Macroeconomics
LECTURE SLIDES SET 4
Professor Antonio Ciccone
Macroeconomics
SET 4
Slide 1
3. Ramsey-Cass-Koopmans
(RCK) Model: Applications
3.1Government expenditures, consumption and interest rate
3.2 Financing government expenditures: bonds vs taxes
Macroeconomics
SET 4
Slide 2
3.1 Government expenditures, consumption and interest rate
- Comparative dynamics in RCK Model
- Permanent unexpected fall in output
- Temporary unexpected fall in output
-Wars, expenditures and interest rate
-The role of expectations
-Permanent anticipated fall in output
-Temporary anticipated fall in output
Macroeconomics
SET 4
Slide 3
RCK Model
c
c-ISOCLINE: CONSUMSUMPTION CONSTANT
k-ISOCLINE: CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 4
c
c-ISOCLINE: CONSTANT CONSUMPTION
k-ISOCLINE: CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 5
Permanent unexpected fall in output for a given k
c
c-ISOCLINE: CONSTANT CONSUMPTION
NEW k-ISOCLINE:
CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 6
Consumption evolution
Permanent unexpectd fall in output
Macroeconomics
SET 4
time
Slide 7
Capital intesity evolution
Permanent unexpected fall in output
Macroeconomics
SET 4
time
Slide 8
--Consumption can JUMP when new information appears
-- But the evolution of consumption must be smooth from
now (following first-order conditions)
 There CAN’T be ANTICIPATED jumps in
consumption
Macroeconomics
SET 4
Slide 9
Temporary unexpected fall in output for a given k: PART I
c
c-ISOCLINE: CONSTANT CONSUMPTION
k-ISOCLINE: CONSTANT CAPITAL
NEW k-ISOCLINE:
CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 10
Temporary unexpected fall in output for a given k: PART II
c
c-ISOCLINE: CONSTANT CONSUMPTION
k-ISOCLINE: CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 11
Temporay unexpected fall in output: Equilibrium response
c
c-ISOCLINE: CONSTANT CONSUMPTION
NEW k-ISOCLINE:
CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 12
Temporary unexpected fall in output: Equilibrium response
c
c-ISOCLINE: CONSTANT CONSUMPTION
k-ISOCLINE: CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 13
Capital intensity evolution
Beginning of
temporary fall
Macroeconomics
End of
temporary fall
SET 4
Time
Slide 14
Real interest rate evolution
Beginning of
temporary fall
Macroeconomics
End of
temporary fall
SET 4
Time
Slide 15
Consumption evolution
Beginning of
temporary fall
Macroeconomics
End of
temporary fall
SET 4
Time
Slide 16
Wars and real interest rate
-- Supose that the Government expenditures caused by a
war are an unexpected and temporary event.
We’ll study capital stock, interest rate and consumption
dynamic responses to the war.
-- Public expenditures associated with wars reduce the
amount of output avalaible for consumption and
investment.
F ( K , L)  G  C  I
 INCREASE IN G Same effect as an output fall
Macroeconomics
SET 4
Slide 17
Real interest rate evolution
Beginning of war
Macroeconomics
End of war
SET 4
time
Slide 18
Militar expenditures and long run interest rate at The
United Kingdom (Barro, 1987)
Macroeconomics
SET 4
Slide 19
-The role of expectations
- Permanent
anticipated fall in output
- Temporary
anticipated fall in output
Macroeconomics
SET 4
Slide 20
Permanent anticipad fall in output: PART I
c
c-ISOCLINE: CONSTANT CONSUMPTION
k-ISOCLINE: CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 21
Permanent anticipated fall in output: PART II
c
c-ISOCLINE: CONSTANT CONSUMPTION
NEW k-ISOCLINE:
CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 22
Permanent anticipated fall in output: Equilibrium response
c
c-ISOCLINE: CONSTANT CONSUMPTION
NEW k-ISOCLINE:
CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 23
Capital intensity evolution
Permanent fall takes place
here
time
INFO about the FUTURE
permanent fall in output
Macroeconomics
SET 4
Slide 24
Consumption evolution
Permanent fall takes place
here
time
INFO about the FUTURE
permanent fall in output
Macroeconomics
SET 4
Slide 25
-The role of expectations
- Permanent
anticipated fall in output
- Temporary
anticipated fall in output
Macroeconomics
SET 4
Slide 26
Temporary anticipated fall in output for a given k: PART I
c
c-ISOCLINE: CONSTANT CONSUMPTION
k-ISOCLINE: CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 27
Temporary anticipated fall in output for a given k: PART II
c
c-ISOCLINE: CONSTANT CONSUMPTION
NEW k-ISOCLINE:
CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 28
Temporary anticipated fall in output for a given k: PART III
c
c-ISOCLINE: CONSTANT CONSUMPTION
k-ISOCLINE: CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 29
Temporary anticipated fall in output: Equilibrium response
c
c-ISOCLINE: CONSTANT CONSUMPTION
k-ISOCLINE: CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 30
Temporary anticipated fall in output: Equilibrium response
c
c-ISOCLINE: CONSTANT CONSUMPTION
k-ISOCLINE:
CONSTANT CAPITAL
0
k
k*
Macroeconomics
SET 4
Slide 31
Capital intensity evolution
INFO about the FUTURE START of temporary
temporary fall in output fall in output
Macroeconomics
SET 4
END of temporary
fall in output
time
Slide 32
Consumption evolution
INFO about the FUTURE START of temporary
temporary fall in output fall in output
Macroeconomics
SET 4
END of temporary
fall in output
time
Slide 33
3. Ramsey-Cass-Koopmans
(RCK) Model: Applications
3.1 Government expenditures, consumption and interest rate
3.2 Financing Government expenditures: bonds vs taxes
Macroeconomics
SET 4
Slide 34
Government expenditures and taxes
Deficit
GOV
t
 Gt  Tt
Government intertemporal budget constraint


0
VP0tTt dt Wealth
GOV
t
Macroeconomics
SET 4

  VP0t Gt dt
0
Slide 35
--Suppose that families believe in Governments intertemporal
budget constraint
-- Government reduces taxes at moment t
-- But there are no evidence that government will reduce also
its expenditures
-- What happens to DISCOUNTED TAXES FLOW?


0
Macroeconomics
VP0tTt dt
SET 4
Slide 36
NOTHING, because:


0

VP0tTt dt  VP0t Gt dt  Wealth
GOV
t
0
And nothing has changed in right-side of the equation
 GOVERNMENT WILL COMPENSATE THE REDUCTION
IN CURRENT TAXES WITH AN INCREASE IN FUTURE
TAXES
Macroeconomics
SET 4
Slide 37
Lets look at families constraint:


0


0
0
VP0t Ct dt   VP0tTt dt  VP0t wt Ldt  Q0
-- The reduction in current taxes DOESN’T affect this
constraint at all. Only matters the PRESENT DISCOUNTED
VALUE OF TAXES
-- and the present value of taxes holds constant if
government expenditures don’t change
Macroeconomics
SET 4
Slide 38
-- TAX REDUCTIONS DOESN’T CHANGE FAMILIES CONSUMPTION
-- AS A RESULT, NATIONAL SAVING RATE DOESN’T CHANGE
St  Yt  Ct  Gt
-- LLAVORS, NO AFECTA:
-- SO, IT DOESN’T AFFECT:
- INVESTMENT (!)
- INTERES RATE (!)
-- FAMILIES INCREASE THEIR SAVINGS, BUT IT IS CANCELED
WITH THE INCREASE IN GONVERNMENT DEBT:
St  (Yt  Tt  Ct )  (Tt  Gt )
Macroeconomics
SET 4
Slide 39
Hence, if government reduces taxes
 It has to issue debt (Bonds)
 El govern ha d’assegurar-se que el tipus d’interès real dels
títols replica el tipus d’interès de mercat (abans d’emetre
nous títols)
 The Government has to make sure that bond’s real interest
rate replies market interest rate (before issuing new bonds)
 Families buy those bonds with their savings.
So:
 Families buy Government bonds using what they “saved” by
the reduction in taxes
Macroeconomics
SET 4
Slide 40
3. Diamond model
1. Overlapping generations model
2. Model specification
1. Technology
2. Families behavior
3. Dynamic equilibrium system
3. Equilibrium growth and optimality
4. Diamond model applications
1. Government expenditures and interest rate
2. Financing government expenditures: bonds vs taxes
Macroeconomics
SET 4
Slide 41
1. Overlapping generations model
-Model in discret time.
- Families live for two periods, and only work in the first
one.
Macroeconomics
SET 4
Slide 42
GENERATION’S LYFE CICLE
Time 1
Time 2
Time 3
Time 4
Generation 1
ACTIVE (or
YOUNG):
Work and
consume
RETIRED (or OLD): Only
consume
Macroeconomics
SET 4
Slide 43
TEMPORAL ORGANIZATION
Time 1
Time 2
Time 3
Time 4
Generation 1
Generation 2
Generation 3
Macroeconomics
SET 4
Slide 44
2.
Diamond model specification
1.
2.
3.
Technology
Families behavior
Dynamic equilibrium system
Macroeconomics
SET 4
Slide 45
1. Technology

1
Yt  K t ( At Lt )
Given by retired
Given by actives
Capital completly depreciates with production: =1
Macroeconomics
SET 4
Slide 46
2.
Families behavior
Generation t
Productions
uses
Generation’s t
work
Production uses
Generation’s t
capital
t+1
t
time
Born
-Earns salary
-Consumes
Macroeconomics
-Earns interest
-Consumes
SET 4
Slide 47
GENERATION t UTILITY
MAXIMIZATION WITH DISCOUNT TAXES AND INTEREST
max U (ct )  (1   )U (ct 1 )
Respect C
Subject to: INTERTEMPORAL BUDGET CONSTRAINT
ct 1
ct 
 wt
1  rt
Macroeconomics
SET 4
Slide 48
First-order conditions for generation t
U ' (ct )  (1   )(1  r )U ' (ct 1 )
“EFFECTIVE TEMPORAL DISCOUNT”
Macroeconomics
SET 4
Slide 49
Suppose the following utility function CES (Constant
Elasticity of Substitution):
C[t ]11/ 
U (C[t ]) 
1 1/ 
with   0
1/ 
U ' (C[t ])  C[t ]
Temporal consumption path will be:
ct 1
 [(1   )(1  rt )]
ct
Macroeconomics
SET 4
Slide 50
If we introduce the consumption path into the budget cosntraint we
obtain:
c t 1
ct 
 wt
1  rt
ct [(1   )(1  rt )]
ct 
 wt
1  rt
ct (1  (1   ) (1  rt ) 1 )  wt
wt
ct 

 1
1  (1   ) (1  rt )
Macroeconomics
SET 4
Slide 51
Consumtion and saving of Generation t (when young)
wt
ct 
1  (1   ) (1  rt ) 1
wt
st  wt  ct  wt 
1  (1   ) (1  rt ) 1
(1   ) (1  rt ) 1
st  wt
1  (1   ) (1  rt ) 1
Macroeconomics
SET 4
Slide 52
r
0
(1   )
s t  wt
1  (1   )
Macroeconomics
SET 4
s
Slide 53
r
0
(1   )
s t  wt
1  (1   )
Macroeconomics
st  wt
SET 4
s
Slide 54
r
0
(1   )
s t  wt
1  (1   )
Macroeconomics
SET 4
s
Slide 55
3.
Dynamic equilibrium system
K t 1  I t  Lt st

 1
(1   ) (1  rt )
 Lt wt

 1
1  (1   ) (1  rt )
Macroeconomics
SET 4
Slide 56

1
Yt  K t ( At Lt )
Given by ACTIVES
Financed with RETIRED
SAVINGS

wt  PMLt  (1   ) K t At
 1
rt  PMK t  1  K t
Macroeconomics
SET 4
1
Lt

1
( At Lt )
1
Slide 57
K t 1  I t  Lt st

 1
(
1


)
(
1

r
)
t
 (1   ) K t ( At Lt )1
1  (1   ) (1  rt ) 1
 1
rt  K t
Macroeconomics
1
( At Lt )
SET 4
1
Slide 58
K t 1

Lt 1 At 1
Lt At K t ( At Lt )1 (1   ) (1  rt ) 1

Lt 1 At 1
Lt At
1  (1   ) (1  rt ) 1
 1
rt  K t
Macroeconomics
1
( At Lt )
SET 4
1
Slide 59
~
kt 1 
~  (1   ) (1  rt ) 1
1
kt
(1  n)(1  a) 1  (1   ) (1  rt ) 1
~ 1
rt  kt  1
Macroeconomics
SET 4
Slide 60
3. Equilibrium growth and
optimality
 1
~
kt 1 

~ (1   )
1
kt

(1  n)(1  a) 1  (1   )
Macroeconomics
SET 4
Slide 61

kt 1  bkt
kt 1
0
k0
Macroeconomics
k1
BGP
SET 4
kt
Slide 62
Optimality
• It’s not clear how different generations should be
weighted.
• The resulting assignation is at least Optimal in
terms of Pareto?
Macroeconomics
SET 4
Slide 63
Dynamic inefficiency
• A situation where resources assignation is not
Pareto effcient.
• In other words, a situation where we can
increase the consumption of at least one
generation without reducing other’s
consumption.
Macroeconomics
SET 4
Slide 64
Consider the case WITHOUT technological progress
a=0
How much is need to invest by person at moment t
to hold the capital intensity of use?
K t 1 Lt ii
it
kt 1 


Lt 1 Lt 1 1  n
kt 1  kt  it  kt (1  n)
Macroeconomics
SET 4
Slide 65
kt (1  n)
yt  f (kt )
Per capita consumption
0
k
kt
gold
Macroeconomics
SET 4
Slide 66
PMK 1  n
Per capita consumption
kt
0
k
gold
Macroeconomics
SET 4
Slide 67

1
(
1


)
kt 1 
kt
(1  n) 1  (1   )
Steady state or BGP

k
k
BGP
BGP
1  (1   )

k
1  n 1  (1   )

 1
(1   )
 

(
1

n
)
1

(
1


)

Macroeconomics
SET 4



1
1
Slide 68
PMK  (1   )k
 1

PMK
BGP
 1
(1   )
 (1   )

 (1  n) 1  (1   )
Macroeconomics
SET 4



1
Slide 69