May 9, 2006
Macro (3):
1
1.1
2006-5-10(Wed) and 12(Fri)
Takao Fujimoto: Fukuoka
Implicit Function Theorem
Implicit Function Theorem (just memorize this)
Theorem: When f (x, y) = k, where k is a real constant, then
∂f
dy
=− ∂x .
∂f
dx
∂y
Proof. From the total differentiation, 0 = df =
Total differentiation: df(x1 , x2 , . . . , xn ) =
Pn
i=1
∂f
∂f
· dx +
· dy, the result follows. QED.
∂x
∂y
∂f
· dxi .
∂ xi
More generally,
Theorem: When f (x1 , x2, . . . , xn ) = k, where k is a real constant, then
∂f
dxj
∂ xi
=−
.
∂f
dxi
∂ xj
Now let us deal with the quadratic
equation in the previous section. f (x, a, b) = ax2 +bx +c = 0.
¯
∂x
x2 ¯¯
¯
From this,
=−
< 0. This is because x∗ > 0, a > 0, and b > 0.
∂a
2ax + b ¯x=x∗
NB. In the use of implicit function¯ theorem, x is the solution (or equilibrium)
x∗ .
¯
¯
¯
∂x
x ¯
∂x
1 ¯
In a similar way,
=−
< 0. And
=−
< 0 , (NB. c < 0).
¯
∂b
2ax + b x=x∗
∂c
2ax + b ¯x=x∗
y = 3x2 + 4x − 2
y = 3x2 + 4x − 1
May 9, 2006
10
8
6
4
2
-3
-2
00
-1
1
x
2
3
-2
-4
Fig.1. two quadratic equations: c changes
x2 + 4y 2 = 16
6
4
y
2
-6
-4
-2
00
2
x
4
6
-2
-4
-6
Fig.2. an ellipse and a tangent line
Exercises:
Q01. Draw diagrams to understand the results above for the quadratic equations.
Q02. Consider the case of a cubic equation ax3 + bx2 + cx + d = 0 with a, b, c > 0, and d < 0.
(Report: not mandatory.)
Math Exercise:
√
Q03. Find out the equation of a tangent line going through the point ( 3, 12 ) on the ellipse
x2 + 4y2 = 4.
May 9, 2006
2
Keynesian One-Sector Model (somewhat repeated)
Equalities among National Product, National Income, and National Expenditures. Net
products are realized when someone buy them, and so basically national product equals national
expenditures. Then, these sales are distributed as salaries, wages, remuneration, dividends,
rents, and so on. Thus, national product equals national income. A problem lies in hoardings,
while another consists in inventory investment.
Exercises:
Q04. Describe how hoardings may be cancelled out.
Q05. State your ideas to increase the national income in an artificial way.
The Simplest Keynes’s Model: Y = C(Y ) + I + G, where Y is the (net) national (domestic)
product or income, C(Y ) the consumption function, I the (net) investment of private sector, G
the expenditures(consumption + (net) investment) by the government.
NB. When “net” is used, the total depreciation is removed, while when “gross” is used, it is not.
Keynes’s one-sector model
dY
(the ratio between the increased national income and the increased
dG
government expenditures)
By the implicit function theorem, we have
Keynes’s Multiplier:
dY
=−
dG
when we assume 0 <
−1
> 1,
dC
1−
dY
dC
dC
< 1. The value
is called the marginal propensity to consume.
dY
dY
NB. Partial derivatives are now written as ordinary ones simply because the related functions
contain only one variable Y .
The Keynesian Open Model: Y = C(Y ) + I + G + E − M(Y ), where E is the export and
M(Y ) the import function.
May 9, 2006
Keynes’s Multiplier for the Open Model:
dY
=−
dG
−1
> 1,
dC dM
1−
+
dY
dY
dC
dM
−
) < 1.
dY
dY
Def. The trade balance B ≡ E − M (Y ).
Exercises:
Q06. Compute dB/dG.
Q07. Compute dY /dI .
Q08. We consider a Keynesian Open Model, Y = C(Y )+I +G+E−M(Y ), where E is the export
and M (Y ) the import function. Suppose the import function is linear, M (Y ) = mY + n, (with
m > 0 and n > 0) and the government encourages the people to import, which is represented by
an increase in the parameter n, wishing to decrease the trade balance B ≡ E − M(Y ). Is the
intention of the government fulfilled?
Answer. We have to know the sign of dB
. Substitute mY + n for M(Y ), and we get
dn
when we can assume 0 < (
Y − C(Y ) − I − G − E + (mY + n) = 0, and
(1)
B = E − (mY + n).
(2)
Thus, from eq.(1) we have
dY
=−
dn
1−
1
,
dC(Y )
dY
+m
+m
−1=
and from eq.(2) we obtain
dY
dB
= −m ·
−1 =
dn
dn
1−
provided 0 <
dC(Y )
dY
m
dC(Y )
dY
1
dC(Y )
−1
dY
dC(Y )
− dY + m
< 0,
< 1. Therefore, the government wish can be accomplished in this model.
Q02. We again consider a Keynesian Open Model, Y = C(Y ) + I + G + E − M (Y ), where E
is the export and M (Y ) the import function, and also suppose the import function is linear,
M(Y ) = mY + n, (with m > 0 and n > 0) and the government encourages the people to import,
which is represented this time by an increase in the parameter m, wishing to decrease the trade
balance B ≡ E − M (Y ). Is the intention of the government fulfilled again?
dB
dY
Hint. In this case, dm
= −Y − m · dm
.