω = 0.1 of being able to update

their prices. 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 aggre-

gate outcomes. Specifically, prices were determined by the evolution of inflation:

Πt = 1 + 0.0016(cmedt − cSS) + 0.0744(hmedt − hSS).

The nominal wage and the output price were then calculated using median realized labor

supply and output consumed as

Pt = Pt−1Πt, and

Wt = Pt−1Πt

(hmedt

)1.5 (cmedt

)0.33.

Thus, wages and prices were unable to adjust fully to accommodate excess aggregate

8

labor supply or output demand.

To induce exponential discounting within an infinite horizon environment, we gener-

ated 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. When a sequence ended,

participants would have either a positive or 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 her debt, and their points would be deducted accordingly. To make this

salient, we provided participants with a hypothetical adjusted score assuming that the

previous period was the last period of a sequence.

3.2 Non-Rationed Equilibrium Predictions

From the economy-wide resource constraint and the production function we have that

Ct = Yt = 10Ht, where Ct =∑n

i=1 ci,t, Yt ≡∑n

i=1 Yi,t, and Ht ≡∑n

i=1 hi,t; n is the

number of households. We can obtain a non-rationed symmetric steady state equilibrium

in which individual-level output demand and labor supply are cSS = 22.37 and hSS =

2.24, respectively.

In the symmetric steady state, the household’s optimal consumption–leisure tradeoff

condition is given by:

h1.5i,t = c−0.33

i,t

Wt

Pt. (1)

The household optimally trades off consumption between periods t and t+1 according

to the following Euler equation:

0.965

[(ct+1

ct

)−0.33(1 + it)

(1 + πt)

]= 1. (2)

9

3.3 Rationing Rules

Note that in the above model markets are assumed to always clear, and thus there is no

concept of rationing. Theoretically, if subjects are not myopic, in the sense that they

do not make decisions based only on the current period, the equilibrium predictions will

not be affected by the choice of a specific rationing rule. In this paper, we explicitly

test this hypothesis by contrasting the economic stability under three different rationing

rules. We focus on rationing rules for which realized outcomes for an individual i are

a random function of her own effective supplies and demands, as well as the aggregate

effective supply and demand on the market. The rules have the the following features:

no individual is forced to trade more than she likes, trades are feasible, and only the

market with excess supply/demand is rationed.

In our environment, there are n households that simultaneously submit their desired

labor supply and output demand. Given aggregate output demand (CDt ) and labor

supply (HSt ), actual consumption (ci,t) and labor (hi,t) will be as follows.

1. If CDt = HS

t , neither output nor labor was rationed, ci,t = cDi,t and hi,t = hSi,t.

2. If CDt > HS

t , subjects obtained the hours of work they requested, hi,t = hSi,t, and

output was rationed, ci,t = min(θRci,t, c

Di,t

).

3. If CDt < HS

t , subjects obtained the output they requested, ci,t = cDi,t, and labor was

rationed, hi,t = min(θRhi,t, h

Si,t

).

Here, θRc (θRh ) is the individual consumption (hours) when output (labor) is rationed

according to a specific rule R ∈ {Random,Equitable, Priority}. Notice that households

do not obtain more than their desired consumption/labor. Below we describe in detail

the rationing rules we consider for our experiments. One feature that all rules have in

common is that they are efficient, in the sense that the aggregate output produced is

consumed by all agents and the total hours of work hired are allocated among them. 1.

Random Queue (Random):Under this rationing rule in each period households are

assigned a position in a queue. Households at the front of the queue have priority for

the scarce hours or output and positions are randomized in each period. When there is

10

excess output demand, expected consumption is given by

E(ci,t) =1

nmin

{ZHS

t , cDi,t

}︸ ︷︷ ︸

First Position

+1

nmin

{max

{ZHS

t −1∑

q=1

[cj,t]q , 0

}, cDi,t

}︸ ︷︷ ︸

Second Position

+

. . .+1

nmin

{max

{ZHS

t −n−1∑q=1

[cj,t]q , 0

}, cDi,t

}︸ ︷︷ ︸

Last Position

,

while in instances of excess labor supply, expected labor is:

E(hi,t) =1

nmin

{CD

t

Z, hSi,t

}︸ ︷︷ ︸

First Position

+1

nmin

{max

{CD

t

Z−

1∑q=1

[hj,t]q , 0

}, hSi,t

}︸ ︷︷ ︸

Second Position

+

. . .+1

nmin

{max

{CD

t

Z−

n−1∑q=1

[hj,t]q , 0

}, hSi,t

}︸ ︷︷ ︸

Last Position

,

where [cj,t]q and [hj,t]q denote consumption and hours of work of agent j 6= i, respec-

tively, and q ∈ {1, n} is the position in the queue.

2. Equitable Rule (Equitable): The second rationing rule we consider involves

all households equally sharing the rationed hours or output up to their desired de-

mand. Households obtain ci = min{ZHS

n, cDi

}when there is excess output demand, and

hi = min{CD/Zn

, hSi

}when there is excess labor supply. Any undesired hours of work

or units of output are allocated in equal shares among those with excess demands for

labor hours or output.

3. Priority Rule (Priority): Finally, we consider a rationing rule that gives house-

holds priority to purchase the output they personally produced: ci = min{ZhSi , cDi }.

Similarly, if labor hours are rationed, participants are given priority to work the hours

associated with their purchased output, hi = min{cDi /Z, hsi}. Any undesired hours of

work or units of output are randomly allocated among those with excess demands for

labor hours or output.

Table 1 shows 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

11

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.

Table 1: Rationing Rules Example

Subject hSi cDi hici

RandomI UT PriorityII

1 2 100 2 100 30 20+x2 5 80 5 20 30 50+y3 1 50 1 0 30 10+z4 4 30 4 0 30 30

Aggregate 12 260 12 120 120 120

(I) Here it is assumed that subject i ∈ /1, 2, 3, 4/ has theith position in the queue.(II) x+ y + z = 10.

The experiment was conducted at Simon Fraser University at the CRABE Labora-

tory. Subjects were undergraduate participants recruited from a wide variety of disci-

plines. We conducted six sessions of the Random and Priority treatments and seven

of the Equitable treatment. Each session consisted of eight or nine inexperienced par-

ticipants and consisted of only one treatment. At the beginning of each session we

conducted a 35-minute instruction phase that involved a discussion of the game, the

rationing rule, and four periods of guided practice. To facilitate learning, we provided

participants with a highly visual interface in which they could move markers represent-

ing their hypothetical decisions and the median subject’s decision to learn how their

points, bank account, and the market wage and price would adjust.3 Payoffs, including

a $7 show-up fee, ranged from $10 to $38.

4 Aggregate Findings

In this section, we summarize our findings across treatments. Our analysis focuses on

participants who have at least 20 rounds of experience to allow participants ample time

3Screenshots of the computer interface can be found in Figures 9- 11 of the Appendix.

12

to learn the decision task. The data from the experienced sequences are treated as one

time series, unless otherwise noted.

4.1 Production

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 labeled SS refers to 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.

Figure 1: Cumulative Distribution Functions of Median Labor Supply and Output De-mand

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Steady State

Hours of Work

Random Equitable Priority

1

(a) CDF Median Labor Supply

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Steady State

Units of Output

Random Equitable Priority

1

(b) CDF Median Output Demand

Note: Decisions of experienced participants.

Mean labor supply under the Priority treatment is 2.71 hours (s.d.=1.68), while it

is modestly higher in the Random treatment, with participants supplying an average

of 2.97 hours (s.d.=2.21). By contrast, mean labor supply is lower in the Equitable

with 2.36 hours (s.d. 1.69). Two-sided Wilcoxon rank-sum tests are unable to reject

the null hypotheses that session-level mean labor supplies are identical across treatments

(p > 0.31 in all pairwise comparisons). However, when we pool labor supply data from all

periods and participants together by treatment, a two-sample Kolmogorov-Smirnov test

13

Figure 2: Cumulative Distribution Functions of Individual Labor Supply and OutputDemand

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Steady State

Hours of Work

Random Equitable Priority

1

(a) CDF Individual Labor Supply

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Steady State

Units of Output

Random Equitable Priority

1

(b) CDF Individual Output Demand

Note: Decisions of experienced participants.

rejects the null hypotheses that the distributions are identical (p = 0.000 in all cases).

Indeed, labor supply in the Priority treatment is heterogeneous but largely centered

around the steady state. By contrast, labor supply decisions in the Random treatment

exhibit greater heterogeneity and a relatively more bipolar distribution. Participants

under a random queue have a tendency to either work very little or work a lot. In

the Equitable treatment, we observe considerably lower labor supplies across the entire

distribution, and especially for possible hours below the steady state.

14

Figure 3: Relative Frequency (First Row: Labor Supply, Second Row: Output Demand,Third Row: Consumption)

0 10 20 30 40 50 60 70 80 90100

0

0.05

0.1

0.15

0.2Steady State

Output Demand

Random

0 1 2 3 4 5 6 7 8 9 10

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Steady State

Labor Supply

Random

0 10 20 30 40 50 60 70 80 90100

0

0.05

0.1

0.15

0.2Steady State

Output Demand

Equitable

0 1 2 3 4 5 6 7 8 9 10

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Steady State

Labor Supply

Equitable

0 10 20 30 40 50 60 70 80 90100

0

0.05

0.1

0.15

0.2Steady State

Output Demand

Priority

0 1 2 3 4 5 6 7 8 9 10

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Steady State

Labor Supply

Priority

0 10 20 30 40 50 60 70 80 90100

0

0.1

0.2

0.3

0.4Steady State

Consumption

Random

0 10 20 30 40 50 60 70 80 90100

0

0.1

0.2

0.3

0.4Steady State

Consumption

Equitable

0 10 20 30 40 50 60 70 80 90100

0

0.1

0.2

0.3

0.4Steady State

Consumption

Priority

1

15

Table 2: Session-Level Statistics on Production, Prices, and PointsI

Treatment Sessions StatisticTotal Output Freq. Excess Output Avg.

Produced Output Demand Volatility Points

Random 6mean 238.44 0.79 0.248 0.914

s.d. 45.11 0.32 0.039 4.349

Equitable 7mean 219.09 0.88 0.242 5.815

s.d. 54.51 0.20 0.112 1.798

Priority 6mean 241.68 0.87 0.184 5.149

s.d. 43.44 0.22 0.032 2.635

Random vs.Equitable p-value 0.391 0.277 0.317 0.010

Random vs. Priority p-value 0.749 0.319 0.025 0.025

Equitable vs. Priority p-value 0.568 0.641 0.116 0.567

(I) Session-level results for experimented subjects are presented: total labor hired, gross inflation rate, theaverage points earned in a period by subjects for consumption and labor decisions, and the average pointstaking into consideration their bank account balances. Total labor is adjusted for the number of subjectsparticipating.

Output demand differs more clearly across treatments. We observe the highest aver-

age demands under the Priority treatment (mean of 48.69, s.d.=30.28), followed by the

Equitable treatment (mean of 44.50, s.d.=29.94) and the Random treatment (mean of

39.00, s.d.=27.81). Median demands follow a similar ordering. While the session-level

mean output demands are not significantly different between the Random and Equitable

(p = 0.198) or the Equitable and Priority (p = 0.475), the differences are significant be-

tween the Random and Priority (p = 0.0782). As with the labor supply decisions,

Kolmogorov-Smirnov tests reject that the pooled distributions of demand decisions are

identical across treatments (p = 0.000 in all cases). 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 Equitable than Priority 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 100 units.

The differences in labor supply and output demand do not translate into significantly

different levels of mean production across treatments. Table 2 presents the session-level

16

summary statistics with two-sided Wilcoxon rank-sum tests provided to denote statis-

tical differences between treatments. Mean total output produced is the lowest in the

Equitable treatment at 219.09 units (s.d.=54.51), and is relatively higher in the Random

with 238.44 units (s.d.=45.11) and Priority with 241.68 (s.d.=43.44). The treatment

differences in mean production at the session level are not statistically significant, with

p > 0.391 in all pairwise comparisons. We observe high frequencies of output rationing

in all treatments, occurring on average between 79 and 88% of the time. Rationing of

labor hours occurs minimally in five of six sessions of the Random (the exception is in

Random2 where labor rationing never occurs), in four of seven Equitable sessions (Eq-

uitable1, Equitable3, Equitable5, and Equitable7), and in only two of six session of the

Priority (Priority4 and Priority6).

Output volatility differs significantly across treatments. The lowest levels of volatility

are in the Priority treatment (mean=0.184, s.d.=0.032), and volatility increases modestly

in the Equitable (mean=0.242, s.d.=0.112, p = 0.116) and quite large and significantly

in the Random (mean=0.248, s.d.=0.039, p = 0.025).

Figure 4: Lorenz Curve

0 10 20 30 40 50 60 70 80 90 100−350

−300

−250

−200

−150

−100

−50

0

50

100

Percentage of Subjects

PercentageofPoints

Random Equitable Priority Line of Equality

1

These are cumulative frequency curves showing the distribution of subjects against their averagepoints (not including the bank account).

17

In terms of average points earned, there is no statistical difference between subjects

participating in Equitable and Priority treatments. Participants in the Random treat-

ment, however, receive significantly less output and earn significantly fewer points on

average. Figure 4 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 inequal-

ity is due to heterogeneity in participants’ preferences for labor and output. Inequality

worsens under the Priority rationing rule. When participants are largely responsible for

their own points 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 a panic impulse to

demand the highest levels of output each period in the hopes of being at the front of the

queue. Such impulses 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

To analyze the convergence of median labor supply and output demand to the steady

state values, we first contrast behavior across treatments graphically. Figures 5.a and

5.b show box-plots of the average median labor supply and output demand for each

sequences, while Figure 5.c and 5.d depict the aggregate outcomes. Figure 6 contrasts

the time series of average labor supply and output demand across treatments. The

horizontal red lines 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 consider-

able heterogeneity in both median and aggregate labor supplies, where some sessions

experience very high levels of labor supply. By contrast, in the Equitable treatment,

18

Figure 5: Labor Supply and Output Demand

Steady State

01

23

45

Hou

rs o

f Wor

k

Random Equitable Priority

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

(a) Average of Median Labor Supply

Steady State2040

6080

Uni

ts o

f Out

put

Random Equitable Priority

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

(b) Average of Median Output Demand

Steady State

1020

3040

50H

ours

of W

ork

Random Equitable Priority

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

(c) Aggregate Labor Supply (Per-Period Aver-age)

Steady State

300

400

500

600

700

200

Uni

ts o

f Out

put

Random Equitable Priority

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

(d) Aggregate Output Demand (Per-PeriodAverage)

Notes: On the x-axis it is displayed i ∈ {1, ..5}; i = 1 represents the range from the first period1 to TotalPeriod× 0.20 of each session, similarly i = 2 represents the range from the first periodTotalPeriod× 0.20 + 1 to TotalPeriod× 0.40. The bottom and top of the box are the first andthird 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 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 thannine participants we adjust the aggregate labor supply (output demand): Adj.AggregateLabor =AggregateLaborSupply × 9/N(Adj.OutputDemand = AggregateOutputDemand× 9/N).

19

Figure 6: Average Labor Supply and Output Demand Over Time

1 2 3 4 5

2

2.2

2.4

2.6

2.8

3

3.2

3.4

Steady State

Period

Random Equitable Priority

1

(a) Average Labor Supply

1 2 3 4 520

25

30

35

40

45

50

Steady State

Period

Random Equitable Priority

1

(b) Average Output Demand

Notes: On the x-axis it is displayed i ∈ {1, ..5}; i = 1 represents the range from the first period1 to TotalPeriod × 0.20 of each session, and i = 2 represents the range from the first periodTotalPeriod× 0.20 + 1 to TotalPeriod× 0.40.

labor supplies in some sessions can 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, Duffy, and Hommes (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.4 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

4If at period t∣∣hmed

t − hss∣∣ > 1/ : (

∣∣cmedt − css

∣∣ > 10) we still count it as a converging period only if

convergence is restored in the following period (∣∣hmed

t+1 − hss∣∣ < 1/ : (

∣∣cmedt+1 − css

∣∣ < 10)), where, hmedt

and cmedt are median labor supply and median output demand at period t, respectively.

20

Equitable 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, respectively).

Table 3: Number of Periods Before

ConvergenceI

Sessions

No. of Periods Before Convergence Total No.

Med. Labor Med. Output of

Supply Demand Periods

Random1 76 74 79

Random2 Never Never 90

Random3 66 Never 79

Random4 1 62 69

Random5 Never 65 73

Random6 80 63 84

Equitable1 Never 84 97

Equitable2 38 Never 75

Equitable3 49 Never 70

Equitable4 Never Never 72

Equitable5 41 Never 81

Equitable6 Never 40 79

Equitable7 61 26 74

Priority1 6 Never 77

Priority2 9 Never 78

Priority3 34 Never 84

Priority4 10 66 100

Priority5 3 65 70

Priority6 Never Never 71

(I) In each period t we compute∣∣cmedt − css

∣∣ and∣∣hmedt − hss

∣∣.We then claim that convergence occurs in the first period in

which∣∣cmedt − css

∣∣ < 10 (∣∣hmedt − hss

∣∣ < 1), and this is pre-

served until the end of the session. If at period t∣∣cmedt − css

∣∣ >10

∣∣(hmedt − hss∣∣ > 1) we still count it as a converging period

if convergence is restored in the following period∣∣cmedt+1 − css

∣∣ <10

∣∣(hmedt+1 − hss∣∣ < 1).

Finally, we formally test convergence, following the econometric procedure first in-

troduced to experimental economics by Noussair, Plott, and Riezman (1995). The re-

gression model for each treatment is the following:

yst =1

t

S∑s=1

αsDs + βt− 1

t+ εst,

where yit is the dependent variable (in this case, median labor supply/output demand),

21

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. As Duffy (2014) pointed out, 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, β 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.

Table 4: Convergence Model EstimatesI,II

TreatmentDependent

α1 α2 α3 α4 α5 α6 α7 βModel

ρVariable Prediction

Random

med ht2.812 1.896 4.107 1.809 3.373 1.756 NA 2.277

2.237 0.7422(0.82) (0.74) (0.89) (0.39) (2.15) (0.72) NA (0.1)

med ct27.061 27.798 38.754 27.203 40.156 68.384 NA 31.9422

22.37 0.3986(9.2) (8.19) (11.46) (7.15) (9.43) (10.71) NA (0.85)

Equitable

med ht2.125 2.387 1.032 2.409 2.606 1.92 3.862 1.97

2.237 0.7957(0.39) (0.53) (1.14) (0.9) (0.7) (0.55) (0.62) (0.09)

med ct16.455 44.48 31.32 88.258 54.44 16.534 16.073 32.73

22.37 0.4041(11.33) (10.42) (9.29) (16.62) (9.85) (6.58) (7.54) (0.8)

Priority

med ht2.025 1.986 3.388 2.241 2.876 4.289 2.286

2.237 0.7043(0.43) (0.33) (0.58) (0.51) (0.43) (1.14) (0.06)

med ct50.066 74.326 29.551 32.694 49.304 39.129 40.792

22.37 0.5244(11.74) (25.87) (10.64) (13.92) (10.39) (15.16) (1.34)

(I) Only takes into account experienced subjects.

(II) Standard errors are in parentheses. Following Noussair, Plott, and Riezman (1995) the standard errors are corrected for het-

erosedasticity and first order autocorrelation.

5 Individual-Level Analysis

Does the nature of rationing influence how participants respond to output rationing?

We conduct a series of panel regressions to study how individual labor supply and

output demand decisions are affected by either past experiences of output rationing or

by the quantity of rationing, and we allow for differentiated treatment effects. We also

22

control for the bank account balance with which the participant enters the period. The

estimations include only submitted decisions. The results are presented in Table 5.

Personally experiencing output rationing in the previous period has a significant

negative effect on labor supply decisions in all treatments. In column 1, we compare

the effects of individual rationing in the Priority to the Equitable and Random. Being

unable to purchase all the requested output leads participants to reduce their labor

hours in the Priority treatment by 0.254 hours. This reaction is statistically significant

at the 1% level. Compared to participants in the Priority, Equitable and Random

participants reduce their labor supplies by an additional 0.276 and 0.371 hours. In

column 2, when the baseline treatment is set to Random, we observe that while the

Equitable labor supplies react somewhat less to output rationing, the difference between

the Random and Equitable treatments is not statistically significant. The extent of

quantity rationing also leads to differentiated effects across treatments. In columns 3

and 4 we present results for labor supply, where the baseline treatment is Priority and

Random respectively. We observe that as the quantity of output rationed increases by

one unit, an individual’s labor supply in the following period under the Priority rule

decreases by 0.012 hours. Participants in the Equitable and Random treatments react

even more negatively to the same degree of quantity rationing. In column 4, we observe

that Equitable participants are significantly more reactive to an increases in the extent

of their rationing than their Random counterparts.

Consumer demand is also sensitive to personal experiences of output rationing. Sim-

ply being unable to satiate last period’s demands leads participants to significantly

increase their labor supply decisions. In column 5, we observe that in the Priority

treatment, participants increase their output demand on average by 14.292 units. The

reaction is significantly muted in the Equitable treatment, with consumption demand

increasing by only 10.027 units. Compared to reactions under the Priority treatment,

Random reactions are also smaller on average but not statistically significantly different.

In column 6, we see that compared to participants in the Random treatment,

23

Table 5: Effects of Rationing Across Treatments on Labor Supply and Output Demand DecisionsI

Labor Supply Output Demand

(1) (2) (3) (4) (5) (6) (7) (8)

Banki,t−1 -0.000*** -0.000*** -0.000*** -0.000*** 0.001* 0.001* 0.001 0.001

(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

CDi,t -0.000 -0.000 0.003 0.003

(0.00) (0.00) (0.00) (0.00)

IndividualOutputRationedi,t−1 -0.254*** -0.624*** 14.292*** 10.378***

(0.08) (0.10) (1.61) (1.98)

IndividualOutputRationedi,t−1 x Equitable -0.276** 0.095 -4.265** -0.351

(0.11) (0.12) (1.85) (2.14)

IndividualOutputRationedi,t−1 x Random -0.371*** -3.914

(0.12) (2.57)

IndividualOutputRationedi,t−1 x Priority 0.371*** 3.914

(0.12) (2.57)

QuantityRationedi,t−1 -0.012*** -0.007*** 0.360*** 0.549***

(0.00) (0.00) (0.04) (0.01)

QuantityRationedi,t−1 x Equitable -0.006** -0.011*** 0.206*** 0.018

(0.00) (0.00) (0.04) (0.03)

QuantityRationedi,t−1 x Random -0.005** -0.189***

(0.00) (0.04)

QuantityRationedi,t−1 x Priority 0.005** 0.189***

(0.00) (0.04)

HSi,t -0.115 -0.115 0.784 0.784

(0.55) (0.55) (0.61) (0.61)

α 2.958*** 2.958*** 2.812*** 2.812*** 38.473*** 38.473*** 32.364*** 32.364***

(0.29) (0.29) (0.30) (0.30) (1.97) (1.97) (1.94) (1.94)

N 9671 9671 9671 9671 9671 9671 9671 9671

χ2 83.11 83.11 114.9 114.9 270.2 270.2 2087.1 2087.1

(I) This table presents results from a series of random effect panel regressions. CDi,t and HSi,t refer to current period output demands and labor supplies.

Banki,t−1 refers to the end-of-period bank account balance. IndividualOutputRationedi,t−1 takes the value of 1 if, in the previous period, participant ireceived less consumption than she demanded, and 0 otherwise. QuantityRationedi,t−1 measures the amount by which the participant was rationed onoutput. Specifications (1) ,(3), (5), and (7) use the Priority treatment as a baseline, while the remaining specifications use the Random treatment. Robuststandard errors are clustered at the session level. *p < 0.10, **p < 0.05, and ***p < 0.01.

24

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 participant

will demand in the following periods. Relative to Priority participants, the increase in

consumer demand is significantly and quantitatively much larger among Equitable and

Random participants (with no significant differences between the latter two).

The above results suggest that rationing does 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 also be experienced

because 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. Consumption decisions are also not as reactive

to output rationing in the Random and Equitable treatments. Random participants

demanding large quantities of output face a greater probability of receiving them (and a

higher consumption bill) than those in the Priority, resulting in more cautious decisions.

Moreover, asking for relatively more output in the Random 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).

6 A Model of Myopic Decisions 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 out-

put demand can be explained by assuming that agents form their decisions myopically,

without regard to their budget constraint and future utility maximization. In this case,

agents find it optimal to undersupply labor while maximizing of much of the consump-

tion demanded from other agents’ production that hey obtain. To derive the equilibrium

of this model, suppose that there are n agents in the economy. The labor supply and

output demand of agent i ∈ {1, ..., n} are denoted by hSi ∈ {0, 10} and cDi ∈ {0, 100},

25

respectively, whereas hours worked and output purchased are denoted by hi and ci,

respectively. Agent i chooses nSi and cDi to maximize:

maxcDi ,h

Si

U (ci, hi) (3)

subject to

ci = min{cDi , θ

Rci(C

D−i, H

S)}, hi = hSi if CD > ZHS

ci = cDi , hi = min{hSi , θ

Rhi(C

D, HS−i)}

if CD < ZHS

where U (ci, hi) =(

11−σ)c1−σi −

(1

1+η

)h1+ηi , and θRc (θRh ) is the individual consumption

(hours) when output (labor) is rationed according to a rule

R ∈ {Random,Equitable, Priority}. Aggregate output demand and labor supply are

denoted by CD ≡∑n

i=1 cDi and HS ≡

∑ni=1 h

Si , respectively. We can compare the

symmetric equilibrium for labor supply under the different rationing rules.5

Random Queue: Under a random queue rationing scheme, the probability of ob-

taining 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

that agents demand the same amount of output, cD1 = cD2 = . . . = cDn = c. Agent i

chooses how many hours to work to maximize her expected utility:

maxhsi

1

n

[U(min

{cDi ,

[θRandomci

]1

}, hsi)

+ U(min

{cDi ,

[θRandomci

]2

}, hsi)

+ . . .+ U(min

{cDi ,

[θRandomci

]n

}, hsi)],

where[θRandomci

]q

= max{ZHS − (q − 1)c, 0

}, and q ∈ {1, . . . , n} is the position of

individual i in the queue. Positions are randomly assigned, and there is 1n

probability of

obtaining each one of the spots.

5We rule out of the analysis the excess labor supply case. Whenever there is excess labor supply,agents would find profitable to deviate by cutting their hours of work, up to the point at which therewould be excess output demand. This is consistent with the data; in the Random treatment there wasno excess labor supply in any of the sessions, while in the Equitable only 12% of the periods saw excesslabor supply (see Table 2).

26

The maximization problem can be solved numerically. There are multiple Nash

equilibria for labor supply. The equilibrium depends on output demand. Assuming that

there are nine agents, if c > 48 then(hSi)Random

= 0.471. For c < 48, the equilibrium

range is(hSi)Random

= [0.471, 0.65].6

Equitable Rule: Under an equitable allocation scheme, each agent receives an equal

share of the total production, θEquitableci = ZHS

n, up to her specified demand. The first

order condition with respect to hSi from equation (3) is

(Z

n

)1−σ(HS)−σ =

(hSi)η.

In a symmetric equilibrium, HS = nhsi . Thus, each agent will choose to work

(hSi)Equitable

=

(1

n1

η+σ

)Z

1−ση+σ .

With n = 9, the Nash equilibrium labor supply is 0.699.

Priority Rule: Under a priority rationing scheme, ci depends on hSi . Specifically,

θPriorityic = ZhSi . The first order condition with respect to hSi from equation (3) is given

by:

Z1−σ (hSi )−σ =(hSi)η.

Thus, the Nash equilibrium labor supply is

(hSi)Priority

= Z1−ση+σ .

Under the parameterization of the experimental environment, the Nash equilibrium labor

supply is 2.32.

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

6If agents’ output demand is high enough. Specifically, if 0 < ZHS < c only the first agent in the

queue will obtain some output. Labor supply equilibrium would be(hSi)Random

=

(1

n1+ση+σ

)Z

1−ση+σ .

27

does not depend on the number of agents. Under the Equitable rule and Random rule,

as the number of agents increase, 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 the predictions of the myopic model. In re-

sponse to past rationing, participants significantly reduce their labor supply and increase

their output demands. Compared to their Priority rule counterparts, participants reduce

their labor supplies by significantly more when they face Random Queue and Equitable

rationing schemes. 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 highest on average under a Priority rule,

the differences across treatments are not significant. The myopic model does, however,

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=49.6, s.d.=27.97) than in the Priority rule (mean=150.98, s.d.=35.95, p =0.004)

and Equitable rule treatments (mean=98.30, s.d.=54.81, p =0.086). While the average

minimum aggregate labor supply is lower under an Equitable rule than under a Priority

rule, the differences are 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 rep-

resentative agents and market clearing both abstract away from these important issues

and consequently miss out on important, realistic dynamics driven by market spillovers.

In our experimental economy populated with heterogeneous participants, distribution

schemes are shown to play an important role in fostering economic stability. Our find-

28

ings suggest that convergence to the steady state equilibrium depends significantly on

the presence of property rights and 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 instead rely 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 this 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 op-

portunities. In this case, aggregate production is significantly less volatile and quickly

and consistently converges to the high-production steady state.

We must be careful not to argue that the undersupplying of labor in our environ-

ments 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 fur-

ther 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 unrea-

sonable 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 outcomes are more likely to occur under rationing rules where

there are minimal or nonexistent property rights to own production.

Rationing schemes have important implications for inequality. Inequality is signifi-

cantly greater when jobs and output are distributed on a first-come, first-serve 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

29

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 equal-

ity, 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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34

Appendix A: Theoretical Model

The framework that we use to implement our experimental design is based on a repre-

sentative agent dynamic general equilibrium model (DGE) with sticky prices and mo-

nopolistic 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 var-

ious 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 cen-

tral banks. In this model households optimally choose their consumption of final goods,

labor supply, and savings. In this economy 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 a la Calvo (1983). Finally, the central bank sets the

nominal interest rate in response to fluctuations in inflation. We begin with a descrip-

tion 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 con-

sumption and labor as follows

Ut =∞∑j=0

βj

(c1−σt+j

1− σ−h1+ηt+j

1 + η

),

where

ct =

(∫ 1

0

cθ−1θ

it di

) θθ−1

.

They obtain utility from the immediate consumption of a bundle of differentiated vari-

eties, each variety denoted by cit, and disutility from working ht hours. The coefficient

35

of relative risk aversion is represented by σ, the elasticity of labor supply by 1/η, and

the elasticity of substitution between different varieties is given by θ.

Equation 4 is the household’s budget constraint that equates expenditures to income:

Ptct +Bt = (1 + it−1)Bt−1 +Wtht + Tt. (4)

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, Wtht; and receives

a transfer, Tt, from the monopolistic firms. The representative household maximizes its

utility stream (7) by making optimal choices on ct, ht, and Bt subject to the budget

constraint (4).

From the household’s first order conditions the following equations are derived:

hηtc−σt

=Wt

Pt, and (5)

β

[(ct+1

ct

)−σ(1 + it)

(1 + πt+1)

]= 1. (6)

Equation 5 describes the labor–leisure intratemporal trade-off taking the real wage

as given. Equation 6 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. (7)

36

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. Each period, only a

fraction 1 − ω of the firms are allowed to adjust the price (Calvo (1983) mechanism).

The prices set by the firms determine the demand for each variety:

Yit =1

I

(PitPt

)−θYt,

where I is the number of firms in the economy. Pt is the aggregate price index and is

defined as

Pt ≡ I1θ−1

{[I∑i=1

(Pit)1−θ]} 1

1−θ

.

The Calvo assumption about price stickiness can be also written as

P 1−θt = (1− ω) (P o

t )1−θ + ω (Pt−1)1−θ . (8)

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.

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.

37

We also impose that total demand for output is financed by income from output pro-

duction:

Ct = Yt.

Steady State Equilibrium

To derive the steady state equilibrium, one can start by solving cost minimization prob-

lem for firm i (equation (7)):

minhit

(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

Pt

1

Z.

It can be shown that the ratio of the price set by the firms when they are able to update

it, P ot , relative to the aggregate price index, Pt, is

P ot

Pt=

θ

θ − 1

∑∞i=1 ω

iβic1−σt+i mct+i

(Pt+iPt

)θ∑∞

i=1 ωiβic1−σ

t+i

(Pt+iPt

)θ−1≡ θ

θ − 1

S1tS2t

,

where

S1t = c1−σt mct + βωEtS1t+1, and

S2t = c1−σt + βωEtS2t+1.

With no shocks in the economy, S1t = S1t+1 and S2t = S2t+1. This implies that

P ot

Pt=

θ

θ − 1mct. (9)

38

Thus, in the steady state,W

P=θ − 1

θZ.

Using the market clearing conditions and equation (5), the steady state values for labor

and consumption are obtained:

hSS =

(θ − 1

θZ1−σ

) 1η+σ

, and

cSS = Z

(θ − 1

θZ1−σ

) 1η+σ

.

From equation 6 the steady state for the interest rate can be obtained:

iSS =1

β− 1.

Derivation of the Inflation Equation

Combining equations (5), (8), and (9), the following equation for inflation is obtained:

πt =ω

11−θ[

1− (1− ω)(

θθ−1

hηt cσt

Z

)1−θ] 1

1−θ− 1. (10)

Linearizing (10), by using a first order Taylor approximation around the steady state,

linearized inflation is obtained:

πt = 1 + γc(cmedt − cSS) + γh(hmedt − hSS),

where

γc =1

(1− θ)Z(cSS)σ−1 (

hSS)ησθ(ω − 1)ω

11−θΨ,

γh =1

(1− θ)Z(cSS)σ (

hSS)η−1

ηθ(ω − 1)ω1

1−θΨ,

39

and

Ψ =

1− (1− ω)

((cSS)σ (

hSS)ηθ

(θ − 1)Z

)1−θ−(1+ 1

1−θ )((cSS)σ (

hSS)ηθ

(θ − 1)Z

)−θ.

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 6). 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.

40

Table 6: Parameters and Steady State Values

Parameter Parameter Description ValueZ Productivity level 10

1− ω Fraction of firms updating 0.1δ Inflation target of the central bank 1.5γ Interest smoothing parameter 0.5θ Measure of substitutability 15β Rate of discounting 0.965ρ Natural nominal rate of return 0.0363

1/σ Elasticity of intertemporal substitution 3.031/η Frisch labor supply elasticity 0.67µ∗ Steady state markup (θ/(θ − 1)) 1.07C∗ Steady state consumption 22.37N∗ Steady state labor 2.237W ∗ Steady state nominal wage 10P ∗ Steady state output price 1.07

41

Appendix C: Aggregate Variables

Figure 7: Median Labor Supply per Session

0 20 40 600

2

4

6

8

10

Period

Random 1

0 20 40 60 800

2

4

6

8

10

Period

Random 2

0 20 40 600

2

4

6

8

10

Period

Random 3

0 20 40 600

2

4

6

8

10

Period

Random 4

0 20 40 600

2

4

6

8

10

Period

Random 5

0 20 40 60 800

2

4

6

8

10

Period

Random 6

0 20 40 600

2

4

6

8

10

Period

Equitable 1

0 20 40 600

2

4

6

8

10

Period

Equitable 2

0 20 40 600

2

4

6

8

10

Period

Equitable 3

0 20 40 600

2

4

6

8

10

Period

Equitable 4

0 20 40 60 800

2

4

6

8

10

Period

Equitable 5

0 20 40 600

2

4

6

8

10

Period

Equitable 6

0 20 40 600

2

4

6

8

10

Period

Equitable 7

0 20 40 600

2

4

6

8

10

Period

Priority 1

0 20 40 600

2

4

6

8

10

Period

Priority 2

0 20 40 60 800

2

4

6

8

10

Period

Priority 3

0 20 40 60 80 1000

2

4

6

8

10

Period

Priority 4

0 20 40 600

2

4

6

8

10

Period

Priority 5

0 20 40 600

2

4

6

8

10

Period

Priority 6

Med. Labor Supply 1st-3rd quartiles Labor Supply Begin Sequence Steady State

1

42

Figure 8: Median Output Demand per Session

0 20 40 600

20

40

60

80

100

Period

Random 1

0 20 40 60 800

20

40

60

80

100

Period

Random 2

0 20 40 600

20

40

60

80

100

Period

Random 3

0 20 40 600

20

40

60

80

100

Period

Random 4

0 20 40 600

20

40

60

80

100

Period

Random 5

0 20 40 60 800

20

40

60

80

100

Period

Random 6

0 20 40 600

20

40

60

80

100

Period

Equitable 1

0 20 40 600

20

40

60

80

100

Period

Equitable 2

0 20 40 600

20

40

60

80

100

Period

Equitable 3

0 20 40 600

20

40

60

80

100

Period

Equitable 4

0 20 40 60 800

20

40

60

80

100

Period

Equitable 5

0 20 40 600

20

40

60

80

100

Period

Equitable 6

0 20 40 600

20

40

60

80

100

Period

Equitable 7

0 20 40 600

20

40

60

80

100

Period

Priority 1

0 20 40 600

20

40

60

80

100

Period

Priority 2

0 20 40 60 800

20

40

60

80

100

Period

Priority 3

0 20 40 60 80 1000

20

40

60

80

100

Period

Priority 4

0 20 40 600

20

40

60

80

100

Period

Priority 5

0 20 40 600

20

40

60

80

100

Period

Priority 6

Med. Labor Supply 1st-3rd quartiles Labor Supply Begin Sequence Steady State

1

43

Appendix D: Computer Interfaces

Figure 9: Main screen

44

Figure 10: Personal history screen

45

Figure 11: Market history screen

46

Appendix E: Instructions

The instructions distributed to subjects in all the treatments (Random, Equitable, and

Priority) are reproduced on the following pages. Subjects received the same set of

instructions with the exception of the tables in Appendix E.

47

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:

(

)

. Similarly, If your bank account has a positive balance at the end of the experiment

you will gain:

(

)

.

**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 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.

52