the overlapping generations model (romer chapter 2, part b) by ole hagen jørgensen, [email protected] 4/10...

The OverlappingGenerations Model

(Romer chapter 2, Part B)

By Ole Hagen Jørgensen, [email protected]

4/10 2006

Copyright 2006 © CEBR, Copenhagen - www.cebr.dk

Introduction

I will teach three lectures:

1. One lecture: on the basic OLG model in Romer (2001), chapter 2, part B

2. One lecture: on a recently developed solution method for the OLG model:

3. One lecture: on the Real Business Cycle literature (RBC)

My own research actually applies RBC-techniques for solving OLG models! There will be a presentation in PowerPoint on each subject - Therefore, you will

receive 3 handouts (or download from Blackboard or www.cebr.dk/oj)


Intergenerational issues

Motivation for life-cycle approach to economic dynamics

1. Life-cycle aspects of human behavior are important to study… We model explicitly the different periods of life

2. Distribution of welfare over generations How the choices of one generation can affect the succeeding generation How different exogenous shocks to the economy affects different

generations (demographic shocks, productivity shocks)

3. Intergenerational transfers Purpose: If the market equilibrium allocates consumption unevenly

across generations there may be a scope for redistribution.

How: Taxes and benefits Case: Pensions, education, health


The model

The OLG model will be presented according to the following outline:

1. Description of the economyA. Basic assumptionsB. DemographicsC. Household utilityD. Life-cycle consumptionE. FirmsF. Resources

2. Dynamics of the economyA. Household utility maximizationB. Capital accumulation and Steady State

3. Case study

4. Efficiency and welfareA. Dynamic efficiencyB. Government policy


Description of the economy

A. Basic assumptionsB. DemographicsC. Household utilityD. Life-cycle consumptionE. FirmsF. Resources


Basic assumptions

Overall assumptions about the economy: Time is discrete There is one good, to be consumed or saved/invested The economy “lives” on forever (no last generation) Individuals have finite lifetime (finite horizon) Infinite number of agents Closed economy Perfect competition Absence of externalities No government sector (could be included easily) No uncertainty (perfect foresight)

Of course the economy has more detailed characteristics – we turn to those when relevant…


Demographics

The life-cycle of generations The lifetime is divided into two periods: young and old

When individuals are young they work When individuals are old they are retired

One period therefore amounts to a half lifetime Who are alive at the same time? (vertical box) We want to inspect the behavior of one specific generation (horizontal box)

We trace generation “0” that is born at time t=0 and is old in t+1

We keep track of generation 0 denoted with t...

time period

generation 0 1 2

-1 old (C2t)

0 young (C1t) old (C2t 1)

1 young (C1t 1) old (C2t 2)

2 young(C1t 2)


Demographics

Assumptions on the demographic structure of the economy Could be modeled in great detail

Different sexes Survival probabilities Different skills by different people

Very simple assumption in this model Fixed growth rate of the population over generational periods:

(Equivalent to the continuous time variant )

where Lt is the number of individuals born at time t, where n is the growth rate of the population

Lt 1 nLt 1

L t nLt


Household utility

Individuals derive utility only from consumption in their two periods of life

Two factors determine how individuals decide to divide consumption over time in a risk-free (certain/perfect foresight) environment

1. The consumption “smoothing” motive, captured by the term ρ2. The consumption “fluctuation” motive, captured by the term θ

We discuss each in turn…

Ut UtC1t ,C2t 1

Ut C 1t1

1 1

1 C 2t 1

1

1


Household utility

The consumption “smoothing” motive Individuals generally like to smooth (evenly divide) their consumption

over periods The degree of impatience towards consuming today is captured by the

discount rate, ρ The discount rate of future consumption is generally 1/(1+ρ) so that

household utility can be represented in present value terms as:

Ut C1t 11

C2t 1


Household utility

Consumption “fluctuation” motive Uncertain environment

In an uncertain environment you might be risk averse, and might not be willing to shift consumption very freely over time.

Say, if you decide to smooth consumption 50/50 over your two periods, and if you are uncertain about how your consumption will vary your tend to stick to the safer level in each period.

If you expect the interest rate to increase in the next period, you would get a higher lifetime consumption if you shift some units of consumption. If you are risk averse you would rather stick to the safer levels of consumption – you then miss out on the extra consumption

θ measures the degree of consumption risk aversion Certain environment

In this case there is no risk (perfect foresight) You can still appreciate stable consumption levels in each period, so

the parameter θ then measures the degree to which you like stable and consumption

Again, if you expect the interest rate to increase in the future period, and you prefer consumption not to fluctuate – you will then not take advantage of the potentially higher lifetime consumption.

θ measures the degree of consumption fluctuation aversion


Household utility

The utility function: utility from consumption features intertemporal consumption smoothing motive, ρ features consumption fluctuation motive, θ

We divide by (1-θ) to ensure positive marginal utility in case θ>1 Note: for ρ>0, second period utility is valued less than first period utility We assume that ρ>-1: weight on second period consumption>0. Also,

Ut C 1t1

1 1

1 C 2t 1

1

1

assume 0 , n 0 , 1


Life-cycle consumption

People live for two periods, as adults and as old, and they need to consume in each period

Adults (workers) The adults work, consume, and save:

where

Old (retirees) The elderly are retired, and consume (they do not work, but live of their

savings and interest earnings)

Intertemporal budget constraint (IBC)

C1t wtA t St

C2t 1 1 rt 1 St

C1t C 2t 1

1 r t 1 wtA t

A t 1 gA t 1


Firms

Firms use two factors in production: labor and capital Firms pay the wage rate, wt , to the labor, Lt, supplied by workers Firms rent capital, Kt, from retirees at a rental price of rt

The production function is generally:

Due to CRS we can restate the capital in efficiency units, where:

The wage rate is the marginal product of labor in production:

Return to capital is defined by marginal product of capital in production (assume no capital depreciation, δ=0)

Total Returns:

Yt FKt ,A tLt

FK,AL F KAL

, ALAL

F KAL

, 1 fk

wt fk t k tf k t

rt f k t

R t 1 rt f k t 1

k t K t

A tL t


Resources (economy-wide)

We have seen the consumption budget constraint for the household (IBC) There is also a constraint on consumption for the economy as a whole: society

cannot prioritize over more than is actually produced (closed economy) In each period people save and consume (to save is to invest):

Resource constraint (RC):

In efficiency units:

Ct LtC1t Lt 1C2t

I t Kt 1 Kt

Yt Ct I t

Yt Ct I t

Yt Kt LtC1t Lt 1C2t Kt 1

Y t

A t 1L t 1 K t

A t 1L t 1 K t 1

A t 1L t 1 L t

A t 1L t 1C1t L t 1

A t 1L t 1C2t

yt k t 1 n1 gk t 1 c1t 11 n

c2t

11 n1 g

yt 11 n1 g

k t k t 1 11 n1 g

c1t 11 n1 n1 g

c2t


The model




3. Case study



Dynamics of the economy

A. Household utility maximizationB. Capital accumulation and Steady State


Household utility maximization

Simple intertemporal utility maximization:

C 1t, C 2t

max Ut C 1t1

1 1

1 C 2t 1

1

1

st. C1t 11 r t 1

C2t 1 A twt

UtC1tUt

C2t 1

C 1t

11

C 2t 1 1 C 2t 1

C 1t

IBCC1t IBC

C2t 1

11

1 rt 1

1 rt 1

C2t+1

C1t

Slope of Utility function (MRS)

Slope of IBC = -(1+rt+1)



Optimal consumption allocation depends on 1. The consumption “smoothing” motive, ρ2. The consumption “fluctuation” motive, θ3. The future interest rate, rt+1

The intertemporal optimality condition (Euler equation)

or

This equation is all we need derive through maximization!! The rest of the solution of the model is only based on simple math (insert/reduce)

C 2t 1

C 1t 1 r t 1

1

1/

C2t 1 1 r t 1

1

1/C1t



To find first and second period consumption, C1t and C2t+1, we just insert the Euler equation into IBC:

- reduce

- The savings rate:

insert the optimal C*1t into The Euler equation to derive C*

2t+1:

C1t 11 r t 1

C2t 1 A twt

C1t 11 r t 1

1 r t 1

1

1/C1t A twt

1 1/ 1 r t 1 1/ 1

1 1/ C1t A twt

C1t 1 1/

1 1/ 1 r t 1 1 / A twt

C1t 1 st A twt

strt 1 1 r t 1 1 /

1 1/ 1 r t 1 1 /

C2t 1 stA twt 1 r t 1 1/

1 1/ 1 r t 1 1 / A twt



We need to derive how much people save, because this determines our consumption in the future.

As such, the intertemporal structure of this model evolves around savings (recall that today’s savings is equal tomorrows capital stock)

Savings can simply derived from first period consumption:

or

lets analyze the dynamics of the savings rate for alt. parameter assumptions…

St strt 1 A twt

St wtA t C1t



The dynamics of the savings rate

Substitution effect: relative price change savings rate increases Income effect: purchasing power savings rate decreases

Special case, θ=1:

No consumption impatience, ρ=0:

half of your lifetime income is:1) consumed in period one,2) and saved for period two

C1

C2

-(1+r’)

-(1+r)

C1C’1

C2

C’2

strt 1 1 r t 1 1 /

1 1/ 1 r t 1 1 /

strt 1 st 12

strt 1 st 12



Summary of households intertemporal choice

The Euler equation:

rt+1 ↑: 2. period consumption becomes relatively more preferable

ρ ↑: 1. period consumption becomes relatively more preferable

θ ↓: For a given change in (1+rt+1)/(1+ ρ) Consumption is shifted more freely over periods (larger increase in C2t+1/C1)

Inverse elasticity of intertemporal substitution is constant (CRRS):

(your rate of marginal substitution, MRS, changes more when your consumption “fluctuation” aversion, θ, is low)

C 2t 1

C 1t 1 r t 1

1

1/

MRSC1,C2 MRSC1,C2

C2C1

C2C1

MRSC1,C2



Summary of the household’s intertemporal choice

The household maximization problem boiled down to deriving the savings rate, st(rt+1).

this concludes the section on household utility maximization – we’ll move on to study the economy’s capital accumulation…

strt 1 1 r t 1 1 /

1 1/ 1 r t 1 1 /


Capital accumulation and Steady State

The equation of motion for the capital stock

Next period’s capital stock is the current period’s investments!

Thus: we know the level of next period’s capital stock in this period…

Current Savings Current Investments strt 1 A twtLt

Next Period Capital Stock Kt 1

Kt 1 strt 1 A twtLt



We can now derive the dynamics of the economy(the way the capital stock evolves over time – thus, also all other variables)

Transform the expression for the motion of the capital stock into efficiency units to derive the Balanced Growth Path: (divide by )

Insert the general expressions for rt+1 and wt:

Kt 1 strt 1 A twtLt

K t 1

A t 1L t 1 s tr t 1 A tw t

A t 1L t 1

k t 1 11 n1 g

strt 1 wt

k t 1 11 n1 g

st fk t 1 fk t k tf

k t

A t 1Lt 1



The general case

If we don’t know exactly how factor returns are determined relative to kt we obtain this expression for the evolvement of the economy:

The economy evolves over time, and households want to save/invest to generate the capital stock that will provide them with the highest possible utility. When they reach this capital stock they will keep their savings at the level that will re-generate this particular capital stock in all future periods. Equilibrium condition for the economy to be in its long run equilibrium:

kt+1 = kt

k t 1 11 n1 g

st fk t 1 fk t k tf

k t

k

k 0



The Steady State capital stock can be determined as illustrated: Given well-behaved preferences, and given Cobb-Douglas technology We will discuss the more general case later!...

kt

kt+1

45o

k*



1. Existence Determined by Inada conditions, where the slope of kt+1 is

approaching 0 for lim. kt ∞, and approaching ∞ for lim. kt 0. This is related to the decreasing marginal product of capital through the production function and the requirement that kt+1=kt

2. Uniqueness Also determined by the Inada conditions. Hence, the slope is falling

for k getting larger and larger – therefore the motion of capital can only cross the 45-degree line once.

3. Stability Determined through inspection of the phase diagram:



kt

kt+1

45o

k* kAkB

Convergence to Steady State – two casesA. Initial over-accumulation:

A too high capital stock is caused by too much savings by households (too high for utility to be maximized over intertemporal consumption allocation). If utility could be higher by changing the consumption allocation then the current level of capital is not sustainable and is not compatible with household utility maximization. Consequently, people will start saving less, spending more in the current period – total savings fall – investments fall – the capital stock decreases until savings has reached its optimal level compatible with household preferences and utility maximizing…

B. Initial under-accumulation:Opposite: People will save more to maximize utility – the capital stock rises…



To summarize: If we do not know the relationship between the factor returns, rt+1 and wt and the level of capital, kt, then the savings can shift up and down. As such, the path of the capital stock can also shift up and down.

This is determined through the expression for the path of the economy: the equation of motion of the capital stock

Different versions of the relationship between the capital stock and factor returns can be illustrated graphically…

k t 1 11 n1 g

st fk t 1 fk t k tf

k t



Summary of the section on Capital accumulation:

We have determined the path of the economy in two ways: Analytically

- through the expression for the dynamic evolution of capital, kt+1(kt)

Graphically- through the steady state condition for capital, kt+1=kt

Recall key relationshipswhen the capital stock has been derived, all other variables in the model can be determined!


The model




3. Case study



Case study

The Diamond OLG model

Three assumptions:1. Logarithmic utility:

2. Cobb-Douglas technology:

3. No capital depreciation:

Ut lnC1t 11

lnC2t 1

yt k t

0

wt 1 k t

rt k t 1


Case study

Recall the general expression for the law of motion of capital (and the economy):

For assumed log utility insert the savings rate (for θ=1):

For assumed Cobb-Douglas technology insert the wage rate:

k t 1 11 n1 g

st fk t 1 fk t k tf

k t

strt 1 1 r t 1 1 /

1 1/ 1 r t 1 1 / 12

k t 1 11 n1 g

12

fk t k tfk t

k t 1 11 n1 g

12 1 k t

k t 1 1 2 1 n1 g

k t


Case study

If we should draw curve for this fundamental difference equation then note that and we have again:

Consequently we can derive a Steady State…

1

kt

kt+1

45o

k*


Case study

Comments on existence, uniqueness, and stability

1. Existence:Is the slope positive and decreasing in k?

2. Uniqueness:Since the expression for kt+1 is unchanged for increasing values for kt, the Inada conditions ensure that for low k’s the curve is very steep and for high k’s the curve flattens

3. Stability:The function is based on well-behaved preferences and Cobb-Douglas technology so the analysis of the phase-diagram before also applies here: thus stability

k t 1 1 2 1 n1 g

k t

d2kt 1

dk2 1 1 1 n1 g2

k t 2 0

dkt 1

dk 1

1 n1 g2 k t

1 0


Case study

Steady State Remove subscripts:

This value for the Steady State capital stock can be calibrated and a numerical estimate can be derived

Hence, all other variables can also be derived numerically (since they all ultimately depend on the capital stock)

One could then make experiments with the model: Change parameter values and trace the effects on variables How would workers consumption change? How would retirees consumption change?

lets do some sensitivity analyses…

k t 1 1 2 1 n1 g

k t

k 1 2 1 n1 g

11


Case study

Experiments

Change parameter values and trace the effects on variables How would workers’ consumption change? How would retirees’ consumption change?


Case study

Sensitivity analyses

1. What if you get less impatient with your consumption (ρ )

2. What if the growth rate of the population increases (n )


Case study

Sensitivity analysis: A fall in ρ:

Why: People want to consume less today and more tomorrow – this increases savings – increases the long run capital stock per effective worker – increases wages – decreases return to capital!

k 1 2 1 n1 g

11 k

kt

kt+1

45o

k* k’*


Case study

Sensitivity analysis: A rise in n:

Why: There are now more workers to share the capital – capital/labor ratio falls – wages fall – returns increase – savings fall – the long run level of capital per effective worker falls!

kt

kt+1

45o

k*k’*

k 1 2 1 n1 g

11 k


The model




3. Case study



Efficiency and welfare

A. Dynamic efficiencyB. Government policy



We have determined the equilibrium capital stock, but we have to ask two questions:

1. Is capital at its efficient level(we need to derive the Golden Rule capital stock)

2. What can be done to achieve the optimal level of capital(we need to consider policy)


Dynamic efficiency

Is the level of long run capital, k*, optimal? How to determined this: The capital stock should be at a level consistent with maximum utility Utility is maximized over lifetime consumption, so when lifetime

consumption is maximized – so must welfare! When is consumption maximized? Since everything in the model

depends on the level of kt then which level must kt have?

We have available resources for allocation to worker’s and retirees’ consumption (i.e. RC). Assume no productivity growth in this example:

Maximize consumption w.r.t. kt in Steady State, where:

yt k t 1 n1 gk t 1 c1t 11 n

c2t

fk k 1 nk c1 11 n

c2

Ct L tC1t L t 1C2t C t

A tL t L tC 1t

A tL t L t 1C 2t

A tL t ct c1 1

1 nc2

c fk nk


Dynamic efficiency

Maximize consumption w.r.t. k:

The Golden Rule capital stock, kGR, that maximizes utility can then be derived if we now the expression for f(kGR)

Recall that:

Golden Rule capital stock:

dcdk

f k n 0

f kGR n

Y Kt A tLt 1 y k t

kGR n

11

kGR 1 n

c

k

nkf(k)

y

kGR

IcMAX


Dynamic efficiency

Key issue: is the market solution for k, which is k*, equal to kGR? If yes, then welfare is maximized automatically! If no, then is the allocation at least Pareto efficient?

Compare k* to kGR: Is it possible that k* ≠ kGR. Check for k* > kGR: Recall that for log-utility, zero capital depreciation, and g=0:

It is definitely possible that . If is the allocation then efficient or inefficient (graphically)k kGR

k kGR

k kGR

1 1 n2

11

n

11

1

1

1

n1 n

1

12 1

for 0 1 , n 0 , 0 , g 0 , 1


Dynamic efficiency

1. If the Golden Rule capital stock, kGR, is lower than the market determined capital stock, k*, is k* then an efficient allocation? YES: because if current workers were somehow forced to save more (to

increase the capital stock) then they would have to give up current consumption in order for future generations to better of (welfare function should value future generations’ utility higher than current generations’)

2. If the Golden Rule capital stock, kGR, is higher than the market determined capital stock, k*, is k* then an efficient allocation? NO: everybody gains…

k

c

k*kGR

nk

f(k)

y

k*

Lost consumption potential

Lost consumption potential I

cMAX


Dynamic efficiency

Think of dynamic in-efficiency in two ways:

1. Pareto efficiency: If we are in an equilibrium where the government can redistribute from young to old so both generations, and all succeeding generations, are better of – then the current equilibrium could clearly not be Pareto efficient!

2. Savings vs. transfers: If the real interest rate is lower than the population growth rate – then it would be more efficient to take 1 unit of consumption from the current young and transfer the 1 unit to the old. Since the current old generation is (1+n) times smaller than the current young the 1 unit from the young can actually be divided to the young so they each get (1+n) units. If this goes on for ever through all generations, that is a way for young to give up one unit of consumption and in turn get (1+n) units in old age. The return on savings would be (1+r), so if r<n, it would actually be more efficient (all generations would get more lifetime utility) to permanently transfer units from young to old instead of saving through capital investments. The government could facilitate this transfer, and thus bring savings down to the Golden Rule level of savings.


Dynamic efficiency

Government policy: redistribution

1. Government transfers x units of consumption from workers

2. Retirees receive x units, but the size of the old generations is smaller by (1+n), so the will receive (1+n)*x units

3. The level of x is determined by the Government

4. This “arrangement” will (must) go on forever…

5. It is clear that workers save less, so k* falls over time, and the Government has fixed x so eventually: k*=kGR

Time

Generation 0 1 2 3 4

-1 C2t

0 C1t C2t+1

1 C1t+1 C2t+2

2 C1t+2 C2t+3

3 C1t+3 C2t+4

-x

(1+n)*x

-x

(1+n)*x

-x

(1+n)*x

-x

(1+n)*x


Dynamic efficiency

Two sources of financing of consumption in old age:1. Savings2. Transfers

We know that savings will yield a return of: (1+r) We know that transferring x=1 units of income will yield a return of (1+n) We know that if there is dynamic inefficiency we have r<n We know that in this situation transfers will yield higher return than savings

Consequently: If we have dynamic inefficiency, transfers will be more efficient than savings – thus the government can improve on the decentralized equilibrium!!



Government participation can be incorporated in various ways? Through the Government budget (see Romer, 2001:section 2.12) Through a pension system, e.g. Pay-As-You-Go

(I will deal with a pension system in my next lecture)

Bottom-line: We can incorporate several different mechanisms that is able to handle intergenerational transfers

More one this in my next lecture!!


The model




3. Case study



Concluding remarks

Intergenerational aspects A model of life-cycle optimization Key dynamic variable of the model is the capital stock per effective worker Importance of population dynamics Importance of productivity

Important propertiesNote that for the economy to be in Steady State: hence:

What is the growth rate for of variables?

k

k 0 y

y 0

Kt k tA tLt

lnKt lnk t lnA t lnLt

lnK t

t ln kt t lnA t

t ln L t

t

K

K k

k A

A L

L

K

K g n

lnX t

t lnX t

X t X t

t X

X


Concluding remarks

For national income:

For income per worker, relative to productivity growth (not per effective worker)

Y

Y g n

Yt ytA tLt

Y

Y y

y A

A L

L

Y

Y k

k A

A L

L

Y

Y L

L 0 A

A

Y

Y L

L A

A

yy

per worker g y

yper worker

yy

per effective worker 0


Concluding remarks

Potential extensions of the model (NEXT LECTURE):1. Government2. Pensions (PAYG)3. Endogenous retirement4. Endogenous labor supply5. Endogenous population (fertility) growth 6. Bequests7. Survival rates

Potential extensions of the solution method (NEXT LECTURE)


The next lecture

An analytical solution method for transitional dynamics of the OLG model

I will show you a recently developed method to solve for the transitional dynamic of the model to a new steady state:

Changing parameters in the model above only gives the new values in the new steady state, while the new method derives the dynamics of the transition towards the new steady state

This solution method is on the frontier of research on OLG models and has never been taught in a lecture before…


The next lecture

t

k

k1

t=0 t=j

k2

Shock to nt (negative)

Which path does the economy follow to the new steady state?

Key issue: A different capital stock for different generations different wages and interest rates a shock produces unequal intergenerational risk sharing

Transitional dynamics

the overlapping generations model (romer chapter 2, part b) by ole hagen jørgensen, [email protected] 4/10...

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