Equilibrium Analysis in Economics

1
Chapter 3

• Equilibrium Analysis in Economics

2
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 analysts 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.

3
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.

4
Partial Market Equilibrium A Linear Model

• Constructing the modelSince 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)

5
Partial Market Equilibrium A Linear Model

• Translated into mathematical statements, the
  model can be written as
• Qs Qd Qd a bP (a,b gt 0) Qs -c
  dP (c,d gt 0)

6
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).

7
Partial Market Equilibrium A Linear Model

• Solution of by Elimination of Variables
• We already defined Q a bP Q -c dP a
  bP -c dP (b d)P a c P
  and Q

8
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 Qs 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.

9
Partial Market Equilibrium A Nonlinear Model

• Quadratic Equation versus Quadratic Function
• Quadratic Equation P2 4P 5 0Quadratic
  Function P2 4P 5One 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.

10
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 x1 , x2
  Applying this formula to our quadratic
  equation, where
• a 1, b 4, and c -5 and x P, the roots
  are found to be P1 , P2
• Now we reject P2 -5 value because Price (P)
  cannot be negative, so our solution will be P
  1, and hence Q 4 P2 3 for P 1.

11
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 ( i1,2, , n) If a
  solution exists, there will be a set of prices
  Pi and corresponding quantities Qi such that
  all the n equations in the equilibrium condition
  will be simultaneously satisfied.

12
General Market Equilibrium

• Two Commodity Market Model
• Let us discuss a simple model in which only two
  commodities are related to each other.

13
General Market Equilibrium

• In the above model, a and b coefficients pertain
  to the demand and supply functions of the first
  commodity, and the a and ß 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

14
General Market Equilibrium

• Numerical Example
• Suppose the demand and supply functions are as
  follows
• With these symbols we find the ci and ?i
• We now find the equilibrium values using the
  formulas given above

15
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
• By solving simultaneously, these n equations can
  determine the n equilibrium prices P and Q.

16
Equilibrium in National-Income Analysis

• As an example, we may cite the simplest
  Keynesian national income model
• 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