Post 9: General Market Equilibrium

The equilibrium condition of an n-commodity market model will involve n equations, one for each commodity, in the form:

                                                            (9.1)

If a solution exists, there will be a set of prices and corresponding quantities such that all n equations in the equilibrium condition will be simultaneously satisfied.

Derivation with Two-Commodity Model
To illustrate the general market equilibrium equation, let us consider a simple 2 commodity model. For simplicity, the demand and supply functions of both commodities are assumed to be linear. In parametric terms, such a model can be written as:


                                  

                                  
                                                                                        (9.2)
                                  
                                  

                                  

where the a and b coefficients pertain to the demand and supply functions of the first commodity, and and coefficients are assigned to those of the second, and P1 and P2 are the prices of the two commodities in consideration.The restrictions and the other signs for the coefficients would be considered later.

From 9.2,
                                                   (9.3)       
                                

There are as many as 12 parameters involved in equations 9.3. Let us therefor define the shorthand symbols,
                                          (i = 0, 1, 2)
                                
gives,
                                                                                                  (9.4)
                                

which may be solved further to get
                       
                                
                                

For these two values to make sense, however, certain restrictions should be imposed on the model. First, since division by zero is undefined, so should be . Second, to assure positivity, the numerator must have the same sign as the denominator.

Also, a negative sign in coefficient before the prices would suggest that the two commodities are substitutes for each other.

 Derivation with n-Commodity
 If all commodities in an economy are included in a comprehensive market model, the result will be a Walrasian type of general-equilibrium model, in which the excess demand for every commodity is considered to be a function of the prices of all the commodities in the economy.

Some of the prices may, of course, carry zero coefficients when they play no role in the determination of the excess demand of a particular commodity. In general, however, with n-commodities in all, we may express the demand and supply functions as follows:

                                        (i = 1, 2, ..., n)
                               

 The equilibrium condition iss itself composed of a set of n equations:

                                     (i = 1, 2, ..., n)

So we there have a 3n equations.

Upon substitution, however, the model can be reduced to a set of n simultaneous equations only:

                       

 Alternatively,
                       
        
Solved simultaneously, these n equations can determine the n equilibrium prices , if a solution indeed exists. And then the may be derived from the demand or supply functions.

Solution of a General-Equation System
For a general-function model containing, say, a total of m parameters () - where m is not necessarily equal to n - the n equilibrium prices can be expected to take the general analytical form of

                     

But an important catch exists: the expression can be justified if and only if a unique solution does exist. Yet, there is no a priori reason to presume that every model will automatically yield a unique solution. So it is important to mention the consistency and functional independence as the two prerequisites for application of the process of counting equations and unknowns. In general, in order to apply that process, make sure that:
1) the satisfaction of any one equation in the model will not preclude the satisfaction of another and
2) no equation is redundant

For simultaneous-equation model, there exists systematic methods of testing the existence of unique solution. These would involve, for linear models, an application of the concept of determinants. In the non-linear models, such a test would also require a knowledge of so-called partial derivatives and a special type of determinant, called Jacobian determinant. Those will be discussed in later posts.