004: Macroeconomic Theory

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004: Macroeconomic Theory

Lecture 20

Mausumi Das

Lecture Notes, DSE

November 5, 2014

Das (Lecture Notes, DSE) Macro November 5, 2014 1 / 16

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Samuelson-Diamond Overlapping Generations Model

We have so far analysed optimizing savings behaviour by the households in the context of an ini…nite horizon R-C-K framework.

We now turn to an alternative framework, where agents optimize over a …nite time horizon.

This framework was …rst developed by Samuelson (1958) in the context of an exchange economy, which was later extended to a production economy by Peter Diamond (1965).

Each agent now lives exactly for 2 periods. For convenience we shall denote these two periods of an agent’s life time by ‘youth’and

‘old-age’respectively.

The agent works only in the …rst period of his life (when young) and is retired in the second period (when old).

Thus he has to make provisions for his old-age consumption from his

…rst period wage income itself (through savings).

The agent optimally decides about his current and future consumption by maximizing his life-time utility (to be speci…ed).

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OLG Model: Production Side Story

Once again, the production side story in the OLG model is identical to that of Solow.

Thus the economy starts with a given stock of capital (K_t) and a given stock of labour force (N_t) at timet.

Notice that since people do not work during their oldage, the current labour force consist only of the current youth.

We also assume that all …rms have access to an identical production technology - which satis…es all standard neoclassical properties.

The …rm-speci…c production functions can be aggregated to generate an aggregate production function such that :

Y_t =F(K_t,N_t).

At every point of time the market clearing wage rate and the rental rate of capital are given by:

wt =FN(Kt,Nt); rt =FK(Kt,Nt).

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OLG Model: Household Side Story

There are H households or dynasties in the economy. In every household, at any point of timet, there are two cohorts of agents - those who are born in period t (‘generation t’- who are the current young) and those who were born in the previous period (‘generation t 1’- who are the current old): hence the name ‘overlapping generations’.

Thus at any point of timet, total population consists of two successive generations of people:

Lt =Nt+Nt 1.

(Notice that although total population is Lt,total labour force at time t in only N_t.)

We shall assume that population in successive generations grows at constant rate n:

N_t+1 = (1+n)N_t.

Hence total population in the economy (L_t) also grows at the same constant rate n.

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Life Cycle of a Representative Member of Generation t:

All agents withina generation are identical. So we can talk in terms of a representative agent belonging to ‘generationt’.

The agent is born at the beginning of period t with an endowment of one unit of labour.

All agents in this model are sel…sh - they care only about their own consumption/utility and not about their childrens’utility. Hence they do not leave any bequest.

No bequest implies that the young agent has only labour endowment and no capital endowment.

The young agent in periodt supplies his labour inelastically to the labour market in period t to earn a wage incomew_t.

Out of this wage income, the agent consumes a part and saves the rest - which becomes his capital stockin the next period and allows him to earn a rental income in the next period.

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Representative Agent’s Utility Function:

Notice that since the agent would not be working in the next period, the savings and the consequent ownership of capital is the only source of income for him in the next period.

Thus an agent is a worker in the …rst period of his life and becomes a capitalist (capital-owner) in the second period of his life.

Out of this wage income, the agent consumes a part and saves the rest - which becomes his capital stockin the next period and allows him to earn a rental income in the next period.

The young agent in periodt optimally decides on his current

consumption (c_t¹) and current savings (s_t) (or equivalently, his current and future consumption- c_t¹ andc_t²₊₁) so as to maximise his lifetime utility:

U(c_t¹,c_t²₊₁) u(c_t¹) +βu(c_t²₊₁); u⁰ >0; u⁰⁰ <0; 0< β<1. (1)

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Representative Agent’s Budget Constraints:

The …rst period budget constraint of the agent:

c_t¹+st =wt.

The second period budget constraint of the agent:

c_t²₊₁ = (1+r_t^e₊₁ δ)s_t.

Combining, we get the life-time budget constraint of the agent as:

c_t¹+ ^c

t2+1

(1+r_t^e₊₁ δ) =w_t. (2) The agent decides on his optimal consumption in the two periods by maximising (1) subject to (2).

Since the agent is taking his savings decision in the …rst period, when second period’s market interest rate is not yet known, he must optimize on the basis of some expected value of the r_t+1.

However, we shall assume that agents have perfect foresight/rational expectations such that r_t^e₊₁ =rt+1 for all t.

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Representative Agent’s Optimal Consumption & Savings:

From the FONC of the optimization exercise:

u⁰(c_t¹)

u⁰(c_t²₊₁) = β(1+rt+1 δ). (3) From the FONC and the life-time budget constraint, we can derive the optimal solutions as:

c_t¹ = ψ(wt,rt+1); c_t²₊₁ = η(wt,rt+1). Corresponding optimal savings:

st =wt ψ(wt,rt+1) φ(wt,rt+1).

Before we proceed further, it is useful to note the signs of the partial derivativess_w ands_r.

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Representative Agent’s Optimal Consumption & Savings (Contd.):

Notice that sw

∂φ(w_t,r_t₊₁)

∂wt

=1 ∂ψ(w_t,r_t₊₁)

∂wt

=1 ∂c_t¹

∂wt

.

Under the assumption that both c_t¹ andc_{t_1}² are normal goods, a unit increase in the wage rate w_t ceteris paribusmust increase both c_t¹ andc_t²₊₁. Thus 0< ^∂c

t1

∂w_t,∂c_t²₊₁

∂w_t <1.

This in turn implies that

0<s_w <1.

The sign ofsr however is ambiguous. Notice that sr

∂φ(w_t,r_t₊₁)

∂rt+1

= ^∂ψ(w_t,r_t₊₁)

∂rt+1

= ^∂c

t1

∂rt+1

.

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Representative Agent’s Optimal Consumption & Savings (Contd.):

Recall that 1

1+r_t+1 δ is the relative price of c_t²₊₁ in terms of c_t¹. Thus a unit increase inrt+1 ceteris paribus decreases the relative price ofc_t²₊₁.As a result

c_t¹ should decrease due to the substitution e¤ect, while c_t¹ should increase due to the income e¤ect of a price change.

The sign of ∂c_t¹

∂rt+1

depends on which e¤ect dominates.

This in turn implies that

s_r R ⁰

according as, substitution e¤ect R income e¤ect.

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Aggregate Consumption & Savings:

Let us now turn our attention to the aggregate economy.

Recall that in every period the total output is distributed as wage income and capital income:

Yt =wtNt+rtKt.

The entire wage income goes to the current young generation. Each of them saves a part of the wage income (st) and consume the rest.

Thus,

w_tN_t =c_t¹N_t+s_tN_t.

On the other hand, the entire interest income goes to the current old generation. Each of them consume not only the interest earnings but the left over capital stock as well. Thus,

c_t²Nt 1 =rtKt + (1 δ)Kt.

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Aggregate Consumption & Savings (Contd.):

Thus aggregate consumption in this economy at timet : C_t = c_t¹N_t +c_t²N_t ₁

= [wtNt stNt] + [rtKt+ (1 δ)Kt]

= wtNt+rtKt [stNt (1 δ)Kt]

= Y_t S_t

Notice that aggregate savings S_t has two components:

1 The positive savings by the young (stNt);

2 The negative savings by the old ( (1 δ)Kt).

As before, all savings are automatically invested, which augments next period’s capital stock:

S_t = I_t K_t+1 (1 δ)K_t ) ^K^t⁺¹ =stNt.

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Dynamics of Capital-Labour Ratio:

Since we know that labour force in this economy is growing at the raten,i.e.,N_t+1 = (1+n)N_t, we can derive the dynamic of capital-labour ratio (k_t) as:

k_t₊₁ = ^s^t

(1+n) = ^φ(w_t,r_t₊₁)

(1+n) ^. ⁽⁴⁾ Again, from the production side of the story, we already know that

w_t = f(k_t) k_tf⁰(k_t); r_t₊₁ = f⁰(k_t₊₁).

Thus we can write (4) as:

k_t₊₁ =Φ(k_t,k_t₊₁). (5) Equation (5) is the basic dynamic equation of the OLG model, which implicitly de…nes k_t₊₁ as a function of k_t.Givenk₀,we should be able to trace the evolution of the capital-labour ratio over time.

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Dynamics of Capital-Labour Ratio (Contd.):

Notice however thatΦ(kt,kt+1)is a nonlinear function of kt and k_t+1.So we have to use the phase diagram technique.

In drawing the phase diagram, …rst note that the slope of the phase line can be determined by total di¤entiating (4):

(1+n)dk_t₊₁ = s_wdw_t

dk_t dk_t +s_rdr_t₊₁ dk_t₊₁dk_t₊₁ i.e., dkt+1

dk_t = ^s^w[ ktf⁰⁰(kt)]

(1+n) s_rf⁰⁰(k_t+1)^.

It is immediately obvious that the slope may not be positive (unless sr >0).

Even when the slope is positive, the curvature is not necessarily concave - since it would involve the third derivative of the utility function and thef(k)function - whose signs are not known.

Hence anything is possible - we may have situations of no steady state; unique unstable steady state; multiple steady states (some stable, some unstable) and so on.

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Stability & Dynamic E¢ ciency in the OLG Model:

In other words, the nice result of the Solow model of a unique and globally stable steady state is no longer gauranteed -despite the production function satifying all the standard neoclassical properties - including diminishing returns and the Inada conditions!

What about dynamic e¢ ciency?

Even that is not gauranteed any more!

However, to discuss the issue we …rst have to de…ne the ‘golden rule’

for the OLG model - which we take up in the next class.

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References:

Reference for the R-C-K Model:

D. Acemoglu: Introduction to Modern Economic Growth, Pages 174-175; 214-221.

Reference for the OLG Model:

D. Acemoglu: Introduction to Modern Economic Growth, Chapter 9, Sections 9.2,9.3, 9.4 (Pages 329-339).