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Working Paper Series
Aggregate Demand Management with
Multiple Equilibria
WP 03-04
Huberto M. Ennis
Federal Reserve Bank of Richmond
Todd Keister
Centro de Investigación Económica
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Aggregate Demand Management with Multiple Equilibria *
Huberto M. Ennis
Research Department, Federal Reserve Bank of Richmond
Todd Keister
Department of Economics and Centro de Investigación Económica
ITAM
Federal Reserve Bank of Richmond Working Paper No. 03-04
June 2003
Abstract
We study optimal government policy in an economy where (i) search frictions create a
coordination problem and generate multiple Pareto-ranked equilibria and (ii) the
government finances the provision of a public good by taxing trade. The government must
choose the tax rate before it knows which equilibrium will obtain, and therefore an
important part of the problem is determining how the policy will affect the equilibrium
selection process. We show that when the equilibrium selection rule is based on the
concept of risk dominance, higher tax rates make coordination on the Pareto-superior
outcome less likely. As a result, taking equilibrium-selection effects into account leads to a
lower optimal tax rate. We also show that public-employment policies that appear to be
inefficient based on a standard equilibrium analysis may be justifiable if they influence the
equilibrium selection process.
JEL Classification Numbers: E61, E62, H21
Keywords: Search Frictions, Coordination, Equilibrium Selection, Government Policy
________________________________________________
* We thank Scott Freeman, Preston McAfee, and seminar participants at Kentucky, Miami,
Queens, Texas, Vanderbilt, the Federal Reserve Bank of Dallas, the 2002 Midwest Macro
Conference, the First Annual University of Texas - ITAM workshop and the Economic Fluctuations
workshop at the University of Essex. Part of this work was completed while Keister was visiting the
University of Texas at Austin, whose hospitality and support are gratefully acknowledged. The
views expressed herein are those of the authors and do not necessarily reflect those of the Federal
Reserve Bank of Richmond or the Federal Reserve System.
E-mail addresses: huberto.ennis@rich.frb.org (H. M. Ennis), keister@itam.mx (T. Keister)
1
Introduction
We present a model where the search and matching process creates externalities and, as
a result, the market for output becomes more efficient as more people enter it. Following
Diamond (1982), we refer to the intensity of participation in the market as the level of
aggregate demand. There is also a government in our model that can tax market transactions
and use the proceeds to provide a public good. Our interest is in Þnding the welfaremaximizing tax rate. If the model has a unique equilibrium for every possible policy choice,
this task is straightforward. The optimal tax rate will balance the marginal beneÞt of the
public good against the marginal cost of providing it, including the decrease in aggregate
demand and hence the loss in output-market efficiency that the tax causes. However, as
is common in models with trading frictions, our model has multiple equilibria for a broad
range of parameter values. We assume that the government must set the tax rate before
knowing the actions of private agents, which implies that the government does not know
which equilibrium will obtain. It seems entirely possible in such a setting that the choice of
tax rate may inßuence which of the equilibria the economy ends up in, an effect that should
be taken into account in determining the optimal policy. In fact, Diamond (1982, p. 882)
states that “... one of the goals for macro policy should be to direct the economy toward
the best [equilibrium] ... after any sufficiently large macro shock.” Hence it is important to
try to understand the effects that government policy can have on the process of equilibrium
selection and how these effects inßuence the government’s optimal choice of policy.
In our model, there is a continuum of identical agents, each of whom must decide whether
or not to produce for the market. If an agent produces, she must exert costly effort in order
to Þnd a trading partner. For a given tax rate there are two possible (symmetric, purestrategy) equilibria: one where all agents produce and search, and one where no one produces
and there is no market activity. Following Howitt and McAfee (1992), we label these the
1
“optimistic” and “pessimistic” outcomes, respectively. The search level that agents choose
in the optimistic outcome is decreasing in the tax rate, and if the tax rate is set too high the
optimistic outcome is no longer an equilibrium. The pessimistic outcome is an equilibrium
regardless of the tax rate; if an individual agent believes that no one else is producing for
the market, then she believes that searching for a trading partner will be in vain and she
will therefore choose not to produce. Whenever the optimistic outcome is an equilibrium, it
Pareto dominates the pessimistic outcome.
What tax rate should the government in this economy choose? If the policy had no
inßuence on the equilibrium selection process, then the simple structure of our model would
lead to a clear answer: the tax rate should be chosen to maximize expected welfare in
the optimistic equilibrium. This is because the tax policy has no effect on welfare in the
pessimistic equilibrium, and therefore the same policy choice is (weakly) optimal in each of
the equilibria. However, we argue that in general the likelihood of agents coordinating on a
speciÞc outcome will depend on the individual payoffs associated with each possible choice
of action, which implies that the equilibrium selection process will be inßuenced by the tax
rate. In particular, we discuss a class of equilibrium selection rules that are based on the
concept of risk dominance. We show that under any of these rules, the fact that a higher
tax rate decreases the beneÞt of producing implies that a higher tax rate also decreases the
likelihood that the optimistic equilibrium will obtain. This is an additional cost of taxation
that a traditional analysis that ignores equilibrium selection would miss. This cost has clear
implications for policy: the optimal tax rate must be lower than the rate that maximizes
welfare in the optimistic equilibrium. The reason for this result is simple: If the tax rate is
decreased slightly from this starting point, there is a small (second order) loss in utility should
the optimistic equilibrium obtain. At the same time, there is a larger (Þrst-order) increase in
the probability of reaching the optimistic equilibrium, and therefore expected utility (across
2
equilibria) increases. Thus the effect of taking equilibrium selection into account is clear:
it lowers the optimal tax rate. In other words, discouraging market activity is more costly
than a standard equilibrium analysis would indicate.
When additional policy tools are introduced, taking equilibrium selection into account can
also change the composition of the optimal policy. We demonstrate this in Section 4 by giving
the government an additional policy tool: it can directly control the actions of a fraction
of the agents in the economy. In other words, it can directly produce the public good by
employing agents and dictating their production and search decisions. We choose parameter
values so that public production is inefficient in the following sense: if government policy had
no inßuence over the equilibrium selection process, this second policy should never be used.
However, we show that when equilibrium selection effects are taken into account, the optimal
policy involves a positive amount of public production. There are two key elements behind
this result. First, taking equilibrium selection into account lowers the optimal tax rate, as we
discussed above. This increases the marginal value of the public good and therefore makes
public production more attractive. Second, unlike taxation, public production does not
necessarily decrease the probability of the optimistic equilibrium obtaining; in our example,
it actually increases that probability. This is because public production increases the level of
aggregate demand and therefore makes a private agent willing to produce for a wider range
of beliefs about the market environment. Therefore the cost of public production is lower
than an equilibrium analysis would indicate, and at the new, higher marginal valuation of
the public good, some public production is optimal. Hence, looking at equilibrium selection
shows that public sector employment/production may be a useful policy for coordination
purposes.
There is a large literature in which models with multiple equilibria are used to discuss
policy issues without imposing an equilibrium selection mechanism as we do here. This lit-
3
erature is based on the idea that sometimes a meaningful comparison can be made between
the sets of equilibria resulting from different policy choices, and hence a policy recommendation can be made independent of the equilibrium selection process. Suppose, for example,
that one policy choice leads to a unique, Pareto optimal equilibrium while another choice
generates that same equilibrium plus some other, inferior equilibria. A strong case can
then be made for choosing the former policy. Early work along these lines includes Grandmont (1986), Reichlin (1986), and Woodford (1986), where the inferior equilibria involve
endogenous ßuctuations and the preferred policy has a natural interpretation as a stabilization policy.1 In many situations, however, bad equilibria cannot be costlessly eliminated by
the proper policy choice. Cooper and Corbae (2002) study a model with a strategic complementarity in the Þnancial intermediation process. Monetary policy can eliminate some
undesirable equilibria, but not all of them. The authors conclude that “without a theory
of equilibrium selection, predictions about the effects of monetary policy are impossible to
make.” (p. 23) The simpler model presented in this paper contains the essential features
that give rise to this problem. We provide a method for dealing with such situations, and in
doing so, we show how models where undesirable equilibria cannot be easily eliminated can
still be brought to bear on policy issues in interesting ways.
In the next section, we describe the model and the properties of equilibrium. In Section 3,
we show how to formulate the optimal policy problem in the presence of multiple equilibria.
For this, we introduce the concept of an equilibrium selection mechanism and derive the
optimal policy under several different mechanisms. In Section 4, we introduce the public1
The “stabilization” approach has since been applied in a wide variety of settings. See,
for example, Smith (1991) and the symposium introduced by Woodford (1994). In an environment where multiple equilibria exist for all policy choices, Keister (1998) shows that
meaningful policy statements can sometimes be made without imposing an equilibrium selection rule.
4
employment policy and show how equilibrium-selection effects can lead the government to
use this policy even though it is inefficient from a standard equilibrium perspective. Finally,
in Section 5 we offer some concluding remarks.
2
The Model
Consider an economy with a [0, 1] continuum of identical agents and a single commodity that
is produced using only labor. Each agent faces a binary production decision; she must either
produce at a Þxed scale or not at all. Producing requires labor input that gives disutility
c ≥ 0. Agents receive no utility from consuming their own output, but have a utility value F
of consuming the output of someone else.2 Finding a trading partner in the output market
requires costly search effort. After observing the production decisions of all other agents, an
agent who has produced chooses a search intensity. This intensity determines the probability
that she will Þnd a trading partner. If a partner is found, the two agents exchange output
and consume; all exchanges are one-for-one trades. If an agent chooses not to produce, she
does not enter the output market and has zero utility from trade (a normalization).
There is also a public good that can be produced using units of output that have been
traded and are therefore consumable. All agents receive utility from the public good. However, because there is a continuum of agents, there is a pure free-rider problem and the public
good will not be privately provided. Instead, there is a benevolent government that can tax
trade at a rate τ ≥ 0 and use the proceeds to produce the public good. Our interest is in how
the policy variable τ should be set. An important aspect of the problem we consider is the
2
We think of this as a form of specialization in production; see Diamond (1982). One
could instead allow agents to consume their own output but assume that doing so yields less
utility than consuming the output of someone else. By relabelling utility values and costs,
this approach is easily shown to be equivalent to the model here.
5
timing of decisions. We study the case where the government must set the tax rate before
observing the actions of agents. In particular, the government must set τ before knowing
which equilibrium will obtain. It is in such settings that our question of interest — how the
policy choice affects the equilibrium selection process — arises. In addition, we assume that
the government must set τ before the exact cost of production c is known. At this point
the cost is a random variable with support [cL , cH ] and density function f , which we take
(without any real loss of generality) to be uniform. In other words, the tax rate must be
set with less information than private agents will have when they decide whether or not to
produce. This uncertainty plays an important role in our formulation of the optimal policy
problem; we discuss this issue in detail in Section 3 below.
Before we can study the problem of an agent deciding whether or not to produce, we
must calculate the beneÞt of producing. For this we need to look at the workings of the
output market.
2.1
Output Market
If agent i has produced, she can search for a trading partner by choosing a search intensity
γ i ∈ [0, 1] . In this setting, externalities naturally arise: An increase in individual search
effort not only makes it more likely that the individual will Þnd a partner, it also makes
it more likely that other agents in the economy will Þnd a partner. The fraction of agents
Þnding a match in the output market is given by an aggregate matching function m.3 Let γ
3
Assuming a matching function simpliÞes matters considerably, but it clearly does not
come without cost. In particular, policy changes may affect the nature of the matching
process, and this effect will be absent in our analysis. See Lagos (2000) for an interesting
study in this direction.
6
denote the average level of search intensity in the economy, so that we have
Z 1
γ=
γ i di.
(1)
0
Note that γ is also the total amount (“units”) of search intensity in the market. We assume
that the value of m depends only on γ (and not on the distribution of γ i across agents). We
assume that m : [0, 1] → [0, 1] is a smooth, strictly increasing bijection, so that m (0) = 0
and m (1) = 1 hold. In addition, we assume that m is strictly convex. This assumption
captures the externality described above and implies that there are increasing returns in
the matching process; if the number of searching agents doubles (holding search intensity
constant), the number of matches formed more than doubles. Hence the efficiency of the
output market is increasing in the number of searching agents and in their search intensities.
Following Diamond (1982), we refer to the aggregate amount of search intensity as the level
of aggregate demand.
Notice that the above assumptions imply that we have m (γ) ≤ γ for all possible values
of γ. Since the search intensity of each agent is less than unity, the number γ is necessarily
lower than the fraction of agents who are searching with positive effort levels. Therefore, the
fraction of agents who are matched, m (γ), is necessarily lower than the fraction of agents
who are searching with positive effort levels. This shows that the matching function always
yields a feasible outcome.
While the function m gives the fraction of agents who Þnd a partner, which agents
actually get matched is random, with the probability of an individual agent being matched
proportional to her search intensity. SpeciÞcally, deÞne
ρ(γ) =
m(γ)
.
γ
(2)
Then ρ is the probability of being matched per unit of search intensity. An individual
agent choosing intensity level γ i will be matched with probability ρ(γ)γ i . Note that our
7
assumptions on m and the upper bound on γ i ensure that this probability is always between
zero and one.
Output that has been traded can be converted one-for-one into units of the public good.
Let x denote the amount of public good produced per agent. This will be equal to the tax
rate multiplied by the number of agents who make a trade (the tax base). Then all agents
receive utility v (x) , where the function v is strictly increasing and strictly concave, and
satisÞes
v(0) = 0
2.2
lim v 0 (x) = ∞.
and
x→0
The Agent’s Problem
Each agent Þrst decides whether or not to produce. If she produces, she then chooses a search
intensity γ i after observing the production decisions of all other agents. We examine the
latter choice Þrst. The cost function for search intensity d (γ i ) is smooth, strictly increasing,
strictly convex, and satisÞes
lim d0 (γ i ) = 0.
γ i →0
If an agent meets a partner, they exchange output. If she does not meet a partner, she does
not consume. We use the variable φi to represent the production choice; φi = 1 corresponds
to producing and φi = 0 to not producing. If the agent produces, she will choose γ i to
maximize expected utility. The agent takes the level of provision of the public good as given
and (since it enters her utility function in an additively separable way) her production and
search decisions are independent of this level. She also takes as given the probability ρ of
Þnding a match per unit of search intensity and the tax rate τ . We use λ (ρ, τ ) to denote
8
the expected value of producing given a particular (ρ, τ ) pair, so that we have
λ (ρ, τ ) = max ργ i (1 − τ ) F − d (γ i ) .
γ i ∈[0,1]
(3)
We assume that the cost function d is such that the unique solution to this problem
γ i = d0−1 (ρ (1 − τ ) F )
is always less than unity.4 Note that γ i is positive as long as ρ and (1 − τ ) are positive.
Since the expected beneÞt of search effort is linear in γ and the marginal cost of effort is zero
when γ is zero, it is optimal to engage in a positive level of search whenever the probability
of Þnding a partner is positive. Also note that γ i is increasing in ρ and therefore in γ. In
other words, there is strategic complementarity in the choice of search intensity (see Cooper
and John, 1988).
An agent will produce if λ (ρ, τ ) deÞned in (3) is greater than the cost of producing. That
is, we have the production decision rule
1
φi =
∈ {0, 1}
0
>
if λ (ρ, τ ) = c.
<
(4)
Note that we do not allow for mixed strategies. We can write the optimal choice of γ i as
γ i = φi d0−1 (ρ (1 − τ ) F ) .
(5)
The boundary condition limγ i →1 d0 (γ i ) = ∞ is sufficient to guarantee this, but is not
necessary for our results.
4
9
2.3
Private-Sector Equilibrium
In equilibrium, the decisions made by agents must generate the market conditions that each
agent takes as given. The individual decisions generate functions γ : [0, 1] → [0, 1] and
φ : [0, 1] → {0, 1} . Recalling the deÞnitions of γ in (1) and ρ (γ) in (2), we have the following
deÞnition of equilibrium for a given government policy.
DeÞnition: A private-sector equilibrium is a list {φ∗ , γ ∗ , ρ∗ , x∗ } such that
(i) given ρ∗ , φ∗i satisÞes (4) and γ ∗i satisÞes (5) for each private agent
(ii) ρ∗ = ρ(γ ∗ )
R1
(iii) x∗ = τ ρ∗ 0 φ∗i γ ∗i di.
The third condition is the government budget constraint: the amount of public good provided
is equal to total tax revenue. We consider only symmetric private-sector equilibria, where
either all agents produce or no agent produces. We label the beliefs associated with these
equilibria “optimistic” and “pessimistic,” respectively. We begin by looking at the former
case.
2.3.1
Optimistic Beliefs
We Þrst consider the situation where all agents produce and search. Because all agents are
identical, they will choose the same search intensity, which we denote γ H . This is then equal
to the total amount of search intensity in the market γ. Using the deÞnition of ρ in (2), the
search intensity in the optimistic equilibrium is the nonzero solution to
µ
¶
m (γ H )
0−1
γH = d
(1 − τ ) F .
γH
We assume that the functions d and m are such that the right-hand side is concave in
γ H . This implies that γ H is decreasing in τ — higher tax rates lead to lower equilibrium
10
search intensities.5 We use λH to denote the equilibrium expected value of producing under
optimistic beliefs. This expected payoff is given by
λH (τ ) = ρH γ H (1 − τ ) F − d (γ H ) ,
where we have
ρH =
m (γ H )
.
γH
Notice that the strict convexity of m implies that ρH is strictly increasing in γ H and is
therefore strictly decreasing in τ . This, in turn, implies that λH is strictly decreasing in τ .
The intuition is straightforward: by taxing more, the government decreases the beneÞt of
Þnding a trading partner, which leads agents to choose lower search intensities. This makes
the output market less efficient, which further decreases the beneÞt of search effort. In other
words, the tax is distortionary for two reasons; not only does it make producing and trading
less valuable, it also makes Þnding a trading partner more difficult. Furthermore, for a given
realization of the cost of production c, if τ is set high enough λH will fall below c and the
optimistic outcome will not be an equilibrium.
In the optimistic outcome, the government budget constraint reduces to x∗ = τ ρH γ H .
Welfare, measured as the expected utility of each agent, is therefore given by
UH (τ , c) = ρH γ H (1 − τ ) F − d (γ H ) − c + v (τ ρH γ H )
(6)
where ρH and γ H are, of course, functions of τ as discussed above. This expression gives the
welfare level associated with the policy choice τ if the optimistic equilibrium obtains. Let
5
The basic requirement for this to hold is that d have greater curvature than m. For
example, if both functions are of the form xα , we require that the exponent on d be larger
than that on m. If the right-hand side were instead convex in γ H , the comparative statics
would be reversed, with increases in the tax rate raising equilibrium search levels. We focus
on the more intuitive case where higher taxes discourage market activity.
11
b
τ denote the value of τ that maximizes this expression (notice that b
τ is greater than zero
and does not depend on the value of the cost c). This would be the optimal tax rate for the
government to set if it were certain that the optimistic outcome would obtain.
2.3.2
Pessimistic Beliefs
If no other agent searches, the value of searching is zero. Therefore λL = 0 holds, regardless
of the tax rate τ . As long as the cost of production is positive, the optimal decision of an
agent facing these market conditions is to not produce, and hence there is an equilibrium
with UL = 0. Notice that in this equilibrium, no public good is provided. The tax policy
is completely ineffective if agents have pessimistic beliefs. In Section 4, we consider an
additional policy tool (public production) that is more effective in this case.
2.3.3
Multiplicity
From the above discussion it should be clear that the pessimistic equilibrium exists for
all values of τ and c. The optimistic equilibrium, on the other hand, only exists when the
(after-tax) expected value of producing is at least as large as the cost. That is, the optimistic
outcome is an equilibrium if and only if
λH (τ ) ≥ c
(7)
holds. For a given cost c, a high enough tax rate will eliminate the optimistic equilibrium
and leave the pessimistic outcome as the only equilibrium. Our interest is in situations where
multiple equilibria exist for all relevant values of the policy parameter, and hence we do not
want to focus on situations where the government needs to worry about the possibility of
eliminating the optimistic equilibrium. We therefore assume that the following condition
holds:
λH (b
τ ) ≥ cH .
12
(8)
Recalling that λH is decreasing in τ , this condition implies that for any τ ≤ b
τ the optimistic
outcome will be an equilibrium regardless of the realization of the cost variable c. In par-
ticular, the government can set the tax rate to maximize expected utility in the optimistic
outcome and be assured that this outcome will be an equilibrium once the cost is realized.
What value of τ should the government set? Our primary interest is in answering this
question. In other words, we want to formulate and solve an optimal policy problem.
3
The Optimal Tax Rate
Formulating an optimal policy problem in general requires calculating the level of welfare that
is generated by each possible policy choice. When there are multiple equilibria associated
with at least some policy choices, it is not immediately clear how to do this. By far the
most common approach in the literature is to essentially ignore the issue of multiplicity by
assuming that the Pareto-best equilibrium will obtain (see, for example, Diamond, 1982).
This gives a clear policy prescription: the policy should be chosen to maximize expected
utility in the Pareto-best equilibrium. In our setting, the optimal policy would then be b
τ.
Because of the simple structure of our model, this approach might even seem to be without
any loss of generality. In the pessimistic outcome, no trade takes place and hence the tax rate
is irrelevant. Therefore, what could be wrong with setting the tax rate to maximize welfare
in the optimistic outcome? In other words, the tax rate b
τ is (weakly) optimal regardless of
which outcome obtains in equilibrium, and thus it seems like this rate must be the optimal
policy. In this section, we show that this prescription is correct only if the process by which
an equilibrium is selected is completely independent of the value to individual agents of
producing and searching. We argue that such selection rules are intuitively unappealing. We
introduce other selection rules, based on the notion of risk dominance, that allow for the
13
selection process to depend on individual incentives, and we show that under such (more
realistic) rules the optimal tax rate is lower than b
τ.
3.1
Formulating the Optimal Policy Problem
In order to have a completely-speciÞed optimal policy problem, the government must be
able to assign a probability distribution over outcomes to each of its possible policy choices.
With these probabilities, expected welfare can be calculated for each policy choice and the
best policy can be determined. We refer to a complete speciÞcation of such probabilities as
an equilibrium selection mechanism.
DeÞnition: An equilibrium selection mechanism (ESM) is a function π : R+ × C → [0, 1]
that assigns, for each pair (τ , c), a probability π (τ , c) to the optimistic outcome generated
by that pair and a probability (1 − π (τ , c)) to the pessimistic outcome. These probabilities
must be consistent with equilibrium, in that:
(i) π (τ , c) = 0 holds if the optimistic outcome is not an equilibrium for that (τ , c) pair,
and
(ii) π (τ , c) = 1 holds if the pessimistic outcome is not an equilibrium for that (τ , c) pair.
The two restrictions say that an outcome should be assigned positive probability only if it is
actually an equilibrium of the economy based on that (τ , c) pair. For the moment we place
no further restrictions on the function π. Suppose, for example, that the government believes
that the Pareto-best equilibrium will always obtain. Then π will equal one whenever the pair
(τ , c) is such that the optimistic equilibrium exists and zero otherwise. Another ESM could
be generated by assuming that private agents base their beliefs (and hence their actions)
on the realization of a two-state sunspot variable. Suppose that whenever τ and c are such
that both equilibria exist, agents produce and search if sunspots appear and stay home if
14
sunspots do not appear. Then, whenever τ and c are such that both equilibria exist, π (τ , c)
would be equal to the probability of sunspots appearing. In general, any rule by which a
private-sector equilibrium is selected can be represented by some function π satisfying the
two restrictions given above.
Putting aside for the moment the question of where the function π would come from,
notice that using an ESM does allow us to formulate an optimal policy problem. Given
π, a benevolent government should choose the tax rate that maximizes the expected utility
of private agents, where the expectation is both across values of the cost c and across the
possible equilibrium outcomes. In other words, the optimal policy problem is given by
Z cH
max
[π (τ , c) UH (τ , c) + (1 − π (τ , c)) UL (τ , c)] f (c) dc,
τ
cL
which, since UL = 0 holds in this model, simpliÞes to
Z cH
π (τ , c) UH (τ , c) f (c) dc.
max
τ
(9)
cL
The best tax rate for the government to choose will clearly depend on the equilibrium
selection mechanism π. For a given function π, we call the solution to (9) the π-optimal tax
rate and denote it by τ ∗π . We provide next an expanded equilibrium concept that explicitly
incorporates the equilibrium selection mechanism and the government’s optimal choice of
policy given this mechanism.
DeÞnition: A selection-based equilibrium is a function π, a tax rate τ π , and two lists
©
ª
φj , γ j , ρj , xj with j = H, L, such that
ª
©
(i) the list φj , γ j , ρj , xj is a private-sector equilibrium for j = H, L,
(ii) the function π is a well-deÞned equilibrium selection mechanism, and
15
©
ª
(iii) given the function π and the lists φj , γ j , ρj , xj , the tax rate τ π solves the policy
problem (9).
Note that if the model allows for a constant function π to be a well-deÞned ESM, then
the above deÞnition with such a function reduces to the deÞnition of a sunspot equilibrium
(Cass and Shell, 1983), expanded to include an optimizing government. In what follows,
when the meaning is clear from the context, we will drop the preÞxes private-sector and
selection-based, and refer simply to an equilibrium. We now turn to a characterization of the
optimal policy under two different classes of equilibrium selection rules.
3.2
Traditional Selection Rules
As we discussed above, the most common approach to dealing with multiple equilibria in a
policy problem is to simply assume that the Pareto-best equilibrium will always obtain. This
selection rule is sometimes called payoff dominance. We argued above that the π-optimal tax
rate under this rule is given by b
τ . We now verify this argument formally. The equilibrium
selection mechanism under payoff dominance is given by
1 if (7) holds
π (τ ; c) =
.
0
otherwise
Under assumption (8), we have that π (b
τ , c) = 1 for all c. Recall that the tax rate b
τ maximizes
UH (τ , c) for all values of c (see (6)). Therefore b
τ maximizes the integrand in (9) for every
value of c, and hence maximizes the integral. This veriÞes that under the payoff dominance
ESM, b
τ is indeed the optimal policy.
Another common approach to equilibrium selection is to assume that all agents base
their actions on the realization of a sunspot variable. Suppose that after the tax rate is set
and the cost of production becomes known, but before private agents choose their actions,
16
one of two states of nature is revealed. With probability π s ∈ (0, 1) sunspots appear and
with probability (1 − π s ) there are no sunspots. Suppose further that all private agents have
the following belief: If the pair (τ , c) is such that both equilibria exist, then the optimistic
equilibrium will obtain if sunspots appear and the pessimistic equilibrium will obtain if they
do not. Then it is individually optimal for each agent to produce and search if sunspots
appear and to stay home if they do not. That is, there is a sunspot equilibrium where
the optimistic outcome obtains with probability π s whenever both equilibria exist. The
equilibrium selection mechanism for the sunspots selection rule is given by
π
if (7) holds
s
π (τ , c)
.
0
otherwise
Notice that the government’s objective function in (9) under the sunspots ESM is equal
to the objective function under the payoff-dominance ESM multiplied by the constant π s .
Therefore, the policy prescription generated by the sunspots selection rule is exactly the
same as that generated by payoff dominance, and we have the following result.
Proposition 1 Under both the payoff-dominance ESM and the sunspots ESM, the π-optimal
tax rate is given by τ ∗π = b
τ.
This result can be interpreted as giving conditions under which the issue of multiplicity
can be essentially ignored. If the policy only affects allocations in one of the equilibria and
the probability assigned to that equilibrium is a constant, unaffected by the policy choice,
then the policy should be chosen to maximize welfare in that equilibrium.
The sunspots selection rule is probabilistic, meaning that the government’s policy does
not completely determine the Þnal outcome. Instead, the policy typically determines a nondegenerate probability distribution over the set of equilibria. We argue below in favor of
probabilistic selection mechanisms because they are intuitively more plausible and because
17
they tend to deliver more robust policy conclusions. For the moment, however, we focus on a
less-appealing aspect that the sunspots approach shares with payoff-dominance: equilibrium
selection is unaffected by the value to individual agents of producing and searching, and hence
by the government’s policy choice. That is, as long as both equilibria exist, the probabilities
assigned to each equilibrium are the same for all values of τ . This approach to equilibrium
selection seems overly simplistic to us. Imagine two scenarios, one where the tax rate is low
and the value of producing λH is much higher than the realized cost c, and another where
the tax rate is so high that λH is just slightly larger than c. In each of these scenarios,
the optimistic outcome is an equilibrium. Thus both the payoff-dominance ESM and the
sunspots ESM assert that the optimistic outcome is equally likely to obtain in each of the two
scenarios. We would argue, however, that intuitively this prediction seems unreasonable. It
is difficult to believe that agents will behave in the same way regardless of whether the gains
from coordinating to the optimistic outcome are very large or very small. Moreover, this
prediction is at odds with the experimental evidence on coordination games. Van Huyck,
Battalio, and Beil (1990), (1991), for example, report that in a series of experiments the
frequency with which subjects converged to each equilibrium varied systematically with the
treatment variables. In other words, the frequency with which each equilibrium was selected
depended on the incentives that subjects faced to choose different actions (according to the
payoff matrix of the game).6 This evidence indicates that it would be more reasonable to
expect that the optimistic equilibrium would be more likely to obtain in the Þrst scenario,
when λH is large relative to c and the potential payoff from producing and searching is much
higher. It also casts serious doubt on the optimality of the tax rate b
τ . If higher taxes make
coordination on the optimistic outcome less likely, then the arguments given above in favor
6
See Cooper (1999) and Ochs (1995) for reviews of the experimental literature on coordination games.
18
of b
τ are no longer valid. We now examine a class of equilibrium selection mechanisms that
embody the above intuition, and we show that under these rules the optimal tax rate is less
than b
τ.
3.3
Risk Dominance
Harsanyi and Selten (1988) have proposed using risk dominance as an equilibrium selection
criterion.7 In models of the type we are considering, the risk-dominance criterion has a simple
interpretation. Suppose an agent believes that the optimistic and pessimistic equilibrium
outcomes are equally likely to occur (that is, she has a diffuse prior that places equal weight
on the possible equilibrium levels of aggregate demand). Then the optimistic equilibrium is
risk dominant if her optimal action given these beliefs is to produce, while the pessimistic
equilibrium is risk dominant if her optimal action is to not produce. Furthermore, if all
agents assign equal probability to the possible equilibrium levels of aggregate demand, all
agents will choose the risk-dominant action and the risk-dominant equilibrium will obtain.
In this section, we show that applying this approach to our model generates a π function
that can be used as an equilibrium selection mechanism, and we derive the implications of
this mechanism for the optimal policy.
Suppose that the government has set a particular tax rate τ . The crucial question is
then: For what values of the cost c will the optimistic outcome be risk dominant (and hence
obtain)? This will happen if an agent who assigns equal probability to the two possible
7
See Harsanyi and Selten (1988, Section 3.9) and the discussion in Young (1998). Carlsson
and van Damme (1993) provide additional justiÞcation for this criterion. They study global
games (where the payoff structure is determined by a random draw) and show that as
the noise vanishes, iterated elimination of dominated strategies leads to the risk dominant
equilibrium as the unique outcome.
19
equilibrium levels of aggregate demand is willing to produce; that is, if we have
1
1
λH (τ ) + λL (τ ) ≥ c.
2
2
Using the fact that λL = 0 holds for all values of τ , we can rewrite this condition as
1
λH (τ ) ≥ c.
2
(10)
Recall that λH is strictly decreasing in τ . Therefore, a higher tax rate implies that an even
lower draw for the cost of production will be required to make the optimistic equilibrium
risk dominant. The risk-dominance equilibrium selection mechanism is given by
1 if (10) holds
π (τ ; c) =
.
0
otherwise
(11)
Notice that condition (10) is clearly stronger than the condition for the existence of the
optimistic equilibrium (7). If λH is only slightly above c, then the optimistic equilibrium
exists but is not risk dominant, and hence under this mechanism the pessimistic equilibrium
will obtain. Only if the incentive to produce is “large enough” does this mechanism predict
that the optimistic equilibrium will obtain.
Before returning to the optimal policy problem, we note that the (unconditional) probability of the optimistic outcome is given by
·
¸
Z cH
1
Π (τ ) ≡
π (τ ; c) f (c) dc = Prob c < λH (τ ) .
2
cL
If 12 λH (τ ) ∈ (cL , cH ) holds, then using the fact that the distribution of c is uniform allows
us to rewrite this expression as
Π (τ ) =
1
λ
2 H
(τ ) − cL
.
cH − cL
20
This expression is clearly strictly decreasing in τ . In other words, setting a higher tax rate
makes it less likely that the optimistic equilibrium will be risk dominant and hence makes it
more likely that the pessimistic equilibrium will obtain.
We now impose two conditions on the support of the cost variable that make the equilibrium selection problem nontrivial. SpeciÞcally, we assume that
λH (0) > 2cL
and
λH (b
τ ) < 2cH
(12)
hold. The Þrst condition says that if the tax rate is set to zero, then it is possible for the
optimistic equilibrium to be risk dominant. If this condition did not hold, the pessimistic
equilibrium would be risk-dominant for all (τ , c) pairs, and hence the choice of policy would
be irrelevant. Recall that λH is independent of the distribution of the cost variable, and
hence this condition can always be made to hold by setting cL close enough to zero. The
second condition says that when the government chooses the tax rate b
τ , it is possible for the
pessimistic equilibrium to be risk-dominant. If this did not hold, the optimistic equilibrium
would always be risk dominant under b
τ , and hence using risk dominance would essentially
reduce to using payoff dominance. This condition can always be made to hold by setting cH
high enough.
We now show that the equilibrium-selection effect of taxation implies that the government
should set the tax rate lower than b
τ.
Proposition 2 If (12) holds, the π-optimal tax rate under the risk-dominance ESM satisÞes
τ ∗π < b
τ.
Proof: We need to consider two cases. First suppose that λH (b
τ ) < 2cL holds. Then if the
government chooses b
τ the pessimistic equilibrium will be risk dominant for all values of c.
In this case, the optimal policy clearly satisÞes τ ∗π < b
τ , since choosing b
τ gives an expected
21
utility of zero while the Þrst condition in (12) implies that a positive expected utility is
possible.
τ ) > 2cL holds. This
The second, and perhaps more interesting, case is where λH (b
condition implies that under the policy b
τ the optimistic equilibrium is risk dominant for
some values of c. Then deÞne
ΛH (τ ) = λH (τ ) + v (τ ρH γ H ) ,
so that we have
UH (τ , c) = ΛH (τ ) − c.
Note that the deÞnition of b
τ implies that Λ0H (b
τ ) = 0 holds. For values of τ close enough to
b
τ , so that 12 λH (τ ) ∈ (cL , cH ) holds, we can use (11) to write the objective function of the
government as
Z
λH (τ )/2
cL
[ΛH (τ ) − c] f (c) dc.
(13)
τ.
From this expression it is apparent that the optimal policy τ ∗π cannot be greater than b
Moving the tax rate above b
τ lowers both ΛH and λH . Such a move would lead to integrating
a strictly smaller function over a strictly smaller domain, which would clearly yield a lower
level of expected utility. Therefore taking equilibrium selection into account must either
lower the optimal tax rate or leave it unchanged. Since the distribution of c is uniform, (13)
is easily integrated to yield
µ
µ
¶
¶
1
1 1
2
2
λH (τ ) − cL ΛH (τ ) −
λH (τ ) − cL .
2
2 4
The derivative of this expression is given by
µ
¶
·
¸
1
1 0
1
0
λH (τ ) − cL ΛH (τ ) + λH (τ ) ΛH (τ ) − λH (τ ) .
2
2
2
At an interior solution, this derivative must be zero. The Þrst term in parentheses is positive
for all τ ≤ b
τ . The term in square brackets is always positive (this follows from the deÞnition
22
of ΛH ). We saw above that λ0H (τ ) is always negative; therefore at an interior solution Λ0H (τ ∗π )
must be positive. This shows that b
τ cannot be the solution, and therefore that τ ∗π < b
τ must
¥
hold.
Another way of seeing this result is to note that decreasing τ slightly from b
τ will cause
only a small (second order) decrease in the utility value of the optimistic outcome, and will
bring a larger (Þrst order) increase in the probability of attaining that outcome. Therefore
expected utility must increase, and the effect of equilibrium selection on the optimal policy
is clear: it lowers the optimal tax rate. The tax on trade decreases the beneÞt of producing
when aggregate demand is high, which makes producing a riskier (or less attractive) activity
when an agent is uncertain about the level of aggregate demand. This makes coordination
on the optimistic outcome less likely. This proposition demonstrates the central idea of the
paper: when equilibrium selection depends on the incentives faced by individual agents, the
optimal tax rate is lower than b
τ . In such situations, ignoring the issue of multiplicity will
lead to the wrong policy conclusion.
3.4
Other Approaches
Using the risk-dominance criterion is a fairly simple and well-accepted way of studying how
changes in payoffs (caused here by changes in the tax rate) affect the process of equilibrium
selection. However, the risk-dominance ESM does have a substantial drawback: it generates
a large discontinuity in the government’s objective function. Suppose that the government
knew the value of the cost of production c when it chose τ . Then the optimal policy would
be to set τ as close to b
τ as possible, subject to the constraint that the optimistic equilibrium
be risk dominant. This is intuitively rather unappealing, particularly in cases where the
constraint is binding, i.e., when the pessimistic equilibrium is risk dominant for τ = b
τ . The
23
government would then be setting a policy that makes the optimistic outcome barely risk
dominant, with arbitrarily close policies leading to the pessimistic outcome. In our model,
the fact that the government faces some uncertainty about the cost of production eliminates
this sharp discontinuity in the government’s objective function, leading to more robust policy
conclusions.
However, in general it seems more realistic to assume that even for a given value of
c, the government faces some uncertainty about which equilibrium will be selected. In
discussing the sunspots ESM above, we referred to mechanisms that have this property as
being probabilistic. We showed that while the sunspots ESM has the advantage of being
probabilistic, is has the disadvantage of assuming that equilibrium selection is completely
unaffected by the value of producing. At this point it seems natural to ask how one could
combine the probabilistic aspect of the sunspots approach with the payoff-sensitive properties
of risk dominance. In order to do this, we build on the intuition discussed above in which
the likelihood of an equilibrium occurring is related to the strength of the incentives that
agents have to choose that action. Following Young (1998), we deÞne the risk factor of the
optimistic equilibrium to be the smallest probability p such that if an agent believes that
with probability p the likelihood of Þnding a match in the output market is given by ρH
(the level in the optimistic equilibrium), and with probability (1 − p) this likelihood is zero
(its level in the pessimistic equilibrium), then she (weakly) prefers producing (and searching
with effort level γ H ) to not producing.8 The risk factor of the pessimistic equilibrium is then
(1 − p). The risk factor measures how willing an agent is to choose an equilibrium action
8
Young’s deÞnition applies to 2 × 2 games, but the extension to our setting is straightforward. Morris, Rob, and Shin (1995) introduce the related notion of p-dominance for
two-player, multi-action games. In the binary-choice environment, the risk factor of an equilibrium is equal to the smallest number p such that the action proÞle where both agents
choose that equilibrium action is p-dominant.
24
when she is unsure of what the market conditions will be. When an equilibrium has a low
risk factor, agents are willing to choose that action for a wide range of beliefs, and therefore
we should expect that the economy is very likely to coordinate on that outcome. Note that
risk-dominance is a particular equilibrium selection rule that is based on risk factors; it says
that the optimistic equilibrium will be selected whenever its risk factor is less than one-half.
Our intuitive argument in favor of probabilistic ESMs can be restated as saying that when
both equilibria exist, the probability of each equilibrium should be strictly between zero and
one and should be a decreasing function of its risk factor.9 In our model, the risk factor of
the optimistic equilibrium is given by
p=
c − λL (τ )
c
=
.
λH (τ ) − λL (τ )
λH (τ )
Here we see that the risk factor is increasing in both the tax rate and the cost of production.
A higher tax rate or a higher cost will imply that an agent needs to be more optimistic
about the output market in order to be willing to produce. If the probability assigned to
the optimistic outcome is a decreasing function of the risk factor, as we have argued that it
should be, then it will also be a decreasing function of the tax rate and of the cost. We now
state two conditions that summarize these desirable properties that an ESM in our model
should possess:
If (τ , c) is such that both (private-sector) equilibria exist, then
(14)
(i) 0 < π (τ , c) < 1 holds
(ii) π is strictly decreasing in both τ and c.
We have shown elsewhere (Ennis and Keister, 2003b, Section 4) how applying the stochastic
learning process of Howitt and McAfee (1992) to a binary-choice economy with identical
9
Ennis and Keister (2003a) derives the implications of risk-factor-based equilibrium selection in a model of bank runs. It also provides a detailed discussion of the relationship
between the risk-factor-based approach and the standard sunspots approach.
25
agents like the one we study here generates an equilibrium selection mechanism with exactly
these two properties. In the learning approach, information about the true value of the
cost c arrives gradually, and agents take actions and learn about market conditions as this
information is arriving. The coevolution of agents’ beliefs about c and about market conditions determines where the economy converges. In other words, which equilibrium agents
end up coordinating on depends on the actual path taken by the beliefs about c en route to
the true value. Equilibrium selection thus depends on the realization of a sequence of random variables, and is therefore random. Rather than presenting the details of the learning
process here, we will derive the policy implications of any equilibrium selection mechanism
that satisÞes the two conditions given in (14). To keep things simple, we will assume that π
is everywhere differentiable in τ . This is stronger than is necessary, but allows us the prove
the proposition in a simple and transparent way.
Proposition 3 Assume condition (8) holds. For any equilibrium selection mechanism π
satisfying (14), the π-optimal tax rate satisÞes τ ∗π < b
τ.
Proof: The optimal tax rate maximizes
Z cH
π (τ , c) [ΛH (τ ) − c] f (c) dc.
cL
Differentiating with respect to τ yields
¶
Z cH µ
∂π
0
(τ , c) [ΛH (τ ) − c] + π (τ , c) ΛH (τ ) f (c) dc.
∂τ
cL
Evaluating the derivative at b
τ gives us
Z cH
∂π
τ ) − c] f (c) dc.
(b
τ , c) [ΛH (b
cL ∂τ
Condition 8 tells us that the optimistic equilibrium exists at (b
τ , c) for all values of c. This
implies that (i) the term in square brackets is positive for all c and (ii) the ∂π/∂τ term is
26
negative for all c. Therefore, the entire derivative is negative and the optimal tax rate must
¥
be less than b
τ.
The point of Propositions 2 and 3 can be summarized as follows. When higher tax rates
adversely affect the process of equilibrium selection, ignoring the issue of multiplicity of
equilibrium will lead to the wrong policy conclusion. In such cases, there is an additional
cost of taxation that a standard equilibrium analysis will not recognize. Even though the tax
rate b
τ is (weakly) optimal in each of the private-sector equilibria separately, the propositions
show that the true optimal tax rate must be less than b
τ.
4
A Public Employment Policy
In previous sections, we showed how taking equilibrium selection effects into account can
change the optimal level of a policy variable. In this section, we show how these effects also
tend to inßuence the types of instruments that are used as part of the optimal policy. To
demonstrate this, we give the government a second policy tool: it can produce the public
good by employing a fraction of the agents in the economy and directing their production
and search decisions. We use ψ to denote this fraction, and we refer to these agents as the
public sector. We assume that the government must choose ψ at the same time that it sets
the tax rate τ , before the cost of production c is known and before private agents make their
production decisions. However, the government chooses the search intensity of public sector
agents γ G later, at the time that private agents set their search intensity. In other words,
while τ and ψ are policies that must be set in advance, the level of γ G can be adjusted to
the ex-post situation in the economy. We think that this treatment of the decision-making
process captures an important feature of the actual situations faced by policy makers: some
policy instruments are more ßexible while others must be set in advance and hence require
27
making predictions about future market outcomes.
If the public sector were assumed to be as efficient as the private sector, the optimal policy
would involve having a large public sector (thereby reducing the fraction of the economy
subject to coordination failures) and not using taxation. This would not be very interesting.
Our goal is to show that even a very inefficient public employment/production policy —
one that would never be used based on a standard equilibrium analysis — can be part of
the optimal policy when equilibrium selection effects are considered. We therefore assume
that public sector agents are relatively inefficient in the production of the public good. In
particular, when a public sector agent is matched in the output market, only a fraction σ < 1
of the output she receives in trade is transformed into the public good; the rest is lost. In
our example below, we will set σ at a fairly low level so that, in general, taxation is the most
efficient way of providing the public good.
An important difference between taxation and public production is the effect these policies have on the value of producing for private agents. As we have shown above, taxation
decreases the value of producing. Public production, on the other hand, can increase this
value, especially in the case where no other private agents produce. When an agent is deciding whether or not to produce, she may be unsure about the actions of other private agents,
but she knows that at least the ψ public agents will be in the market. Thus the public
employment policy can achieve something that the tax policy cannot: it can decrease the
potential loss associated with having produced when no one else has. Because these losses
never happen in equilibrium, this beneÞt of the policy is not captured in a simple equilibrium analysis or when equilibrium selection follows a payoff-insensitive rule. However, when
equilibrium selection is sensitive to individual incentives (as reßected in the risk factors),
this beneÞt implies that public production increases rather than decreases the probability of
reaching the optimistic equilibrium outcome. Combined with the lower optimal tax rate (see
28
the previous section) and the resulting higher marginal utility of the public good, this property makes public production much more attractive, and in our example makes the optimal
level of such production positive. Hence the set of policy instruments used when equilibrium
selection issues are taken into consideration can be different than those prescribed by an
analysis that ignores the selection problem.
For the sake of simplicity, we assume that the government randomly chooses the agents
who will comprise the public sector and that it does not compensate them. In a more realistic
situation the government would compensate public sector agents to make them indifferent
between public-sector and private-sector work. We work with the former case because it is
notationally much simpler, and does not change the main message. In either case, a natural
objective for the government is to maximize the integral of all agents’ utilities (where all
agents are equally weighted), which in our setting is the same as maximizing the expected
utility of an agent before she is assigned to the public or private sector.
It is worth pointing out here that the public production policy makes it possible for the
government to eliminate the pessimistic outcome as an equilibrium. If the public sector is
made large enough, there will be enough output-market activity to induce a private agent
to produce and search regardless of the actions of other private agents. However, having a
large public sector is costly, both because public production is inefficient and because there
is diminishing marginal utility of the public good. For most parameter values, eliminating
the pessimistic equilibrium is not optimal; expected utility is higher when the Pareto-inferior
outcome is allowed to occur with positive probability.10
10
This point seems relevant for the type of research undertaken by Chatterjee and Corbae
(2000). They study the potential gains of eliminating the possibility of events like the Great
Depression in the U.S. The authors correctly point out that their calculation does not take
into account the cost of implementing the policies that would eliminate these Depressionlike episodes. We show that when the choice of policies is studied explicitly, it may not be
29
The timing of events is as follows. The government Þrst chooses the pair (ψ, τ ). Then the
cost of production c is revealed and agents decide whether to produce or not. Finally both
the government and the agents chose their search intensities. If the economy has a unique
private-sector equilibrium, this is equivalent to saying that the government chooses ψ and
τ before uncertainty about c is revealed and then chooses γ G as a Stackelberg leader in the
determination of the public and private levels of search intensity. If there is more than one
such equilibrium, it also means that the government must choose the policies (ψ, τ ) before
it knows what the level of aggregate demand will be for each possible value of c.
In the Appendix, we describe how to compute the two equilibrium outcomes for given
values of (ψ, τ ) and c. We denote the welfare level in the optimistic outcome by UH (ψ, τ ; c)
and that in the pessimistic outcome by UL (ψ; c). These functions reßect the optimal decision
rules of: (i) private agents, who choose whether to produce or not, and if producing, the
optimal level of search intensity, and (ii) the government, which chooses the search intensity
γ G to maximize welfare in a given private-sector equilibrium.
We now discuss how the government determines the optimal policy pair (ψ, τ ). We
concentrate on this stage because it is where the equilibrium selection issues arise. Suppose
that the government completely ignores the question of equilibrium selection and simply
assumes that the optimistic outcome will always obtain. Then the government chooses ψ
and τ to maximize
Z
cH
UH (ψ, τ ; c) f (c)dc.
cL
b b
Let (ψ,
τ ) be the solution to this problem.11 Alternatively, when the government believes
optimal to fully eliminate the possibility of these types of events. See Dagsvik and Jovanovic
(1994) for a study where a coordination failure of the type we examine here is considered to
represent a Depression-like episode.
11
b b
Later, we will choose parameter values such that λH (ψ,
τ ) > cH holds, and therefore the
30
that equilibrium selection is driven by sunspots, the optimal policy problem is
Z cH
[π s UH (ψ, τ ; c) + (1 − π s ) UL (ψ; c)] f (c)dc,
max
ψ,τ
(15)
cL
where π s is the Þxed probability of the optimistic outcome. Let (ψ s , τ s ) be the solution to
this problem.
We want to compare the solutions to these two problems with the optimal policy when
risk dominance determines equilibrium selection.12 The cut-off value of c under the riskdominance equilibrium selection mechanism is now given by:
e
c (ψ, τ ) ≡
1
[λH (ψ, τ ) + λL (ψ, τ )] ,
2
and the optimal policy problem is similar to (15) but where the constant π s is now replaced
by the function
1
π (ψ, τ ; c) =
0
if c < e
c (ψ, τ ) holds
.
otherwise
Note that this policy problem can be re-written as
Z
Z ec(ψ,τ )
max
UH (ψ, τ ; c) f (c)dc +
ψ,τ
cH
UL (ψ; c) f (c)dc.
e
c(ψ,τ )
cL
b = ψs = 0
Let (ψ ∗ , τ ∗ ) be the solution to this problem. We now provide an example where ψ
and ψ ∗ > 0 hold. In other words, in the following example if the government ignores the
inßuence of individual-level incentives on the equilibrium selection process, it chooses not to
run a public sector. However, when it takes risk-dominance-based equilibrium selection into
account, the government chooses a positive level of public employment.
optimistic outcome is indeed an equilibrium for all values of c.
12
Similar results can be obtained with a probabilistic ESM based on the risk factors of the
equilibria, as presented in Section 3.4
31
Example. There is a quadratic matching function m (x) = x2 , a cubic cost function d (x) =
1
x3 , and the public-good utility function is given by v (x) = x 2 . The cost of production is
uniformly distributed with support [0, 0.15]. We use parameter values F = 1.5 and σ = 0.1.
Table 1: The Equilibrium Selection Effect on Policy
Ignoring Eqm. Selection Risk Dominance ESM
ψ
0.000
0.096
τ
0.087
0.026
λH
0.190
0.263
λL
0.000
0.004
In this example, when the government ignores equilibrium selection and believes that the
optimistic outcome will always obtain, it chooses not to run an active public sector. The
public sector is relatively inefficient and taxation is a lower-cost way of providing the public
good. (Note that λH > cH holds, so that the optimistic outcome is always an equilibrium.)
Similarly, for all values of π s greater than 0.35 a government solving problem (15) would
choose not to run a public sector, i.e., ψ s = 0.13 It is easy to see that in such a case the
optimal tax rate under sunspots τ s will also equal b
τ . However, when the government is
aware of equilibrium selection effects that are driven by risk dominance, the optimal policy
involves an active public sector that employs 9.6% of the population and produces 37% of
13
One way to get a sense of what would be a reasonable value for π s is to note that in
this example, when the government sets the policy at (ψ∗ , τ ∗ ) , the probability of observing
the optimistic equilibrium under risk-dominance selection is about 0.89. Furthermore, the
threshold value for π s depends on the value of σ, the efficiency of public employment, and
with a small enough value of σ the government would choose not to run a public sector for
any value of the probability π s .
32
the public good in the optimistic equilibrium. This demonstrates how taking into account
(risk-factor-based) incentives in the equilibrium selection process can have a signiÞcant effect
on the types of policies that are optimal for a government to use.
5
¤
Concluding Remarks
The above analysis has illustrated what we think is a fairly general property of equilibrium
selection in environments where aggregate demand matters: Policies that reduce the value
to an individual agent of participating in the market will make it less likely that agents
coordinate on a high-market-activity equilibrium. We have shown that risk-factor-based
equilibrium selection mechanisms have clear policy implications: Taking equilibrium selection
into account reveals policies like taxation to be more costly than a simple equilibrium analysis
would indicate, and therefore such policies should be used less. If the government has a
variety of tools available, taking selection effects into account will lead it to substitute away
from the ones that decrease the value of producing and toward the ones that do not, even
if the latter are inefficient from the standard equilibrium viewpoint. We also intend for
the analysis to make a larger point: By taking a stand on the properties of the equilibrium
selection process, one can derive meaningful predictions and policy prescriptions from models
with multiple equilibria.
Our result in Section 4 with respect to public employment is reminiscent of a colorful
suggestion by Keynes (1936, p. 129).
If the Treasury were to Þll old bottles with banknotes, bury them at suitable depths
in disused coal mines which are then Þlled up to the surface with town rubbish, and
leave it to private enterprise on well-tried principles of laissez-faire to dig the notes
up again (the right to do so being obtained, of course, by tendering for leases of the
note-bearing territory), there need be no more unemployment and, with the help of
the repercussions, the real income of the community, and its capital wealth also, would
33
probably become a good deal greater than it actually is. (Emphasis added.)14
In other words, Keynes suggests that the government undertake a costly activity solely in
order to increase the beneÞt to private agents of engaging in market activity. This seems
difficult to justify from a traditional equilibrium point of view. While encouraging agents
to choose higher search intensities does improve the efficiency of the output market, it is
hard to believe that the multiplier effect could possibly be large enough to justify the initial
(entirely wasteful) outlay. It seems likely to us that Keynes instead believed that government
policy could be used to direct the economy toward a good equilibrium (as in the quote from
Diamond (1982) in our introduction). Viewed in this light, such a policy recommendation
seems less absurd. Perhaps it is just an exaggerated way of making our same point: policies
that encourage market activity are more likely to bring about coordination on outcomes with
high levels of market activity.
Appendix: Computing Equilibria with Public Employment
In this appendix, we describe how to compute an equilibrium of the economy for a given
policy (ψ, τ ) and a cost of production c. The economy’s average level of search intensity is
given by
γ = ψγ G +
Z
1−ψ
φi γ i di,
0
and the total amount of the public good provided in equilibrium is equal to
Z 1−ψ
ρφi γ i di.
x = ψργ G σ + τ
0
14
We thank Tom Humphrey for bringing this passage to our attention.
34
The optimal decision rules of a private agent are still given by (4) and (5). We again only
consider symmetric outcomes, so that either all private agents produce and search at intensity
γ H or no private agent produces.
In the optimistic outcome, when private agents produce, the total amount of search
intensity in the market is given by γ = (1 − ψ) γ H + ψγ GH , and the probability of being
matched per unit of search intensity is given by
ρ(γ H , γ GH , ψ) =
m ((1 − ψ) γ H + ψγ GH )
.
(1 − ψ) γ H + ψγ GH
We can Þnd the equilibrium search intensity by inserting ρ(γ H , γ GH , ψ) into (5) and solving
for γ H . This intensity is a function of the policy parameters, and we denote it γ H (γ G , ψ, τ ).
DeÞne also the function ρH (γ G , ψ, τ ) ≡ ρ(γ H (γ G , ψ, τ ), γ G , ψ). Welfare in this case is given
by
UH = (1 − ψ) [λH − c] + ψ [−d (γ G ) − c] + v (xH ) ,
where we have xH = (1 − ψ) τ ρH γ H + ψσρH γ G , and λH = ρH γ H (1 − τ )F − d(γ H ). Using
the deÞnitions of γ H (γ G , ψ, τ ) and ρH (γ G , ψ, τ ) given above, we can obtain the function
UH (γ G , ψ, τ ; c) that gives agents’ expected utility as a function of the policy parameters.
The government then chooses the public sector’s search intensity in the optimistic outcome
to maximize the function UH (γ G , ψ, τ ; c). The solution is given by
γ GH (ψ, τ ) = arg max UH (γ G , ψ, τ ; c).
The optimistic outcome is an equilibrium as long as λH (ψ, τ ) > c holds.
In the pessimistic outcome, when no private agent produces, the total level of search
intensity is given by γ = ψγ G and the average probability of being matched is given by
ρL (γ G , ψ) =
35
m (ψγ G )
.
ψγ G
¡ ¡
¢
¢
Welfare is then given by UL (γ G , ψ; p) = ψ (−d (γ G ) − c) + v ρL γ G, ψ ψγ G σ . Since private
agents do not produce, welfare is equal to the utility provided by the public good minus the
production and search costs of the public sector agents. The government chooses the search
intensity γ G according to
γ GL (ψ) = arg max UL (γ G , ψ; c) .
Note that γ GL is only a function of the size of the public sector ψ. Since private agents are
not producing, the tax policy is irrelevant in this outcome.
For given values of (ψ, τ ) and c, the pessimistic outcome is an equilibrium whenever
a private agent who believes that the average probability of being matched is equal to
ρL (ψ) ≡ ρL (γ GL (ψ), ψ) decides not to produce. If she were to produce, she would set her
search intensity to
γ L (ψ, τ ) = d0−1 (ρL (ψ) (1 − τ )F ) ,
and her expected beneÞt from producing would be given by
λL (ψ, τ ) = ρL (ψ) γ L (ψ, τ )(1 − τ )F − d (γ L (ψ, τ )) .
If λL (τ , ψ) < c holds, then the expected beneÞt of producing is lower than the cost, she
would choose not to produce, and the pessimistic outcome is an equilibrium. However, if
the government sets (τ , ψ) so that λL (τ , ψ) > c holds for a given c, then even an agent with
pessimistic beliefs would choose to produce and therefore such beliefs are not self-fulÞlling.
In other words, the policy has eliminated the pessimistic outcome as an equilibrium and only
the optimistic outcome is possible. However, notice that even if the government recruits all
of the agents in the economy (so that ψ = 1), the value of ρ only reaches
m(γ G )
γG
< 1. Hence
there is a bound on the ability of the government to increase λL , and it may be the case that
36
no level of ψ can eliminate the pessimistic equilibrium for some realizations of c.
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