Download Capital Accumulation and International Trade

Capital Accumulation and International Trade
Fernando Alvarez and Robert E. Lucas, Jr.1
University of Chicago and NBER
November, 2007
Abstract
Capital accumulation is introduced into a version of Eaton-Kortum model of international trade. The effects of tariff changes on world steady states and transition
dynamics are studied. A calibrated version of the model is used estimate the shortand long-run gains of a world-wide trade elimination of tariffs. The importance of
capital in income convergence is estimated.
Keywords: Trade theory, capital accumulation, economic growth, tariff policy.
1
We thank people who helped us.
1
1. Introduction
We introduce physical capital into the Eaton-Kortum (2002) model of trade,
modified as in Alvarez-Lucas (2007). The Eaton-Kortum model has a single primary
input: non-tradeable labor. In our paper, we interpreted labor as “equipped labor”
and identified the income of this factor as value added. This broadened view enabled
us to calibrate the model realistically to U.S. national income and product data, but
did not give us a framework for analyzing any genuine dynamics. In this paper we
add physical capital as a second primary input, also assumed non-tradable, and add
capital goods as a second final good, along with consumption. We take intertemporal preferences and a law of motion for capital from standard growth theory. The
combined model describes a world of n economies, evolving over time as a system of
autonomous differential equations.
In the next section we introduce capital into an a single, autarchic economy, obtaining the two-sector model of economic growth that is the basis for the analysis
to follow. This model is calibrated using national accounts data from the U.S. In
Section 3 we set up the notation for a world economy of n such economies, define an
equilibrium, and discuss the general mathematical structure of the theory. A central
feature of this theory–discussed more fully in the concluding section–is that no international borrowing or lending is permitted: trade in goods is balanced at every
date.
We impose a very tight parameterization in formulating the theory, but even so
there is little that can be said in general about the qualitative behavior of the dynamic
system. In the rest of the paper we two different specializations, each directed at a
specific substantive question. In Section 4 we study the dynamics of tariff reform in
a symmetric world of identical economies. We study the full dynamics of this world,
2
and reproduce a thought-experiment carried out in a static context in Alvarez and
Lucas (2007). We compare the estimated effects of tariff reduction with and without
transition dynamics, and so are able to isolate the importance of incorporating capital
into the theory and to relate the results to analogous findings in public finance.
In Sections 5, 6, and 7 we turn to the study of income “convergence” is a world of
different economies, under the assumption of costless trade. In this context, the existence and uniqueness of equilibrium time paths and steady states can be established
quite generally, as we show in Section 5. In Section 6 we examine the dynamics of
a small, open economy. In Section 7, we examine stability of the n-country steady
state. The analysis lets us quantify, at least provisionally, the importance of capital
accumulation and trade in the reduction of international income inequality.
2. A Closed Economy Growth Model
We begin by introducing capital goods into the closed economy described in
Section 2 of Alvarez-Lucas (2007) (which we abbreviate as A/L). In this paper, as
in the earlier one, there is a continuum of produced goods, each produced using a
common, Cobb-Douglas production function but with different intercepts. With an
eye toward later sections, we refer to these goods as “tradeables” The intercept for any
given tradeable good is given by x−θ , where θ > 0 and where x is a random variable,
independently drawn for each good from an exponential distribution with parameter
λ. We identify each good by its productivity level, referring to “good x.” If q(x)
units of each good x are produced at date t, define q to be the Spence-Dixit-Stiglitz
aggregate
q=
∙Z
∞
−λx
λe
1−1/η
q(x)
0
¸η/(η−1)
dx
.
(2.1)
This aggregate has three uses: as an input in the production of the non-tradeable
consumption good, qc , as an input in the production of non-tradeable capital, qk , and
3
as an input into the production of each tradeable intermediate good x, qm (x). Any
individual tradeable x affects production only through its affect on the aggregate q.
In the closed economy we have
qm ≡
Z
∞
λe−λx qm (x)dx
(2.2)
0
and
q = qm + qf + qk .
(2.3)
Let capital per person be k = K/L. The law of motion for capital is assumed to be
dk
= z − δk,
dt
(2.4)
where z is gross investment.
Each person has one unit of labor, of which sc is employed in the production of the
consumption good, sk is employed in the production of the capital good, and sm (x)
is employed in the production of good x. Define
Z ∞
sm =
λe−λx sm (x)dx.
0
We assume that
sc + sk + sm = 1.
(2.5)
The production functions for the two final goods involve labor, capital, and intermediate goods:
and
¡
¢α
c = s1−ϕ
kcϕ qc1−α ,
c
¡
¢γ
z = s1−ϕ
kkϕ qk1−γ .
k
The production functions for each intermediate good x involve the same three inputs:
£
¤β
q(x) = x−θ (sm (x))1−ϕ km (x)ϕ [qm (x)]1−β .
4
The assumption that the relative share parameter ϕ is the same in all these technologies implies that
k=
kc
kk
km (x)
=
=
sc
sk
sm (x)
for all x, and we can write
c = (sc kϕ )α qc1−α ,
(2.6)
z = (sk kϕ )γ qk1−γ ,
(2.7)
q(x) = x−θ [sm (x)(k(x))ϕ ]β [qm (x)]1−β .
(2.8)
and
We treat capital as owned by households and rented to firms. Then we can think
of a household at date t as endowed with kϕ units of a composite “equipped labor”,
rented to firms at a competitive price w. In the temporary equilibrium established at
each date t, the prices p, pm (x) , and pm of the final good, of each tradeable x, and of
the tradeable aggregate, are all determined as multiples of w by the cost-minimizing
behavior of competitive firms. In particular, the price of the tradeable aggregate is
pm = χm λ−θ/β w,
(2.9)
and the prices of the two final goods are
pc = χc wα pm 1−α ,
(2.10)
pk = χk wγ pm 1−γ
(2.11)
and
where the parameters χi depend on the technology parameters α, β, and γ.
The intertemporal decisions in this economy are made entirely by households. We
assume that everyone has the preferences
Z ∞
e−ρt u (c(t)) dt,
0
5
(2.12)
where
u (c) =
for σ > 0.
c1−σ
1−σ
In each period households sell kϕ units of equipped labor at price w, and buy
z units of new capital goods at price pk and c units of final non-tradeable consumption
goods at price pc . Thus in the absence of taxes and transfers,
c=
w ϕ pk
k − z.
pc
pc
(2.13)
must hold for all t. The equilibrium relative prices w/pc and pk /pc depend only on
technological parameters. The problem for the household is to maximize (2.13) by
choice of paths for consumption c, investment z, and capital k, subject to (2.4) and
(2.13), and given the initial capital holdings k(0).
The solution of this problem must satisfy the familiar Euler equation
µ
∙
¶¸
1 dc
pc d(pk /pc )
1 w ϕ−1
ϕk
− ρ+δ−
=
.
c dt
σ pk
pk
dt
For later purposes, it is more convenient to express consumption in units of the
investment good,
pc
c,
pk
¸ µ
∙
¶
1 de
1 pc d(pk /pc )
1 w ϕ−1
ϕk
− (ρ + δ) − 1 −
=
.
e dt
σ pk
σ pk
dt
e=
(2.14)
From (2.4) and (2.15) we have the companion equation
dk
w
= k ϕ − δk − e.
dt
pk
(2.15)
In the present application, the price ratios pc /pk and w/pk are constants, independent
of (c, k). Equations (2.14)-(2.15) are autonomous, and their properties are familiar.
The pair (k, c) given by
k=
µ
pk ρ + δ
w ϕ
6
¶−1/(1−ϕ)
(2.16)
and
w ϕ
k − δk.
(2.17)
pk
and to restate this Euler equation in the equivalent form is the unique, globally
e=
stable steady state. For future reference we write the linearize system of (2.‘14)(2.15) around the steady state given by (2.16)-(2.17) as
⎡ ¡
⎤
⎡ ¡
¢
¢ ⎤
d k − k̄ /dt
k − k̄
⎣
⎦=N ⎣
⎦
d (e − ē) /dt
(e − ē)
(2.18)
¡ ¢
where the matrix N contains the relevant derivatives evaluated at ē, k̄ . For instance
the expression for N11 is
N11
∂
=
∂k
µ
¶
w ϕ
k − δk − e
pk
k̄,ē
We present in an appendix the remaining entries of the matrix N.
We finish this section by calibrating the steady state of this economy to U.S. observations, paralleling Section 5 of A/L. Here we follow Cooley and Prescott (1995)
in dividing government purchases into capital and consumption goods. We identify
pc Lc as the value of private plus government consumption, and pk Lz as gross investment, private and government. On average over the period 1987-2003, investment in
this sense was 0.20 of GDP and the value of the corresponding capital stock was 2.85
times GDP. The share of capital in national income for this broad measure of capital
is about 0.40 (see Cooley and Prescott (1995). These numbers imply the parameter
values ϕ = 0.4 and ρ = δ = 0.07.
Value added in nontradeables–defined as all of manufacturing, mining, and
agriculture–is about 0.75 of GDP in the U.S. and other wealthy economies. The
shares of both labor and capital devoted to non-tradeable goods production is also
about 0.75. The above estimates of ϕ, ρ, and δ imply that to be consistent with this
target value, α and γ must satisfy
.75 = (0.8)α + (0.2)γ.
7
Roughly half of the investment expenditures are comprised of nontradeables. Equipment and software, which is about 0.10 of GDP, is more tradeable than the other
investment expenditures, structures and residential housing.
2
We use γ = 0.5, which
then implies α = 0.81.
These parameters do not suffice to calculate the roots of the characteristic equation
associated with N: A value for the intertemporal elasticity of substitution 1/σ is
needed. Once σ is specified we can compute μN , its stable root. For σ = 2 and
σ = 1 the implied half-lives on the stable path are 8.8 and 5.7 years respectively.
3. Equilibrium with n Countries
In this section we consider the general case of n open economies, concluding
with a definition of equilibrium. The state of the world economy at any date is the
vector of per capita capital stocks k = (k1 , ..., kn ) .The constant populations are L =
(L1 , ..., Ln ), so that the units of equipped labor of country i are Li kiϕ . The constant
productivities are λ = (λ1 , ..., λn ) and the iceberg transportation costs are described
by K = [κij ]. Tariffs that are specific to country pairs (i, j) can be introduced, as in
Eaton-Kortum (2002) and A/L. To simplify the exposition, we omit this complication
here. Of course, “throwaway” tariffs–tariffs with revenues that are discarded and
not rebated to consumers–can be incorporated into the iceberg factors κij .
In such a world economy, the situation of individual households in country i
2
See Eaton and Kortum (2001). Burstein, Neves and Rebelo (2004) analyze input/output tables
for the OECD and four developing countries. They conclude that for the average country about
half of investment expenditures are non-tradeable. In their Table 1 construction services used in
investment are 50.8% of investment. Since construction itself has 32.4% of tradeable inputs in gross
ouput, we take 34.3% (=.508 (1-.324)) as the fraction of nontradeable investment due to construction.
They estimate the fraction of wholesale, retail and transportation in investment sector purchases in
investment to be 17.1% . Thus a fraction of 51.4% (=.171+34.3) of investment expenditures are
non-tradeables.
8
continues to be described by the Euler equations (2.15)and (2.16), except that the
prices are no longer determined internally and in general they are no longer constant.
To define an equilibrium, then, we need to relate the prices wi , pci , pki and pmi to
the variables ki and ci . We call this relation a temporary equilibrium. As in AL, the
construction of a temporary equilibrium falls into two parts. We first take as given
the vectors k, c, and wages w, and solve for the price vectors pc , pk and pm . Then we
formulate a fixed point problem in w, given k and c.
The process by which tradeable goods prices are established, given wages, is
precisely the same as in the static theory described in AL. Goods are now labelled
by the vector x = (x1 , ..., xn ) of productivity draws in all n countries. Buyers of any
good x in country i buy at the lowest price, taking transportation costs into account.
One can then show (AL, equation (3.8)) that the tradeable aggregate prices satisfy
pmi
⎛
à β 1−β !−1/θ ⎞−θ
n
X
wj pmj
= AB ⎝
λj ⎠ , i = 1, ..., n.
κ
ij
j=1
(3.1)
It is shown in Theorem 1 of AL that under the restrictions on parameter values
imposed above, for any w ∈ Rn++ there is exactly one solution pm = pm (w) ∈ Rn++
to (3.1), and that the function so defined is continuously differentiable, homogeneous
of degree one, strictly increasing in w, and strictly decreasing in the parameters κij .
Once pmi is expressed as a function of the wage vector w, equations (2.11) and (2.12)
can be used to express the final goods prices pci and pki as functions of w too.
In terms of this function pm , we can can also express the fraction Dij of i’s spending
on tradeables that is spent on goods produced in j as a function of w:
−1/θ
Dij = (AB)
Ã
wjβ pmj 1−β
pmi κij
!−1/θ
λj .
(3.2)
This is (3.11) in AL. In terms of these functions pm and Dij the condition for trade
9
balance is:
Li qi = Di
n
X
Lj qj
(3.3)
j=1
The left side of (3.3) is country i’s receipts from the sale of tradeables (including
home sales); the right side is total world purchases of tradeables multiplied by the
fraction Di that is spent on i’s goods.
One can show that (3.3) implies
Li wi (1 − sci −
ski )kiϕ
= Di
n
X
j=1
Lj wj (1 − scj − skj )kjϕ
(3.4)
where the employment shares in capital and consumption goods are given by
¶
µ
pki ei
(3.5)
ski = γ 1 −
wi kiϕ
sci = α
pki ei
.
wi kiϕ
(3.6)
The left side of (3.4) is the income of labor and capital employed in country i’s
production of tradeables. This income is equal to β times i’s receipts from the sale of
tradeables because of the assumed Cobb-Douglas technology in the tradeables sector.
In the same way, the sum on the right of (3.4) is equal to β times total world purchases
of tradeables.
We use the formulas (3.1), (3.2), (3.5) and (3.6) to express all of the variables
pm , Dij , ski and sci as known functions of the wage vector w. Thus we can view (3.4)
as a system of n equations in the wage vector w, given k and e. We summarize this
fact in the
Definition. A temporary equilibrium is a triple (w, k, e) that satisfies (3.4).
Suppose it can be shown that for any (k, e) ∈ R2n
++ there is a unique temporary
equilibrium (w, k, e) with w in the set
∆ = {w ∈ Rn+ :
Pn
10
i=1
Li wi = 1}.
Then the wage rate can be expressed as a function w = (k, e),
: R2n
++ → ∆, the
vector of tradeables prices pm can in turn be expressed as a function of w by Theorem
1 of A/L, and the prices pc and pk can be expressed as functions of pm and w. Then
repeating (2.14) and (2.15) for each country i we obtain an autonomous dynamic
system in the 2n variables (ki , ei ), i = 1, ..., n.
A bounded solution to these equations–if one exists–is a complete mathematical
description of the evolution of the world economy from any initial distribution k(0) =
(k1 (0), ..., kn (0)) of capital stocks, for any configuration of populations L, technologies
λ, and transportation costs κ. Our aim in the rest of the paper is to use specific
examples to understand the structure of this system.
4. Tariff Reform in a World of Identical Countries
In Section 6 of A/L we studied the effects of tariffs on trade volumes and welfare using a hypothetical world of identically sized countries. The symmetry of this
world makes the determination of equilibrium wages trivial–they are all equal–yet
the implied volume predictions turn out to be surprisingly good approximations to
the predictions of more realistic specifications. In this section, we pursue the same
simplifying idea in a dynamic context. In this case, we specialize the equilibrium
defined in the last section to a world with identical populations, Li = 1, identical
technologies, λi = λ, and identical initial capital stocks, ki (0) = k(0), i = 1, ..., n.
The iceberg transportation factors are assumed to be symmetric too, in the sense
that κij = κ for i 6= j and κii = 1.
In this section we add tariffs, since our objective is to study a tariff reform. For
this purpose, we replace κij with a product κij ωij , where κij is the fraction of goods
shipped from i to j that actually reach j, and ω ij is the fraction of dollars paid by
buyers in j that actually reach producers in i. What happens to the fraction 1 − ω ij
of dollars that do not reach i–the tariff revenues? For the purposes of this section
11
we follow the analysis in A/L and assume that these revenues are rebated lump sum
to consumers in j. We also impose symmetry on tariffs, simply replacing κ with κω.
In such a world, wages and prices will be uniform across countries at levels w,
pm , and p, say. The formulas (3.1) and (3.2) become
(AB)1/β
w
pm = ³
´θ/β
1/θ
θ/β
1 + (n − 1) (κω)
λ
and
Dii =
1
1 + (n − 1) (κω)1/θ
.
(4.1)
(4.2)
With symmetry, we do not need to make explicit use of the trade balance condition.
To restate the Euler equations in this application, the consumer budget constraint
needs to be modified. In place of (2.13) we write
wkϕ + T = pk e + pk z.
(4.3)
where T is tariff revenues in dollars. The tax base for the tariff is imported tradeables,
equal to pm qm (1 − Dii ). As described in detail in A/L, total spending on tradeables
can be expressed in terms of factor costs using share formulas. This reasoning leads
to a second formula expressing T in terms of e, k, and the constant prices. Using this
formula to eliminate T permits us to restate the budget constraint (4.3) as
z = (1 + (1 − γ) ζ)
w ϕ
k − (1 + (α − γ) ζ)e,
pk
(4.4)
where the factor
ζ=
(1 − Dii )(1 − ω)
β − (β − γ) (1 − Dii )(1 − ω)
(4.5)
corrects for the effects of rebated tariff revenues. We arrive, then, at the pair of
autonomous equations
∙
¸
1 de
1 w ϕ−1
=
ϕk
− (ρ + δ)
e dt
σ pk
12
(4.6)
and
dk
w
= (1 + (1 − γ) ζ) k ϕ − δk − (1 + (α − γ) ζ)e.
dt
pk
(4.7)
In the event that tariffs are 0 (ω = 1) or not rebated, we have ζ = 0 and these
equations reduce to the Euler equations for autarchy, (2.14) and (2.15), and hence its
linearization corresponds to the one in (2.18), and its speed of convergence is given
by and μN .
The unique steady state for (4.6) and (4.7) is readily calculated. We used the
approximate linear system to establish (local) stability. To compute the half-lives for
this world economy, we used the parameter values given in Section 2, together with
the values β = 0.5, θ = 0.15, and κ = 0.75 taken from A/L. We will refer to these as
our benchmark parameter values. We find that the half-lives are similar to those in
the closed economy: Varying n vary between 1 and 100 and varying ω between 1 and
.8, the half lives do not vary more than half a year.
Trade volumes, defined as the steady state value of imports relative to GDP, are
given by
v (ω) =
wkϕ
I
+ I (1 − ω)
where I are imports and T = I (1 − ω) . For the case α = γ, where consumption and
capital goods are equally tradeable, this formula implies
v (ω) =
β
³
Dii
1−Dii
1−α
´
.
+ ω + (1 − ω)
(4.8)
Increasing country size here means a decrease in n, which from (4.2) implies an
increase in home purchases Dii . The volume formula (4.8) thus implies that trade
volumes are decreasing in country size, as was the case in the static model. In fact,
(4.8) holds exactly in both the static and dynamic versions. In the more realistic case
when α > γ, using benchmark parameter values, the implications for trade volume
in the symmetric case in the steady state of the model with capital are almost the
13
same as they are the static model. Both models are thus broadly consistent with the
observed volume-size relation.
We now repeat in this dynamic context the thought experiment we carried out
in A/L in which all economies reduce their tariffs from a common value 1 − ω to zero.
Table 2.1 shows the effects of such tariff reforms on the steady state capital stock, for
worlds of 3, 10, and 100 identical economies, and for initial tariff rates ranging from 5
to 20 percent. (Recall from A/L that the effects of tariff changes in a small economy
in a symmetric world–the last row in the table–are very good approximation to the
calculated effects in a realistically configured world.)
TABLE 2.1 Percentage change in the Steady State
Capital Stock due to the removal of tariffs
n \ ω .95 .9 .85 .8
3
2
3
4
5
10
4
8
12 14
100
8
16 24 32
One sees that removing even small tariffs has a large effect on the steady state level
of capital, especially for a small economy.
We simulated the transition dynamics implied by a world-wide removal of 10
percent tariff beginning from a steady state in which the tariff is imposed. In the
calculation we used the approximate linear system valid near the new, untaxed steady
state, the benchmark parameter values, and a value of σ = 2. In Table 2.2 we report
the long run effect on trade volume, by comparing the steady state trade volume with
14
and without tariffs, as well as a summary statistic of the transition. We use
TABLE 2.2
Long and short run effects on Trade Volume of removing a 10% tariff
S. S. trade volume
S. S. trade volume
trade volume
10% tariff
0% tariff
at impact
3
6.3
11.
11.5
10
19.1
28.
28.9
100
40.3
47.2
47.6
n
Note that in all cases the trade volume attains its new steady state immediately after
the tariff is removed, even overshooting slightly in two cases. (This last effect is due
the increased importance of capital goods during the transition combined with the
high tradeables intensity of capital goods.)
Finally, Figure 4.1 [old Figure 4.2] illustrates the computed the welfare gains
of permanently removing a 10 percent, the experiment described in Table 2.2, as a
function of country size. These gains were calculated in three ways. The lowest curve,
labelled “Λ static,” was obtained by setting ϕ = δ = 0, and calibrating to the same
statistics as in Table 1 of A/L, so that α = 0.75. These estimates are comparable to
those reported in A/L. The top curve, labelled “Λss steady state,” compares steady
state consumption levels with and without the 10% tariff. The middle curve, labelled
“Λd w/transition,” compares the lifetime utilities of two consumption paths: the
constant path corresponding to a steady state with tariffs, and the variable path corresponding to a free-trade economy which starts with the capital stock corresponding
to the steady state level of the economy with tariffs. That is, Λd is defined as
¡
¢ Z ∞
u c (ω) eΛd (ω)
=
e−ρt u (c (t)) dt,
(4.9)
ρ
0
where c (ω) is consumption in the taxed steady state and c (t) is the equilibrium
consumption path under the tariff reform we have specified. One can see that the
15
welfare gains of the static case are close to dynamic gains estimated with transition
costs taken into account.
5. Equilibrium with Costless Trade
We turn next to the quite different special case of costless trade: κij = ω ij = 1 for
all i, j. We permit the endowment, technology parameters, and initial capital stocks
to differ across countries. In this section, we study the existence and uniqueness of
temporary equilibria and steady states. Section 6 will study stability of the steady
state in a small, open economy. Section 7 studies local stability of the steady state
in a symmetric world economy in which initial capitals differ across economies.
In the absence of trade costs, there will be a single world price pm (say) of tradeables,
and (3.1) can be solved explicitly for this price:
−1/β
pm = (AB)
à n
X
−β/θ
wj λj
j=1
!−θ/β
.
(5.1)
Given a common tradeables price, producers in all countries will distribute spending
across source countries in the same way. The fraction Dj of spending on tradeables
that is spent on goods produced in j is the same for all countries i,
−β/θ −β
pm λj .
Dj = (AB)−1/θ wj
(5.2)
We will normalize prices and wages by setting the common tradeables price pm = 1
for all dates. We use (2.10 ) to write pci = χc wiα and pki = χk wγ . Then ei and ci are
related by
ei =
pc
χ
ci = c wiα−γ ci .
pk
χk
(5.3)
In this notation, (3.5) and (3.6) imply
Li wi (1 − sci − ski ) kiϕ = Li [wi kiϕ (1 − γ) − (α − γ) χk wiγ ei ] .
16
(5.4)
Substituting from (5.2) and (5.4) into the trade balance condition (3.4) gives
Li [wi kiϕ (1 − γ) − (α − γ) χk wiγ ei ]
−1/θ
= (AB)
−β/θ
wi λi
n
X
j=1
£
¤
Lj wj kjϕ (1 − γ) − (α − γ) χk wjγ ej .
(5.5)
Given (k, e), we construct the unique temporary equilibrium in two steps. Note
first that the summation on the right of (5.5) does not depend on i, so if we define
the number M by
−1/θ
M = (AB)
n
X
j=1
£
¤
Lj wj kjϕ (1 − γ) − (α − γ) χk wjγ ej
(5.6)
we can re-state (5.6) more compactly as
−β/θ
wi kiϕ (1 − γ) − (α − γ) χk wiγ ei = M wi
λi
.
Li
(5.7)
The constant M must then be chosen to satisfy (5.5). Notice that if M > 0, (5.5)
and (5.6) together imply
1/θ
1 = (AB)
n
X
−β/θ
wi
λi ,
i=1
consistent with the normalization pm = 1.
Lemma 1 : Assume α > γ. For any strictly positive values of ki , ci ,and ξ i ≡
M λi /Li , equation (5.7) has a unique solution wi =
(ki , ei , ξ i ) . This function
is
continuously differentiable on the domain R3++ , strictly increasing in ξ i and ei and
strictly decreasing in ki .
Proof : Rewrite (5.7) as
−β/θ−1
kiϕ (1 − γ) − (α − γ) χk wiγ−1 ei = ξ i wi
17
(5.8)
As wi varies from 0 to ∞, the right side (5.7) is strictly decreasing from ∞ to 0.
The left side is strictly increasing from −∞ to kiϕ (1 − γ) > 0 while the right side is
decreasing from ∞ to 0. The properties of the solution function
Lemma 2 : Assume α > γ. With
R4n
++ , either M = 0 and
wi =
µ
are evident. ¤
as constructed in Lemma 1, for any (k, e, λ, L) ∈
α − γ χk ei
1 − γ kiϕ
¶1/(1−γ)
, i = 1, ..., n,
is the only solution to (5.5) and (5.6), or else
1/θ
1 = (AB)
¶−β/θ
µ
n
X
λi
ki , ei , M
λi
Li
i=1
(5.9)
has a unique, continuously differentiable solution
M = Φ(k, e, λ, L).
(5.10)
Proof : By Lemma 1, if α > γ the right side of (5.9) is strictly decreasing in M,
approaching 0 as M → ∞. If the right side of (5.8) is greater than or equal to 1,
(5.9) has exactly one solution. Otherwise, M = 0 is the only solution. If M > 0, the
differentiability of Φ is clear. ¤
We summarize in
n
Proposition 1 : If α > γ, for all (k, e, λ, L) ∈ R4n
++ there is a unique w ∈ R++ such
that (w, k, e) is a temporary equilibrium.
The case of α = γ is easy to treat separately.
In a world where wages and prices are always in temporary equilibrium, the Euler
equations for country i are
¸ µ
∙
¶
1 dei
1
1 1 1−γ ϕ−1
1 dwi
wi ϕki − (ρ + δ) + 1 −
=
(α − γ)
,
ei dt
σ χk
σ
wi dt
18
(5.11)
dki
1
= wi1−γ kiϕ − δki − ei ,
dt
χk
(5.12)
where the wage rate wi in country i is expressed in terms of the vectors underlying
state and costate variables (k, e) and country-specific parameters L and λ as
wi = (ki , ei ,
We have characterized the functions
λi
Φ(k, e, λ, L)).
Li
(5.13)
and Φ above. We note that neither of these
functions has an i subscript (though the arguments do), and that Φ is symmetric in
the coordinates (ki , ei , Li , λi ).
¡
¢
A steady state of the system (5.11)-(5.13) is a solution to k, w, M to
1 1−γ ϕ−1
w ϕki = ρ + δ,
χk i
−β/θ
wi kiϕ (1 − α) + (α − γ) χk wiγ δki = wi
1 = (AB)1/θ
n
X
−β/θ
wi
M
λi
,
Li
(5.14)
(5.15)
(5.16)
λi .
i=1
We establish
Proposition 2 : If α ≥ γ, there is a unique steady state.
Proof. Use (5.14) to express ki in terms of wi :
(1−γ)/(1−ϕ)
ki = Cwi
where
∙
¸1/(ϕ−1)
χk
C=
(ρ + δ)
.
ϕ
Then (5.15) implies
1+[ϕ(1−γ)/(1−ϕ)]
(1 − α) C ϕ wi
γ+[(1−γ)/(1−ϕ)]
+ (α − γ) χk δCwi
−β/θ
= wi
M
λi
.
Li
(5.17)
If α ≥ γ there is evidently a unique, positive solution wi = Ψ(Mλi /Li ) to (5.17), and
this function Ψ is a continuously differentiable, strictly increasing function, ranging
19
from 0 to ∞ on (0, ∞). We substitute into (5.16) to obtain
¶−β/θ
µ
n
X
λi
1/θ
Ψ M
λi
1 = (AB)
L
i
i=1
which has a unique solution M. Then wi = Ψ(Mλi /Li ), i = 1, ..., n, are the steady
state wages and the steady state capital stocks are given by (5.14). ¤
We finish this section with a result about aggregation.
Proposition 3. Assume that 2 of the n countries, say i = 1 and 2, have the same
technology, λ1 = λ2 , the same size, L1 = L2 , and the same initial per capita capital,
k1 (0) = k2 (0) . In an equilibrium, e1 (t) = e2 (t) , k1 (t) = k2 (t) and w1 (t) = w2 (t) for
all t ≥ 0.
To see why Proposition 3 must hold, just replace the result in the system given by
(5.11)-(5.13), using the symmetry of
and Φ under the assumption that L1 = L2 and
λ1 = λ2 . Since the initial capital are the same, the resulting systems are identical for
countries 1 and 2. The importance of this lemma is that we can regard countries 1 and
2 as two indistinguishable parts of the same country. In other words, one can regard
the word as having instead of n countries, n−1 countries, with one country of size L1 +
L2 and productivity parameter λ1 + λ2 . Proposition 3 implies that this new country
has the same per capita consumption and capital, as well as the same wages that
countries 1 and 2. Thus, in the costless trade case, differences in populations Li are
easy to accommodate, we can view the world economy as comprising a large number
of “countries” and aggregate them into units of the desired size. This aggregation
result relies on the assumption that trade costless, where national boundaries cease
to matter.
The next sections studies the behavior of the dynamical system given by (5.11)(5.13). We have not obtained useful sufficient conditions for the steady state of the
20
system (5.11)-(5.13) to be globally saddle-path stable. We focus instead on two useful
special cases. In section 6 we examine the dynamics of one small economy, while the
rest of the world is at steady state. In section 7 we examine the local dynamics of
a world made of n countries where we assume that countries can differ on their size
Li , but they all have the same ratio λi /Li .
6. Dynamics : A Small Economy
In this section we study an economy that is small, in sense that events in that
country do affect the value of the variable M = Φ(k, e, λ, L). In fact, we assume that
M is constant at the steady state value derived in Section 5. Formally, we assume
that the size of this economy tends to zero, Li → 0, while we are keeping λi /Li =
λ̄/L̄. constant. To see that the values of (ki , ei ) have no effect on M = Φ(k, e, λ, L), as
Li → 0, use the right hand side of (5.6). The function
is given by the values of
wi that solves (5.8) setting λi /Li = λ̄/L̄.
Under these two assumptions, we drop the i subscript in (5.11)-(5.12), suppress the
³
´
λ̄
arguments M̄λi /Li in the function , writing (k, e) in place of
k, e, M̄ L̄ . The
Euler equations for the small economy is
¸ µ
∙
¶
1 de
1
1 1
1 d (k, e)
1−γ
ϕ−1
(k, e)
ϕki − (ρ + δ) + 1 −
=
(α − γ)
(6.1)
e dt
σ χk
σ
(k, e) dt
and
dk
1
=
(k, e)1−γ kϕ − δk − e.
dt
χk
(6.2)
This autonomous system in (k, e) can be analyzed with standard methods.
An instructive special case arises when γ = α, or when the tradeables content of
investment and consumption goods are exactly equal. In this case, the wage function
does not depend on c and we can solve (5.8) directly for
µ
¶1/(1+β/θ)
ξ
w = (k, ξ) =
.
(1 − α) kϕ
21
(6.4)
Substituting into (6.1)-(6.2) and simplifying yields
#
" µ
¶(1−α)/(1+β/θ)
1 de
ξ
1 1
ϕkϑ−1 − (ρ + δ) ,
=
e dt
σ χk 1 − α
dk
1
=
dt
χk
where
µ
ξ
1−α
¶(1−α)/(1+β/θ)
ϑ=ϕ
ϕkϑ − δk − e,
(6.5)
(6.6)
α + β/θ
.
1 + β/θ
The dynamics of (6.5)-(6.6) can be worked out on a standard phase diagram, such as
Figure 2.1 in Barro and Sala-i-Martin (200?)). The unique steady state is globally
stable.
Comparing (6.5) and (6.6) with their autarchy counterparts, it is as if costless trade
replaces the capital coefficient ϕ with the smaller value ϑ. With trade, increases in
capital are associated with the technological diminishing returns familiar from the
closed economy case and also with diminishing returns due to deterioration in the
terms of trade. This effect increases the speed of convergence to the steady state. As
can be readily seen from the expression for ϑ, if α is close to one, ϑ is close to ϕ and
the differences with the close economy model will be small.
Returning to the case α ≥ γ, equation (6.1) becomes
∙
¶
¸ µ
1 de
1 1 1−γ ϕ−1
1d
1
=
(α − γ)
ϕki − (ρ + δ) + 1 −
e dt
σ χk
σ
dt
∙
¸
1 1 1−γ ϕ−1
ϕki − (ρ + δ)
=
σ χk
¶
∙
¸
µ
1
k (k, e) dk
e (k, e) de
(α − γ)
+
.
+ 1−
σ
dt
dt
Putting time derivatives on the left, (6.1) and (6.2) can now be written
¸
¶
¸
∙
∙
µ
e e (k, e) 1 de
1 1 1−γ ϕ−1
1
ϕk
− (ρ + δ)
(α − γ)
=
1− 1−
σ
e dt
σ χk
¶
∙
¸
µ
1
1
1−γ ϕ
k (k, e)
(α − γ)
(k, e)
k − δk − e
+ 1−
σ
χk
22
(6.7)
and
dk
1
(k, e)1−γ kϕ − δk − e.
=
dt
χk
(6.8)
From Proposition 2, we know that if α > γ the system (6.7)-(6.8) has a unique
steady state, the solution to
1
(k, e)1−γ ϕk ϕ−1 = ρ + δ
χk
and
1
(k, e)1−γ kϕ = δk + e.
χk
To study the local stability of this steady state, we examine the characteristic roots
¡ ¢
of the linear system at the steady state values k, e . For future reference we write
the linearized version of (6.7)-(6.8) as
⎡ ¡
⎤
⎡ ¡
¢
¢ ⎤
d k − k̄ /dt
k − k̄
⎣
⎦=S ⎣
⎦
d (e − ē) /dt
(e − ē)
(2.18)
where S contains the relevant derivatives evaluated at the steady state values. For
instance,
S11
where
k
∂
=
∂k
µ
µ 1−γ ϕ
¶
¶
w1−γ kϕ
w k
∂
− δk − e
+
− δk − e
χk
∂w
χk
k̄,ē,w̄
k̄,ē,w̄
k
is the derivatives of (k, e, (λ/L) M) evaluated at the steady state values of
k̄, ē and M̄. The expressions for all the entries of S are in the Appendix. For future
reference we have written the expressions for Sij as the sum of the direct effect of k or
e, and the indirect effect through wi . The following lemma, whose proof we relegate
to an appendix, characterizes the elasticities of the function
so defined.
Lemma A1. The the elasticities of the function , keeping M fixed, and evaluated
at steady state satisfy:
k
k
1 + δ ke
1
h³
´
i ≤ −ϕ
¡
¢ < 0,
= −ϕ
¡
¢
1−α
k
α − γ + 1 + βθ
(α − γ) + 1 + βθ
+
δ
1−γ
e
23
0<
e
e
=
1
1
´
i≤
³
¡
¡
¢ h¡
¢
¢ ³ 1−α ´ .
β
1−γ
β
k
(1 − γ) + 1 + θ
(1 − γ) + 1 + θ α−γ
1 + δ e α−γ − 1
Thus, as in the case of α = γ, we have
k
< 0, so that an increase in capital,
and hence output and exports, depresses the terms of trade, introducing an extra
source of decreasing returns. When α > γ, the is an extra effect:
e
> 0. Increasing
consumption e while keeping capital constant, requires a decrease in investment, and
since investment is more tradeable than consumption, it improves the terms of trade.
We also notice that this elasticities do not depend on the values of M̄ or λ̄/L̄. This
is so because these quantities affect the country only trough the steady state level
of w, which, given our Cobb-Douglas assumption on production, has no effect the
steady state value of the ratio of k/e. That is to say, if M̄ or λ̄/L̄ are such that the
steady sate wage w of the country is higher, then both its steady state capital and
consumption e, increase in the same proportion. Using this lemma we can examine
the matrix S and conclude
Proposition 4 : If α ≥ γ, the unique steady state of (6.7)-(6.8) is locally saddle path
stable: the matrix S has two real roots of opposite sign.
The proof consists on showing that the determinant of S is negative. It is given
in the appendix. We call μS the negative root of S. Figure 6.1 computes the half
life associated with the root μs , as a function of the Armington elasticity, given by
1/θ + 1. Figure 6.1 plots four such functions, for the combinations of two values for
intertemporal elasticity of substitution, namely 1/σ = 1 and 1/σ = 0.5, as well as two
values of the share parameters of tradeables in the production of consumption and
investment, namely (α, γ) = (0.75, 0.75) and(α, γ) = (0.81, 5) . The two combinations
of values for (α, γ) are chosen so that the share of tradeables in GDP is 0.75, a value
close to the one in the US economy. The difference between the two cases is that in
one consumption and investment have the same share, and in the other, as it is in
the US, investment has a higher tradeable share.
24
Insert Figure 6.1 about here.
As we show for the particular case of α = γ above, in each of the four curves in
Figure 6.1 lower Armington elasticity, i.e. higher values of θ, correspond to higher
convergence rate, i.e. to lower half life. Indeed, the limit as the Armington elasticity
goes to infinity is the close economy model of section 2. For instance, for σ = 2 and
(α, γ) = (0.81, 0.5) , the half life corresponding to θ = 0.15 is about 7.5 years, while
the half life of the corresponding close economy is about 8.8 years. Fixing the values
of (α, γ) , the different curves shows that, as in the close economy case, a higher
intertemporal elasticity of substitution 1/σ implies a faster convergence rate, i.e. a
smaller half life. Finally, the higher tradeability of investment represented by the
case where α > γ, decreases the half life, especially for low value of the Armington
elasticity (this is to be expected, since as the elasticity goes to infinity, as seen in
section 2, the half life is independent of the value of α and γ). Thus, the higher
tradeability of investment reinforces the effect of the terms of trade on the rate of
return on capital captured in the α = γ case, producing an even faster convergence
rate.
7. Dynamics: A Symmetric World Economy
Here we return to the n-country case, retaining the costless trade assumption of
Section 5 and specializing the system (5.11)-(5.13) to the case of countries with the
same productivity in tradeables λi /Li = λ̄/L̄, but with an arbitrary distribution of
sizes Li . The ratio λi /Li has the interpretation of the productivity of tradeables
relative to productivity in all goods. The size Li has the interpretation of number of
workers measured in efficiency units.3 We show that the system is locally saddle path
3
In Alvarez and Lucas (2006) we explored the implications of the differences of λ/L across coun-
tries. We find that, while this differences are important to explain cross country differences in the
25
stable. The speed of adjustment of country i to its steady state is given by a simple
function of the speed of adjustment for the the world economy studied in section 4,
the speed of adjustment of the small economy studied in section 6, and the size of
the country.
We start first considering the case of countries of the same size, letting Li = L̄
and same productivity parameter λi = λ̄, all i. The countries can differ arbitrarily in
their initial capital stocks ki (0). Using the functions
and Φ defined in Section 5 by
Lemmas 1 and 2, in a temporary equilibrium, these wage rates are given by
wi = (ki , ei ,
λ̄
Φ(k, e)),
L̄
(7.1)
where we have suppressed the constant arguments λ, L from Φ (·) . Here ki and ei are
scalars while k and e are the n-vectors describing the world economy. Thus, the time
derivatives of (7.1)
µ
µ
¶
¶
dwi
λ̄
λ̄
dki
dei
= k ki , ei , Φ(k, e)
+ e ki , ei , Φ(k, e)
(7.2)
dt
dt
dt
L̄
L̄
#
µ
¶ "X
n
n
dkj X
dej
λ̄
λ̄
+
Φk (k, e)
Φei (k, e)
+ Φ ki , ei , Φ(k, e)
dt
dt
L̄
L̄ j=1 i
j=1
Inserting this into (5.11) we obtain the 2n system of differential equations:
"
#
µ
¶1−γ
λ̄
dei
ei 1
=
ki , ei , Φ (k, e)
ϕkiϕ−1 − (ρ + δ)
dt
σ χk
L̄
(7.3)
¶
¶
∙ µ
¶
µ
¸
µ
dki
dei
λ̄
λ̄
1
k
e
(α − γ) ei
ki , ei , Φ(k, e)
+
ki , ei , Φ(k, e)
+ 1−
σ
dt
dt
L̄
L̄
#
¶
µ
¶ "X
µ
n
n
X
λ̄
dk
de
λ̄
1
j
j
+
(α − γ) ei Φ ki , ei , Φ(k, e)
Φk (k, e)
Φei (k, e)
+ 1−
σ
dt
dt
L̄
L̄ j=1 i
j=1
1
dki
=
dt
χk
µ
¶1−γ
λ̄
ki , ei , Φ (k, e)
kiϕ − δki − ei
L̄
(7.4)
relative prices of tradables to non tradables, they do not affect much other features of the model,
such as volume and gains of trade.
26
for i = 1, 2, ..., n. Each country contributes two of these equations, and these take
the same form for all countries. But their capital stocks ki differ, and so must their
costate values ei and, due to the terms of trade effect, their wage rates, wi .
The unique steady state of this system is symmetric, with ki , ei , and wi equal to
common values k̄, ē, and w̄ for all i.
To study the local dynamics of this system
it is useful to refine our characterization of the derivatives of wi , to which we turn
next. Evaluated at the symmetric steady state, then, the derivatives ∂Φ/∂kj are the
same for all j, and similarly for ∂Φ/∂ej . We denote these two numbers by Φk and Φe .
The steady state derivatives of wages in i with respect to capital stocks are then
∂wi
=
∂ki
k (ki , ei ,
λ̄
Φ(k, e)) +
L̄
Φ (ki , ei ,
λ̄
λ̄
Φ(k, e)) Φk (k, e) =
L̄
L̄
k
+
Φ
λ̄
Φk
L̄
and
∂wi
λ̄
=
∂kj
L̄
The values
k , Φ,
Φ (ki , ei ,
λ̄
Φ(k, e))Φk =
L̄
Φ
λ̄
Φk
L̄
if j 6= i.
and Φk do not vary with i, nor do their costate counterparts
e
and Φe . In light of these facts, the linear approximation to (7.4) can be written
¢
¡
X ¡
¡
¢
¢
d ki − k
= a ki − k + b
a
(7.5)
kj − k
j6=i
dt
X
(ei − e)
+b (ei − e) + bb
j6=i
where the constants a, b
a, b, and bb, readily calculated, do not vary with j.
Sum-
ming (7.5) over i, it follows that the deviations of the average capital stock, K =
P
(1/n) i ki , from k follow
¢
¡
i
¢ h
¡
d K −k
(7.6)
= [a + (n − 1)b
a] K − k + b + (n − 1)bb (E − e) .
dt
Using symmetry, the linear expansion of the costate equations (7.3) in terms can
27
be treated in a similar way obtaining
X ¡
¡
¢
¢
d (ei − e)
= d ki − k + db
kj − k
j6=i
dt
X
(ei − e)
+f (ei − e) + fb
j6=i
¢
¡
¢
¡
X d kj − k
d ki − k
+b
g
+g
j6=i
dt
dt
X
d (ej − ē)
d (ej − ē) b
+h
.
+h
j6=i
dt
dt
Summing (7.7) we obtain:
i
h
i¡
¢ h
d (E − e)
= d + (n − 1)db K − k + f + (n − 1)fb (E − e) +
dt
¢ h
¡
i d (E − e)
d K−k
b
+ h + (n − 1)h
.
[g + (n − 1)b
g]
dt
dt
(7.7)
(7.8)
For future reference we notice that the linearization around steady state gives the
following expression for g and h :
λ̄
Φk , ĝ =
L̄
λ̄
e + Φ Φe , ĥ =
L̄
g =
k
h =
+
Φ
The next lemma explores the fact that
λ̄
Φk , and
L̄
λ̄
Φ Φe .
L̄
Φ
are relative prices, with numeraire pm =
1, so that in the context of a symmetric case, a proportional increase in all capital
stocks keeps the relative prices unaltered.
Lemma 3. At the symmetric steady state:
0=
+n
k
Φ Φk
λ̄
and 0 =
L̄
e
+n
Φ Φe
λ̄
L̄
Proof. From (5.9) differentiating
1/θ
pm = 1 = (AB)
w.r.t. ki :
1/θ
0 = (AB)
µ
¶−β/θ
n
X
λ̄
λ̄
ki , ei , Φ (k, e)
L̄
i=1
µ
¶−β/θ−1
λ̄
λ̄
(−β/θ)
ki , ei , Φ (k, e)
L̄
28
(7.9)
µ
µ
µ
¶
¶
¶
λ̄
λ̄
λ̄
∂Φ
(k, e)
Φ (k, e) + Φ ki , ei , Φ (k, e)
+
k ki , ei ,
∂ki
L̄
L̄
L̄
µ
¶−β/θ−1 µ
¶
n
X
λ̄
λ̄
λ̄
∂Φ
1/θ
kj , ej , Φ (k, e)
(k, e) λ̄
(AB) (−β/θ)
Φ (k, e)
Φ kj , ej ,
∂kj
L̄
L̄
L̄
j6=i
Evaluating at the symmetric steady state and simplifying we obtain the desired expression. The proof for the other expression follows from a similar derivation. QED
Using Lemma 3 into (7.8) we have
i
i¡
¢ h
d (E − e) h
= d + (n − 1)db K − k + f + (n − 1)fb (E − e) .
dt
(7.10)
The combination of (7.6) and (7.10) implies that the local dynamics of the world
wide capital stock and consumption can be studied as an autonomous two-dimensional
system with a single state variable.
Next we obtain a second, one-state variable system, also autonomous, in the deviations of any one country’s capital stock from the average. We have ki − K =
¡
¢
ki − k − K − k , so that
X ¡
¢
¢ ¡
¢
¡
kj − k =
kj − k − ki − k
j6=i
j
¢ ¡
¢
¡
= n K − k − ki − k
X
Also
X
j6=i
Then (7.5) implies
Now we write
(ej − e) = n (E − e) − (ei − e) .
¢
¡
¡
¢
¡
¢
d ki − k
= (a − b
a) ki − k + b
an K − k
dt
³
´
+ b − bb (ei − e) + bbn (E − e)
¢
¢
¡
¡
d ki − k
d K −k
d (ki − K)
=
−
dt
dt
dt
29
and using (7.6):
³
´
d (ki − K)
= (a − b
a) (ki − K) + b − bb (ei − E) .
dt
Likewise for the costate variables we have, using (7.8):
³
´
³
´
d (ei − E)
= d − db (ki − K) + f − fb (ei − E) +
dt
¶ ³
µ
´ ∙ d (e − E) ¸
d (ki − K)
i
+ h − ĥ
+ (g − ĝ) ,
dt
dt
rearranging, replacing the expression for d (ki − K) /dt, and solving for d (ei − E) /dt we
obtain
d (ei − E)
³
´
³
´
³ dt ´
f − fˆ + (g − ĝ) b − bb
d − db + (g − ĝ) (a − b
a)
³
´
³
´
=
(ki − K) +
(ei − E)
1 − h − ĥ
1 − h − ĥ
We sum up the implications of these facts in a Proposition.
Proposition 5 . Under the assumption of costless trade ( κij = ωij = 1) and
n symmetric countries (λi = λ, Li = L) , the Euler systems for the state/costate
averages (E (t) , K (t)) and the deviations from average (ei (t) − E(t), ki (t) − K(t)),
i = 1, 2, ...n, and can be written as
⎡ ¡
⎤ ⎡
⎤ ⎡
⎤
¢
d K − K̄ /dt
a + (n − 1) â b + (n − 1) b̂
K − K̄
⎣ ¡
⎦=⎣
⎦ ⎣
⎦,
¢
ˆ
ˆ
d E − Ē /dt
d + (n − 1) d f + (n − 1) f
E − Ē
and
⎡
⎤
⎡
a − â
d(ki − K)/dt
⎦ = ⎣ d−db +(g−ĝ)(a−ba)
⎣
( )
d (ei − E) /dt
1−(h−ĥ)
b − b̂
(f −fˆ)+(g−ĝ)(b−bb)
1−(h−ĥ)
⎤ ⎡
⎦ ⎣
ki − K
ei − E
⎤
⎦
(7.11)
(7.12)
The relevance of this Proposition is that due to symmetry instead of analyzing a
2n × 2n system we can analyze two distinct 2 ×2 systems. We will turn next to the
30
characterization of these coefficient that define these two systems, where we will show
that they are both saddle-path stables. Hence, the local dynamics for the i − th country can be obtained by first solving for the stable solution for the world-wide averages
(K (t) , E (t)) using (7.11) given the initial value of Ki (0), and by solving for the stable
path of deviations from the world wide averages (ki (t) − K (t) , ei (t) − E (t)) using
(7.12) given the initial condition (ki (0) − Ki (0)). The path of (ki (t) , ei (t)) is obtained by adding the two paths.
For instance, the coefficients of (7.11), the linearization of (7.4) with respect to
ki and kj for j 6= i are:
∙
∙
∙
¸
¸
∂ w1−γ k ϕ
∂ w1−γ kϕ
− δk − e
+
− δk − e
a =
∂k
χk
∂w
χk
w̄,k̄,ē
w̄,k̄,ē
¸
∙ 1−γ ϕ
∂ w k
λ̄
â =
− δk − e
Φ Φk
∂w
χk
L̄
w̄,k̄,ē
λ̄
k + Φ Φk
L̄
¸
Here we have decomposed the derivatives in the direct effect of ki ei and their indirect
effect, through their effect on wi . We can similarly compute the expressions for b, b̂, d,
ˆ f and fˆ. We collect the expressions for all this term in Lemma 4 in the appendix.
d,
Using the expression for a and â it is easy to see that
∙
¸
¸
∙
∙
∂ w1−γ kϕ
∂ w1−γ kϕ
− δk − e
+
− δk − e
a+(n − 1) â =
∂k
χk
χk
w̄,k̄,ē ∂w
w̄,k̄,ē
and using lemma 3 we have
∙
¸
∂ w1−γ kϕ
− δk − e
a + (n − 1) â =
∂k
χk
w̄,k̄,ē
By using the expressions in Lemma 4 we can also readily verify that
¸
∙
∂ w1−γ kϕ
− δk − e
b + b̂ (n − 1) =
∂e
χk
w̄,k̄,ē
and that
∙ µ
¶¸
∂ e 1 1−γ ϕ−1
w ϕk
− (ρ + δ)
∂k σ χk
∙ µ
¶¸ w̄,k̄,ē
∂ e 1 1−γ ϕ−1
f + (n − 1) fˆ =
w ϕk
− (ρ + δ)
∂e σ χk
w̄,k̄,ē
d + (n − 1) dˆ =
31
λ̄
k + n Φ Φk
L̄
¸
We collect our result in a proposition.
Proposition 6. The matrix for the linear approximation of the Euler systems for
the state/costate averages (E (t) , K (t)) of Proposition 5 is identical to the one for
the closed economy neoclassical growth model:
⎡
⎤
a + (n − 1) â b + (n − 1) b̂
⎦
N =⎣
d + (n − 1) dˆ f + (n − 1) fˆ
Proof. This follows from Lemma 4 and inspection of the terms of the matrix N.
Now we turn to the analysis of the system for the deviations of the world-wide
averages (7.12). Let’s examine the equation for d (ki − K) /dt. Using the expressions
previously derived for a and â we have
¸
¸
∙
∙
∂ w1−γ kϕ
∂ w1−γ k ϕ
a − â =
− δk − e
+
− δk − e
∂k
χk
∂w
χk
w̄,k̄,ē
w̄,k̄,ē
Notice that these terms do not involve
Φ
k
so they don’t take into account the effect
of k It can be readily seen that a − â equals the terms S11 . From similar calculations
we can verify the same result for the remaining terms in the matrix S, as can be seen
in the appendix. We collect this result in a Proposition.
Proposition 7. The matrix for the linear approximation of the Euler systems for the
state/costate deviations from the world-wide averages (k (t) − K (t) , e (t) − E (t)) of
Proposition 5 is identical to the one for the small open economy:
⎡
⎤
a − â
b − b̂
S = ⎣ (d−db)+(g−ĝ)(a−ba) (f −fˆ)+(g−ĝ)(b−bb) ⎦
1−(h−ĥ)
1−(h−ĥ)
Taking Proposition 6 and 7 together, using well known results for the neoclassical
growth model and our result for the small open economy in Proposition 4, we have
established the saddle path stability of the systems in (7.11)-(7.12).
32
Now we state our main result for a world with free trade and countries with the same
ratios λi /Li but with an arbitrary distribution of sizes Li . Let K−i denote the average
capital in countries different from i, and let country i have population Li > 0, where
P
we normalize the population of the world to one, so that i Li = 1. Formally, we let
the average capital in the rest of the world be:
K−i ≡
X
j6=i
Lj
kj
1 − Li
so that Lj / (1 − Li ) is the share of country j in the rest of the world.
Proposition 8 . Under the assumption of costless trade ( κij = ωij = 1) and that
all countries have the same ratio λi /Li , the (linearized) equilibrium path of capital
of country i is
ki (t) − k̄ =
¤ £
¡
¢
¤ £
¤
£ μ t
e S (1 − Li ) + eμN t Li ki (0) − k̄ + eμN t − eμS t (1 − Li ) K−i (0) − k̄
where μS and μN are the stable (negative) eigenvalues of the matrices S and N.
These expression shows that the speed of convergence to steady state is a simple
function of the size of a country Li , the speed of convergence of world-wide economy
μN , and the speed of convergence of the small open economy, μS . Recall that, almost
by definition, the speed of convergence μN and μS are independent of size Li , hence
size as an independent effect. Recall also that the common level of λ̄/L̄ does not affect
μS or μN . The expression in Proposition 8 says that the effect of the gap between the
£
¤
initial capital and the steady state capital of one country ki (0) − k̄ depends on a
population weighted average of the speed of convergence of the small open economy
and the world-wide economy. The effect of the average capital in the rest of the
¡
¢
world K−i (0) − k̄ , depends on the size of the rest of the world, (1 − Li ) , times the
difference in the speed of convergence between the close and small open economy.
33
Notice that letting Li → 0 we obtain the case of the small open economy analyzed in
section 5, extending it to cover the case where the rest of the world is outside steady
state. Finally, notice that Figure 6.1 provides enough information to quantify the
effects of μS and μN , since it plots for four different combination of parameter values,
the half life implied by μS for a range of values of 1/θ + 1, and since the μS → μN as
θ → 0.
Proof of Proposition 8 . Consider the case where the world is divided into ñ identical
cities. Each city has a fraction 1/ñ of the world population, with the same value of
λ. Furthermore assume that these identical "cities" can be assigned into n countries.
P
We use mi to denote the number of cities that belong to country i, so that ni=1 mi =
ñ. The size of country i, defined as the fraction of the world population located in
P
this country, is given by Li = mi /ñ. Clearly ni=1 Li = 1, We assume that the initital
per-capita capital of each city j, denoted by k̃j (0) in country i is the same, so that
ki (0) = k̃j (0) .
For each of the identical ñ cities we have
¡
¢
k̃j (t) − k̄ = (kj (t) − K (t)) + K (t) − k̄
where K (t) is the average per-capita capital in the world. Using Propositions 5, 6
and 7 for the linearized system of ñ identical cities, the per-capita capital of city
j evolves as:
μS t
so
k̃j (t) − K (t) = e
¡
¢
K (t) − k̄ = eμN
k̃j (t) − k̄ = eμS
t
t
i
h
k̃j (0) − K (0) ,
£
¤
K (0) − k̄
i
h
k̃j (0) − K (0) + eμN
34
t
£
¤
K (0) − k̄ >
This set-up satisfies the hypothesis of the aggregation result of Proposition 3, hence
we can regard country’s i has having λi = mi λ̄, and we have λi /Li = λ̄ ñ. Moreover,
by Proposition 3 we have that each city j per capita capital k̃j (t) equals the country
per-capita capital ki (t) for all t ≥ 0. Additionally, we can write K (t) as
X mi
X
1X
K (t) =
ki (t) =
k̃j (t) =
Li ki (t)
ñ j=1
ñ
i=1
i=1
ñ
n
n
Hence, using the evolution for k̃j (t) , setting ki (t) = k̃j (t) , and using K−t (t)
K (t) = Li ki (t) + (1 − Li ) K−i (t)
we have
or
¤
£
ki (t) − k̄ = eμS t ki (0) − k̄
¢
¡
¢¤
¤£ ¡
£
+ eμN t − eμS t Li ki (0) − k̄ + (1 − Li ) K−i (0) − k̄
¡
¤
¢ ¤ £
£
= eμS t + eμN t − eμS t Li ki (0) − k̄
¡
¢
¤
£
+ eμN t − eμS t (1 − Li ) K−i (0) − k̄
ki (t) − k̄ =
QED.
¤ £
£ μ t
¤
e S (1 − Li ) + eμN t Li ki (0) − k̄
¡
¢
¤
£
+ eμN t − eμS t (1 − Li ) K−i (0) − k̄
In the symmetric case treated here, μS is the same for all countries. If cross-country
differences in λi /Li were admitted, we would have n − 1 different roots μiS describing
the evolution of the system, and the determination of the evolution of each country
will depend on the whole distribution of capital across the world.
With more effort, the method and results of this section can be extended to accommodate symmetric transportation costs (κij = κ < 1 for all i 6= j). In this case a
35
result like Proposition 5 and, 6 holds, and a modified version of Proposition 7. In this
case the matrix describing the deviation from world-averages also has one stable and
one unstable root, but the matrix depend on the number of countries n. Because of
this, the aggregation result used to extend the analysis to countries of different size
does not hold, this should be no surprise with transportation cost across countries,
but with costless trade within a country, a large country is not equivalent to the union
of small countries. Also, as indicated in Section 6, imposing a common tradeables
intensity for capital and consumption goods (α = γ) turns our two-sector model into
the more familiar and tractable one sector model. Stronger stability results can be
obtained for this case, see Moll (2007).
8. Conclusion
This paper is a contribution to the research on integrated models of trade and
growth stimulated by Ventura’s (1997) paper. Ventura studied a Heckscher-Ohlin
type model with a technology that effectively ensures that factor price equalization
always holds. In his model, the path of the world stock of capital is independent of
the distribution of capital across countries, and coincides with the path of a closed
neoclassical growth model, which Ventura calls the “integrated economy. We obtain a
similar decomposition of the dynamics, but for reasons that have nothing to do with
factor mobility, either direct or indirect.
In Ventura’s model, capital markets are complete and there is a continuum of steady
states, corresponding to redistributions of capital across countries that do not change
the steady state value of the world capital. Bajona and Kehoe (2004) examine the ncountry dynamics in a context similar to Ventura’s, but they assume continuous trade
balance–no international borrowing–and postulate a technology under which factor
price equalization is possible but by no means ensured. As capital stocks evolve, a
36
country may pass in and out of the “cone of diversification.” The resulting system has
dynamic possibilities that cannot be explored with the kind of local approximations
this paper applies.
Our paper uses the same market context–continuous trade balance–as Bahona
and Kehoe (2004), and addresses similar issues. But the gains from trade in our model
arise from the intraindustry trade that arises from the Eaton-Kortum technology.
Differences in factor endowments can affect these gains, but are not their main source.
Factor price equalization can be approached asymptotically, but when and whether
it holds exactly is a matter of no special importance. The behavior of factor prices
becomes a quantitative issue, not a qualitative one.
I THINK WE NEED A REFERENCE TO VENTURA AND ACEMOGLU. I AM
NOT EXACTLY SURE HOW TO WRITE THAT REFERENCE, SINCE THE SET
UP IS CLOSE TO OURS (THEY CONSIDER THE CASE OF LOG UTILITY,
NO DEPRECIATION, AND α = γ), THEIR CONCLUSIONS ARE BASICALLY
CORRECT (THE MODEL IS STABLE), BUT THEIR ANALYSIS IS INCORRECT.
We have used our formulation to address two substantive questions. In Section 4,
we re-examined a tariff reform simulation carried out in a static context in our earlier
paper. With explicit dynamics introduced, in a realistic way, it becomes possible
to study the transition dynamics of tariff reduction quantitatively. We found that
this analysis confirmed the accuracy of the static analysis as a description of steady
states, before and after reform, but implied much reduced welfare gains. Although
these results were obtained under the assumption of identical economies, there is good
reason to believe–as spelled out in A/L–that they apply much more broadly.
The rest of the paper addressed the classic question of the convergence of economies
to common levels of per capita capital and production. In Section 7 we concluded that
introducing trade into the neoclassical growth model contributes essentially nothing
to this question. The transition dynamics of a single economy to its long run capital
37
level in a world with costless trade are quantitatively the same as the dynamics of the
same economy under autarchy. Certainly this conclusion would be drastically altered
if we were to permit international borrowing, but doing so would imply capital flows
that exceed by far the flows we observe now and at any time in the past. For this
reason we follow Bahona and Kehoe (2004), not Ventura (1997) but it would be useful
to have models with capital markets that are in between these extremes.
REFERENCES
[1] Alvarez, Fernando, and Robert E. Lucas, Jr. 2007. “General Equilibrium Analysis
of the Eaton-Kortum Model of International Trade.” Journal of Monetary Economics
[2] Bajona, Claustre, and Timothy J. Kehoe. 2006. “Trade, Growth, and Convergence in
a Dynamic Heckscher-Ohlin Model.” NBER Working Paper #12567.
[3] Burstein, Ariel, João C. Neves, and Sergio Rebelo. 2004. “Investment Prices and Exchange Rates: Some Basic Facts.” Journal of the European Economic Association,
2: 302-309.
[4] Cooley, Thomas F., and Edward C. Prescott (1995). “Economic Growth and Business Cycles”, Chapter 1 in Thomas F. Cooley, ed., Frontiers of Business Cycle
Research. Princeton: Princeton University Press.
[5] Eaton, Jonathan, and Samuel Kortum. 2001. “Trade in Capital Goods.” NBER Working Paper #8070.
[6] Eaton, Jonathan, and Samuel Kortum. 2002. “Technology, Geography, and Trade.”
Econometrica, 70: 1741-1779.
38
[7] Moll, Benjamin. 2007.
[8] Ventura, Jaume. 1997. “Growth and Interdependence.” Quarterly Journal of Economics, 112: 57-84.
39
Appendix
A1. Expressions for the linearized version of the close economy:
µ
¶
w ϕ
∂
k − δk − e
,
N11 =
∂k pk
k̄,w̄,ē
µ
¶
∂ w ϕ
k − δk − e
,
N12 =
∂e pk
k̄,w̄,ē
¸¶
µ ∙
∂ 1 w ϕ−1
ϕk
− (ρ + δ)
ē,
N21 =
∂k σ pk
k̄,w̄,ē
¸¶
µ ∙
∂ 1 w ϕ−1
N22 =
ϕk
− (ρ + δ)
ē.
∂e σ pk
k̄,w̄,ē
A2. Expressions for the Linearized version of the small open economy:
µ
µ 1−γ ϕ
¶
¶
w k
∂ w1−γ kϕ
∂
S11 =
− δk − e
+
− δk − e
∂k
χk
∂w
χk
k̄,ē,w̄
k̄,ē,w̄
k
µ 1−γ ϕ
¶
¶
w1−γ kϕ
w k
∂
S12
− δk − e
+
− δk − e
e
χk
∂w
χk
k̄,ē,w̄
k̄,ē,w̄
³ h
i´
∂
e
1
1−γ
ϕ−1
w
ϕk
−
(ρ
+
δ)
∂k σ χk
k̄,ē,w̄
¡
¢
S21 =
+
ē e
1
1 − 1 − σ (α − γ) w̄
³ h
h
i ¡
i´
¢
∂
e
1
1
1
1−γ
ϕ−1
1−γ ϕ
k
w
ϕk
−
(ρ
+
δ)
+
1
−
w
k
−
δk
−
e
(α
−
γ)
∂w σ χk
σ
χk
k̄,ē,w̄
¡
¢
ē e
1
1 − 1 − σ (α − γ) w̄
³ h
i´
∂
e
1
1−γ
ϕ−1
w
ϕk
−
(ρ
+
δ)
∂e σ χk
k̄,ē,w̄
¡
¢
S22 =
+
1 − 1 − σ1 (α − γ) ēw̄e
i ¡
i´
³ h
h
¢
ϕ−1
∂
e
1
1
1
1−γ
1−γ ϕ
k
w ϕki − (ρ + δ) + 1 − σ (α − γ)
w k − δk − e
∂w σ χk
χk
k̄,ē,w̄
¡
¢
ē e
1
1 − 1 − σ (α − γ) w̄
∂
=
∂e
µ
40
k
.
e
.
Proof of Proposition 4.
Lemma A1. The the elasticities of the function , keeping M fixed, and evaluated
at steady state satisfy:
0<
k
k
= −ϕ
e
e
=
1 + δ ke
1
´
i ≤ −ϕ
h³
¡
¢ < 0,
¡
¢
1−α
k
α − γ + 1 + βθ
+
δ
(α − γ) + 1 + βθ
1−γ
e
1
1
´
i≤
³
¡
¡
¢ h¡
¢
¢ ³ 1−α ´ .
β
1−γ
β
k
(1 − γ) + 1 + θ
(1 − γ) + 1 + θ α−γ
1 + δ e α−γ − 1
Proof of Lemma A1. Differentiating
β
(k, e)γ−1 e = ξ (k, e)−(1+ θ )
kϕ (1 − γ) − (α − γ) χk
keeping ξ fixed we have :
ϕ
ϕk (1 − γ) − (α − γ) (γ − 1) χk
γ−1
e
k
k
µ
¶
β k
=− 1+
θ
k
ξ
−(1+ βθ )
and rearranging:
k
k
= −ϕ
(1/χk ) 1−γ kϕ
´ i
³
¡
¢h
β
α−γ
1−γ
ϕ
(α − γ) e + 1 + θ (1/χk )
k − 1−γ e
and evaluating it at steady state:
k
k
= −ϕ
e
k
(α − γ)
Differentiating w.r.t. e :
− (α − γ) (γ − 1) χk
γ−1
e
e
e
e
k
+δ
´
i
¡
¢h ³
+
δ
+ 1 + βθ ke 1−α
1−γ
− (α − γ) χk e
γ−1
¶
µ
β e e
ξ
=− 1+
θ
−(1+ βθ )
or
e
e
=
(α − γ) e
¢
¡
β
(α − γ) (1 − γ) e + 1 + θ [(1/χk ) 1−γ kϕ (1 − γ) − (α − γ) e]
and evaluating it at state:
e
e
=
1
h¡
i
¡
¢
¢ ³ 1−γ ´
(1 − γ) + 1 + βθ
1 + δ ke α−γ
− 1
41
To obtain the bounds for the elasticities, notice that since α ≥ γ and δk/e ≥ 0
1 + δ ke
1
1
´
i≥³
´ ¡
h³
¡
¢
≥
¡
¢
¢
β
β
α−γ
β
1−α
k
(α
−
γ)
+
1
+
(α − γ) + 1 + θ
+ δe
+ 1+ θ
θ
1−γ
1+δk/e
which gives the bound on k k / . Using δk/e ≥ 0 we obtain the bound on e e / . QED.
Proof of Proposition 4 (The local dynamics of the small open economy are saddle
path stable) The dynamic system can be written as:
k̇ = sk (k, e)
ė = se (k, e) ≡ be (k, e, sk (e, k))
2
2
2
where the functions sk : R+
→ R, se : R+
→ R and bk : R+
× R → R, where
sk (e, k) = 1/χk
(e, k)1−γ kϕ − e − δk,
³
´
e/σ
be k, e, k̇ = h
¡
¢
1
1 − 1 − σ (α − γ) e
e
e
+h
¡
¢
1 − 1 − σ1 (α − γ) e
¤
£
i (1/χk ) (e, k)1−γ ϕkϕ−1 − (δ + ρ)
µ
¶
1
1
i 1−
(α − γ)
σ
e
k
k̇
Linearizing the functions se and sk around the steady state values, we have
⎡ ⎤ ⎡
⎤⎡ ¡
⎡ ¡
¢ ⎤
¢ ⎤
k̇
s
s
k − k̄
k − k̄
⎣ ⎦ = ⎣ kk ke ⎦ ⎣
⎦≡S ⎣
⎦
ė
sek see
(e − ē)
(e − ē)
The steady state is a saddle iff the eigenvalues of S are of different sign, which is
equivalent to det (S) < 0. Instead we have that
det (S) = skk bee − ske bek < 0.
where bee and bek are the derivatives of the function be evaluated at the steady state
values. To see why, differentiating ae :
sek = bek + bek̇ skk ,
see = (bee + bek̇ ske )
42
We can express det (S) as
det (S) = skk see − ske sek
= skk (bee + bek̇ ske ) − ske (bek + bek̇ skk )
= skk bee − ske bek + bek̇ (skk ske − ske skk )
= skk bee − ske bek .
The rest of the proof consists on deriving explicit expressions for skk , bee , ske , bek and
replacing them to verify that det (S) < 0. We have:
skk = ρ + (1 − γ)
(1/χk )
1−γ ϕ
k
µ
k
¶ µ
k
1 + δk/e
= ρ + (1 − γ)
k/e
bee
and
bek
µ
ske = −1
∙
(1/χk )
i
e/σ
= h
¡
¢
1 − 1 − σ1 (α − γ) e
1/σ
= h
¡
¢
1 − 1 − σ1 (α − γ) e
e/σ
= h
¡
¢
1
1 − 1 − σ (α − γ) e
1/σ
= h
¡
¢
1 − 1 − σ1 (α − γ) e
where k, c and
e,
k
e
e
1−γ
k
k
k
¶
¶
,
ϕkϕ−1
(1 − γ)
e
i (ρ + δ) (1 − γ)
e
e
e
e
¸
,
∙
ϕ (1/χk ) 1−γ kϕ−1 ϕ (1/χk ) 1−γ kϕ−1
k
i (ϕ − 1)
+
(1 − γ)
k
k
e
∙
¸
³e´
k k
i
(ρ + δ) (ϕ − 1) + (1 − γ)
k
e
are evaluated at their steady state values. Then
det (S) = skk bee − ske bek
∙
µ
¶ µ
1 + δk/e
k
= ρ + (1 − γ)
k/e
1/σ
+h
¢
¡
1 − 1 − σ1 (α − γ) e
¶¸
1/σ
e
h
i (ρ + δ) (1 − γ)
¡
¢
1 − 1 − σ1 (α − γ) e e
∙
¸
³e´
k k
i
(ρ + δ) (ϕ − 1) + (1 − γ)
k
e
k
43
e
k
¸
Using Lemma A.1 we have
0 < (α − γ)
e
e
<
α−γ
≤1
1−γ
thus
1/σ
h
¡
¢
1 − 1 − σ1 (α − γ) e
e
Then we write
h
i
¡
¢
e e
1
1 − 1 − σ (α − γ)
det (S)
Ŝ ≡
(ρ + δ) /σ
∙
µ
¶ µ
¶¸
k e k k
e
= ρ + (1 − γ) 1 + δ
(1 − γ)
e k
i >0.
e
³e´∙
k
+
(ϕ − 1) + (1 − γ)
k
k
¸
or
∙
µ
¶
µ
k
k
k
k
ρ+ 1+δ
(1 − γ)
Ŝ =
e
e
e
If
∙
then we are done since
k
¶¸
(1 − γ)
µ
¶
µ
k
k
k
ρ+ 1+δ
(1 − γ)
e
e
e
> 0 and
k
e
k
e
¶¸
+ ϕ − 1 + (1 − γ)
k
k
<0
< 0 . So assume otherwise. Then, using the
bounds on the elasticities of Lemma A1:
#
"
µ
¶
e
k
k
(1 − γ)
1−γ
k
¡
¡
¢ (1 − γ) e + ϕ − 1 − ϕ
¢
≤
ρ− 1+δ
ϕ
Ŝ
β
e
e
e
α−γ + 1+ θ
α − γ + 1 + βθ
#
"
µ
¶
1
k
(1 − γ)
k
´
³
¡
¢
ρ− 1+δ
ϕ
≤
¡
¢
β
1
1−α
e
e
α−γ + 1+ θ
1 + 1 + βθ α−γ
1−γ
+ϕ − 1 − ϕ
(1 − γ)
¡
¢
α − γ + 1 + βθ
using that at steady state
ϕρ
k
ρ=
≤1
e
ρ + δ (1 − ϕ)
44
we have
k
≤
Ŝ
e
#
µ
¶
k
(1 − γ)
1
´
³
¡
¢
1− 1+δ
ϕ
¡
¢
β
1
e
(α − γ) + 1 + βθ
1 + 1 + θ α−γ 1−α
1−γ
"
+ϕ − 1 − ϕ
(1 − γ)
¡
¢
(α − γ) + 1 + βθ
using that δk/e ≥ 0
#
"
1
(1 − γ)
(1 − γ)
k
´
³
¡
¡
¢
¢
+ϕ−1−ϕ
Ŝ ≤ h (ϕ) ≡ 1 − ϕ
¡
¢
β
β
1
1−α
e
(α − γ) + 1 + θ
(α − γ) + 1 + βθ
1 + 1 + θ α−γ
1−γ
rearranging
¡
¢ ³ 1−α ´ ⎤
2 (1 − γ) + 1 + βθ α−γ
1
⎦
³
¡
¢
h (ϕ) = ⎣1 − ϕ
¡
¢
β
(α − γ) + 1 + θ
1+ 1+ β
⎡
θ
1
α−γ
We have
h (0) =
´
1−α
1−γ
+ (ϕ − 1)
1
´
³
−1<0
¡
¢
1
1−α
1 + 1 + βθ α−γ
1−γ
and since h is linear in ϕ, it suffices to show that h (1) ≤ 0 .Direct computation gives
⎡
¢ ³ 1−α ´ ⎤
¡
2 (1 − γ) + 1 + βθ α−γ
1
⎦
´
³
¡
¢
h (1) = ⎣1 −
¡
¢
β
1
1−α
(α − γ) + 1 + θ
1+ 1+ β
θ
and thus
or
or
or
¡
¢ ³ 1−α ´
2 (1 − γ) + 1 + βθ α−γ
¡
¢
h (1) ≤ 0 iff
≥1
(α − γ) + 1 + βθ
α−γ
µ
¶µ
¶
µ
¶
β
1−α
β
2 (1 − γ) + 1 +
≥ (α − γ) + 1 +
θ
α−γ
θ
¶µ
¶
µ
1−α
β
− 1 ≥ (α − γ) − 2 (1 − γ)
1+
θ
α−γ
¶µ
¶
µ
1 − 2α + γ
β
≥α+γ−2
1+
θ
α−γ
45
1−γ
But
and
µ
¶µ
¶ µ
¶
β
1 − 2α + γ
1 − 2α + γ
1+
≥
≥ 1 − 2α + γ
θ
α−γ
α−γ
1 − 2α + γ ≥ α + γ − 2
which is equivalent to 3 ≥ 3α . This establishes the desired inequality. QED.
Lemma 4. The coefficients of (7.11), the linearization of (7.4) are given by
∙
¸
¸
¸
∙
∙
λ̄
∂ w1−γ k ϕ
∂ w1−γ kϕ
− δk − e
+
− δk − e
a =
k + Φ Φk
∂k
χk
∂w
χk
L̄
w̄,k̄,ē
w̄,k̄,ē
¸
∙ 1−γ ϕ
∂ w k
λ̄
â =
− δk − e
Φ Φk
∂w
χk
L̄
w̄,k̄,ē
∙
¸
¸
∙
∙
∂ w1−γ kϕ
∂ w1−γ k ϕ
b =
− δk − e
+
− δk − e
∂e
χk
∂w
χk
w̄,k̄,ē
w̄,k̄,ē
∙ 1−γ ϕ
¸
∂ w k
λ̄
− δk − e
b̂ =
Φ Φe
∂w
χk
L̄
w̄,k̄,ē
λ̄
e + Φ Φe
L̄
¸
The coefficients of (7.12), the linearization of (7.3) are given by
∙
∙ µ
∙ µ
¶¸
¶¸
∂ e 1 1−γ ϕ−1
∂ e 1 1−γ ϕ−1
d =
w ϕk
− (ρ + δ)
+
w ϕk
− (ρ + δ)
∂k σ χk
∂w
σ
χ
k
w̄,k̄,ē
w̄,k̄,ē
∙
¸
¶¸
∙ µ
λ̄
∂ e 1 1−γ ϕ−1
w ϕk
− (ρ + δ)
dˆ =
k + Φ Φk
∂w σ χk
L̄
w̄,k̄,ē
∙
¶¸
¶¸
∙ µ
∙ µ
∂ e 1 1−γ ϕ−1
∂ e 1 1−γ ϕ−1
f =
w ϕk
− (ρ + δ)
+
w ϕk
− (ρ + δ)
∂e σ χk
∂w
σ
χ
k
w̄,k̄,ē
w̄,k̄,ē
∙ µ
¶¸
λ̄
∂ e 1 1−γ ϕ−1
w ϕk
− (ρ + δ)
fˆ =
Φ Φe
∂w σ χk
L̄
w̄,k̄,ē
Proof of Proposition 7. For the derivatives w.r.t. (e − E) of d (ki − K) /dt we have
:
∙
∙
¸
¸
∂ w1−γ kϕ
∂ w1−γ k ϕ
b − b̂ =
− δk − e
+
− δk − e
∂e
χk
∂w
χk
w̄,k̄,ē
w̄,k̄,ē
46
e
λ̄
k + Φ Φk
L̄
λ̄
e + Φ Φe
L̄
¸
¸
A similar argument hold for the equation for d (ei − E) /dt, although the algebra
is more complicated. For instance we have
³
´
b
d − d + (g − ĝ) (a − b
a)
∙ µ
∙ µ
¶¸
¶¸
∂ e 1 1−γ ϕ−1
∂ e 1 1−γ ϕ−1
=
w ϕk
− (ρ + δ)
+
w ϕk
− (ρ + δ)
k
∂k σ χk
∂w σ χk
w̄,k̄,ē
w̄,k̄,ē
!
¶ Ã ∙ 1−γ ϕ
∙
µ
¸
¸
∂ w k
1
∂ w1−γ kϕ
+ (α − γ) ē 1 −
− δk − e
+
− δk − e
k
k
σ
∂k
χk
∂w
χk
w̄,k̄,ē
w̄,k̄,ē
and
¶
µ
³
´
ē e
1
(α − γ)
1 − h − ĥ = 1 − 1 −
σ
w̄
Thus
∂
∂k
S21 =
³ h
e
σ
³
´
d − db + (g − ĝ) (a − b
a)
³
´
= S21
1 − h − ĥ
1
w1−γ ϕkiϕ−1
χk
i´
− (ρ + δ)
k̄,ē,w̄
¡
¢
ē e
1
1 − 1 − σ (α − γ) w̄
i ¡
i´
³ h
h
¢
ϕ−1
∂
e
1
1
1
1−γ
1−γ ϕ
k
w
ϕk
−
(ρ
+
δ)
+
1
−
w
k
−
δk
−
e
(α
−
γ)
i
∂w σ χk
σ
χk
k̄,ē,w̄
¡
¢
+
ē e
1
1 − 1 − σ (α − γ) w̄
In the same manner we can verify that
³
´
³
´
f − fˆ + (g − ĝ) b − bb
³
´
= S22
1 − h − ĥ
QED.
47
k