Walrasian General Equilibrium Allocations and
Dynamics in Two-Sector Growth Models
Bjarne S. Jensen, Copenhagen Business School*
Abstract
This paper analyses and solves miniature Walrasian general equilibrium
systems of momentary and moving equilibria. The Walrasian framework
encompasses the fundamental neoclassical and classical two-sector growth
models; the regimes (families of solutions) of steady-state and persistent
growth per capita in various competitive two-sector economies are parametrically characterized. Moreover, the endogenous behavior of relative prices
and the sectorial allocation of primary factors are analyzed in detail. The
technology parameters of the capital good industry are decisive for obtaining
long-run per capita growth in closed (global) economies. A review of the
literature complements the theorems on the general equilibrium allocations,
the dynamic systems, and the time paths of Walrasian two-sector economies.
To be presented at the Conference:
Dynamics, Economic Growth and International Trade, VI
Vienna, June 22-23, 2001.
JEL Classification Number(s): F11, F43, O40, O41
Abbreviations: None.
Number of Figures: 4
Number of Tables: 0
∗) Bjarne S. Jensen, Department of Economics, Copenhagen Business School,
Solbjerg Plads 3, DK-2000 Frederiksberg, Denmark. Tel: (+45) 3815 2583;
Fax: (+45) 3815 2576; E-mail: [email protected].
1. Introduction
The aim of pure theory and models is to deduce definite conclusions from
explicit premises. In terms of theory construction, assumptions should be
fundamental (general), and from a technical (mathematical) point of view,
they should be tractable. The concepts (terms and parameters) used in the
theory must be capable of exact interpretation and refer to observable facts as
closely as possible. In this vein, the generality and tractability of assumptions
constitute the fascinating simplicity of celestial mechanics, Gauss [1809], Landau & Lipschitz [1976]. Simplicity in this double sense has also motivated the
research in economic growth theory. Great emphasis was naturally first given
to labor and capital accumulation in aggregate (one-sector) growth models. In
the endogenous growth literature, the scope of aggregate models has recently
been enlarged by increasing the number of primary factors, such as human
and intangible capital (education, useable knowledge, learning by doing) as
well as the number and quality of intermediate inputs, through innovations.
But if varieties (qualities) of the final product are introduced, the production process is mostly assumed to use only a single factor, e.g., labor. Hence,
factor allocation issues are mainly omitted , and economic generality (fundamentals) is impaired. Although the new aggregate growth models in the
last decade have made valuable contributions in the understanding of economic development, it must be recognized that, De Long & Summers [1991,
p. 486], ”the importance of disaggregation calls into question the utility of
research programs directed at spelling out alternative mechanisms driving
all of aggregate growth in single-good models, as if relative prices and relative quantities of different products did not matter. Economists’ emphasis
on single-good models is odd given that these models offer almost no scope
for the relative price effects economists stress in most contexts”.
An extensive literature on two(multi)-sector growth models, however, began in the 1960s, but debate and controversy reigned and still linger. The
seminal work on two-sector growth models with flexible sector technologies
was done by Uzawa [1961-62, 1963], Inada [1963], Drandakis [1963]. A framework for efficient factor allocation using the price mechanism – ”miniature
Walrasian general equilibrium system,” Solow [1961-62, p. 48-50] – was established. But the theoretical and empirical relevance of various sufficient
conditions for stability of steady states was deemed to be of modest value.
In a well-known survey about the current state of the art, Hahn & Matthews
[1965, p. 39] said: ”Two-sector growth models of the type we have been
considering do not represent any great advance in realism over one-sector
models. As far as conclusions are concerned, while those of two-sector models do point to certain complications absent from one-sector models, the
modifications that need to be made to the broad results found in one-sector
models are not very fundamental.” If this verdict on the economic insights
1
from the general analysis of two(multi)-sector growth models was correct, the
historical relevance and future of growth models would be dim. In contrast
to growth theory, the main interest of international trade theory is above
all directed to factor allocation, output composition, and their dualities for
commodity and factor prices, Jones [1956-57], Kemp [1969], Dixit & Norman
[1980], Gandolfo [1998]. A systematic and elegant treatment of the basic twofactor, two-sector general equilibrium model was given by Jones [1965], and
the static general equilibrium analysis was extended to growing two-sector
economies, where the conditions for convergence to balanced growth were
also examined.
Soon, Hahn returned to the two-sector growth story and first noted that
its connection with ordinary general equilibrium analysis had not been given
the prominence it deserved; next, Hahn [1965, p. 339] stated: ”The story
starts with a given stock of capital inherited from the past. Since there are
constant returns to scale we might just as well start with a given capitallabor ratio. The first question, familiar to general equilibrium theorists, is:
does there exist a momentary equilibrium for any given capital-labor ratio?
The second question asks: is such an equilibrium unique? The answer to
this is important for the simple reason that multiple momentary equilibria
would make it impossible without further postulates to predict the subsequent
development of the system from the initial conditions. Supposing momentary
equilibrium to be uniquely determined by the capital-labor ratio we now ask:
where is the system going? Here we want to know whether a steady state is
approached or not. Answering this question involves an examination of the
existence of a steady state solution, possibly also the uniqueness of this and
of course rather straight-forward stability analysis. The questions asked may
be answered in varying degrees of generality” (italics ours).
Though these questions and the answers to them could seem straightforward, the truth is that the issues indicated have dominated subsequent
research and are still not resolved. Our purpose is now to provide definite
answers to these static and dynamic issues of existence, uniqueness, and stability. As we shall see, in solving even standard two-sector growth models, the
uniqueness concept of static general equilibria is blurred and confounded with
the uniqueness of general equilibrium dynamics (initial value problems). An
inconvenient or intractable choice of the state variable in general equilibrium
dynamics has created profound delusions (including artifacts of limit cycles) about the solutions to basic neoclassical and classical two-sector growth
models. We shall mathematically eliminate the ”causal indeterminacy” problem. For both steady state and persistent (endogenous) per capita growth,
the critical importance of technology parameters for the capital good (rather
than consumer good) industry has been routinely missed .
Moreover, the technological lack of generality in the widely adopted
Inada [1963, 1964] conditions eventually impeded further advances in pure
2
theory and reduced the observable relevance of neoclassical and other growth
models, cf. Aghion & Howitt, [1998, p. 11]. In the structural relations and the
dynamic systems of this paper, the size of sectorial substitution elasticities
and ”total productivity coefficients” are among the key parameters. For extensive empirical/theoretical evidence on the role of substitution elasticities
and other parameters, see Easterly & Fischer [1995], Prescott [1998], Jensen
& Wang [1997, 1999], Klump & Grandville [2000]. The main expositions and
references to the early two-sector growth literature are: Stiglitz & Uzawa
[1969], Burmeister & Dobell [1970], Wan Jr. [1971], Gandolfo [1980].
The organization of the paper is as follows. In section 2, we present and
solve the structural equations for supply and demand, which uniquely give
the timeless (static and comparative static), competitive general equilibrium
(CGE) solutions for all the variables as distinct composite functions of the
factor endowments. Section 3 analyzes the dynamic system and alternative
evolutions of the two-sector general equilibria. Some of our results resemble
Rebolo [1991], although our capital good sector also employs labor, and the
subfamily of persistent (endogenous) growth solutions does not presume a
linear technology for the core or composite capital good. Final comments are
offered in section 4.
In the general equilibrium framework, two-sector growth models should
present the pure theory (abstract model, ideal story) of the intricate economic
interaction between man (labor) and machinery (capital), both momentarily
and in the long-run historical perspective.
2. General Equilibrium of Two-Sector Economies
2.1. Supply side
Consider an economy consisting of a capital good industry (sector) and a
consumer good industry, labeled 1 and 2, respectively. Sector technologies,
Fi (Li , Ki ), i = 1, 2, are described by nonnegative smooth concave homogeneous production functions with constant returns to scale in labor and capital
Yi = Fi (Li , Ki ) = Li Fi (1, Ki /Li ) ≡ Li Fi (1, ki ) ≡ Li fi (ki ) ≡ Li yi , Li = 0 (1)
(2)
Fi (0, Ki ) = Ki Fi (0, 1), Fi (Li , 0) = Li Fi (1, 0) = Li fi (0)
Fi (0, 0) = 0, Fi (0, 1) ≥ 0, Fi (1, 0) ≡ fi (0) ≥ 0
(3)
where the function fi (ki ), is a strictly concave monotonic increasing function
in the single ratio variable ki ∈ [0, ∞[, i.e., fi has the properties
∀ki > 0 : fi (ki ) = dfi (ki )/dki > 0,
fi (ki ) = d 2fi (ki )/dki2 < 0
(4)
lim fi (ki ) ≡ β̄i ≤ ∞, lim fi (ki ) ≡ β i ≥ 0, fi (ki ) ∈ Ji ≡ β i , β̄i . (5)
ki →0
ki →∞
3
The domain of Fi may not always include the axes, and hence, the boundary
values (2-3) are not defined. The limit symbols, β̄i , β i , (5), impose no effective
restrictions on marginal products of capital. The sectorial output elasticities
with respect to marginal and proportional factor variation are, cf. (1),
M P Li
∂Yi Li
ki fi (ki )
=
=1−
> 0, ki = 0,
∂Li Yi
APLi
fi (ki )
M PKi
∂Yi Ki
ki fi (ki )
Ki ≡ E(Yi , Ki ) ≡
=
=
= E(yi , ki ) > 0,
∂Ki Yi
APKi
fi (ki )
∂Yi λ
i ≡ E(Yi , λ) ≡
= Li + Ki = 1,
∂λ Yi
∂Ki ki
L (σi − 1)
∂Li ki
K (1 − σi )
= i
, E(Li , ki ) ≡
= i
E(Ki , ki ) ≡
∂ki Ki
σi
∂ki Li
σi
Li ≡ E(Yi , Li ) ≡
(6)
(7)
(8)
(9)
where Li (Ki ) is the sectorial output elasticity of labor (capital); i , the output elasticity with respect to scale variation, is independent of ki . The elasticities Li and Ki – which have the same signs as M PLi and M PKi and
hence by (4) are always positive – are homogeneous functions in Li , Ki of
degree zero, regardless of the degree of F (Li , Ki ), (1). Therefore, as to the
behavior of output in response to factor variation, the elasticities Li , Ki , by
only depending on input ratios, provide succinct expressions to characterize analytically the variational properties of the sector production functions.
The general relationship between the sectorial factor output elasticities, (67), and the sectorial substitution elasticities between labor and capital, σi , is
given by formulas, (9), which are monotonic functions in ki for constant σi .
At any point of the isoquants (1), the rates of technical substitution for
Fi ∈ C 2 -class is measured by the ratio of the marginal products of labor and
capital, denoted the marginal rates of substitution, ωi (ki ), which by (4) are
positive monotonic functions. Hence, by (4-7), we obtain
ωi (ki ) =
M P Li
L
fi (ki )
− ki = i ki > 0,
= M PKi
fi (ki )
Ki
∀ki > 0.
(10)
The factor endowments, total labor force (L) and the total capital stock (K),
are inelastic supplied and are fully employed (utilized), i.e.,
L = L1 + L 2 ,
K = K1 + K 2 ,
L1 /L + L2 /L ≡ λL1 + λL2 ≡ l1 + l2 ≡ 1,
K1 /K + K2 /K ≡ λK1 + λK2 ≡ 1,
(11)
(12)
or the full employment assumption (11-12) may be rewritten as
k ≡ K/L ≡ l1 k1 + l2 k2 ≡ k2 + (k1 − k2 )l1 .
4
(13)
For later reference, we note that factor allocation fractions λLi , λKi , (11-12),
are related, and λL1 , λKi , are given by, cf. (13),
λL1 ≡ l1 ≡
k − k2
ki
, λKi ≡ li , k1 = k2 ;
k1 − k2
k
(14)
Clearly, the expressions (14) and k1 = k2 dictate that
[0 < λLi < 1] ⇔ [0 < λKi < 1], [k1 = k = k2 ] ⇔ [λLi = λKi ] i = 1, 2 (15)
[ki > k > kj ] ⇔ [λKi > λLi ] ⇔ [λKj < λLj ], {i, j} = {1, 2}
(16)
Free factor mobility between the two industries and efficient factor allocation
impose the common MRS condition, cf. (10),
ω = ω1 (k1 ) = ω2 (k2 ),
(17)
From the efficiency of factor allocation follows – for any production functions
(1, 4), cf. (17), (10-8),
k2 (ω)
>
<
k1 (ω) ⇔ K2
>
<
K1 ⇔ L1
>
<
L2 .
(18)
For the variables k1 and k2 to satisfy (17), it is, beyond (4), further required
that the intersection of the sectorial range for ω1 (k1 ) and ω2 (k2 ) is not empty,
ωi (ki ) ∈ Ωi = [ω i , ω i ],
Ωi ⊆ R+ ,
i = 1, 2
ω ∈ Ω ≡ Ω1 ∩ Ω2 = [ω, ω] = ∅,
(19)
(20)
where ω is the common wage-rental ratio, (17). With variable sector capitallabor ratios, k1 (ω), k2 (ω), the diversification cone becomes, cf. (20),
CK = {(L, K) ∈ R2+ | k1 (ω) < k < k2 (ω) ∨ k2 (ω) < k < k1 (ω); ∀ω ∈ Ω} (21)
The two industries are assumed to operate under perfect competition (zero
excess profit); absolute (money) factor prices (w, r) are the same in both industries; and absolute (money) output (product, commodity) prices (P1 , P2 )
represent unit cost. The model will have no money circulation, so money only
serves as a numeraire (unit of account) for the price variables. Of course, only
relative money prices matter in the model here; but we prefer to retain money
prices rather than use one of the commodities as the numeraire.
Hence, we have the competitive producer equilibrium equations,
Pi = 0.
w = Pi · M PLi , r = Pi · M PKi ; ω = w/r,
Yi = Lyi li , Pi Yi = wLi + rKi , Li = wLi /Pi Yi , Ki = rKi /Pi Yi
5
(22)
(23)
or, equivalently,
w
r
yi L i
ω
ki
= fi (ki ) ; Li =
, K i =
. (24)
= y i Li ,
=
Pi
Pi
ω
ki + ω
ki + ω
M PK2
y 2 L2
M P L2
f (k2 )
f2 (k2 ) − k2 f2 (k2 )
P1
=
= 2
=
(25)
p≡
=
=
P2
M PK1
f1 (k1 )
y 1 L1
f1 (k1 ) − k1 f1 (k1 )
M P L1
Remark 1. As our object is indeed miniature Walrasian general equilibrium
economies, we may quote, for (22), Walras [1954, p. 385]:
”1. Free competition brings the cost of production down to a minimum.
2. In a state of equilibrium, when cost of production and selling price are
equal, the prices of the services are proportional to their marginal productivities, i.e., to the partial derivatives of the production function.
These two propositions taken together constitute the theory of marginal
productivity. This is a cardinal theory in pure economics-”
∇
The connection between relative factor prices (service prices) and relative
commodity (output) prices follows from (10, 17, 25),
p(ω) =
M PK2 [k2 (ω)]
f [k2 (ω)]
P1
(ω) =
= 2
,
P2
M PK1 [k1 (ω)]
f1 (k1 (ω)]
ω = w/r.
(26)
In the general equilibrium context, the factor price-commodity price (FPCP)
correspondence p(ω), (26), is quite remarkable by being robust [invariant,
hinging on stipulations (8) and (23)] to any demand-side specifications and
to the amount of factor supplies; the locus p(ω) is invariant, but the actual
general equilibrium point of (26) depends on all supply- and demand-side
parameters and initial factor endowment proportions, cf. sections below. The
exact form of the composite function (26) needs particular attention.
Remark 2. The elasticity of (26) is generally, by the composite rule:
E (P1 /P2 , ω) = E (M PK 2 , k2 ) E(k2 , ω) − E (M PK 1 , k1 ) E(k1 , ω)
= (−L2 /σ2 )σ2 − (−L1 /σ1 )σ1 = L1 − L2 = K 2 − K 1
(27)
(28)
which is positive/negative according to (18). Evidently, p(ω) is always inelastic, as E(P1 /P2 , ω), (28), is numerically less than unity, and hence,
E(ω, P1 /P2 ) = 1/ (K 2 − K 1 ) > 1
(29)
i.e., everywhere numerically larger than unity. This, along with the percentage change of any competitive commodity price always being a convex
combination of percentage changes in factor prices, P̂i = Li ŵ + (1 − Li )r̂,
is in accordance with the Stolper-Samuelson theorem and the price version
of the ”magnification effects” in Jones [1965]; cf. Deardorff [1994].
∇
Gross domestic product (GDP) Y , is the monetary value of outputs from
both sectors,
Y ≡ P1 Y1 + P2 Y2 = L(P1 y1 l1 + P2 y2 l2 ) ≡ Ly
6
(30)
and is with (22-25) equal to the total factor incomes
Y = wL + rK = L(w + rk) = L(ω + k)Pi fi (ki ) = Ly,
(31)
which defines the factor income distribution shares, δK + δL = 1,
δK ≡ rK/Y = rk/y, δL ≡ wL/Y = w/y; δK /δL = k/ω, δK = k/(ω + k). (32)
CES sector technologies, boundary values, and Inada conditions
The general CES forms of Fi (Li , Ki ), (1), γi > 0, 0 < ai < 1, σi > 0, are
i
Kiai ≡ Li fi (ki )
Yi = Fi (Li , Ki ) = γi L1−a
i
σ σ−1
i
σi −1
σi −1
i
σi
σi
Yi = Fi (Li , Ki ) = γi (1 − ai )Li
+ ai K i
≡ Li fi (ki )
(33)
(34)
where CES forms of fi (ki ) and fi (ki ) are
σi = 1,
σi = 1,
σi = 1,
fi (ki ) = γi kiai , fi (ki ) = γi ai kiai −1
σ /(σi −1)
(σ −1)/σi i
fi (ki ) = γi (1 − ai ) + ai ki i
,
1/(σi −1)
−(σ −1)/σi
fi (ki ) = γi ai ai + (1 − ai )ki i
(35)
(36)
(37)
By evaluating (36-37), the limits of fi (ki ) and fi (ki ) become,
σ /(σ −1)
(∀i : σi ≷ 1 ⇒ ai i i
≶ 1),

σi
σi

 lim fi (ki ) = lim γi aiσi −1 ki = 0,
lim fi (ki ) = γi (1 − ai ) σi −1
ki →0
ki →0
ki →∞
σi
(38)
σi < 1 :
σi −1

 lim fi (ki ) = γi ai ,
lim fi (ki ) = 0
ki →0
σi > 1 :

σi

 lim fi (ki ) = γi (1 − ai ) σi −1 ,
ki →0

 lim fi (ki ) = ∞,
ki →0
ki →∞
σi
σ −1
lim fi (ki ) = lim γi ai i ki = ∞
ki →∞
ki →∞
σi
(39)
σ −1
lim fi (ki ) = γi ai i
ki →∞
A major problem in both one- and two-sector growth models has been a clear
delineation of the class (parametric family) of technologies Fi that comply
with Inada [1963, p. 120] and Uzawa [1962-63, p. 108] boundary conditions
fi (0) = 0,
fi (0) = ∞,
fi (∞) = ∞, i = 1, 2
fi (∞) = 0, i = 1, 2
(40)
(41)
It is a paradox that regularity properties of ”neoclassical ” production functions are often codified by combining the concavity properties (4) with the
boundary (limit) values in (40-41), because - as the comparison with (38-39)
7
shows - neither (40) nor (41) are satisfied by the general CES family, σi = 1.
Although, e.g., Drandakis [1963, p. 27] and especially Wan [1971, pp. 37-40,
p. 118] noted that the CD form (35) is the only CES member that fulfills the
Inada-Uzawa conditions (40-41), these conditions acquired a prominent role
in the growth literature, where they were relied on to guarantee the existence
of at least one balanced growth path. Relevant sufficient stability conditions,
however, usually involved sectorial substitution elasticities larger or smaller
than one, (σi ≶ 1), which with CES technologies was then inconsistent with
the use of (40-41) as the existence condition. Hence, latent contradictions
pervade much of the standard dynamic analysis of neoclassical and classical
two-sector growth models. Moreover, the possibility of persistent (”endogenous”) growth per capita was inherently precluded, since the limit fi (∞)
in (41) must be relaxed, as demonstrated in general for the Solow model in
Jensen & Larsen [1987]; cf. Durlauf & Quah [1999, p. 257].
Incidentally, the generality problem with (40-41) appears in various versions. Caballé & Santos [1993, p. 1046], Ladrón-de-Guevara et al. [1999, p.
612] impose – to guarantee interiority conditions in intertemporal optimization – the unbounded marginal productivity conditions at the lower boundary:
lim M PL (L, K̄) = ∞,
L→0
lim M PK (L̄, K) = ∞,
(42)
F (0, K̄) = 0
(43)
K →0
as well as both factors being essential:
F (L̄, 0) = 0,
For general CES functions with σi = 1, (42) and (43) are also contradictory,
as the derivative limit (42) requires σi > 1, which implies that each production factor is inessential , i.e., nonzero limits in (43), cf.(34). When factors are
essential (43), marginal products cannot intuitively be expected to become
infinitely large, no matter how little of the factor is employed (42). Accordingly, as CES with σi > 1 demonstrates, the assumptions (42-43) cannot be
maintained simultaneously.
The question remains whether the conditions (40-41), (42-43) can apply
to any other broad class of technologies, e.g., VES (variable elasticity of substitution). From a mathematical point of view, (40-41) look rather innocent,
since they are satisfied by any power function; but in economic terms, they
are extravagant asymptotic productivity assumptions. When labor itself is so
important as to be essential, fi = 0, then labor productivity is not likely to
increase beyond any upper bound, even with an infinitely large capital/labor
ratio. By the way, it makes economic sense that the salient property of the
marginal product of capital f (∞), being always bounded away from zero,
is related to labor being inessential : Fi (0, Ki ) > 0, which haplessly cannot
even be symbolized by the function fi . But with CES and σi > 1, the similar
8
property of capital being inessential, fi (0) > 0, also implies: Fi (0, Ki ) > 0,
∀Ki > 0, since Fi (0, 1) > 0 ⇔ Fi (1, 0) = fi (0) > 0, cf. (34), (3). In standard
dynamic models, the critical condition for persistent growth per capita is
actually that labor is inessential, Fi (0, Ki ) > 0, as shown in Jensen & Wang
[1997, p. 114]. For more on CES, see Klump & Preissler [2000].
For the CES technologies, we have the well-known monotone relations
between marginal rates of substitution and factor proportions
σ
1 − ai 1/σi
1
1 − ai i
σi
ki ,
ki = (ωi ) , ci =
i = 1, 2.
(44)
ωi =
ai
ci
ai
With (35-37), and irrespective of the numerical size of σi , we always have,
cf. (20-21): Ω = R+ = [0, ∞[.
As is well known, for σi > 1, the CES isoquants will cut the (L, K)-axes;
but their isoquant tangents will have the slopes, 0 and −∞, at the K-,
L-axis intercepts. No extra restrictions need to be imposed on Fi (34) to
avoid zero employment of factors for any conceivable finite positive values
of the wage-rental ratio, ω. However, with σi > 1, we can initially start and
continue along the coordinate axes. In fact, the qualitative dynamics along the
coordinate axes is topologically equivalent to the dynamics on the interior
factor space. Variants of Ak-models provide such examples, cf. Jensen &
Larsen [1987, pp. 135]. Such topologically equivalent dynamics evidently does
not hold when σi < 1 (factors are essential). The literature contains rather
misleading graphics of CES isoquants for production and utility functions,
especially in the case of σi > 1, see Inada [1964, p. 140]. Despite different
visual appearances and shapes of CES level curves (33-34), the formulas (44)
apply with no restrictions on σi .
With two-sector models and CES technologies, it is apparent from (44)
that sectorial factor ratio (”intensity”) reversals can only be avoided if and
only if σ1 = σ2 and a1 = a2 . Hence, with σ1 = σ2 , there will be a intersection
(reversal ) point, (ki , ωi ) = (k̄, ω̄):
a1 (1 − a2 )
k̄ =
a2 (1 − a1 )
σ2
σσ1−σ
2
1
c σ1
= 2σ2
c1
σ
1
2 −σ1
,
c2
ω̄ =
c1
σ
1
2 −σ1
(45)
Another lack of generality of premises in two-sector growth and trade models
- similar to (40-41) – is due to the traditional capital-intensity ranking:
∀ω ∈ Ω : k2 (ω) > k1 (ω) ⇔ K2 < K1 ⇔ L1 < L2 ,
(46)
which with CES is equivalent to severe technology parameter restrictions
∀ω ∈ Ω : k2 (ω) > k1 (ω) ⇔ σ1 = σ2 ∧ a2 > a1
9
(47)
Supply and price correspondence with CES technologies
With (35), (37) and (44), the relative commodity prices (comparative costs)
(26) become, with σi = 1 and σi = 1, respectively,
f2 [k2 (ω)]
γ2 aa22 (1 − a2 )1−a2 a2 −a1
γ2 a2 k2 (ω)a2 −1
=
ω
=
f1 [k1 (ω)]
γ1 a1 k1 (ω)a1 −1
γ1 aa11 (1 − a1 )1−a1
1/(σ2 −1)
γ2 a2 a2 + (1 − a2 )k2 (ω)−(σ2 −1)/σ2
f2 [k2 (ω)]
=
p(ω) = 1/(σ −1)
f1 [k1 (ω)]
γ1 a1 [a1 + (1 − a1 )k1 (ω)−(σ1 −1)/σ1 ] 1
p(ω) =
=
(1 + c2 ω 1−σ2 )
σ /(σ1 −1)
(1 + c1 ω 1−σ1 )1/(σ1 −1)
γ1 a1 1
(49)
1/(σ2 −1)
σ /(σ2 −1)
γ2 a 2 2
(48)
(50)
and, with σ1 = σ2 = σ :
γ2
p(ω) =
γ1
a2
a1
σ
1 + c2 ω 1−σ
1 + c1 ω 1−σ
1/(σ−1)
,
1 − ai
ci =
ai
σ
i = 1, 2.
(51)
Hence, cf. (45), and with σ1 = σ2 ,
σ /(σ −1)
γ2 a2 2 2
p̄ = p(ω̄) =
σ 1−1
σ1−σ
2
2
2 −σ1
c
2
1 + c2 c1
σ 1−1
σ1−σ
1
1
−σ1
σ1 /σ1 −1
2
c
1 + c1 c21
γ1 a1
(52)
Since the CES marginal rate of substitution ωi , (44), always has the limit
values zero and infinity, we need, for precise geometry and intuition, to know
the limits of the relative prices p(ω), (50) for ω going to zero and to infinity.
To this end, let
p∗ ≡
σ /(σ −1)
γ2 a 2 2 2
,
γ1 aσ1 1 /(σ1 −1)
p∗∗ ≡
γ2 (1 − a2 )σ2 /(σ2 −1)
γ1 (1 − a1 )σ1 /(σ1 −1)
(53)
Lemma 1. The graph of the function p(ω), (50) – the CES factor price-commodity price (FPCP) correspondence – has finite and infinite limits, according to the following CES classification:
σ1 < 1,
σ2 < 1 :
σ1 > 1,
σ2 > 1 :
σ1 > 1,
σ2 < 1 :
σ1 < 1,
σ2 > 1 :
lim p = p∗
ω→0
lim p = p∗∗
ω→0
lim p = 0
ω→0
lim p = ∞
ω→0
10
lim p = p∗∗
(54)
lim p = p∗
(55)
ω→∞
ω→∞
lim p = 0
(56)
lim p = ∞
(57)
ω→∞
ω→∞
At the single reversal point (k̄, ω̄), (45), diversification cones Ck , (21), degenerate to one point. The reversal price ratio, P1 /P2 = p̄ ≡ p(ω̄) is always
a maximum (iff σ1 > σ2 ) or a minimum (iff σ2 > σ1 ), cf. Figure 1.
For the substitution elasticities (54)-(56), the range of p(ω), (50) is bounded.
With γ1 = γ2 , and both substitutions elasticities, either small (54) or large
(55), the range of p(ω) becomes a narrow interval, and there will be only
small differences between the values of p∗ and p∗∗ (53) - especially when ai ,
i = 1, 2 have similar size, cf. (50).
Iff σ1 = σ2 = 1, the functions, p(ω), (51), are always monotonic, bounded,
and increasing between p∗ and p∗∗ , iff a2 > a1 . Only the CD relative prices,
p(ω), (48), are both monotonic and unbounded.
Proof. The convergence of (50) to (54-57) is rather easily established; proof
is given in Jensen et al. [2001].
The factor-price equalization (FPE) theorem in trade theory is traditionally obtained from the monotonicity of the FPCP graph, which evidently
requires extreme parameter restrictions on sector technologies, see Fig.1; cf.
Minhas [1962], Kemp [1969, p. 9], Dixit & Norman [1980, p. 16], Gandolfo
[1998, p. 84].
Insert figure 1 about here
2.2. Demand side
As to aggregate demand (expenditure shares) decomposition between consumption and investment (saving), we shall employ two macro saving relations, viz., proportional saving and classical saving, which have been the
standard opposites in much of the growth and trade literature. In an extensive
empirical search for linkages between long-run economic growth and a variety
of economic, political, and institutional indicators, Levine & Renelt [1992]
found that, while almost all empirical results are fragile, a robust connection
between growth and the share of investment in GDP could be identified. We
proceed along this line with (58) below.
The case of different saving propensities for ”capitalists” and ”workers”
will not be considered here, as we should then have to introduce, besides the
factor income distribution, also the wealth (ownership, asset) distribution and
the personal income distribution. To ignore the wealth distribution and just
to consider different saving propensities for the capital and labor components
of the factor income distribution easily creates problems and paradoxes, see
Pasinetti (1962), Samuelson & Modigliani (1966), Stiglitz (1967). Through
history and across countries, households, businesses, and government have
contributed in different proportions to national saving. It is immaterial for
our purposes whether investment is controlled by owners or managers. The
classical assumption with saving only out of capital income allows us to study
11
the consequences of various demand side specifications for the character of the
momentary general equilibrium solutions and the evolution of two distinct
macrodynamic systems. Intertemporal optimization should not be allowed to
obscure and burden the analysis in this paper on basic two-sector dynamics.
Hence, we use two (”neoclassical and classical”) monetary saving functions
(I) S = sY,
0 < s < 1;
(II) S = sK δK Y = sK rK,
0 < sK ≤ 1. (58)
The saving assumptions are alternatively expressed by the expenditure condition for commodity 1 (newly produced capital goods), cf. (30)
(I)
(II)
P1 Y1 = sY ; y/P1 = l1 y1 /s, y/P2 = l2 y2 /(1 − s).
P1 Y1 = sK δK Y, y/P1 = l1 y1 /sK δK , δK = 0.
(59)
(60)
and obviously from (30), (59-60) follows the equivalence of the value of consumer good production and consumption expenditures, P2 Y2 = (1 − s)Y ,
P2 Y2 = (1 − sK δK )Y . Using (59-60), (32), (22), (11-12), (6-7), the saving
ratios, s and sK , can alternatively be decomposed as
(I)
(II)
s = P1 Y1 /Y = δK λK1 /K1 = δL l1 /L1
sK = P1 Y1 /(rK) = s/δK = λK1 /K1
(61)
(62)
2.3. General equilibrium
The ”demand side” of the economy is summarized by, respectively,
(I)
P1 Y1
s
=
,
P2 Y2
1−s
(II)
sK δK
P1 Y1
=
.
P2 Y2
1 − sK δK
(63)
The ”supply side” of the economy – operating under constant returns to
scale and with full (11-12) and efficient factor utilization (17-24) – is always
summarized by
L 1 L2
l1 L2
K1 K2
λ K K
P1 L1 y1
P1 Y1
=
=
=
=
= 1 2
P2 Y2
P2 L2 y2
L 2 L1
l2 L1
K2 K1
λK2 K1
(64)
Hence, competitive general equilibrium (CGE) states (Walrasian equilibria),
Debreu [1954] – with market clearing prices on the commodity/factor markets
and Pareto efficient endowments allocation – require, respectively,
(I)
(II)
λK1 K2
l1 L2
s
=
,
=
1−s
(1 − l1 )L1
(1 − λK1 )K1
l1 L2
λK1 K2
sK δK
=
=
.
1 − sK δK
(1 − l1 )L1
(1 − λK1 )K1
12
(65)
(66)
From (65-66), we have, cf. (61-62),
sL1
sK1
, λK1 =
sL1 + (1 − s)L2
sK1 + (1 − s)K2
sK L1 (1 − L2 )
, λK1 = sK K1 ,
l1 =
sK L1 + (1 − sK )L2
l1 =
(I)
(II)
(67)
(68)
Next, from (67-68) and (61-62), we get
(I)
(II)
δL = sL1 + (1 − s)L2 ,
δK = sK1 + (1 − s)K2 .
sK L1 + (1 − sK )L2
K2
δL =
,
δK =
.
1 + sK (L1 − L2 )
1 + sK (K2 − K1 )
(69)
(70)
Finally, by (32) and (69-70), we obtain, ∀k ∈ CK ⇔ ∀ω ∈ Ω, (20-21)
(I)
(II)
ω[sK1 + (1 − s)K2 ]
ωδK
=
= ΨI (ω).
δL
sL1 + (1 − s)L2
ωK2
ωδK
=
= ΨII (ω),
k=
δL
sK L1 + (1 − sK )L2
k=
(71)
(72)
The formulas of ΨJ (71-72) are ”reduced form” expressions derived from the
”structural equations” of the respective systems, J = I, II, which relate
the ”endogenous” two-sector general equilibrium values (solutions) of the
wage-rental ratio ω = w/r to the ”exogenous” factor endowments (aggregate
capital-labor) ratio, k = K/L. Then, having obtained ω from (71-72), we can
go back through (67-70), (18), (25) to get the associated general equilibrium
values of all other endogenous variables (sector outputs, allocation fractions
of inputs, income shares, relative commodity prices).
Remark 3. The ”reduced form” expressions (71-72) were represented in equivalent forms, Uzawa [1961-62, p. 43; 1963, p. 108], Drandakis [1963, p. 220],
Inada [1963, p. 124], Burmeister & Dobell [1970, p. 123], Gandolfo [1980, p.
487],
(I)
(II)
s(k2 + ω)k1 + (1 − s)(k1 + ω)k2
= ΨI
s(k2 + ω) + (1 − s)(k1 + ω)
(k1 + ω)k2
k=
= ΨII
ω + (1 − sK )k1 + sK k2
k=
(73)
(74)
While demand and technology components are always involved, the technology representation in (71-72) by the bounded output elasticities (6-8), (22-23)
is intuitively and mathematically convenient in further economic analysis. ∇
13
General equilibria and the factor endowment-factor price correspondence
The competitive general equilibrium, (CGE) functions, ΨJ (ω), J = I, II – and
their graphs, as loci of timeless general equilibrium values and as trajectories of motion – are of paramount importance for inquiring into the statics,
comparative statics, and dynamics of two-sector economies, and they will
be called the factor endowment-factor price (FEFP) correspondence. Alternatively, a CGE function ΨJ may be dubbed the Walrasian kernel , since it
selects in the Edgeworth box diagram the relevant Walrasian equilibrium allocation as a specific allocation (point) within the core of the contract curve,
cf. Mas-Colell et al. [1995, p. 654]. The graph of a Walrasian kernel ΨJ constitutes the seed (nucleus elements) of Walrasian allocations, since the subset
of CGE components (ω, k) generates the remaining components of Walrasian
allocations (product prices and sectorial factor allocation and production).
The locus of the Walrasian kernel, k = ΨJ (ω), effectively links the Pareto
optimal allocations – extracted from information about technology and expanding endowments, represented by either outward shifting transformation
curves or by changing contract curves of expanding Edgeworth boxes - to the
relevant Walrasian equilibrium price vector (p, ω) [on the locus of the FPCP
correspondence p(ω)].
Regarding the shape of the graphs of ΨJ , it is evident that, if both substitution elasticities are larger than one, σi > 1, i = 1, 2, then the numerator
(denominator) expression in (71-72) will increase (decrease), cf. Ki , Li (9),
which always ensures that the two CGE (Walrasian) kernels ΨJ are monotonic increasing. When one or both σi are less than one, only a detailed
examination will reveal the global and local shape of the CGE graphs.
To this end, we calculate CGE elasticities and summarize results in
Lemma 2. The elasticity, EI (k, ω) = (dΨI /dω)(ω/ΨI ), ∀ω ∈ Ω, of the CGE
function ΨI (ω), (71), is given by
EI (k, ω) = 1 + (1/δL ) [l1 K1 (σ1 − 1) + l2 K2 (σ2 − 1)] ,
1 s
2
(1 − s)(K1 − K2 ) + l1 K1 σ1 + l2 K2 σ2 > 0,
=
δL δK
(75)
(76)
By the global positivity (76), ΨI (ω) is monotonic increasing, and the inverse
ω(k) = ΨI−1 (k) exists (not necessarily in closed form) for every k ∈ CK .
Thus, in the two-sector general equilibrium, with structural relation (58, I),
factor endowments with a higher capital-labor ratio k are always accompanied
by a single and higher general equilibrium value of the wage-rental ratio ω,
and vice versa, i.e.,
∀k ∈ CK : E I (ω, k) = 1/E I (k, ω) > 0.
14
(77)
The elasticity, EII (k, ω), of the CGE function ΨII (ω), (72), is given by:
EII (k, ω) =
sK L1 (K2 − K1 + K1 σ1 ) + (1 − sK K1 )L2 σ2
≷ 0.
sK L1 + (1 − sK )L2
(78)
By (78), ΨII (ω) is not unfailingly monotonic, and Ψ−1
II (k) will not always
exist as a single-valued inverse mapping; but the necessary and sufficient
condition for a monotonic ΨII (ω) is:
EII (k, ω) > 0 ⇔ sK L1 (K2 − K1 + K1 σ1 ) + (1 − sK K1 )L2 σ2 > 0.
(79)
An overly strong sufficient condition for EII (k, ω) > 0, ∀ω ∈ Ω, is immediately seen (79) to be the traditional global sectorial capital intensity ranking,
∀ω ∈ Ω : k2 (ω) > k1 (ω) ⇔ K2 > K1 ⇔ L1 > L2 .
(80)
A far less restrictive sufficient condition for monotonicity of EII (k, ω) is
∀ω ∈ Ω :
σ1 (ω) + σ2 (ω) ≥ 1.
(81)
Corollary 2.1. With CES technologies, EII (k, ω), (78), becomes:
K
+ c1 ω 2(1−σ1 ) ] + sK ω 1−σ1 [( s2K − 1)σ2 + σ1 − 1]
sK σ1 ω σ2 −σ1 + c2 σ2 [ 1−s
c1
ω(c1 + ω −σ1 [(1 − sK ) cc21 + sK ω σ2 −1 + c2 ]
(82)
By (82), a milder sufficient condition for EII (k, ω) > 0 is
(2/sK − 1) σ2 + σ1 > 1
(83)
which, with sK = 1 (Kalecki), complies with the stronger requirement, (81).
Proof. The capital intensity (80) was given in Uzawa [1961-62, p. 44] for the
special case sK = 1. The substitution elasticity conditions (81), (83) were
derived in Drandakis [1963, p. 222]. The generalized condition (79) as well as
(78), (76) was proved in Jensen, [1994, p. 128]; see Gandolfo [1980, pp. 488].
The CES expression (82) follows from (78) and (99) below.
Remark 4. The CGE elasticity of ΨII (72) was introduced by Drandakis [1963,
p. 221]. The CGE elasticity of ΨI (71) was called the ”aggregate elasticity
of substitution (an economy-wide elasticity of substitution between factors)”
by Jones [1965, p. 562-64], and he gave EI (k, ω), with σD = 1, as
EI (k, ω) = |λ||θ| (σS + σD ) = (l1 − λK1 ) (L1 − L2 ) (σS + 1) > 0,
(84)
where σS represents (p. 563) ”the elasticity of substitution between commodities on the supply side (along the transformation schedule),” and this
positive σS is given by a composite expression of σi , li , λKi , Ki ; the positive
15
σD is the (p. 562) ”elasticity of substitution between two commodities on
the demand side” that occurs with homothetic community preferences. The
proportional saving rate (expenditure share) assumption implies σD = 1.
The elasticity (84) is always positive, cf. (14), (18), i.e., the Walrasian kernel
(76) is monotonic increasing with any common, strictly quasiconcave homogeneous utility function for all economic agents, Hahn [1965, pp. 343].
∇
Multiple Walrasian equilibria and the shape of Walrasian kernels
Without common preferences, the demand pattern (expenditure shares), (63,
II) – admitting ”paradoxical” cases of negative elasticity of the CGE function ΨII (72) – creates an analog to the Giffen paradox of a positive price
elasticity (substitution effects are adversely dominated by income effects)
for a demand function of a budget-constrained utility-maximizing consumer.
However, the general equilibrium paradox is more difficult to ”decompose”
and to formulate succinctly than the partial (consumer) equilibrium paradox, because both supply side (with two industries) and demand side effects
determine the sign of the CGE elasticities. The latter can sometimes be positive even with zero intrasectorial substitution elasticities σi = 0, cf. (76),
while the ranking (80) is decisive for the sign of (78) with σi = 0. The paradoxical sign of price elasticities in the Giffen and the CGE case are only local
properties, restricted to a finite part of commodity (endowment) space, cf.
Varian [1992, p. 118], Wold [1948], Wold [1952, p.100-103], and for ΨJ , see
Lemma 3 below.
But a local region with negative sign of (78) and positive sign elsewhere
implies that at least three CGE values of the wage-rental ratio w/r in this
local region can coexist with a constant (exogenous) factor endowment ratio K/L. This phenomenon has been interpreted as creating both a CGE
”uniqueness” problem and a ”dynamic causality” problem, cf. ”multiple equilibria” in section 1 and Burmeister & Dobell [1970, p. 113]: ”uniqueness of
static (momentary) equilibrium at all times and causality are equivalent concepts in two-sector models”.
As the circumstances of multiple wage-rental ratios are elusive, the subject matter is carefully explained for the classical case to eliminate these
purported problems from two-sector growth models. The possibilities for the
coexistence of two CGE values of ω with the same k must involve the following conditions. Raising the common ω will certainly increase both sectorial
capital-labor ratios ki (”intrasectorial substitution effect”). Hence, the two
ki must differ, and the allocation fractions li must be distinct for k to remain
unchanged, cf. (13). As the supply side is the same for (76) and (78), the
demand side and hence output composition are crucial for the actual local
multiplicity of ω in the classical case (72, 78).
16
With unchanged k, a larger w/r evidently implies a smaller (larger) factor
income share of capital δK (labor, δL ), cf. hyperbola (32), which reduces (increases) the demand for capital good Y1 (consumer good Y2 ). We have already
mentioned that negative CGE elasticity is only possible with both σi less than
one. The latter implies that the ”intrasectorial substitution” effects may be
small , and that changes in output composition will be accomplished by intersectorial ”substitution” (factor reallocations, factor mobility). With σ1 < 1,
it is seen from (68), (9) that λK1 decreases and so does l1 = (k/k1 ) λK1 , (14),
for constant k. Thus, sector 1 loses both factors, and its production declines,
as did its demand, whereas sector 2 expands. Accordingly, the output composition (ratio) Y1 /Y2 falls. With the sectorial ”capital intensity” assumption,
k1 > k2 , and larger w/r, the price ratio (P1 /P2 ) falls, cf. (27). With both
(Y1 /Y2 ) and (P1 /P2 ) reduced, the expenditure share (P1 Y1 ) / (P2 Y2 ) falls, as
it also must as a consequence of a smaller δK , cf. hyperbola (63, II); thus,
with k1 > k2 and small σi , it is consistent with CGE to have both a low and
higher wage-rental ratio for a constant overall capital-labor ratio. [Incidentally, constant overall saving ratios do not allow for such necessary change in
expenditure shares, cf. (63, I), hence the sign in (76)].
Now if we keep the small σi and just change the sectorial ranking k2 > k1 ,
then all the implications of a larger w/r will again be those mentioned above,
except that here the price ratio (P1 /P2 ) will increase. But with constant endowments K/L, the output ratio (Y1 /Y2 ) and the price ratio (P1 /P2 ) cannot
move in opposite directions and preserve competitive general equilibrium
states. It violates profit maximization (Pareto efficiency in production), cf.
Arrow & Hahn [1971, p. 260]; or just consider tangents to the transformation
curve in a diagram of two outputs with the concavity assumptions, (4); see
also Wan [1971, p. 124-27].
As mentioned above, when multiple wage-rental rations occur, three ωvalues, and hence, three Walrasian equilibrium allocations must exist with a
given endowment ratio, k. Accordingly, both small and large intrasectorial
substitution effects (large increase in ki , high mechanization of both industries) are possible together with necessary accompanying changes in income
distribution and demand/output composition, as outlined in detail above
(the ranking k2 > k1 is still precluded by the same arguments).
Such large shifts to another Walrasian equilibrium allocation can be associated with ”big push” and coordinated investments as a basis for industrialization, as done by, e.g., Murphy et al. [1989, p. 1004; 1989, p.540]: ”we
chiefly associate the big push with multiple equilibria of the economy and
interpret it as a switch from the cottage production equilibrium to industrial
equilibrium.” Although the static (factor reallocation with constant endowment) big push and the dynamic (pace of factor endowment accumulation)
big push are related phenomena, they are distinct, as seen in section 3 below.
The big push (”take-off ”) approach is mentioned here to indicate the theoret17
ical/empirical scope of the Walrasian kernel in the abstract CGE two-sector
growth model. For extensive treatment of CGE locus, see Balasko [1988].
Example. The widely used numerical example of multiple wage-rental
ratios has been the special classical case of sK = 1 and the following CES
sector technologies, cf. Uzawa [1961-62, p. 45], Burmeister [1968, p. 198; 1970,
p. 127],
y1 = f1 (k1 ) = 0.001(k13 + 7−4 )1/3 ;
y2 = f2 (k2 ) = (k23 + 1)1/3
(85)
The CES parameter values of ai , σi , γi can be retrieved from (85), (36), as
a1 = 0.9996,
a2 = 0.5 σ1 = σ2 = 0.25 γ1 = 0.001 γ2 = 1.26
(86)
Although a1 is extreme, other values might do as well; what matters is that
both σi are much less than 0.5, cf. (83). The numerator expression of (82)
3
3
becomes, with (86): 0.25 + 0.035ω 2 − 0.5ω 4 , which clearly can be negative for
0.4 < ω < 33. Thus, ΨII can relate multiple (three) values of ω for a given:
3 < k < 4.5. Incidentally, note that multiple CGE values of ω for a given
k has nothing to do with factor intensity reversals, cf. (45), as the sectorial
σi are the same in (85); cf. Fig. 1, Case 1.5. Multiple CGE with ΨII may
occur on both sides of the reversal point in Cases 1.1.1-1.1.4, but only where
∇
k1 > k2 .
It must be emphasized that the CGE possibility of multiple w/r for given
K/L has theoretically nothing to do with any sector producing a capital
good. Both outputs could be consumer goods without altering the CGE results above. The crux of the matter is that (with a particular group of factor
owners who demand only one output) multiple wage-rental ratios can coexist,
if it is possible to create additional demand for the good that heavily uses
the higher-priced factor and efficiently (profitably) expands its production.
Nevertheless, such increases in a factor price ratio for a given K/L cannot
persist, since the parallel intrasectorial substitution (”own”) effects can continue, whereas major intersectorial factor reallocations (”cross”) effects are
transient and will dwindle to minor importance for the subsequent CGE
solutions of the sectorial output composition (ratios).
From the discussion above and the quoted (and other) literature on twosector growth models, a cardinal methodological issue becomes manifest:
Is non-monotonicity (allowing multiple values of w/r for any locally given
range of K/L) of the CGE functions ΨJ an insuperable dilemma for economic
disaggregation and rigorous mathematical construction and solution of static
and dynamic competitive general equilibrium models?
In theoretical and empirical scientific work, it often happens that, instead of deducing ”effects from causes,” we wish to find the ”causes from
the effects.” Rather than to discover (predict) events from some imperfect
18
knowledge of laws, the events (effects) may be known (observed), and we
want to deduce the underlying causes behind events. Thus, despite locally
admitting multiple Walrasian equilibria by a non-monotone Walrasian kernel ΨII , the latter as an FEFP correspondence, k = ΨII (ω), always allows
us to go back from any observed wage-rental ratio to a unique capital-labor
ratio. That three different ω-values can be related by ΨII (ω) to the same k,
is no problem, as each ω determines other distinct properties (output prices,
etc.) of each CGE allocation. But evidently, an ”exogenous” (endowment)
variable like k is not necessarily a convenient ”state” variable in solving the
static or dynamic equations of a Walrasian two-sector economy. To insist on
always using k as a state variable is just defective mathematical procedure,
and the purported ”causality” problem (in the literature quoted above and
in the introduction) mathematically disappears by adopting the wage-rental
ratio ω as a substitute state variable. ”Keine Hexerei, nur Behendigkeit”
(no black magic, just cunning). For notion of causality, cf. Jensen & Larsen
[1987, p. 221].
Moreover, by observing the FEFP and FPCP correspondences in Fig. 1,
it is intuitively not surprising that the wage-rental ratio ω turns out to be
the key variable (”parametrization”) to describe and unlock the dynamics
in two-sector growth models: the factor price ratio ω is the critical intrinsic
link (”bridge”) connecting factor markets (endowments) and the commodity
markets. Indeed, as we shall see, even with a monotonic ΨI (ω), (76), the
endowment ratio k will in most cases be an intractable dynamic state variable.
Walrasian general equilibrium allocations with CES technologies
CD case. From (35), (44) with σi = 1, and (67-70), we get
Ki = ai ,
Li = 1 − ai ;
(I) δK = sa1 + (1 − s)a2 , δL = s (1 − a1 ) + (1 − s) (1 − a2 )
= 1 − a2 + s (a2 − a1 )
(I)
= a2 − s (a2 − a1 ) ,
a2
1 − a2 + sK (a2 − a1 )
,
δL =
(II) δK =
1 + sK (a2 − a1 )
1 + sK (a2 − a1 )
s(1 − a1 )
sa1
(I) l1 =
, λK1 =
1 − a2 + s(a2 − a1 )
a2 − s(a2 − a1 )
sK (1 − a1 )a2
(II) l1 =
, λK1 = sK a1 ,
1 − a2 + sK (a2 − a1 )
(87)
(88)
(89)
(90)
(91)
(92)
Then, (71-72) and (89-90) give linear competitive general equilibrium (CGE)
19
relations between the factor endowment ratio and the wage-rental ratio:
(I)
(II)
ωδK
a2 − s (a2 − a1 )
=ω·
= νI ω
δL
1 − a2 + s (a2 − a1 )
ωδK
ωa2
= νII ω
k = ΨII (ω) =
=
δL
1 − a2 + sK (a2 − a1 )
k = ΨI (ω) =
(93)
(94)
It follows from (75), (79) and is immediately seen by (93-94) that
σ1 = σ2 = 1 ⇔ EI (k, ω) = EII (k, ω) = 1
(95)
The Walrasian kernels ΨJ (93-94), located between the CD ωi -lines, (44), are,
ω1 =
1 − a1
k1 ,
a1
ω2 =
1 − a2
k2 ;
a2
1 − a2
1 − a1
≤ νJ ≤
,
a2
a1
a2 > a1 . (96)
From (35), (93-94), we get, with ω(k) = ΨJ−1 (k), J = I, II,
a
a i νJ i a i
a i νJ
ki [ω(k)] =
· k; yi = fi (ki [ω(k)]) = γi
k
1 − ai
1 − ai
(97)
Hence, we have by (97), (23), (28), and (95),
(I − II) : E (Y1 /Y2 , k) = E (y1 /y2 , k) = a1 − a2 ; E (P1 /P2 , k) = a2 − a1 (98)
Recapitulating, the Walrasian general equilibria of two-sector economies with
CD technologies always display - irrespective of the demand side - constancy
of sectorial output elasticities (cost shares), factor allocation fractions, and
factor income distributions. Certainly, the distributional outcomes (88-90)
are no surprise with σi = 1, but global invariance (91-92) of sectorial factor allocation emphasizes the bifurcation (”knife edge”) status of the CD
functional form. Although the sectorial capital-labor ratios (97), wage-rental
ratios (96), and the factor endowment ratio k are changing, they all do it at
the same rate and leave the allocation fractions (14) unchanged, cf. (91-92).
The factor allocation aspects of a two-sector CD economy are the opposite of the sectorial ”fixed coefficient” (Leontief or small open) economy.
The latter only allows for intersectorial factor transfers and no intrasectorial factor substitution; the CD economy generates no intersectorial factor
transfers, whereas intrasectorial substitution (”capital deepening”) entirely
accommodates for changes in output composition and relative output prices,
(97-98). The invariance of the ”occupational industrial structure,” li (k), is
evidently counterfactual consequences of CD sector technology assumptions.
The factor allocation invariance (irrespective of steady states or not) carry
over to general equilibrium systems of multisectorial (J = I, ., N ) economies
with CD technologies, see Dhrymes [1962, p. 268]. CD assumptions serve us
mainly in providing an important benchmark case within the CES family.
20
CES case. Using (6-7), (36-37) and (44), we obtain
i −1
1 − ai 1−σ
1
ci ω 1−σi
σi
ki
=
,
=
Ki = 1 +
L
i
ai
1 + ci ω 1−σi
1 + ci ω 1−σi
From (67-70), (99), we get
−1
−1
(I) δK = s 1 + c1 ω 1−σ1
+ (1 − s) 1 + c2 ω 1−σ2
1 + (1 − s)c1 ω 1−σ1 + sc2 ω 1−σ2
=
,
δL = 1 − δK
1 + c1 ω 1−σ1 + c2 ω 1−σ2 + c1 c2 ω 2−σ1 −σ2
1 + c1 ω 1−σ1
,
(II) δK =
1 + (1 + sK )[c1 ω 1−σ1 + c2 ω 1−σ2 ] + c1 c2 ω 2−σ1 −σ2
1 + c2 ω 1−σ2
1 + c2 ω 1−σ2
(I) l1 =
,
λ
=
K
1
c2 σ1 −σ2
1
1 + 1−s
ω
+ cs2 ω 1−σ2
+ 1−s
c ω 1−σ1 + c2 ω 1−σ2
s c1
s
s 1
1
sK
,
λK1 =
(II) l1 =
1−sK c2 σ1 −σ2
c2 1−σ2
1 + c1 ω 1−σ1
1 + sK c 1 ω
+ sK ω
(99)
(100)
(101)
(102)
(103)
(104)
and hence finally by (100-102)
ωδK
ω σ1 +σ2 + sc2 ω 1+σ1 + (1 − s)c1 ω 1+σ2
=
δL
sc1 ω σ2 + (1 − s)c2 ω σ1 + c1 c2 ω
ωδK
ω σ1 +σ2 + c1 ω 1+σ2
(II) k = ΨII (ω) =
=
δL
sK c1 ω σ2 + (1 − sK )c2 ω σ1 + c1 c2 ω
(I) k = ΨI (ω) =
(105)
(106)
Lemma 3. The CES graphs of the Walrasian kernels, ΨJ , J = I, II, (105106) are located between the monotone CES ωi -curves, (44), and for any
value of the saving ratios, s and sK , the graphs of ΨJ pass through the intersection (reversal) point, (45), which is entirely determined by the technology
parameters. For any size of the sectorial substitution elasticities, σi , i = 1, 2,
the CGE functions, ΨJ , have the limit properties:
lim ΨJ (ω) = 0, lim ΨJ (ω) = ∞;
ω→0
ω→∞
J = I, II
lim ΨJ (ω)/ΨJ (ω) = 0, lim ΨJ (ω)/ΨJ (ω) = ∞
ω→0
ω→∞
(107)
(108)
With (105-106), the CGE elasticities (76), (78) have finite limits as follows:
σ1 < 1, σ2 < 1 :
σ1 > 1, σ2 > 1 :
σi > 1 > σj :
σ1 > 1 > σ2 :
σ2 > 1 > σ1 :
lim EJ (k, ω) = lim EJ (k, ω) = max{σ1 , σ2 }(109)
ω→0
ω→∞
lim EJ (k, ω) = lim EJ (k, ω) = min{σ1 , σ2 } (110)
ω→0
ω→∞
lim EI (k, ω) = lim EI (k, ω) = 1
(111)
lim EII (k, ω) = 1, lim EII (k, ω) = σ2
(112)
lim EII (k, ω) = σ2 , lim EII (k, ω) = 1
(113)
ω→0
ω→∞
ω→0
ω→∞
ω→0
ω→∞
21
Proof. The verification of the logical economic test (for any s and sK ) of
the passage of ΨJ (ω) through the reversal point follows by inserting ω̄, (45),
into (105-106), which gives k̄ (45) identically. The limits (107-108) are seen
immediately from (105-106). The limits (109-113) follow from the formula,
EJ (k, ω) = ΨJ (ω)ω/ΨJ (ω), by straightforward calculations. These finite limits are due to the same exponents of the dominating polynomials in the
numerator and the denominator.
From (99) we note that [in the limits ω may be replaced by k, cf. (107)]
σi < 1 :
σi > 1 :
lim Ki = 1,
ω→0
lim Ki = 0,
ω→0
lim Li = 0;
ω→0
lim Li = 1;
ω→0
lim Ki = 0,
ω→∞
lim Ki = 1,
ω→∞
lim Li = 1 (114)
ω→∞
lim Li = 0 (115)
ω→∞
Lemma 4. For the CES two-sector competitive general equilibrium economy,
the limits of the factor allocation fractions and factor income shares are, with
a demand side specification according to proportional saving:
σi
σ1 < σ2 < 1 :
1 > σ1 > σ2 :
σ1 > σ2 > 1 :
1 < σ1 < σ2 :
σ1 > 1 > σ2 :
σ1 < 1 < σ2 :
k→0
l1 → 0 λK1 → s
l1 → 1 λK1 → s
l1 → s λK1 → 0
l1 → s λK1 → 1
l1 → 1 λK1 → 0
l1 → 0 λK1 → 1
δK → 1
δK → 1
δK → 0
δK → 0
δK → 1−s
δK → s
l1 → s
l1 → s
l1 → 0
l1 → 1
l1 → 0
l1 → 1
k→∞
λK1 → 0
λK1 → 1
λK1 → s
λK1 → s
λK1 → 1
λK1 → 0
(116)
δK → 0
(117)
δK → 0
(118)
δK → 1
(119)
δK → 1
(120)
δK → s
(121)
δK → 1−s (122)
and are, with classical saving, [δ̄K = 1/ (1 + sK )]:
σi
σ1 < σ2 < 1 :
1 > σ1 > σ2 :
σ1 > σ2 > 1 :
1 < σ1 < σ2 :
σ1 > 1 > σ2 :
σ1 < 1 < σ2 :
k→0
l1 → 0 λK1 → sK δK → 1
l1 → 1 λK1 → sK δK → 1
l1 → 0 λK1 → 0 δK → 0
l1 → 0 λK1 → 0 δK → 0
l1 → 1 λK1 → 0 δK → δ̄K
l1 → 0 λK1 → sK δK → 0
l1 → 0
l1 → 0
l1 → 0
l1 → 1
l1 → 0
l1 → 1
k→∞
λK1 → 0 δK → 0
λK1 → 0 δK → 0
λK1 → sK δK → 1
λK1 → sK δK → 1
λK1 → sK δK → 0
λK1 → 0 δK → δ̄K
(123)
(124)
(125)
(126)
(127)
(128)
(129)
If the sector technologies have the same σi , i = 1, 2, then the limits of (117122) become, [l¯1 = sc1 / (sc1 + (1 − s)c2 ), λK1 = sc2 / (sc2 + (1 − s)c1 )]:
σi
k→0
σ1 = σ2 < 1 : l1 → l¯1 λK1 → s δK → 1
σ1 = σ2 > 1 : l1 → s λK1 → λK1 δK → 0
22
k→∞
(130)
l1 → s λK1 → λK1 δK → 0 (131)
l1 → l¯1 λK1 → s δK → 1 (132)
and, (124-129) become, [l¯1 = sK c1 /sK c1 + (1 − sK )c2 ]:
σi
k→0
¯
σ1 = σ2 < 1 : l1 → l1 λK1 → sK δK → 1
σ1 = σ2 > 1 : l1 → 0 λK1 → 0 δK → 0
k→∞
(133)
l1 → 0 λK1 → 0 δK → 0 (134)
l1 → l¯1 λK1 → sK δK → 1 (135)
Proof. Using (100-104), most of the formulas follow directly from (114-115),
and the rest via l’Hospital.
The CGE implications of sectorial substitution elasticities (parameters)
σi by Lemma 4 can be expressed as
Proposition 1. If both σi < 1, then the industry with the highest σi will
(asymptotically) completely absorb the abundant factor, but it will not necessarily entirely dispense with the relatively scarce factor.
If both σi > 1, then the industry with the highest σi will (asymptotically)
completely dispense with the scarce factor, but it will not necessarily entirely
absorb the relatively abundant factor.
If σi > 1 > σj , then the industry with the higher σi will (asymptotically)
entirely absorb the abundant factor and completely dispense with the scarce
factor.
Only if σ1 = σ2 will the two industries asymptotically employ a fraction
of both factors.
The comparative static analysis of exogenous factor endowment variations
for Walrasian equilibria with CES technologies in Lemma 4 is helpful for
the economic understanding of the stock evolution of the primary factors,
when one of the outputs is a capital good, which in turn affects the overall
factor endowment ratio. In this way, static, comparative static, and dynamic
analysis of Walrasian equilibria are inherently related; Meade [1962], Amano
[1964], Takayama [1963]; but only dynamic analysis gives the final states.
Without technical progress, the ”conditions of economic progress are selfevident” to Walras [1954, p. 386]. Besides additional labor services, ”capital
goods must evidently be created out of savings before their services can
become available for use” [p. 387]. Saving (investment) rates, labor and the
macrodynamic role of the capital good (machinery) in the two industries
next need to be carefully examined.
3. The dynamic systems and evolution of two-sector economies
The equations of factor accumulation for neoclassical and classical two-sector
growth models with flexible sector technologies are formally, Uzawa [1963, p.
106], given by (δ is the depreciation rate of capital),
dL/dt ≡ L̇ = nL,
dK/dt ≡ K̇ = Y1 − δK = Ly1 l1 − δK = L {f1 (k1 )l1 − δk} .
23
(136)
(137)
In the general equilibrium models of two-sector economies above, k1 and
l1 were with (76), (79) through ω (17), uniquely determined by the factor
endowments ratio k, cf. (71), (72), and hereby the accumulation equations
(136-137) become genuine autonomous (time invariant) differential equations
in the state variables L and K and represent a standard homogeneous dynamic system,
L̇ = Ln ≡ Lf(k),
K̇ = L f1 (k1 [ΨJ−1 (k)])l1 [ΨJ−1 (k)] − δk ≡ LgJ (k);
(138)
J = I, II. (139)
As gJ (k), (139), are intricate functions of k, we rewrite gJ (k) in alternative
forms by (137), (58-60), (31-32), cf. Burmeister & Dobell [1970, p. 111], Wan
[1971, pp. 119], Gandolfo [1980, pp. 490],
sf1 (k1 )
−δ} = LgI (k)(140)
δK
K̇ = sK δK Y/P1 −δK = sK Kr/P1 −δK = Lk{sK f1 (k1 )−δ} = LgII (k) (141)
K̇ = sY/P1 −δK = Ls(ω + k)f1 (k1 )−δK = Lk{
From the governing functions gJ (k), (140-141), J = I, II, the director functions, hJ (k) ≡ gJ (k) − kf(k), of (138-139) that control dk/dt ≡ k̇ become,
sf1 (k1 [ω(k)])
− (n + δ) ; ω(k) = ΨI−1 (k) (142)
k̇ = hI (k) = k
δK [ω(k)]
k̇ = hII (k) = k [sK f1 (k1 [ω(k)]) − (n + δ)] ;
ω(k) = Ψ−1
II (k)
(143)
Remark 5. In a note, Solow [1961-62] obtained the differential equation
k̇ = sf1 (k1 ) − (n + δ)k + s(k − k1 )f1 (k1 )
(144)
which can be rewritten as, cf. (10), (17),
k̇ = s[f1 (k1 ) − k1 f1 (k1 )] + skf1 (k1 ) − (n + δ)k
= sf1 (k1 )(ω + k) − (n + δ)k = k [sf1 (k1 )/δK − (n + δ)]
(145)
(146)
which is the neoclassical director function hI (k), (142).
∇
The dynamic systems (142-143) in k are difficult to evaluate quantitatively
and generally intractable; e.g., if σ1 = σ2 , then, k = ΨI (ω), (105) cannot be
exists) in closed form. But k = ΨJ (ω), (71-72), are
inverted (although Ψ−1
I
continuously differentiable functions of ω, and dynamics in k can, whenever
convenient, be converted into a dual autonomous dynamics in ω,
ω̇ =
hJ (k)
hJ (ΨJ [ω])
k̇
=
=
≡ J (ω)
dk/dω
dk/dω
ΨJ (ω)
24
(147)
Hence we get, cf. (142-143),
ΨI (ω) sf1 [k1 (ω)]
ω
sf1 [k1 (ω)]
−n−δ =
− n − δ (148)
I (ω) = ΨI (ω)
δK (ω)
EI (k, ω)
δK (ω)
ΨII (ω)
ω
II (ω) = [sK f1 [k1 (ω)] − n − δ] =
[sKf1 [k1 (ω)] − n − δ] (149)
ΨII (ω)
EII (k, ω)
With CES technologies, we have, from (148-149), (100), (50),
1/(σ −1)
σ /(σ −1)
ΨI (ω)
sγ1 a1 1 1 (1 + c1 ω 1−σ1 ) 1
I (ω) = − (n + δ) (150)
ΨI (ω) s (1 + c1 ω 1−σ1 )−1 + (1 − s) (1 + c2 ω 1−σ2 )−1
1/(σ1 −1)
ΨII (ω) σ /(σ −1) 1 + c1 ω 1−σ1
− (n + δ)
(151)
sγ1 a1 1 1
II (ω) = ΨII (ω)
with ΨI (ω) and ΨII (ω) given in (105-106); for EII (k, ω) of CES, see (82).
Existence and uniqueness of steady states or persistent growth
The complete set (family) of k(t) solutions to the dynamic systems (142-143)
is qualitatively described and classified in:
Theorem 1. The neoclassical and classical two-sector growth models (136143) have no positive, stationary k(t)-solution [k(t) = 0 is attractor], iff
∀k > 0 : (I) β¯1 < ([n + δ] /s)δK (k),
(II) β¯1 < (n + δ) /sK
(152)
and have at least one steady state [ray path in (L, K)-space], iff, cf. (5)
∃k > 0 : (I) ([n + δ]/s)δK (k) ∈ J1 , (II) (n + δ)/sK ∈ J1 , J1 = β 1 , β¯1 (153)
The stationary capital-labor ratios ∀t : k(t) = κ ∈ CK , (21), are obtained by
(I) f1 [k1 (κ)] = ([n + δ]/s)δK (κ),
(II) f1 [k1 (κ)] = (n + δ) /sK
(154)
With existence (153), a sufficient condition for a unique root of hI (k) is
∀k > 0 : E(hI (k)/k, k) < 0 ⇔ −λK 1 L1 σ1 − λK 2 [L1 + L2 (σ2 − 1)] < 0 (155)
With (79), and (153), the root of hII (k) in (154) is always a unique attractor.
The time paths of the growth model solutions, k(t), display persistent growth
– limt→∞ k(t) = ∞ – if and only if
∀k > 0 : (I) β 1 > ([n + δ] /s) δK (k), (II) β 1 > (n + δ) /sK
25
(156)
Corollary 1.1. With existence assured (153), a weak sufficient condition for
a unique neoclassical attractor by (154) is,
∀ω ∈ Ω : EI (δK , ω) ≥ 0
⇔
∀ω ∈ Ω : EI (k, ω) ≥ 1,
(157)
and strong sufficient conditions for satisfying (155) are
∀ω ∈ Ω : σ2 (ω) ≥ 1,
∀ω ∈ Ω : σ1 (ω) = σ2 (ω) ∧ L1 > L2 ⇔ k2 (ω) > k1 (ω).
(158)
(159)
Proof. The family of solutions to (142-143) depends entirely on the shape
of the director function, h(k), and the number of roots of h(k). The existence
of nonzero roots requires that, respectively, ([n + δ]/s)δK (k) and [n + δ]/sK
belongs to the range of f1 as stated in (153). If no root exists, we have
either the case (152) with origo as attractor, or the case (156) with persistent
growth.
If it exists, a unique attractor in the interval stated in (153) always occurs
with a global negative sign of the elasticity, E(h(k)/k, k) < 0, that can be
derived from (142); the necessary and sufficient condition for such a negative
sign is shown explicitly in (155); cf. Jensen [1994, p. 138, p. 129]. A bit
stronger, sufficient condition (157) is that δK (k), is monotonically increasing
or constant, which is a special way of satisfying the criterion (155). The
elasticity conditions in (157) are equivalent by k = ωδK /(1 − δK ), cf. (71). Remark 6. The CGE elasticity condition EI (k, ω) ≥ 1 was given by Jones
[1965, p.42, σD = 1], Burmeister & Dobell [1970, p. 122]. But Jones mentioned that (157) is overly strong, and it is sufficient that (p.567), cf. (84),
∀ω ∈ Ω : EI (k, ω) > (θF /θL ) |θ|σD = [(1 − s)/δL ] (L1 − L2 ) · 1
(160)
The sector substitution elasticity condition was derived in Drandakis [1963,
p. 225]. Uzawa [1963, p. 112] obtained the capital intensity condition (159).
The dynamic implications of the capital intensity ranking (159) in the
neighborhood of k = κ (where by continuity k̇ 0) can intuitively be seen
by equating the time derivative k̇ from (13) to zero, which gives
k = κ ⇔ k̇ = 0 ⇔ l˙1 (k2 − k1 ) = l1 k̇1 + (1 − l1 )k̇2
(161)
Since k̇i = ki (ω)ω̇ always have the same sign, we get, near k = κ by (161)
and (14), the following sign implications for l˙1 and λ̇K1 ,
k2 > k1 : k̇i ≷ 0 ⇔ l˙1 ≷ 0 ∧ λ̇K1 ≷ 0;
k1 > k2 : k̇i ≷ 0 ⇔ l˙1 ≶ 0
(162)
Thus, k2 > k1 ensures that positive (negative) k̇i are accompanied by intersectorial transfers of both factors to increase (reduce) the output of the
26
capital good. Thus, with k2 > k1 , the factor reallocation stabilizes the economy around k = κ, whereas the opposite ranking renders κ into a repeller.
Similarly, the parameter condition σ2 ≥ 1, (158), implies that the CGE
two-sector dynamics operates (through sector production and allocation of
factors) in such a way that the consumer good industry will release/absorb
resources conducive to reaching and maintaining the steady state k = κ, if
κ > 0 exists.
The size of the CGE elasticity EI (k, ω), (157) implies that the monotone Walrasian kernel ΨI (ω), (71), is concave in the diagrams of Fig. 1.
Thus, larger changes in the endowment ratio k can be accommodated in the
Walrasian equilibrium economy with smaller effects on the wage/rental ratio, ω. This is equivalent to saying that flexible factor prices easily create
larger intrasectorial factor substitutions and intersectorial factor reallocations. Consequently, such an economy is not prone to have several steady
states generated by local curvature properties of hI .
The neoclassical demand side (savings) was not affected by factor income
distribution, cf. (58). It may then appear surprising that the capital share
δK (k) is a very prominent general equilibrium variable in stating the criteria
that classify the family of solutions in Theorem 1. However, to succinctly express and decompose the governing functions of capital accumulation (140),
the bounded variable δK (k) is mainly a formal auxiliary term helpful in evaluating concrete cases.
CD case. The absence of closed form solutions to nonlinear dynamic systems
also applies to the general neoclassical and classical systems (136-139). However, closed form general equilibrium dynamics with CD sector technologies
can be easily given.
Hence by (140-143), (97), (93-94), (89), bI = 1 + ν1 , bII = 1,
gJ (k) = sJ bJ γ1 a1 [a1 νJ /(1 − a1 )]a1 k a1 − δk ≡ sJ µJ k a1 − δk,
k̇ = hJ (k) = sJ µJ k a1 − (n + δ)k, sI = s, sII = sK
(163)
(164)
The unique root of hJ (k), (164), gives the steady state (ray) path, cf.(136)
1
∀t > 0 K(t)/L(t) = k(t) = κJ = [sJ µJ /(n + δ)] 1−a1 ;
L(t) = L0 ent (165)
The family of non-ray solutions of k(t) is given by explicitly solving the
Bernoulli equation (164), cf. Gandolfo [1997, p. 181], Jensen [1994, p. 80] as:
1
k(t) = κJ 1 − 1 − (k0 /κJ )1−a1 e−(n+δ)(1−a1 )t 1−a1 .
(166)
The system of trajectories, (166), are as loci represented by the equation
1
1−a1
1
K = κJ [L1−a
(k
/κ
)
−
1
+ L1−a1 ] 1−a1 .
0
J
0
27
(167)
with their convexity/concavity properties d2 K/dL2 ≷ 0 for ∀k0 ≶ κJ . The
phase portrait (167)) is parabolic, and hence the trajectories will geometrically not have the attractor (165) as an asymptote. For any solution (166)
along the trajectories (167) or Walrasian kernels (93-94), the global invariance of sectorial factor allocation prevails.
CES case. The qualitative properties of the family of solutions k(t) in the
Walrasian general equilibrium growth models with CES sector technologies
are summarized in
Proposition 2. For the two-sector growth models (136-139) with CES sector
technologies, the sufficient conditions for the existence of at least one positive
steady-state solution are [no positive, attractive, steady state solution κ (154)
exists with the RHS inequalities of (168-170) reversed ] :
σ1
(I) σ1 < 1, σ2 < 1 : β¯1 = γ1 a1σ1 −1 > (n + δ)/s
(168)
(I) σ1 < 1, σ2 > 1 : β¯1 = γ1 a1
> (n + δ)
(169)
> (n + δ)/sK
(170)
σ1
σ1 −1
(II)
σ1 < 1 :
σ1
σ1 −1
β¯1 = γ1 a1
With σ1 ≤ 1 (sufficient condition), persistent growth of k(t) is impossible.
With σ1 > 1, necessary and sufficient conditions for limt→∞ k(t) = ∞ are:
σ1
(I) σ1 > 1, σ2 > 1 : β 1 = γ1 a1σ1 −1 > (n + δ)/s
(171)
(I) σ1 > 1, σ2 < 1 : β 1 = γ1 a1
> (n + δ)
(172)
> (n + δ)/sK
(173)
σ1
σ1 −1
(II)
σ1 > 1 :
σ1
σ1 −1
β 1 = γ1 a1
except that (172) is occasionally not sufficient for small initial values.
Proof. The proof proceeds with the dual version, J (ω). The term
ΨJ (ω)/ΨJ (ω) has no influence on these limit analyses, cf. Lemma 3. Singularities of ΨII are dealt with below, see Fig. 3.
Ad σ1 < 1: The large fraction in the bracket (150) goes to zero for ω →
∞, hence no permanent increasing solutions of ω(t) exist. If (168-169) are
satisfied, then the large fraction passes at least once monotone through the
constant, when ω goes from zero to infinity. The difference s in the constant
comes from the denominator taking values 1 or s for ω → 0, depending of
the size of σ2 . If the inequalities are reversed, then I (ω) is always negative.
Ad σ1 > 1: The large fraction in the bracket (150) goes to infinity for
ω → 0. If and only if (171-172) are satisfied, then the large fraction eventually
remains above the constant, when ω goes from zero to infinity. The difference
s in the constant comes from the denominator going towards the values 1 or
s for ω → ∞, depending of the size of σ2 . Hence I (ω) is positive for large
28
values of k. If the inequalities are reversed, then I (ω) eventually becomes
negative. The necessary conditions (171), (173) are also sufficient, since kI (k)
and k( II)(k) are, with respectively σ2 > 1 and σ1 > 1, monotone decreasing,
cf. (158), (83), but remain above RHS values in (171), (173).
The conditions (170), (173) follow from (151), (38-39), and as done above,
(152), (156).
An intuitive and independent check of σ1 < 1 preventing persistent
growth is [in addition to the zero limit of f1 (k1 )] simply given by the upper finite value of f1 (k1 ), (38).
Proposition 2 also follows immediately from (152), (156) of Theorem
1, combined with the appropriate δK limits in (117-118, 122), (119-121) of
Lemma 4. But note that the proof of Lemma 4 was actually performed by
using the ”dual” state variable ω, which was also used for the limits of the
Walrasian kernels ΨJ , (107). The critical (”watershed”) value of σ1 = 1 is
due to the limits of β 1 , (38-39, 41).
It comes out explicitly from Proposition 2 that the global issues of the
existence of any steady state or persistent growth depend on the size of the
key parameters: σi , ai , γ1 , s, sK , n, δ. While the accumulation parameters
(s, sK , n, δ) are significant, the particular and fundamental importance of
the technology parameters (σ1 , γ1 , a1 ) in the capital good sector for deciding
the types of the long-run evolution (family of solutions) in the CGE growth
models complies with observation and economic intuition for many scholars,
and also corresponds to the crucial role allotted to capital good industries
by economic historians and the general public. Nevertheless, the conclusions
in Proposition 2 contrast rather sharply with the standard literature on twosector growth models.
But first, let us briefly comment on the relative importance of the individual parameters. The most important parameter in Proposition 2 is the
substitution elasticity in the capital good sector, σ1 – with the CD technology
as the fundamental bifurcation value. The ”total productivity” parameter γ1
in the capital good sector matters in all the stated conditions (168-173), and
they can all be violated with giving γ1 any value between 0 and ∞. It is
beyond the scope of this paper to enter a discussion of the possible dispersion of this parameter; see hereto Prescott [1998]. But if we here restrict it
to γ1 = 1 and if σ1 2, then (171) will usually be satisfied for other relevant
parameters, in particular with high saving rates. Evidently, the decisive role
is played by σ1 rather than σ2 in Proposition 2.
This key role played by the technology of the capital good industry has
failed to be properly recognized in the ”mainstream” literature of the twosector growth models. In an extensive summary results, Stiglitz & Uzawa
[1969] reported, e.g., in the case of neoclassical (proportional) saving, that
sufficient conditions for uniqueness and stability of (convergence to) balanced
(steady state) growth paths are (p.407): ”1. substitution elasticity in each
29
sector greater than or equal to one. 2. capital intensity in capital goods
sector ≤ capital intensity in consumption goods sector.”
These two conditions are neither necessary nor sufficient for the long-run
steady state family. Indeed, very high value of σ1 would preclude the existence
of steady state growth, cf. (171-172); the ”capital intensity condition” cannot
be maintained when σ1 = σ2 , cf. (45), and does not address the global (rather
than local stability and uniqueness) issue of the existence of a steady state or
persistent growth of k(t). Indeed, none of the conditions (157-159) or even
(155) ensure the existence of a steady state. A high σ1 as well as a high σ2
contributes to satisfying the monotonicity properties (155), but if σ1 is high
enough (σ1 2), then no steady state exists, and we have persistent growth
(171-172).
In an influential paper, Inada [1963, p. 124] used the capital intensity
condition, k2 > k1 , to ensure momentary uniqueness of Walrasian equilibria
(monotonicity of CGE loci, ΨJ ) in a ”generalized Uzawa model” with different constant saving parameters for capital and labor income. With such
generalized saving (demand side), the Walrasian kernel nevertheless accords
in several respects with the CGE locus of the classical model, ΨII , (72), (74).
Certainly, multiple Walrasian equilibria were avoided by k2 > k1 , cf. Lemma
2. Next he considered the process of capital accumulation, and the existence,
uniqueness and stability of a steady state depended on the behavior of this
expression, Inada [1963, p. 125], sr = sK ; Burmeister & Dobell [1970, p. 111],
φ = (f1 (k1 )/k) [sw w + sr k] = f1 (k1 ) [(sw /δK ) + sr − sw ]
(174)
where the form with δK follows from (32). For the cases without multiple
CGE, the function φ (174) is qualitatively similar to gI (k)/k (140), and
hI (k)/k (142). Hence, the qualitative dynamics and the families of solutions
to the growth model with ”mixed savings” correspond to the neoclassical
parts (I) of Theorem 1 and Proposition 2. But to obtain monotonicity and
limits of φ, Inada used the derivative conditions (assumptions) of production functions, (41). With (41), φ (174) will have the same limits as (41),
e.g., φ(∞) = 0, which implies that hI has a root and hence an attractive
steady state solution will always exist. Despite their extreme lack of generality, it was through these purported regularity (”well-behaved”) properties
of production functions (41) that the fundamental importance of the capital
good technology parameters was largely overlooked, lost or ignored by the
”mainstream” economic growth literature.
Drandakis (1963) studied the same ”generalized Uzawa model” as Inada
above. The momentary uniqueness conditions (81), (83) were here important steps towards understanding the properties of the ΨJ functions (71-74).
Drandakis always assumed that existence issues were taken care of by directly imposing conditions on the range cf. Drandakis [1963, p. 226] of the
functions: ωi , (19-20), ΨJ (107), and φ, (174). Accordingly, a steady state
30
solution with φ, (174), was taken for granted by Drandakis [1963, p. 223]
without invoking (41). Thus, the existence condition for a steady state was
not actually (parametrically) specified . Theorem 1 and Proposition 2 (with
the dual dynamics approach not hitherto commonly used) supply this missing
information in general and for the CES case.
Incidentally, let it here be noted that our two-sector framework fully encompasses one-sector growth models. If the two industries have the same
production function, f1 = f2 , then the Walrasian kernels, ΨJ , (71-72) coincide with the last expression for the wage-rental curves (10).
Remark 7. The ”planning (operation) approach” to economic development
and economic historians have often given primary attention to promoting the
capital good industry, e.g., Mahalanobis [1955, p. 51] emphasized that ”for
rapid industrialization of an underdeveloped country it would be desirable to
keep the cost of capital as low as possible. The further removed the type of
capital goods under consideration is from the production of final consumer
goods the greater is the need of keeping the price low. Heavy machinery which
would manufacture machinery to produce investment goods is the furthest
removed from the consumption end.”
Rosenberg [1963, p. 263] maintained that ”a major handicap of underdeveloped countries, then, is located in their inability to produce investment goods at prices sufficiently low to assure a reasonable rate of return on
prospective investments. Reasoning symmetrically, one of the most significant propelling forces in the growth of currently high-income countries has
been the technological dynamism of the capital goods industries, which has
maintained the marginal efficiency of capital at a high level.”
∇
Regarding the relative price of the capital good and consumer good, p =
P1 /P2 , any particular size of this price ratio can inherently (an endogenous
variable in our context) neither be a necessary nor a sufficient condition for
persistent growth to occur; but for small open economies with exogenous
prices, the price ratio (terms of trade) is one of the growth determinants
(parameters), see Jensen and Wang [1999, p. 391].
It is evident from a comparison of the conditions (171-172) that (172) is
easier to satisfy. In this situation with σ2 < 1, persistent growth of k(t) goes
together with a long-run declining relative price of the capital good, cf. (56),
Fig. 1, Case 1.3. The persistent growth of (171), (173) can be accompanied
by either relatively rising or relatively falling capital good prices, cf. Fig.
1, Case 1.2.1-1.2.4; see motion along the trajectories ΨJ in Fig. 2 below.
Note that when persistent growth conditions (171-173) hold, then multiple
Walrasian equilibria are ruled out, since then ΨII is by σ1 > 0, cf. (79), (81)
also monotonic in Fig. 1.
Although declining relative prices of capital goods may go together with
long-run growth per capita, it is not low and falling capital good prices per se
that keep up the rate of return (marginal efficiency) of capital and persistent
31
growth of k(t). Fundamentally, it is the high technological substitution elasticity in a capital good industry that must continuously preserve a sufficiently
high marginal productivity, (return to capital), in this critical important industry. If σ1 meets the conditions, (171-173), then high values of σ2 will also
contribute to mechanizing and maintaining the growth rate of the consumer
goods and thereby increase the welfare per capita: see Proposition 3 below.
Singularities, multiple equilibria, steady states, and development traps
As already mentioned, it has been maintained that two-sector growth models
occasionally have k(t) solutions asymptotically approaching a limit cycle,
with perpetual oscillations in the capital-labor ratio, cf. Inada [1963, p. 112].
Burmeister & Dobell [1970, pp. 131] followed up on this possibility in the
classical growth model with CES technologies. We will briefly eliminate these
dynamic anomalies from being consequences of multiple Walrasian equilibria.
Some of the intricate issues involved are concerned with various singularities
of the director functions, hJ (k) and J (ω).
Let us first consider the neoclassical (proportional) saving case with a
monotone Walrasian kernel, ΨI . The shape of hI (142), I (148), (150) and
the character of its singularities critically depend on the respective locations
of the graphs of the bounded function δK and the monotone decreasing f1 .
With σ1 < 1, a constellation of δK (k) and f1 (k) (and associated hI ) is
shown on the left side of Figure 2, which give two steady states, of which
the lower one is a repeller. This case [with intercepts relaxing the inequality,
e.g. (169), cf. (122)], giving an implosion interval below κ1 , is hardly of much
practical interest. The conditions for a unique positive attractive root hI
– either with σ1 < 1 (118-119) or otherwise not satisfying the inequalities
(171-172) – were already given above (155), (157), (159).
With σ1 > 1 and satisfying the necessary condition (172), persistent
growth may yet fail (when also multiple small roots of hI exist) for a range
of initial values around a lower attractor. Cases like (172) and (121) are
illustrated on the right side of Fig. 2. Thus, with current factor allocation
and initial endowments below κ2 , a take-off into persistent growth is impossible, even though the necessary technological opportunities for sustaining
the growth process are available in the capital good sector. The momentary
allocation problem is not caused by multiple Walrasian equilibria, cf. (78).
To avoid the stalemate of the development trap (initial values between κ1
and κ2 ), the remedy is to transfer (reallocate) adequate resources to the capital good sector, either by increasing (at least temporarily) the overall saving
(investment) rate s (”forced saving”) and/or by imposing restraints on population growth, both of which contribute to lifting hI above the axis, see right
side of Fig. 2. See also Galor & Zeira [1993]. A larger TFP parameter of the
capital good sector γ1 may give a similar ”big push” upwards of hI in Fig. 2;
32
cf. Murphy et al. [1989], Parente & Prescott [1999], Prescott [1998].
Insert figure 2 about here
If the classical (always monotone) function hII (k) locally does not exist,
(which can only occur for: σ1 < 1 (81), (83), and EII (k, ω) < 0), then the
dual (which always exists) director function II (ω) must take over the deterministic (dynamic) control of the classical solutions, ω(t) and k(t), and
outline the trajectories (motions) on the CGE locus ΨII . However, the dual
function II (ω) has singularities of its own that need concise explanation.
The diagrammatic exposition in Fig. 3 gives the graphs of II (ω), (149),
(151) for the three possible cases. The Walrasian kernel ΨII in local regions
with the multiple Walrasian equilibria is shown at the top of Fig. 3. The
derivatives ΨII (ω) and f1 (ω) follow next in Fig. 3. The three possible locations
of the steady state value, ωκ (compared with ω ∗ and ω ∗∗ ), and their associated
sign of the bracket expression (149) give – combined with the sign of ΨII –
the actual sign of ω̇ along the ω-axis, cf. (149, 147). Such collection of sign
information makes the Cases 1-3 on the left side of Fig. 3, which represents
the complete description of possible graphs for the dual II (ω). These three
phase diagrams of ω̇ immediately give the corresponding solutions ω(t) on
the right side of Fig. 3. Note that solutions starting between ω ∗ and ω ∗∗
cannot leave this interval, and solutions outside cannot enter it. Accordingly,
no time path connects the multiple Walrasian equilibria allowed for in the
classical growth model; i.e., these three different momentary (static) general
equilibrium states (for a given k) remain isolated from each other over time.
The trajectories of the family of solutions ω(t), as well as the attractors/repellers from Cases 1-3, Fig. 3, are depicted on their Walrasian kernels
ΨII in Fig. 4, Cases 4.1-4.3. The associated time paths and attractors of k(t)
follow immediately in Fig. 4. The Case 4.4 in comparison with Case 4.2 illustrates very different k(t) solutions from similar initial k(0), but with distinct
initial ω(0) values.
Insert figure 3 about here
Insert figure 4 about here
The conclusion from analyzing the graphs of the dual functions II (ω)
is evidently that limit cycles of ω(t) [and k(t)] as well as instantaneous
jumps, (discontinuous trajectories) for the sectorial capital-labor ratios ki (t),
Burmeister & Dobell [1970, p. 135], do not occur in the classical growth model
on the assumptions hitherto upheld.
Note that in contrast to the critical values ωκ , one of the singularities of
II (ω) [with vertical tangents at ω ∗ or ω ∗∗ ] will be approached with infinite
speed and reached in finite time, as illustrated in Fig. 3.
33
Persistent economic growth and asymptotic growth rates
To complement the persistent growth solutions of the state variable k(t) or
ω(t) with disaggregate information about the general equilibrium evolution
for sectorial and other endogenous per capita variables, we characterize the
respective time paths by their pertinent long-run (asymptotic) growth rates
[ω̂(t) ≡ ω̇/ω(t), etc.]:
Proposition 3. With (171-173), the long-run growth rates of k(t) and ω(t)
in Walrasian two-sector growth models (136-151) with CES technologies are:
sβ 1 −(n+δ)
;
min{σ1 ,σ2 }
(I) σi > 1 : lim ω̂ =
t→∞
(II) σi > 1 : lim ω̂ =
t→∞
lim k̂ = sβ 1 − (n + δ)
t→∞
sK β 1 −(n+δ)
; lim k̂
min{σ1 ,σ2 }
t→∞
= sK β 1 − (n + δ)
(175)
(176)
With (175-176), the long-run sectorial and per capita growth rates are
ˆ i ) = lim (y/P
ˆ i ), i = 1, 2
lim k̂i = lim ŷi = lim (w/P
t→∞
t→∞
t→∞
t→∞
(177)
where
σ1
[sβ − (n + δ)]; lim k̂2 = sβ 1 − (n + δ) (178)
t→∞
t→∞
σ2 1
σ2
(I) 1 < σ1 < σ2 : lim k̂1 = sβ 1 − (n + δ); lim k̂2 = [sβ 1 − (n + δ)] (179)
t→∞
t→∞
σ1
σ1
(II)
σi > 1 :
lim k̂1 = [sK β 1 −(n + δ)]; lim k̂2 = sK β 1 −(n + δ)(180)
t→∞
t→∞
σ2
(I) σ1 > σ2 > 1 : lim k̂1 =
If only the capital good sector has a high substitution elasticity, we have
(I) σ1 > 1, σ2 < 1 : lim ω̂ = lim k̂ = β 1 − (n + δ)
t→∞
(II) σ1 > 1, σ2 < 1 : lim ω̂ =
(181)
t→∞
sK β 1 − (n + δ)
t→∞
; lim k̂ = sK β 1 − (n + δ) (182)
t→∞
σ2
With (181-182), the long-run sectorial and per capita growth rates are
(I − II)
(I − II)
ˆ 1 ) = lim (y/P
ˆ 1 ),
lim k̂1 = lim ŷ1 = lim (w/P
(183)
ˆ 2 ) = lim (y/P
ˆ 2 ) = 0,
lim ŷ2 = lim (w/P
(184)
t→∞
t→∞
t→∞
t→∞
t→∞
t→∞
t→∞
where
(I)
(II)
(185)
lim kˆ1 = σ1 [β 1 − (n + δ)]; lim k̂2 = σ2 [β 1 − (n + δ)]
t→∞
lim kˆ1 = σ1 /σ2 sK β 1 − (n + δ) ; lim k̂2 = sK β 1 − (n + δ) (186)
t→∞
t→∞
t→∞
34
Proof. Proposition 3 follows immediately from Proposition 2: (171-173),
combined with (148-149), (110-113), and next using (115), (59-60), (44), (39),
(30) and (17).
Thus by (148-149): lim ω̂ = [ lim EJ (k, ω)]−1 [sJ β 1 −(n+δ)]; sI = s, sII = sK .
t→∞
ω→∞
The FEFP correspondence k = ΨJ (ω) next gives k̂ = EJ (k, ω)ω̂; k̂i = σi ω̂
and ŷi = ki k̂i , holds generally with CES. These relations and limits establish
the relevant asymptotic growth rates in Proposition 3.
With both σi > 1, and σ1 = σ2 , the growth patterns (175-180) imply
the long-run factor allocations given in (119-120), (126-127), that are mainly
distinguished by the long-run allocation of labor, which eventually will be
employed in the sector with the lower σi and relatively higher cost/output
prices. But the sectorial ranking of σi is not critically important for the
evolution of the ”standard of living” (consumption per capita). The latter
will continue to grow either way. But if maximum growth of per capita consumption is the goal, then the highest k̂2 will be preferred, i.e., the ranking
σ2 > σ1 > 1, cf. (178-179). The long-run growth rates of k̂ and k̂i in the
classical model are the same, irrespective of σ2 .
The growth scenario (181), (183-185) offers the most rapid (even unaffected by the saving parameter s) expansion of the capital good industry
and the overall capital-labor ratio. The relative prices of capital goods are
declining, cf. Fig. 1, Case 1.3., and the capital stock will eventually be used in
the production of machinery; but per capita consumption is bounded. Thus,
an abnormal growth scenario with a backward or neglected modernization
(with low capital/labor ratios) of the consumer good industry evolving together with an ”advanced” (highly mechanized) capital good industry, is
nevertheless within the scope of the neoclassical two-sector growth model
with CES parameters (181).
The favorable growth pattern (175-180) of increasing welfare per capita in
any numeraire that occurs with both σi > 1, cf. Fig. 1, Case 1.2.1-1.2.4, Case
1.7.-1.8, accord with the classical (or early neoclassical, 1870) view (”virtue”)
of savings/investment as a crucial determinant of economic growth.
As to empirical evidence, the theoretical general equilibrium predictions
of Proposition 3 tally with some observations and studies of long-run growth
conducted by De Long & Summers, [1991] and Jones [1994]. In particular, high rates of equipment investment (”mechanization”) are prime determinants for national growth performance (productivity, per capita growth)
and, De Long & Summers, [1991, p. 470], ”consider the joint behavior of
equipment prices and quantities is regarded as the strongest of the pieces
of evidence: Fast growth goes with high quantities and low prices of equipment investment.” From the benchmark data of Penn World Tables (1991),
Jones constructs relative price levels of producer durables by dividing the
PPP-adjusted deflators of producer durables by the PPP-adjusted price of
35
consumption goods. Then by careful estimation on a 65-country sample from
1960-1985, Jones [1994, p. 378] finds a robust and strong negative relationship
between the relative price of capital and the rate of economic growth.
We may tentatively use this sample statistics on growth and relative price
series as further evidence, cf. Remark 7, for regarding the Cases 1.2.3-1.2.4 in
Fig. 1 as perhaps the best overall choice for exhibiting the historical record of
sectorial (industrial) growth paths. The parametric range of Case 1.8 is too
narrow. Cases 1.2.3-1.2.4 imply (for smaller initial values) sectorial reversals
of capital-labor ratios and relative output prices. Such reversals seem rather
common and fit historical descriptions of the capital good industry as having
the largest potential for increasing its capital-labor ratio. The production of
machinery may initially be rather labor-intensive, but eventually becomes
highly mechanized by making various engines (steam, combustion, electric)
” cheap as well as good,” cf. Mokyr [1990, p. 87]. This supports factor substitution and mechanization subsequently in the consumer good industry. In
this way, the capital good (multi-purpose machinery/equipment) is a ”Lever
of Riches” (productivity and per capita growth) in both sectors with the capital good industry itself and its technology parameters naturally being of
primary importance for the economic growth process. The Walrasian twosector growth models and Proposition 3 confirm this logic and economically
demonstrate the mutual relations between man and machinery in the proper
setting of a Walrasian general equilibrium framework.
36
4. Final comments
Decentralization with dispersion of factor endowment and decision-making
authority is the main principle underlying the general economic assumption
of competitive markets with individual (profit, utility) maximizing agents.
Some may find it regrettable that it is not always benevolent intentions that
deliver good results. But one can argue, as did Adam Smith, that it is very
important to understand that private interests (as an invisible hand) can
guide market processes and nevertheless lead to social good.
However, competitive markets and the resource allocation were not yet
seen as being explicitly steered by commodity and factor prices. The full
recognition of the price system and several primary factors rather than only
one (land or labor), as well as the concept of a competitive general market equilibrium, can be ascribed to Leon Walras. Momentary and moving
equilibria of commodity and factor markets in ”miniature Walrasian general
equilibrium systems” have been the subject matter in all the analyses of
two-sector growth models and their solutions.
Walras completed his work in June 1900 with the vision that sometime
”mathematical economics will rank with the mathematical sciences of astronomy and mechanics; and on that day justice will be done to our work” [1954,
p. 48]; a reminder to us at the turn of this century. The catching-up remains
to be seen.
Acknowledgement. For many invaluable discussions and comments on
an earlier draft, I am deeply indebted to Mogens Esrom Larsen, Institute for
Mathematical Sciences, University of Copenhagen.
37
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42
Case 1.1.1, σ1 < σ2 < 1, p∗ > p∗∗
w/r
Case 1.1.2, σ1 < σ2 < 1, p∗ < p∗∗
ΨJ
ω1
w/r
ω2
P1/P2
P1/P2
k
p
w/r
p* p
k
Case 1.1.4, σ2 < σ1 < 1, p∗ < p∗∗
ΨJ
ω2
k
p**
k
Case 1.1.3, σ2 < σ1 < 1, p∗ > p∗∗
w/r
ω1
P1/P2
k
p
p**
Case 1.2.1, 1 < σ1 < σ2, p∗ > p∗∗
p** p*
w/r
ω1
ω
k
P1/P2
k
Case 1.2.3, 1 < σ2 < σ1, p∗ > p∗∗
p
k
Case 1.2.4, 1 < σ2 < σ1, p∗ < p∗∗
w/r
ω2
ω
ω2
ω
ΨJ
ω1
P1/P2
k
k
p
Case 1.3, σ1 > 1, σ2 < 1
p** p*
k
Case 1.4, σ1 < 1, σ2 > 1
w/r
w/r
ω2
ω1
ΨJ
ω
P1/P2
ΨJ
ω
ω1
P1/P2
k
p
k
Case 1.5, σ1 = σ2 < 1, a2 < a1
w/r
ω2
ω1
P1/P2
k
p*
p**
k
w/r
Case 1.7, σ1 = σ2 > 1, a2 > a1
P1/P2
k
p
ω1
p**
w/r
ω2
Case 1.6, σ1 = σ2 < 1, a2 > a1
ΨJ
P1/P2
p*
ΨJ
ω1
P1/P2
k
p* p**
ΨJ
ω2
k
p** p*
w/r
p
ω1
ω
ΨJ
ω2
P1/P2
p
k
Case 1.2.2, 1 < σ1 < σ2, p∗ < p∗∗
w/r
p* p**
ω1
P1/P2
k
p*
ΨJ
ω2
ω
ω
p
ω2
ω
ω
p* p**
ΨJ
ω1
ΨJ
ω2
k
p**
p*
Case 1.8, σ2 = σ1 > 1, a2 < a1
w/r
ω1
ΨJ
ω2
P1/P2
k
p**
p*
ω2
ΨJ
ω1
k
Figure 1: FPCP correspondence: p(ω), (50); and Walrasian kernels, k =
ΨI (ω), (71-72)
43
n+δ
_
β1
δκ
(n+δ)δΚ /s
f1´
β1
f1´
(n+δ)δΚ /s
n+ δ
κ1
κ2
k
κ1
k
κ2
.
k
.
k
hI(k)
κ1
κ2
k
κ1
κ2
k
hI(k)
ω
ω
ΨΙ
ΨΙ
ωκ
ωκ
ωκ
2
ωκ
1
2
1
κ1
κ2
k
κ1
κ2
k
Figure 2: Singularities of neoclassical director function, hI (k), (142), with
(169) and (172)
44
ω
k
ΨII (ω)
ΨII (ω)
ω**
k*
ω*
k**
ω*
ω**
ω
k
k** k*
Ψ'II
f1'
Ψ'II
f1'
f1'
n+δ
sK
ω*
ω**
ω
.
ω(t)
ω
ωκ ω* ωκ ω** ωκ
ω
ωκ
Case 1
ω**
ω*
ω
ω** ωκ
ω*
t
ω(t)
.
ω
ω**
Case 2
ωκ
ω*
.
ω
ω*
ωκ
ω**
ω
t
ω(t)
ω**
Case 3
ω*
ωκ
ωκ ω*
ω**
ω
t
Figure 3: Dual dynamics and singularities of the classical director function
II (ω), (149)
45
Case 4.1
Case 4.2
II
II
**
**
*
*
k**
k*
k
k** Case 4.4
Case 4.3
II
**
**
*
*
k
k*
II
k**
k*
k
k** k*
k
Figure 4: The Walrasian kernel, ΨII , (72), with trajectories of [k(t), ω(t)]
given by II , (149).
46