San José State University
Department of Economics
& Tornado Alley
Most of economic theory consists of comparative statics analysis. Comparative Statics is the determination of the changes in the endogenous variables of a model that that will reusult from a change in the exogenous variables or parameters of that model. A crucial bit of information is the sign of the changes in the endogenous variables.
There is very limited opportunity to establish the signs of the impacts of changes in macroeconomics or any field that does not have an explicit maximization or minimization operation involved. But in microeconomics comparative statics is a powerful tool for establishing important deductions of theories.
First consider the case without maximization or minimization being involved, such as occurs in macroeconomics. The simplest case is situation in which one variable y is determined by some variable x. Suppose the value of y is determined as the solution to an equation,
This equation holds for all valuees of x so it holds that the differential dy and dx satisfy the equation
This relationship can be solved for dy; i.e.,
In order to know the sign of the impact of a change in x on y we need to know the signs of both derivatives, fx and fy.
Now consider the case in which y is determined such as to maximize some function g(x,y), where x has a value outside of the control of the decision maker. The first order condition for g(x,y) to be a maximum with respect to y is:
The second order condition is that:
If the value of x changes then
We know because y is chosen so as to mazimize g that the second order condition requires that gyy > 0. The sign of the impact of a change in x on y depends only upon the sign of gyx.
Consider a comparative statics analysis of monopoly pricing for a monopolist facing a market with a demand function of the form:
where N is the population in the market area, y is the percapita disposable income and p is the price of the product. a and b are positive parameters.
The total cost C for the firm is given by:
where F is fixed cost and c is the constant marginal variable cost.
A comparative statics analysis tells how the monopoly price would be affected by changes in the exogenous variables N and y and in the parameters F and c.
From the demand function Q = N(ay - bp), the inverse demand function (price as a function of quantity sold) is
The profit function for the monopolist is then
The first order condition for a maximum is
which says that a meaningful solution exists only if (a-bc)>0.
The second order condition for a maximum is
Since b and N are postivie the second order condition is automaticallly satisfied.
The comparative statics results can be determined in this case by simply differentiating the first order condition with respect to the parameters; i.e.,
In the above example the second order condition was automatically satisfied. Now suppose the cost function is
This is the case of U-shaped marginal and average costs.
In this case the first and second conditons for a profit maximum reduce to:
The second order condition is satisfied only if
The first order condition is a quadratic equation in Q. It will have two solutions. One solution will be for a profit minimum and the other for a profit maximum. The solution that is greater than (e-1/bN)/3 will be for the profit maximum.
The partial derivative of the first order condition with respect to a is
The denominator of the fraction involves positive and negative terms so without further information it would not be possible to determine the sign of the ratio. But the second order condition tells us that 3Q>(e-1/bN) so the numerator has to be positive and thus the ratio is positive. Therefore (∂Q/∂a)>0. Likewise the signs of the effects of changes in the other parameters can be determined.
Now consider a couple of cases in which the economic variables are not determined from an optimization procedure.
It should be noted that when variables are not determined by the results of optimization less can be said about the sign of the comparative statics effects.
An important application of comparative statics analysis is in macroeconomics. This is a nonoptimizing application so the opportunity to make theoretical deductions as to the sign of the impact of changes in the exogenous variables is more limited.
A macroeconomic model is given in terms of a set of equations. The simplest macroeconomic model is the following in which
The model is then:
This model is so simple it hardly requires any analysis. It can be solved explicitly for the endogenous variables Y and C; i.e.,
Despite the simplicity of the above model it is worthwhile going through the general procedure which would have to be applied to more complicated models. First we need the differential form of the model, which in the above case is:
The next step is put all the exogenous variables, in this case the differentials of Y and C, on the left side of the equations, leaving the right side for the differentials of the exogenous variables; i.e.,
Then the necssarily linear equations for the differentials are written as a matrix equation.
|| 1||-1 ||
|| -b||1 ||
|| dY ||
|| dC ||
|| 1||1||1 ||
|| 0||0||0 ||
|| dI ||
|| dG ||
|| dNX ||
If the vector of differentials of the endogenous variables is denoted as dZ and the vector of differentials of the exogenous variables as dX then the matrix equations can be expressed in the form
The solution is then
The comparative statics analysis consists of finding the elements of the matrix Γ-1B
While the matrix formulation has certain advantages for the purpose of an introduction to comparative statics it is better to obtain the solutions to the system of equations by way of Cramer's Rule. Cramer's Rule says that the solutions for the dependent variable can be expressed as a ratio of determinants. The denominator of the ratio is the determinant of the matrix of coefficients of the dependent variables. The numerator is the determinant of the matrix constructed by replacing the column of the coefficient matrix by the column of the constants on the RHS of the system of equations.
For example, if the effect of a change in I on Y is sought, then in the above equations dG and dNX are set equal to 0. The system of equations is then
|| 1||-1 ||
|| -b||1 ||
|| dY ||
|| dC ||
|| dI ||
|| 0 ||
To obtain dY in terms of dI take the ratio of the determinants of two matrices. One matrix is the coefficient matrix
|| 1||1 ||
|| -b||1 ||
Note that dY corresponds to the first column of the coefficient matrix so the other matrix is the above matrix with the first column replaced
|| dI||1 ||
|| 0||1 ||
Their determinants are (1-b) and dI, respectively, so the solution for dY by Cramer's Rule is
The value of 1/(1-b) is called the multiplier.
The numerator in the ratio for dC is
|| 1||dI ||
|| -b||0 ||
and thus dC = bdI/(1-b) and hence
This is also the value for ∂C/∂G and ∂C/∂NX
An extension of the analysis for the above macroeconomic model is one which is the same as above except that
If there are no changes in the parameters a and b then the analysis is the same as the previous model with b replaced with b(1-t).
(To be continued.)
This example deals with the interesting aspect of exports and imports being money values rather than physical units so exports and imports are expenditures rather than quanitities.
Suppose exports depend upon the exchange rate E. Let E be the number of foreign currency units per dollar, say 100 yen per dollar. Suppose the demand function for American timber by Japanese users is:
where Q is in physical units per year, say board-feet/yr, and P is the price of timber in yen, say yen per board-foot. If p is the U.S. price of timber, $ per board-foot, the price to Japanese buyers is pE. Thus the physical quantity of timber sold as a function of E is
But for macroeconomic analysis what is needed is the dollar value of the sales; i.e.
Thus the dollar value of the level of exports is negatively related to E; i.e.,
The comparative statics analysis for this case gives effects on the dollar value of exports of the various variables and parameters:
Now suppose we have the demand function for some import to the U.S., say laptop's from Japan,
where Q is the number of laptops per year and p is the price of laptops in dollars. If the price of laptops in Japan is P yen then the price in dollars is P/E. Thus the relationship between physical units of imports and the exchange rate is
But again we want the dollar value of the imports, pQ rather than physical units. Therefore the level of imports is
a more complicated relationship than occured in Example 3 for exports.
Now consider the marginal effects on the dollar value of imports M of a change in the parameters of the demand function, the price of laptops in Japan and the exchange rate E.
An interesting comparative statics problem can now be formulated making use of the ideas presented above. Suppose a Japanese producer has monopoly for television sets in the U.S. as well as Japan. It can set the price for TV's in Japan. Given the exchange rate E the price for TV's in the U.S. is then determined. Let the cost function be
Consider the following:
(To be continued.)
The quintessential economics problem is constrained optimization. Likewise the most interesting comparative statics analysis involves constraints. Consider the problem of mazimizing utility with respect to the consumption of two goods, x1 and x1 subject to a budget constraint, p1x1 + p2x2 = Y. The first order conditions for such a constrained maximzation problem are:
The second order conditions are that the relevant bordered Hessian matrix is negative definite.
Now consider changes in p1 and p2, say dp1 and dp2. and a change in consumer income y, say dY. As a result of the changes in the parameters the rates of consumption will undergo some infintesimal changes, dx1 and dx2. These infinitesimal changes must satisfy the condition
The first order conditions must be satisfied at any values for the parameters. Thus it is valid to differentiate the first order conditions with respect to the parameters. (In differentiating it must be remembered that the Lagrangian multiplier λ is now also a dependent variable like x1 and x2 and a function of the parameteres p1, p2 and Y.) The result is a set of equations that must be satisfied by the infinitesimal changes; i.e.,
These equations are combined with the equation from the budget constraint
These equations form a system which can be represented in matrix form as:
(To be continued.)
Example 6: This is a numerical example of the general case dealt with in the previous materical. Let U=x1x2, with p1=2, p2=1 and Y =12. The values of x1 and x2 and of λ can be determined which maximize utility. Values of x1, x2 and λ can be determined which satisfy the first order conditions. The values of the second derivatives of U at the critical level can also be determined. The second order conditions require that the principal subdeterminants of the bordered Hessian matrix made up of the second derivatives and the prices should have specified signs.
The equations satisfied by effects of changes in the parameters can be created from the first order conditions. This solutions for the effects of the changes in the parameters can be expressed in terms of Cramer's rule as the ratio of determinants. The denominator of these ratios is a determinant whose sign is known from the second order conditions. Thus in many cases comparative statics results can be established with the combined use of the first order and second order conditions.
(To be continued.)
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