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 7: Example of a Liner Model - Partial Market Equilibrium

The analytical procedure of solving an economic model is applied to what is known as static or equilibrium analysis. An equilibrium is a collection of selected interrelated variables so adjusted to one another that no inherent tendency to change prevails in the model which they constitute.

Constructing the Model
In a static equilibrium analysis, the standard problem is that of finding the set of values of the endogenous variables which will satisfy the equilibrium analysis of the model.

a) Identifying the variables: We will consider one commodity model and so it is necessary to include only three variables in the model: the quantity demanded (Qd), the quantity supplied (Qs), and its price (P).

b) Equilibrium condition of the identified variables: Having chosen the variables, our next order of business is to specify an equilibrium condition. The standard assumption is that equilibrium occurs in the market if and only if the excess demand is zero (i.e., Qd - Qs = 0), that is, if and only if the market is cleared. But this raises the question of how Qd and Qs themselves are determined. To answer this, we assume that Qd is a decreasing linear function of P (as P increases, Qd decreases). On the other hand, Qs is postulated to be an increasing linear function of P, with the condition that no quantity is supplied unless the price exceeds a particular positive level.

So the model will contain one equilibrium condition (Qd = Qs) plus two behavioral equations (behavior of Qd and Qs) which govern the demand and supply sides of the market.




Translated into mathematical statements, the model can be written as,

Qd = Qs
Qd = a -bP               (a, b > 0)
Qs = -c + dP            (c, d > 0)

For the demand function (Qd), the vertical intercept is at a and its slope is -b, which is -ve and downward sloping. The supply function (Qs) has the slope d being +ve as it is upward sloping, and intercept is at -c. By intercepting the supply curve at a -ve value in y axis, we made sure that the provision stated earlier, that supply will not be forthcoming unless the price is positive (P1 in this case), is satisfied.

Solution by Elimination of Variables
We can set Q = Qd = Qs and rewrite the model equivalently as:

Q = a - bP
Q = -c + dP

=> a -bP = c +dP
=> P* = (a+c)/(b+d)

So the solution value of P, P*, is expressed entirely in terms of the parameters, which represent given data for the model.

To find the value of equilibrium quantity, Q* (= Qd* = Qs*) that correspond to the value P*, simply substitute the demand or supply equation with P* value, and we get,

Q* = a -b * (a+c)/(b+d)
=> Q* = (ad-bc)/(b+d )

The positivity of Q* requires requires that the numerator (ad -bc) be positive as well. Hence, to be economically meaningful (i.e., to have Q* > 0), the present model should contain the additional restriction that ad > bc.

This can rewritten in Set notation as:

D = {(P, Q) | Q = a - bP}
S = {(P, Q) | Q = -c + dP}
D ∩ S = (P*, Q*).

The market equilibrium is unique at (P*, Q*)

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.