The Diamond coconut model is an
economic model
In economics, a model is a theoretical construct representing economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified, often mathematical, framework desig ...
constructed by the American economist and 2010
Nobel laureate
The Nobel Prizes ( sv, Nobelpriset, no, Nobelprisen) are awarded annually by the Royal Swedish Academy of Sciences, the Swedish Academy, the Karolinska Institutet, and the Norwegian Nobel Committee to individuals and organizations who make out ...
Peter Diamond
Peter Arthur Diamond (born , 1940) is an American economist known for his analysis of U.S. Social Security
In the United States, Social Security is the commonly used term for the Federal government of the United States, federal Old-Age, ...
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 ''Journal of Political Economy'' is a monthly peer-reviewed academic journal published by the University of Chicago Press. Established by James Laurence Laughlin in 1892, it covers both theoretical and empirical economics. In the past, the ...
''. 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
The natural rate of unemployment is the name that was given to a key concept in the study of economic activity. Milton Friedman and Edmund Phelps, tackling this 'human' problem in the 1960s, both received the Nobel Memorial Prize in Economic Scienc ...
may not be unique (in fact there may exist a
continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal number ...
of "natural rates") and even if it is unique, it may not be
efficient. Diamond's model was of interest to
New Keynesian economists
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New or NEW may refer to:
Music
* New, singer of K-pop group The Boyz
Albums and EPs
* ''New'' (album), by Paul McCartney, 2013
* ''New'' (EP), by Regurgitator, ...
who saw it as potential source of
coordination failure, which could cause markets to fail to clear.
["Mankiw, N Gregory and Romer, David. "Introduction." ''New Keynesian Economics''. Vol. 1. 1991.]
The model takes its name from the abstract set up imagined by Diamond. He envisioned an island (a
closed economy
Autarky is the characteristic of self-sufficiency, usually applied to societies, communities, states, and their economic systems.
Autarky as an ideal or method has been embraced by a wide range of political ideologies and movements, especially ...
) 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 obtaining coconuts is not worth climbing the tree. Hence, what individuals believe others will do plays a crucial role in determining the overall outcome. As a result, people's (fully
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
) 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 one of 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
externality
In economics, an externality or external cost is an indirect cost or benefit to an uninvolved third party that arises as an effect of another party's (or parties') activity. Externalities can be considered as unpriced goods involved in either co ...
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
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
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
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
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
dividend
A dividend is a distribution of profits by a corporation to its shareholders. When a corporation earns a profit or surplus, it is able to pay a portion of the profit as a dividend to shareholders. Any amount not distributed is taken to be re-in ...
payment), and the
capital gain
Capital gain is an economic concept defined as the profit earned on the sale of an asset which has increased in value over the holding period. An asset may include tangible property, a car, a business, or intangible property such as shares.
...
(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
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
with non-negative
support
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* Supporting character
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, 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.
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 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 state
In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...
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).
See also
*
Search and matching theory
In economics, search and matching theory, is a mathematical framework attempting to describe the formation of mutually beneficial relationships over time. It is closely related to stable matching theory.
Search and matching theory has been es ...
References
{{reflist
*Diamond, Peter A., 1981. “Mobility Costs, Frictional Unemployment, and Efficiency,” Journal of Political Economy 89(4), 798-812.
*Diamond, Peter A., 1982. “Aggregate Demand Management in Search Equilibrium,” Journal of Political Economy 90(5), 881-894
*Rod Cros
pgs 71-72
Labour economics
New Keynesian economics