202: Dynamic Macroeconomic Theory

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202: Dynamic Macroeconomic Theory

Inequality, Political Economy & Growth: Alesina-Rodrik (1994)

Mausumi Das

Lecture Notes, DSE

April 8-9 & 15, 2015

Das (Lecture Notes, DSE) Dynamic Macro April 8-9 & 15, 2015 1 / 19

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Inequality, Political Economy & Growth

Alesina-Rodrik ("Distributive Politics and Economic Growth", QJE, 1994) provides an outline of a model which analyses how inequality may hamper growth - working through the political economy channel.

Alesina & Rodrik use an endogenous growth framework provided by Barro ("Government Spending in a Model of Endogenous Growth", JPE,1990) to discuss the issue of taxation and political economy.

So before we go to the Alesina-Rodrik model, let us discuss the Barro model …rst.

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Digression: Barro (1990) - A Prelude to Alesina-Rodrik

The Barro model is a pure endogenous growth modelwithout any distributional considerartion. (In fact all households/…rms are identical here. So the question of inequality does not arise.) It is very similar to the simpli…ed Romer model presented earlier, except that now the engine of growth is di¤erent.

In Romer, growth occured through knowldege accumulation which, due to externality, enhanced aggregate productivity.

In Barro growth occurs due to infrastructural investment made by the government (in terms of improved roads, improved law and order etc.), which again enhances aggregate productivity.

However these infrastructural inputs are …nanced by taxing the household/…rm income.

This mode of …nancing the infrastructural inputs creates an externality across households/…rms which (as we shall see) is very similar to Romer.

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Barro Model of Endogenous Growth: Economic Structure

Economic structure is almost identical to Romer.

A single …nal commdity is produced - which can be either consumed or invested in physical capital.

The economy has S identical …rms and H identical households.

Each household consists on a single in…nitely livedmember.

There is no population growth, which implies that the size of labour force in every period:N =H.

Also,S =H=N so that there is no di¤erence between the per capita, per household and per …rm value of a variable and these also coincide with the economy-wide average.

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Barro Model: Production Side Story

Each …rm is endowed with a production technology which uses labour (l), physical capital (k) and a governement- provided infrastructural input (g).

We shall assume a speci…c functional form given by:

F(k_i,g,l_i) = (k_i)^α(gl_i)¹ ^α, 0<α<1.

Since g is provided by the government, the …rms treat this as

exogenous and choose the optimal level of the …rm speci…c inputs (ki

andl_i) so as to maximise pro…t.

Notice that just like the Romer model, the production function F is concave and CRS in the …rm speci…c inputs,k_i andl_i, but actually exhibits IRS when we consider all the three factors: k,l andg . But the …rms do not internalise the increasing returns; in their perception the production function is CRS.

This allows perfect competition to prevail in the market economy.

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Barro Model: Production Side Story (Contd.)

Each …rm take the market wage rate (wt) and the rental rate for capital (r_t) as given and employ capital and labour such that

wt = (1 α)g_t¹ ^α(kit)^α(lit) ^α r_t = αg_t¹ ^α(k_it)^α ¹(l_it)¹ ^α

Once again we shall assume that the total labour input is constant and is equally distributed over all the …rms and normalized to unity, such that:

lit =^¯l =1.

Then the …rm-speci…c output is given by

F(k,g,1) = (k)^α(g)¹ ^α f(k;g)

(Since all …rms are identical, we shall also ignore the …rm-speci…c subscript (i) from now on.)

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Barro Model: Production Side Story (Contd.)

Accordingly,

wt = (1 α)g_t¹ ^α(kt)^α r_t = αg_t¹ ^α(k_t)^α ¹

Notice that …rms are competitive who earn zero pro…ts. Thus the entire output of a …rm is distributed to a household in the form of wage and rental income.

Thus income of a household:

yt =wt +rtkt

Plugging back the actual values ofwt and rt, one can see that a household’s income and a …rm’s output are identical (which is consistent with the assumption that …rms are competitive and earn zero pro…ts. ):

wt +rtkt =f(kt,gt) =g_t¹ ^α(kt)^α.

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Barro Model: Financing of the Infrastructural Input

The government provides the infrastructural input g upfront(before the production takes place) and imposes a tax post-production(in the same period) to recover the cost that it had incurred in providing the input.

The government can either choose the level ofg, and set the tax rate accordingly; or it can choose the tax rate, and determine the level of g to be provided residually. We shall assume the latter.

We shall assume that the government …nances g_t by taxing the household’s income (post-production)y_t at some pre-determined tax rateτ.

Since houshold’s income and …rm’s output are identical, equivalently we can assume the the government imposes a production tax on the

…rms (post production). However we shall stick to the household income taxation interpretation.

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Barro Model: Financing of the Infrastructural Input (Contd.)

It therefore follows that

g_t = τy_t =τg_t¹ ^α(k_t)^α ) ^g_k^t

t α

= τ

) ^g^t = (τ)¹^αk_t.

Notice that this relationship between gt and kt is known to the ominiscientsocial planner, but not to the atomistic…rms (or households).

Thus for …rms the production function is still given by F(k_t,g_t,1) =g_t¹ ^α(k_t)^α f(k_t,g_t).

But for the social planner the per capita production function looks as follows

(τ)¹^αk_t ¹ ^α(k_t)^α = (τ)¹^α^αk_t φ(k_t).

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Barro Model: Financing of the Infrastructural Input (Contd.)

Accordingly, private (market) return to capital (under perfect foresight) is given by:

r_t = αg_t¹ ^α(k_t)^α ¹= α(τ)¹^α^α And the corresponding social return is given by:

φ⁰(k_t) = (τ)¹^α^α

Notice that once again the social return to capital formation is higher that the private (market) return.

Again this is due to externality, although, unlike Romer, here the externality is not - it works through the taxation scheme.

An increase in capital stock generates more output - which in turn generates more tax revenue (through the proprtional income tax) - which creates more infrastructure - which in turn generates even more output.

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Barro Model: Household Preferences

Preferences of the single-member in…nitely-lived representative household is denoted by the following life-time utility function:

U₀ =

Z∞ t=0

log(c_t)exp ^ρtdt; ρ>0.

It is easy to very that the log speci…cation of the utility function satis…es all the standard properties, namely,

u⁰(c)>0; u⁰⁰(c)<0; lim

c!⁰u⁰(c) =∞; lim

c!∞u⁰(c) =0.

In the market economy each household maximises the above utility function subject to its bugdet constraint.

The social planner in benevolent; so he maximises the same utility function, but his budget constarint would be di¤erent than that of the household.

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Romer Model: Social Planner’s Problem

The dynamic optimization problem of the social planner is given by:

Z∞ t=0

log(ct)exp ^ρtdt (I)

subject to its post-tax(per capita) budget constraint (after deducting the cost of provision of g from the …nal output):

k˙ = (1 τ) (τ)¹^α^αk_t c_t; k(t)=^0, ^k⁰ ^given.

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Romer Model: Problem of the Market Economy

The corresponding problem for a household operating in the market economy is given by:

Z∞ t=0

log(ct)exp ^ρtdt (II)

subject to its post-taxbudget constraint:

k˙ = (1 τ) [w_t +r_tk_t] c_t; k(t)=^0, ^k⁰ ^given.

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Solution to the Social Planner’s Problem:Characterization of the Optimal Path

It can be easily shown from the FONCs that the dynamic equations for the Social Planner would be given by:

˙ c

c = (1 τ) (τ)¹^α^α ρ; (1) k˙

k = (1 τ) (τ)¹^α^α ^c^t

k_t (2)

These two equations along with the TVC now characterise the optimal path for the social planner.

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Social Planner’s Problem: Characterization of the Optimal Path (Contd.)

Once again we shall focus on the balanced growth path: the path where all variable in the economy grow at constant rates.

Is it easy to argue (as we did for the simpli…ed Romer model) the along a balanced growth path ct andkt would grow at the same rate.

Hence for this planned economy, the balanced growth path is characterized by:

˙ c c = ^k^˙

k = (1 τ) (τ)¹^α^α ρ.

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Corresponding Problem for the Competitive Market Economy:

Recall that the only di¤erence between the social planner problem and the household’problem in the market economy is in terms of the per capita production function: The social planner knows that the per capita output is givenφ(k)while the household/…rm reads the per capita output asf(k,g).

It can easily shown from the FONCs that the dynamic equations for the market economy would be given by:

˙ c

c = (1 τ)r_t ρ; (3)

k˙

k = (1 τ)^w^t+rtkt

k_t

ct

k_t (4)

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The Competitive Market Economy: Characterization of the Optimal Path

Assuming perfect foresight on the part of the households, and thereby substituting g_t = (τ)¹^αk_t as well as the actual values ofw_t andr_t in the equations above:

˙ c

c =α(1 τ) (τ)¹^α^α ρ; (5) k˙

k = (1 τ) (τ)¹^α^α ^c^t kt

(6) These two equations along with the TVC now characterise the optimal path for the competitive market economy (under perfect foresight).

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Competitive Market Economy: The Optimal Path (Contd.)

Once again one can …nd out the balanced growth path for this economy.

Arguing as before, it can be shown that along the optimal balanced growth trajectory for this competitive market economy:

˙ c c = ^k^˙

k =α(1 τ) (τ)¹^α^α ρ.

Thus clearly the growth rate of per capita consumption/output is lower in the competetive market economy than in the planned economy (but the initiallevel of consumption ishigher in the competetive market economy than in the planned economy).

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Barro Model: Some Extra Insights (as compared to Romer)

Notice that even though the Barro model structure-wise look very similar to Romer, the Barro model provides a more direct role of the government in the growth process.

In the Barro modelthe government can directly in‡uence the growth rate (both for the planned economy and well as for the market economy) by changing the tax rateτ.

In fact the relationship between tax rate and growth rate is not monotonic.

One can easily calculate the ‘growth maximixing’tax rate (for both the economies) as:

τ =1 α

Can you calculate the Welfare-maximing tax rate for both the economies? Is it the same as τ ?