004: Macroeconomic Theory

004: Macroeconomic Theory

Lecture 16

Mausumi Das

Lecture Notes, DSE

October 28, 2014

Solow Model: Golden Rule & Dynamic Ine¢ ciency

In the last class we have de…ned the ‘golden rule’savings ratios_g and the correspondingsteady state capital-labour ratio k_g in the context of Solow model.

The ‘golden rule’k_g represents one particular steady state (among all the possible steady states - corresponding to various values of the savings ratio s 2 [0,1]) which maximises the steady state level of per capita consumption, given by:

c (s) = f(k (s)) sf(k (s))

= f(k (s)) (n+δ) k (s). [Using the de…nition of k ] Accordingly, the ‘golden rule’capital-labour ratio that maximises the above expression is de…ned by the following equation:

k_g :f⁰(k ) = (n+δ).

Digrammatic Representation of the Golden Rule Steady State:

The point (k_g,c_g) in some sense represents the ‘best’or the ‘most desirable’steady state point (although in the absence of an explicit utility function, such quali…cations remain somewhat vague).

Alternative Digrammatic Representation of the Golden Rule Steady State:

There are many possible steady states to the left and to the right of k_g - associated with various other savings ratios.

Golden Rule & the Concept of ‘Dynamic Ine¢ ciency’

Importantly, all the steady states to the rightof k_g are called

‘dynamically ine¢ cient’steady states.

From any such point one can ‘costlessly’move to the left - to a lower steady state point - and in the process enjoy a higher level of current consumption as well as higher levels of future consumption at all subsequent dates. (How?)

Notice however that the steady states to the leftofk_g arenot

‘dynamically ine¢ cient’. (Why not?)

Cause of ‘Dynamic Ine¢ ciency’in Solow Model

Notice that dynamic ine¢ ciency occurs because people oversave.

This possibility arises in the Solow model because the savings ratio is exogenously given - it is not chosen through households’optimization process.

If the steady state of an economy indeed happens to be dynamically ine¢ cient, then it justi…es an active, interventionist role of the government in the Solow model - even though government cannot a¤ect the long run rate of growth of the economy.

Limitations of the Solow Growth Model:

Even though the Solow model is supposed to be a growth model - it cannot really explain long run growth:

The per capita income does not grow at all in the long run;

The aggregate income grows at an exogenously given raten, which the model does not attempt to explain.

The steady state in the Solow model might be dynamically ine¢ cient.

It is not clear why households will not correct this ine¢ cinency by choosing their savings ratio optimally. But this latter possibility is simply not allowed in the Solow model.

Extensions of Solow Growth Model:

The basic Solow growth model has subsequently been extended to counter some of these critisisms.

The primary challenge is to retain the basic result of the Solow model (namely, existence of a unique and globally stable steady state) while relaxing various restrictive assumptions.

We now look at two such extensions:

1 Solow Model with Technological Progress: This extension allows the per capita income to grow in the long run; developed by Solow himself (Solow (1957)).

2 Neoclassical Growth Model with Optimizing Households: This extension allows the households to choose their consumption/savings behaviour optimally over in…nite horizon; developed independently by Cass (1965) & Koopmans (1965).

Solow Model with Exogenous Technological Progress:

Let us now introduce a productivity-speci…c parameter into our Solovian …rm-speci…c production function:

Yit =F(Kit,Nit,At);F_A >0,

whereF satis…es all the standard Neoclassical properties speci…ed earlier.

The term At captures the state of the technology at time t.Since all

…rms have access to identical technology, this technology-speci…c parameter is the same for all …rms (hence noi-subscript here).

The assumption of identical …rms and CRS implies that the aggregate production function will also take similar form:

Y_t =F(K_t,N_t,A_t);F_A >0.

Technological progress implies that the productivity-speci…c term, A_t, increases in value over time. Thus with the same amount of labour and capital, the economy can now produce greater amount of output.

Di¤erent Types of Technological Progress:

Technological progress can be of three types:

1 Labour Augmenting or Harrod-Neutral: Technical improvement enhances the productivity of labour alone:

Yt =F(Kt,AtNt).

2 Capital Augmenting or Solow-Neutral: Technical improvement enhances the productivity of capital alone:

Yt =F(AtKt,Nt).

3 Capital & Labour Augmenting or Hicks-Neutral: Technical

improvement enhances the productivity of both capital and labour in equal proportion and thus augments total factor productivity:

Y_t =F(A_tK_t,A_tN_t) =A_tF(K_t,N_t).

Notice that with Cobb-Douglas Production Function, all the three notions of technical progress are equivalent. (Prove this.)

Characterization of Technological Progress in terms of Isoquants:

Technological progress implies that the isoquant for some given level of output will shrink inward (same output can now be produced with less inputs). The exact nature of the shift depends on the type of technological progress.

With Harrod-neutral technological progress, along any ray through the origin (say along the 45^O line) the curve becomes steeper. (Prove this.)

Characterization of Technological Progress in terms of Isoquants (Contd.):

With Solow-neutral technological progress, along any ray through the origin (say along the 45^O line), the curve becomes ‡atter. (Prove this.)

Characterization of Technological Progress in terms of Isoquants (Contd.):

With Hicks-neutral technological progress, along any ray through the origin (say along the 45^O line), the slope of the curve remains unchanged. (Prove this.)

Technological Progress and Balanced Growth:

Modern Growth Theory often focuses on Balanced Growth Path: a scenario when every variable in the economy grows at some constant rate (not necessarily equal for all variables).

Recall that steady state is also a special case of balanced growth (when the growth rate is constant at 0).

It can be shown that only Harrod-neutral technological progress can generate balanced growth path. (Proof is available in Acemoglu, pages 59-64. Will not be discussed in class; is not part of the syllabus).

Henceforth, we shall therefore restrict our analysis only to Harrod-neutral technological progress.

(Question: Does the steady state of the original Solow model (without technological progress) satisfy the requirements of a balanced growth path?)

Solow Model with Harrod-Neutral Technological Progress:

Suppose all assumptions of the original Solow model remain unchanged, except that we now have a …rm-speci…c production technology, given by:

Y_it =F(K_it,AtN_it).

The corresponding aggregate production technology is given by Y_t =F(K_t,A_tN_t) F(K_t,Nˆ_t),

where we denote the productivity-augmented labour term: A_tN_t Nˆ_t (e¤ective labour).

Labour productivity increases automatically over time (like ‘manna from heaven’) at an exogenous rate m:

At+1 = (1+m)At.

Each …rm now equates the marginal product of ‘e¤ective labour’with the wage rate per unit of ‘e¤ective labour’(wˆ) and the marginal product of capital with the rental rate of capital (r).

Solow Model with Harrod-Neutral Technological Progress (Contd.):

CRS and identical …rms implies that the …rm-speci…c marginal product and the (social) marginal product for the aggregate economy are the same.

Thus we get the aggregate demand functions for ‘e¤ective labour’

and capital as:

F_Nˆ(Kit,Nˆit) = F_Nˆ(Kt,Nˆt) =wˆt; FK(Kit,Nˆit) = FK(Kt,Nˆt) =rt.

At the beginning of any time period t, the economy starts with a given stock of population (Nt), a given stock of capital (Kt) and a given level of labour productivity (A_t).

The wage rate and rental rate adjust so that that there is full employment of all the factors at every point of time t.

Capital- E¤ective Labour Ratio & Output per unit of E¤ective Labour:

Using the CRS property, we can write:

ˆ y_t Y_t

Nˆt

= ^F(K_t,A_tN_t)

A_tN_t =F K_t

A_tN_t,1 f(k^ˆ_t), whereyˆ_t represents output per unit of e¤ective labour, and kˆ_t

represents the capital-e¤ective labour ratio in the economy at time t.

Using the relationship that F(K_t,Nˆ_t) =N^ˆ_tf(k^ˆ_t), we can easily show that:

F_Nˆ(N^ˆ_t,K_t) = f(k^ˆ_t) k^ˆ_tf⁰(k^ˆ_t) =wˆ_t; F_K(N^ˆ_t,K_t) = f⁰(k^ˆ_t) =r_t.

[Derive these two expressions yourselves].

Properties of the Reduced Form Production Function:

Once again, given the properties of the aggregate production

function, one can derive the following properties of the reduced form production function (in terms of e¤ective labour) f(k^ˆ):

(i) f(0) = 0;

(ii)f⁰(k^ˆ) > 0; f⁰⁰(k^ˆ)<0;

(iii) Lim

kˆ!⁰

f⁰(k^ˆ) = _∞; Lim

kˆ!∞f⁰(k^ˆ) =0.

Finally, using the de…nition that kˆt Kt

AtNt,we can write kˆt+1

Kt+1

A_t+1N_t+1

= ^sF(Kt,AtNt) + (1 δ)Kt

(1+m)A_t(1+n)N_t ) ^kˆt+1= ^sf(k^ˆ_t) + (1 δ)k^ˆ_t

(1+m)(1+n) ^g˜(k^ˆ_t). (1)

Dynamics of Capital-E¤ective Labour Ratio:

Equation (1) represents the basic dynamic equation in the discrete time Solow model with technological progress. Once again we use the phase diagram technique to analyse the dynamic behaviour ofkˆ_t. In plotting the g˜(k^ˆ_t) function, note:

˜

g(0) = ^sf(0) + (1 δ).0 (1+n) =0;

˜

g⁰(k) = ¹

(1+m)(1+n) ^sf

0(k^ˆt) + (1 δ) >0;

˜

g⁰⁰(k) = ¹

(1+m)(1+n)^sf

00(k^ˆ_t)<0.

Dynamics of Capital-E¤ective Labour Ratio (Contd.):

Moreover, as before, Limˆ kt!⁰

˜

g⁰(k^ˆ_t) = ∞;

ˆLim

kt!∞g˜⁰(k^ˆt) = (1 δ)

(1+m) (1+n) <1.

We can now draw the phase diagram for kˆ_t :

Dynamics of Capital-E¤ective Labour Ratio (Contd.):

From the phase diagram we can identify two possible steady states:

(i)kˆ = 0 (Trivial Steady State);

(ii)kˆ = k^ˆ >0 (Non-trivial Steady State).

As before, ignoring the non-trivial steady state, we get a unique non-trivial steady state, kˆ , which is globally asymptotically stable:

starting from any initial capital-e¤ective labour ratiokˆ0 >0, the economy would always move to kˆ in the long run.

Implication:

In the long run, output per unit of e¤ective labour: yˆ_t f(k^ˆ_t)will be constant atf(k^ˆ ).

Long Run Growth Implications of Solow Model with Technological Progress:

How about the long run growth rate of per capita income and aggregate income?

Notice that now per capita income grows at a constant ratem, While aggregate income grows at a constant rate (m+n).

Thus incorporating technical progress in the Solow model indeed allows us to counter the critisism that per capita income does not grow in the long run.

Notice however that both the growth rates are still exogenous; we still do not know what determines mand n.Thus Solow model still does not tell us what determines long run economic growth!!

Long Run Growth of Per Capita Income without Technological Progress?

Exogenous technological progress is not a very satisfactory way to generate long run growth of per capita income in the Solow model.

Without a proper theory to explain this phenomenon, it remains a mere technical exercise.

Can we have long run growth of per capita income in the Solow model - even without exogenous technical progress?

The answer is: yes, but only if you allow some of the Neoclassical properties of the production function to be relaxed.

Recall that the long run constancy of the per capita income in the Solow model arises due to the strong uniqueness and stability property of the steady state - which in turn depends on two key assumptions:

the property of diminishing returns and the Inada Conditions.

The simplest way to generate long run growth of per capita income in the Solow model is to allow for relaxation of one of the Inada

conditions.

An Exercise:

Find an example of a well-known production function which satis…es all the standard Neoclassical properties except the Inada conditions and explore the possibility of perpetual (long run) growth in per capita income in the context of this

production function.