004: Macroeconomic Theory

004: Macroeconomic Theory

Lecture 21

Mausumi Das

Lecture Notes, DSE

November 10, 2014

Samuelson-Diamond Overlapping Generations Model (Contd.)

In the last class we had developed the Samuelson-Diamond Overlapping Generations model.

We have seen that the Solow result of a unique and globally stable steady state - which was retained in the R-C-K framework - may no longer hold in the OLG framework.

In fact, a non-trivial steady state may not exist. Even if it exists, it may not be unique or stable.

And this divergence from the Solow model arises - despite the production function satifying all the standard neoclassical properties - including diminishing returns and the Inada conditions.

Moreover, even when a unique and globally stable steady state exits in the OLG model, it may not be dynamically e¢ cient (as we shall see soon).

Dynamics of the OLG Model:

In the last class we had derived the basic dynamic equation of the OLG model as:

kt+1 = ^st

(1+n) = ^φ(w(kt),r(kt+1)) (1+n) ^, such that

dkt+1

dk_t = ^sw[ ktf⁰⁰(kt)]

(1+n) s_rf⁰⁰(k_t+1)^.

Of course we cannot analyse the dynamics any further unless we assume some speci…c functional forms.

In this class we are going to look at the dynamics of the OLG model with speci…c functional forms.

In the process we shall also show that the steady state in an OLG model may violate dynamic e¢ ciency.

Golden Rule & Dynamic E¢ ciency in the OLG Model:

Before we turn to the speci…c functional forms, let us …rst de…ne the golden rule in the context of the OLG model.

As before let us de…ne the ‘golden rule’as that particular steady state which maximises the steady state level of per capita (average)

consumption.

Notice however that now there are two sets of people at any point of time t - current young (N_t) and current old (N_{t 1}).

Hence per capita (average) consumption at any point of time t would be de…ned as:

c_t = ^c

t1N_t+c_t²N_{t 1} Lt

= ^c

t1N_t +c_t²N_{t 1} Nt +Nt 1

= (1+n)c_t¹+c_t² 1+ (1+n) ^. In other words, the per capita consumption in periodt is the weighted average of the consumption of the current young and that of the current old.

Characterization of the Steady State in OLG Model:

So what would be the steady state value of per capita consumption?

Recall that the basic dynamic equation is the OLG model is given by:

kt+1= ^st

(1+n) = ^φ(w(kt),r(kt+1)) (1+n)

Accordingly, the steady state(s) of the OLG model is de…ned as:

k = ^s

(1+n) = ^φ(w(k ),r(k )) (1+n)

) ^s = (1+n)k (1) On the other hand, from the optimal solutions of c¹ andc², we know that at steady state:

c¹ = ψ(w(k ),r(k )) =w(k ) s ; (2) c² = η(w(k ),r(k )) = [1+r(k ) δ]s . (3)

Steady State Per Capita Consumption in the OLG Model:

Hence from (1), (2) & (3) steady state per capita consumption:

c = (1+n) c¹ + c² 1+ (1+n)

= (1+n) [w(k ) s ] + [(1+r(k ) δ)]s 1+ (1+n)

= (1+n)w(k ) + [r(k ) δ n)]s 1+ (1+n)

= (1+n) [f^w(k ) +r(k )k g (n+δ)k ] 1+ (1+n)

= ¹+n

1+ (1+n)[f(k ) (n+δ)k ].

Maximizing c with respect tok ,we would still get the golden rule condition as:

k_g :f⁰(k ) = (n+δ)

Alternative De…nition of Golden Rule:

An alternative de…nition of the ‘golden rule’in the context of the OLG model can be as follows: it is that particular steady state which maximises the steady state level of utility of any agent.

When the economy is at a steady state bothw and r become

constants. Thus for any agent belonging to any generation (t 1 ort or t+1),the life-time consumption pro…le look exactly the same:

c_t¹ ₁ = c_t¹=c_t¹₊₁ =ψ(w(k ),r(k )) c¹ ; c_t² = c_t²₊₁ =c_t²₊₂ =η(w(k ),r(k )) c² . Hence steady state utility of any generation is given by:

U( c¹ , c² ) u( c¹ ) +βu( c² ).

It is easy to check that the k that maximisesU( c¹ , c² )is still given by:

kg :f⁰(k ) = (n+δ).

Thus the two de…nitions of the golden rule are equivalent. (Verify.)

OLG Model with Speci…c Functional Forms:

Let us now assume speci…c functional forms.

Let

U(c_t¹,c_t²₊₁) logc_t¹+βlogc_t²₊₁; f(k_t) = (k_t)^α; 0<α<1.

Also let the rate of depreciation be 100%, i.e., δ=1.

Thus the life-time budget constraint of the representative agent of generation t is given by:

c_t¹+^c

t2+1

r_t+1 =w_t. The corresponding FONC:

c_t²₊₁

c_t¹ = βr_t+1. (4)

OLG Model: Speci…c Functional Forms (Contd.)

From the FONC and the life-time budget constraint, we can derive the optimal solutions as:

c_t¹ = ¹ 1+βwt; s_t = ^β

1+βw_t; c_t²₊₁ = βrt+1

1

1+βwt . Corresponding dynamic equation:

k_t+1 = ^st

(1+n) = ¹ 1+n

β 1+βw_t . Finally, given the production function,

w_t = (1 α) (k_t)^α

OLG Model: Dynamics with Speci…c Functional Forms

Thus the dynamics of this speci…c case is simple:

k_t+1 = ¹ 1+n

β

1+β (1 α) (k_t)^α.

It is easy to see that the above phase line will generate a unique non-trivial steady state which is globally stable.

The corresponding steady state solution is de…ned as:

k = ¹

1+n

β

1+β (1 α) (k )^α

i.e.,k = ¹

1+n

β

1+β (1 α)

1 1 α

. Is this steady state dynamically e¢ cient? Not necessarily!

OLG Model: Dynamic Ine¢ ciency

Notice that given the speci…c functional form, the ‘golden rule’value of the capital-labour ratio can be derived as:

k_g : f⁰(k ) = (n+δ) i.e.,k_g : α(k )^{α 1} =1+n i.e.,k_g = ^α

1+n

1 1 α

.

It is easy to verify that the steady state under this speci…c example will be dynamically ine¢ cient whenever

β

1+β > ^α 1 α.

Example: β= ¹

2;α= ¹ 5.

Reason for Dynamic Ine¢ ciency in the OLG Model:

Thus we see that dynamic ine¢ ciency may arise in the OLG model - despite optimizing savings behaviour by households.

This apparently paradoxical result stems from the fact that agents are

‘sel…sh’in the OLG model; they do not care for their children’s utility/consumption.

Thus when they optimise they equate the MRS with (the discounted value of) the actual future return (r_t+1+1 δ):

u⁰(c_t¹)

u⁰(c_t²₊₁) = β(r_t+1+1 δ).

On the other hand in the R-C-K model an altruisticagent equates the MRS with (the discounted value of) the future return net of population growth rt+1+1 δ

1+n :

u⁰(c_t)

u⁰(c_t+1) = β 1

1+n (r_t+1+1 δ).

Reason for Dynamic Ine¢ ciency in OLG Model (Contd.)

Now notice that u⁰(c_t)

u⁰(c_t+1) T¹)^u0(c_t)T^u0(c_t+1))^ct S^ct+1. This implies that in either framework,

c_t <c_t+1 i.e, consumption would rise over time if the corresponding (future) return is greater than the subjective cost 1

β ;

ct >ct+1 i.e, consumption would fall over time if the corresponding (future) return is less than the subjective cost 1

β ;

ct >ct+1 i.e, consumption would remain unchanged over time if the corresponding (future) return is exactly equal to the subjective cost

1 β .

Reason for Dynamic Ine¢ ciency in OLG Model (Contd.)

But how does consumption increase (decrease) over time? It increases (decreases) through the act of saving (dis-saving).

The earlier statement can therefore be rephrased as follows:

In either framework,

households will go on saving if the corresponding (future) return is greater than the subjective cost 1

β ;

households will go on dis-saving if the corresponding (future) return is less than the subjective cost 1

β ;

households will neither save nor dissave (and maintain a constant consumption) if the corresponding (future) return is exactly equal to the subjective cost 1

β .

Since the perceived return for the households is greater under the OLG framework than under the R-C-K framework, households have a tendency to oversave under the OLG model compared to the R-C-K model.

Reason for Dynamic Ine¢ ciency in OLG Model (Contd.)

This tendency to oversave persists even when the economy has moved into a dynamically ine¢ cient region (i.e.,rt+1 <δ+n ).

In particular, consider a scenario where 1

β <(r_t+1+1 δ)<(1+n).

(Notice that in this scenario, rt+1 <δ+n).

Now under the R-C-K model, households will surely reduce saving (or will dis-save), because their percieved return is now less than unity where as the subjective cost 1

β is greater than unity.

But there is no reason why under the OLG model households would stop saving, because even though the gross marginal product (r_t+1+1 δ) has fallen below (1+n)- it is still greater than the subjective cost 1

β .

Dynamic Ine¢ ciency & Scope for Government Intervention in the OLG Model:

Since under the OLG framework, the steady state of the decentralized market economy may be dynamically ine¢ cient (despite rational expectations on part of the agents), this again justi…es a role of government in improving e¢ ciency.

Thus the conclusions of the OLG model are diametrically opposite of that of the R-C-K model - even though both are based on strictly neoclassical production function and optimizing agents.

The di¤erence arises primarily due to the absence of parental altruism in the OLG framework.

It can be shown that if we introduce parental altruism in the OLG model (by incorporating a bequest term that each parent leaves to his child at the end of his life time), then the OLG framework will be very similar to the R-C-K framework. .