Diamond coconut model
Encyclopedia
The Diamond coconut model is an economic model
constructed by the American economist and 2010 Nobel laureate
Peter Diamond
which analyzes how a search economy
in which traders cannot find partners instantaneously operates. The model was first presented in a 1982 paper published in the Journal of Political Economy
. The main implication of the model is that people's expectations as to the level of aggregate activity play a crucial role in actually determining this level of aggregate economic activity. A frequent interpretation of its conclusion, as applied to the labor market, is that the so called Natural Rate of Unemployment
may not be unique (in fact there may exist a continuum
of "Natural Rates") and even if it is unique, it may not be efficient. Diamond's model was of interest to new Keynesian economists
who saw it as potential source of coordination failure
, which could cause markets to fail to clear.
The model takes its name from the abstract set up imagined by Diamond. He envisioned an island (a closed economy) populated by individuals who only consume
coconuts. Coconuts are obtained by being picked (they are "produced
") from palm trees at a cost. Because of a particular taboo existing on this island a person who has picked a coconut cannot consume it themselves but must find another person with a coconut. At that point the two individuals can trade their respective coconuts and eat them. The key point is that when an individual finds a palm tree, because climbing the tree is costly, they will only be willing to climb it to get a coconut if there are a sufficiently high number of other individuals who are willing to do likewise. If nobody else is obtaining coconuts then there won't be any potential trading partners and climbing the tree is not worth it. Hence, what individuals believe others will do plays a crucial role in determining the overall outcome. As a result people's (fully rational
) expectations become a self-fulfilling prophecy and the economy can wind up with multiple equilibria, most if not all of them characterized by inefficiency.
(for "employed") and they find trading partners at the rate 
at which point they trade coconuts, earn income 
and become "searchers".
The fact that the probability of finding a trading partner is increasing in the number of people who already have coconuts - mathematically
- represents a "thick market externality"; the "thicker" the market in the sense of more potential traders, the more trades occur. It involves an externality
because each person who chooses to pick a coconut does so with only their own self interest in mind, but the fact that they do so has an effect on the overall social outcome.
People who are currently looking for coconut palm trees find these at a random rate
. This means that the finding of palm trees follows a Poisson process
characterized by the parameter
. If total population is normalized to 1 (hence, 
is the share of the population that is employed) then the number of searchers in this economy is 
.
The figure above illustrates the population flows in this economy.
(or loss) involved in switching states when a trade or coconut-picking occurs. Additionally, out of steady state, the value of the asset may fluctuate over time.
Mathematically, the present discounted value of having a coconut is given by
where
is the value of having a coconut, 
is the value of being in the state "looking for a palm tree", 
is the gain to be realized upon finding a trading partner and 
is the discount rate
which measures individual's impatience. Likewise, the present discounted value of searching for palm trees is given by
where
is the rate at which searchers find palm trees, and 
is the expected cost
(hence it enters with a minus sign) of climbing a palm tree when one is found.
In the general version of the model, the cost of climbing a palm tree is a random draw from some (publicly known) Probability distribution
with non-negative support
, for example the Uniform distribution
on
. This means that on the island "some trees are tall and some are short", and as a result picking coconuts from them can be hard or easy.
. Additionally the cost of obtaining a coconut when one finds a palm tree is constant, at 
(this is the "all trees are of the same height" assumption).
The evolution of the proportion of people who are currently carrying coconuts and looking for trading partners is given by:
if every searcher who finds a palm tree chooses to climb it and obtain a coconut, and
if every searcher who finds a palm tree chooses not to obtain a coconut when coming upon the opportunity of doing so.
In the first equation
is just the number of searchers who happen to find a palm tree at a particular time 
(the "inflow" of coconut carriers), while 
is the number of previous coconut-carriers who managed to successfully find a trading partner and hence reverted back to being searchers (the "outflow"). In the second equation, since nobody ever bothers to climb a tree and obtain coconuts, the number of coconut-carriers simply declines over time. The two potential adjustment paths are illustrated in the figure below.
of this economy, the number of searchers and the number of coconut carriers has to be constant,
. Hence there are two possible steady state in the simple version of the model. The "bad" outcome where nobody who finds a palm tree picks a coconut so that 
and an interior equilibrium where 
. The bad results occurs if everyone who finds a palm tree believes that not enough other people will pick coconuts and as a result it is not worth it to pick the coconut themselves. This then becomes a pessimistic self-fulfilling belief.
Whether or not the good outcome is possible depends on parameter values, and as these determine the value of each asset in steady state. In this case the value of the assets will be constant so that
and we can solve for the difference between 
and 
:
For it to be worth it to climb a palm tree this difference has to be greater than the cost of climbing a tree. If
we have 
which means no one will want to pick coconuts. Hence 
is indeed an equilibrium. Otherwise we need 
. Note that this 
is independent of 
while the 
given above is a function of 
only. This means that the critical value of 
could be below or above the "good" steady state value. If costs of climbing the tree are high, or the agents are very impatient (high 
) then 
will be the only equilibrium. If 
and 
are low then there will be two equilibria, and which one the economy winds up at will depend on initial conditions (the level of employment that the economy begins with).
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
constructed by the American economist and 2010 Nobel laureate
Nobel Memorial Prize in Economic Sciences
The Nobel Memorial Prize in Economic Sciences, commonly referred to as the Nobel Prize in Economics, but officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel , is an award for outstanding contributions to the field of economics, generally regarded as one of the...
Peter Diamond
Peter Diamond
Peter Diamond was an English actor who had trained at the Royal Academy of Dramatic Art and remembered as a stuntman on television or film....
which analyzes how a search economy
Search theory
In microeconomics, search theory studies buyers or sellers who cannot instantly find a trading partner, and must therefore search for a partner prior to transacting....
in which traders cannot find partners instantaneously operates. The model was first presented in a 1982 paper published in the Journal of Political Economy
Journal of Political Economy
The Journal of Political Economy is an academic journal run by economists at the University of Chicago and published every two months by the University of Chicago Press. The journal publishes articles in both theoretical economics and empirical economics...
. The main implication of the model is that people's expectations as to the level of aggregate activity play a crucial role in actually determining this level of aggregate economic activity. A frequent interpretation of its conclusion, as applied to the labor market, is that the so called Natural Rate of Unemployment
Natural rate of unemployment
The natural rate of unemployment is a concept of economic activity developed in particular by Milton Friedman and Edmund Phelps in the 1960s, both recipients of the Nobel prize in economics...
may not be unique (in fact there may exist a continuum
Continuum (theory)
Continuum theories or models explain variation as involving a gradual quantitative transition without abrupt changes or discontinuities. It can be contrasted with 'categorical' models which propose qualitatively different states.-In physics:...
of "Natural Rates") and even if it is unique, it may not be efficient. Diamond's model was of interest to new Keynesian economists
New Keynesian economics
New Keynesian economics is a school of contemporary macroeconomics that strives to provide microeconomic foundations for Keynesian economics. It developed partly as a response to criticisms of Keynesian macroeconomics by adherents of New Classical macroeconomics.Two main assumptions define the New...
who saw it as potential source of coordination failure
Coordination failure (economics)
In economics coordination failure is a concept that can explain recessions through the failure of firms and other price setters to coordinate. In an economic system with multiple equilibria, coordination failure occurs when a group of firms could achieve a more desirable equilibrium but fail to...
, which could cause markets to fail to clear.
The model takes its name from the abstract set up imagined by Diamond. He envisioned an island (a closed economy) populated by individuals who only consume
Consumption (economics)
Consumption is a common concept in economics, and gives rise to derived concepts such as consumer debt. Generally, consumption is defined in part by comparison to production. But the precise definition can vary because different schools of economists define production quite differently...
coconuts. Coconuts are obtained by being picked (they are "produced
Production function
In microeconomics and macroeconomics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs...
") from palm trees at a cost. Because of a particular taboo existing on this island a person who has picked a coconut cannot consume it themselves but must find another person with a coconut. At that point the two individuals can trade their respective coconuts and eat them. The key point is that when an individual finds a palm tree, because climbing the tree is costly, they will only be willing to climb it to get a coconut if there are a sufficiently high number of other individuals who are willing to do likewise. If nobody else is obtaining coconuts then there won't be any potential trading partners and climbing the tree is not worth it. Hence, what individuals believe others will do plays a crucial role in determining the overall outcome. As a result people's (fully rational
Rational expectations
Rational expectations is a hypothesis in economics which states that agents' predictions of the future value of economically relevant variables are not systematically wrong in that all errors are random. An alternative formulation is that rational expectations are model-consistent expectations, in...
) expectations become a self-fulfilling prophecy and the economy can wind up with multiple equilibria, most if not all of them characterized by inefficiency.
Population flows in the model
The agents in the model are always in two "states"; they are either currently carrying a coconut and looking for someone to trade it with, or they are searching for a palm tree in order to possibly pick a coconut. The number of agents who are carrying a coconut at time t is denoted by (for "employed") and they find trading partners at the rate at which point they trade coconuts, earn income and become "searchers".The fact that the probability of finding a trading partner is increasing in the number of people who already have coconuts - mathematically
- represents a "thick market externality"; the "thicker" the market in the sense of more potential traders, the more trades occur. It involves an externalityExternality
In economics, an externality is a cost or benefit, not transmitted through prices, incurred by a party who did not agree to the action causing the cost or benefit...
because each person who chooses to pick a coconut does so with only their own self interest in mind, but the fact that they do so has an effect on the overall social outcome.
People who are currently looking for coconut palm trees find these at a random rate
. This means that the finding of palm trees follows a Poisson processPoisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
characterized by the parameter
. If total population is normalized to 1 (hence, is the share of the population that is employed) then the number of searchers in this economy is .The figure above illustrates the population flows in this economy.
The value of having a coconut or looking for one
Each state can be thought of as a form of an asset, for example, the asset "having a coconut". The present discounted value of this asset depends on the benefit or cost incurred when a person finds a trading partner or a palm tree (this is like a one time divident payment), and the capital gainCapital gain
A capital gain is a profit that results from investments into a capital asset, such as stocks, bonds or real estate, which exceeds the purchase price. It is the difference between a higher selling price and a lower purchase price, resulting in a financial gain for the investor...
(or loss) involved in switching states when a trade or coconut-picking occurs. Additionally, out of steady state, the value of the asset may fluctuate over time.
Mathematically, the present discounted value of having a coconut is given by
where
is the value of having a coconut, is the value of being in the state "looking for a palm tree", is the gain to be realized upon finding a trading partner and is the discount rateTime preference
In economics, time preference pertains to how large a premium a consumer places on enjoyment nearer in time over more remote enjoyment....
which measures individual's impatience. Likewise, the present discounted value of searching for palm trees is given by
where
is the rate at which searchers find palm trees, and is the expected costExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
(hence it enters with a minus sign) of climbing a palm tree when one is found.
In the general version of the model, the cost of climbing a palm tree is a random draw from some (publicly known) Probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
with non-negative support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
, for example the Uniform distribution
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
on
. This means that on the island "some trees are tall and some are short", and as a result picking coconuts from them can be hard or easy.Simple mathematical version of the model
In the most simple version of Diamond's model, the probability of finding a trading partner - another person who's carrying a coconut - is exactly equal to the share of the population that is currently in possession of a coconut,. Additionally the cost of obtaining a coconut when one finds a palm tree is constant, at (this is the "all trees are of the same height" assumption).The evolution of the proportion of people who are currently carrying coconuts and looking for trading partners is given by:
if every searcher who finds a palm tree chooses to climb it and obtain a coconut, and if every searcher who finds a palm tree chooses not to obtain a coconut when coming upon the opportunity of doing so.In the first equation
is just the number of searchers who happen to find a palm tree at a particular time (the "inflow" of coconut carriers), while is the number of previous coconut-carriers who managed to successfully find a trading partner and hence reverted back to being searchers (the "outflow"). In the second equation, since nobody ever bothers to climb a tree and obtain coconuts, the number of coconut-carriers simply declines over time. The two potential adjustment paths are illustrated in the figure below.The steady state
In the steady stateSteady state
A system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...
of this economy, the number of searchers and the number of coconut carriers has to be constant,
. Hence there are two possible steady state in the simple version of the model. The "bad" outcome where nobody who finds a palm tree picks a coconut so that and an interior equilibrium where . The bad results occurs if everyone who finds a palm tree believes that not enough other people will pick coconuts and as a result it is not worth it to pick the coconut themselves. This then becomes a pessimistic self-fulfilling belief.Whether or not the good outcome is possible depends on parameter values, and as these determine the value of each asset in steady state. In this case the value of the assets will be constant so that
and we can solve for the difference between and :For it to be worth it to climb a palm tree this difference has to be greater than the cost of climbing a tree. If
we have which means no one will want to pick coconuts. Hence is indeed an equilibrium. Otherwise we need . Note that this is independent of while the given above is a function of only. This means that the critical value of could be below or above the "good" steady state value. If costs of climbing the tree are high, or the agents are very impatient (high ) then will be the only equilibrium. If and are low then there will be two equilibria, and which one the economy winds up at will depend on initial conditions (the level of employment that the economy begins with).