In
economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying
exogenous
In a variety of contexts, exogeny or exogeneity () is the fact of an action or object originating externally. It contrasts with endogeneity or endogeny, the fact of being influenced within a system.
Economics
In an economic model, an exogen ...
parameter.
As a type of ''
static analysis
Static analysis, static projection, or static scoring is a simplified analysis wherein the effect of an immediate change to a system is calculated without regard to the longer-term response of the system to that change. If the short-term effect i ...
'' it compares two different
equilibrium states, after the process of adjustment (if any). It does not study the motion towards equilibrium, nor the process of the change itself.
Comparative statics is commonly used to study changes in
supply and demand when analyzing a single
market, and to study changes in
monetary
Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, such as taxes, in a particular country or socio-economic context. The primary functions which distinguish money are a ...
or
fiscal policy when analyzing the whole
economy. Comparative statics is a tool of analysis in
microeconomics (including
general equilibrium
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an o ...
analysis) and
macroeconomics
Macroeconomics (from the Greek prefix ''makro-'' meaning "large" + ''economics'') is a branch of economics dealing with performance, structure, behavior, and decision-making of an economy as a whole.
For example, using interest rates, taxes, an ...
. Comparative statics was formalized by
John R. Hicks (1939) and
Paul A. Samuelson
Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he "h ...
(1947) (Kehoe, 1987, p. 517) but was presented graphically from at least the 1870s.
For models of stable equilibrium rates of change, such as the
neoclassical growth model,
comparative dynamics is the counterpart of comparative statics (Eatwell, 1987).
Linear approximation
Comparative statics results are usually derived by using the
implicit function theorem to calculate a
linear approximation to the system of equations that defines the equilibrium, under the assumption that the equilibrium is stable. That is, if we consider a sufficiently small change in some exogenous parameter, we can calculate how each endogenous variable changes using only the
first derivatives of the terms that appear in the equilibrium equations.
For example, suppose the equilibrium value of some endogenous variable
is determined by the following equation:
:
where
is an exogenous parameter. Then, to a first-order approximation, the change in
caused by a small change in
must satisfy:
:
Here
and
represent the changes in
and
, respectively, while
and
are the partial derivatives of
with respect to
and
(evaluated at the initial values of
and
), respectively. Equivalently, we can write the change in
as:
:
Dividing through the last equation by d''a'' gives the comparative static derivative of ''x'' with respect to ''a'', also called the
multiplier of ''a'' on ''x'':
:
Many equations and unknowns
All the equations above remain true in the case of a system of
equations in
unknowns. In other words, suppose
represents a system of
equations involving the vector of
unknowns
, and the vector of
given parameters
. If we make a sufficiently small change
in the parameters, then the resulting changes in the endogenous variables can be approximated arbitrarily well by
. In this case,
represents the
×
matrix of partial derivatives of the functions
with respect to the variables
, and
represents the
×
matrix of partial derivatives of the functions
with respect to the parameters
. (The derivatives in
and
are evaluated at the initial values of
and
.) Note that if one wants just the comparative static effect of one exogenous variable on one endogenous variable,
Cramer's Rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of ...
can be used on the
totally differentiated system of equations
.
Stability
The assumption that the equilibrium is stable matters for two reasons. First, if the equilibrium were unstable, a small parameter change might cause a large jump in the value of
, invalidating the use of a linear approximation. Moreover,
Paul A. Samuelson
Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he "h ...
's correspondence principle states that stability of equilibrium has qualitative implications about the comparative static effects. In other words, knowing that the equilibrium is stable may help us predict whether each of the coefficients in the vector
is positive or negative. Specifically, one of the ''n'' necessary and jointly sufficient conditions for stability is that the
determinant of the ''n''×''n'' matrix ''B'' have a particular sign; since this determinant appears as the denominator in the expression for
, the sign of the determinant influences the signs of all the elements of the vector
of comparative static effects.
An example of the role of the stability assumption
Suppose that the quantities demanded and supplied of a product are determined by the following equations:
:
:
where
is the quantity demanded,
is the quantity supplied, ''P'' is the price, ''a'' and ''c'' are intercept parameters determined by exogenous influences on demand and supply respectively, ''b'' < 0 is the reciprocal of the slope of the
demand curve
In economics, a demand curve is a graph depicting the relationship between the price of a certain commodity (the ''y''-axis) and the quantity of that commodity that is demanded at that price (the ''x''-axis). Demand curves can be used either for ...
, and ''g'' is the reciprocal of the slope of the supply curve; ''g'' > 0 if the supply curve is upward sloped, ''g'' = 0 if the supply curve is vertical, and ''g'' < 0 if the supply curve is backward-bending. If we equate quantity supplied with quantity demanded to find the equilibrium price
, we find that
:
This means that the equilibrium price depends positively on the demand intercept if ''g'' – ''b'' > 0, but depends negatively on it if ''g'' – ''b'' < 0. Which of these possibilities is relevant? In fact, starting from an initial static equilibrium and then changing ''a'', the new equilibrium is relevant ''only'' if the market actually goes to that new equilibrium. Suppose that price adjustments in the market occur according to
:
where
> 0 is the speed of adjustment parameter and
is the
time derivative of the price — that is, it denotes how fast and in what direction the price changes. By
stability theory, ''P'' will converge to its equilibrium value if and only if the
derivative is negative. This derivative is given by
:
This is negative if and only if ''g'' – ''b'' > 0, in which case the demand intercept parameter ''a'' positively influences the price. So we can say that while the direction of effect of the demand intercept on the equilibrium price is ambiguous when all we know is that the reciprocal of the supply curve's slope, ''g'', is negative, in the only relevant case (in which the price actually goes to its new equilibrium value) an increase in the demand intercept increases the price. Note that this case, with ''g'' – ''b'' > 0, is the case in which the supply curve, if negatively sloped, is steeper than the demand curve.
Without constraints
Suppose
is a smooth and strictly concave objective function where ''x'' is a vector of ''n'' endogenous variables and ''q'' is a vector of ''m'' exogenous parameters. Consider the unconstrained optimization problem
.
Let
, the ''n'' by ''n'' matrix of first partial derivatives of
with respect to its first ''n'' arguments ''x''
1,...,''x''
''n''.
The maximizer
is defined by the ''n''×1 first order condition
.
Comparative statics asks how this maximizer changes in response to changes in the ''m'' parameters. The aim is to find
.
The strict concavity of the objective function implies that the Jacobian of ''f'', which is exactly the matrix of second partial derivatives of ''p'' with respect to the endogenous variables, is nonsingular (has an inverse). By the
implicit function theorem, then,
may be viewed locally as a continuously differentiable function, and the local response of
to small changes in ''q'' is given by
:
Applying the chain rule and first order condition,
:
(See
Envelope theorem
In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, ...
).
Application for profit maximization
Suppose a firm produces ''n'' goods in quantities
. The firm's profit is a function ''p'' of
and of ''m'' exogenous parameters
which may represent, for instance, various tax rates. Provided the profit function satisfies the smoothness and concavity requirements, the comparative statics method above describes the changes in the firm's profit due to small changes in the tax rates.
With constraints
A generalization of the above method allows the optimization problem to include a set of constraints. This leads to the general
envelope theorem
In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, ...
. Applications include determining changes in
Marshallian demand in response to changes in price or wage.
Limitations and extensions
One limitation of comparative statics using the implicit function theorem is that results are valid only in a (potentially very small) neighborhood of the optimum—that is, only for very small changes in the exogenous variables. Another limitation is the potentially overly restrictive nature of the assumptions conventionally used to justify comparative statics procedures. For example, John Nachbar discovered in one of his case studies that using comparative statics in general equilibrium analysis works best with very small, individual level of data rather than at an aggregate level.
Paul Milgrom and Chris Shannon pointed out in 1994 that the assumptions conventionally used to justify the use of comparative statics on optimization problems are not actually necessary—specifically, the assumptions of convexity of preferred sets or constraint sets, smoothness of their boundaries, first and second derivative conditions, and linearity of budget sets or objective functions. In fact, sometimes a problem meeting these conditions can be monotonically transformed to give a problem with identical comparative statics but violating some or all of these conditions; hence these conditions are not necessary to justify the comparative statics. Stemming from the article by Milgrom and Shannon as well as the results obtained by Veinott and Topkis an important strand of
operational research was developed called
monotone comparative statics. In particular, this theory concentrates on the comparative statics analysis using only conditions that are independent of order-preserving transformations. The method uses
lattice theory and introduces the notions of quasi-supermodularity and the single-crossing condition. The wide application of monotone comparative statics to economics includes production theory, consumer theory, game theory with complete and incomplete information, auction theory, and others.
[See: Topkis, D. M. (1998): Supermodularity and Complementarity, Frontiers of economic research, Princeton University Press, ; and Vives, X. (2001): Oligopoly Pricing: Old Ideas and New Tools. MIT Press, .]
See also
*
Model (economics)
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 de ...
*
Qualitative economics
Qualitative economics is the representation and analysis of information about the direction of change (+, -, or 0) in some economic variable(s) as related to change of some other economic variable(s). For the non-zero case, what makes the change ...
Notes
{{Reflist, 2
References
*John Eatwell et al., ed. (1987). "Comparative dynamics," ''The
New Palgrave: A Dictionary of Economics'', v. 1, p. 517.
*John R. Hicks (1939). ''
Value and Capital''.
*Timothy J. Kehoe, 1987. "Comparative statics," ''The New Palgrave: A Dictionary of Economics'', v. 1, pp. 517–20.
*
Andreu Mas-Colell
Andreu Mas-Colell (; born 29 June 1944) is an economist, an expert in microeconomics and a prominent mathematical economist. He is the founder of the Barcelona Graduate School of Economics and a professor in the department of economics at Pompeu ...
, Michael D. Whinston, and Jerry R. Green, 1995. ''Microeconomic Theory''.
*Paul A. Samuelson (1947). ''
Foundations of Economic Analysis''.
*Eugene Silberberg and Wing Suen, 2000. ''The Structure of Economics: A Mathematical Analysis'', 3rd edition.
External links
San Jose State University Economics Department - ''Comparative Statics Analysis''AmosWeb Glossary(from the
American Stock Exchange
NYSE American, formerly known as the American Stock Exchange (AMEX), and more recently as NYSE MKT, is an American stock exchange situated in New York City. AMEX was previously a mutual organization, owned by its members. Until 1953, it was known ...
glossary
Comparative statics,