Editing Comparative statics (section)

==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 [[Derivative|first derivatives]] of the terms that appear in the equilibrium equations.

For example, suppose the equilibrium value of some endogenous variable <math>x</math> is determined by the following equation:
:<math>f(x,a)=0 \,</math>

where <math>a</math> is an exogenous parameter.  Then, to a first-order approximation, the change in <math>x</math> caused by a small change in <math>a</math> must satisfy:
:<math>B \text{d}x + C \text{d}a = 0.</math>

Here <math>\text{d}x</math> and <math>\text{d}a</math> represent the changes in <math>x</math> and <math>a</math>, respectively, while <math>B</math> and <math>C</math> are the partial derivatives of <math>f</math> with respect to <math>x</math> and
<math>a</math> (evaluated at the initial values of <math>x</math> and <math>a</math>), respectively. Equivalently, we can write the change in <math>x</math> as:
:<math>\text{d}x = -B^{-1}C \text{d}a .</math>

Dividing through the last equation by d''a'' gives the '''comparative static derivative''' of ''x'' with respect to ''a'', also called the [[multiplier (economics)|multiplier]] of ''a'' on ''x'':

:<math>\frac{{\text{d}x}}{{\text{d}a}} = -B^{-1}C.</math>

===Many equations and unknowns===
All the equations above remain true in the case of a system of <math>n</math> equations in <math>n</math> unknowns.  In other words, suppose <math>f(x,a)=0</math> represents a system of <math>n</math> equations involving the vector of <math>n</math> unknowns <math>x</math>, and the vector of <math>m</math> given parameters <math>a</math>.  If we make a sufficiently small change <math>\text{d}a</math> in the parameters, then the resulting changes in the endogenous variables can be approximated arbitrarily well by <math>\text{d}x = -B^{-1}C \text{d}a</math>.  In this case, <math>B</math> represents the <math>n</math>×<math>n</math> [[Jacobian matrix|matrix of partial derivatives]] of the functions <math>f</math> with respect to the variables <math>x</math>, and <math>C</math> represents the <math>n</math>×<math>m</math> matrix of partial derivatives of the functions <math>f</math> with respect to the parameters <math>a</math>. (The derivatives in <math>B</math> and <math>C</math> are evaluated at the initial values of <math>x</math> and <math>a</math>.) Note that if one wants just the comparative static effect of one exogenous variable on one endogenous variable, [[Cramer's Rule]] can be used on the [[Total differentiation#The total derivative via differentials|totally differentiated]] system of equations <math>B\text{d}x + C \text{d}a \,=0</math>.

===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 <math>x</math>, invalidating the use of a linear approximation. Moreover, [[Paul A. Samuelson]]'s '''correspondence principle'''<ref>Samuelson, Paul, "The stability of equilibrium: Comparative statics and dynamics", ''[[Econometrica]]'' 9, April 1941, 97-120: introduces the concept of the correspondence principle.</ref><ref>Samuelson, Paul, "The stability of equilibrium: Linear and non-linear systems", ''Econometrica'' 10(1), January 1942, 1-25: coins the term "correspondence principle".</ref><ref>Baumol, William J., ''Economic Dynamics'', Macmillan Co., 3rd edition, 1970.</ref>{{rp|pp.122–123.}} 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 <math>B^{-1}C</math> 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 <math>B^{-1}</math>,  the sign of the determinant influences the signs of all the elements of the vector <math>B^{-1}C\text{d} a</math> 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:

:<math>Q^{d}(P) = a + bP</math>

:<math>Q^{s}(P) = c + gP</math>

where <math>Q^{d}</math> is the quantity demanded, <math>Q^{s}</math> 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]], 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 <math>P^{eqb}</math>, we find that

:<math>P^{eqb}=\frac{a-c}{g-b}.</math>

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

:<math>\frac{dP}{dt}=\lambda (Q^{d}(P) - Q^{s}(P))</math>

where <math>\lambda</math> > 0 is the speed of adjustment parameter and <math>\frac{dP}{dt}</math> 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]] <math>\frac{d(dP/dt)}{dP}</math> is negative.  This derivative is given by

:<math> \frac{d(dP/dt)}{dP} = - \lambda(-b+g).</math>

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.