ITFD Growth and Development
UPF 2008-2009
LECTURE SLIDES SET 3
Professor Antonio Ciccone
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 1
II. ECONOMIC GROWTH
WITH ENDOGENOUS
SAVINGS
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Slide 2
1. Household savings behavior
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Slide 3
1. “Keynesian theory” of savings and
consumption
1. The Keynesian consumption (savings) function
• So far we assumed a “Keynesian” savings
function
• where s is the marginal propensity to save.
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Slide 4
Because of the BUDGET CONSTRAINT
this implies the “Keynesian” consumption
function
where c is the marginal propensity to
consume.
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Slide 5
2. Limitations
CONCEPTUAL
The consumption behavior is assumed to be “mechanic” and “shortsighted”:
– Are households really only looking at CURRENT income when
deciding consumption?
Not really. Many households borrow from banks in order to be able
to consume more today because they know they will be able to pay
the money back in the future.
– If people save, presumably they are doing this for future
consumption. Hence, savings is a FORWARD-LOOKING decision
and must take into account what happens in the future.
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Slide 6
Assuming savings as a function of current income
therefore appears to contradict the use that households
make of their savings.
EMPIRICAL
“Consumption smoothing:”
– Empirically, we observe that households smooth
consumption. To put it differently, the income of
households is often more volatile than their
consumption.
This suggests that households look forward and try to
stabilize consumption (their standard of living) as much as
they can.
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Slide 7
FIGURE 1: CONSUMPTION SMOOTHING: A
VOLATILE INCOME PATH
HOUSEHOLD INCOME OF FARMER
time
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Slide 8
FIGURE 2: INCOME AND "KEYNESIAN CONSUMPTION"
HOUSEHOLD INCOME OF FARMER
HOUSEHOLD CONSUMPTION OF FARMER (“KEYNESIAN” theory)
time
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Slide 9
FIGURE 3: CONSUMPTION SMOOTHING
HOUSEHOLD INCOME OF FARMER
HOUSEHOLD CONSUMPTION OF FARMER
(EMPIRICAL OBSERVATION)
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time
Slide 10
FIGURE 4: SAVINGS AND DIS-SAVINGS IN
CONSUMPTION SMOOTHING MODELS
HOUSEHOLD INCOME
CONSUMPTION SMOOTHING
DIS-SAVE TO MAINTAIN
CONSUMPTION LEVELS
SAVE FOR
“RAINY DAYS”
time
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Slide 11
INTERESTINGLY:
The Keynesian theory of consumption seems to do better
at the aggregate level than at the level of individual
households. For example:
– Keynesian theory does well in describing relationship between
consumption and income of a country at different in different years
– Theory does also well in describing relationship between
consumption and income across different countries
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Slide 12
A PUZZLE?
CONSUMPTION
AGGREGATE LEVEL
Germany 1980
Or Country 3
INDIVIDUAL HOUSEHOLD LEVEL
Ms B
Mr C
Mr A
Ms D
Germany 1960
Or Country 2
Germany 1950
Or Country 1
INCOME
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Slide 13
2. The permanent income theory
of consumption and savings
1. Basic idea and two-period model
Households make consumption decisions:
• LOOKING FORWARD to future
• USING SAVINGS AND LOANS from BANKS to
maintain their living standards STABLE in time to
the extent possible
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Slide 14
SIMPLEST POSSIBLE formal model (2 PERIODS)
INGREDIENTS:
– Household lives 2 periods and tries to maximize
INTERTEMPORAL utility
U (C[0])  (1   )U (C[1])
– Understands that will earn LABOR income Lw[0] in
period 0 and Lw[1] in period 1
– Starts with 0 WEALTH
– Can save and borrow from bank at interest rate r
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MATHEMATICAL MAXIMIZATION PROBLEM:
max U (C0 )  (1   )U (C1)
by choosing C0 and C1
subject to
S=Lw0-C0
C1=Lw1+(1+r)S
DISCOUNT APPLIED TO FUTURE UTILITY
NOTE that S can be NEGATIVE (which means the
household is BORROWING or DISSAVING)
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Slide 16
MATHEMATICAL FORMULATION
Maximize INTERTEMPORAL UTILITY
max U (C0 )  (1   )U (C1)
by choosing C
subject to INTERTEMPORAL BUDGET
CONSTRAINT
C1=Lw1+(1+r)S= Lw1+(1+r)(Lw0-C0)
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INTERTEMPORAL BUDGET CONSTRAINT
can also be written:
C1
Lw1
C0 
 Lw0 
1 r
1 r
IMPORTANT TERMINOLOGY:
Lw1
Lw0 
1 r
PERMANENT INCOME (PI)
PRICE OF FUTURE CONSUMPTION
RELATIVE TO CURRENT CONSUMPTION
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Slide 18
GRAPHICALLY: INCOME LEVELS AND CONSUMTION
C[1]
Lw[1]
Lw[0]
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C[0]
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Slide 19
C[1]
THE INTERTEMPORAL BUDGET CONSTRAINT
Lw[1]
1+r
Lw[0]
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C[0]
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Slide 20
INTERTEMPORAL UTILITY MAXIMIZATION
C[1]
Lw[1]
1+r
Lw[0]
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C[0]
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Slide 21
C[1]
Lw[1]
C[1]
1+r
Lw[0]
C[0]
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C[0]
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Slide 22
BORROWING FOR CURRENT CONSUMPTION
C[1]
Lw[1]
REPAY
C[1]
BORROW
1+r
Lw[0]
C[0]
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C[0]
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Slide 23
2. Closed form solution in a simple case
SUPPOSE THAT
INTEREST RATE is ZERO: r = 0
FUTURE UTILITY DISCOUNT is ZERO:
MAXIMIZATION PROBLEM BECOMES:
max U (C0 )  U (C1 )
with respect to C
subject to
C0  C1  Lw0  Lw1  PI
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FIRST ORDER MAXIMIZATION CONDITIONS:
C1
First-order conditions can be obtained from
max U (C0 )  U ( PI  C0 )
with respect to C0
where we have substituted the budget constraint.
TAKE DERIVATIVE WITH RESPECT TO C[1] AND SET EQUAL
ZERO:
U (C0 ) U ( PI  C0 )

(1)  0
C0
C1
U (C0 ) U (C1 )

C0
C1
OR
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U '(C0 )  U '(C1 )
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Slide 25
EQUALIZE MARGINAL UTILITY AT DIFFERENT POINTS IN
TIME
THIS IMPLIES 
C0  C1
“PERFECT CONSUMPTION SMOOTHING”
Using the INTERTEMPORAL BUDGET CONSTRAINT yields
consumption as a function of PERMANENT INCOME
Y0  Y1
C0  C1 
 PI / 2
2
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Slide 26
"CONSUMPTION FUNCTION"
C[0]
0.5*Lw[0]+0.5*Lw[1]
0.5*Lw[1]
Lw[0]
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Slide 27
THE EFFECT OF AN INCREASE IN INITIAL-PERIOD INCOME ON
C[0]
C[0]
“TEMPORARY” INCREASE IN INCOME
0.5*Lw[0]+0.5*Lw[1]
0.5*Lw[1]
INCREASE
In first-period income
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Lw[0]
Slide 28
THE EFFECT OF AN INCREASE IN INITIAL AND FUTURE INCOME
“PERMANENT” INCREASE IN INCOME
INCREASE Lw[1]
C[0]
0.5*Lw[0]+0.5*Lw[1]
INCREASE Lw[0]
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Lw[0]
Slide 29
DISCOUNTING OF FUTURE UTILITY, AND INTEREST
MAXIMIZATION WITH DISCOUNTING&INTEREST
max U (C0 )  (1   )U (C1)
with respect to C
subject to INTERTEMPORAL BUDGET CONSTRAINT
C1
Lw1
C0 
 Lw0 
1 r
1 r
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Slide 30
FIRST-ORDER CONDITIONS
U '(C0 )  (1   )(1  r )U '(C1)
“EFFECTIVE TIME DISCOUNTING”
 CONSTANT CONSUMPTION
DISCOUNTING OF FUTURE UTILITY AND POSTITIVE
INTEREST RATE JUST OFFSET
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Slide 31
UPWARD SLOPING CONSUMPTION PATHS IN TIME:
 INCREASING CONSUMPTION
OVER TIME
POSITIVE INTEREST MORE THAN OFFSETS UTILITY
DISCOUNTING
DOWNWARD SLOPING CONSUMPTION PATHS IN TIME:
 DECREASING CONSUMPTION
OVER TIME
UTILITY DISCOUNTING MORE THAN OFFSETS
POSITIVE INTEREST
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Slide 32
AN EXAMPLE
Take the following utility function:
with
FIRST-ORDER CONDITION BECOMES
1/ 
C0
1/ 
 (1   )(1  r )C1
or
C1

  (1   )(1  r ) 
C0
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Slide 33
3. The case of 3 and more periods
-- Timing
-- Intertemporal budget constraint
-- Optimality conditions
-- Time consistency
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Slide 34
PRESENT-VALUE INCOME AND CONSUMPTION
YOU ARE HERE
C[0]
C[1]
t=0
t=1
interest
discounting
interest
discounting
Q[0]
w[0]L
- PERMANENT
INCOME
- PRESENT VALUE
CONSUMPTION
C[2]
t=2
interest
discounting
w[1]L
w[2]L
Lw0
Lw1
Lw2
Q0 


1  r0 (1  r0 )(1  r1 ) (1  r0 )(1  r1 )(1  r2 )
C0
C1
C2


1  r0 (1  r0 )(1  r1 ) (1  r0 )(1  r1 )(1  r2 )
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Slide 35
INTERTEMPORAL BUDGET CONSTRAINT
Lw0
Lw1
Lw2
Q0 


1  r0 (1  r0 )(1  r1 ) (1  r0 )(1  r1 )(1  r2 )

C0
C1
C2


1  r0 (1  r0 )(1  r1 ) (1  r0 )(1  r1 )(1  r2 )
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Slide 36
BUDGET CONTRAINT AND TIME EVOLUTION OF WEALTH
t=0
Q[0]
t=1
t=2
C[0]
C[1]
C[2]
w[0]L
w[1]L
w[2]L
Q1  (1  r0 )Q0   Lw0  C0 
Q2  (1  r1 )Q1   Lw1  C1 
Q3  (1  r2 )Q2   Lw2  C2 
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Slide 37
INTERTEMPORAL BUDGET CONSTRAINT
Q0 GIVEN
Qt  (1  rt 1)Qt 1   Lwt 1  Ct 1 
EndOfPeriod
QT
0
IF T FINAL PERIOD
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Slide 38
OPTIMAL SOLUTION OF CONSUMPTION PROBLEM
MAXIMIZE BETWEEN ADJACENT PERIODS
U '(Ct )  (1   )(1  rt 1 )U '(Ct 1 )
plus BUDGET CONSTRAINT WITH EQUALITY
EoP
QT
(1  r0 )(1  r1 )(1  r2 )
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0
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Slide 39
Shortest way from A to B?
B
A
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Shortest way from A to B
B
A
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Must be the shortest way
between ANY two points
B
C
D
A
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Must be the shortest way
between ANY two points
B
C
D
A
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INFINITE HORIZON
1
PV0t 
(1  r0 ) *(1  r1 ) *...*(1  rt )
=TIME ZERO (PRESENT) VALUE OF 1 EURO PAID AT
(end of) PERIOD t
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Slide 44
INTERTEMPORAL BUDGET CONSTRAINT
Q0 GIVEN
Qt  (1  rt 1)Qt 1   Lwt 1  Ct 1 
lim
T 
EoP
PV0T QT
0
NO-PONZI-GAME condition
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Slide 45
WHAT IF:
NO PONZI GAME CONDITION VIOLATED?
EoP
PV0T QT
0
e
TIME T
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Slide 46
INTERTEMPORAL BUDGET CONSTRAINT
WITH EQUALITY
Q0 GIVEN
Qt  (1  rt 1)Qt 1   Lwt 1  Ct 1 
lim
T 
EoP
PV0T QT
= 0
NO-PONZI-GAME condition
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Slide 47
WHAT IF:
lim
T 
EoP
PV0T QT
b0
EoP
PV0T QT
e
0
TIME T
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Slide 48
CAN INCREASE TIME-0 CONSUMPTION
 CONSUMPTION PLAN NOT OPTIMAL!
NECESSARY FOR OPTIMALITY:
lim
T 
EoP
PV0T QT
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0
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Slide 49
TIME CONSISTENCY of
HOUSOLD CONSUMPTION PLANS
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Slide 50
TIME 0 CONSUMPTION PLANS
C[0]
YOU ARE HERE
C[1]
t=0
t=1
interest
discounting
interest
discounting
Q[0]
w[0]L
C[2]
t=2
interest
discounting
w[1]L
w[2]L
TIME 1 CONSUMPTION PLANS (NO NEW INFO)
YOU ARE HERE
t=0
Q[0]
C[1]
C[2]
t=1
Q(1)
interest
discounting
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t=2
w[1]L
interest
discounting
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w[2]L
Slide 51
***** TIME CONSISTENCY *****
C[0]
YOU ARE HERE
C[1]
t=0
t=1
interest
discounting
interest
discounting
Q[0]
w[0]L
C[2]
t=2
interest
discounting
w[1]L
w[2]L
TIME 1 CONSUMPTION PLANS (NO NEW INFO)
YOU ARE HERE
t=0
C[1]
C[2]
t=1
Q(1)
interest
discounting
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t=2
w[1]L
interest
discounting
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w[2]L
Slide 52
3. Optimal consumption and savings in
continuous time
1. Infinite horizon
  t
e
0
max 
U (Ct )dt


0
0
subject to
 PV0t Ct dt  Q0   PV0t ( Lwt )dt
= TIME ZERO (PRESENT) VALUE
OF 1 EURO PAID AT TIME t
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Slide 53
2. Intertemporal budget constraint
Qt  (1  rt 1)Qt 1   Lwt 1  Ct 1 
Wealth in
discrete time
Qt  (1  rt 1)Qt 1   Lwt 1  Ct 1 
Qt  Qt 1  rt 1Qt 1   Lwt 1  Ct 1 
Wealth in
continuous time
Qt  rt Qt   Lwt  Ct 
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Slide 54
Intertemporal budget constraint in continuous time
satisfied with equality if
Q0 given
Qt  rt Qt   Lwt  Ct 
lim PV0t Qt =0
t 
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Slide 55
3. Interpretation of  and r
r is the interest rate that is received between two very close
periods in time
is the discount rate applied PER UNIT OF TIME between
two very close periods in time
TO SEE THAT is the discount rate applied PER UNIT OF
TIME between two very close periods in time
1) Note that the utility discount between period 0 and t is:
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Slide 56
2) Hence the utility discount per unit of time is:
3) What is the limit as t0?
Hopital’s rule yields
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Slide 57
4. First-order condition
where:
is INTERTEMPORAL RATE OF TIME
PREFERENCE and measures how IMPATIENT people are
is the INTERTEMPORAL ELASTICITY OF
SUBSTITUTION and measures how much future
consumption increases when the interest rate goes up (how
much people “respond to interest rates”)
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CONSTANT CONSUMPTION IN TIME
OPTIMAL CONSUMPTION PATH r = 
C(t)
C(0)
TIME
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Slide 59
INCREASING CONSUMPTION IN TIME
OPTIMAL CONSUMPTION PATH r > 
C(t)
C(0)
TIME
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Slide 60
DEACREASING CONSUMPTION IN TIME
OPTIMAL CONSUMPTION PATH r < 
C(0)
C(t)
TIME
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Slide 61
5. Closed form solution in special case
ASSUME
(consumers have an INFINITE HORIZON)
SOLUTION CHARACERIZED BY
 PEOPLE WANT CONSTANT CONSUMPTION OVER
TIME (“PERFECT CONSUMPTION SMOOTHING”
CASE)
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Slide 62
THE INTERTEMPORAL BUDGET CONSTRAINT
without initial wealth

  e rt Lw[t ]dt  PERMANENT INCOME
0
HENCE
C[t ]
 PERMANENT INCOME
r
C[t ]  r *PERMANENT INCOME
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Slide 63
6. Deriving the continuous time first-order condition
• MAXIMIZATION BETWEEN ANY TWO
PERIODS SEPARATED BY TIME x
• subject to
= TOTAL SPENDING IN TWO PERIODS
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Slide 64
Take the following utility function:
with
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Slide 65
FIRST ORDER CONDITIONS FOR THE
TWO PERIODS IN TIME
making use of the utility function
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Slide 66
REWRITING THIS CONDITIONS YIELDS
subtracting 1 from both sides
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Slide 67
DIVIDE BY x (the TIME BETWEEN THE TWO
PERIODS) to get CONSUMPTION GROWTH
PER UNIT OF TIME
What happens when the two periods get closer and
closer (x0)?
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Slide 68
• Apply Hopital’s rule
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Slide 69
HENCE as two periods become VERY
CLOSE
WHICH IS WHAT WE WANTED TO SHOW
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Slide 70
SUMMARIZING
QUESTION: What characterizes the optimal
consumption PATH that solves
  t
e
0
max 
  t
e
0
U (Ct )dt  
Ct11/ 
dt
1  1/ 
subject to


0
0
 PV0t Ct dt  Q0   PV0t Lwt dt
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Slide 71
ANSWER:
Ct ˆ
 Ct   (rt   )
Ct
and
or


0
0
 PV0t Ct dt  Q0   PV0t Lwt dt
Qt  rt Qt  (1  rt )  Lwt  Ct 
lim PV0t Qt =0
t 
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Slide 72
2. The Ramsey-Cass-Koopmans
model
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Slide 73
1. Equilibrium growth with infinite-horizon households
We will now integrate a household that chooses consumption optimally over an
infinite horizon in the Solow model. The results is often refereed to as the CassKoopmans model.
The Cass-Koopmans model is exactly like the SOLOW MODEL only that the
household does NOT behave mechanically but instead chooses consumption
and savings to maximize:
  t
e
0
max 
subject to


0
0
U (C[t ])dt
 PV0t C[t ]dt   PV0t w[t ]Ldt  Q[0]
where
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Slide 74
In order to NOT complicate things too much
we will simplify the model by assuming:
1. no technological changes (i.e. a=0 in Solow
model)
2. no population growth (i.e. n=0 in Solow model)
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Slide 75
1. Technology and the capital market
WHAT WE CAN KEEP FROM THE SOLOW MODEL
PRODUCTION FUNCTION
CONSTANT RETURNS PRODUCTION FUNCTION
E(1)
E(2)
CAPITAL ACCUMULATION EQUATION
E(3)
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Slide 76
CAPITAL MARKET EQUILIBRIUM
E(4)
E(5)
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Slide 77
2. Household behaviour
WHAT WE CANNOT KEEP IS
INSTEAD:
E(6)
E(7)
INTERTEMPORAL BUDGET
CONSTRAINT WITH EQUALITY
where c[t] is CONSUMPTION per PERSON
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Slide 78
3. Dynamic equilibrium system
WE WILL TRY TO CHARACTERIZE THE EQUILIBRIUM
OF THIS ECONOMY IN TERMS OF THE EVOLUTION
OF c and k.
The goal is to reduce the equations above to a TWODIMENSIONAL DIFFERENTIAL EQUATION SYSTEM
WHERE
CHANGE in CONSUMPTION c=FUNCTION OF k and c
CHANGE IN CAPITAL k=FUNCTION OF k and c
(E6) and (E5) imply
E(8)
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Slide 79
(E3) and (E4) imply
recall that there is NO population growth
and therefore
E(9)
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 80
SO WE HAVE OUR TWO EQUATIONS:
and
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 81
2. Equilibrium growth and optimality
THESE CAN BE BEST ANALYZED IN A PHASE DIAGRAM
Start with capital accumulation equation
FIRST: Find ISOCLINE, which are the (c, k) combinations such
that
INTERPRETATION: capital per worker does NOT grow IF the
economy consumes all of the output net of capital depreciation.
In this case, investment is just enough to cover the depreciation
of capital.
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 82
k-ISOCLINE
c
k-ISOCLINE: CAPITAL DOES NOT GROW
k
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 83
c
CHANGES IN k in PHASE DIAGRAM
k-ISOCLINE: CAPITAL DOES NOT GROW
k
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 84
Continue with the optimal consumption equation
FIRST: Find ISOCLINE, which are the (c, k)
combinations such that
or
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 85
c-ISOCLINE
c-ISOCLINE: CONSUMPTION
DOES NOT GROW
c
0
k*
is the k such that f’(k)=d
BGSE Growth and Development, 2008-09
SLIDE SET 3
k
Slide 86
CHANGES IN c in PHASE DIAGRAM
c-ISOCLINE: CONSUMPTION
DOES NOT GROW
c
0
k*
is the k such that f’(k)=d
BGSE Growth and Development, 2008-09
SLIDE SET 3
k
Slide 87
CHANGES IN c in PHASE DIAGRAM
c
c-ISOCLINE: CONSUMPTION
DOES NOT GROW
0
k*
is the k such that f’(k)=d
BGSE Growth and Development, 2008-09
SLIDE SET 3
k
Slide 88
PUTTING CHANGES in k and c TOGETHER
c
c-ISOCLINE: NO CONSUMPTION GROWTH
k-ISOCLINE: CAPITAL DOES NOT
GROW
0
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 89
c
c-ISOCLINE: NO CONSUMPTION GROWTH
k-ISOCLINE: NO CAPITAL
GROWTH
0
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 90
c
c-ISOCLINE: NO CONSUMPTION GROWTH
k-ISOCLINE: NO CAPITAL
GROWTH
0
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 91
All these paths satisfy by construction:
-period-by-period consumer maximization
-capital market equilibrium
They DO NOT necessarily satisfy constraints like:
-non-negative capital stock k[t]>=0
-intertemporal budget constraint with EQUALITY
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 92
PATHS that violate NON-NEGATIVE capital stock (consume too
much in beginning)
c-ISOCLINE: NO CONSUMPTION GROWTH
c
k-ISOCLINE: NO CAPITAL
GROWTH
0
k(0)
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 93
PATHS THAT DO NOT SATISFY BUDGET CONSTRAINT WITH
EQUALITY (consume too little in beginning)
c-ISOCLINE: NO CONSUMPTION GROWTH
c
k-ISOCLINE: NO CAPITAL
GROWTH
k_bar
0
k(0)
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 94
(1) Wealth=Capital
Q(t)=K(t) or q(t)=k(t)
(2) Intertemporal budget constraint with equality
lim PV0t qt = lim PV0t kt =0
t 
t 
t
PV0t =e
BGSE Growth and Development, 2008-09
  r d
0
SLIDE SET 3
Slide 95
PATHS THAT DO NOT SATISFY BUDGET CONSTRAINT WITH
EQUALITY
c-ISOCLINE: NO CONSUMPTION GROWTH
c
f’(k)-d=r=0
f(k)-dk
k_bar
k(0)
k*
POSITIVE INTEREST NEGATIVE INTEREST RATE
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 96
k
lim PV0t qt = lim PV0t  (k_bar )
t 
t 
t
PV0t =e
  r d
0
NEGATIVE INTEREST RATE
BGSE Growth and Development, 2008-09
SLIDE SET 3
time t
Slide 97
PATHS THAT DO NO SATISFY BUDGET CONSTRAINT WITH
EQUALITY
c-ISOCLINE: NO CONSUMPTION GROWTH
c
YOU ARE NOT SPENDING
ALL YOUR PERMANENT
INCOME!!!!!!!
k_bar
0
k(0)
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 98
EQUILIBRIUM (“SADDLE”) PATH
c
c-ISOCLINE: NO CONSUMPTION GROWTH
k-ISOCLINE: NO CAPITAL
GROWTH
0
k(0)
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 99
SADDLE PATH SATISFIES INTERTEMPORAL
BUDGET CONSTRAINT WITH EQUALITY
Capital market equilibrium:
kt  f (kt )  ct  d kt
Income per worker=Labor income + Capital income:
kt   (rt  d )kt  Lwt   ct  d kt
kt  rt kt  Lwt  ct
Hence:
kt  qt  qt  rt qt  Lwt  ct
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 100
Moreover:
lim PV0t qt = lim PV0t kt = lim PV0t k * =0
t 
t 
As:
t 
t
lim PV0t = lim e
t 
t 
  r d
0
0
given that interest rates>0 for k<=k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 101
OPTIMALITY
-- What would social planner do?
- Social planner: dictator who decides
allocation according to HH welfare
subject to physical contraints
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 102
The GLOBALLY OPTIMAL PATH MUST SATISFY
MRS=MRT
(A)
If not satisfied, the planner could increase utility between
adjacent periods by either:
-- consuming one unit less today, investing that unit, and
consuming the resulting additional output tomorrow
-- consuming one unit more today, invest one unit less today,
and reducing future consumption accordingly
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 103
The GLOBALLY OPTIMAL PATH MUST SATISFY
RESOURCE CONSTRAINT (B)
k[t ]  f (k[t ])  c[t ]  d k[t ]
To see why, suppose first that k[t ]  f (k[t ])  c[t ]  d k[t ]
-- in this case the planner must be throwing away goods
(investment goods) because the increase in the number of
machines k[t ] is LESS THAN the machines built less
depreciation f (k[t ])  c[t ]  d k[t ] : BUT THROWING AWAY
GOODS CANNO BE OPTIMAL!!
Now suppose instead k[t ]  f (k[t ])  c[t ]  d k[t ]
-- now the planner is a REAL MAGICIAN!! as the number of
machines in the economy goes up by k[t ] which is GREATER
THAN machines built less depreciation f (k[t ])  c[t ]  d k[t ]
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 104
ALL THE PATHS THAT SATISFY CONDITIONS (A) and (B)
c
c-ISOCLINE: NO CONSUMPTION GROWTH
k-ISOCLINE: NO CAPITAL
GROWTH
0
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 105
NOW NOTE:
-- Starting the allocation by jumping ABOVE the SADDLE
PATH CANNOT BE OPTIMAL because you end up violating
the non-negativity constraint for capital
-- Starting the allocation by jumping BELOW the SADDLE
PATH CANNOT BE OPTIMAL either. The proof is to
construct another path—that is clearly not optimal either—
but that still is BETTER THAN the paths starting out below
the saddle path. How to do that is explained on the next
slides.
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 106
We are trying to show that the RED PATH CANNOT BE
GLOBALLY OPTIMAL
c-ISOCLINE: NO CONSUMPTION GROWTH
c
k-ISOCLINE: NO CAPITAL
GROWTH
0
k(0)
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 107
CONSIDER THE ALTERNATIVE GREEN PATH, which:
-- concides with RED PATH until k* is reached and then
JUMPS UP to the green dot where is stay forever
c-ISOCLINE: NO CONSUMPTION GROWTH
c
k-ISOCLINE: NO CAPITAL
GROWTH
0
k(0)
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 108
-- The GREEN PATH CANNOT POSSIBLY BE OPTIMAL
because consumption JUMPS and therefore the green
path violates CONSUMPTION SMOOTHING, which was
CONDITION A above.
-- Still, the GREEN PATH is certaintly better than the RED
PATH because it has the same consumption until k* and
MORE consumption from there onwards!!!
-- For all RED PATHS (that is, all paths starting below the
saddle path), there is a GREEN PATH. So no paths
starting below the saddle path can be optimal (despite
the fact that it satisfies conditions A and B).
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 109
HENCE:
The only path starting at k[0] that :
-- satisfies CONDITIONS A and B, which are necessary
for optimality
-- satisfies non-negativity of capital
-- satisfies that there is NO OTHER PATH we can
construct that is better
 IS THE SADDLE PATH
 EQUILIBRIUM AND OPTIMAL ALLOCATIONS ARE
EQUAL
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 110
OPTIMAL AND EQUILIBRIUM ALLOCATION
c
c-ISOCLINE: NO CONSUMPTION GROWTH
k-ISOCLINE: NO CAPITAL
GROWTH
0
k(0)
k
k*
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 111
In the steady state:
• Savings rate is constant, just like in the
Solow model
• But it is endogenous in the sense of
depending on “fundamentals” like time
preference etc.
• In the simplest case: S=I=dK
Combined with: r+d=MPK and r=
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 112
In the steady state with
technological change:
• With technological change and
population growth: S=I=(d+a)K
• Growth of consumption=(r-)
 What is the relationship between the
SS savings rate S/Y and the rate of
technological change a?
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 113
Comparative statics
• Greater impatience (discount rate)?
(effects on income, capital, wages,
interest rates)
• Capital income taxation?
• A temporary cut of lump-sum taxes?
BGSE Growth and Development, 2008-09
SLIDE SET 3
Slide 114