May 24, 2005
Macro (4):
2005-5-18(Wed), 20(Fri), 25(Wed), 27(Fri)
Takao Fujimoto: Fukuoka
Reference:
Alpha C. Chiang, Fundamental Methods of Mathematical Economics, 2nd ed. McGraw-Hill,
1967, 1974. (Chapters 4 to 8.) (A Japanese translation available.)
English dictionaries: home pages
http://www.m-w.com/
(Merriam-Webster: etymology)
http://dictionary.cambridge.org/ (Cambridge Advanced Learner’s Dictionary: examples)
http://thesaurus.reference.com/ (thesaurus)
Math dictionaries: home pages
http://mathworld.wolfram.com/ (Mathematica:(advanced))
http://planetmath.org/encyclopedia/ (PlanetMath:)
http://thesaurus.maths.org/ (Math thesaurus:)
Ch.2. IS-LM Model by J.R.Hicks
References:
Alpha C. Chiang, Fundamental Methods of Mathematical Economics, 2nd ed. McGraw-Hill,
1967, 1974. (Chapters 4 to 8.) (A Japanese translation available.)
John Maynard Keynes, The General Theory of Employment, Interest and Money, Macmillan,
1936. (also in The Collected Writings of John Maynard Keynes, vol.VII, Macmillan, 1973.)
John Maynard Keynes, “The General Theory of Employment”, Quarterly Journal of Economics,
vol.51, pp.209-23, 1937.
John Richard Hicks, “Mr. Keynes and the ‘Classics’; A Suggested Interpretation”, Econometrica,
vol.5, 1937, pp.147-59
David C. Colander: Macroeconomics, 4th ed. Irwin/McGraw-Hill, 1995, 2001.
David C. Colander, Edward Gamber, Macroeconomics, Prentice Hall, 2002. ISBN: 0130303720.
Rudiger Dornbusch, Stanley Fischer, Richard Startz: Macroeconomics, 7th ed.,
McGraw-Hill, 1977, 1998.
May 24, 2005
Joseph E. Stiglitz, Carl Walsh: Principles of Macroeconomics, 3rd ed., Norton & Co., 1996, 2002.
NB. In the Keynes’s model, unemployment is understood to be caused by insufficient
effective demand, i.e., ex ante, C + I + G. The same is true in the IS-LM model. There is,
however, no labor market in these models, and so, there is no way to associate the increase
in Y with that of employment.
1. IS-LM Model (or Two-Market Model) by J. R. Hicks
This is a two-market model, while Keynes’s is a one-market or one-sector model. One is for goods
or commodities as in Keynes’s, the other for money. The equilibrium conditions are given as
(
Y = C(Y, r) + I(Y, r) + G
M = L(Y, r)
(1), the market for goods & services,
(2), the market for money.
where Y and r are two endogenous variables. New dramatis personae are:
r: the rate of interest; I(Y, r): the investment function in the private sector; M: a given money
stock; L(Y, r): the liquidity preference function (= the demand function for money stock).
The first equation (1) describes the equilibrium condition for the goods & services market, while the
second (2) for the money market.
Policy variables: G and M.
From eq.(1), we derive the IS-curve, a relation between Y and r , while from eq.(2) the LM-curve.
The name IS comes from rewriting eq.(1) as (Y − C − T ) + (T − G) = I , where T stands for total
tax. The equation tells us that the savings of the private sector plus the savings by the government
equals the investment of the private sector.
May 24, 2005
Fig. the IS-curve and the LM-curve
Exercises:
Q01. Show that the IS-curve is downward-sloping, while LM-curve is upward-sloping.
Q02. Explain the meaning of two areas on the Y-r plane: one above the IS-curve and the other
below. Also explain the two areas on the Y-r plane separated by the LM-curve.
Q03. Explain the significance of the liquidity trap proposed by Keynes.
2. Mathematical Treatment for Two Variable Case
Preliminary: Systems of Linear Simultaneous Equations of 2 variables:
(
a11 · x1 + a12 · x2 = b1
a21 · x1 + a22 · x2 = b2
(1)
(2)
Multiply (1) by a22 , and (2) by a12 , and eliminate x2 by subtracting the latter from the former
result of multiplication. We get
(a11 · a22 − a12 · a21 ) x1 = b1 · a22 − b2 · a12 .
Thus, we have
x1 =
b1 · a22 − b2 · a12
b2 · a11 − b1 · a21
, and x2 =
.
a11 · a22 − a12 · a21
a11 · a22 − a12 · a21
( This expression for x2 can be obtained by changing the subscript 1 to 2, and 2 to 1 in the solution
of x1 .)
The original simultaneous equations are rewritten, using matrix and vector notation as
Ax = b, where A ≡
Ã
a11 a12
a21 a22
!
,x=
Ã
x1
x2
!
, and b =
Ã
b1
b2
!
.
And when we introduce a symbol |M |, which is called the determinant of a matrix M ,
¯Ã
¯ m
¯
11
|M | ≡ ¯¯
m21
m12
m22
!¯
¯
¯
¯ ≡ m11 · m22 − m12 · m21 ,
¯
the solutions of the above simultaneous equations can be represented as follows:
¯Ã
¯Ã
!¯
!¯
¯ b a
¯ a
¯
¯
¯
¯
¯
¯
1
12
11 b1
¯
¯
¯
¯
¯ b2 a22 ¯
¯ a21 b2 ¯
¯
¯
Ã
Ã
!¯
!¯
x1 = ¯
¯ , and x2 = ¯ a
¯.
¯ a11 a12 ¯
¯
¯
11 a12
¯
¯
¯
¯
¯ a21 a22 ¯
¯ a21 a22 ¯
This is the Cramer’s rule for the case of two variables.
Preliminary 2: Total differentiation:
f(x1 , x2 , . . . , xn ) → df =
n
X
∂f
i=1
∂xi
· dxi .
May 24, 2005
3-a. Comparative Statics in IS-LM Model
IS-LM Model (restated):
(
Y = C(Y, r) + I(Y, r) + G
M = L(Y, r)
(1)
.
(2)
NB. All variables are in nominal terms, i.e., in money terms.
NB. There has not yet appeared any variable to explain the employment directly.
Assumptions:
∂C
∂I
(1 − ∂Y
− ∂Y
) > 0 (the sum of marginal propensity to consume and to invest is less than unity);
∂L
> 0 (the higher is income, the greater demand for money (stock));
∂Y
∂L
<
0 (the higher is the interest rate, the less demand for money (stock));
∂r
∂C
< 0 (the higher is the interest rate, the less desire to consume); and
∂r
∂I
< 0 (the higher is the interest rate, the less desire to invest.).
∂r
First, transfer the RHS’s of eqs.(1) and (2) to the LHS’s to have the form f () = 0. By total
differentiation, we get
(
∂I
(1 − ∂C
− ∂Y
) · dY + (− ∂C
− ∂∂Ir ) · dr + (−1) · dG = 0
∂Y
∂r
∂L
· dY + ∂L
· dr + (−1) · dM = 0
∂Y
∂r
(1)0
.
(2)0
dr
When we examine the Keynesian fiscal policy, we calculate ∂Y
and/or dG
, and set dM = 0. (In
∂G
the partial derivative notation, we regard M also as a variable, but fixed while considering the effect
of changes in G. When we use the ordinary differentiation, simply we think of M as constant.)
Dividing the above eqs.(1)0 and (2)0 by dG, gives us
(
∂I
(1 − ∂C
− ∂Y
) · dY
+ (− ∂C
−
∂Y
∂r
dG
∂L d r
∂L dY
· dG + ∂ r · dG = 0
∂Y
∂I
)
∂r
·
dr
dG
This is a system of simultaneous equations with two variables,
Cramer’s rule to have
dY
= ¯¯
dG
¯ (1 −
¯
¯
¯
¯ 1, (− ∂C − ∂I )
¯
∂r
∂r
¯
∂L
¯ 0,
∂r
∂C
∂I
− ∂Y
∂Y
∂L
∂Y
¯ =
¯
∂I
(1 −
), (− ∂C
−
)
∂r
∂ r ¯¯
∂L
¯
,
dr
dG
¯
¯
dY
dG
−
∂I
∂Y
¯
∂C
∂I
− ∂Y
), 1 ¯¯
∂Y
¯
∂L
, 0 ¯
∂Y
∂C
∂I
− ∂Y
), (− ∂C
− ∂∂Ir )
∂Y
∂r
∂L
∂L
,
∂Y
∂r
> 0, because we assume (1 −
Exercises:
Q01. Show that
∂C
∂Y
¯
¯ (1 −
¯
¯
¯
dr
= ¯¯
dG
¯ (1 −
> 0, and
dr
dG
∂C
∂Y
−
∂I
∂Y
dY
dG
)·
and
∂L
∂r
∂L
−
∂r
dr
.
dG
Thus we can apply
(− ∂C
−
∂r
∂I
)
∂r
·
∂L
∂Y
.
∂r
Thus, dY
> 0, because we assume (1 −
dG
Similarly,
Thus,
¯
¯
¯
¯
¯
(1)∗
.
(2)∗
=1
∂C
∂Y
−
∂I
∂Y
) > 0,
∂L
∂Y
¯ =
¯
(1 −
¯
¯
¯
) > 0,
∂L
∂Y
> 0,
∂C
∂Y
∂L
∂r
< 0,
∂C
∂r
< 0, and
∂L
− ∂Y
∂I
− ∂Y
) · ∂L
− (− ∂C
−
∂r
∂r
> 0,
∂L
∂r
< 0,
∂C
∂r
∂I
∂r
∂I
)
∂r
< 0, and
·
∂I
∂r
< 0.
∂L
∂Y
< 0.
> 0, using the diagram with the IS-curve and the LM-curve.
May 24, 2005
dY
Q02. Derive the formulas for dM
and
of the increase in money stock.
Hint:
Q01. Figure
dr
.
dM
These are the effects on income and interest rate
Fig. IS-LM Curves
3-b. Comparative Statics in IS-LM Model : Another Approach
IS-LM Model (restated):
(
Y = C(Y, r) + I(Y, r) + G
M = L(Y, r)
(1)
.
(2)
Assumptions:
∂C
∂I
(1 − ∂Y
− ∂Y
) > 0 (the sum of marginal propensity to consume and to invest is less than unity);
∂L
> 0 (the higher is income, the greater demand for money (stock));
∂Y
∂L
<
0 (the higher is the interest rate, the less demand for money (stock));
∂r
∂C
< 0 (the higher is the interest rate, the less desire to consume); and
∂r
∂I
< 0 (the higher is the interest rate, the less desire to invest.).
∂r
(1) the first method: total differentiation (in Section 3-a).
(2) the second method: direct differentiation wrt G.
NB. Remember that Y and r are the functions of G, thus Y (G) and r(G).
We carefully differentiate eqs.(1) and (2) wrt G , noting that Y and r are the functions of G. We
have
(
dY
dG
∂C dY
= ∂Y
· dG + ∂C
·
∂r
∂L dY
∂L
0 = ∂Y · dG + ∂ r ·
dr
dG
dr
dG
+
∂I
∂Y
·
dY
dG
+
∂I
∂r
·
dr
dG
+1
(1)0
.
(2)0
May 24, 2005
A little manipulation leads to the same system of equations as we had through the first method:
(
∂I
(1 − ∂C
− ∂Y
) · dY
+ (− ∂C
−
∂Y
∂r
dG
∂L d r
∂L dY
· dG + ∂ r · dG = 0
∂Y
∂I
)
∂r
·
dr
dG
=1
(1)∗
.
(2)∗
Then, we can apply Cramer’s rule again.
Shift of the Curves
Let us regard eq.(1) as a constraint with three variables, Y , r, and G, in the 3 dimensional Euclidean
◦
space. So, this may show a hyper-surface in that space. At a particular value of G , a cross-section
of the hyperplane gives a IS-curve.
Now when G is changed, this IS-curve may shift. To which direction? To answer the question,
we cut the hyper-surface horizontally at a particular value of r, say, at r◦ , and examine the sign of
dY
.
dG
From eq.(1) as usual, we get
Y − C(Y, r) − I(Y, r) − G = 0.
By the implicit function theorem,
dY
−1
=−
∂C
dG
1 − ∂Y −
∂I
∂Y
> 0.
(∗∗)
Therefore, at a fixed rate of interest, the GDP shifts rightward as G increases. Since the IS-curve is
downward sloping, we may also say the IS-curve shifts upward.
NB. The equation (**) above shows dY
, when the rate of interest is fixed, and so this is not the
dG
Keynesian multiplier in the IS-LM model.
Fig. Shift of the IS-curve when G increases
May 24, 2005
Q03. Explain how the LM-curve shifts when the money stock M increases.
Q04. Make your analysis of the IS-LM model with international trade and/or income tax.
Weak points of the IS-LM model:
(1) There is contained no explicit variable describing employment and the total labour force.
This weak point is descended from the simple Keynes’ one-market model.
(2) Related to the above, no real GDP can be found in the model.
4. Four Regions in IS-LM Model
Please understand well the economic states represented by four regions formed by the two curves, ISand LM-curve. The interest rate may be regarded as the “price of money”, and when the demand for
money is greater than its supply, the interest rate goes up, and vice versa. The equilibrium seems to
be stable in the diagram below. It may, however, be unstable depending on the speeds of adjustment
in each market, and the slopes of two curves.
Fig. Seemingly stable, but not necessarily
Balanced Budget Theorem in the IS-LM Model
(
Y = C(Y − T, r) + I(Y − T, r) + G
M = L(Y − T, r)
(1)
.
(2)
NB. We now use the disposable income YD ≡ Y − T , in all three functions C, I, and L.
Total differentiation of the above system gives
(∗)
(
dY −
∂C
∂YD
·
∂YD
∂Y
· dY −
∂C
D
· ∂Y
·
∂YD
∂T
∂YD
∂L
· ∂Y
∂YD
∂I
∂I
D
D
dT − ∂C
· dr − ∂Y
· ∂Y
· dY − ∂Y
· ∂Y
· dT −
∂r
∂Y
∂T
D
D
∂YD
∂L
∂L
· dY + ∂YD · ∂T · dT + ∂r · dr − dM = 0.
∂I
∂r
· dr − dG = 0,
May 24, 2005
When we consider the Keynes’s policy, i.e., the increase of G, we set dT = 0 and dM = 0. Note here
D
D
that ∂Y
= 1 and ∂Y
= −1. Then divide the whole system by dG and this leads to
∂Y
∂T
(
(1 −
∂C
∂YD
∂I
) · dY
+ (− ∂C
∂YD
∂r
dG
∂L
dY
∂L
dr
·
+
·
∂YD
∂r
dG
dG
−
− ∂I
)·
∂r
= 0.
dr
dG
= 1,
Therefore, by the Cramer’s rule we obtain
dY
=
dG
(1 −
∂C
∂YD
−
∂I
)
∂YD
·
∂L
∂r
∂L
−
∂r
(− ∂C
−
∂r
∂I
)
∂r
·
∂L
∂YD
.
∂C
This is exactly the same as the formula in “macro I 05.pdf” with ∂C
now being ∂Y
. And when
∂Y
D
∂L
∂L ∂L
is very large compared with ∂YD , as in or near the liquidity trap, i.e., ( ∂YD / ∂r ) ' 0, then
dY
=
dG
(1 −
∂C
∂YD
−
∂I
)
∂YD
1
− (− ∂C
−
∂r
∂I
)
∂r
·
∂L ∂L
( ∂Y
/ ∂r )
D
'
1
(1 −
∂C
∂YD
−
∂I
)
∂YD
∂L
∂r
> 1.
Next, when the government insists to maintain dG = dT , then replace dT by dG in the system
(*) and divide this system by dG, and we get
(
(1 −
∂C
∂YD
−
∂I
)
∂YD
dY
+ (− ∂C
− ∂I
)·
∂r
∂r
dG
∂L
dY
∂L
dr
· dG + ∂r · dG =
∂YD
·
dr
=
dG
∂L
.
∂YD
1−
∂C
∂YD
−
∂I
,
∂YD
Again by the Cramer’s rule,
(1 −
dY
=
dG
(1 −
∂C
∂YD
∂C
∂YD
−
−
∂I
)
∂YD
∂I
)
∂YD
·
·
∂L
∂r
∂L
∂r
− (− ∂C
−
∂r
− (− ∂C
−
∂r
∂I
)
∂r
∂I
)
∂r
·
·
∂L
∂YD
∂L
∂YD
= 1, and
dr
= 0.
dG
Remark 1. As was explained in the class, whether the Keynes’s multiplier, dY
, is greater than
dG
unity, or whether it is very large has no explicit meaning on real production and employment. Just
∂C
think of those countries where ∂Y
is near unity.
D
Remark 2. The balanced budget theorem in these simple models simply tells us that when the
government wishes to keep 4G = 4T , then the Keynes’s fiscal policy might have little effect on
employment.
5. General Treatment of n Variable Case
We consider the model described by the following system of simultaneous equations. There exist n
equations and n variables, x1 , x2 , . . ., xn , and two parameters, α and β. (The number of parameters
can be any finite positive integer.)

f1 (x1 , x2, . . . , xn ; α, β) = 0,








f2 (x1 , x2, . . . , xn ; α, β) = 0,
···
fn (x1 , x2, . . . , xn ; α, β) = 0.
May 24, 2005
We wish to know how the solutions x∗1 , x∗2, . . . , and x∗n shift when a given parameter α changes a
i
little, i.e., dx
.
dα
First, take the total differentiation of each equation in the above system.

df1 =




 df =
2





dfn =
∂f1
∂x1
∂f2
∂x1
· dx1 +
· dx1 +
∂f1
∂x2
∂f2
∂x2
∂fn
∂x1
· dx1 +
∂fn
∂x2
∂f1
1
1
· dx2 + . . . + ∂x
· dxn + ∂f
· dα + ∂f
· dβ = 0,
∂α
∂β
n
∂f2
∂f2
∂f2
· dx2 + . . . + ∂xn · dxn + ∂α · dα + ∂β · dβ = 0,
···
∂fn
n
n
· dx2 + . . . + ∂x
· dxn + ∂f
· dα + ∂f
· dβ = 0.
∂α
∂β
n
Then, divide each equation by dα, (putting dβ = 0), and transfer the terms
obtaining











∂f1
∂x1
∂f2
∂x1
∂fn
∂x1
∂f1
+ ∂x
·
2
∂f2
+ ∂x2 ·
···
dx1
n
· dα + ∂f
·
∂x2
·
·
dx1
dα
dx1
dα
dx2
dα
dx2
dα
+ ... +
+ ... +
∂f1
∂xn
∂f2
∂xn
·
·
dxn
dα
dxn
dα
dx2
dα
+ ... +
∂fn
∂xn
·
dxn
dα
∂fi
’s
∂α
to the RHS,
1
= − ∂f
,
∂α
∂f2
= − ∂α ,
= ···,
n
= − ∂f
.
∂α
(1)
This is now a familiar system of linear equations: Ax = b.
Thus, by use of the Cramer’s rule for the case of n variables,
dxi
=
dα
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
∂f1
∂x1
∂f2
∂x1
..
.
..
.
∂fn
∂x1
¯ ∂f
¯
1
¯ ∂x1
¯ ∂f2
¯
¯ ∂x1
¯ .
¯ ..
¯
¯ ∂fn
¯
∂x1
1
− ∂f
∂α
∂f2
− ∂α
..
.
n
− ∂f
∂α
∂f1
∂xn
∂f2
∂xn
..
.
..
.
∂fn
∂xn
∂f1
∂x2
∂f2
∂x2
···
···
...
∂f1
∂xn
∂f2
∂xn
∂fn
∂x2
···
∂fn
∂xn
..
.
..
.
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
,
where in the numerator the i-column is replaced by the RHS vector of eq.(1). This vector often
turns out very 
simple.

∂f1
∂f1
∂f1
· · · ∂x
∂x
∂x
1
2
n
 ∂f2 ∂f2
∂f2 


 ∂x1 ∂x2 · · · ∂xn 
 is called the Jacobian matrix of f ’s with respect to xi ’s, and
The matrix 
.
.
.
.
 ..
..
. . .. 


∂fn
∂x1
∂fn
∂x2
∂fn
· · · ∂x
n
is often written as Jf (x). The determinant of Jacobian matrix is simply called the Jacobian.
6. General Treatment of Curves and Their Shifts in n Variable Case
We consider the following system of n equations, but there are (n + 1) variables:

f1 (x1, x2 , . . . , xn ; y; α) = 0








f2 (x1, x2 , . . . , xn ; y; α) = 0
,
...
fn (x1, x2 , . . . , xn ; y; α) = 0
where x1, x2 , . . . , xn , and y are variables, and α is a parameter.
(In the case of IS-curve of IS-LM model, n is 1, x1 is the rate of interest r, and y is the national
income Y . The parameter α can be either G or M . In the case of AS-curve of AS-AD model (in
May 24, 2005
Chapter 4), n is 3, and x1 , x2, and x3 are LE , P, and W respectively, y the gross domestic product
with α being P e .)
If we are lucky enough, we can eliminate n − 1 variables, and can get a single equation containing
only xi and y (and α), which represent a y-xi curve. When the Jacobian of the system is not zero,
i
we can calculate dx
in a neighborhood of an equilibrium like this.
dy
First, carry out total differentiation to have

f11 · dx1 + f12 · dx2 + . . . + f1n · dxn + f1y · dy + f1α · dα = 0




where
f21 · dx1 + f22 · dx2 + . . . + f2n · dxn + f2y · dy + f2α · dα = 0
,

...



fn1 · dx1 + fn2 · dx2 + . . . + fnn · dxn + fny · dy + fnα · dα = 0
fij =
(2)
∂fi
∂fi
∂fi
, fiy =
, and fiα =
.
∂xj
∂y
∂α
Putting dα = 0, and dividing each equation by dy, we get a linear equation system
Ax = −b,
where



A≡


f11 f12
f21 f22
..
..
.
.
fn1 fn2


. . . f1n


. . . f2n 

, x ≡ 

.
...
.. 



. . . fnn
dx1
dy
dx2
dy
..
.
dxn
dy
Thus, by using the Cramer’s Rule, it follows



f1y

 f 

 2y 


 , and b ≡ 
 ..  .



.

fny
dxi
|A(i : −b)|
=
, where A(i : −b) is the matrix A with its i-column replaced by −b.
dy
|A|
i
By the sign of dx
, we can tell whether xi is increasing or decreasing as y increases.
dy
Now, when the parameter α changes a little, in which direction does the y-xi curve shift? This
can be examined by calculating
dxi
, setting dy = 0.
dα
From eqs.(1), by setting dy = 0, and dividing every equation by dα, we have
Az = −d,
where



A≡


f11 f12
f21 f22
..
..
.
.
fn1 fn2


. . . f1n

. . . f2n 



..  , z ≡ 
...


. 
. . . fnn
Again by the Cramer’s Rule, we can compute
dx1
dα
dx2
dα
..
.
dxn
dα






 , and d ≡ 




f1α
f2α
..
.
fnα



.


dxi
|A(i : −d)|
=
, where A(i : −d) is the matrix A with its i-column replaced by −d.
dα
|A|