2. THE NATURE OF COMPARATIVE STATISTICS
Comparative statics, as the name suggests, is concerned with the
comparison of different equilibrium positions associated with the
different values of the exogenous variables and the parameters in
the model.
Comparative statics answers the question “how will the
equilibrium value of an endogenous variable change when there
is a change in any of the exogenous variables or parameters.”
is a change in any of the exogenous variables or parameters.”
Comparative statics allows economists to estimate such things
as:
the responsiveness of consumer demand to a projected excise tax, tariff,
or subsidy;
the effect on national income of a change in investment, government
spending, or the interest rate; and
the likely price of a commodity given some change in weather conditions,
price of inputs, or availability of transportation.
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3. EXAMPLE: THE MARKET MODEL
Consider the simple one-commodity market model:
At equilibrium Qd=Qs, using this the market clearing
solutions to the problem will be:
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4. EXAMPLE: THE MARKET MODEL
These solutions are referred to as a reduced form
equation
the two endogenous variables have been reduced to explicit
expressions of the four mutually independent parameters
a, b, c, and d.
To find how an infinitesimal change in one of the
parameters will affect the value of P*, we simply
parameters will affect the value of P*, we simply
differentiate (1) w.r.t each of the parameters.
Similarly, we can draw qualitative or quantitative
conclusions from the partial derivatives of Q* w.r.t
each parameter, such as:
To avoid misunderstanding, however, a clear
distinction should be made between the two
derivatives and
4
*
Q
*
Q
Q
5. CONT’D
The later derivative is a concept appropriate to the
demand function taken alone, and without regard to
the supply function.
The derivative pertains, on the other hand, to the
equilibrium quantity in (2), which takes into account
the interaction of demand and supply together.
Concentrating on P* for the time being, we can get the
*
Q
Concentrating on P* for the time being, we can get the
following four partial derivative from equation 1 given
as:
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7. EXAMPLE: THE MARKET MODEL
Since all the parameters are restricted to being
positive in the present model, we can conclude that:
For a fuller appreciation of the results in (3), let us
For a fuller appreciation of the results in (3), let us
look at Figure 1, where each diagram shows a change
in one of the parameters.
Notice that we are plotting Q (rather than P) on the
vertical axis.
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9. EXAMPLE: THE NATIONAL-INCOME MODEL
Consider a simple national-income model with two
exogenous variables, investment (I0) and government
expenditure (G0):
This model can be solved for Y by substituting the
third equation of (4) into the second and then
substituting the resulting equation into the first.
The equilibrium income (in reduced form) is
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10. EXAMPLE: THE NATIONAL-INCOME MODEL
Similar equilibrium values can also be found for the
endogenous variables C and T, but we shall
concentrate on the equilibrium income.
From (5), there can be obtained six comparative-
equilibrium derivatives.
Among these, the following three have special policy
significance:
significance:
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11. COMPARATIVE STATICS OF GENERAL FUNCTION MODELS
In the comparative static problems considered before,
equilibrium values of endogenous variables of the
model could be explicitly expressed in terms of the
exogenous variables,
Accordingly, the technique of simple partial
differentiation was all we need to obtain the desired
differentiation was all we need to obtain the desired
comparative static information.
However, when a model contains functions expressed
in general form, explicit solutions are not available.
In such cases, a new technique must be employed that
makes use of such concepts as implicit function rule to
find the comparative static derivatives directly from
the given general function model. 11
12. CONT’D
Example :- Consider a market model:
At equilibrium
0
,
0
,
0
,
0
0
1
p
s
p
s
Qs
y
D
p
D
y
p
D
Qd
At equilibrium
Where , p= endogenous
Y0=exogenous
We know that every equation price is a function of
income. i.e. .
Therefore, the equilibrium condition can be taken to be
an identify in the equilibrium solution.
0
)
(
)
,
(
)
(
)
,
( 0
0
P
S
y
P
D
P
S
y
P
D
0
*
y
p
p
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13. CONT’D
The comparative static analysis of this model will
therefore be concerned with how a change in y0 will
affect the equilibrium position of the model, i.e.
i. what is the effect of a change in Y0 on p*?
0
,
0
,
0
*
*
0
*
y
p
F
p
s
y
p
D
i. what is the effect of a change in Y0 on p ?
0
)
(
)
( *
*
0
*
0
0
*
p
s
p
D
y
D
p
F
y
F
dy
dp
13
14. CONT’D
Thus, increase y P*or the other way.
What is the effect of a change in Y0 on Q* ?
At equation , Q*=Qd=Qs
We can write ,
0
*
*
*
.
, y
P
P
and
P
S
Q
Thus, the comparative static results convey the proposition
that an up –ward shift of the demand curve ( due to a rise in
income) will result in a higher equilibrium price as well as a
higher equilibrium quantity.
0
.
0
*
0
*
dy
dp
dp
ds
dy
dQ
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15. LIMITATIONS OF COMPARATIVE STATIC ANALYSIS
By its very nature, comparative statics has the
following limitations.
Ignores the process of adjustment from the old equilibrium
to the new one.
Neglects the time element (length of time ) involved in the
adjustment process from one to another equilibrium.
Assumes that a new equilibrium can be defined and
Assumes that a new equilibrium can be defined and
attained after a disequilibrating change in a parameter,
i.e. disregards the possibility that the new
equilibrium may not be attained ever because of
the inherent instability of the model all these
limitation are addressed by dynamic analysis
which will be dealt in the next chapter.
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16. Reading assignment
Differentiating systems of equations
The Jacobian and hessian determinants
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