Section 15.4: Comparative Statics Using the Implicit Function Theorem
I assigned Section 15.4 because I think it's a good idea for us to see an economic
application of the implicit function theorem. However, I find that this section
is somewhat difficult to follow. Let me give you some big-picture comments to
help you understand what is going on.
- For those who saw it in intermediate micro, this is exactly the Edgeworth
Box framework. You have two consumers (each with their own utility function)
and two goods. To begin with, consumer 1 has all of good 1 and consumer 2
has all of good 2. Consumer 1 can exchange some good 1 with consumer 2 for
some good 2, thus making both consumers better off.
- We have five endogenous variables: four of them tell us each consumer gets
of each good (x1, x2, y1, y2), and one (p) that tells us how much good 1 can
be obtained (by consumer 2 from consumer 1) for a unit of good 2.
- We have three exogenous variables (also known as parameters):
- the parameter (alpha) telling us the relative weight of u(x) and u(y)
in each consumer's utility function.
- the total endowment (e1) of good 1.
- the total endowment (e2) of good 2.
- We have five equations in our five unknowns. Two equations (41) and (42)
come from consumer 1's maximization problem and his budget constraint. Two
more equations (43) and (44) come from consumer 2's maximization problem and
her budget constraint. The last two equations (45) and (46) come from the
total consumption of each good being equal to the total endowment of each
good in the economy. That's six equations, but one of them is redundant: we
can ignore (44) because it is implied by three of the other equations. That
means we have five equations determining five endogenous variables.
- Now that we understand the model, the whole point of the application is
to show how the Implicit Function Theorem can be useful to help us do comparative
statics. In particular, we ask "Suppose the total endowment e2 of good
2 increases. How does this affect each consumer's consumption bundle and the
price of the two goods?" In other words, if we change the exogenous variable
e2 but hold alpha and e1 constant, by how much do x1, x2, y1, y2, and p change?
- The answers, derived from the Implicit Function Theorem, are given in (53).
We can see that the third equation and the fifth equation show us derivatives
that are unambiguously positive, no matter what the shapes of the consumer
utility functions u1(x) and u2(x): the price of good 1 and the quantity of
good 2 consumed by consumer 1 both go up when e2 goes up. The other three
expressions have ambiguous signs that depend on the amounts of concavity of
the two consumer utility functions.
- Note that the quantities r1, r2, R1, R2, and D are defined for notational
convenience, so that the expressions in (53) are simpler to write. This is
a standard technique. Note in particular that we have defined them in ways
that guarantee that each of them are positive, which makes it easier to examine
the expressions in (53) to see if they are positive or negative. These five
quantities all depend on the first and second derivatives of u1(x) and u2(x).