Download Equation (6.2) gives so

6.1
Equation (6.2) gives d Y C (d Y dT ) d I d G
dT 2dG ,
dY dI 0 so dY C dY 2 C d G d G
1 2 C 1C
dG > 0 if C >
1
since d G < 0
2
6-1
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.2
Y
C Y T I G so dY
C cdY C cdT dI dG . If dI
0 and the
government wants dY to also equal 0, then 0 0 C cdT 0 dG or dT dG C c .
So taxes should increase, by more than government purchases (since the marginal
propensity to consume is positive but less than 1).
6.3
Equation (6.8) gives
1 C I
LY
Lr
dY
dr
(1 C ) dG
0
so
1 C (1 C ) dG
dr 0
LY
1 C I
> 0
LY (1 C ) d G
( 1 C )Lr L Y I
LY
Lr
6-3
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.4
The interesting result of this problem is that the balanced-budget multiplier is
the same with a progressive tax system as with the lump-sum tax used in the
example in the body of the text. If the tax function is going to be changed in order
to raise more revenue to offset an expenditure increase, we need to introduce
another parameter, say T , in the tax rate function: let the average tax rate be
t Y , T . When the government increases its purchases of goods and services G, it
can keep the budget deficit unchanged by changing the parameter T in such a
way that tax revenues T, which now equal T t Y , T Y , increase by the same
amount as purchases have increased. Adding the tax revenue equation to the
system of equations (6.5), the total differentials are
dY C c dY dT I cdr dG
dT
tY YdY tdY tTYd T
dM
LY dY Lr dr
or, in matrix notation,
ª 1 C c C c I cº ª dY º ª dG º
« t Y t 1
0 »» «« dT »» ««tTYd T »» .
« Y
«¬ LY
0 Lr »¼ «¬ dr »¼ «¬ dM »¼
Setting dM 0 and using Cramer’s rule to solve for dT,
1 Cc
dG I c
tY Y t tTYd T
LY
dT
0
0
tY Y t Lr dG tTYd T Lr 1 C c I cLY Lr
J
J
where the Jacobian is
C c Lr tY Y t Lr 1 C c I cLY
J
C cLr tY Y t Lr 1 C c I cLY .
So if dT is going to equal dG then
tY Y t Lr dG tTYd T Lr 1 C c I cLY dG
C cLr tY Y t Lr 1 C c I cLY
C cL t Y t L 1 C c I cL dG t Y t L dG t Yd T L 1 C c I cL 1 C c L 1 t Y t I cL dG t Y L 1 C c I cL d T
1 C c L 1 t Y t I cL
dT
dG.
t Y L 1 C c I cL r
Y
r
r
Y
r
T
Y
Y
Y
Y
r
Now solving for dY,
T
Y
Y
r
r
T
Y
r
Y
dG C c I c
0
tTYd T 1
dY
0
0
J
Lr
Lr dG C ctTYd T .
J
Substituting in for dT ,
dY
§
1 C c Lr 1 tY Y t I cLY ·
Lr ¨ dG C ctTY
dG ¸
¨
¸
c
c
t
Y
L
1
C
I
L
r
Y
T
©
¹
J
§ Lr 1 C c I cLY C c 1 C c Lr I cLY 1 C c Lr tY Y t ·
Lr ¨
¸¸
¨
c
c
L
C
I
L
1
r
Y
©
¹ dG
J
Lr 1 C c Lr 1 C c I cLY
§ C cLr tY Y t Lr 1 C c I cLY
¨¨
© C cLr tY Y t Lr 1 C c I cLY
·
¸¸ dG
¹
Lr 1 C c dG
Lr 1 C c I cLY
which is the same as equation (6.9), the result in the textbook example.
6.5
Equation (6.6) gives
1 C I
LY
Lr
dY
dr
dG C dT
dM
dG
dM
so
1 C dG
LY
dr dM
1 C I ( 1 C ) d M LY dG
( 1 C ) Lr LY I > 0
LY
Lr
if dr = 0 then d M LY d G / (1 C ) .
dG I dY dM
Lr
1C
I
Lr dG I dM
( 1 C ) Lr LY I Lr dG I LY dG / ( 1 C )
( 1 C )L r L Y I LY
dG
Lr
( 1 C ) Lr I d Y / ( 1 C )
( 1 C ) Lr I d Y
dG
1 C
As the marginal propensity to consume C increases, dY increases. The higher is dG, the
higher is dY. When the Fed makes r constant, we are in the simple Keynesian world with
fixed interest rates. This is called accomodating monetary policy.
6-5
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.6
The AD-AS model is made up of the three equations (6.10)-(6.12). Taking
differentials and setting dP E = dY F = dT = 0 yields the matrix equation
1 − C ′ − I ′

 LY
Lr

 −g′
0

0 
 dG 
 dY  

M     dM 
.
dr =
P2     P 
 dP 
1     0 
So
1 − C′
dG
dM
LY
P
0
−g′
dr =
1 − C′ −I ′
LY
Lr
−g′
0
0
M
P2
1
M  dM

− dG  LY + g ′ 2  +
(1 − C ′)
P  P

=
0
M 
g ′  2 I ′  + Lr (1 − C ′ ) + LY I ′
P 
M
2
P
1
so if dr = 0 then
dM =
P 
M
LY + g ′ 2  dG.

P 
1 − C′ 
Using Cramer’s rule to solve for dY and then substituting in for dM,
dG − I ′ 0
dM
M
Lr
P
P2
I′  P 
M 
Lr dG + 
 LY + g ′ 2  dG 
0
0
1
′
P  1− C 
P  
=
dY =
1 − C′ −I ′ 0
M 
g ′  2 I ′  + Lr (1 − C ′ ) + LY I ′
P 
M
LY
Lr
2
P
−g′
0
1
I′ 
M 

 Lr + 1 − C ′  LY + g ′ P 2  


= 
dG.
M 

g ′  2 I ′  + Lr (1 − C ′ ) + LY I ′
P 
Both the numerator and denominator are negative, so equilibrium GDP will
increase. Furthermore, inspection reveals that the denominator equals (1 − C ′ )
times the numerator, so
dY =
1
dG .
1 − C′
Equilibrium output rises by the same amount as if government purchases had
increased in the simple Keynesian model of Section 6.2.1. What has happened is
that the monetary policy, by keeping interest rates constant, has avoided the
crowding out that ordinarily occurs when expansionary fiscal policy increases the
demand for money, raising interest rates and reducing investment.
Using Cramer’s rule (and substituting in for dM) to solve for dP,
1 − C′ −I ′
dP =
LY
Lr
−g′
0
1 − C′ −I ′
LY
Lr
−g′
0
dG
dM
P
0
dM


−g′  −I ′
− Lr dG 
P


=
0
M


g ′  2 I ′  + Lr (1 − C ′ ) + LY I ′
P


M
2
P
1

I′ 
M 
g ′  Lr +
LY + g ′ 2  

1 − C′ 
P 
= 
dG
M 

g ′  2 I ′  + Lr (1 − C ′ ) + LY I ′
P 
So dP = g ′dY , which could have obtained more easily from equation (6.12).
Ordinarily this increase in the price level would have real effects since it reduces
the real money supply and should therefore raise interest rates. But the Fed has
ensured that interest rates do not changes, so the inflation doesn’t cause any
crowding out of investment. In more complex models, higher prices also affect
consumption and net exports, so the change in equilibrium output would be
different than dG (1 − C ′ ) .
6.7
Adding real wealth to equation (6.10), taking total differentials, and setting the
differentials of all exogenous variables except W to zero yields the matrix
equation
W

 CW

1 − CY − I ′ CW P 2 
dW 

dY

 
P

M   
 L
Lr
dr  =  0 
Y
2


P 
 dP   0 
 −g′
0
1    






where CY is now the notation for the marginal propensity to consume and CW is
the derivative of the consumption function with respect to real wealth.
Using Cramer’s rule to solve for dY,
CW
W
dW − I ′ CW 2
P
P
M
Lr
0
P2
0
0
1
dY =
1 − CY
W
P2
M
P2
1
− I ′ CW
LY
Lr
−g′
0
=
Lr CW P
dW .
M
W 

g ′  2 I ′ + Lr CW 2  + Lr (1 − C ′ ) + LY I ′
P 
P
Both the numerator and denominator are negative, so an increase in nominal
wealth increases equilibrium GDP (by shifting the aggregate demand curve to the
right). The increased wealth increases consumption and, though some of the
resulting increase in aggregate demand will be crowded out by higher interest
rates and prices, in the end equilibrium GDP will rise.
From equation (6.13),
dP = g ′dY =
g ′Lr CW P
dW ,
M
W 

g ′  2 I ′ + Lr CW 2  + Lr (1 − C ′ ) + LY I ′
P 
P
a result we could also have derived using Cramer’s rule. So the equilibrium
aggregate price level will increase, since aggregate demand has increased but the
aggregate supply curve has not shifted.
Using Cramer’s rule to solve for dr,
1 − CY
CW
dW
P
LY
0
−g′
0
dr =
1 − CY
LY
−g′
W
P2
M
P2
1
CW
W
− I ′ CW 2
P
M
Lr
P2
0
1
−
=
CW
M

dW  LY + g ′ 
P
P  −(+ )(+ )

=
dW
( −)
J
so the increase in nominal wealth will increase equilibrium interest rates, because
the higher equilibrium GDP will increase the demand for money, plus the higher
equilibrium price level will reduce the supply of real balances.
6.8
The AD-AS model is made up of the three equations (6.10)-(6.12). Taking
differentials and setting dP E = dY F = dT = dG = 0 yields the matrix equation
1 − C ′ − I ′

 LY
Lr

 −g′
0

Using Cramer’s rule to solve for dY,
0
−I ′ 0
dM
M
Lr
P
P2
0
0
1
=
dY =
1 − C′ −I ′ 0
=
LY
Lr
−g′
0
M
P2
1
0 
 0 
 dY  

M     dM 
dr =
.
P2     P 
 dP 
1     0 
−
1
( − I ′)
P
 M
g ′  I ′ 2  + Lr (1 − C ′ ) + I ′LY
 P 
dM
I′ P
( −)
dM =
dM .
( −)
g ′ ( I ′M P ) + Lr (1 − C ′ ) + I ′LY
2
Since dM < 0 , equilibrium GDP falls. If money demand becomes more sensitive
to interest rates, Lr becomes more negative, so the denominator increases in
absolute value and GDP falls by less. (This is because the interest rate will not
have to rise as much to restore equilibrium in the money market, so investment
will not fall by as much.) To see how the result depends on the sensitivity of
investment to interest rates, divide both numerator and denominator by I ′ :
1P
dY =
dM . Both numerator and denominator are
2
g ′ ( M P ) + Lr (1 − C ′ ) I ′ + LY
now positive. If I ′ becomes larger in absolute value (that is, more negative), the
denominator becomes smaller and GDP will fall by more (because higher interest
rates will now cause a larger decrease in investment).
From equation (6.13), dP = g ′dY so dP =
g ′I ′ P
dM , a
g ′ ( I ′M P ) + Lr (1 − C ′ ) + I ′LY
2
result we could also have obtained using Cramer’s rule. The impacts of money
demand and investment sensitivities to interest rates are similar to their impacts
on output, since the change in price is a movement along an unchanging aggregate
supply curve. So if money demand is more sensitive to interest rates, then output
changes by less and so do prices; if investment is more sensitive to interest rates,
output changes by more and so do prices.
Finally, turning to the equilibrium interest rate, using Cramer’s rule yields
1 − C′ 0
0
dM M
LY
P
P2
1
(1 − C ′)
0
1
−g′
(+)
P
dr =
dM =
dM .
=
1 − C′ −I ′ 0
M
(
)
−

g ′  I ′ 2  + Lr (1 − C ′ ) + I ′LY
 P 
M
LY
Lr
2
P
′
0
1
−g
Since the money supply is lower, the equilibrium interest rate is higher. Increases
in the absolute values of both I ′ and Lr make the denominator larger in absolute
value, so interest rates rise by less. (In the case of money demand sensitivity, this
is because interest rates don’t have to rise as much to restore equilibrium in the
money market. When investment is more sensitive to interest rates, investment
will fall by more when interest rates rise, which in turn means output and the
demand for money will fall by more as well, meaning the money market will be
easier to restore to equilibrium.)
6.9
W

Y = C  Y − T (Y ) ,
P

M
= L (Y , r )
P
P = P (Y − Y F )

F
 + I ( r ) + G (Y − Y )

The total differentials of these three equations are
W
1

dY = C1 ( dY − T ′dY ) + C2  dW − 2 dP  + I ′dr + G′ ( dY − dY F )
P
P

1
M
dM − 2 dP = LY dY + Lr dr
P
P
dP = P′ ( dY − dY F )
Putting these total differentials in matrix notation,
W 

 1
F 
1 − C1 (1 − T ′ ) − G ′ − I ′ C2 P 2 
 C2 P dW − G′dY 

  dY  

1
M
  


 . So
LY
Lr
dr =
dM


P 2    
P
dP




F


0
P′
−1 
P′dY









1 − C1 (1 − T ′ ) − G′ − I ′ −G′ 
∂P* 

=
Ly
Lr
0  J
F
∂Y

P′
P′ 
0

=
(
(
P′ G′Lr + Lr (1 − C1 (1 − T ′ ) − G′ ) + I ′LY
))
M
W 

P′  − I ′ 2 − Lr C2 2  − Lr (1 − C1 (1 − T ′ ) − G′ ) + I ′LY
P
P 

The two terms in the numerator including G′ cancel each other out, so
(+)
( −) (+ )  
( +)   ( −)
P′   Lr (1 − C1 (1 − T ′ ) )+ I ′ LY  


( − ) < 0.
∂P*


=
=
F
∂Y
 (+)

 
M
W



P′  − I ′ 2 − Lr C2 2  − Lr  (1 − C1 (1 − T ′ ) ) − G′ + I ′ LY 
( +)  ( −) P
(−)  (−) (+) 
( −) ( + ) P  
( −) 

(−)
(+)  
( −)



As full-employment GDP goes up, prices are driven down since the aggregate
supply curve has shifted to the right. Additionally, what is new in this model is
that as full-employment GDP goes up, government purchases increase to try to
reduce the gap between GDP and full-employment GDP. This gap does indeed
decrease, which puts upward pressure on prices and means that the aggregate
price level does not decrease by as much as it otherwise would (since
P′G ′Lr > 0 ). But the math shows that the price level does, in the end, decrease.
(
)
6.10
C2
∂Y *
=
∂W
=
1
P
W
P2
M
P2
−1
− I ′ C2
0
Lr
0
0
J
− Lr C2
1
P
M
W 

P′  − I ′ 2 − Lr C2 2  − Lr (1 − C1 (1 − T ′ ) − G′ ) + I ′LY
P
P 

(
)
=
( + ) > 0.
(+)
(The denominator was shown to be positive in the answer to Problem 6.9.)
As G ′ increases in absolute value (becomes more negative) nothing will happen
to the numerator. But the denominator will change since the term
1 − C1 (1 − T ′ ) − G′ becomes more positive. This makes the denominator more
positive as well, making the magnitude of ∂P ∂Y P smaller. (The derivative
becomes less negative.) The economic explanation is that as the sensitivity of
government purchases to changes in GDP gets larger, then as the aggregate price
level falls due to the rightward shift of the AS curve, government purchases are
simultaneously increasing by more than they used to, which spurs GDP and keeps
the aggregate price level from falling as much.
6.11
As shown in the answer to Problem 6.9, the total differentials of the system of
equations, when written in matrix notation, are
W 

 1
F 
1 − C1 (1 − T ′ ) − G ′ − I ′ C2 P 2 
 C2 P dW − G′dY 

  dY  

M
1
  


.
LY
Lr
dr =
dM


P 2    
P

 dP


P′
0
P′dY F
−1    








If dW and dM are nonzero but dYF = 0, then
1
1 − C1 (1 − T ′ ) − G ′ − I ′ C2 dW
P
1
LY
Lr
dM
P
0
0
1
1
P′


P′  − I ′ dM − Lr C2 dW 
P
P
.
dP =
= 
J
J
So if dP = 0, then − I ′
1
1
LC
dM − Lr C2 dW = 0 or dM = − r 2 dW (which is
P
P
I′
negative if dW is positive, which makes sense).
6.12
Rewriting equation (6.26), the impact of monetary policy on interest rates is
∂r *
1
=
.
∂M Lr + LY I r (1 − C ′ − IY )
If the marginal propensity to consume C ′ or the derivative of investment with
respect to income IY increase, then each round of the expenditure multiplier
process induces more extra aggregate demand and therefore a larger increase in
the demand for money. This will make interest rates fall less than they otherwise
would in response to expansionary monetary policy, so ∂r * ∂M becomes smaller
in absolute value (less negative): the second term in the denominator becomes
larger in absolute value, driving down the absolute value of the derivative as a
whole.
If money demand becomes more responsive to changes in interest rates, then Lr
gets larger in absolute value (more negative), so ∂r * ∂M becomes smaller in
absolute value (less negative). When the money supply is increased, interest rates
do not have to fall by as much to restore equilibrium in the money market.
If investment becomes more responsive to changes in interest rates, then I r gets
larger in absolute value (more negative), so ∂r * ∂M becomes smaller in
absolute value (less negative). When the money supply is increased, interest rates
fall, which leads to a larger increase in investment, which in turn makes money
demand rise more and so equilibrium in the money market is restored more easily.
If money demand becomes more responsive to changes in income, then LY gets
larger in absolute value (more negative), so ∂r * ∂M becomes smaller in
absolute value (less negative). When the money supply is increased, interest rates
fall, which leads to an increase in investment, which raises income; then the
greater sensitivity of money demand to income means money demand rises by
more, so equilibrium in the money market is restored more easily.
6.13
Rewriting equation (6.25), the impact on output of monetary policy is
∂Y *
1
=
.
∂M Lr (1 − C ′ − IY ) I r + LY
If the marginal propensity to consume C ′ increases, then each round of the
expenditure multiplier process induces more extra aggregate demand and
therefore an expansionary monetary policy will increase equilibrium output by
more: the first term in the denominator becomes smaller in absolute value, driving
up the value of the derivative as a whole.
If money demand becomes more responsive to changes in interest rates, then Lr
gets larger in absolute value (more negative), so ∂Y * ∂M becomes smaller.
When the money supply is increased, interest rates do not have to fall by as much
to restore equilibrium in the money market, so there is less induced increase in
investment..
If money demand becomes more responsive to changes in income, then LY gets
larger in absolute value (more negative), so ∂Y * ∂M becomes smaller. When
the money supply is increased, interest rates fall, which leads to an increase in
investment, which raises income; then the greater sensitivity of money demand to
income means money demand rises by more, so equilibrium in the money market
is restored more easily. Hence interest rates fall by less and there is less induced
investment.
6.14
Y = C (Y ) + I ( r ) + G
M = L (Y , r )
N = e (Y )
dY = C ′dY + I ′dr + dG
so dM = LY dY + Lr dr
dN = e′dY
or
1 − C ′ − I ′ 0   dY   dG 
 L
Lr 0   dr  =  dM 
 Y
0 1   dN   0 
 −e′
dY =
dG
dM
0
−I ′ 0
Lr 0
(−)
0 1
Lr dG + I ′dM
Lr
∂Y *
=
so
=
>0.
J
Lr (1 − C ′ ) + I ′LY
∂G Lr (1 − C ′ ) + I ′LY
(−)
This is the standard IS-LM result because the third equation, relating the state of
the environment to GDP, does not affect the equilibrium values of Y and r since
there is no feedback from N to those variables.
6.15
1 − C ′ − I ′ dG
LY
Lr dM
(+)
0
0
−e′
−e′ ( − I ′dM − Lr dG )
∂N *
e′Lr
=
< 0 and
dN =
=
so
Lr (1 − C ′ ) + I ′LY
∂G
J
Lr (1 − C ′ ) + I ′LY
(−)
(+)
∂N *
er′ I ′
=
< 0.
Lr (1 − C ′ ) + I ′LY
∂M
(−)
An increase in either government purchases or the money supply will be
expansionary and therefore lead to an increase in equilibrium output. This degrades
the environment.
6.16
For dN = 0, − e′ ( − I ′dM − Lr dG ) = 0 ⇒ I ′dM + Lr dG = 0 or dM = −
Lr
dG .
I′
6.17
Applying Cramer’s rule to matrix equation (6.36),
0
XY
f′
0 1 − C ′ − IY − X Y − I r
∂e * 1
=
∂M
=
LY
J
Lr
=
− I r X Y − f ′ (1 − C ′ − IY − X Y )
J
− ( I r − f ′ ) X Y − f ′ (1 − C ′ − IY ) −(−)(−) − (+ )
=
<0
J
(+)
(The Jacobian is written out and signed on page 151 of the text.) The
expansionary monetary policy lowers interest rates, leading to smaller net capital
inflows, and increases GDP and therefore imports. Higher imports and smaller net
capital inflows both lower the value of the dollar versus other currencies since US
consumers will want to convert dollars into other currencies in order to buy
imports and more dollars will be converted into other currencies (and fewer other
currencies will be converted into dollars) due to capital flows.
If investment is more sensitive to interest rates, then I r gets larger in absolute
value (more negative). This makes the numerator of ∂e * ∂M larger in absolute
value, but the Jacobian also becomes greater (see page 151). There will be more
“crowding in” of investment when interest rates fall, so equilibrium GDP and
therefore imports will rise by more (see equation (6.40)). On the other hand,
interest rates will fall by less (see equation (6.41)), since the higher increase in
GDP raises money demand by more so interest rates don’t have to fall as far to
restore equilibrium in the money market. The two effects work in opposite
directions in the foreign exchange market: the supply of dollars coming from US
consumers who want to import will rise, but the net demand for dollars from
capital markets will also rise. So overall it’s impossible to say in general whether
exchange rates will change by more or by less when investment becomes more
sensitive to interest rates.
6.18
The effect of various parameters on the change in equilibrium output caused by an
increase in the money supply can be seen from equation (6.40):
∂Y *
1
=
.
∂M  Lr (1 − C ′ − IY ) ( I r − f ′ ) + LY 
If investment is more sensitive to interest rates, then I r gets larger in absolute
value (more negative). This makes the denominator of ∂Y * ∂M smaller and
therefore equilibrium GDP will rise by more: there will be more “crowding in” of
investment when interest rates fall.
If money demand is more sensitive to output, then LY gets larger and so the
denominator of ∂Y * ∂M becomes larger and therefore equilibrium GDP will rise
by less: as GDP starts to rise, money demand increases, restoring equilibrium in
the money market more easily and therefore leading to smaller changes in interest
rates, investment, and output.
If the marginal propensity to consume C ′ increases, then the denominator of
∂Y * ∂M gets smaller and therefore equilibrium GDP will rise by more: there
will be more induced consumption on each round of the multiplier process.
6.19
With fixed exchange rates, the endogenous variables are Y, r, and ∆F . The
exogenous variables are r F , G, M, and e. In this problem, only M is changing so
we can write equations (6.26)-(6.28) as a system of identities, with each
endogenous variable written as a function of M:
X ( Y * ( M ) , e ) + f ( r * ( M ) − r F ) − ∆F * ( M ) ≡ 0
Y * ( M ) − C (Y * ( M ) ) − I (Y * ( M ) , r * ( M ) ) − G − X (Y * ( M ) , e ) ≡ 0
M + ∆F * ( M ) − L (Y * ( M ) , r * ( M ) ) ≡ 0
Implicitly differentiating with respect to M and putting the result in matrix
notation,
XY
f ′ −1  ∂Y * ∂M   0 

1 − C ′ − I − X
− I r 0   ∂r * ∂M  =  0  .
Y
Y


− LY
− Lr 1  ∂∆F * ∂M   −1
∂Y *
=
∂M
0
f′
−1
0
−Ir
0
−1 − Lr
XY
1
f′
−1
1 − C ′ − IY − X Y
−Ir
0
− LY
− Lr
1
=
Ir
− ( − Lr (1 − C ′ − IY − X Y ) − I r LY ) − I r X Y − f ′ (1 − C ′ − IY − X Y )
=
Ir
( −)
=
> 0.
( Lr − f ′ )(1 − C ′ − IY − X Y ) + I r ( LY − X Y ) (−)(+) + (−)(+ )
∂r *
=
∂M
XY
0
1 − C ′ − IY − X Y
0
−1
0
− LY
XY
−1
f′
1
−1
−Ir
− Lr
0
1 − C ′ − IY − X Y
− LY
1
=
1 − C ′ − IY − X Y
− ( − Lr (1 − C ′ − IY − X Y ) − I r LY ) − I r X Y − f ′ (1 − C ′ − IY − X Y )
=
1 − C ′ − IY − X Y
(+)
=
< 0.
( Lr − f ′ )(1 − C ′ − IY − X Y ) + I r ( LY − X Y ) (−)(+) + (−)(+ )
If capital is perfectly mobile, f ′ → ∞ and monetary policy is completely
ineffective since the domestic interest rate cannot deviate from the world interest
rate, which is assumed to stay constant. Moreover, net exports cannot change
either since the trade balance must equal zero. Since the interest rate and
exchange rates don’t change, neither does investment or output. (When f ′ → ∞
the Jacobian gets infinitely large in absolute value, driving both comparative static
derivatives to zero.)
If capital is perfectly immobile, f ′ = 0 and the comparative static derivatives
equal the corresponding derivatives from an IS-LM model where net exports
depend on output, except for the last term in the denominator. In an IS-LM model,
even one where net exports depend on output, the term I r would be multiplied by
LY . In the Mundell-Fleming model with fixed exchange rates, I r is multiplied by
( LY − X Y ) . This makes the denominator larger in absolute value and therefore the
changes in equilibrium output and interest rates are smaller. The change in the
denominator of the comparative static derivatives reflects the fact that when the
money supply is increased, domestic income will rise and therefore imports will
rise. With no capital flows, the only way to keep the exchange rate constant is for
the central bank to buy dollars in the foreign exchange market to offset the dollars
being sold in order to increase imports. This effectively reduces the money
supply, offsetting some of the original expansion of the money supply.
6.20
With fixed exchange rates, the endogenous variables are Y, r, and ∆F . The
exogenous variables are r F , G, M, and e. In this problem, only G is changing so
we can write equations (6.26)-(6.28) as a system of identities, with each
endogenous variable written as a function of G:
X (Y * ( G ) , e ) + f ( r * ( G ) − r F ) − ∆F * ( G ) ≡ 0
Y * ( G ) − C (Y * ( G ) ) − I (Y * ( G ) , r * ( G ) ) − G − X (Y * ( G ) , e ) ≡ 0
M + ∆F * ( G ) − L (Y * ( G ) , r * ( G ) ) ≡ 0
Implicitly differentiating with respect to M and putting the result in matrix
notation,
XY
f ′ −1  ∂Y * ∂G   0 

1 − C ′ − I − X
− I r 0   ∂r * ∂G  = 1  .
Y
Y


− LY
− Lr 1  ∂∆F * ∂G   0 
0
1
0
∂Y *
=
XY
∂G
1 − C ′ − IY
− LY
=
f′
−Ir
− Lr
− XY
−1
0
1
f′
−Ir
− Lr
−1
0
1
− ( f ′ − Lr )
− ( − Lr (1 − C ′ − IY − X Y ) − I r LY ) − I r X Y − f ′ (1 − C ′ − IY − X Y )
=
Lr − f ′
( Lr − f ′)(1 − C ′ − IY − X Y ) + I r ( LY − X Y )
=
1
(1 − C ′ − IY − X Y ) + I r ( LY − X Y ) ( Lr − f ′ )
=
1
> 0.
(+ ) + ( (−)(+ ) (−) )
If the marginal propensity to consume C ′ or the derivative of investment with
respect to output IY gets larger, the denominator of the comparative static
derivative gets smaller and expansionary fiscal policy has a larger impact on
equilibrium output because there is more induced aggregate demand on each
round of the multiplier process. If investment becomes more sensitive to interest
rates, then I r gets larger in absolute value, the denominator gets larger, and
expansionary fiscal policy becomes less effective since there is more crowding
out. If money demand is more sensitive to changes in output, LY gets larger, the
denominator gets larger, and expansionary fiscal policy becomes less effective
since increases in output create larger increases in money demand, raising interest
rates more and crowding out more investment. When money demand is more
sensitive to interest rates, Lr gets larger in absolute value, the denominator gets
smaller, and expansionary fiscal policy becomes more effective since smaller
increases in interest rates are needed to restore equilibrium in the money market,
leading to less crowding out of investment. The larger is f ′ , the more net capital
flows in when interest rates rise. To keep exchange rates fixed, the central bank
has to sell dollars in the foreign exchange market, effectively increasing the
money supply and reducing the rise in equilibrium interest rates. So there is less
crowding out of investment and equilibrium GDP will rise by more.
Algebraically, the denominator of the comparative static derivative gets smaller in
absolute value, so the derivative gets larger. When net exports change by more in
response to changes in GDP, X Y becomes larger in absolute value and there are
two effects on the comparative static derivative, both of which make the
denominator algebraically larger. So fiscal policy is less effective. The first effect
is that less aggregate demand is induced in each round of the multiplier process,
because more of the induced consumption is of imported goods. The second effect
is that the increase in imports would tend to lower the value of the dollar in
foreign exchange markets, since there is an increased supply of dollars being
converted to other currencies. To keep exchange rates fixed, the central bank must
buy these extra dollars, effectively reducing the money supply, which raises
interest rates and leads to more crowding out of investment.
6.21
To the Mundell-Fleming model with flexible exchange rates of Section 6.3.2, add
as an argument of the net capital inflows function ψ , and index of political
stability. Assume that this index becomes more positive when there is more
stability and becomes less positive when there is more instability. Since more
stability increases the attractiveness of financial investments in the country, net
capital inflows should increase when the stability index increases. The three
equations of the model are
X ( Y , e ) + f ( r − r F , ψ ) = ∆F = 0
Y = C (Y ) + I (Y , r ) + G + X (Y , e )
M + ∆F = M = L (Y , r )
and implicit differentiation with respect to G yields, in matrix notation,
XY
f r   ∂e * ∂G   0 
 Xe
− X 1 − C′ − I − X
− I r  ∂Y * ∂G  = 1  .
Y
Y
 e
 0
LY
Lr   ∂r * ∂G   0 
(Note that f r is a function of the interest rate differential and also of the political
stability index. Although it is clear that f ψ > 0 , it is not quite as clear whether
greater political stability also increases the net capital inflow response to changes
in domestic interest rates. For answering this problem we will assume that it does,
so f rψ > 0 .)
The effect of fiscal policy on output can be seen by solving for the comparative
static derivative ∂Y * ∂G . Using Cramer’s rule,
XY
 Xe
− X 1 − C′ − I − X
Y
Y
 e
 0
LY
Xe 0
−Xe
0
f r   ∂e * ∂G   0 
− I r  ∂Y * ∂G  = 1 
Lr   ∂r * ∂G   0 
fr
1 −Ir
0 Lr
XY
∂Y *
=
Xe
∂G
− X e 1 − C ′ − IY − X Y
fr
−Ir
=
X e Lr
.
X e ( Lr (1 − C ′ − IY ) + LY ( I r − f r ) )
0
LY
Lr
(The derivation of the denominator is shown on page 151 of the text.)
When there is more political instability, the stability index ψ gets smaller, which
we have assumed makes f r smaller as well. (Even when interest rates increase,
the instability means foreign investors are reluctant to send their capital to the
country and domestic investors are reluctant to invest their capital domestically.)
A smaller value of f r makes the term ( I r − f r ) less negative (smaller in absolute
value), so the denominator becomes smaller and the derivative as a whole
becomes larger: fiscal policy is more effective. This is because, with net capital
inflows increasing by less as expansionary fiscal policy raises interest rates, the
dollar will not rise in value as much compared to other currencies. So net exports
will not fall by as much, and equilibrium output rises by more.
6.22
(Note that this problem should refer to Section 6.4.)
For the LM curve, the money market equilibrium condition is
f (Y , r ) = L (Y , r ) − M ( r ) = 0 . So the slope of the LM curve is given by
∂r
∂f ∂Y
LY
LY
=−
=−
=
.
∂Y
∂f ∂r
Lr − M r M r − Lr
The LM curve is steeper ( ∂r ∂Y is greater) the larger is the sensitivity of money
demand to income LY , the smaller is the effect of interest rates on the money
supply M r , and the smaller in absolute value (less negative) is the sensitivity of
money demand to interest rates Lr . For the LM curve to be steep, it must take a
large increase in interest rates to restore equilibrium when an increase in income
increases money demand. This will be the case if money demand is very sensitive
to income and is not very sensitive to interest rates, or if money supply is not very
sensitive to interest rates.
For the IS curve, the goods market equilibrium condition is
g (Y , r , G ) = Y − C (Y − T (Y ) ) − I ( r , Y ) − G = 0 so the slope of the IS curve is
1 − C ′ (1 − T ′ ) − IY 1 − C ′ (1 − T ′ ) − IY
∂r
∂g ∂Y
=−
=−
=
.
∂Y
∂g ∂r
−Ir
Ir
The IS curve is steeper ( ∂r ∂Y is greater in absolute value) the greater is the
marginal tax rate T ′ and the smaller (in absolute value) are the marginal
propensity to consume C ′ , the sensitivity of investment to income IY , and the
sensitivity of investment to the interest rate I r . As the interest rate falls,
investment increases and so does output, through the multiplier process. A higher
tax rate reduces the size of the multiplier, while a higher marginal propensity to
consume increases it, as does a greater tendency for investment to increase when
output increases. For the IS curve to be steep, there should be a small multiplier
since Y should not have to increase much to restore goods market equilibrium
when r falls. A small sensitivity of investment to interest rates yields a steep IS
curve because a decrease in the interest rate will not lead to much increase in
investment, and therefore not much increase in output.
6.23
Π = P ( Q ) Q (1 − t ) − C ( Q )
FOC : P′Q (1 − t ) + P (1 − t ) − C ′ = 0
SOC : P′′Q (1 − t ) + P′ (1 − t ) + P′ (1 − t ) − C ′′ < 0
Implicit differentiation of the first-order condition with respect to t yields
*
− ( P′Q + P ) = 0
( P′′Q (1 − t ) + P′ (1 − t ) + P′ (1 − t ) − C ′′) dQ
dt
dQ *
P′Q + P
=
dt
P′′Q (1 − t ) + P′ (1 − t ) + P′ (1 − t ) − C ′′
the numerator of which is negative by the second-order condition. The numerator
is positive if marginal cost is positive since the first-order condition implies that
P′Q + P = C ′ (1 − t ) . So dQ * dt < 0 . Since dP * dt = P′ ( dQ * dt ) , dP * dt > 0
as long as the demand curve is downward sloping. The formula for dP * dt is


dP *
P′Q + P
= P′ 
 .
′′
′
′
′′
dt
P
Q
1
t
P
1
t
P
1
t
C
−
+
−
+
−
−
(
)
(
)
(
)


6.24
By the envelope theorem,
∂Π * ∂Π d Π
=
=
= − PQ < 0
∂t
∂t
dt dQ =0
6.25
(Note this problem should refer to the parameter c.)
Differentiating equations (6.73) with respect to c,
∂q1* ∂q2*
∂q1*
−2
−
− 6cq1
− 3q12 = 0
∂c ∂c
∂c
*
*
∂q
∂q
∂q*
− 1 − 2 2 − 6cq2 2 − 3q22 = 0
∂c
∂c
∂c
 −2 − 6cq1
 −1

−1   ∂q1* ∂c  3q12 

= 
−2 − 6cq2  ∂q2* ∂c  3q22 
3q12 ( −2 − 6cq2 ) + 3q22
∂q1*
=
.
∂c ( 2 + 6cq1 )( 2 + 6cq2 ) − 1
Using symmetry,
2
∂q1* ∂q2* 3q ( −2 − 6cq + 1) (−)
=
=
=
<0
2
∂c
∂c
( 2 + 6cq ) − 1 (+)
 3q 2 ( −2 − 6cq + 1) 
∂Q * ∂q1* ∂q2*
=
+
= 2
<0
 ( 2 + 6cq )2 − 1 
∂c
∂c ∂c


2
 3q ( −2 − 6cq + 1) 
∂q* ∂q*
∂P *
= − 1 − 2 = −2 
>0
 ( 2 + 6cq )2 − 1 
∂c
∂c ∂c


6.26
If TCi = bi qi3 then the FOC for firm i is a − 2qi − q j − 3bi qi2 = 0 . Implicitly
differentiating with respect to a yields 1 − ( 2 + 6bi qi )
*
∂qi* ∂q j
−
= 0 . Doing this for
∂a ∂a
both firms and putting the results in matrix notation,
−1
 − ( 2 + 6b1q1 )
 ∂q1* ∂a   −1


=
−1
− ( 2 + 6b2 q2 )  ∂q2* ∂a   −1

so
∂q1*
∂q*
2 + 6b1q1 − 1
2 + 6b2 q2 − 1
.
=
and 2 =
∂a ( 2 + 6b1q1 )( 2 + 6b2 q2 ) − 1
∂a ( 2 + 6b1q1 )( 2 + 6b2 q2 ) − 1
We can’t say for sure which is greater because, while b1 > b2 , q1* will end up
being smaller than q2* . It seems like the higher-cost firm should both sell less
output and increase its output by less when the demand curve shifts up, but this
result can’t be shown from these formulas. When the demand curve shifts up,
marginal revenue for both firms increases. So both firms will want to make more
output to restore equilibrium between marginal revenue and marginal cost. The
low-cost firm has a flatter marginal cost curve, so it will take a bigger increase in
output to make MC rise to equilibrium with the higher MR. On the other hand, the
high-cost firm is also going to be increasing output, and that will affect the lowcost firm’s MR, so the story is more complicated than it may first appear.
6.27
a
i 3
q i c q i so the first-order condition is
q1 q2
a q1 q2 a qi
q1 q 2
2 a qj
SOC:
q1 q 2
3
2
a qj
3c q i2 0 or
q1 q 2
6 c q i < 0 or 2 a q j q 1 q 2
3 c q i2 0
2
3
6 c qi < 0
Differentiating through the first-order condition with respect to a,
qj
( q1 q 2 ) 2
1
q j
( q1 q 2 ) 2
a
qj ( q 1 q 2 ) ( q 1 q 2 )
q j
a
2a q j
q i
( q1 q 2 ) 3
a
2a q j
q i
a
q j
a
q j
a
6c q i
q i
a
( q 1 q 2 ) 6c q i
3
0 or
qi
a
0
Combining the equations for the two firms (i = 1, 2) yields
2 a q 2 ( q 1 q 2 ) 3 6c q 1
( q 1 q 2 ) 2a q 2
q 1 / a
( q 1 q 2 ) 2 aq 2
2 a q 1 ( q 1 q 2 ) 3 6c q 2
q 2 / a
q 1
a
q 2 ( q1 q 2 )
q 1 ( q1 q 2 )
q 2 ( q 1 q 2 ) ( 2a q 1 ( q 1 q 2 ) 3 6 cq 2 ) q 1 ( q 1 q 2 ) ( q 1 q 2 2 aq 2 )
( 2a q 2 ( q 1 q 2 ) 3 6c q 1 ) ( 2 aq 1 ( q 1 q 2 ) 3 6c q 2 ) ( q 1 q 2 2 aq 2 ) 2
Using symmetry,
6-23
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
q 1
a
a
q ( 2 q ) 4 6 cq q ( 2 q ) 2
2 aq ( 2 q ) 3 6 cq
96 c q 6 4 q 3
2 a q 48 cq 4
2
q 2
2q ( 1 a )
2
2
( 2 q 2 aq ) 2
24 q 3 c q 3 1
4 a 2 q 2 192 a cq 5 48 2 c 2 q 8 4 q 2 ( 1 a ) 2
24 q 3 cq 3 1
4 a 2 q 2 192 ac q 5 48 2 c 2 q 8 4 q 2 1 2a a 2
24 q 3 c q 3 1
192 ac q 5 48 2 c 2 q 8 4 q 2 8 aq 2
as long as a >
()
>0
192 a c q 5 48 2 c 2 q 8 4 q 2 ( 2a 1 ) ( )
24 q 3 cq 3 1
1
.
2
6-24
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.28
i P (a,Q ) qi C (q i ) so the first- and second-order conditions for firm i are
i
qi
2i
qi2
Let i 2i
q i2
P
P qi 1 Q
q j
C 0
q i
q
P
1 j
qi
Q
2
2
q
P
1 j
qi Q
qi
2
C <0
. Then differentiating through the FOC with respect to a yields
P qi
2 P 1 q j
i
qi
Q a
a
a
qi
2
q
P
qi 2 1 j
q i
Q
qj
0
a
Putting together these equations for the two firms (i = 1,2) yields
2P 1 q2
q1
q1
Q 2
1
q2
q1
a
2P 1 q1
q2
Q 2
q2
a
2
2
q
P
P
1 2
q1 Q a
a
q1
q1
a
2
q
P q2 P 1 1
Q a
a
q2
1 2 q1 q2
2P
Q 2
2
1
q1
2
q
P q1 P 1 2
Q a
a
q1
2
q
P
P
1 1
q2 Q a
q2
a
2
2P 1 q2
q1
Q 2
q1
q2
1
q2
q1
6-25
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
If q i / q j 0 then q1 q2 and
q1
a
q2
a
2
P q P
Q a
a
2
2
2
2
P
q P2 C P q P q P2
Q a
Q
a
Q
Q
2
P
2 P
2
q 2 C Q
Q
q
Q 2
2
P q P
a
2
Q a
P
2 P
2
q 2 C Q
Q
2
2P
2
P
C Q
2
q
2P
2
2
Q 2
If q i / q j 1 / 2 then q1 q2 and
q a
2
P 1.5 q P
a
Q a
3
P
2 P
P
2P
2P
q
1.5 q 2 C 1.5
1.5q
Q a
Q
a
Q
Q 2
2
P
3
1.5q P2 C Q
Q
2
2.25q
2
P 1.5 q P
a
3
Q a
2
P
3
1.5 q P2 C Q
Q
2
2P
Q 2
2
P
C Q
2
2.25 q
2
2P
Q 2
2
6-26
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
If q1 / q2 1 and q2 / q1 0 then
q1
a
2
P q1 P
Q a
a
4
2
2
2
P
2 q2 P2 C P 2q1 P q1 P2
Q a
Q
a
Q
Q
P
2 P
2
q1 2 C Q
Q
2
P q1 P
a
Q a
2
P
2
q1 P2 C Q
Q
P
2 P
2P
4
2 q2 2 C 2 q1 q2
Q
Q
Q 2
4
P
C Q
2
P 2q2 q1 P2
a
2
P
4
2 q 2 P2 C Q
Q
Note that if there is a parallel shift in the demand curve,
2
Q
2 q1 q2
2P
2
Q 2
2P
0 so all of those terms fall out
Q a
of the expressions above.
6-27
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.29
Firm 1 chooses L1 to maximize Π1 = R1 F 1 ( L1 ) , F 2 ( L2 , α ) − w1 L1 , treating L2 as fixed.
(
)
The FOC is R11 FL1 − w1 = 0 and, since both R11 and FL1 are functions of L1 , the SOC is
1
1
R11 FLL
+ FL1 R111 FL1 < 0 or R11 FLL
+ R111 ( FL1 ) < 0 . Similarly, the FOC for firm 2 is
2
R22 FL2 − w2 = 0 and the SOC is R22 FLL2 + R222 ( FL2 ) < 0 .
2
The equilibrium of this model is defined by the simultaneous solution to the two first
order conditions (like in a Cournot model), so to find comparative static derivatives we
need the total differentials of the two first order conditions:
(R F
1
1
(R
2
21
1
LL
)
+ R111 ( FL1 ) dL1 + ( R121 FL2 FL1 ) dL2 + ( R121 Fα2 FL1 ) d α − dw1 = 0
2
(
F F ) dL + R F + R
2
L
1
L
1
 R1 F 1 + R1 ( F 1 )2
L
11
 1 LL

R212 FL2 FL1

1
2
2
2
LL
(F )
2 2
L
2
22
) dL + ( R F
2
2
2
2
Lα
+R F F
2
22
2
α
2
L
) d α − dw
or
2
=0

1


− ( R121 Fα2 FL1 ) d α + dw1
  dL  = 
 . So
 2
2 2
2
2 2
2
2 2
2
2 2  dL

−
+
α
+
R
F
R
F
F
d
dw

  ( 2 Lα
R2 FLL + R22 ( FL )  
22 α L )

R121 FL2 FL1
R121 FL1 FL2
2 2
2
2
∂L1* 0 R2 F22 + R22 ( FL )
=
.
∂w1
J
2
Note that the denominator is a Jacobian but it is not a Hessian since the two first order
conditions do not come from the same maximization problem (so the Jacobian is not the
determinant of the matrix of second derivatives of a single objective function). Therefore
the denominator cannot be signed by some second order condition. In fact, it cannot in
general be signed:
1
+ R111 ( FL1 )
R11 FLL
J =
(
2
R22 FLL2 + R222 ( FL2 )
R212 FL2 FL1
1
= R11 FLL
+ R111 ( FL1 )
R121 FL2 FL1
2
)(R F
2
2
2
LL
2
)
+ R222 ( FL2 ) − ( R212 FL2 FL1 )( R121 FL2 FL1 ) .
2
Both of the first two terms are negative, from the two firms’ FOC, so their product is
positive. But this positive value then has another positive value subtracted from it, since
the last two terms in parentheses will presumably have the same sign since R212 and R121
should have the same sign (probably negative since if one firm’s quantity increases, that
should drive down the price and therefore the revenue of the other firm). So we cannot
even be sure that the first firm will lower its equilibrium use of labor when its wage
increases. The economic explanation is as follows: when firm 1’s wage increases, it will
want to increase marginal revenue since its marginal cost is now higher. But its marginal
revenue depends on both its own output and the output of its competitor. As firm 1 starts
to reduce its output, this will affect firm 2’s revenue. If the cross-partial derivative is high
enough, firm 2’s marginal revenue function will shift enough to create a large change in
its quantity, which then in turn has a large effect on firm 1’s revenue (and marginal
revenue function). In a weird enough case, these cross-price effects might be large
enough compared to the own-price effects that firm 1 will end up using more labor after
its wage increases.
6.30
From equation (6.86),
L w
a ( I C0 )
aw 2
( wL I C 0 ) 2
( wL I C 0 ) 2
From equation (6.80), L aT So
( 1 a ) ( I C0 )
w
1
1a
( T L ) 2
.
wL wa T ( 1 a ) (I C0 ) ,
wL I C 0 wa T a ( I C 0 ) a (wT I C 0 ) , and
TL T aT (1 a ) ( I C 0 )
w
(1 a )
wT I C 0
w
So
a ( I C0 )
L w
aw 2
a ( wT I C 0 )
a ( I C0 )
a ( wT I C 0 )
a ( wT I C 0 )
2
aw 2
a (wT I C 0 )
2
a ( I C0 )
aw 2
( 1 a )
wT I C 0
2
2
w
1
w2
(1 a) wT I C 0
a w 2 (1 a) a 2 w 2
a ( wT I C 0 )
a ( I C 0 ) ( 1 a)
2
2
1
1 a
2
1
a 2 (1 a) (wT I C 0 ) 2
( 1 a ) ( I C0 )
w2
which is equation (6.82).
6-28
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.31
We want to maximize W ( x ,y ) xy subject to P x x P y y I , or to maximize
x
FOC:
I Px x
I 2 Px x / P y 0
Py
<
I x Px x 2
Py
x I / (2 P x )
SOC: 2 P x / Py < 0
6-32
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.32
The reason the utility-maximizing choices of x and y are the same is that V is a
Monotonic function of U: V = eU .
The utility-maximizing choices of x and y from
U ( x, y ) = a ln ( x − x0 ) + b ln ( y − y0 ) come from the first-order conditions of
optimizing the Lagrangian L = a ln ( x − x0 ) + b ln ( y − y0 ) + λ ( I − px x − p y y ) :
a
− λp x = 0
x − x0
FOC:
b
− λp y = 0
y − y0
I − px x − p y y = 0
So
a
b
=
( x − x0 ) px ( y − y0 ) p y
⇒ py y =
b ( x − x0 ) px
+ p y y0 .
a
Substituting into I − px x − p y y = 0,
b ( x − x0 ) px
+ p y y0 = 0
a
bp x
 b
I − x 0 + p y y0 = px x 1 + 
a
 a
bpx x0

 a 
+ p y y0  = p x x

 I −
a
 a + b 

 1  a 
bpx x0

+ p y y0  = x *
 
 I −
a

 px   a + b 
I − px x −
 1   a 
 1 
bp x

I − x 0 + p y y0  
y* =   ( I − px x *) =    I − 
 p y    a + b  
 py 
a

 
 
For the utility function V ( x, y ) = ( x − x0 )
L = ( x − x0 )
a
( y − y0 )
b
a
( y − y0 )
b
,
+ λ ( I − px x − p y y ) so the FOC are
a ( x − x0 )
( y − y0 ) − λ p x = 0
a
b −1
( x − x0 ) b ( y − y0 ) − λp y = 0
a −1
I − px x − p y y = 0
Thus
b
a
b
a −1
b
a
b −1
( x − x0 ) ( y − y0 ) = ( x − x0 ) ( y − y0 )
px
py
ap y
bpx
=
x − x0
y − y0
⇒ py y =
bpx ( x − x0 )
+ p y y0
a
This is the same result as obtained using the utility function U, and from this point
on the derivations of x * and y * are identical.
6.33
The marginal rate of substitution MRS is defined by
MRS (a)
A x y 1
y y
x
x
U / y b / ( y y0 ) and U / x a / (x x 0 )
MRS U / y A
1
x ( 1 ) y U / x A
A
MRS (d)
A x 1 y (c)
U / y A x y 1 and U / x A x 1 y MRS (b)
dy (x)
U / x
dx
U / y
A
1
1
1
a / (x x0 )
b / (y y0 )
( 1 ) / x ( 1 ) y ( 1 ) / a ( y y0 )
b ( x x0 )
(1 ) y 1 and
x (1 ) y x (1 ) y ( 1 ) / x 1
x 1
(1 ) / ( 1 ) y 1
x
1 y
1
1 U / y b and U / x a
MRS a
a
b
b
6-33
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
x
y
1
6.34
(a)
(b)
y y < 0 and
x
x
2 y
2
x
a ( y y0 )
y < 0 and
b ( x x0 )
x
2y x 2
x ( y / x ) y
x2
a
b
( ) ( ) ( ) > 0 so
()
( x x0 )( y/ x ) ( y y0 )
( x x0 ) 2
( ) ( ) ( ) > 0 so
()
6-34
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
(c)
2 y x 2
(d)
y 1
x
( 1)
1 x
y
1
x
y
< 0 if 0 < < 1
2
y a / b < 0 and
x
and
1 x ( y / x)
( ) ( )( ) ( ) ( ) > 0 if 0 < < 1 so
y
y2
2y x 2
0 so
6-35
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.35
The marginal rate of technical substitution MRTS is defined as
MRTS FL A
FK A
A
MRTS 1
1
1
F
dK ( L )
L
FK
dL
L ( 1 ) K
L ( 1 ) K
L ( 1 ) K
A
1
1 L 1
(1 ) K 1
L 1
1
1 L ( 1 ) K
1 ( 1 ) K
1
1 L
K
1
6-36
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.36
(a)
α + β1 ( λp1 ) + β2 ( λp2 ) + β3 ( λI ) = α + λ (β1 p1 + β 2 p2 + β3 I )
Not homogeneous unless α = 0 , in which case it’s homogeneous of degree 1.
(b)
α + β1 ( λp1 ) + β2 ( λp2 ) + β3 ( λI ) = α + λ (β1 p1 + β 2 p2 + β3 I )
Not homogeneous. Even if α = 0 , ln x increases by a factor of λ , so x increases
by a factor of e λ which does not equal λ k for any k.
(c)
α + β1 ln ( λp1 ) + β2 ln ( λp2 ) + β3 ln ( λI )
= α + β1 ln λ + β1 ln p1 + β2 ln λ + β 2 ln p2 + β3 ln λ + β3 ln I
= α + β1 ln p1 + β2 ln p2 + β3 ln I + (β1 + β 2 + β3 ) ln λ
which implies that the new value of x is αp1β1 p2β2 I β3 λβ1 +β2 +β3 while the old value of
x was αp1β1 p2β2 I β3 :
Homogeneous of degree β1 + β2 + β3 .
(d)
α ( λp1 ) 1 ( λp2 )
β
β2
( λI )
β3
= αλβ1 +β2 +β3 p1β1 p2β2 I β3
Homogeneous of degree β1 + β2 + β3 . This is the same answer as part (c) because
the two demand functions are mathematically identical, just written differently.
(e)
 λp 
 λp 
λI 
 α + β ln  1  + γδ  1 
λp1 
 λp2 
 λ p2 

Homogeneous of degree 0.
δ
δ
 I 
 p1 
 p1  
 =  α + β ln   + γδ   
 p1 
 p2 
 p2  


6.37
From equation (6.102),
so
P (1) AL 2 K P AL 1 K 1
dL
P AL 1 K 1
P ( 1) AL K 2
dK
dK dw AL 1 K dP
dw AL 1 K dP
dr AL K 1 dP
P ( 1) AL K 2
J
dr AL K 1 dP P AL 1 K 1
J
Now let dw / w dr / r dP / P. The first term in parentheses in the numerator of the
expression for dK equals w AL 1 K P w P AL 1 K 0 by the first-order
conditions. Similarly, the third term in parentheses in the numerator of the expression for dK
equals r AL K 1 P r P AL K 1 0 by the first-order conditions.
So dK = 0.
6-38
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.38
F ( L, K ) AL K , MP L F L ( L, K ) A L 1 K , and M P K FK (L , K ) A L K 1
F ( L , K ) A( L ) ( K ) AL K F (L , K )
FL ( L , K ) A ( L ) 1 ( K ) A L 1 K 1 1 FL (L , K )
FK ( L, K ) A ( L ) ( K ) 1 A L K 1 1 1 F K ( L, K )
6-39
Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company.
6.39
FOC: D′ − Cθ = 0, or D′ = Cθ : the marginal damage done by pollution needs to equal
the marginal cost of pollution control
SOC: D′′ + Cθθ > 0
Here are reasonable assumptions about the signs of the derivatives:
D′ > 0 : as more pollution is allowed, the total damages due to pollution increase.
Cθ > 0 : as more pollution is controlled, the total costs of control increase
Ct < 0 : if it’s better control technology, then total costs should fall
Cθθ > 0 : as more pollution is controlled, the marginal costs of control increase
Ctt might be > 0, < 0, or = 0 depending on the nature of the technological change
Cθt < 0 : the better technology should reduce the marginal cost of control (this one is
arguable; if you assumed differently then the sign of ∂x * ∂t will be different)
Cθt
∂x *
∂x *
∂x *
+ Cθθ
− Cθt = 0 so
=
<0
D′′ + Cθθ
∂t
∂t
∂t
since Cθt < 0 by assumption and the denominator is positive by the SOC. Note that
only the sign of Cθt needs to be assumed; but the FOC requires that the signs of D′
and Cθ must be the same, and the SOC does put some restraints on Cθθ .
Implicitly differentiating the FOC, D′′
Assuming that Cθt < 0 , better technology lowers the marginal cost of pollution
control, so more control is needed to make marginal damage = marginal cost of
control; thus less pollution results.