Slide 1

ITFD Growth and Development

LECTURE SLIDES SET 3

Professor Antonio Ciccone

Slide 2

II. ECONOMIC GROWTH WITH ENDOGENOUS

SAVINGS

Slide 3

1. Household savings behavior

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

][][ tYstS

Slide 5

Because of the BUDGET CONSTRAINT

this implies the “Keynesian” consumption function

where c is the marginal propensity to consume.

][][][ tYtCtS

][][)1(][ tcYtYstC

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.

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.

Slide 8

time

HOUSEHOLD INCOME OF FARMER

FIGURE 1: CONSUMPTION SMOOTHING: A VOLATILE INCOME PATH

Slide 9

time

HOUSEHOLD INCOME OF FARMER

HOUSEHOLD CONSUMPTION OF FARMER (“KEYNESIAN” theory)

FIGURE 2: INCOME AND "KEYNESIAN CONSUMPTION"

Slide 10

time

HOUSEHOLD INCOME OF FARMER

HOUSEHOLD CONSUMPTION OF FARMER (EMPIRICAL OBSERVATION)

FIGURE 3: CONSUMPTION SMOOTHING

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

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

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

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

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

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

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

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

r

LwLw

11

0

r1

1

Slide 19

C[0]

C[1]

Lw[0]

Lw[1]

GRAPHICALLY: INCOME LEVELS AND CONSUMTION

Slide 20

C[0]

C[1]

Lw[0]

Lw[1]

1+r

THE INTERTEMPORAL BUDGET CONSTRAINT

Slide 21

C[0]

C[1]

Lw[0]

Lw[1]

1+r

INTERTEMPORAL UTILITY MAXIMIZATION

Slide 22

C[0]

C[1]

Lw[0]

Lw[1]

1+r

C[0]

C[1]

Slide 23

C[0]

C[1]

Lw[0]

Lw[1]

1+r

C[0]

C[1]

BORROWING FOR CURRENT CONSUMPTION

BORROW

REPAY

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

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

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

Slide 27

Lw[0]

C[0]

0.5*Lw[1]

0.5*Lw[0]+0.5*Lw[1]

"CONSUMPTION FUNCTION"

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]

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

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

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

)1)(1( r

1)1)(1( r

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

(1-β)(1+r) > 1

(1-β)(1+r) < 1

Slide 33

AN EXAMPLE

Take the following utility function:

FIRST-ORDER CONDITION BECOMES

or

1/ 1/0 1(1 )(1 )C r C

1

0

(1 )(1 )C

rC

/1

/11

][])[('

0/11

][])[(

tCtCU

withtC

tCU

Slide 34

3. The case of 3 and more periods

-- Timing-- Intertemporal budget constraint-- Optimality conditions-- Time consistency

Slide 35

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

Slide 36

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

Slide 37

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[0] C[1] C[2]

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

Slide 38

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

Slide 39

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

Slide 40

Shortest way from A to B?

A

B

Slide 41

Shortest way from A to B

A

B

Slide 42

Must be the shortest way between ANY two points

A

B

C

D

Slide 43

A

B

C

D

Must be the shortest way between ANY two points

Slide 44

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

Slide 45

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

Slide 46

TIME T

0EoP

T TPV Q

0

WHAT IF: NO PONZI GAME CONDITION VIOLATED?

Slide 47

INTERTEMPORAL BUDGET CONSTRAINT WITH EQUALITY

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

Slide 48

0lim 0EoPT T

TPV Q b

TIME T

0EoP

T TPV Q

0

WHAT IF:

Slide 49

CAN INCREASE TIME-0 CONSUMPTION

CONSUMPTION PLAN NOT OPTIMAL!

NECESSARY FOR OPTIMALITY:

EoP0lim 0T T

TPV Q

Slide 50

TIME CONSISTENCY ofHOUSOLD CONSUMPTION PLANS

Slide 51

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)

Slide 52

***** 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)

Slide 53

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

tdssr

t eVP 0)(

0

Slide 54

2. Intertemporal budget constraint

1 1 1 1(1 )t t t t tQ r Q Lw C

t t t t tQ r Q 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

Slide 55

Intertemporal budget constraint in continuous time satisfied with equality if

0lim =0t tt

PV Q

t t t t tQ r Q Lw C

0 givenQ

Slide 56

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:

te 1

Slide 57

2) Hence the utility discount per unit of time is:

3) What is the limit as t0?

Hopital’s rule yields

0

01

t

e t

1lim

)1(

lim1

00

t

t

t

t

t e

ttte

t

e

t

e t )1(

Slide 58

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”)

)][(][

][

][

/][

trtC

tC

tC

ttC

Slide 59

TIME

OPTIMAL CONSUMPTION PATH r =

C(t)

C(0)

CONSTANT CONSUMPTION IN TIME

Slide 60

TIME

OPTIMAL CONSUMPTION PATH r >

C(t)

C(0)

INCREASING CONSUMPTION IN TIME

Slide 61

TIME

OPTIMAL CONSUMPTION PATH r <

C(t)C(0)

DEACREASING CONSUMPTION IN TIME

Slide 62

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)

][tr

T

0)][(][

][ tr

tC

tC

Slide 63

THE INTERTEMPORAL BUDGET CONSTRAINTwithout initial wealth

HENCE0

[ ] PERMANENT INCOMErte Lw t dt

[ ]PERMANENT INCOME

C tr

[ ] *PERMANENT INCOMEC t r

0 0

][r

CCdtedttCe rtrt

Slide 64

6. Deriving the continuous time first-order condition

• MAXIMIZATION BETWEEN ANY TWO PERIODS SEPARATED BY TIME x

• subject to

= TOTAL SPENDING IN TWO PERIODS

)][(][

][

][

/][

trtC

tC

tC

ttC

])[(])[(max )( xtCUetCUe xtt

][][ )( xtCetCe xtrrt

Slide 65

Take the following utility function:Take the following utility function:

with

/11

][])[(

/11

tCtCU

Take the following utility function:

with

/1][])[(' tCtCU

1/1][)/1(])[('' tCtCU

Slide 66

FIRST ORDER CONDITIONS FOR THE TWO PERIODS IN TIME

making use of the utility function

])[('])[('

])[('])[(')( xtCUetCU

xtCUeetCUxr

rxx

/11

][])[(

/11

tCtCU

/1)(/1 ][][ xtCetC xr

Slide 67

REWRITING THIS CONDITIONS YIELDS

subtracting 1 from both sides

xrexTC

tC )(

/1

][

][

xretC

xtC )(

][

][

1][

][

][

][ )( xre

tC

tC

tC

xtC

Slide 68

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)?

x

e

tCx

tCxtCxr 1

][

][][)(

0

01)(

x

e xr

Slide 69

• Apply Hopital’s rule

)(1

)(lim

)1(

lim

1

)(

0

)(

0

)(

rer

xxx

e

x

e

xr

x

xr

x

xr

Slide 70

HENCE as two periods become VERY CLOSE

WHICH IS WHAT WE WANTED TO SHOW

)][(][

][

][

][

][

][][

trtC

tC

tCttC

tCx

tCxtC

Slide 71

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

Slide 72

ANSWER:

and

or

ˆ ( )tt t

t

CC r

C

0 0 00 0

t t t tPV C dt Q PV Lw dt

0lim =0t tt

PV Q

)( ttttt CLwQrQ

Slide 73

2. The Ramsey-Cass-Koopmans model

Slide 74

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

dssr

t

t

eVP

0)(

0

Slide 75

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)

Slide 76

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

)('),(

kfK

LKFPMK

)()1,()1,(),( kfkFL

KF

L

YLKFY

][][][][

tKtItKt

tK

Slide 77

CAPITAL MARKET EQUILIBRIUM

E(4)

E(5) ][])[('][ trtkftPMK

][][][][ tCtYtStI

Slide 78

WHAT WE CANNOT KEEP IS

INSTEAD:

E(6)

E(7) INTERTEMPORAL BUDGET CONSTRAINT WITH EQUALITY

where c[t] is CONSUMPTION per PERSON

2. Household behaviour

][][ tsYtS

)][(][

][ trtc

tc

Slide 79

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

)])[('()][(][

][ tkftrtc

tc

Slide 80

(E3) and (E4) imply

recall that there is NO population growth

and therefore

E(9)

L

tK

L

tI

L

tK ][][][

][][])[(][

][][][][

][][

][][

][][][][

tktctkftk

tdktctytk

L

tKtc

L

tY

L

tK

L

tK

L

tCtY

L

tK

Slide 81

SO WE HAVE OUR TWO EQUATIONS:

][][])[(][

)])[('(][

][

tktctkftk

and

tkftc

tc

Slide 82

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

][][])[(][ tktctkftk

0][ tk

0][][])[(][ tktctkftk

][])[(][ tktkftc

Slide 83

k

c k-ISOCLINE: CAPITAL DOES NOT GROW

k-ISOCLINE

Slide 84

k

c

k-ISOCLINE: CAPITAL DOES NOT GROW

CHANGES IN k in PHASE DIAGRAM

Slide 85

Continue with the optimal consumption equation

FIRST: Find ISOCLINE, which are the (c, k) combinations such that

)])[('(][

][ tkftc

tc

0)])[('(][

][ tkf

tc

tc

0])[(' tkf

][])[(' trortkf

Slide 86

k

c c-ISOCLINE: CONSUMPTION DOES NOT GROW

k*is the k such that f’(k)=

c-ISOCLINE

0

Slide 87

k

c c-ISOCLINE: CONSUMPTION DOES NOT GROW

k*is the k such that f’(k)=

CHANGES IN c in PHASE DIAGRAM

0

Slide 88

k

c c-ISOCLINE: CONSUMPTION DOES NOT GROW

k*is the k such that f’(k)=

CHANGES IN c in PHASE DIAGRAM

0

Slide 89

k

c

k-ISOCLINE: CAPITAL DOES NOT GROW

c-ISOCLINE: NO CONSUMPTION GROWTH

k*

PUTTING CHANGES in k and c TOGETHER

0

Slide 90

k

c

k-ISOCLINE: NO CAPITAL GROWTH

c-ISOCLINE: NO CONSUMPTION GROWTH

k*0

Slide 91

k

c

k-ISOCLINE: NO CAPITAL GROWTH

c-ISOCLINE: NO CONSUMPTION GROWTH

k*0

Slide 92

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

Slide 93

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

Slide 94

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

Slide 95

(1) Wealth=Capital

(2) Intertemporal budget constraint with equality

Q(t)=K(t) or q(t)=k(t)

0 0lim = lim =0t t t tt t

PV q PV k

t

sdsrt eVP 0

0

Slide 96

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

Slide 97

time tNEGATIVE INTEREST RATE

0 0lim = lim ( _ )t t tt t

PV q PV k bar

t

sdsrt eVP 0

0

Slide 98

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!!!!!!!

Slide 99

k

c

k-ISOCLINE: NO CAPITAL GROWTH

c-ISOCLINE: NO CONSUMPTION GROWTH

k*k(0)

EQUILIBRIUM (“SADDLE”) PATH

0

Slide 100

SADDLE PATH SATISFIES INTERTEMPORALBUDGET CONSTRAINT WITH EQUALITY

Capital market equilibrium:

Income per worker=Labor income + Capital income:

Hence:

( )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

t t t t tq r q Lw c t tk q

Slide 101

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*

Slide 102

OPTIMALITY

-- What would social planner do?

- Social planner: dictator who decides allocation according to HH welfare subject to physical contraints

Slide 103

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)

)])[('(][

][ tkftc

tc

Slide 104

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)

Slide 105

k

c

k-ISOCLINE: NO CAPITAL GROWTH

c-ISOCLINE: NO CONSUMPTION GROWTH

k*0

ALL THE PATHS THAT SATISFY CONDITIONS (A) and (B)

Slide 106

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.

Slide 107

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

Slide 108

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

Slide 109

-- 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).

Slide 110

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

Slide 111

k

c

k-ISOCLINE: NO CAPITAL GROWTH

c-ISOCLINE: NO CONSUMPTION GROWTH

k*k(0)

OPTIMAL AND EQUILIBRIUM ALLOCATION

0

Slide 112

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=K

Combined with: r+=MPK and r=

Slide 113

In the steady state with technological change:

• With technological change and population growth: S=I=(+a)K

• Growth of consumption=(r-)

What is the relationship between the SS savings rate S/Y and the rate of technological change a?

Slide 114

Comparative statics

• Greater impatience (discount rate)?

(effects on income, capital, wages, interest rates)

• Capital income taxation?

• A temporary cut of lump-sum taxes?