Download The textbook starts with IS, LM, and the Phillips curve altogether in a

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Topic 1. Comparative Statics
I. The Simplest Macroeconomic Model
-Goods Market only;
-Static Expectations
1.
Introduction
The textbook starts with IS, LM, and the Phillips curve altogether in a model. We will
start simply with the IS curve, and then proceed to work with the IS and LM. Finally we
will take all three, i.e., IS, LM, (Aggregate Demand which combines IS and LM), and the
Aggregate Supply into a consideration.
Here, we are interested in qualitative natures of solutions and comparative statics. Why
are we less interested in the quantitative computation of the equilibrium national income?
It is because that in reality we usually say “Suppose that right now the economy is at
equilibrium. If the government decides to spend extra, say, $1billion, what would be the
impact on the national income?: How much will there be an increase in the national
income?” The question presumes that we are at the equilibrium. The only remaining
question about the equilibrium is whether that equilibrium is stable or not. It is not a
quantitative question, but a qualitative question: What is the characteristic of the current
equilibrium? Then we move onto to address the Comparative Statics question in the
simplest model with the goods market only and the static expectations.
What do you mean by the comparative statics? It was a change in Y* or i* in response to
a change in exogenous variables or independent variables such as Co, To, Io, To, Go, or
MS. In math, it is dY/dGo, for example. Thus we use the general functional form, as
opposed to a specific functional form such as of a linear form.
In this section of our note, we will focus on the Comparative Statics, and also on the
qualitative nature of stability of the equilibrium.
First, let’s work on the IS curve.
Recall that the IS curve summarizes the equilibrium conditions of the goods market in
terms of interest rates and national income level. In other words, the IS curve shows the
combinations of (i, and Y) which bring the equilibrium in the goods market. Of course,
the equilibrium in the goods market is established when the demand for goods and the
supply of goods are equal to each other.
Essentially, we are dealing with the same question of comparative statics as we have
done in the review of IS-LM. However, we would like to gear up to deal with more
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complex comparative statics in a more convenient mathematical way. The catch is that
“convenience” calls for an advanced mathematical skill: Total Differentials plus
Cramer’s Rule in this case.
Beside these new topics, what more complications do we have to know?

We are going to use general functional forms, not specific functions for
the components for IS-LM.
For example, we are no longer using C = Co + c1 (Y- T).
C is a function of Disposable Income, which is a function of Income and
Tax. Thus,
C = f(Y-T) .
If we take a simplifying assumption that Tax is a lump-sum and fixed or
T = To for now, we get
Thus C = f( Y) .
The main advantage of this general functional form over a specific
function at hand is that the consumption function does not have to be
specified, and it can be non-linear in Y (although we do not see an
immediate merit for this case).
Recall when we write the function, we do not write down parameters
(coefficients) or constants(exogenous variables). If you want to carve up
the exogenous variables, you may write down:
C = f(Y; Co, To).
Instead of using the notation for “function” in the order of f, g, h, etc, as in
the math convention, we may use the same letter of the variable for the
functional form:
C =C(Y; Co, To).
In the past, we used c1 for the marginal propensity to consume. Now we
may use dC/d(Y-T) as the marginal propensity to consume of disposable
income.
C = C(Y)
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In this special case, where T is an exogenous variable set by the
government or T = To, dC/d(Y-T) = dC/ dY.
dC/d(Y-T) is the MPC. Instead of this cumbersome notation, we may use
CY-T. dC/dY is the derivative of C with respect to Y. Instead of this
cumbersome notation, we may use CY.
In this special case, dC/dY = CY-T. or MPC = CY
You may carve up the parameter explicitly such as
C = C ( Y; Co, To: CY ).
However, usually we do not write down the exogenous variables or
parameters except in our mind.
How about the general case, where Tax is not all fixed but has an element
proportional to income: T =T(Y), such as in the case where T = To + t1 Y
(it does not have to linear like this), and TY is the marginal tax rate.
C = C( Y-T ) = C ( Y- T( Y ) ), where
In the general case,
dC/dY
= dC/d(Y-T) x d(Y-T) /dY + dC/d(Y-T) x d(Y-T)/dT x dT/dY
= CY-T x 1 + CY-T x (-1) x TY
= CY-T (1 - TY ) = MPC times (1-MTR).
Note: dC/dY with proportional tax < dC/dY with only lump-sum tax as
1-MTR < 1.
Now with all this background information and some simplifying assumptions such as:
i)
ii)
iii)
ii)
There is only the goods market in the economy. No money market.
Price level is fixed.
Thus i=r: nominal interest rate = real interest rate
In this model for now, the interest rate is given from outside, and thus it is
exogenous; To be exact, here the interest rate is ‘real interest rate’ and
thus I am using ‘r’ for its notation.
Investment is a decreasing function of interest rates;
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I = I( r ) and dI/dr = Ir <0.
Recall Ir = - b
iv)
Government sets its expenditures G; G is exogenous; G = Go
If the government changes G from Go to G1. It is called ‘Expansionary
Fiscal Policy’.
v)
Tax is lump-sum only: T = To.
vi)
Consumption is a function of disposable income Y, and C = C(Y-To)
=C(Y; Co, To) = C(Y), where the marginal propensity to come being
equal to dC/d(Y-To) =dC/dY = CY.
2. Model
Y C  I G
C  C (Y ),0  CY  1
I  I (r ), I r  0
___
G  Go  G
(T  To : lmplicitly )
3. Key Issues:
i)
Existence of Equilibrium;
There is an equilibrium, and as will be shown, it is stable.
ii) Comparative Statics;
4
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In this model with the interest rate being held constant, what will happen to the
equilibrium national income Y if there is a change in G, I, or any other exogenous
variables?
Y
dY
 0 ?" , or “
?”
G
dG

“Does Fiscal Policy work?”: " G  Y ?" , "

“Does Monetary Policy work?:

“Out of the two monetary instruments, i and MS, which one should be
used for monetary policy?, or Does it make a difference?”
dY
?
dMS
The answer crucially depends on a set of adopted assumptions.
4. Comparative Statics for the model with Goods Market and Static Expectation
exogenous variables (G , r , To, Co)
endogenous variables (Y , C , I )  P does not matter.
How did we solve the question of comparative statics in the review of IS-LM?
Going back to the question again: The autonomous government expenditure multiplier
"
dY
 ? or  0 ,  0 " : “Is Fiscal Policy effective with respect to Y?”; Mathematically
dG
it is the (total) derivative, the (total) impact of a changing G on Y. Basically, the
derivative is the coefficient of the exogenous variable in the solution for the endogenous
variable:
What we did in the review was so called “Substitution Method:
Substitution Method goes this way:
We can reduce the number of equations by substituting functions for variables:
Y C  I G ;
C  C (Y ) ;
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I  I (r )
Becomes one equation:
Y = C(Y) + I (r ) + G.
Solver for Y* such as
Y – C(Y) = I(r ) + G
And get the coefficient of Y in the solution, which is Y as a function of all exogenous
variables of r, G, To, and Co.
Alternatively, we may get the total derivative: What if C(Y) is non-linear? So you
cannot factor Y out from C(Y) in an easy way.
Do not solve for Y*. Transform the given equation(s) into total differential (system). The
total derivative comes from total differential such as
dY 
dC
dI
dY  dr  dG
dY
dr
Divide the both sides by dG to solve for dY/dG
dY dC dY dI dr dG



dG dY dG dr dG dG
where dr/dG = 0 and dG/dG=1: r is an exogenous variable for now and is
independent of all variables here, and thus dr/dG =0
dY/dG = dC/dY dY/dG + 1
dY dC dY

1
dG dY dG
(1 
dC dY
)
1
dY dG
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dY
1

dG 1  dC
dY
1

1  CY
________________________________________________________________________
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Mathematical Review on Comparative Statics with Matrix Algebra (pp.8-11)
 Digress 

 Why " dY  1?" from Y  C  I  G
Pause


dG
 dG  dY (Partial Direct Effect)
dC dY
dI
dY
}
In other words,
(Indirect Effect)
dC
dI
 0 nor
 0 .
dG
dG
**How to get the total derivative? Back to Calculus:
Suppose that
z = f(x, y):
o Partial Derivatives:
z f ( x, y ) f


 fx
x
x
x
z f ( x, y ) f


 fy
y
y
y
o Total Differential:
(We have to first get total differentials
to get total derivatives.)
o Total Derivative:
f
f
dx 
 dy
x
y
 fxdx  fydy
z 
dz
dx
dy
 fx
 fy
dx
dx
dx
dy
 fx  fy
dx
“Partial” “Indirect Effect”
(Direct effect)
__________________________________________________
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The best way to solve for comparative statics is by 1) working on the system of
equation (no substitution) as a set, 2) getting the total differentials, which includes
dY and dG along with others, and 3) get the specific (total) derivatives dY/dG
through Cramer’s rule.
From “ Y  C  I  G ” , we get get the total differential for the system of all equations:
dY  dC  dI  d G ;
(i)
In the same manner,
From C  CY  , we get dC 
 
From I  I r, Y , we get dI 
dC
dY  C y dY
dY
dI
d r  Ird r
dr
(ii)
(iii)
Let’s write them down again,
-
(i)
(ii)
→ dY  dC  dI  d G
→ dC  C y dY
(iii)
→ dI  I r d r
Second, Cramer’s rule requires the following format of matrix multiplication:
[endogenous variables dY , dC, dI  ; exogenous variables d r, d G ]
 dY 
 
Coeff . Martix A dC   Coeff . Martix B d G 
 dr 
 dI 
Column vector of end. Var.
Column of exo. Var.
Rewriting the above system of total differentials for IS-LM from (i) to (iii), we
get:
dY  dC  dI  d G  0

 C y dY  dC  0
 0
 0

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0
 dY  0
 dC  dI  0
  I rd r ,
Thus, by figuring out ‘inner product’ of matrix-vector multiplication, we can
express the above in the matrix-vector multiplication format in the following
manner:
A x
Endo
= B x Exo
1  1  1  dY 

 C
1
0   dC  
y

 
0
1  dI 
 0
a
1 0 
0 0 


0 I r 
d G 
 
dr 
b
To get the solution for a comparative statics of dY/dG, we have to use Cramer’s
rule:
A'
dY

,
dG
A
where
1  1  1


1 0  , and A’ =
A =  C y
 0
0
1
 1  1  1
0
1 0 

0
0
1
Note that in A’, b replaces a of A.
dY/dG is obtained by the ratio of two determinants where the denominator is the
determinant of the original coefficient matrix for the endogenous variable, and the
numerator is the determinant of a newly made matrix with the column of the
coefficients for dY replaced by the column of the coefficients for dG.
You may review Crammer’s Rule in general by clicking here. The reference is
from Alpha Chiang’s book.
In this concrete case,
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 1  1  1
0
1 0 

0 0
1
dY
1
 

d G  1  1  1 1  C y
 C
1 0 
 y
 0 0
1
Note again that, in the numerator determinant, a is replaced with b.
Now, how to calculate the value for the determinant?
One way is ‘La place expansion’ for a high order matrix(more than 2x2):
Choose the column or row with the largest number of zero;
The determinant is equal to the sum of the product of the components
of the column or vector and their respective cofactors; the cofactor is made of the
sign equal to (-1) power of the sum of the ranks of column and row of the
component and the new components which exclude the original components of
the corresponding column and the row. See Alpha Chiang, Chapter 5.
For instance, if we expand along the 3rd row and 3rd column,
the determinant of the denominator is equal to
1 x (-1) power (3+3) x the determinant of the matrix with the components of 1 -1
-Cy 1 (=1 – Cy). That is 1- Cy.
In the same manner, if we calculate the determinant of the numerator along the
first row and the first column, it is equal to 1 x (-1) power (1+1) x the determinant
of the matrix (1 0 0 1) that is 1. Thus it is equal to 1.

dY
1

 0 as 0  C y  1
dG 1  Cy
Advantage of this Cramer’s rule over the substitution method is that once it is set up, you can get
all other comparative statics very easily without starting all over as is the case in the substitution
method.
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5. Revised Assumption about the Investment Function:
So far we have taken a very simple and unrealistic assumption about investment. Can we
try to set up an investment function and thereby to explain investment in a systematic
way?
First, some thoughts on the investment behaviors by the business sector:
Along with
Y C  I G
0  C y  1 ,
C  C( y)
in sum, we may come up with
I  I ( r, y )
I r  0 , I y  0 → Accelerator Model Theory.
Now investment is a function of both real interest rates and the national income. It is a
crude but somewhat realistic approach: Investment is a function of interest rate and
income. This can be illustrated as follows:
r
I ( y0 )
I ( y1 )
r0
I(r0, y1)
I(r0, y0 )
In exactitude, the interest rate of the current period straddles between now and the future,
and the income variable here is the future expected income.
By assuming that people have static expectations, these can be simplified as the future
variables can be replaced with the current variables.
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Solve for the total derivative dY/dG by using Cramer’s rule:
First, getting the total differentials for the system of equations:
dY  dC  dI  d G
dC  C y dY
dI  I r d r  I y dY
Second, rearranging with the endogenous variables on the left-hand side of the
equal sign and the exogenous variables on the right-hand side:
dY  dC  dI  d G  0

 C y dY  dC  0
 0
 0

 I y dY  0  dI  0  I rd r
Third, rewriting the above in the matrix format:

1  1  1  dY 


1 0   dC  
 C y
 
  I y
0
1  dI 
1 0 
0 0  d G 

 dr 
0 I r   
Fourth, by applying Cramer’s rule and LaPlace expansion, we get
 1  1  1
0 1 0 



0 0 1
dY
1
 

d G  1  1  1 1  I y  C y


1 0 
 C y
  I y 0 1


Our question is:
dY
 or  ?
dG
The sign of dY/dG depends on the sign of 1 – Iy – Cy:
If 1 - Cy > Iy, then dY/dG is positive, and desirable.
However, if 1 - Cy < Iy, then dY/dG is negative.
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What does this mean?
Alternatively, if 1 > Cy + Iy, then dY/dG is positive, and desirable.
If 1 < Cy + Iy, then dY/dG is negative, and undesirable.
Can you illustrate this point with the Keynesian cross diagram?
Suddenly, with a small modification in investment function, the entire comparative statics
seems to change, leading to a uncertainty of effectiveness of fiscal policy.
How likely could this happen?
It can happen under a certain circumstances.
We note that the larger Cy , the more likely 1 < Cy + Iy.
Policy Implications and Theoretical/Historical Background
-Keynes’s mistrust of the businessmen in general, and necessity of government’s control
of investment at the national level; the legacies of Keynes’s ideas
-Applications/Implilcations
Under what circumstances (about the investment) would an expansionary fiscal policy
would be a bad idea as a way to get the economy out of recession?