MACROECONOMICS I UPF

UPF Macroeconomics I, 2008-09 SLIDE SET 4 SLIDE 1

MACROECONOMICS IUPF

LECTURE SLIDES SET 4Professor Antonio Ciccone


3. Applications of the Ramsey-Cass-Koopmans (RCK) model

3.1 Government spending, consumption, and interest rates

3.2 Bond versus tax financed government spending



- Comparative “dynamics” in the RCK model

- Permanent, surprise drop in output

- Temporary, surprise drop in output

- Wars, government expenditures and interest rates

- The role of expectations- Permanent, anticipated drop in output

- Temporary, anticipated drop in output


k

c

k-ISOCLINE: NO CAPITAL GROWTH

c-ISOCLINE: NO CONSUMPTION GROWTH

k*0

The RCK model


k

c



k*0


k

c

NEW k-ISOCLINE: NO CAPITAL GROWTH


k*0

Permanent, surprise fall in output for given k


timePermanent, surprise fall in output

Evolution of consumption


timePermanent, surprise fall in output

Evolution of capital intensity


-- consumption can JUMP at the time newinformation arrives

-- but consumption must be smooth (follow thefirst-order condition) from than onward:

There CANNOT BE an ANTICIPATED jump in consumption


k

c



k*0

Temporary, surprise fall in output for given k: PART I



k

c



k*0

Temporary, surprise fall in output for given k: PART II


k

c



k*0

Temporary,surprise fall in output: Equilibrium response


k

c



k*0

Temporary,surprise fall in output: Equilibrium response


timeSTART of Tempfall in output

END of Tempfall in output

Evolution of the capital intensity




Evolution of real interest rate






Wars and real interest rates

-- Suppose government expenditures associated withwars are surprise, temporary events

-- Study the dynamic response of: capital, interest rates, and consumption to wars

-- Government expenditures associated with warsdecrease output available for consumption andinvestment

( , )F K L G C I− = +

INCREASE G Same effect as temporary fall in output


timeSTART of War END of War

Evolution of real interest rate



- The role of expectations

- Permanent, anticipated drop in output



k

c



k*0

Permanent, anticipated fall in output: PART I


k

c



k*0

Permanent, anticipated fall in output: PART II


k

c



k*0

Permanent, anticipated fall in output: Equilibrium response


timeINFO of permanent FUTUREfall in output

Evolution of capital intensity

Output actually falls


timeINFO of permanent FUTUREfall in output


Output actually falls


- The role of expectations

- Permanent, anticipated drop in output



k

c



k*0

Temporary, anticipated fall in output for given k: PART I


k

c



k*0

Temporary, anticipated fall in output for given k: PART II


k

c



k*0

Temporary, anticipated fall in output for given k: PART III


k

c



k*0

Temporary, anticipated fall in output: Equilibrium response


k

c



k*0

Temporary, anticipated fall in output: Equilibrium response


timeINFO of FUTURETemp fall in output


Evolution of the capital intensity

START of FUTURETemp fall in output


time


INFO of FUTURETemp fall in output


START of FUTURETemp fall in output


3. Application of the Ramsey-Cass-Koopmans (RCK) model


3.2 Bond versus tax financed government spending


Government expenditures and taxes

t t tGDEFICIT G T= −

Government intertemporal budget constraint

0 00 0

t t t t tPV T dt GWEALTH PV G dt∞ ∞

+ =∫ ∫


-- Suppose that households believe in government budgetconstraint

-- The government cut taxes at time t

-- But there is no indication that the government cutsexpenditures

00

t tPV T dt∞

∫

-- WHAT HAPPENS TO DISCOUNTED FLOW OF TAXES?


Nothing, because:

0 00 0

t t t t tPV T dt PV G dt GWEALTH∞ ∞

= −∫ ∫

and the right-hand side of this equation has not changed.

Government will have to compensate current tax cut by taxincrease sometime in the future.


Now let’s look at household intertemporal budget constraint:

0 00 0

0 00

t t t t

t t

PV C dt PV T dt

PV w Ldt Q

∞ ∞

∞

+

= +

∫ ∫

∫

-- current tax cut does NOT affect this constraint at all as onlythe DISCOUNTERD PRESENT VALUE OF TAXES MATTERS

-- and present value of taxes remains constant if expendituresdo not change


-- TAX CUT DOES NOT CHANGE HH CONSUMPTION

-- AS A RESULT IT DOES NOT CHANGE THE NATIONALSAVINGS RATE:

t t t tS Y C G= − −

-- DOES NOT AFFECT:- INVESTMENT(!) - AND INTEREST RATES (!)

-- HH SAVINGS INCREASES, BUT IS OFFSET BY AN INCREASE IN GOVERNMENT DEFICIT:

( ) ( )t t t t t tS Y T C T G= − − + −


Hence, government cuts taxes

Has to issue debt (government bonds)Government ensures that real interest rate on bond mimics

market interest rate (before issue of new bonds)Households buy these new bonds with their tax savings

Hence,Household use to buy government bonds what they “save”

in current taxes


3. The Diamond model

1. Overlapping generations models2 Setup of the Diamond model1. Technology2. Household behavior3. Dynamic equilibrium system

3. Equilibrium growth and optimality4. Applications of the Diamond model1. Government spending, consumption, and interest rates2. Bond versus tax financed government spending


-- Discrete time model-- Households live for two periods, and only work in the first

1. Overlapping Generations models


Time 3 Time 4Time 2Time 1

Generation 1

YOUNG:Workand

Consume

RETIRED:Consume

THE LIFE CYCLE OF A SINGLE GENERATION


TIMING

Time 3 Time 4Time 2Time 1

Generation 1

Generation 2

Generation 3


2. Setup of the Diamond model

1. Technology2. Household behavior3. Dynamic equilibrium system


1. Technology

( )1 t t t tY K A L αα −=

Owned/Supplied by RETIRED Supplied by YOUNG

Capital fully depreciates during production: 1δ =


time

t t+1

Born -Earn Wage-Consume-Save

- Earn Interest- Consume

2. Household behavior

GENERATION tProduction usesgeneration t labor

Production usesgeneration t capital


UTILITY of GENERATION t

MAXIMIZATION WITH DISCOUNTING&INTEREST

with respect to C

subject to INTERTEMPORAL BUDGET CONSTRAINT

11

tt t

t

cc wr++ =+

1max ( ) (1 ) ( )t tU c U cβ ++ −


FIRST-ORDER CONDITIONS FOR GENERATION t

1'( ) (1 )(1 ) '( )t tU c r U cβ += − +

“EFFECTIVE TIME DISCOUNTING”


Take the following CES utility function:

with

CONSUMPTION PROFILE BECOMES

[ ]1 (1 )(1 )tt

t

c rc

σβ+ = − +


11

tt t

t

cc wr++ =+

[ ](1 )(1 )1

t tt t

t

c rc w

r

σβ− ++ =

+

( )11 (1 ) (1 )t t tc r wσ σβ −+ − + =

11 (1 ) (1 )t

tt

wcrσ σβ −=

+ − +

Consumption profile intoBudget constraint


11 (1 ) (1 )t

tt

wcrσ σβ −=

+ − +

11 (1 ) (1 )t

t t t tt

ws w c wrσ σβ −= − = −

+ − +

1

1(1 ) (1 )

1 (1 ) (1 )t

t tt

rs wr

σ σ

σ σββ

−

−− +

=+ − +

Consumption and savings of generation t (when young)


s

r

(1 )1 (1 )t ts w

σ

σββ

−=

+ −

0


s

r

(1 )1 (1 )t ts w

σ

σββ

−=

+ − t ts w=

0


s

r

(1 )1 (1 )t ts w

σ

σββ

−=

+ −

0


11

1(1 ) (1 )

1 (1 ) (1 )

t t t t

tt t

t

K I L s

rL wr

σ σ

σ σββ

+−

−

= =

− +=

+ − +

3. Dynamic equilibrium system

Savings per young person


( )1(1 )t t t t tw MPL K A L ααα −= = −

( )111 1t t t t tr MPK K A L ααα −−= − = −

( )1 t t t tY K A L αα −=

Financed by SAVINGSof the now RETIRED

Supplied by YOUNG


( )

11

11

(1 ) (1 )(1 )1 (1 ) (1 )

t t t t

tt t t

t

K I L s

rK A Lr

σ σαα

σ σβαβ

+−

−−

= =

− += −

+ − +

( )11 1t t t tr K A L ααα −−= −


( )

1

1 11 1

11 1

(1 ) (1 )1 (1 ) (1 )

t

t t

t t tt t t

t t t t t

KL A

K A LL A rL A L A r

αα σ σ

σ σββ

+

+ +− −

−+ +

=

− +=

+ − +

( )11 1t t t tr K A L ααα −−= −


11

1(1 ) (1 )1

(1 )(1 ) 1 (1 ) (1 )

t

tt

t

k

rkn a r

σ σα

σ σββ

+−

−

=

− +=

+ + + − +

%

%

1 1t tr kαα −= −%


3. Equilibrium growth and optimality

1

1 (1 )(1 )(1 ) 1 (1 )

t

t

k

kn a

σα

σββ

+ =

−=

+ + + −

%

%

1σ =


0

1tk +%

0k%

1t tk bkα+ =% %

tk%1k% BGP


Optimality

• How to weight different generationsunclear.

• Is the allocation at least Pareto efficient?


Dynamic Inefficiency

• A situation where the allocation is not evenPareto efficient

• I.e. we can increase consumption of atleast one generation without decreasingconsumption of all others


Consider case without technological progress a=0.

How much do we need to invest per person at time tto keep capital intensity constant?

11

1 1 1t t t t

tt t

K L i ikL L n

++

+ += = =

+

1t tk k+ = (1 )t ti k n= +


0 GoldRulek

( )t ty f k=

tk

(1 )tk n+

Consumption per capita


0

tkGoldRulek

Consumption per capita

1MPK n= +


11 (1 )

(1 ) 1 (1 )t tk kn

σα

σββ+

−=

+ + −

1 (1 )(1 ) 1 (1 )

k kn

σα

σββ

−=

+ + −

111 (1 )

(1 )1 (1 )BGPk

n

σ α

σββ

−⎛ ⎞−= ⎜ ⎟+ + −⎝ ⎠

BGP


11 (1 )(1 )

(1 )1 (1 )BGPMPK

n

σ

σβαβ

−⎛ ⎞−

= − ⎜ ⎟+ + −⎝ ⎠

1(1 )MPK kαα −= −

1 (1 )(1 ) (1 )(1 )

BGPMPK nσ

σβα

β⎛ ⎞+ −

= − +⎜ ⎟−⎝ ⎠

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