SAN JOSÉ STATE UNIVERSITY
ECONOMICS DEPARTMENT
Thayer Watkins

Macroeconomic Models

Model 0

Solution:

Y = C0 + bY + I + G + Xn
(1-b)Y = C0 + I + G + Xn
Y = C0/(1-b) + (1/(1-b))(I + G + Xn)
or
Y = kC0 + k(I + G + Xn)
where k = 1/(1-b)

The results of the analysis can be put in the form of a table that shows the effect of a change in each exogenous variable on each endogenous variable. This is called the reduced form of the model.

Endogenous
Variable		Exogenous VariablesParameters
 IGXnC0
Ykkkk
Ckbkbkb1+kb

The analysis indicates that Y is a linear function of (I+G+Xn); i.e.,

Y = K0 + k(I + G + Xn)
where k = 1/(1-b) and
K0 = kC0

It is better to represent the reduced form of the model as the following table:

Endogenous
Variable		Exogenous VariableParameters
 I + G + XnC0
Ykk
Ckb1+kb

The graphs of Y versus (I + G + Xn) for the U.S. and Japan are shown below.

Regression of Y on (I+G+Xn) for the U.S. Data gives:

Y = -226.476 + 3.05(I+G+Xn)
R2 = 0.954

The negative value for the constant is a problem.

For the Japanese data the result is:

Y = 3.897 + 2.309(I+G+Xn)
R2 = 0.993

MODEL 0.1

The model above ignores the endogenous nature of Imports. A better formulation is:

Solution:

Y = C0 + bY + I + G + X - M0 - mY
(1-b + m)Y = C0 + I + G + X - M0
Y = (C0- M0)/(1-b + m) +
(1/(1-b + m))(I + G + X)
or
Y = k(C0- M0) + k(I + G + X)
where k = 1/(1-b + m)

Again,the results of the analysis can be put in the form of a table that shows the effect of a change in each exogenous variable on each endogenous variable:

Endogenous
Variable		Exogenous VariablesParameters
 IGXC0M0
Ykkkk-k
Ckbkbkb1+kb-kb
Mkmkmkmkm1-km

The abbreviated version of the reduced form of the model is:

Endogenous
Variable		Exogenous VariableParameters
 I + G + XC0M0
Ykk-k
Ckb1+kb-kb
Mkmkm1-km

As in Model 0 the analysis indicates that Y is a linear function of (I+G+X); i.e.,

Y = K0 + k(I + G + X)
where k = 1/(1-b+m) and
K0 = k(C0-M0)

The graphs of Y versus (I + G + X) for the U.S. is shown below.

Regression of Y on (I+G+X) for the U.S. Data gives:

Y = 108.315 + 2.24(I+G+X)
R2 = 0.978

A multiplier of 2.24 corresponds to b-m = 0.5536.

Model 1

The previous models do not take into account explicitly tax policy. Let YD be aggregate disposable income. Then the model takes the form:

Solution: The first task in the solution is to obtain an equation that gives C in terms of Y.

YD = Y(1-t) - T0 + Tr
C = C'0 - b'T0 + b'(1-t)Y + b'Tr
Y = C'0 - b'T0 + b'(1-t)Y + b'Tr +
I + G + X - M0 - mY
(1-b'(1-t) + m)Y =
(C'0 - b'T0 + b'Tr - M0) + I + G + X
Y =
(C'0-M0-b'T0+b'Tr)/(1-b'(1-t) + m) +
(1/(1-b'(1-t) + m))(I + G + X)
or
Y = k(C'0 -M0-b'T0)
+ kb'Tr + k(I + G + X)
where k = 1/(1-b'(1-t) + m)

Endogenous
Variable		 Exogenous VariablesParameters
              I          G         X                     Tr      C'0         M0         T0
Ykkkkb'k-k-kb'
YDk(1-t)k(1-t)k(1-t)1+kb'(1-t)k(1-t)-k(1-t)-k(1-t)
Ckb'(1-t)kb'(1-t)kb'(1-t)b'(1+kb'(1-t))1+kb'(1-t)-kb'(1-t)-b'(1+k)
Mkmkmkmkb'mkb'm1 - kb'm-kb'm
Tktktktkb'tkb't-kt1-kb't

In abbreviated form:

Endogenous
Variable		 Exogenous VariablesParameters
      I + G + X                   Tr     C'0         M0         T0
Ykkb'k-k-kb'
YDk(1-t)1+kb'(1-t)k(1-t)-k(1-t)-k(1-t)
Ckb'(1-t)b'(1+kb'(1-t))k-k-kb'
Mkmkb'mkb'm1 - kb'm-kb'm
Tktkb'tkb't-kt1-kb't

Model 2

The previous models do not take into account the endogenousness of Investment. It is also necessary to take into account the role of the money supply.

The first steps in the solution are to express C and I as functions of only exogenous variables and Y. This has already been accomplished in the for C in the previous model,

C = C'0 - b'T0 + b'(1-t)Y + b'Tr

For investment we start by expressing the intererest rate R as:

R = (q/j)Y + (L0 - MS)/j

When we substitute this expression for R in the investment function the result is:

I = I0 - (g/j)(L0 - MS) + [h -gq/j]Y

Whether Y has a positive or negative coefficient in the investment function depends upon the relative magnitudes of h and (gq/j).

For later convenience let us express the investment function as:

I = I'0 + (g/j)MS + h'Y

When the derived consumption function and the derived investment function are substituted into the equilibrium condition the result is:

Y = C'0 - b'T0 + b'(1-t)Y + b'Tr +
I'0 + (g/j)MS + h'Y +
G + X - M0 - mY

The solution for Y is then:

Y = k[C'0 - b'T0 + I'0 - M0] + kb'Tr + (kg/j)MS + k(G + X)

where k = 1/(1 - b'(1-t) - h' + m).

The reduced form table is then:

Endogenous
Variable		 Exogenous VariablesParameters
        G+X              Tr                   MS             C'0          M0            T0               L0
Ykkb'kg/jk-k-kb'-kg/j
YD k(1-t)1+kb'(1-t)kg(1-t)/jk(1-t)-k(1-t)-1-kb'(1-t)-kg(1-t)/j
Ckb'(1-t)b'(1+kb'(1-t))kb'g(1-t)/j1+kb'(1-t)-kb'(1-t)b'(1+kb'(1-t))-kb'g(1-t)/j
Mkmkb'mkgm/jkm1-km-kb'm-kgm/j
Ikh'kb'h'(g/j)(1+kh')kh'-kh'-kb'h'-kgh'/j
Tktkb'tkgt/jkt-kt1-kb't-kgt/j
Rkq/jkb'q/j(-1+kgq2/j)/jkq/j-kq/j-kb'q/j-kgq/j2

Model 3

The following model allows us examine the macroeconomic effects of a shift in Net Foreign Financial Investment. Note that taxes and transfer payments are combined into net taxes, NT.

G0, Tr and MS0 are exogenous.

The method of solution is the same as in the previous models. The equations giving C and I in terms of only Y and exogenous variables are:

C = C'0 - b'(T0-Tr) + b'(1-t)Y
I = I0 - (g/j)(L0 - MS) + [h -gq/j]Y

Now we can make use of the balance of payments equation to find net exports, X-M, without determining X and M separately. Thus,

X-M = -FFI = -FFI0 - sR
but R = (q/j)Y + (L0 - MS)/j
so X-M = -FFI0 - (sq/j)Y - (s/j)L0 + (s/j)MS

The equilibrium condition is then:

Y = C'0 - b'(T0-Tr) + b'(1-t)Y
+ I0 - (g/j)(L0 + (g/j)MS) + [h -gq/j]Y + G -FFI0 - (sq/j)Y - (s/j)L0 + (s/j)MS

Combining terms and solving for Y gives:

Y =
k[C'0 - b'(T0-Tr) + I0 - ((g+s)/j)L0 - FFI0]
+ k((g+s)/j)MS + kG

where k =
1/[1- b'(1-t)- (h -gq/j)+ (sq/j)]
or equivalently
k = 1/[1- b'(1-t)- (h -(g+s)q/j)].

A notable implication of the model is that changes in X0 and M0 do not affect the level of GDP.

We also see then that when foreign investors autonomously increase their financial investment Y decreases (at a rate equal to the multiplier k).

The effect of a change in any of the exogenous variables on any other endogenous variable can be found by making use of the equation giving that variable as a function of exogenous variables and Y. For example, the equation for the interest rate is:

R = (q/j)Y + (L0 - MS)/j
which means that
R/MS = (q/j) (Y/MS) - (1/j).

Since we known that Y/MS = k((g+s)/j), this means that

R/MS = (q/j) k((g+s)/j) - (1/j)
= [qk(g+s)-1]/j.

It might seem that we do not known a priori; i.e., without empirical measurement, what the sign of qk(g+s)-1 is, and therefore we cannot say what the sign of R/MS is. One effect of an increased money supply is to lower the interest rate, but another is to increase Y which increases the demand for money which has the effect of increasing the interest rate. However an increase in the money supply increases Y only to the extent that it decreases the interest rate so it there is no decrease in R there is no increase in Y. The relationship of k and q(g+s) can be seen by noting that

(1/k) = 1- b'(1-t)- (h -gq/j)+ (sg/j)
= 1- b'(1-t)- h + q(g+s)/j
and hence
1- b'(1-t)- h = (1/k) - q(g+s)/j
= - [kq(g+s)/j - 1]/k.

Thus

[kq(g+s)/j - 1] = - k(1- b'(1-t)- h).

Since
j>0 and (1- b'(1-t)- h)>0 it follows that

[kq(g+s)/j - 1]<0
and hence
R/MS <0

The lesson we learn from the above is that if a quantity is found to be the difference of two positive terms we cannot automatically conclude that the sign is not determinable. There may be additional information in the model that allows us to make an unambiguous determination of sign.

The full solution; i.e., the solution for all of the endogenous variables, is considerably more complex for Model 3 than it is for Model 2. The reason for this is the exchange rate variable E. Although exports and imports both depend upon E, the net exports X-M can be determined without knowing E. The exchange rate E can be determined but its solution is complicated and hence the solutions for X and M are also complicated. The reduced form table for Model 3 without X, M and E is:

Endogenous
Variable		 Exogenous VariablesParameters
 
     G       
     Tr       
       MS     
             C'0          M0          X0            T0                L0               FFI0
Ykkb'k(g+s)/jkk(1-t) 0-kb'-kg/j-k
YD k(1-t)1+kb'(1-t)k(g+s)(1-t)/jk(1-t) 00-1-kb'(1-t)-kg(1-t)/j -k(1-t)
Ckb'(1-t)b'(1+kb'(1-t))kb'g(1-t)/j 1+kb'(1-t)00b'(1+kb'(1-t)) -kb'g(1-t)/j-kb'(1-t)
X-M-ksq/j-ksqb'/j -sqk(g+s)/j2-sqk/j0 0 sqkb'/j(s/j)(1+kqg/j)-(1+sqk/j)
Ikh'kb'h'(g/j)(1+kh')kh'0 0-kb'h'-kgh'/j-kh'
Tktkb'tkgt/jkt001-kb't-kgt/j-kt
Rkq/jkb'q/j-(1-k(g+s)q/j)/j)/j kq/j 00-kb'q/j(1-kgq/j)/j-kq/j

Although it is not necessary to determine the equilibrium levels of X and M separately or the exchange rate E their values are determined by the model. If we know the equilibrium real interest rate R then we have

X - M = - FFI = FFI0 - sR
X0 - vE - M0 - pE - mY = -FFI0 -sR
thus
(v+p)E = X0 - M0 - mY + FFI0 + sR
and hence
E = (X0 + FFI0 - M0)/(v+p)
- (m/(v+p))Y + (s/(v+p))R
.

This says that whatever will increase the real interest rate while leaving Y constant will increase the value of the dollar and whatever increases Y without increasing R will decrease the value of the dollar.

Model 3.1

An exogenous inflation rate can be incorporated into the analysis with the following model in which Investment depends upon the real rate of interest R but the demand for money depends upon the nominal rate of interests R + H0.

G0, MS0 and H0 are exogenous.