Chapter 3
Equilibrium Analysis in Economics
The Meaning of Equilibrium
Equilibrium can be defined in various ways.
Equilibrium is “a constellation of selected interrelated variables so
adjusted to one another that no inherent tendency to change
prevails in the model which they constitute”.
Several keywords are important to define:
First, the word selected underscores the fact that there do exist
variables which, by the analyst’s choice, have not been included in
the model.
Second, the word interrelated suggests that, in order for equilibrium
to occur, all variables in the model must simultaneously be in a state
of rest.
Third, the word inherent implies that, in defining an equilibrium, the
state of rest involved is based only on the balancing of the internal
forces of the model, while the external factors are assumed fixed.
The Meaning of Equilibrium
In essence, an equilibrium for a specified model is a situation
characterized by a lack of tendency to change. It is for this reason
that the analysis of equilibrium is referred to as statics.
The desirable variety of equilibrium, which we shall refer to as goal
equilibrium, will be treated later. In this chapter, the discussion will
be confined to nongoal equilibrium, resulting from an impersonal or
suprapersonal process of interaction and adjustments of economic
forces.
Examples of this are
– the equilibrium attained by a market under given demand and supply conditions
– the equilibrium of national income under given conditions of consumption and
investment patterns.
Partial Market Equilibrium – A Linear Model
Constructing the model:
Since one commodity is being considered, it is necessary to include only three
variables in the model:
1. the quantity demanded of the commodity (Qd)
2. the quantity supplied of the commodity (Qs)
3. its Price (P).
The quantity is measured , say, in pounds per week, and price in dollars.
First, we must specify an equilibrium condition. The standard assumption is that
equilibrium occurs in the market if and only if the excess demand is zero
(Qd - Qs = 0).
We assume that Qd is a decreasing function of P, (as P increases, Qd decreases)
and Qs is postulated to be an increasing function of P, (as P increases, Qs
increases)
Partial Market Equilibrium – A Linear Model
Translated into mathematical statements, the model can be written
as:
Qs = Qd
Qd = a – bP
(a,b > 0)
Qs = -c + dP
(c,d > 0)
Partial Market Equilibrium – A Linear Model
Four parameters a,b,c, and d appear in the two linear
functions, and all of them are specified to be positive.
When demand function is graphed (graph above), the vertical
intercept is at a and its slope is –b, which is negative, as
required.
On the other hand, when supply function is graphed, the
vertical intercept is seen to be negative at -c; whereas, the
required type of slope is d being positive.
The equilibrium solution of the model may simply be denoted
by an ordered pair (P*, Q*).
Partial Market Equilibrium – A Nonlinear Model
Let the linear demand in the isolated market model be replaced by
a quadratic demand function, while the supply function remains
linear. Then a model such as the following may emerge:
Qd = Q s
Qd = 4 – P2
Qs = 4p – 1
This can be reduced to 4 – P2 = 4p - 1  P2 + 4P – 5 = 0.
This is a quadratic equation. A major difference between a
quadratic equation and the linear equation is that the former will
yield two solution values.
Partial Market Equilibrium – A Nonlinear Model
Quadratic Equation versus Quadratic Function
Quadratic Equation: P2 + 4P – 5 = 0
Quadratic Function: P2 + 4P – 5
One may legitimately consider each ordered pair in the table- such as (-1,0) and (-5,0) as a
solution to the quadratic function. This can be shown graphically can be as well.
Partial Market Equilibrium – A Nonlinear Model
The Quadratic Formula
In general, given a quadratic in the form: ax2 + bx + c = 0
There are two roots, which can be obtained from the quadratic
formula:
1
2
 b  ( b  4 ac ) 2
x*1 , x*2 =
2a
Applying this formula to our quadratic equation, where
a = 1, b = 4, and c = -5 and x = P, the roots are found to be
1
P*1 , P*2 =
 4  (16  20 ) 2
2

46
 1,  5
2
Now we reject P*2 =-5 value because Price (P) cannot be negative,
so our solution will be P* = 1, and hence Q* = 4 – P2 = 3 for P* = 1.
General Market Equilibrium
For every commodity, there would normally exist many substitutes
and complementary goods. Thus a more realistic depiction of the
demand function of a commodity should take into account the effect
of not only price itself but also the prices of related commodities.
The same holds true for supply function.
The equilibrium condition of an n-commodity market model will
involve n equations, one for each commodity, in the form
Ei = Qdi – Qsi = 0 ( i=1,2, …, n)
If a solution exists, there will be a set of prices P*i and
corresponding quantities Q*i such that all the n equations in the
equilibrium condition will be simultaneously satisfied.
General Market Equilibrium
Two Commodity Market Model
Let us discuss a simple model in which only two commodities
are related to each other.
General Market Equilibrium
In the above model, a and b coefficients pertain to the
demand and supply functions of the first commodity, and the α
and β coefficients are assigned to those of the second. The
model is then reduced to two variables:
We now define the shorthand symbols and derive the
following formula:
General Market Equilibrium
Numerical Example:
Suppose the demand and supply functions are as follows:
Q d 1  10  2 P 1  P 2
Q S 1   2  3 P1
Q d 2  15  P1  P2
Q S 2   1  2 P2
With these symbols we find the ci and γi:
c 0  10  (  2 )  12
  15  (  1)  16
0
c1   2  3   5
c2  1  0  1


1
 1 0  1
2
 1  2  3
We now find the equilibrium values using the formulas given above:
P
P
*
1
*
2

52
, and
14

92
14
, and
Q
Q
*

7
1
*
2
64

85
7
General Market Equilibrium
n-commodity case
The previous discussion of the multicommodity market has been limited to the case of
two commodities. As more commodities enter into the model, there will be more
variables and more equations, and the equations will get longer and complicated. With
n - commodities, we express the demand and supply function as follows:
Q di  Q di ( P1 , P2 ,...., Pn )
Q si  Q si ( P1 , P2 ,...., Pn )
( i  1, 2 , 3 ,..., n )
By solving simultaneously, these n equations can determine the n equilibrium
prices P* and Q*.
Equilibrium in National-Income Analysis
As an example, we may cite the simplest Keynesian national – income
model:
Y  C  I0  G0
C  a  bY
Where Y and C are the endogenous variables national income, and
consumption expenditure, respectively, and the I0 and G0 represent the
exogenously determined investment and government expenditures.
The equilibrium national income, Y* and the equilibrium consumption
expenditure, C* is given by the following equation:
a  I0  G0
*
Y 
1 b
*
C 
a  b(I0  G0 )
1 b