Discussion Paper No. 525

GOVERNMENT SPENDING IN
A MODEL OF ENDOGENOUS GROWTH
WITH PRIVATE AND PUBLIC CAPITAL

Dipankar Dasgupta

January 2001

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Government Spending in a Model of
Endogenous Growth with Private and Public
Capital∗Dipankar Dasgupta†
Institute of Social & Economic Research
Osaka University, 6-1 Mihogaoka, Ibaraki
Osaka 567-0047, Japan (Till March 31, 2001)
andEconomic Research Unit
Indian Statistical Institute203 B.T. Road
Calcutta 700 035, India (From April 1, 2001)
e—mail : ojaantrik@yahoo.com

February, 2001

JEL classification: C61; C62; H21; H41; H54; O4.
Keywords: Decentralization, endogenous growth, ineﬃciency, mixed economy, optimal income tax, public input, transitional dynamics.
∗

This is a revised version of ISER Discussion Paper No. 525, Osaka University, Japan.
The author is particularly indebted to Micaela Notarangelo for discussions and Mathematica related help. Comments by Koichi Futagami, Yasushi Iwamoto, Akihisa Shibata,
Tomoichi Shinotsuka and Kenji Yamamoto during seminar presentations were most helpful.
†Government Spending in a Model of Endogenous
Growth with Private and Public Capital

ABSTRACTThe paper constructs a two-sector model of endogenous growth for aMixed Economy characterized by two private inputs, labor and capital, andthe services of an accumulable pure public input (of which an important example is infrastructure). Final goods are produced by a competitive private
sector with the help of the three inputs, the public input being supplied freeof charge by the Government. The Government accumulates infrastructural
stocks noncompetitively with the same inputs, meeting the cost of accumulation from income tax revenues. The policy instruments for the Government
are the tax rate and the revenue shares spent on the private inputs. The
paper describes the equilibrium of the Mixed Economy and compares the
optimum choices of the two instruments under alternative policy objectives.
It also demonstrates the transitional stability of steady state paths. Ineﬃciencies of the Mixed Economy make its equilibria socially suboptimal. The
best growth path of the economy is the unique solution to the Command
Economy problem. However, despite the existence of suitably defined eﬃciency prices for a fictitious private economy that supports the Command
Economy path, the latter is not decentralizable. The reason for this lies in
the inability of the private economy to resolve the joint eﬀect of labor andpublic capital into their individual eﬀects. Paradoxically, the highest, though
ineﬃcient, growth rate for the Mixed Economy can be reasonably close to
the socially best growth rate of the Command Economy. This gives riseto an identification problem, constituting thereby a dilemma for developing
economies in search of market friendly systems.

Government Spending in a Model of Endogenous
Growth with Private and Public Capital
1

Introduction

The importance of the State in the process of economic development is hardto underestimate. This is particularly evident from the fact that an economy’s growth path is often conditioned by the availability of inputs that
qualify as pure public goods. The nonexcludability of such inputs implies
that it will be incentive incompatible for private entrepreneurs to produce
and supply a class of vital inputs necessary for economic growth. Consequently, there emerges a natural role for the Government to supplement theprivate sector, especially in developing economies.
Barro (1990) qualifies as the seminal paper in the field.1 The novelty ofthis paper lay in endogenizing the rate of growth of an economy by linking
it to the Government’s fiscal exercises. The Government in Barro subsidizedeach household producer by providing a free infrastructural or public input,
but covered the subsidy by levying an equivalent tax on the final output produced. This balanced budget procedure connected up the tax rate chosen
by the Government to the post tax marginal productivity of capital, andhence, the rate of growth endogenously desired by the Representative Household in the economy. The link between fiscal policy and growth allowed for
a comparison of the tax rate that maximized steady state welfare with theone that maximized steady state growth. In particular, for a Cobb-Douglas
production structure (though not for a more general one), Barro found thatthe two rates were identical.
Barro’s one-sector exercise was extended by Futagami et al (henceforth,1

Unlike the approach of the present paper, Barro is skeptical about the pure publicnessof inputs in the real world. Hence, he concentrates only on rival public inputs. In spite of acontinued skepticism regarding the issue, Barro & Sala-i-Martin (1999) present a model ofnonrival public input also. Extensions of Barro’s paper have appeared since its publication.
Alesina and Rodrik (1992), , Dasgupta (1999, 2000), Futagami et al (1993) and Grenier& Semmler (2000) constitute a representative sample.

1 INTRODUCTION

2F M S) to a two-sector model, where infrastructural services were visualized
as flowing out of an accumulable stock. F M S justified the necessity for theextension on the following grounds:

First, many public infrastructures such as highways, airports, and electrical and gas
facilities are stock variables in nature. · · · Second, and more importantly, there areseveral empirical studies supporting the importance of public capital in private
production. For example, Aschauer (1988) finds empirically that public capital raises themarginal productivity of private capital. Estimating a version of the Cobb-Douglas
production function with public capital for the Japanese economy, Iwamoto (1990) also
finds a significant eﬀect of public capital.

The F MS position is supported by the World Development Report, 1994,
which found that a 1 % rise in the stock of infrastructure leads to a 1 %improvement in the GDP across countries. It also estimated that the annual
economic costs of inadequate transport infrastructure in China was at least
1 % of its GNP. The India Infrastructure Report (1996) expressed a similarsentiment in discussing the fast-growing East and South East Asian countries.
The two-sector extension led F M S to consider the transitional dynamics
for the model, which they proved to be stable. They also looked into the relationship between growth rate maximizing and welfare maximizing income
tax rates in the presence of infrastructural stocks. As opposed to the Barro
result in this context, they found that for a Cobb-Douglas production structure, the two tax rates were not equal. F MS argued that the implication ofthis result was that

· · · the policy maker’s work is more complex than simply maximizing the growth rate of
the economy.

In spite of these significant diﬀerences between Barro and F M S, there
were certain respects in which the economies they studied were similar. First,
both were concerned with household producers with no explicit role for laborin the production function. Secondly, the public input was viewed as a rival
commodity, though supplied freely by the Government. Thirdly, there being1 INTRODUCTION

3

a single private input, capital, neither paper was concerned with market
determined distribution of factor incomes. Finally, taxes were levied directlyon the final good produced by the economy. In subsequent work, Barro &Sala-i-Martin (1999) (henceforth, BS) re-established the equality between
between the growth and the welfare maximizing tax rates for an alternative
one-sector version of Barro (1990), in which the household producer was
replaced by a competitive entrepreneur, (b) the technology recognized laboras well as private capital and public infrastructure services, (c) the public
input was allowed to be nonrival and (d) returns to labor and private capitalwere determined in competitive factor markets. To the extent, however, thatthe model had a single sector, the public input was a pure flow as in Barro(1990). Moreover, taxes continued to be raised on the final output.
In the present paper, I reinvestigate the Barro-BS-F M S issues by adding
on to the BS framework infrastructural stocks as in F MS.2 However, unlike
the BS model, where taxation of output and incomes might well be equivalent instruments, I allow for taxes to be imposed directly on incomes, since
there does not seem to be large-scale evidence of direct taxation of output
in developing economies to create infrastructure necessary for growth anddevelopment. The model has a profit maximizing, competitive private sector producing a consumable and accumulable final output and a government
sector producing changes in the public capital stock as well as supplying theservice flows emanating from it. The service in question is assumed to be apure public input, which prevents the Government sector from functioning
on competitive principles. Consequently, there arises a divergence between
marginal productivities and factor returns in the public sector, which is amajor source of ineﬃciency in the Mixed Economy under consideration.
Private capital and labor employed in the public sector are assumed to be
paid according to their opportunity costs, viz, the factor returns prevailingin the private sector, while the Government’s tax revenue determines thebudget available for purchasing these inputs in the factor markets. As in
Barro-BS-F M S, the Government’s budget is balanced. Unlike these papers,
however, the budget has to be distributed over the two factors of production.
2

An alternative interpretation of the endogenous growth model of the present work
would be to view it as a generalization of the two sector, two factor Rebelo (1991) economy
to a three factor world through the introduction of a (third) pure public (i.e. nonrival andnonexcludable) input into both production functions.

1 INTRODUCTION

4

Thus, there are two choice parameters for the Government. The first, like
Barro-F M S-BS, is a tax rate. The second is yet another ratio representing
the allocation of the tax revenue amongst private inputs (capital and labor
services) used for accumulating infrastructure. (See Section 2 below for theexact definition of the ratio.) The necessity of choosing the two parametersat their optimal levels simultaneously calls for a fresh consideration of theBarro-F M S-BS issues and justifies the exercise to follow.3
In the context of the Mixed economy, I demonstrate that a tax rate—
expenditure ratio pair that maximizes the steady state growth rate is nonwelfare maximizing. This confirms the robustness of the F M S conclusion
stated above. However, I also show that growth and welfare maximization
would be equivalent policies under the additional constraint that factor sharesin the public sector, and not merely the factor returns, mimic the ones prevailing in the private sector. Consequently, a limited version of the Barro-BS
result survives in the two-sector extension also. Quite apart from these issues,
the paper proves, like F M S, that the economy is transitionally stable. Thus,
given any choice of the two parameters and starting from any set of initialvalues of the private and public capital stocks, the dynamic path converges
over time to the steady state value of the ratio of the two stocks. Without
this result, policy choices of the Government with reference to steady state
growth will amount to empty theorizing.
The ineﬃciencies in the Mixed Economy imply that its welfare maximizing equilibrium is not an overall social optimum. Hence, the paper goes on
to prove the existence of a socially optimum (or, first best) steady growth
path characterizing a centrally planned or Command Economy and suggests
a scheme for decentralizing it. The scheme in question, however, treats theservices of labor and public capital as a joint input. Since a market for suchan input cannot function in the real world, the decentralization exercise isoperationally vacuous, although the Command Economy optimum exhibitsa higher growth rate than the highest achievable by the Mixed Economy. On
the one hand, therefore, a decentralized economy that can support the first
best growth path of the system is nonfunctional. On the other hand, a func3

Both Dam (1997) and Dasgupta & Dam (2000) introduced the idea of the second
choice parameter, but stopped short of the optimization exercise as well as the transitional
dynamics of this paper.

2 THE MODEL

5

tional market oriented system is grossly suboptimal. Interestingly enough,the highest growth rate achievable by the Mixed Economy for certain parameter values falls only marginally short of the Command Economy’s growthrate, though the Mixed Economy can achieve its highest growth rate only at
the cost of eﬃciency, and hence welfare. This poses a dilemma for developing economies engaged in the process of transition from planned to market
economies.
I describe the model of the paper and set out the equilibrium solutions
for a static and a dynamic steady state economy in Section 2. Sections 3 and4 discuss respectively the problems of growth and welfare maximization withrespect to the two policy parameters. I establish transitional stability of theeconomy in Section 5. Issues relating to the social optimum are discussed in
Sections 6 and 7. Section 8 concludes the paper.

2

The Model

There are two productive sectors, referred to as the Y and the G-sector
respectively. The Y -sector is privately controlled and produces Y , which
is used for consumption as well as capital accumulation. Following Solow(1956), K denotes both the stock of capital and the flow of its services (i.e.
the stock-flow ratio is assumed to be unity).
The G-sector is state owned and engaged in accumulating the infrastructural capital, denoted by G. The stock generates a flow that enters all production (including its own) as an essential input. As with K, the stock-flow
ratio is a constant, so that G represents a flow also. The flow of G is a purepublic good by assumption. Hence, both sectors use the same flow of G.4
The Government supplies it free of user charge to all producers, including
itself.
4

This means of course that both are assumed to be nonsatiated in the use of the public
input. This is particularly evident for infrastructure, which constitutes a bottleneck fordeveloping economies.

2 THE MODEL

6

As in BS, the technology for Y is given by5

Y = A Kyα L1−α
G1−α , 0 < α < 1,
y

(2.1)

where Ky , Ly and G are the flows of capital, labor and the pure public
input used to produce Y and A is parametrically specified. Thus, production
exhibits constant returns to scale in private inputs, Ky and Ly . For fixed G,
the economy faces diminishing returns to private capital. However, for fixed
Ly , (2.1) implies constant returns to G and Ky . Commodity Y will be usedas the numeraire in what follows.
The technology for Ġ, the change in the stock of G, is similar to (2.1)
except for the parameters of the production function. Thus,Ġ = B Kgβ L1−β
G1−β , 0 < β < 1,
g

(2.2)

where Kg , Lg and G are the flows of capital, labor and the public input used
to produce Ġ and B is parametrically specified.6 As with Alesina & Rodrik(1992), Barro and Sala-i-Martin (1999) and BS, I assume the aggregate labor
force to be constant for the economy.7
Society’s welfare is identical with that of the Representative Household
and given by
5

While all the variables appearing below are functions of time, the time index will often
be dropped for notational simplicity.
6
The form of both functions implies that G enters production in a labor augmenting, Harrod-neutral fashion. Harrod neutrality is of course unavoidable for steady state
analysis. An alternative specification of the production functions might be (for example)
Y = K α Lβ G1−α−β and Ġ = K δ L² G1−δ−² . In per capita terms, these reduce to(Y /L) = (K/L)α (G/L)1−α−β and (Ġ/L) = (K/L)δ (G/L)1−δ−² . Since steady state fixes
the values of K/L and G/L in such a model, the endogeneity of the growth rate is lost in
the process. The model then becomes inappropriate for analyzing the eﬀect of policy on
economic growth. See BS and Lucas (1988) on these issues.
7
This is an innocuous assumption. Introducing an exponentially growing labor force isstraightforward and does not aﬀect the endogeneity of the growth rate under the paper’s
model specifications.

2 THE MODEL

7

W =

Z ∞

ln C(t) e−ρt dt,

(2.3)

0

where C(t) represents consumption at point of time t and the constant ρ isa positive discount parameter.8
The Household owns all private capital and is the sole supplier of its
services as well as labor. The final product Y is produced by a Representative Firm that maximizes instantaneous profit, assuming all prices to beparametrically specified. It is charged a price r and w respectively by theHousehold per unit flow of capital and labor services consumed. Further,
there is a proportional tax τ on household income. The tax proceeds areused to pay for the competitive prices of capital and labor services used up
in the production of Ġ.
Profit maximization gives rise to demands for K and L services by the
Y -sector, but not for G services, because a demand function for G does notexist in this model.9 The Y -sector demands for K and L are added to thecorresponding requirements by the G-sector to yield the aggregate demands
for K and L. Like Solow (1956), the available aggregates of the services
are thrown inelastically on the factor market at each point of time, theirprices being determined by equating the inelastic supplies to the aggregate
demands.
To accumulate infrastructure, the G-sector pays K and L the competitive
prices and, like the competitive Firm, has free use of G services. Since thetax proceeds are spent on the two factors capital and labor, it is necessary
to specify a rule for distributing the expenditure over the factors. I assume
that the expenditures are allocated to satisfy the condition that their ratio8

It is more common in Endogenous Growth Theory to deal with the welfare function
Z ∞C(t)1−σ − 1 −ρt
dt.
e
1−σ
0

where σ > 0 is the constant elasticity of instantaneous marginal utility. As σ → 1, thefunction reduces to (2.3). I choose the special case in the paper for notational simplicity.
All results reported here carry through for the more general form.
9This analytical feature derives from Samuelson (1954).

2 THE MODEL

8equals a constant (to be chosen optimally) in steady state.
The mechanics of the model falls into two parts: Static Equilibrium ateach t and the Dynamic Steady State Equilibrium. I describe these in turn.

2.1

Static Equilibrium

For the Y -sector to behave competitively, the factor returns must equal the
marginal products. Thus,r = α A Kyα−1 L1−α
G1−α
y
w =

(2.4)

G1−α ,(1 − α) A Kyα L−α
y

(2.5)

α
w
Ly .
1−α
r

(2.6)

or, alternatively,

Ky =Writing K and L respectively for the aggregate supplies of capital and labor, the G-sector employment of these factors is governed by a budget constraint,10 viz,r Kg + w Lg = τ (r K + w L)

(2.7)

and the rule that10

I assume a uniform tax rate on profits and wages, partly to concentrate attention on
the interplay between the two policy parameters mentioned earlier. Besides, there is no
obvious theoretical reason that recommends diﬀerential taxation of the two factors in the
Representative Household macro model I am concerned with. See, however, Alesina andRodrik (1992) for the political economy of diﬀerential taxation in a model that recognizes
agent heterogeneity in terms of factor endowments.

2 THE MODEL

9r Kg= constant
w Lg

(2.8)

in steady state. The constancy of rKg /wLg in steady state implies thatrKg /(r Kg + w Lg ) as well as wLg /(r Kg + w Lg ) are themselves constants.
I denote the first of these by γ and the second by 1 − γ, so that (2.8) reduces
to
r Kg
γ
=
,
w Lg
1−γ

(2.9)

where 0 < γ < 1.11 The diﬀerent sources of ineﬃciency characterizing theMixed Economy may be noted at this stage. First, while capital and laborin the G-sector are paid the same returns that prevail in the private sector,
these need not equal their marginal products as in equations (2.4) and (2.5).
Secondly, the shares γ and 1 − γ of capital and labor in the G-sector may
not equal the competitive shares β and 1 − β. Thirdly, there is of course
the usual distortion introduced by the tax rate. Lastly, while the marginal
productivity of G is strictly positive in both sectors, the price paid for it iszero. These imply that the Mixed Economy can, at best, reach an ineﬃcientsolution of the dynamic resource allocation problem under consideration.
Equation (2.9) is written in a form similar to (2.6) as

Kg =

γ
w
Lg .
1−γ
r

(2.10)

Equations (2.6), (2.7) and (2.9) together imply
11

As in Dam (1997), one possible choice of the expenditure ratio is the ratio of the
exponents of capital and labor in the G-sector production function. In other words, theexpenditure allocation rule might mimic a necessary condition satisfied by factor returns
under perfect competition. I shall show below (Remark 2 and Proposition 4.2) that γ = β
maximizes neither welfare nor the growth rate of the Mixed Economy.

2 THE MODEL

10

Ly = λ(τ, γ) Lg .

(2.11)

where λ(τ, γ) = ((1 − α)/(1 − γ)) ((1 − τ )/τ ). Assuming now that the total
supply of labor is normalized to unity for all time, it follows from market
clearing that
L = Ly + Lg = 1.
Substituting from (2.11), we see thatLy =λ(τ,γ)
1+λ(τ,γ)Lg =1.
1+λ(τ,γ)(2.12)

Using (2.6), (2.10) and (2.12), the aggregate demand for K turns out to be

Ky + Kg = (αλ(τ, γ)γ
1w+) .
1 − α 1 + λ(τ, γ) 1 − γ 1 + λ(τ, γ) r

(2.13)For every pair (τ, γ), the equilibrium value of w/r is determined by equating
the aggregate demand for K to the aggregate supply, K(t), at each t.12 Thisis shown in Figure 1. The expression (α/(1 − α)) (λ(τ, γ)/(1 + λ(τ, γ))) +(γ/(1 − γ)) (1/(1 + λ(τ, γ))) will appear repeatedly in the paper. To makethe notation simpler, I shall refer to it simply as χ. Note, however, that
χ depends on α, τ and γ. Since χ lies between α/(1 − α) and γ/(1 − γ),Figure 1 here.
12

Note from the derivation of (2.12) that the labor market equilibrium condition isbuilt into (2.13) through the requirement that Ly + Lg = [λ(τ, γ)/(1 + λ(τ, γ))] + [1/(1 +λ(τ, γ))] = 1.

2 THE MODEL

11

which are both positive and finite, there is a well-defined non-zero solution
for w/r corresponding to each specification of τ and K(t). Equations (2.6),(2.10) and (2.12) now give the equilibrium values of Ky and Kg . Equation(2.13) and the condition of market clearance at each point of time implies
that (τ, γ) fixes the equilibrium value of w/K at
w
r
= .
K
χ

(2.14)

Since r is a constant in steady state equilibrium, (2.14) shows that w and K
change at the same rate in steady state.

2.2

Dynamic Steady State Equilibrium

A steady state equilibrium involves:

Ġ
Ẏ
=
G
Y
K̇
=
K
Ċ
=
C
= constant.
In what follows, Ġ/G will be referred to as the supply rate of growth. On
the other hand, Ċ/C is the demand rate of growth chosen by the Household.
Using (2.2) and the derivations of the previous subsection, the supply
rate of growth turns out to be

g s = B ((1 − α) αα/1−α A1/1−α )βγ β (1 − γ)1−β τ
1. (2.15)(1 − γ) τ + (1 − α)(1 − τ ) rβ/1−α2 THE MODEL

12

Thus, given the size of the Government’s budget and its distribution over
the two factors, a rise in r lowers the employment of capital for producing Ġ.
Consequently the supply rate of growth falls. The Representative Household
is endowed with perfect foresight and is assumed to take the time paths ofτ , r and w as parametrically given. Typically therefore, the Household’s
optimization problem over time is to maximize (2.3) subject toK̇ = (1 − τ ) (r K + w) − C.

(2.16)

This is a standard Ramsey problem, whose solution is the demand rate of
growthg d = (1 − τ ) r − ρ.

(2.17)The intersection of (2.17) and (2.15) determines the corresponding equilibrium growth rate and interest rate. For any given (τ, γ) ∈ (0, 1) × (0, 1),
g s → ∞ as r → 0 and g s → 0 as r → ∞.
Similarly,

g d → −ρ as r → 0 and g d → ∞ as r → ∞.
This guarantees an intersection between the two curves, and hence the existence of an equilibrium pair (r, g) for each (τ, γ) ∈ (0, 1) × (0, 1).
Proposition 2.1 For any admissible tax rate and ratio of input expenditure
on infrastructural accumulation, there exists a positive equilibrium steady
growth rate and a corresponding positive equilibrium rate of interest.

3 GROWTH MAXIMIZATION

13

Figure 2, where the equilibrium g and r are represented by g(τ, γ) andr(τ, γ) respectively, illustrates the proposition. The corresponding value of
w is denoted by w(τ, γ). I will refer to g(τ, γ) as the equilibrium steady state
growth rate (ESSGR).
Figure 2 here.
The aggregate welfare for the economy in dynamic steady state equilibrium may be found by integrating (2.3) and substituting from (2.16) and(2.17). Thus,W (τ, γ) = ln [K(0){ρ +(1 − τ ) w(τ, γ) 1 g(τ, γ)
}] +
,
K
ρ
ρ2

(2.18)

where K(0) is the value of the initial capital stock for the economy. Using insuccession equation (2.14), equation (2.17) and the fact that g d = g(τ, γ) in
dynamic steady state equilibrium, the value of equilibrium welfare reduces
to

W (τ, γ) = ln [K(0){ρ +g(τ, γ) + ρ 1 g(τ, γ)
}] +
.
χ
ρ
ρ2

(2.19)

While welfare maximization is an obvious objective for the Government topursue, it is of interest to compare this policy with the alternative policy ofgrowth rate maximization. In particular, it is important to know if growthrate maximization is a proxy for welfare maximization. I shall begin withthe problem of growth maximization.

3

Growth Maximization

Using (2.4) and inverting (2.17) for τ 6= 1,

3 GROWTH MAXIMIZATION

14

G Ly
ρ + gd=()1/(1−α) .
Ky(1 − τ ) α A(3.1)

g s 1/β
Kg
=(
) .
Lg G
B Lg

(3.2)

Similarly, (2.2) yields

Multiplying equations (3.1) and (3.2), using (2.6), (2.10), (2.12) and substituting g d = g s = g,g
ρ + g 1/(1−α) 1 − α γ1( )1/β ()()1/β (1 − τ )1/1−α .
=BαAα 1 − γ 1 + λ(τ, γ)(3.3)

Equation (3.3) gives a necessary condition that is satisfied when g d = g s = g.
It is easily checked that the LHS of (3.3) is strictly convex and increasing
in g. Hence, a unique solution to (3.3) exists for any (τ, γ) ∈ (0, 1) × (0, 1).
This is identically the same as the ESSGR g(τ, γ) established in Proposition
2.1.
Maximizing the growth rate amounts to maximizing the solution of (3.3)
with respect to τ and γ. However, (3.3) being a polynomial equation in g, I
solve the maximization exercise in two steps. The first involves maximization
with respect to τ for any given γ ∈ (0, 1).

3.1

Growth Maximizing Tax Rate

It is not hard to show that for each γ ∈ (0, 1), ∃ a unique τ ∈ (0, 1) for which
the expression
1−α γ
1
(
)1/β (1 − τ )1/1−α
α 1 − γ 1 + λ(τ, γ)(3.4)

3 GROWTH MAXIMIZATION

15

on the RHS of (3.3) reaches an interior maximum with respect to τ . Thisfact, coupled with the behavior of the LHS of (3.3), implies that for eachγ ∈ (0, 1), ∃ τ (γ) ∈ (0, 1) maximizing the ESSGR. Expression (3.4) monotonically rises with γ, thereby implying that τ (γ) rises as γ rises. Moreover,τ (γ) → τ > 0 as γ → 0. The following proposition summarizes the conclusions:

Proposition 3.1 For each proper fraction representing the ratio of expenditures on infrastructural inputs, there exists a unique fraction representing the
ESSGR maximizing tax rate. The growth maximizing tax rate is a strictlyincreasing function of the expenditure ratio and approaches a positive limit
as the latter approaches zero.

Remark 1: The intuitive reasoning underlying the behavior of τ (γ) is asfollows. First, given τ , r and w, a rise in γ raises Kg and lowers Lg . These
are two opposing eﬀects on g s . The joint eﬀect is to raise the latter unambiguously in the region γ ≤ β, but possibly beyond also, as the term 1 − γ inthe denominator of equation (2.15) indicates. Moreover, g s rises unambiguously with τ , since more revenue is spent on the accumulation of G. On theother hand, g d is unaﬀected by γ and falls with τ (as (2.17) shows), because
the Household demands a smaller rate of growth as its post tax income from
capital falls. The joint positive eﬀect of γ and τ on g s turns out to be larger
than the isolated negative eﬀect of τ on g d . Consequently, a simultaneous
rise in γ and τ is likely to raise g(τ, γ) and τ (γ). The behavior of the function
τ (γ) is shown in Figure 3.

Figure 3 here.

The strict monotonicity of τ (γ) allows the function to be inverted. Thisis done by solving the first order condition for a maximum of (3.4) for fixedγ. Denote the resulting expression by γ̃(τ ), where(1 − α)(1 − τ ) (1 − α)2 (1 − τ )γ̃(τ ) = 1 +
−
.
τ
βτ 2

(3.5)

3 GROWTH MAXIMIZATION

16This expression will be useful in locating the joint maximum with respect toτ and γ. I proceed now to a maximum with respect to γ given τ .

3.2

Growth Maximizing Expenditure Ratio

Once again, the existence of a growth rate maximizing γ is governed byequation (3.3). For τ ∈ (0, 1), the behavior of the RHS depends on the
behavior of (γ/(1 − γ)) (1/(1 + λ(τ, γ)))1/β , or, on its positive monotonic
transformation (γ/(1 − γ))β (1/(1 + λ(τ, γ))). Denoting the latter by φ(γ),
the following properties are easily derived:

φ0 (γ)
φ0 (γ)
φ0 (γ)> 0 for γ small and positive ;→ 0 as γ → 0;∼
= − ∞ for γ ∼= 1.

These observations imply that for any τ ∈ (0, 1), ∃ γ ∈ (0, 1) such that the
ESSGR is maximized. Diﬀerentiating φ(γ), equating to zero and solving, it
follows thatγ=βτ + β(1 − α)(1 − τ ).
βτ + (1 − α)(1 − τ )(3.6)

Thus, the ESSGR maximizing γ is unique for each τ ∈ (0, 1). I denote therelationship by γ(τ ). Diﬀerentiating γ(τ ), it follows that γ 0 (τ ) > 0. Also,γ(τ ) → β as τ → 0. It is possible now to state a result parallel to Proposition
3.1:
Proposition 3.2 For each proper fraction representing the tax rate, there
exists a unique fraction representing the ESSGR maximizing expenditure
ratio. The growth-maximizing ratio is an increasing function of the tax rate
and approaches the exponent of capital in the infrastructural sector as the tax
rate approaches zero.

3 GROWTH MAXIMIZATION

17

Remark 2: For the Cobb-Douglas technology under consideration, eﬃcient
allocation of resources calls for the ratio of capital and labor incomes in eachsector to equal the ratio of factor exponents in the corresponding production
function. The condition is always satisfied by the competitive Y -sector. The
last proposition shows this to be the case for the G-sector also as the tax
rate approaches zero, i.e. when there is no tax induced distortion in the system. The noncompetitive behavior of the G-sector along with the distortion
brought about by the positive tax rate makes the Mixed Economy’s optimal
choice of γ larger than β. On the other hand, γ = β is not realizable sinceτ = 0 rules out production in the G-sector and therefore positive steady state
growth.

3.3

Growth Maximizing

τ and γI will state the conclusions of this subsection in the form of a Remark.
Remark 3: A necessary condition for the existence of an interior growthrate maximizing pair (τ ∗ , γ ∗ ) is that the two functions τ (γ) and γ(τ ) intersect
in the region (0, 1) × (0, 1). Looking for an intersection of τ (γ) and γ(τ ) is
equivalent to looking for an intersection of γ̃(τ ) and γ(τ ). The functionsγ̃(τ ) and γ(τ ) are described completely by the two parameters α and β, theexponents of capital in the two production functions (equations (3.5) and(3.6)). In order to get a feel for the figures involved, it is interesting to test
for the possibility of existence numerically. I do this for two diﬀerent choicesof the parameters (α, β).
Case 1: Using a narrow concept of physical capital (structures and equipment), α and β should be around 0.33. (See, for example, Denison (1962),
Maddison (1982), Jorgenson, Gollop and Fraumeni (1987)). For α = 0.3 andβ = 0.35, Figure 4 (drawn by Mathematica) shows that the two curves do intersect in (0, 1) × (0, 1). However, the coordinates of the point of intersection
exceed 0.8 for both parameters. Thus, the tax rate exceeds 80%. Similarly,γ ∗ > 0.8 is large compared to β = 0.35, the share of capital income in the
G-sector that would emerge under competitive conditions. This raises thepossibility that capital earns substantially more than its marginal productin the G-sector. The exceptionally high values of τ ∗ and γ ∗ suggest severe4 WELFARE MAXIMIZATION

18

social costs. Hence, (τ ∗ , γ ∗ ) is not expected to maximize welfare in a Mixed
Economy. This will be confirmed by Proposition 4.2 below.
Case 2: Following Barro and Sala-i-Martin (1999), however, and using abroader definition of capital (to include, for example, human capital), thetwo parameters should have values close to 0.75. Using this value, the tax
rate falls to a more reasonable figure of 28%, while γ ∗ still hovers arounda value slightly exceeding 0.8. This is shown in Figure 4 also. Once again,γ ∗ > β indicates a source of ineﬃcieny.

Figure 4 here.

The stage is now set for a discussion of welfare maximization.

4

Welfare Maximization

As with the analysis of a growth maximizing pair, I approach the question of
welfare maximization in two stages, by considering in succession the partial
problems of with respect to γ and τ .

4.1

Eﬀect of the Expenditure Ratio on Welfare

Diﬀerentiating the RHS of (2.19) with respect to γ,

4 WELFARE MAXIMIZATION

19∂W (τ, γ)1=×∂γ[K(0)(ρ + (1−τK) w )] ρ1∂g(τ,γ)(χ) − (g(τ, γ) + ρ) z∂γ(χ)2

1 ∂g(τ, γ)
+ 2
,
ρ
∂γ

(4.1)

where
z =

∂χ
∂γ

τ 2 + α τ (1 − τ ) + (1 − α)(1 − τ )τ((1 − γ)τ + (1 − α)(1 − τ ))2> 0.
=

Since ∂g(τ, γ)/∂γ is nonpositive for γ ≥ γ(τ ) ( by virtue of Proposition 3.2),
the strict positivity of z implies that ∂W/∂γ < 0 whenever γ ≥ γ(τ ). The
next proposition states this fact.

Proposition 4.1 For each proper fraction representing the tax rate, lowering the expenditure ratio below its growth maximizing value will increase
welfare.

It follows from Proposition 4.1 that a welfare maximizing pair cannot
lie on the curve γ(τ ). Consequently, it cannot be identical with a growth
maximizing pair. The next proposition states this fact.

Proposition 4.2 A pair of policy parameters maximizing the growth rateis distinct from a pair that maximizes welfare.

4 WELFARE MAXIMIZATION

20

Remark 4: In this context, it is worth considering Barro’s observation(Barro (1990), p S107):
As long as the government and the private sector have the same production functions,the results would be the same if the government buys private inputs and does its own
production, instead of purchasing only final output from the private sector, as I assume.

Proposition 4.2 of the present paper shows that Barro’s (as well as the BS)
finding on the equivalence of growth and welfare maximization is untrue forall choices of the parameters of the production functions, including the casewhere the production functions are identical. As noted in the Introduction,
F M S reach a similar conclusion for a two-sector model. However, unlike thepresent paper, the F M S model does not fit the economy described by thequote from Barro. The two policies are equivalent in my two-sector extension
under the additional constraint that income distribution in the sectors match.
(See Remark 6.) In BS, of course, the public sector is absent and income
distribution automatically follows the pattern of the private sector. Hence,in that model, the policies for growth and welfare maximization merge.

4.2

Eﬀect of the Tax Rate on Welfare

In order to narrow down the search for a welfare maximizing policy, diﬀerentiate the RHS of (2.19) with respect to τ and use (2.18) to get

1∂W (τ, γ)=×∂τρ [K(0)(ρ + (1−τ )Kw(τ,γ) )]∂g(τ,γ)(g(τ,γ)+ρ) (γ−α)(χ) − ((1−γ)τ
∂τ+(1−α)(1−τ ))2(χ)2

1 ∂g(τ, γ)
+ 2
.
ρ
∂τ

(4.2)

4 WELFARE MAXIMIZATION

21

The behavior of the RHS of (4.2) depends on the relative values of γ and α.
Consider first the case γ = α. Then χ = α/(1−α), a constant. Consequently,
the sign of the RHS of (4.2) depends on the sign of ∂g(τ, γ)/∂τ , which iszero at τ = τ (γ), positive for τ < τ (γ) and negative for τ > τ (γ). Hence,
welfare is maximized at τ = τ (γ).
For γ > α, sgn ∂W/∂τ = −ve whenever τ ≥ τ (γ), since ∂g(τ, γ)/∂τ < 0.
Hence, welfare improves with a marginal fall in τ below τ (γ) in this case.
Similarly, when γ < α, sgn ∂W/∂τ = +ve for τ ≤ τ (γ). Consequently,
welfare rises with a mariginal rise in τ above τ (γ).
The following result is accordingly established:

Proposition 4.3 Welfare can be increased by reducing (raising) the tax rate
below (above) the growth maximizing tax rate when the expenditure ratio isgreater (less) than the share of capital in the private sector. When the expenditure ratio equals the share of capital in the private sector, the growth
maximizing tax rate equals the welfare maximizing tax rate.

Remark 5: It might appear that it is the value of γ relative to α alone thatmatters in determining the behavior of welfare as τ varies relative to τ (γ),
with β playing no role in the matter at all. This, however, is not the case.
To appreciate the fact, consider expression (2.19) for the equilibrium valueof steady state welfare. The numerators of both terms on the RHS of (2.19)
involve g(τ, γ), which, being the equilibrium growth rate, depends on both αand β. On the other hand, the denominator of the first term on the RHS of(2.19) derives from equation (2.14), which depends on the static equilibrium
at each t. Since the static equilibrium ignores the marginal productivities
of both private factors in the G-sector, (2.14) involves α and γ, but not β.
Going over now to (4.2), the sign of ∂W/∂τ is determined by (i) the sign of∂g(τ, γ)/∂τ and (ii) the sign of ∂χ/∂τ . The first of these signs depends onthe value of τ relative to τ (γ) and the latter depends on α as well as β. The
second sign, however, depends on the value of γ relative to α only. Thus,
conclusions such as “For γ > α, sgn ∂W/∂τ = −ve whenever τ ≥ τ (γ)” thatappear above involve both α and β.

4 WELFARE MAXIMIZATION

4.3

Joint Eﬀcet of22τ and γ on Welfare

Figure 5 shows the plausible relative positions of growth and welfare maximizing pairs. Following Figure 4, the growth maximizing pair is shown by
the coordinates (τ ∗ , γ ∗ ) of the intersection of τ (γ) and γ(τ ). The horizontally
striped areas represent the possible locations of welfare maximizing τ ’s. Similarly, welfare maximizing γ’s must lie in the vertically striped zone. A (τ, γ)pair jointly maximizing welfare can belong then only to the crosshatched
area, i.e. the set theoretic union of the areas marked P and Q. It should be
S
noted that P Q has no point in common with the segment of γ(τ ) connecting the point (0, β) and the point of intersection (τ ∗ , γ ∗ ) (including (τ ∗ , γ ∗ )itself). It also excludes all points on the segment of τ (γ) joining (τ , 0) to thepoint (τ ∗ , γ ∗ ), except the point ω.

Figure 5 here.

Remark 6: The point ω = (α, τ (α)) is of special interest. First of all, it
falls in the region to which a welfare maximizing pair (τw∗ , γw∗ ) would belong.
Hence, ω = (τw∗ , γw∗ ) is not ruled out. Even if this is not the case, ω has aproperty worth noting. It says that a Government that is committed to asimilar pattern of income distribution in the public and private sectors, viz,γ = α, will choose (α, τ (α)) whether it wishes to maximize the growth rateor welfare. However, even when ω = (τw∗ , γw∗ ), this is a second best growthrate maximization policy, since it fixes γ arbitrarily at the level α. Once γ isallowed to be chosen optimally also, Proposition 4.2 will apply, making the
growth maximizing parameters distinct from the welfare maximizing pair.
When ω 6= (τw∗ , γw∗ ), the point ω is second best from the points of view ofboth maximization problems.
Remark 7: Assuming A = 0.2, B = 0.2, ρ = 0.2, the steady state growth
rates associated with ω and (τ ∗ , γ ∗ ) are as follows. For Case 1 of Remark
3, the growth rate at ω is 6.4%, while that at (τ ∗ , γ ∗ ) is 8.1%. For Case 2of the same Remark, the respective growth rates turn out to be 7.7% and7.9%. Remark 8 in Section 6 reports on the growth rates for the Command
Economy.

5 TRANSITIONAL DYNAMICS

5

23

Transitional Dynamics

As F M S point out, the two sector exercise remains incomplete without adiscussion of the transitional dynamics properties.13 Towards this end, I
define two new variables, k = K/G and c = C/K. Using (2.4), (2.5), (2.9),(2.14), (2.15) and (2.16), it follows thatk̇
= R1 k α−1 − R2 k β − ckċ= −R3 k α−1 + c − ρ,
c

(5.1)
(5.2)

where
R1 = α A (α 1 α−1)×
1−α χ(1 − τ ) (1 +R2 = B(

1
);
χ

γ 1 β
1
)
;
1 − γ χ 1 + λ(τ, γ)

R3 = α A (α 1 α−1
)
×
1−α χ

1
(1 − τ ) .
χThis system of diﬀerential equations describes the transitional dynamics ofthe economic system under consideration. The stationary state for the systemcorresponds to the dynamic equilibrium described earlier in terms of thedemand and supply rates of growth for each possible choice of (τ, γ). Suchan equilibrium must have associated with it admissible stationary values ofk and c. These stationary values will simultaneously solve

13c = R1 k α−1 − R2 k β = Ψ(k)(5.3)

The exercise considers out of steady state dynamics of k and c for each fixed pair (τ, γ).

5 TRANSITIONAL DYNAMICS

24

and
c = R3 k α−1 + ρ = Φ(k).

(5.4)

It is easily checked that both Ψ(k) and Φ(k) are monotone decreasing, strictly
convex functions. Again, the function Φ(k) → +∞ as k → 0 and Φ(k) →ρ as k → ∞. On the other hand, Ψ(k) → +∞ as k → 0 and Ψ(k) →−∞ as k → ∞. For meaningful solutions to the equations, eliminate c toget
(R1 − R3 ) k α−1 = R2 + ρ.
Taking account of the definitions of R1 , R2 and R3 , the equation admits
a unique, positive solution, say k ∗ . The uniqueness of k ∗ implies a corresponding unique solution c∗ for c. To see that c∗ > 0, consider Ψ(k) − Φ(k),which approaches ∞ as k → 0. Thus, Ψ(k) > Φ(k) for small k. SinceΨ(k) → −∞ as k → ∞ and Φ(k) > ρ ∀ k, it follows that Ψ(k) < Φ(k) forlarge enough k. Hence, 0 < c∗ = Ψ(k ∗ ) = Φ(k ∗ ), as shown in the phase
diagram of Figure 6 below.
Figure 6 here.
In order to study the local stability of the dynamic system, I rewrite
equations (5.1) and (5.2) ask̇ = (R1 k α−1 − R2 k β − c) kċ = (−R3 k α−1 + c − ρ) c.
Linear approximation around (k ∗ , c∗ ) givesk̇ = (α R1 k ∗ α−1 − (1 + β) R2 k ∗ β − c∗ )(k − k ∗ )−k ∗ (c − c∗ )ċ = (1 − α) R3 k ∗α−2 c∗ (k − k ∗ )+(2c∗ − R3 k ∗ α−1 − ρ)(c − c∗ ).

(5.5)

6 COMMAND ECONOMY

25

Using (2.15) and (5.4), the relevant determinant simplifies to(α − (1 + β)) g(τ, γ) c∗ + (α − 1) ρ c∗ < 0,where g d = g s = g(τ, γ). The sign of the determinant establishes that thestationary equilibrium is a saddle point. In other words, for each possible
choice of the initial value of k(0) in a neighborhood of k ∗ , ∃ a choice of cin a corresponding neighborhood of c∗ , such that the system (5.5) converges
to (k ∗ , c∗ ). The thick arrowheads in Figure 6 indicate the stable path moreclearly. The results are summarized by the next proposition, which parallels
Proposition 2 of F MS.

Proposition 5.1 For each specification of the values of the two policy instruments, the steady state equilibrium values of the private—public capitalratio and consumption—private capital ratio are unique. Further, there exists
a unique stable path of these ratios converging to the steady growth equilibrium.

6

Command Economy

So far, I have considered the case of a Mixed Economy only. Quite apart fromthe tax rate induced distortion and the non-competitive behavior of the Gsector, the growth rate of such an economy was socially suboptimal because
the agents were not concerned with optimality conditions surrounding thestate variable G. The socially optimum path on the other hand may be foundby solving the corresponding Command Economy’s optimization exercise. I
proceed now to prove the existence of a social optimum and construct pricesthat may support it as the competitive equilibrium of a fictitious two-sector
economy.14
14

The existence of a unique steady state growth path for the Command Economy wasproved by Dam (1997) and Dasgupta & Dam (2000) also. However, these papers were notconcerned with the decentralization exercise to follow.

6 COMMAND ECONOMY

26

Resource allocation in the Command Economy is carried out by solving
a grand optimization exercise by an altruistic social planner organizing production in both sectors and allocating resources between them. The welfare
function of the planner is identically the same as that of the Representative
Household. The planner maximizes (2.3) subject to15

K̇ = A(φK)α (θG)1−α − C(6.1)

Ġ = B((1 − φ)K)β ((1 − θ))G1−β ,

(6.2)

and

where φ and 1 − φ are respectively the shares of K and θ and 1 − θ are theshares (as well as the absolute amounts) of labor employed in the Y andG sectors respectively.16 The problem is solved by maximizing the current
value Hamiltonian

H = ln C + η(A(φK)α (θ G)1−α − C)+ ξ(B((1 − φ)K)β ((1 − θ) G)1−β ),where η and ξ are the costate variables associated with K̇ and Ġ. The steadystate solution for this problem may be called the Command Equilibrium andthe resulting rate of steady growth denoted g ∗ .
The first order optimality conditions for the maximization of H are

∂H
= 0,
∂C
15

(6.3)

The calculations for the more general utility function mentioned in footnote 9 may befound in Dasgupta & Dam.
16
Mino (1996) considers a two sector production structure similar to, though moregeneral than, (6.1) and (6.2). However, the thrust of Mino’s paper is totally diﬀerent fromthe present exercise.

6 COMMAND ECONOMY

27

∂H
= 0,
∂φ
∂H
= 0,
∂θ
η̇
ξ˙

∂H
+ ηρ,
∂K
∂H
= −
+ ξρ.
∂G
= −

(6.4)
(6.5)
(6.6)
(6.7)

Since both the instantaneous utility function and the two production functions are strictly concave, it follows from Cass (1965) that (6.3) through (6.7),
along with the transversality conditions

η(t) K(t) e−ρt → 0 as t → ∞andξ(t) G(t) e−ρt → 0 as t → ∞,are a suﬃcient characterization of the unique optimum path solving the planner’s problem.
To facilitate the arguments of the next section, we note that conditions(6.3) through (6.7) lead respectively to the following equations:C −1 = η(6.8)

ηαA(φK)α−1 (θG)1−α = ξβB((1 − φ)K)β−1 ((1 − θ)G)1−β(6.9)

η(1 − α)A(φK)α (θG)−α = ξ(1 − β)B((1 − φ)K)β ((1 − θ)G)−βη̇ = −Aηα(φK)α−1 (θG)1−α + ηρξ˙ = −B(1 − β)ξ((1 − φ)K)β ((1 − θ)G)−β + ξρ.

(6.10)(6.11)(6.12)

6 COMMAND ECONOMY28

The steady state optimum values of all variables associated with equations(6.8), (6.9), (6.10), (6.11) and (6.12) will be denoted by asterisks.
Using (6.8) and (6.11),

gd =

Ċ
= Aα(φK)α−1 (θ G)1−α − ρ.
C

(6.13)

Analogous to a Market Equilibrium,
Ċ
K̇
Ġ
Ẏ
=
=
=
C
K
G
Y

(6.14)

in steady state. Denote the common growth rate by g. Consistency between
the demand and the supply rate requires (using (2.2)) thatg = B((1 − φ)K β) (1 − θ).
(1 − θ) G(6.15)

Diﬀerentiating (6.5) and using (6.3)η̇ξ˙= = −g.
ηξ(6.16)

It can be shown that equation (6.16) implies that the transversality conditions are satisfied. Using (6.16) in (6.12),(1 − β) B ((1 − φ) K)β ((1 − θ) G)− β = g + ρEquations (6.17) and (6.15) lead to(6.17)

6 COMMAND ECONOMY

29(1 − θ)1/β = ((1 − β) g 1/β) .
g+ρ

(6.18)

Inverting (6.13) and (6.15) and multiplying out,gg + ρ 1/(1−α)θ 1−φ( )1/β ()=(1 − θ)1/β .
B
αA
1−θ φ

(6.19)

Finally, manipulating (6.3), (6.5), (6.19) and (6.18) in succession, we get
g + ρ 1/(1−α) 1 − α βg(1 − β) g 1/β
=
)
(
) .
( )1/β (
B
αA
α 1−β
g+ρ

(6.20)The behavior of the curves in (6.20) is shown in Figure 7. The LHS is theconvex, increasing curve Λ(g), while the RHS is the concave, increasing curveΓ(g). The curve Γ is bounded above by ((1 − α)/α) β (1 − β)(1−β)/β , thereby
guaranteeing a unique intersection in the positive orthant. The solution g ∗to (6.20) represents the command optimum steady state growth rate for the
economy. This gives rise to

Proposition 6.1 A unique, strictly positive welfare maximizing growth rate
exists for the Command Economy.

Figure 7 here.

Remark 8: For Case 1 of Remark 3, the Command Economy growth rate
is g ∗ = 8.5%. This is only slightly higher than the maximal growth rateof 8.1% associated with (τ ∗ , γ ∗ ). Clearly, the welfare associated with theCommand Economy growth rate is higher than that associated with a welfare maximizing (τw∗ , γw∗ ) in the Mixed Economy. Moreover, Proposition 4.2
has established that (τw∗ , γw∗ ) itself dominates (τ ∗ , γ ∗ ) welfare wise. Quite

7 COMMAND SOLUTION THROUGH MARKETS?

30

obviously therefore, it is not merely the best possible growth rate that matters. The instruments that are used to achieve it are equally important.
As the present example demonstrates, a Mixed Economy might be capableof growing at a rate close to the best possible growth rate for the system(i.e. the Command solution) and yet enjoy a low level of welfare. Thus, theproximity of the highest growth rate of the Mixed Economy and the Command Economy may give rise to an identification problem. Going over toCase 2 of Remark 3, the Command Economy growth rate turns out to be
9.3%, which is significantly higher than that achievable by the Mixed Economy. In this case the Command Economy dominates the Mixed Economy bya wide margin, both in terms of the growth rate as well as welfare. Nonetheless, the degree of centralization called for by the Command Economy has its
well-established social costs and cannot constitute a viable choice in a freesociety. It is worth considering, therefore, the possibility of decentralizing
the Command solution.

7

Command Solution through Markets?

From a purely formal point of view, the current value Hamiltonian H for
the Command Economy treats G as a private rather than a public input.
This is evident from the fact that while θ and 1 − θ are defined to be flowsof labor into the two sectors, the Hamiltonian is unable to distinguish thisfrom an alternative scenario, where G is a private input with allocations θ G
and (1 − θ) G into the two sectors. Strictly speaking therefore, the Hamiltonian, recognizes only two factors, viz, K and G, though the corresponding
Mixed economy model involves three. As a result, the equilibrium of the
Command Economy resembles two sector, two factor models of endogenous
growth (such as Rebelo (1991) or Mino (1996)). Any attempt to decentralize this equilibrium must therefore establish a correspondence between
the Command Economy and an artificially constructed two-sector economy
whose equilibrium imitates the Command Economy path.
Accordingly, the Decentralized Economy visualized below does not distinguish between L and G. Instead, it deals with a surrogate for the two,denoted by the fictitious factor of production N . As with G, the factor N is

7 COMMAND SOLUTION THROUGH MARKETS?

31

accumulable, but unlike G, it generates a rival and excludable flow of services entering the production functions of the two sectors. Thus, the economy
produces Y and Ṅ by means of the two rival inputs K and N . Hence thetwo production functions reduce to

Y = A Kyα Ny1−α , 0 < α < 1(7.1)

Ṅ = B Kgβ Ng1−β , 0 < β < 1,

(7.2)

and

where Ny and Ng are the respective flows of N into the two sectors.
The Decentralized Economy is assumed to be made up of four agents,
viz, the representative Household, two aggregative firms and the Government. The first three agents have well-defined objective functions which
they maximize at parametrically specified prices. The government provides
the initial stock of N and finances its acquisition of Ṅ from the revenue generated by the sale of N -services to the firms. The Household has a dynastic
structure and maximizes (2.3) subject to an instantaneous budget constraint.
Its budgetary resources fall into two parts. First, it has an income, rK(t)from private capital holdings, where r stands for the constant rate of interest
on K. Secondly, it receives a subsidy equal to ∆(t) from the Government
at each t as a compensation for the labor it supplies to the two firms. The
amount of the subsidy, which is beyond the control of the Household, will be
specified later. The level of ∆(t) for each t may also be viewed as a wage
rate, since L(t) = 1 ∀ t. The firms, however, do not pay wages directly to
the Household.17 Hence, the Household’s budget constraint isC(t) + K̇(t) = rK(t) + ∆(t),
17

(7.3)

Alternatively, ∆(t) could be distributed over the two firms, in a manner that will beobvious from below, to purchase labor from the Household.

7 COMMAND SOLUTION THROUGH MARKETS?

32

where K(0) = K0 has the same value as the initial capital stock for theCommand Economy. Firm I produces Y and is assumed to maximize theinstantaneous profit function

Π1 = AKyα Ny1−α − rKy − νNy ,(7.4)

where ν represents the price charged by the Government per unit use of Nservices. Commodity Y is treated as the numéraire. Firm II produces Ṅ bymaximizing instantaneous profits given byΠ2 = µBKgβ Ng1−β − rKg − νNg ,(7.5)

where µ is the price per unit charged by Firm II for its product, viz, Ṅ .18
The Government accumulates N by purchasing Ṅ from Firm II.19 The
net revenue accruing to the Government on the N -account is the diﬀerence
between its sales revenue from N services and its investment cost for creating additional N. By definition of ν and µ, the Government’s revenue andexpenditure at each t are

νN

(7.6)

µBKgβ Ng1−β .

(7.7)

and

Thus, the Government has a net revenue from infrastructure accumulationand sale equal to18

It may be noted that while ν is the price charged for the flow of N -services, the priceµ applies to the stock of N .
19
Without loss of generality, Firm II could be operated by the Government itself, inwhich case the relevant prices are used for bookkeeping.

7 COMMAND SOLUTION THROUGH MARKETS?

D = νN − µBKgβ Ng1−β .

33

(7.8)

This completes the description of the decentralized economy.
The next and last proposition of the paper demonstrates the existence ofa price-subsidy scheme for the Decentralized Economy that supports the first
best steady growth path of the Command Economy. By the very constructionof the subsidy, the Government’s budget will be seen to be balanced.

Proposition 7.1 There exist time dependent values of the Household subsidy and time invariant values of the rate of interest, the price of the fictitious
factor of production and the price of the output of Firm II such that(a) the Decentralized Economy chooses identically the same growth rate as
the Command Economy;(b) the fraction of the fictitious input employed by the first (second) firm isidentically the same as the fraction of labor force employed by the Command
Economy in producing the final (incremental stocks of the public) good;(c) the growth path of the fictitious commodity is identical with the growth
path of the stock of the public good in the Command Economy; and(d) the Government maintains a balanced budget along the chosen growth
path.

Proof: Chooser = r∗ = αA(φ∗ K ∗ )α−1 (G∗ )1−α ,(7.9)

where αA(φ∗ K ∗ )α−1 (G∗ )1−α is the marginal product of K along the optimal
steady state path of the Command Economy. Since both K and G grow at
the same rate, r∗ is a constant for all t. Similarly, letµ=ξ ∗ (t)∀tη ∗ (t)(7.10)

7 COMMAND SOLUTION THROUGH MARKETS?

34

and

ν = (g ∗ + ρ)µ.

(7.11)

Given the uniqueness of the optimal path chosen by the Command Economy,
it follows that µ and ν are well-defined. Further, µ is a constant from (6.16).
Hence, ν is a constant also.
The Household’s optimization exercise leads to the maximization of thecurrent value HamiltonianH h = ln C + ηh (r∗ K + ∆ − C),(7.12)

where ηh is the relevant costate variable associated with K̇. A set of necessary
conditions for the optimum solution to the problem is

C −1 = ηh(7.13)andη˙h = −ηh r∗ + ηh ρ = −ηh αA(φ∗ K ∗ )α−1 (G∗ )1−α + ηh ρ.

(7.14)The equation on the extreme right of (7.14) follows from (7.9).
The instantaneous profit maximization problems of Firms I and II leadto the following two conditions each:αAKyα−1 Ny1−α = r∗
(1 − α)AKyα Ny−α = ν(7.15)(7.16)

7 COMMAND SOLUTION THROUGH MARKETS?

35

and

µβBKgβ−1 Ng1−β = r∗

(7.17)µ(1 − β)BKgβ Ng−β = ν.

(7.18)

Equations (7.15) and (7.17) together imply

αAKyα−1 Ny1−α = µβBKgβ−1 Ng1−β = r∗ .

(7.19)

Similarly, (7.16) and (7.18) imply(1 − α)AKyα Ny−α = µ(1 − β)BKgβ Ng−β = ν(7.20)

Using (6.16) and the definitions of µ and ν, (7.19) and (7.20) reduce toη ∗ αAKyα−1 Ny1−α = ξ ∗ βBKgβ−1 Ng1−β ,(7.21)

η∗ (1 − α)AKyα Ny−α = ξ ∗ (1 − β)BKgβ Ng−β(7.22)ξ˙∗ = −ξ ∗ (1 − β) BKgβ Ng−β + ξ ∗ ρ.

(7.23)

andEquations (6.9), (6.12) and the definitions of φ∗ , K ∗ and G∗ imply that
Ky = φ∗ K ∗ , Kg = (1 − φ∗ )K ∗ , Ny = θ ∗ G∗ and Ng = (1 − θ∗ ) G∗ satisfy(7.21), (7.22) and (7.23).
Going over now to the Household’s problem, we may diﬀerentiate (7.13)
and substitute from (7.14) to obtain7 COMMAND SOLUTION THROUGH MARKETS?

36

Ċ= r∗ − ρ.
C
From (6.13) and (7.9), the Household chooses the same growth rate as thatof the Command Economy. However, this still does not determine the levelof C(t). To fix the latter, choose ∆(t) = D(t) for each t. That is, thesubsidy provided to the Household is exactly equal to the net revenue of
the Government as defined in (7.8). Hence, the Government’s Budget isbalanced. Moreover, D(t) > 0 ∀ t. For, using (7.10) and (7.11),

νG∗ − µB((1 − φ∗ )K ∗ )β ((1 − θ∗ )G∗ )1−β = µ((g ∗ + ρ)
(1 − φ∗ )K ∗ β−B() (1 − θ ∗ ))G∗
(1 − θ∗ )G∗= µ(θ∗ g ∗ + ρ)G∗
( using (6.2) )> 0.
(7.24)The positivity of D(t) > 0 allows it to be treated as compensation for labor,or its imputed wage rate.
Equations (7.15) through (7.18) imply then that the RHS of (7.3) reduces
to Y (t). Rewriting (7.3) asCK̇
φ∗ K ∗+= φ∗ A( ∗ )α−1 ,K K
Gbalanced growth requires C and K to grow at the same rate. In other words,
K̇/K = g ∗ . This in turn means that

C(t) = K(t)(φ∗ A(

φ∗ K ∗ (t) α−1)− g ∗ ) ∀ t.
G∗ (t)

8 CONCLUSIONS

37

Hence, (6.1) implies that the choice of C(t) by the Household is the same asthat for the Command Economy. Therefore, comparing (6.8) and (7.13), we
see that ηh (t) = η ∗ (t) ∀ t.
Collecting the results, it follows that equations (7.13), (7.14), (7.21),(7.22)and (7.23) describing the evolution of the Decentralized Economy are
the same as (6.8), (6.9), (6.10), (6.11) and (6.12) characterizing the Command Economy path. The solution to the Command Economy problem
being unique, the Decentralized Economy chooses identically the same path.
Q.E.D.
Remark 9: Even though the above proposition establishes support prices
for the Command Economy, it does so in terms of an economy which behavesvery diﬀerently from the economy I started out with. First, the Household
supplying labor freely to the firms in exchange for a Government subsidy
may not be an operationally feasible idea. Secondly, it is diﬃcult to imagine
profit maximizing entrepreneurs treating the two factors of production, laborand infrastructure, as a single factor and coming out with a market demandfor it. A private economy requires prices of inputs separately to define thecorresponding demand and supply curves. This calls for a separation oflabor from the public input, which the decentralized economy prevents.20
Hence, the Mixed Economy’s ineﬃcient equilibrium may be the only feasible
growth path for the system. As pointed out in the Introduction, this posesa dilemma for developing economies trying to adapt their economies to freemarket structures.

8

Conclusions

In this paper I have attempted to take a fresh look into the issues raised byBarro-F M S-BS in the context of optimal taxation for infrastructure devel20
As should be obvious, the cause underlying market failure in this economy is somewhat
diﬀerent from the standard externality issues associated with public goods (Samuelson(1954)).

8 CONCLUSIONS

38

opment and growth by constructing a full-fledged two-sector model producing
a final good and changes in the infrastructural stock respectively. The final
good is privately produced. The flow of infrastructure being a pure public
input in the model, the sector producing infrastructure cannot function competitively and is viewed as being under the Government’s control. Factor
shares are not determined by marginal productivities in this sector. Thiscalled for the introduction of a new variable into the model, the share of
expenditure out of tax revenues on the diﬀerent private inputs used for infrastructure accumulation. Thus, the optimal taxation problem had to beconsidered in tandem with the problem of optimal choice of the share. I
established that growth maximization is not equivalent to welfare maximization in the economy considered. The equivalence can hold, however, in asecond best sense, under the further constraint that factor shares be thesame in the public and the private sectors.
The two-sector structure of the model made it imperative to study thetransitional stability of the equilibrium growth path. Parallel to F M S therefore, I demonstrated the dynamic steady state path of the Mixed Economy
to be transitionally stable.
Finally, the paper considered the nature of the first best steady statepath for the system. This is the solution to the Command Economy exercise. Given the parameters of the technology and preferences, the Command
Economy simply calculates the economy’s best growth potential. I established that the highest growth rate of a Mixed Economy (for certain parameter values) might not fall too short of the Command Economy growth rate.
The fact that a partially marketized economy may be capable of growing at
a rate close to that of the Command Economy is no cause for rejoice; for thehigh growth rate may be sustained at the cost of eﬃciency and welfare. Thus,
performance wise, a Mixed Economy growing at an impressively high rate
might give rise to an identification problem. Nonetheless, the centralization
implied by a Command Economy being an unpractical choice, I looked into
the possibility of decentralizing the first best growth path. Though apparently decentralizable, I argued that it is actually not so, because the private
economy in question (that sustains the Command Economy path) needs totreat the public service and labor as a joint input. Markets for such inputs
cannot exist in the real world. The conclusion that emerges is that the existence of accumulable pure public inputs imposes severe constraints on the8 CONCLUSIONS

39

growth paths available to a developing economy, especially those engaged in
the process of liberalization.
I have adhered all through to the Barro-F M S-BS specification that theGovernment’s budget is balanced at each point of time. An extension of thepaper might consider the possibility that the Government has recourse torevenue raising instruments other than income taxes. In particular, it is ofsome interest to consider the possibility of deficit financing through borrowing from the public under the constraint that the Government’s budget isbalanced over time. A second extension would be to follow BS and introduce public services with congestion. Yet another possibility is to look into
Alesina and Rodrik (1994) type political economy issues by introducing agent
specific endowments and nonlinear taxation. Some of these will be taken up
in future work.

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