A Search Model of
Political and Social Exchange
Brian C. Albrecht
March 1, 2017
1
Economics as the Science of Exchange
Buchanan was clear in what he thought economists should do as scientists. He used his presidential addresses for the Southern Economic Association, the not-so-subtly-titled “What
Should Economists Do?” to lay out his vision for economics as a field (Buchanan 1964).
Buchanan was looking for a complete reorientation of economics away from the optimization/allocation paradigm that he saw as plaguing economists going back to Robbins (1945).
In place of the optimization paradigm, Buchanan suggested that economists should focus
their “intellectual attention on exchange” (Boettke 2014). Buchanan even suggested economics could be renamed “catallactics” or “symbiotics” (Buchanan 1964, p. 217). Such a
focus, according to Buchanan, would place economists back in the seat of Adam Smith, who
started his analysis on the “propensity to truck, barter, and exchange”(Adam Smith 1776,
p. 17). That seat is where economists should try to be.
Buchanan was unique and had a broad definition of exchange, including even cutting
edge topics within economic theory that he was developing with Gordon Tullock (Buchanan
and Tullock 1962). “The approach to economics that I am advancing extends to cover the
emergence of a political constitution” (Buchanan 1964, p. 220). This paper takes Buchanan’s
challenge of developing a broad theory of social exchange seriously. First, I argue that social
scientists have advanced exchange theory since 1964, both outside and inside economics.
1
Outside of economics that development is called “social exchange theory” and focuses on
non-economic exchange, where exchange does not involve buying or selling goods. Within
economics, the development of exchange theory is called “search theory,” which has focused
on economic exchanges, such and buying or selling goods.
After touching on some of these developments within exchange theory, I bring together
non-economic exchange and search theory. To do this, I develop an explicit model of political/
social exchange using search theory. Search theory is a fruitful way forward in the exchange
tradition that Buchanan recommends for two reasons. First, search theory has developed to
model exchange explicitly, just as Buchanan wants, although the focus has been relatively
narrow up to this point. Major economists in the foundation of search theory had the same
concerns that Buchanan had with the optimization paradigm. Some of the things that
Buchanan thought were necessary for a good model, such as assuming only individuals act,
arise naturally in a search model.
Besides being possible to model exchange, search theory has a second benefit. Search
theory is sufficiently well-developed in economics to model broad questions of political and
social exchange right now. Search theory does not need to remain solely about economic
exchange. As evidence of that argument, I present an “off the shelf” search model to examine
social insurance, which is a popular topic in Buchanan’s original field of public finance. In
this search model, there is no state that exists outside of the actions of individuals. “The
government represents only the collective will of individuals” (Buchanan 1949, p. 505).
Such an approach to public finance was advocated as the right way to model government by
Buchanan throughout his career.
Buchanan’s model of the state is contrary to the standard public finance model, which
ignores Buchanan’s call for modeling exchange. Instead, public finance generally assumes
there exists a government that can tax and spend (Chari and Kehoe 1999) and it does so in
Arrow-Debreu-McKenzie general equilibrium framework (Arrow and Debreu 1954; McKenzie
2
1954). There is no exchange between people or with the government in standard public
finance. However in search theory, social insurance instead emerges in a search model from
the actions of individuals who want to exchange goods. In particular, people want to give
away goods to their neighbors when they are lucky, so that they can receive goods from their
neighbors when they are unlucky. Exchange takes places between specific individuals across
states of the world. Because search theory is so common in economics, formulating ideas in
a search model allows for a language to converse with the broader public finance literature1 .
It is a language that macroeconomists are becoming familiar with, just as price theory and
game theory have long provided a common language for microeconomists.
The remainder of the paper is organized as follows. In Section 2, I lay out what Buchanan
meant by political exchange, a major focus of Buchanan’s academic work. I then show
how political exchange has expanded outside of economics to discuss “social exchange.” In
Section 3, I argue that economists already developed models of explicit exchange, particularly
within the realm of search theory. However, search theory has focused on narrowly on
standard economic exchanges. In Section 4, which is the core of the paper, I present a
simple model of social insurance that extends search theory to include a broader concept of
political and social exchange. Section 5 concludes.
2
Modeling Political Exchange
When Buchanan called for a focus on exchange, he did not simply mean exchanges of apples
for oranges at a local market. His meaning was much broader, including more complex
exchanges. “On several occasions, I have suggested that a necessary component in the
public choice perspective is a conception of politics as a complex exchange process in which
1. The search-theoretic approach here is certainly not the first attempt to bring together traditional public
finance and Buchanan’s work in public choice. For explicit recent attempts, see Dixit (1996), Persson and
Tabellini (2002), and Besley (2007).
3
individuals participate in some sense analogous to their participation in markets” (Buchanan
1993, p. 11). Political exchange is one aspect of a more complex exchange process in society.
In its most abstract formulation, political exchange takes place among all
members of the set of individuals who share a common objective and who can
secure this objective more effectively through joint action... The exchange involved here is complex because of the necessary inclusion of all participants in
the demand enterprise. There is no possible factoring of the exchange into a
single-buyer/single-seller relationship... It its idealized limits, this is a model
of purely voluntary exchange, analogous in important respects to exchange in
private goods markets... (p. 12)
In addition to an exchange model of politics, Buchanan claimed “an exchange model
becomes a necessary starting point for meaningful analysis” of taxation (Buchanan 1976, p.
17-8). On how to model political exchanges, Buchanan spent a lifetime laying the groundwork
for such models, going back to Buchanan (1949).
In that article, Buchanan argues that any model of political exchange (taxation being one
form) requires a model of the state. In particular, he distinguishes two possible approaches
for modeling the state (p. 496). The first he labels organismic, where “the state, including
all individuals within it, is conceived as a single organic entity.” However, an organismic
model of the state does not allow for any discussion of political exchange. Since there is only
one entity, there is no exchange. The second viewpoint, which he labels individualistic, sees
the state as “the sum of its individual members acting in a collective capacity.” Only in this
formulation of a state is it possible to discuss political exchange directly. Buchanan urges a
more explicit delineation of these two approaches.
In Buchanan and Musgrave (1999, p. 17), Buchanan says that his 1949 paper was
“nothing more than a plea to my fellow economists to be more explicit about their political
4
presuppositions.” This original article laid a groundwork for much of Buchanan’s future
work on public finance, work that would fall within the individualistic viewpoint and directly
address political exchange. Buchanan’s 1949 article is worth focusing on because, as Atkinson
(1987) says, that article “sets out his (Buchanan’s) position with great clarity; indeed, it reads
like a manifesto for his life.” It lays out how Buchanan thinks political exchange should be
modeled.
For Buchanan, individualistic theories of the state, and therefore of political exchange,
rests on three necessary presuppositions:
1. Only individuals act.
2. Taxes are exchanged for services provided.
3. Political exchange is quid pro quo.
All of these presuppositions are closely connected and Buchanan does not make this explicit
separation. Yet, Buchanan repeatedly stresses them. The last two presuppositions directly
relate to Buchanan’s emphasis on exchange and the first indirectly relates. Buchanan’s call
in 1964 for economists to focus on exchange is already grounded in his first article from 1949.
Only individuals act. Buchanan argues that “the individual replaces the state as the
basic structural unit.” Instead of assuming that a government just exists, taxes, and spends,
an individualistic theory of the state focuses on individual action. Individuals pay taxes.
Individuals collect taxes. Individuals operate courts. When the focus is on individual action,
the social scientist can start talking about exchange in a way that the organismic approach
does not allow.
Taxes are exchanged for services provided. Buchanan goes on to explain how for the
individualistic theory, “the income of the state represents payment made by individuals out
of their economic resources in exchange for services provided... In no way does it imply that
each particular service is ‘sold’ to individuals...” Instead of focusing only on taxing or only
5
on spending, an individualistic theory keeps those connected. Taxes are for governmental
services, although not directly, like payments in a market.
Political exchange is quid pro quo. Drawing on the work of Wicksell (1967, 1934),
Buchanan argues that an individualistic theory sees the relationship between the individual
and the state as quid pro quo. “There should exist the quid pro quo relationship between
government and all individuals taken together. This is represented by a balancing of the net
taxes paid by certain individuals against the net benefits received” (Buchanan 1949, p. 499).
Any other exchange is a market exchange, even if one party to the exchange is a government,
such when governments sell goods.
2.1
Social Exchange
While Buchanan (1949) laid the ground work for an exchange model of political action,
Buchanan developed a broader concept of exchange by the time he wrote “What Should
Economists Do?”. Exchange is everywhere and manifests in different ways.
What I should stress is the potentiality of exchange in those socio-political
institutions that what we normally consider to embody primarily coercive or
quasi-coercive elements. To the extent that man has available to him alternatives
of action, he meets his associates as, in some sense, an "equal," in other words,
in a trading relationship (Buchanan 1964, p. 220-1)
Buchanan’s focus on exchange, broadly defined, has been carried forward within the
social sciences. One major area is outside of economics, in the sociology, where it is referred
to as “social exchange”2 . At the same time that Buchanan was calling on economics to be
a science of exchange, Blau (1964) was developing an alternative theory of social exchange.
For Blau, social exchange is any voluntary exchange. It differs from economic exchange
2. Some economists were also involved. See Landa (1994) and Leeson (2014) for book length treatments
of social exchange.
6
because social exchange involves an unspecified benefit in the future. “Social exchange, in
contrast, involves the principle that one person does another a favor, and while there is a
general expectation of some future return, its exact nature is definitively not stipulated in
advance” (Blau 1964, p. 93). In contrast, economic exchange involves a specified (possibly
future) benefit, such as in a contract.
Blau’s theory of social exchange, well within the Buchanan exchange tradition, was further advanced in the sociology literature. By the time of Coleman (1990), social exchange
had developed to be versatile enough to talk about barter exchange and monetary exchange
(p. 119-124), gift exchange (p. 125-6), political exchange (p. 126-8), and social status exchange (p. 129-131). This focus on exchange is possibly why Buchanan (2003, p. 6) calls
Coleman’s research program part of public choice in general. In fact, as Buchanan (1993,
p. 11) notes “it is not surprising that he (Coleman) was a contributor to Public Choice
(Coleman 1968) and an active participant in the Public Choice Society...”. For the continuity
between Buchanan’s work in particular, Schwartz (1999, p. 144) sees Coleman’s work as a
“contribution to the same intellectual enterprise.” The unifying theme of that enterprise is
a focus on exchange.
3
Search Theory as Exchange Theory
While Buchanan’s particular interest in non-economic (political and social) exchange was
mainly carried forward by political scientists and sociologists, economists did turn their
attention to models of economic exchange. Such models developed out of a frustration, shared
with Buchanan (1964, p. 218), that “the refinements in the theoretical model of perfectly
competitive general equilibrium have been equally, if not more, productive of intellectual
muddle.” While not citing Buchanan, this section argues that economic models of exchange
fall within the Buchanan exchange tradition.
7
The frustration, almost going back to Buchanan (1964), with the intellectual muddle of
general equilibrium led to a desire for explicit models of exchange. Ostroy (1973, p. 597)
writes that “if there is one theme which distinguishes the present treatment from the standard
theory, it is that exchange is a do-it-yourself affair. Individuals will not exchange with ‘the
market’.” Like Buchanan urged, although not referencing Buchanan explicitly, Ostroy was
looking beyond maximizing utility subject to a budget constraint, that is beyond Robbins
(1945). Ostroy was not alone. Many economists recognized that a better model of economic
exchange was needed.
The most dominant approach that emerged within economics to deal with exchange
directly is search theory. Search theory explicitly models the exchange process and has
become ubiquitous in economics. It is the standard approach in labor economics (Mortensen
and Pissarides 1994) and is becoming standard in monetary economics (Kiyotaki and Wright
1993; Lagos and Wright 2005) and finance (Duffie, Garleanu, and Pedersen 2005).
Search theory moves beyond maximizing utility subject to a budget constraint in a particular way. In a search model, compared to an Arrow-Debreu-McKenzie model, constraints
on the physical environment restrict possible trades. Generally, the physical environment is
restricted so that two people must meet before they can exchange. In labor models, workers
must find jobs and explicitly exchange their labor for a wage with a particular firm. Workers
do not simply supply an amount of labor “to the market.” In money or finance models, people must find explicit trading partners, where each trading partner wants the other person’s
good or asset. People do not simply buy and sell goods “to the market.” At every interaction
in a search model, the exchange is explicit. The model says that this person exchanges with
that person3 . In that way, search is an attempt to focus on exchange, as Buchanan (1964)
suggests.
3. One notable exception is Lagos and Wright (2005) which combines a search model with an ArrowDebreu-McKenzie competitive market.
8
Economists are self-aware of their focus on exchange. In a forthcoming Journal of Economic Literature survey article, Lagos, Rocheteau, and Wright (2016, p. 3) argue that one
goal of using search theory is to model “the transactions process explicitly, in the sense that
agents trade with each other. That is not true in GE (general equilibrium) theory...” The
recent finance search literature also focuses on one trader explicitly finding another trader
(Duffie, Garleanu, and Pedersen 2005, p. 1818-9). Like Buchanan, Arrow-Debreu-McKenzie
general equilibrium leaves many economists uncomfortable (Nosal and Rocheteau 2011, p.
3-11).
Search theory provides economists with one way out of the muddle of general equilibrium.
In the same way that “modern game theory has shown in analyzing games with mixed
interests (where the players’ interests are similar in some respects and are dissimilar in other
respects), the same theoretical model can handle both conflict and cooperation without any
difficulty” (Harsanyi 1969, p. 515n), search theory (a subset of game theory) can handle
both conflict and cooperation. This versatility makes search theory a useful tool to study
all exchange. Buchanan was well aware of the beneficial uses of game theory for modeling
these questions and knew that it could be used to construct individualistic theories of the
state. Instead of using cooperative game theory, as used in Buchanan and Tullock (1962, p.
149-189), a search model is a non-cooperative game. Each person acts independently.
All search models can be translated into a more formal non-cooperative game and provide
the versatility of game theory. Search theory is well-developed, common in the economics
literature, and comes in all varieties of formality. However, unlike more standard gametheoretic models, such as the prisoner’s dilemma, search theory has found limited applications
outside of those mentioned above: labor, money, and finance. In particular, search theory
has basically not touched public finance and political economy. One notable exception is
Dixit (2003), which lays out a framework for analyzing contract enforcement in a search
model. The next section presents a possible model to remedy the lack of search within
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public finance and political economy and shows some of its versatility in modeling exchange.
4
Modeling Political/Social Exchange
4.1
Social Insurance as Exchange
To further develop search as a tool for exchange theory, this section presents an informal
model of social insurance. The formal model is in Appendix A. The topic of social insurance
is chosen, not because it needs to be the most important decision facing members of a
collective, nor because Buchanan was especially concerned with social insurance. Instead,
social insurance is chosen because it is the simplest model to lay out how search theory can
be used to model exchange within public finance and political economy. The whole model
is so simple that it collapses down to a decision whether lucky people with lots of goods will
give some of their goods to unlucky people with fewer goods4 .
In the model, each person lives, makes decisions, and maximizes their discounted expected
utility. The model employs homo economicus, as Buchanan (2000, p. 21) urges. This
assumption ensures methodological symmetry between this model and other models in search
theory. At all stages, people choose that which they believe is best for them. More than just
assuming a certain structure to preferences, I also assume a certain structure to beliefs. In
particular, I assume rational expectations. Each person in the economy can properly forecast
all future probabilities. They have the correct model of endowments, matching, and other
players’ strategies.
To be more specific about endowments and matching, but still keep things simple, assume
people only eat apples5 . People want to eat as many apples as possible. Each day everyone
4. Another reason to look at social insurance is that the topic is popular within public finance. For some
examples, see Kocherlakota (1996), Chetty and Saez (2010), Chetty and Finkelstein (2013), and Golosov,
Troshkin, and Tsyvinski (2016) and the references therein. Redistributive taxation is often justified as a
form of social insurance, although that literature is too large to address.
5. Assume apples can only be eaten whole. This avoids complications of how to divide any one apple into
10
wakes up and finds that the tree in their front yard gives them either zero apples or two
apples. Half the people wake up to zero apples. Half of the people wake up to two apples.
People do not have to work for their apples. Right after people see how many apples they
have (and before they can eat their apples), they search around and meet one of their fellow
citizens at random. Search is costless for match these random matchings. All other matches
are impossible. That is, each person meets one and only one person per period. If a person
with two apples meets another person with two apples, they are identical and don’t benefit
from any trade. The same thing happens if both people have zero apples.
There is room for a possible exchange if a person with two apples meets a person with
zero apples. The person with two apples can freely give one of his apples to the person with
zero apples. People consume whatever apples they have and go to sleep. Each person wakes
up the next day to find a new crop of apples and repeats. Apples cannot be stored to the
next time period.
Again, the model is rigged so that giving one apple to the person with zero is the only
possible trade within this economy. Should the person with two apples give one apple to the
person with zero apples, even though he receives nothing in return? What institutions can
be used to ensure trade happens? If people can write a binding contract at the beginning of
time, they would agree to have the person with two apples give one of those apples away every
time they meet a person with zero apples. This is the best agreement for everyone at the
beginning of time. It satisfies (trivially since everyone is identical) the unanimity criterion
of Wicksell (1967) and Buchanan and Tullock (1962, p. 85-96). People would rather have
one apple than a coin-flip between zero and two apples. Therefore, if they could enforce this
agreement, they would. People want some form of “social insurance” from their neighbors if
possible.
But what if people cannot force their future-selves to give away an apple without receiving
smaller units.
11
something in return? What if there is no commitment? There is no government in the model
to act as a deus ex machina and enforce insurance. Each time a person faces this situation,
his immediate incentives are to hold onto his apple. Two apples are better than one. If
there is no enforcement mechanism, then this social insurance agreement is not possible,
even though everyone prefers it. The problem is that as soon as one person wakes up with
two apples, he can decide to keep both. It is in his immediate incentive to do so.
Again, to keep the model as simple as possible, assume the only punishment or reward
that is possible is future insurance. If Person A does not give his second apple to the Person
B, the only thing that Persons B, C, D, etc can do is to refuse to trade in the future. There
is no outside government to come in and enforce insurance. It may seem strange to talk
about political exchange without talking about coercion. However, this is what Buchanan
recommends. “The concept of rationally chosen ‘self-government’ is a necessary starting
point for any analysis of ‘government’ in a many person setting” Buchanan (1975, p.119).
The self-government involves refusing future trades, not coercion.
Following a standard approach in money-search, assume that if Person A refuses to give
his second apple to Person B, the choice becomes public information with some random
chance. “Random monitoring” becomes a means of enforcement. With some probability,
everyone finds out the Person A did not keep up his part of the agreement. Suppose the preconstitutional agreement is “give an apple to everyone with zero apples, but not if you have
heard that they have previously refused to give an apples to other people.” If monitoring
occurs often enough, this social insurance agreement can be maintained.
To see how social insurance can be maintained, consider a person with two apples when
he meets a person with zero apples. His immediate incentive is to keep his apple. He pays
a cost by giving up an apple. The question is whether the benefit of maintaining the social
insurance agreement exceeds this cost. If a person gives away his apple, he has the benefit
of not losing out from future social insurance. He will lose those benefits if he does not give
12
away his apple and is caught. Those future benefits may be enough to get people to maintain
the social insurance agreement.
A few comparative statics follow immediately. Communities will be more likely to sustain social insurance if (1) people are more patient, (2) the community is smaller, and (3)
monitoring is more likely. All of these will enter the pre-constitutional decision of whether
to agree to social insurance. These results match results about achieving cooperation in a
repeated prisoner’s dilemma. The key result is that an observable institution, called social
insurance, arises from the actions of individuals.
4.2
Search-Theoretic, Individualistic State
While overly simple, the above search model is an attempt to show how the “economist can
look on politics, and on the political process, in terms of the exchange paradigm” (Buchanan
2000, p. 17). In this section, I argue that the search model above is build on the presuppositions that make up an individualistic theory for Buchanan. It is therefore useful as a
starting framework for modeling political and social exchange.
Only individuals act. In the model above, everything that could be called “collective
action,” which is only the social insurance, arises directly from individual action. Each
individual takes his best action. Instead of taking governmental institutions as given, the
model has the institution of social insurance emerge. Such a framework allows us to avoid
talking about “the government” as a single entity and forces us to talk about individuals
making decisions.
Because only individuals act, policy must be part of an equilibrium, not outside of it.
Just as Wallace (1998) argues that “Money should not be a primitive in monetary theory– in
the same way that firm should not be a primitive in industrial organization theory or bond
a primitive in finance theory,” nor should taxes be a primitive in public finance theory. Here
the primitives are an environment and individuals acting.
13
Taxes are exchanged for services provided. Because the final policy must arise from the
action of individuals, benefits must outweigh costs for the people involved. To determine
whether that is true, the search model forced a connection between costs and benefits, that
is, between the apples given and the apples received. If we only considered the apples given,
like a model that only looks at taxes paid, the social insurance would never occur.
Political exchange is quid pro quo. Since there is no direct production of goods, the above
model does not capture what Buchanan (1949) meant by quid pro quo. For Buchanan, quid
pro quo means that all citizens combined received as much in services as they pay in taxes.
Without public production, the model does not address Buchanan’s point. But if by quid
pro quo, we mean a trade that is not money for goods, the above agreement is quid pro
quo6 . In the model of social insurance, a person who gave up his apple in any one match,
did not get a direct benefit for that apple. Instead, the agreement is that the person “pay”
today with some chance they will get paid tomorrow. However, there is no guarantee that
anyone will repay the favor in the future. This uncertainty is a key part of social exchange
(Blau 1964, p. 93), even if it is not one of Buchanan’s three necessary presuppositions for
an individualistic state.
In addition to these presuppositions laid out in Buchanan (1949), the search model above
has all three foundational elements of public choice that Buchanan attributes to Wicksell:
methodological individualism, homo economicus, and politics-as-exchange (Buchanan 1987).
By sharing that foundation, the model also shares a problem with the Wicksellian approach.
The next subsection discusses these problems and possible solutions.
6. This is the way the money-search literature generally uses the term. See Lagos, Rocheteau, and Wright
(2016, p. 13)
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4.3
Constitutional Extension of the Model of Exchange
As stated above, modeling political exchange without coercion might seem odd. Buchanan
was aware of this problem, and recognized it in Wicksell’s foundational work. “The Wicksellian approach is rightly termed the voluntary-exchange theory of the public economy. But
the Wicksellian world is far removed from the worlds we inhabit and observe, where transactions costs and free-rider problems are ubiquitous” (Brennan and Buchanan 1980, p. 7).
Buchanan recommends getting around these problems by extending the model to constitutional choices. To do this, Buchanan makes the distinction between the pre-constitutional
and post-constitutional level of analysis. While the above model is about post-constitutional
decisions, there is the possibility “the exchange model is modified by shifting attention to
the level of constitutional choice” (Buchanan 1993, p. 15).
The Wicksellian problems (which the model in this paper inherits) may be addressed
by discussing a pre-constitutional level. In addition to the presuppositions from Buchanan
(1949) highlighted above, search models allow for a separation of the pre- and post-constitutional
stage. The constitution here would be simple, because the decision problem facing the individual is simple. The only question to answer is whether people with two apples should give
one away. A more subtle model would capture other concerns at the constitutional level,
but the main point is that the topic is at least in the discussion with a search model. In
standard public finance, that is not true.
Instead of checking whether a certain agreement is an equilibrium in the above model,
consider a world with many people that are identical at some stage in time, such as the
constitutional phase in Buchanan and Tullock (1962, p. 63-84). They are deciding on an
agreement for how people should behave once people lift the veil of uncertainty(Brennan
and Buchanan 1985, p. 33-36). People need to come to an agreement because they will be
interacting with each other through their lives. There may be mutually beneficial agreements
to make at the pre-constitutional stage. The question facing the group is, how should
15
people behave when they meet each other and how can it be enforced? This type of preconstitutional decision can be discussed within the framework already developed and is where
Buchanan starts his analysis.
Therefore, the constitutional decision is not merely whether to have social insurance, but
how to make it self-enforcing. How can institutions be designed so that people ignore their
temporary self-interest to the benefit of the whole community? With the assumption of
self-interest, the institutions must give people future incentives that are stronger than their
present incentives. The design of these incentives can all be discussed in a more general
pre-constitutional stage than discussed here.
5
Conclusion
Buchanan used his pulpit as the President of the Southern Economic Association to urge
economists to focus on exchange (Buchanan 1964). This paper argues that a subset of social
scientists took Buchanan’s plea seriously and continue to advance exchange theory in areas
not normally associated with Buchanan and public choice. Landa (1994, p. 38) goes as far
as defining social science as “good exchange theory.” Buchanan continued to develop models
of politics-as-exchange. Sociologists further developed the idea to include ideas such as trust
and status (Blau 1964; Coleman 1990).
Independent from the public choice movement that Buchanan, Coleman, and Landa were
part of, exchange theory also developed within narrowly defined economics. It manifested
as search theory. This was a self-conscious effort to model exchange. To repeat a citation
of Lagos, Rocheteau, and Wright (2016, p. 3), the goal of search theory is to model “the
transactions process explicitly, in the sense that agents trade with each other.” However,
the trades that search theory has focused on have been exclusively traditional economic
exchange, such as barter exchange.
16
The main goal of this paper is to show by example that search theory can be expanded
and harnessed to take on broader questions of political and social exchange. To this end, I
constructed a model of social insurance. In the model, exchange is more complex, involving
the whole community as Buchanan (1993, p. 11) argues is part of political exchange. Also,
each person gives up something today for an unspecified benefit in the future. This is
a feature of social exchange (Blau 1964, p. 93). While the model was overly simple, it
highlights some possibilities for search theory as exchange theory.
17
A
Appendix
Time is discrete and continues forever. There is a continuum of ex ante identical agents,
indexed by superscript i, that live forever and have unit measure. There is a single, nonstorable consumption good. The good can be divided only into integer units. Let cit ∈ Z+
be the consumption of agent i in period t. At the beginning of each period, each agent is
endowed with a random quantity of the consumption good. Denote the endowment of agent
i in time t as eit . With a probability of 1/2, eit = 0, otherwise eit = 2. Call eit = 0 a low
endowment and eit = 2 a high endowment.
After receiving their endowment each period, with a probability of α ∈ [0, 1], each agent
is matched bilaterally and at random with another agent. The matching probability is independent of either agent’s endowment. Therefore, any agent is unmatched with probability
1 − α, matched with an agent with a low endowment with probability α/2, and matched
with an agent with a high endowment with probability α/2. In any match, each agent’s
endowment is public information. After being matched, each agent can decide whether to
give part of his endowment to the other agent and whether to accept. A gross transfer to
agent i from agent j is denoted by τtij . Then the match is terminated.
Finally, each agent consumes the amount of the good that he has remaining, cit = eit +
τtij − τtji . Agents have identically utility functions over consumption, u(cit ), which is strictly
increasing and strictly concave. After consuming, agents discount by a common factor
β ∈ (0, 1) and period t + 1 starts. Each agent has preferences over lifetime consumption,
given by
u(ct ) + Et
"∞
X
#
β
r
u(cit+r )
.
r=1
We will focus in the next section of equilibria in symmetric stationary strategies, where
everyone who has the same endowment chooses the same strategy. To simplify notation,
refer to each agent by his endowment i ∈ {L, H} = I. A strategy for each agent is a choice
transfers given that an agent has an endowment of i and is matched with an agent with an
endowment of j in period t. That is a strategy, τ : {I × I}∞ → Z+ . A stationary strategy
does not depend on t, so τ : {I × I} → Z+
Let Vt be the (recursive) value function for an agent at time t, before his endowment is
realized. That is,
18
1 H
1
1
1
1
1
LH
Vt = (1−α) u(0) + u(2) +α u(0) + u(2) + u(eLt + tLH
t ) + u(et − tt ) +βVt+1 ,
2
2
4
4
4
4
where the only possible transfers are when a person with a high endowment is matched with
a person with a low endowment. Otherwise, each agent consumes his endowment, regardless
of whether he was not matched or matched with someone of the same endowment.
The question of interest in this paper is under what circumstances will a person with has
a high endowment give away one unit of the consumption good every time he is matched
with a person with a low endowment. Call this endogenous outcome social insurance. The
next subsections give conditions where social insurance is feasible.
A.1
Allocations
An allocation is a consumption pair (ci , cj ) for every matching of a type i and j and a consumption decision for every unmatched agent. An allocation is feasible if for every matching
if ci + cj = ei + ej , for every unmatched agent ci = ei , and individually rational. Call an
allocation a social insurance allocation if it is feasible and for every matching ci = cj .
A.2
Equilibrium with Commitment
With commitment, we can focus on the set of feasible allocations and when social insurance
is feasible. Consider a recursive value function for the social insurance, where every time
there is a match, the endowment is split. It is given by
1
1
1
1
1
1
u(0) + u(2) + α u(0) + u(2) + u(0 + 1) + u(2 − 1) + βV SI
2
2
4 4 4
4
α
α
α
1
1
SI
1−
u(0) +
1−
u(2) +βV .
u(1) +
|2 {z }
|2
{z2
} |2
{z2
}
V SI = (1 − α)
=
Insurance Match
No Insurance Type 0
No Insurance Type 2
The individual rationality constraint is determined by the value of not participation in the
social insurance. Call this the value of autarky, given by
1
1
V A = u(0) + u(2) + βV A .
2
2
19
From the concavity of u(·), it follows immediately that V SI > V A . Therefore, with commitment social insurance is feasible.
A.3
Equilibrium without Commitment
Without commitment, we need to look for allocations that are incentive feasible (Nosal and
Rocheteau 2011, p. 24). An allocation is incentive feasible if it is feasible and incentive
compatible. To check for incentive compatibility of social insurance, suppose actions become
public with some known probability µ ∈ [0, 1]. Suppose a public announcement of reneging
on the social insurance triggers global autarky. If the IC for a person with high endowment is
matched with a low endowment person is satisfied, then all other ICs are satisfied. Therefore,
to be incentive feasible, it must be that
h
i
u(1) + βV SI ≥ u(2) + β µV A + (1 − µ)V SI .
Some rearranging gives
β
1−β
| {z }
µα
4
|{z}
Discount Monitoring &
Matching
Technology
[2u(1) − u(0) − u(2)] ≥ u(2) − u(1) .
|
{z
“Risk Aversion”
}
|
{z
}
Deviation Gains
Therefore, social insurance is incentive feasible if people are patient enough, monitoring
or matching is frequent enough, or people are risk-averse enough. This a manifestation of
the folk theorem with discounting (Fudenberg and Maskin 1986).
20
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