BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3Slide 1 MACROECONOMICS BGSE/UPF 2008-2009 LECTURE...

BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3 Slide 1

MACROECONOMICSBGSE/UPF 2008-2009

LECTURE SLIDES SET 3

Professor Antonio Ciccone


II. ECONOMIC GROWTH WITH ENDOGENOUS

SAVINGS


1. Household savings behavior


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


Because of the BUDGET CONSTRAINT

this implies the “Keynesian” consumption function

where c is the marginal propensity to consume.


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.


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.


time

HOUSEHOLD INCOME OF FARMER

FIGURE 1: CONSUMPTION SMOOTHING: A VOLATILE INCOME PATH


time


HOUSEHOLD CONSUMPTION OF FARMER (“KEYNESIAN” theory)

FIGURE 2: INCOME AND "KEYNESIAN CONSUMPTION"


time


HOUSEHOLD CONSUMPTION OF FARMER (EMPIRICAL OBSERVATION)

FIGURE 3: CONSUMPTION SMOOTHING


time

HOUSEHOLD INCOME

CONSUMPTION SMOOTHING

SAVE FOR “RAINY DAYS”

DIS-SAVE TO MAINTAINCONSUMPTION LEVELS

FIGURE 4: SAVINGS AND DIS-SAVINGS IN CONSUMPTION SMOOTHING MODELS


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


INCOME

CONSUMPTIONA PUZZLE?

AGGREGATE LEVEL

INDIVIDUAL HOUSE-HOLD LEVEL

Germany 1950Or Country 1



Mr A

Ms B

Mr CMs D


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


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


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


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


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


C[0]

C[1]

Lw[0]

Lw[1]

GRAPHICALLY: INCOME LEVELS AND CONSUMTION


C[0]

C[1]

Lw[0]

Lw[1]

1+r

THE INTERTEMPORAL BUDGET CONSTRAINT


C[0]

C[1]

Lw[0]

Lw[1]

1+r

INTERTEMPORAL UTILITY MAXIMIZATION


C[0]

C[1]

Lw[0]

Lw[1]

1+r

C[0]

C[1]


C[0]

C[1]

Lw[0]

Lw[1]

1+r

C[0]

C[1]

BORROWING FOR CURRENT CONSUMPTION

BORROW

REPAY


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


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


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


Lw[0]

C[0]

0.5*Lw[1]

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

"CONSUMPTION FUNCTION"


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]


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


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


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


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


C[0]

C[1]

Lw[0]

Lw[1]

1+r

C[0]

C[1]

INCREASE IN INTEREST RATE

HIGH INTEREST RATE

LOW INTEREST RATE


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


3. The case of 3 and more periods

-- Timing

-- Intertemporal budget constraint

-- Optimality conditions

-- Time consistency


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


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


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


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]



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

0 GIVENQ

0

IF FINAL PERIOD

EndOfPeriodTQ

T


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


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


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



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


0lim 0EoPT T

TPV Q b

TIME T

0EoP

T TPV Q

0

WHAT IF:


CAN INCREASE TIME-0 CONSUMPTION

CONSUMPTION PLAN NOT OPTIMAL!

NECESSARY FOR OPTIMALITY:

EoP0lim 0T T

TPV Q


TIME CONSISTENCY ofHOUSOLD CONSUMPTION PLANS


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)


***** 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)


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


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


Intertemporal budget constraint in continuous time satisfied with equality if

0lim =0t tt

PV Q


0 givenQ


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:


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

3) What is the limit as t0?

Hopital’s rule yields


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”)


TIME

OPTIMAL CONSUMPTION PATH r =

C(t)

C(0)

CONSTANT CONSUMPTION IN TIME


TIME

OPTIMAL CONSUMPTION PATH r >

C(t)

C(0)

INCREASING CONSUMPTION IN TIME


TIME

OPTIMAL CONSUMPTION PATH r <

C(t)C(0)

DEACREASING CONSUMPTION IN TIME


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)


THE INTERTEMPORAL BUDGET CONSTRAINTwithout initial wealth

HENCE0

[ ] PERMANENT INCOMErte Lw t dt

[ ]PERMANENT INCOME

C tr

[ ] *PERMANENT INCOMEC t r


6. Deriving the continuous time first-order condition

• MAXIMIZATION BETWEEN ANY TWO PERIODS SEPARATED BY TIME x

• subject to

= TOTAL SPENDING IN TWO PERIODS


Take the following utility function:

with


FIRST ORDER CONDITIONS FOR THE TWO PERIODS IN TIME

making use of the utility function


REWRITING THIS CONDITIONS YIELDS

subtracting 1 from both sides


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)?


• Apply Hopital’s rule


HENCE as two periods become VERY CLOSE

WHICH IS WHAT WE WANTED TO SHOW


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


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



2. The Ramsey-Cass-Koopmans model


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


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)


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


CAPITAL MARKET EQUILIBRIUM

E(4)

E(5)


WHAT WE CANNOT KEEP IS

INSTEAD:

E(6)

E(7) INTERTEMPORAL BUDGET CONSTRAINT

where c[t] is CONSUMPTION per PERSON

2. Household behaviour


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


(E3) and (E4) imply

recall that there is NO population growth

and therefore

E(9)


SO WE HAVE OUR TWO EQUATIONS:

and


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


k

c k-ISOCLINE: CAPITAL DOES NOT GROW

k-ISOCLINE


k

c

k-ISOCLINE: CAPITAL DOES NOT GROW

CHANGES IN k in PHASE DIAGRAM


Continue with the optimal consumption equation

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

or


k

c c-ISOCLINE: CONSUMPTION DOES NOT GROW

k*is the k such that f’(k)=

c-ISOCLINE

0


k



CHANGES IN c in PHASE DIAGRAM

0


k

c

k-ISOCLINE: CAPITAL DOES NOT GROW

c-ISOCLINE: NO CONSUMPTION GROWTH

k*

PUTTING CHANGES in k and c TOGETHER

0


k

c

k-ISOCLINE: NO CAPITAL GROWTH


k*0


k

c



k*0


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


k

c



k*k(0)

PATHS that violate NON-NEGATIVE capital stock (consume too much in beginning)

0


k

c



k*k(0)

PATHS THAT DO NOT SATISFY BUDGET CONSTRAINT WITH EQUALITY (consume too little in beginning)

0

k_bar


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


k

c

f(k)-k


k*k(0)

PATHS THAT DO NOT SATISFY BUDGET CONSTRAINT WITH EQUALITY

f’(k)-=r=0

NEGATIVE INTEREST RATEPOSITIVE INTEREST

k_bar


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

PV q PV k bar

00 =

tr d

tPV e

time tNEGATIVE INTEREST RATE


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!!!!!!!


k

c



k*k(0)

EQUILIBRIUM (“SADDLE”) PATH

0


( )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


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*


OPTIMALITY

-- What would social planner do?

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


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)


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)


k

c



k*0

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


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.


k

c



k*k(0)

We are trying to show that the RED PATH CANNOT BE GLOBALLY OPTIMAL

0


k

c



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


-- 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).


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


k

c



k*k(0)

OPTIMAL AND EQUILIBRIUM ALLOCATION

0

BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3Slide 1 MACROECONOMICS BGSE/UPF 2008-2009 LECTURE...

Documents

Transcript of BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3Slide 1 MACROECONOMICS BGSE/UPF 2008-2009 LECTURE...