We’re now ready to talk about comparative statics.
In comparative static analysis,
what you do is first you have a model, an economic model,
that you solve for the endogenous variables,
the equilibrium values of your endogenous variables.
And, in comparative static analysis, once
you’ve done that,
you then see how a change in one of the parameters or the exogenous variables
affect the equilibrium values of your endogenous variables.
And you do that by using partial derivatives.
Let me illustrate the comparative static analysis using the simple demand and supply model.
Recall this model.
You have this demand function
where quantity demanded is a linear function
of price.
You have a supply function,
which is assumed to be linear.
And then you have this
equilibrium condition.
In equilibrium, quantity demanded equals quantity supplied.
Now, in this model, there are three equations, three endogenous variables:
Qd, Qs, and P.
Of course, in equilibrium, Qd equals Qs, so you could say,
taking that into account, that there are
only two endogenous variables, price and quantity.
The first step in comparative static analysis is to solve the model for the endogenous variables.
And it’s easy to get the expression for equilibrium price here.
If you just substitute these two
equations, Qd equals Qs,
you get a minus bP equals minus c plus dP.
And adding c to both sides and adding bP to both sides, you get a plus c
equals bP plus dP.
Factoring out the price, you get b plus d times price.
And then, dividing both sides by b plus d, you get your expression for equilibrium price:
a plus c over b plus d equals P star.
And I’m going to put a star on this price to denote that this is the equilibrium price.
And let me just write
it over again and put a plus c over b plus d on the right-hand side of the equation.
So we have
That’s the expression for equilibrium price.  This is just repeating stuff that we did in Chapter 3, a long time ago.
If we wanted the equilibrium quantity,
then all we would have to do is go back
to either the demand curve or the supply curve and substitute in for P.
If we go back to the demand curve and substitute in for P, we get Q star equals a – bP star.
Just go to this demand curve and substitute in Q star for this and P star for that.
And we know what P star is.  We’ve already figured that out.
So, Q star must be that.
Which we can simplify a little bit and rewrite this,
putting b plus d in the denominator and the
numerator here, multiplying by 1.
This is equal to a.
Then, minus b times a plus c all divided by b plus d.
Now these two terms have a common denominator.
And if we write this expression
out, we get ab minus ab, okay, that cancels.
So this numerator just becomes ad minus bc over b plus d.
So, this is our expression for Q star.  Let me write it up here.
Q star, the equilibrium quantity, is
just ad minus bc over b plus d.
And let’s enclose it in a rectangle.
So, here’s our expressions for equilibrium price and equilibrium quantity.
Notice both of these
expressions are expressed in terms of parameters.
In some models, you would have parameters
and exogenous variables.
In this particular model, there are no exogenous variables.  So, it’s just parameters on the right-hand side.
Once we have these expressions,
then we can do comparative static analysis.
And we can say,
okay, what happens to the equilibrium price
if this parameter a changes?
What happens to the
equilibrium price if the parameter b changes?
What happens to the equilibrium price if c
changes?
What happens to the equilibrium price if d changes?
By looking at these comparative static derivatives.
If we’re interested in what happens to the
equilibrium price, then we’re interested in these derivatives.
We’re interested in how the equilibrium price changes as one of these parameters changes, then we’re interested in these four partial derivatives.
And, usually, what we’re looking at is the sign
of these partial derivatives.
Sometimes, we would look at the magnitude.
But, oftentimes, all
we’re interested in is the sign.
What is the sign of this?
Is this greater than or less than zero?
Is that greater than or less than zero?
Is this derivative greater or less than zero?
And notice what this is, this partial derivative.
The sign of it is telling what happens to P star as
a changes.
If partial P star partial a is positive,
that means a and P star move in the same
direction.
As a goes up, P star goes up.
As a goes down, P star goes down.
If partial P star partial a is negative,
that means a and P star move in the opposite direction.
A goes up, P star goes down.
A goes down, P star goes up.
So, that’s all that we have to do for comparative static analysis. That’s all it is.
It’s just, first, solving the model for all the endogenous variables,
expressed in terms of parameters and exogenous variables.
And then, checking all the signs of the partial derivatives.
