Comparative statics
Encyclopedia
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"...
, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous
Exogenous
Exogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....
parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
.
As a study of statics
Statics
Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...
it compares two different equilibrium
Economic equilibrium
In economics, economic equilibrium is a state of the world where economic forces are balanced and in the absence of external influences the values of economic variables will not change. It is the point at which quantity demanded and quantity supplied are equal...
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
Supply and demand
Supply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...
when analyzing a single market
Market
A market is one of many varieties of systems, institutions, procedures, social relations and infrastructures whereby parties engage in exchange. While parties may exchange goods and services by barter, most markets rely on sellers offering their goods or services in exchange for money from buyers...
, and to study changes in monetary
Monetary policy
Monetary policy is the process by which the monetary authority of a country controls the supply of money, often targeting a rate of interest for the purpose of promoting economic growth and stability. The official goals usually include relatively stable prices and low unemployment...
or fiscal policy
Fiscal policy
In economics and political science, fiscal policy is the use of government expenditure and revenue collection to influence the economy....
when analyzing the whole economy
Macroeconomics
Macroeconomics is a branch of economics dealing with the performance, structure, behavior, and decision-making of the whole economy. This includes a national, regional, or global economy...
. The term 'comparative statics' itself is more commonly used in relation to microeconomics
Microeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...
(including general equilibrium
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...
analysis) than to macroeconomics
Macroeconomics
Macroeconomics is a branch of economics dealing with the performance, structure, behavior, and decision-making of the whole economy. This includes a national, regional, or global economy...
. Comparative statics was formalized by John R. Hicks (1939) and Paul A. Samuelson (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 theoremImplicit function theorem
In multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is the entire relation, but there may be such a function on...
to calculate a linear approximation
Linear approximation
In mathematics, a linear approximation is an approximation of a general function using a linear function . They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.-Definition:Given a twice continuously...
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
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
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:
.The factor of proportionality
is sometimes called the multiplierMultiplier (economics)
In economics, the fiscal multiplier is the ratio of a change in national income to the change in government spending that causes it. More generally, the exogenous spending multiplier is the ratio of a change in national income to any autonomous change in spending In economics, the fiscal...
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 change in the endogenous variables can be approximated arbitrarily well by 
. In this case, 
represents the 
-by-
matrix of partial derivatives of the functions 
with respect to the variables 
, and 
represents the 
-by-
matrix of partial derivatives of the functions 
with respect to the parameters 
. (The derivatives in 
and 
are evaluated at the initial values of 
and 
.)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's correspondence principle states that stability of equilibrium has testable implications about the comparative static effects. In other words, knowing that the equilibrium is stable may help us predict whether the coefficient 
is positive or negative.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 curveDemand curve
In economics, the demand curve is the graph depicting the relationship between the price of a certain commodity, and the amount of it that consumers are willing and able to purchase at that given price. It is a graphic representation of a demand schedule...
, 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 derivativeTime derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t\,.-Notation:...
of the price — that is, it denotes how fast and in what direction the price changes. By stability theory
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...
, P will converge to its equilibrium value if and only if the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
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.
Comparative statics 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_i,...,x_n.The maximizer
is defined by the 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. By the implicit function theorem
Implicit function theorem
In multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is the entire relation, but there may be such a function on...
, 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
Envelope theorem
The envelope theorem is a theorem about optimization problems in microeconomics. It may be used to prove Hotelling's lemma, Shephard's lemma, and Roy's identity...
).
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 firms profit due to small changes in the tax rates.Comparative statics with constraints
A generalization of the above method allows the optimization problem to include a set of constraints. This leads to the general envelope theoremEnvelope theorem
The envelope theorem is a theorem about optimization problems in microeconomics. It may be used to prove Hotelling's lemma, Shephard's lemma, and Roy's identity...
. Applications include determining changes in Marshallian demand
Marshallian demand function
In microeconomics, a consumer's Marshallian demand function specifies what the consumer would buy in each price and wealth situation, assuming it perfectly solves the utility maximization problem...
in response to changes in price or wage.
Monotone comparative statics
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. Another limitation is that results are cardinalCardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
rather than ordinal
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
; that is, results are not robust to a monotone transformation of the objective function. For economic applications, ordinal results are preferred. In particular, monotone strictly increasing transformations of a utility function represent the same preference relation.
Paul Milgrom and Chris Shannon developed a theory and method for comparative statics analysis using only conditions that are ordinal. The method uses lattice theory
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
and introduces the notions of quasi-supermodularity and the single-crossing condition. The central theorem of monotone comparative statics is:
Suppose
and let 
. Suppose 
, 'p' is quasi-supermodular in 'x' and satisfies the single-crossing property. Then
External links
- San Jose State University Economics Department - Comparative Statics Analysis
- AmosWeb Glossary
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glossary) http://www.amex.com/