BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 1
MACROECONOMICSBGSE/UPF 2008-2009
LECTURE SLIDES SET 3
Professor Antonio Ciccone
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 2
II. ECONOMIC GROWTH WITH ENDOGENOUS
SAVINGS
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 3
1. Household savings behavior
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 4
1. “Keynesian theory” of savings and consumption
• So far we assumed a “Keynesian” savings function
• where s is the marginal propensity to save.
1. The Keynesian consumption (savings) function
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 5
Because of the BUDGET CONSTRAINT
this implies the “Keynesian” consumption function
where c is the marginal propensity to consume.
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 6
2. Limitations
CONCEPTUAL
The consumption behavior is assumed to be “mechanic” and “short-sighted”:
– 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.
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 7
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.
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 8
time
HOUSEHOLD INCOME OF FARMER
FIGURE 1: CONSUMPTION SMOOTHING: A VOLATILE INCOME PATH
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 9
time
HOUSEHOLD INCOME OF FARMER
HOUSEHOLD CONSUMPTION OF FARMER (“KEYNESIAN” theory)
FIGURE 2: INCOME AND "KEYNESIAN CONSUMPTION"
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 10
time
HOUSEHOLD INCOME OF FARMER
HOUSEHOLD CONSUMPTION OF FARMER (EMPIRICAL OBSERVATION)
FIGURE 3: CONSUMPTION SMOOTHING
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 11
time
HOUSEHOLD INCOME
CONSUMPTION SMOOTHING
SAVE FOR “RAINY DAYS”
DIS-SAVE TO MAINTAINCONSUMPTION LEVELS
FIGURE 4: SAVINGS AND DIS-SAVINGS IN CONSUMPTION SMOOTHING MODELS
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 12
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
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 13
INCOME
CONSUMPTIONA PUZZLE?
AGGREGATE LEVEL
INDIVIDUAL HOUSE-HOLD LEVEL
Germany 1950Or Country 1
Germany 1960Or Country 2
Germany 1980Or Country 3
Mr A
Ms B
Mr CMs D
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 14
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
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 15
SIMPLEST POSSIBLE formal model (2 PERIODS)
INGREDIENTS:
– Household lives 2 periods and tries to maximize INTERTEMPORAL utility
– 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
( [0]) (1 ) ( [1])U C U C
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 16
MATHEMATICAL MAXIMIZATION PROBLEM:
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)
0 1max ( ) (1 ) ( )U C U C
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 17
MATHEMATICAL FORMULATION
Maximize INTERTEMPORAL UTILITY
by choosing C
subject to INTERTEMPORAL BUDGET CONSTRAINT
C1=Lw1+(1+r)S= Lw1+(1+r)(Lw0-C0)
0 1max ( ) (1 ) ( )U C U C
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 18
INTERTEMPORAL BUDGET CONSTRAINT can also be written:
IMPORTANT TERMINOLOGY:
PERMANENT INCOME (PI)
PRICE OF FUTURE CONSUMPTION RELATIVE TO CURRENT CONSUMPTION
1 10 01 1
C LwC Lw
r r
10 1
LwLw
r
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 19
C[0]
C[1]
Lw[0]
Lw[1]
GRAPHICALLY: INCOME LEVELS AND CONSUMTION
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 20
C[0]
C[1]
Lw[0]
Lw[1]
1+r
THE INTERTEMPORAL BUDGET CONSTRAINT
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 21
C[0]
C[1]
Lw[0]
Lw[1]
1+r
INTERTEMPORAL UTILITY MAXIMIZATION
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 22
C[0]
C[1]
Lw[0]
Lw[1]
1+r
C[0]
C[1]
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 23
C[0]
C[1]
Lw[0]
Lw[1]
1+r
C[0]
C[1]
BORROWING FOR CURRENT CONSUMPTION
BORROW
REPAY
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 24
2. Closed form solution in a simple case
SUPPOSE THAT
INTEREST RATE is ZERO: r = 0 FUTURE UTILITY DISCOUNT is ZERO:
MAXIMIZATION PROBLEM BECOMES:
with respect to C
subject to 0 1 0 1C C Lw Lw PI
0 1max ( ) ( )U C U C
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 25
FIRST ORDER MAXIMIZATION CONDITIONS:
First-order conditions can be obtained from
with respect to C0
where we have substituted the budget constraint.
TAKE DERIVATIVE WITH RESPECT TO C[1] AND SET EQUAL ZERO:
OR
0 0max ( ) ( )U C U PI C
0 0
0 1
( ) ( )( 1) 0
U C U PI C
C C
0 1
0 1
( ) ( )U C U C
C C
0 1'( ) '( )U C U C
C1
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 26
EQUALIZE MARGINAL UTILITY AT DIFFERENT POINTS IN TIME
THIS IMPLIES
“PERFECT CONSUMPTION SMOOTHING”
Using the INTERTEMPORAL BUDGET CONSTRAINT yields consumption as a function of PERMANENT INCOME
0 10 1 / 2
2
Y YC C PI
0 1C C
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 27
Lw[0]
C[0]
0.5*Lw[1]
0.5*Lw[0]+0.5*Lw[1]
"CONSUMPTION FUNCTION"
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 28
Lw[0]
C[0]
0.5*Lw[1]
0.5*Lw[0]+0.5*Lw[1]
“TEMPORARY” INCREASE IN INCOME
INCREASEIn first-period income
THE EFFECT OF AN INCREASE IN INITIAL-PERIOD INCOME ON C[0]
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 29
Lw[0]
C[0]
0.5*Lw[0]+0.5*Lw[1]
“PERMANENT” INCREASE IN INCOME
INCREASE Lw[0]
INC
RE
AS
E L
w[1
]
THE EFFECT OF AN INCREASE IN INITIAL AND FUTURE INCOME
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 30
DISCOUNTING OF FUTURE UTILITY, AND INTEREST
MAXIMIZATION WITH DISCOUNTING&INTEREST
with respect to C
subject to INTERTEMPORAL BUDGET CONSTRAINT
1 10 01 1
C LwC Lw
r r
0 1max ( ) (1 ) ( )U C U C
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 31
FIRST-ORDER CONDITIONS
“EFFECTIVE TIME DISCOUNTING”
CONSTANT CONSUMPTION
DISCOUNTING OF FUTURE UTILITY AND POSTITIVE INTEREST RATE JUST OFFSET
0 1'( ) (1 )(1 ) '( )U C r U C
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 32
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
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 33
C[0]
C[1]
Lw[0]
Lw[1]
1+r
C[0]
C[1]
INCREASE IN INTEREST RATE
HIGH INTEREST RATE
LOW INTEREST RATE
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 34
AN EXAMPLE
Take the following utility function:
with
FIRST-ORDER CONDITION BECOMES
or
1/ 1/0 1(1 )(1 )C r C
1
0
(1 )(1 )C
rC
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 35
3. The case of 3 and more periods
-- Timing
-- Intertemporal budget constraint
-- Optimality conditions
-- Time consistency
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 36
TIMING
t=0 t=1
Q[0]
w[0]L w[1]L w[2]L
C[1] C[2]
INITIAL WEALTH
t=2
- interest r[0]- utility discount
- interest r[1]- utility discount
C[0]
YOU ARE HERE
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 37
PRESENT-VALUE INCOME AND CONSUMPTION
0 1 20
0 0 1 0 1 21 (1 )(1 ) (1 )(1 )(1 )
Lw Lw LwQ
r r r r r r
0 1 2
0 0 1 0 1 21 (1 )(1 ) (1 )(1 )(1 )
C C Cr r r r r r
- PERMANENTINCOME
- PRESENT VALUECONSUMPTION
t=0 t=1
Q[0] w[0]L w[1]L w[2]L
C[0] C[1] C[2]
t=2
YOU ARE HERE
interestdiscounting
interestdiscounting
interestdiscounting
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 38
INTERTEMPORAL BUDGET CONSTRAINT
0 1 20
0 0 1 0 1 2
0 1 2
0 0 1 0 1 2
1 (1 )(1 ) (1 )(1 )(1 )
1 (1 )(1 ) (1 )(1 )(1 )
Lw Lw LwQ
r r r r r r
C C C
r r r r r r
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 39
BUDGET CONTRAINT AND TIME EVOLUTION OF WEALTH
t=0 t=1 t=2
Q[0] w[0]L w[1]L w[2]L
C[1] C[2] C[3]
1 0 0 0 0(1 )Q r Q Lw C
2 1 1 1 1(1 )Q r Q Lw C
3 2 2 2 2(1 )Q r Q Lw C
C[0]
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 40
INTERTEMPORAL BUDGET CONSTRAINT
1 1 1 1(1 )t t t t tQ r Q Lw C
0 GIVENQ
0
IF FINAL PERIOD
EndOfPeriodTQ
T
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 41
0 1 20
0 0 1 0 1 21 (1 )(1 ) (1 )(1 )(1 )
Lw Lw LwQ
r r r r r r
THE “PRESENT-VALUE BUDGET SURPLUS”
= PERMANENT INCOME minus PRESENT VALUE CONSUMPTION
0 1 2(1 )(1 )(1 )
EoPTQ
r r r
0 1 2
0 0 1 0 1 21 (1 )(1 ) (1 )(1 )(1 )
C C Cr r r r r r
= PRESENT VALUE OF END-OF-LIFE WEALTH
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 42
MAXIMIZE BETWEEN ADJACENT PERIODS
1 1'( ) (1 )(1 ) '( )t t tU C r U C
OPTIMAL SOLUTION OF CONSUMPTION PROBLEM
0 1 2
0(1 )(1 )(1 )
EoPTQ
r r r
plus BUDGET CONSTRAINT WITH EQUALITY
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 43
INFINITE HORIZON
=TIME ZERO (PRESENT) VALUE OF 1 EURO PAID AT (end of) PERIOD t
00 1
1
(1 )*(1 )*...*(1 )tt
PVr r r
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 44
INTERTEMPORAL BUDGET CONSTRAINT
1 1 1 1(1 )t t t t tQ r Q Lw C
0 GIVENQ
0lim 0EoPT T
TPV Q
NO-PONZI-GAME condition
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 45
0lim 0EoPT T
TPV Q b
TIME T
0EoP
T TPV Q
0
WHAT IF:
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 46
CAN INCREASE TIME-0 CONSUMPTION
CONSUMPTION PLAN NOT OPTIMAL!
NECESSARY FOR OPTIMALITY:
EoP0lim 0T T
TPV Q
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 47
TIME CONSISTENCY ofHOUSOLD CONSUMPTION PLANS
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 48
TIME 0 CONSUMPTION PLANS
t=0 t=1
Q[0] w[0]L w[1]L w[2]L
C[0] C[1] C[2]
t=2
YOU ARE HERE
interestdiscounting
interestdiscounting
interestdiscounting
t=0 t=1
Q[0] Q(1) w[1]L w[2]L
C[1] C[2]
t=2
interestdiscounting
interestdiscounting
YOU ARE HERE
TIME 1 CONSUMPTION PLANS (NO NEW INFO)
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 49
***** TIME CONSISTENCY *****
t=0 t=1
Q[0] w[0]L w[1]L w[2]L
C[0] C[1] C[2]
t=2
YOU ARE HERE
interestdiscounting
interestdiscounting
interestdiscounting
t=0 t=1
Q(1) w[1]L w[2]L
C[1] C[2]
t=2
interestdiscounting
interestdiscounting
YOU ARE HERE
TIME 1 CONSUMPTION PLANS (NO NEW INFO)
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 50
3. Optimal consumption and savings in continuous time1. Infinite horizon
subject to
= TIME ZERO (PRESENT) VALUE OF 1 EURO PAID AT TIME t
0max ( )t
te U C dt
0 0 00 0
( )t t t tPV C dt Q PV Lw dt
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 51
2. Intertemporal budget constraint
1 1 1 1(1 )t t t t tQ r Q Lw C
(1 )t t t t t tQ r Q r Lw C
1 1 1 1(1 )t t t t tQ r Q Lw C
Wealth in discrete time
1 1 1 1 1t t t t t tQ Q r Q Lw C
Wealth incontinuous time
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 52
Intertemporal budget constraint in continuous time satisfied with equality if
0lim =0t tt
PV Q
(1 )t t t t t tQ r Q r Lw C
0 givenQ
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 53
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:
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 54
2) Hence the utility discount per unit of time is:
3) What is the limit as t0?
Hopital’s rule yields
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 55
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”)
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 56
TIME
OPTIMAL CONSUMPTION PATH r =
C(t)
C(0)
CONSTANT CONSUMPTION IN TIME
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 57
TIME
OPTIMAL CONSUMPTION PATH r >
C(t)
C(0)
INCREASING CONSUMPTION IN TIME
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 58
TIME
OPTIMAL CONSUMPTION PATH r <
C(t)C(0)
DEACREASING CONSUMPTION IN TIME
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 59
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)
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 60
THE INTERTEMPORAL BUDGET CONSTRAINTwithout initial wealth
HENCE0
[ ] PERMANENT INCOMErte Lw t dt
[ ]PERMANENT INCOME
C tr
[ ] *PERMANENT INCOMEC t r
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 61
6. Deriving the continuous time first-order condition
• MAXIMIZATION BETWEEN ANY TWO PERIODS SEPARATED BY TIME x
• subject to
= TOTAL SPENDING IN TWO PERIODS
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 62
Take the following utility function:
with
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 63
FIRST ORDER CONDITIONS FOR THE TWO PERIODS IN TIME
making use of the utility function
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 64
REWRITING THIS CONDITIONS YIELDS
subtracting 1 from both sides
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 65
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 (x0)?
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 66
• Apply Hopital’s rule
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 67
HENCE as two periods become VERY CLOSE
WHICH IS WHAT WE WANTED TO SHOW
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 68
SUMMARIZING
QUESTION: What characterizes the optimal consumption PATH that solves
subject to
1 1/
0 0max ( )
1 1/t t t
tC
e U C dt e dt
0 0 00 0
t t t tPV C dt Q PV Lw dt
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 69
ˆ ( )tt t
t
CC r
C
0 0 00 0
t t t tPV C dt Q PV Lw dt
ANSWER:
and
or
0lim =0t tt
PV Q
(1 )t t t t t tQ r Q r Lw C
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 70
2. The Ramsey-Cass-Koopmans model
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 71
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 Cass-Koopmans 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:
subject to
where
1. Equilibrium growth with infinite-horizon households
0 00 0
[ ] [ ] [0]t tPV C t dt PV w t Ldt Q
0max ( [ ])te U C t dt
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 72
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)
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 73
WHAT WE CAN KEEP FROM THE SOLOW MODEL
CONSTANT RETURNS PRODUCTION FUNCTION
E(1)
E(2)
CAPITAL ACCUMULATION EQUATION
E(3)
PRODUCTION FUNCTION
1. Technology and the capital market
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 74
CAPITAL MARKET EQUILIBRIUM
E(4)
E(5)
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 75
WHAT WE CANNOT KEEP IS
INSTEAD:
E(6)
E(7) INTERTEMPORAL BUDGET CONSTRAINT
where c[t] is CONSUMPTION per PERSON
2. Household behaviour
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 76
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 TWO-DIMENSIONAL DIFFERENTIAL EQUATION SYSTEM WHERE
CHANGE in CONSUMPTION c=FUNCTION OF k and cCHANGE IN CAPITAL k=FUNCTION OF k and c
(E6) and (E5) imply
E(8)
3. Dynamic equilibrium system
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 77
(E3) and (E4) imply
recall that there is NO population growth
and therefore
E(9)
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 78
SO WE HAVE OUR TWO EQUATIONS:
and
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 79
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.
2. Equilibrium growth and optimality
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 80
k
c k-ISOCLINE: CAPITAL DOES NOT GROW
k-ISOCLINE
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 81
k
c
k-ISOCLINE: CAPITAL DOES NOT GROW
CHANGES IN k in PHASE DIAGRAM
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 82
Continue with the optimal consumption equation
FIRST: Find ISOCLINE, which are the (c, k) combinations such that
or
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 83
k
c c-ISOCLINE: CONSUMPTION DOES NOT GROW
k*is the k such that f’(k)=
c-ISOCLINE
0
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 84
k
c c-ISOCLINE: CONSUMPTION DOES NOT GROW
k*is the k such that f’(k)=
CHANGES IN c in PHASE DIAGRAM
0
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 85
k
c c-ISOCLINE: CONSUMPTION DOES NOT GROW
k*is the k such that f’(k)=
CHANGES IN c in PHASE DIAGRAM
0
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 86
k
c
k-ISOCLINE: CAPITAL DOES NOT GROW
c-ISOCLINE: NO CONSUMPTION GROWTH
k*
PUTTING CHANGES in k and c TOGETHER
0
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 87
k
c
k-ISOCLINE: NO CAPITAL GROWTH
c-ISOCLINE: NO CONSUMPTION GROWTH
k*0
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 88
k
c
k-ISOCLINE: NO CAPITAL GROWTH
c-ISOCLINE: NO CONSUMPTION GROWTH
k*0
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 89
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/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 90
k
c
k-ISOCLINE: NO CAPITAL GROWTH
c-ISOCLINE: NO CONSUMPTION GROWTH
k*k(0)
PATHS that violate NON-NEGATIVE capital stock (consume too much in beginning)
0
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 91
k
c
k-ISOCLINE: NO CAPITAL GROWTH
c-ISOCLINE: NO CONSUMPTION GROWTH
k*k(0)
PATHS THAT DO NOT SATISFY BUDGET CONSTRAINT WITH EQUALITY (consume too little in beginning)
0
k_bar
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 92
Q(t)=K(t) or q(t)=k(t)
0 0lim = lim =0t t t tt t
PV q PV k
00 =
tr d
tPV e
(1) Wealth=Capital
(2) Intertemporal budget constraint with equality
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 93
k
c
f(k)-k
c-ISOCLINE: NO CONSUMPTION GROWTH
k*k(0)
PATHS THAT DO NOT SATISFY BUDGET CONSTRAINT WITH EQUALITY
f’(k)-=r=0
NEGATIVE INTEREST RATEPOSITIVE INTEREST
k_bar
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 94
0 0lim = lim ( _ )t t tt t
PV q PV k bar
00 =
tr d
tPV e
time tNEGATIVE INTEREST RATE
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 95
k
cc-ISOCLINE: NO CONSUMPTION GROWTH
k*k(0)
PATHS THAT DO NO SATISFY BUDGET CONSTRAINT WITH EQUALITY
0
k_bar
YOU ARE NOT SPENDINGALL YOUR PERMANENTINCOME!!!!!!!
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 96
k
c
k-ISOCLINE: NO CAPITAL GROWTH
c-ISOCLINE: NO CONSUMPTION GROWTH
k*k(0)
EQUILIBRIUM (“SADDLE”) PATH
0
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 97
( )t t t tk f k c k
( )t t t t t tk r k Lw c k
t t t t tk r k Lw c
SADDLE PATH SATISFIES INTERTEMPORALBUDGET CONSTRAINT
Capital market equilibrium:
Income per worker=Labor income + Capital income:
Hence:
t t t t tq r q Lw c t tk q
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 98
Moreover:
*0 0 0lim = lim = lim =0t t t t t
t t tPV q PV k PV k
00lim = lim 0
tr d
tt t
PV e
As:
given that interest rates>0 for k<=k*
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 99
OPTIMALITY
-- What would social planner do?
- Social planner: dictator who decides allocation according to HH welfare subject to physical contraints
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 100
MRS=MRT
The GLOBALLY OPTIMAL PATH MUST SATISFY
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
(A)
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 101
RESOURCE CONSTRAINT
The GLOBALLY OPTIMAL PATH MUST SATISFY
To see why, suppose first that
[ ] ( [ ]) [ ] [ ]k t f k t c t k t
[ ] ( [ ]) [ ] [ ]k t f k t c t k t
[ ] ( [ ]) [ ] [ ]k t f k t c t k t
-- in this case the planner must be throwing away goods (investment goods) because the increase in the number of machines is LESS THAN the machines built less depreciation : BUT THROWING AWAY GOODS CANNO BE OPTIMAL!!
[ ]k t
( [ ]) [ ] [ ]f k t c t k t
Now suppose instead
-- now the planner is a REAL MAGICIAN!! as the number of machines in the economy goes up by which is GREATER THAN machines built less depreciation
( [ ]) [ ] [ ]f k t c t k t
[ ]k t
(B)
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 102
k
c
k-ISOCLINE: NO CAPITAL GROWTH
c-ISOCLINE: NO CONSUMPTION GROWTH
k*0
ALL THE PATHS THAT SATISFY CONDITIONS (A) and (B)
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 103
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/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 104
k
c
k-ISOCLINE: NO CAPITAL GROWTH
c-ISOCLINE: NO CONSUMPTION GROWTH
k*k(0)
We are trying to show that the RED PATH CANNOT BE GLOBALLY OPTIMAL
0
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 105
k
c
k-ISOCLINE: NO CAPITAL GROWTH
c-ISOCLINE: NO CONSUMPTION GROWTH
k*k(0)
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
0
BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 106
-- 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/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 107
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/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 108
k
c
k-ISOCLINE: NO CAPITAL GROWTH
c-ISOCLINE: NO CONSUMPTION GROWTH
k*k(0)
OPTIMAL AND EQUILIBRIUM ALLOCATION
0
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