Spring 2005
John Rust
Economics 425
University of Maryland
The Implicit Function Theorem and its use in Economics
1
Overview
The
implicit function theorem
is an important result in calculus, since it tells us conditions under which
certain variables in a relationship (i.e. the “dependent variables”) are “implicit functions” of other vari-
ables (the “exogenous, or shifter variables”), and it provides a formula for showing how the endogenous
variables change when the exogenous shifter variables change (i.e. it allows us to calculate the derivative
of the endogenous variables with respect to a change in the exogenous shifter variables).
1.1
Comparative Statics
Thus, the implicit function theorem is a basis for
comparative static exercises
in economics: in such
exercises we want to Fnd out how certain “endogenous variables” change when the exogenous variables
change.
1.2
Supply/Demand Example
Before actually presenting the implicit function theorem, consider the following supply/demand exam-
ple. Consider the price of wheat,
p
which is determined in competitive equilibrium by the intersection
of the aggregate supply and demand curves for wheat. Let
p
*
be the equilibrium price, and let
r
be the
amount of rainfall. Let’s assume that rainfall affects the supply of wheat but not the demand for wheat,
so we can write
S
(
p
*
,
r
) =
D
(
p
*
)
(1)
as the basic “supply=demand” equilibrium condition determining the equilibrium price of wheat,
p
*
.
Now in this example, the price of wheat,
p
*
, is the “endogenous variable” (since the price is being
determined to set supply equal to demand) and the amount of rainfall,
r
, is the “exogenous” or shifter
variable.
The amount of rainfall can change, but the amount of rainfall does not have to obey any
supply/demand equilibrium condition. Rainfall is determined by the weather, and this is why we call it
“exogenous”, meaning that this variable is determined “outside of the system”. However the price of
wheat is “endogenous” because it is determined by our theory of economics, namely that prices adjust
to set supply equal to demand. Thus
p
*
is determined “inside the system” (in this case, the “system” is
speciFed by the supply and demand equations), and this is why we refer to it as the endogenous variable.
Now if rainfall changes, prices will adjust so that for each possible level of rainfall, there will be a
new equilibrium price that will depend on the level of rainfall. Thus we can write
p
*
(
r
)
to emphasize
that the equilibrium price of wheat is an
implicit function
of the level of rainfall. This implicit function
must satisfy, for each possible level of rainfall
r
S
(
p
*
(
r
)
,
r
) =
D
(
p
*
(
r
))
(2)
Now a common
comparative static exercise
is to ask, “does
p
*
(
r
)
increase or decrease as rainfall
increases?” That is, we seek to Fnd out whether the derivative,
∂
p
*
(
r
)
/
∂
r
, is positive or negative (or
1