Post 10: Equilibrium in National-Income Analysis

The static analysis can be applied to other areas of economics, like Keynesian national-income model too. If we consider the simplest Keynesian national-income model below:

                                  
                                  

where Y = Endogenous variable National Income
           C = Endogenous variable Planned Consumption Expenditure
           = Exogenous variable Investment
          = Exogenous variable Government Expenditure
           a = Parameter variable Autonomous Consumption
           b = Parameter variable Marginal Propensity to Consume

The equations above give us the Y* and C* values as:

                                  
                                  

Both Y* and C* have the expression (1 - b) in the denominator, thus the restriction b 1 is necessary, to avoid division by zero.

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.

Post 6: Steps for solving a mathematical economic model

A mathematical economic model will consist of a system of equations, which may be definitional, behavioral, or in the nature of equilibrium condition. The behavioral equations are usually in the form of functions, which may be linear, or nonlinear, numerical or parametric, and with one or many independent variable(s).

In solving an analytical problem, the first step is to select the appropriate variables - exogenous (variables which are assumed to be determined by forces external to the model) as well as endogenous (variables whose solution we seek from the model) - for inclusion in the model. Next, we must translate into equations the set of chosen analytical assumptions regarding the human (e.g., aggregate consumption pattern in relation to national income), institutional, technological (e.g., production function), legal (e.g., tax structure), and other behavioral aspects of the environment affecting the working of the variables. Only then can we attempt to derive a set of conclusions through relevant mathematical operations and manipulations and to give them appropriate economic interpretations.

The next post (Post 7) on partial market equilibrium would give an example of these steps and how they are used to solve an economic model.

Post 3: Types of Economic Equations

There are 3 types of equations: definitional, behavioral, and conditional.

Definitional Equation: Sets up an identity between two alternate expressions that have exactly the same meaning. For such an equation, the identical-equality ≡ . As an example, total profit is defined as the excess of total revenue over total cost; we can can therefor write:
     

                                  π ≡ R - C

 Behavioral Equation: Specifies the manner in which a variable behaves in response to changes in other variables. This may involve either human behavior (such as aggregate consumption pattern in relation to national income) or nonhuman behavior (such as how total cost of a firm reacts to output changes). Before a behavioral equation can be written, however, it is always necessary to adopt definite assumption regarding the behavior of the variable in question.

Conditional Equation: States a requirement to be satisfied. For example, in a model involving the notion of equilibrium, we must set up an equilibrium condition, which describes the prerequisite for the attainment of equilibrium. Two of the most familiar equilibrium conditions in economics are:

  [quantity demanded = quantity supplied]

and S = I   [Intended saving = Intended investment]