A Study of Even Development and Sustainable Development
M ASAAKI

A BE

Ⅰ.I NTRODUCTION
There are many definitions of sustainable development, and they often
incompatible with each other. At the risk of oversimplification, we candistinguish two broad ideological groups in environmentalism. One grouppursues expansion of consumption for economic growth, and the other groupaims to limit consumption for conservation of environment. The efforts to the
opposite direction from each other are equal to doing nothing. That is why weare not able to progress to sustainable development.
We need to find out the goal of sustainable development. To discuss thegoal as a whole is beyond me. We advance an argument focusing on economic
growth. Some people regard sustainable development as continuative economic
growth. The other people argue steady-state economy is necessary forsustainable development. There are some people, moreover, who think
reductions in the scale of economy are required for sustainable development.
Which position is right?
The purpose of this paper is to find out the goal of sustainable
development and possibility of even-development between the North-South byanalyzing the simple dynamic model of economic growth in the very long run
within the extended framework of Krugman (1981). The model portrays atwo-region world in which the industrial sectors of regions grow through theaccumulation of capital. There are two crucial assumptions. One is external
economies in the industrial sector. Therefore a small ‘head start’ for one region
will cumulate over time, with exports of manufactures from the leading region
crowding out the industrial sector of the lagging region. Second is essentialnessof renewable resource for manufactured production.In this model, renewable
resource plays crucial role. It is the source of manufactured production and it
restrict eternal growth.
Ⅱ.THE MODEL
We consider a world consisting of two regions, North and South. To sharpen
the analysis, we assume that these regions are identical in the sense thattechnological and behavioral relationships are the same. We also assume that theregions have equal labor forces L and that these labor forces do not grow overtime. Thus we have(1)
L N = LS = L .

Each region is able to produce two goods, a manufactured good M and anagricultural product A, and to trade at zero transportation cost. There will thus
be a single world price of manufactured goods in terms of agricultural products,
P M . Agricultural products are produced by labor alone; we will choose units sothat one unit of labor produces one unit of agricultural goods.
The growth sector, however, is manufacturing. Manufacturing requires
capital K, labor L and renewable resource R. We assume that, from the point ofview of an individual firm, the unit capital, labor and renewable resource
requirements are fixed. In the aggregate, however, unit capital, labor andrenewable resource requirements are not constant; instead, in each region they
are decreasing functions of the region's aggregate capital stock. Letting c, v andr are the unit capital, labor and renewable resource requirements, respectively,
we have(2)c N = c(KN ),c S = c(KS ) ,v N = v(K N ) ,v S = v(KS ),rN = r(KN ),rS = r(KS ) ,Where c', v', r' < 0. However, we assume that the absolute value of the elasticity
of unit input requirements with respect to output is less than one, so that total
input requirements rise as manufacturing output rises. In order to simplify ananalysis, we also assume neutral technical growth(c: v: r =constant).
At the initial point, we assume that labor force and renewable resource for
manufacturing are abundant. Given the relationships (2), then, together with full
employment of factors, we can determine the pattern of output. In each region
the output of manufactured goods depends on the capital stock:
KN
KS
(3)
MN =
,
MS =
.
c(K N )c(KS )
Given the output of manufactured goods, then, total requirements of labor andrenewable resource with respect to output are given by the following forms:
v N M N , v S M S and r N M N , r S M S .
Output of agricultural goods can then be determined from the agricultural
sector's role as a residual claimant on labor:(4)
A N = L − vN MN ,A S = L − vSMS .
Next, we define the nature of the renewable resources. With the initial point
when both regions just began accumulation of capital, We assume that Initialendowment of the renewable resource R 0 is abundant and upper limit. The
amount of the renewable resource of both regions does not exceed R max . To
simplify the analysis, the endowment of both regions is equal at the initial point:00(5)
R N = R S = R max .
The stock of renewable resource will be decreased by input to industrial
production and will increase by reproduction of these selves. To keep the model
simple, we assume that the renewable resource has a certain fixed growth rate a(0<a<1) and it is used only for industrial production. When the amount of
resource is R, the increment of next term becomes aR. Hence, if the inputbecomes larger than aR, the amount of renewable resource of next term will
decrease. Thus, we can describe the change of each region's stock of renewable
resources asÝ = aR − r M .
(6)RÝN = aRN − rN M N ,
R
S
S
S
S
Note that the conditions of R N , R S ≦ R max exist in (6).
There are two upper limits K MAXL , K MAXR . K MAXL is the amount of capital,
which comes when the region is completely specialized in manufacturing and no
more labor can be drawn out of agriculture. K MAXR is the amount of capital,
which comes when the input of renewable resources reaches the upper limit R max
and no more resources can be gathered. We can define K MAXL , K MAXR as follows:v(KMAXL ) ⋅ KMAXLr(K MAXR ) ⋅ KMAXR=L ,= RMAX(7)c(K MAXL )c(K MAXR )Consider next the distribution of income. There are two cases: the case inwhich at least some labor is used in agricultural production, and the case ofcomplete specialization in manufacturing. If some labor is used in agriculture,
this ties down the wage rate, which is 1 in terms of agricultural goods, 1/P M interms of manufactures. We can then determine the rental per unit of capital as aresidual. For simplicity, let us assume (though it is not essential) that capital
goods are produced by labor alone, i.e., we include them as part of ‘agricultural’
output. Using the price of manufactured good P M and the cost of gathering
renewable resource C R , then the rental per unit of capital, measured inagricultural (or wage) units, is also the profit rate, and we haveP − v N − rNCRN
P − vS − rS CRS
ρN = M, ρS = M(8)cN
cS
Where ρ N , ρ S are profit rates North and South. We assume that the input costof renewable resources in each region C R depends on the amount of renewable
resources in each region, and they are decreasing functions of the amount of
renewable resources in each region.
(9)
CRN = C( RN ) , C ′(RN ) < 0
CRS = C(RS ) , C ′(RS ) < 0
Since c and v are functions of the capital stocks and C R is a function of theamount of renewable resources, we can also write (8) as a pair of reduced form
equations(10)ρ N = ρ (PM ,KN ,RN ) , ρ S = ρ(PM ,K S ,RS )where ∂ρ / ∂ P M >0, ∂ρ / ∂ K>0 and ∂ρ / ∂ R>0, the second expression iscaused by increasing returns to scale in manufacturing.
When a region is completely specialized in manufacturing, (10) no longer
holds. Instead the rate of profit is determined in Kaldorian fashion by therequirement that savings equal zero, if there is no foreign investment, or by therate of profit on foreign investment if there is such investment. In the latter casethe wage rate is residually determined.
To close the model we need to specify the demand side. We will make twostrong assumptions for the sake of easy algebra; the conclusions of the model
could be derived under weaker but less convenient assumptions. First, saving
behavior is classical: all profits and only profits are saved. Second, a fixedproportion μ of wages will be spent on manufactures, 1- μ on agricultural
goods.
The savings assumption means that, if there is no international investment,
the rate of growth of the capital stock in each region will just equal the rate ofprofit.
KÝNKÝS= ρN ,
= ρS
(11)
KN
KS
The relative price of manufactured goods will be determined by world
demand and supply. Since a fraction μ of wages is spent on manufactures,provided that both countries produce some agricultural goods, and we have
(12)
PM [MN + MS ] = µ[LN + LS ]which can be rewritten as
2µL
(13)
PM =
≡ P (K ,K )[K N / c (KN ) + KS / c (KS )] M N S
This gives us a relationship between the two capital stocks and P M ; it isapparent that P M is decreasing in both capital stocks. Note also that K N and K S
enter symmetrically, so that where K N =K S , ∂ P M / ∂ K N = ∂ P M / ∂ K S .
Finally, we can combine (10), (11) and (13) to express the rate of change
each region’s capital stock as a function of the levels of both capital stocks andrenewable resources:KÝN= ρ (PM , KN , RN ) ≡ gN (KN , KS , RN )
(14)
KN
KÝS
= ρ( PM ,K S , RS ) ≡ gS (KS ,K N , RS )KS
We can confirm the characters of these functions as follows:∂ gN∂ρ ∂PM=⋅<0(15)∂K S ∂PM ∂KS∂gS∂ρ ∂PM=⋅<0∂K N ∂PM ∂KN∂ gN∂ρ ∂PM∂ρ=⋅+∂K N ∂PM ∂KN ∂K N
∂ gS
∂ρ ∂PM ∂ρ=⋅+∂K S ∂PM ∂KS ∂K S
∂gN
∂ρ ∂CRN=⋅>0∂RN ∂CRN ∂RN∂gS ∂ρ ∂CRS=⋅
>0
∂RS ∂CRS ∂RS
The effect of an increase in the other region’s capital stock must be to turnthe terms of trade against manufactures and thus reduce profits; so g 2 <0. The

effect of an increase in the amount of renewable resource must also increases
profits of own region through a decrease of the gathering cost; so g 3 >0. The
effect of an increase in the domestic capital stock is, however, ambiguous, since
there are two effects: a worsening of the terms of trade and a reduction in unit
input requirements. If the scale effects which is positive to profits stronger thanthe price effects which is negative to profit, the equilibrium in this dynamic
growth model is consistently unstable in local (A formal proof is given in the
appendix). Therefore, in the case that the price elasticity of manufactured good
is low, we have a conclusion that uneven development is a necessary outcome inthis model. So we proceed with the analysis by taking assumption that external
economies are relatively weak: g 1 <0.
∂ gN∂ρ ∂PM
∂ρ
=
⋅
+
<0
(16)
∂K N ∂PM ∂KN ∂K N
∂ gS
∂ρ ∂PM ∂ρ
=⋅+<0∂K S ∂PM ∂KS ∂K S
We have now set out a complete dynamic model in which the evolution of
the two regions’ industrial sectors can be followed from any initial position. The
next step is to trace out and interpret the path of the world economy over time.
Ⅲ . THE DYNAMICS
The equilibrium of this model is given by the point that g N =0, g S =0. Thelines g N =0, g S =0 indicate combinations of K N and K S for which profits in Northand South respectively are zero. Given the assumptions in section 2, each line
has a downward sloping.
dKN∂g /∂KS=− N<0(on the line of g N =0)
(17)
dK S
∂gN /∂K N
dKN∂g / ∂K S=− S<0(on the line of g S =0)
dK S∂gS / ∂KN

And it is clear that the line g N =0 is steeper than the line g S =0. When the point
which represent the capital of each region is inner from the each line g=0 at theK S K N -plane, then g>0. Reversely, when the point is outer from the each line g=0,
then g<0.
The line g=0 is affected by the amount of renewable resource. We knowthat the effect of an increase in the amount of renewable resource must also
increases profits of own region through a decrease of the gathering cost.
Therefore, the line g=0 will move outside (inside) if the amount of renewable
resources increase (decrease).
Next, we find out the level of capital K, which remains amount of
renewable resources in next team at every given level of renewable resources.
From the definition of the renewable resources, we havecc
K N = a N RN ≡ K˜ N (RN ),K S = a S RS ≡ K˜ S (RS )
(18)
rN
rS

We call this line “h=0”. Now, we can check changes of the amount of renewable
resources on the K S K N -plane. It is clear that the amount of the renewable
resources will increase (decrease) when the point, which represents the capital ofeach region, is inner (outer) from the line h=0 at the K S K N -plane.
We consider the relative locations of the two kind of lines g=0, h=0. It isnatural to think that the amount of renewable resources is affected bymanufactured production. Actually, we are having experience of depletion allover the world. Therefore, we assume that line h=0 is located inner from the line
g=0.
Now, we have a figure (FIGRE 1) that illustrates the essential point of thisdynamic growth model, and that shows us the level of capital accumulation andamount of renewable resources in each region.
KN
g N =0K˜ N

h S =0h N =0g S =0KS
K˜ S

FIGRE 1The line of profit zero (g=0) and the line of resource change zero (h=0)
Note that the lines in this figure continuously be moved. For example, the lines
are not moved during the capital in north region K N is less than K˜ N . For theamount of renewable resources is on an upper limit at the initial point and it
stays unchanged. But once K N exceed K˜ N , which means inputs for manufactured
goods exceed the regeneration of renewable resources, the lines g N =0, h N =0 will
move inside for the decrease of renewable resources. This movement represents
that the progress of capital accumulation causes increase of renewable resources
input for manufactured goods, and decrease of renewable resources will happen
after K N = K˜ N . After that the profit rate from manufactures good and theregeneration ability of renewable resources will decrease resulting from a rise ofgathering cost of renewable resources.
The point of figure 1 is that the direction of capital accumulation is
directly controlled by the relative location of capital stocks with the line ofprofit zero (g=0). Furthermore, the line of resource change zero (h=0) affects
indirectly the direction of capital accumulation through the movement of the lineof profit zero (g=0).
Level ofKN
K N > K˜ N
K N < K˜ N

Movement ofline h N =0
downward
up wardMovement ofline g N =0
down ward
up wardChange ofresources
decreaseincrease

Level ofKS
K S > K˜ S
K S < K˜ S

Movement ofline h S =0
left ward
right ward

Movement ofline g S =0
left ward
right ward

Change ofresources
decreaseincrease

TABLE 1
Movement of the figure 1 relative with the location of capital stocks
Ⅳ . Dynamics of sustainable development
In this chapter, we analyze the dynamic process of capital accumulation in
two regions since the beginning of a situation, which has abundant renewable
resources and slight capital accumulation in both regions.
First, we consider a case of no renewable resource trade. There are two
representative dynamic processes, which depend on manufactured good priceelasticity of renewable resources. If the elasticity is sufficiently high, the changeof profit rate is larger than the change of renewable resource. That is the line
g=0 moves faster than the line h=0. Conversely, if the manufactured good priceis inelastic, the line h=0 moves faster than the line g=o. Figure 2 and figure 3
show the dynamic process of these two cases.
In the case of elastic (FIGURE 2), we began to analyze from the point 0,
at that point the North has a small head start to the South. The profit rate is
positive in each region. So, both regions progress the capital accumulation. Notethat the small head start cumulates for the present for scale economy. After that,
the North arrives at the line of resource change zero (h N =0). And the renewable
resources in the North began to decrease (the line h N =0 be moved to downward).
It causes that decrease of the profit rate in the North (the line gN =0 be moved to
downward) and the capital accumulation in the North gradually gets slowdown.
Finally, the point of capital accumulation in the North arrives at the line of profit
zero (g N =0), and the capital turn to decrease from this point.
The other side, the South comes later to the North in same process. Now,
the point in figure 2, which represents the level of capital accumulation in eachregion, takes a process from northeast direction to southwest direction. It meansthat both regions accumulate their capitals at first, and after that, their capitals

turn to be decreased by declines of profit rate for decrease of renewable
resources in each region.
Next, it happens that a change of direction again. As the decrease ofcapital, the point arrives at the line h=0, and the renewable resources turns toincrease. After that, profit rate is increased and it is started to accumulate
capital.
As mentioned above, the following process is repeated.
1. Decrease of renewable resources by overreach accumulation of capital.
2. Decrease of capital.
3. Increase of renewable resources.
4. Accumulation of capital.
And, the point of capital accumulation spirally convergence to the central
equilibrium point. This equilibrium is stable.
In the case of inelastic (FIGURE 3), the decrease of capital does notenough speedy to recover the renewable resource. After all, Capital in eachregion decreases until it becomes zero.
We consider next a tradable case of renewable resource. We assume trade
cost is zero. In this case, the input costs of renewable resources in both regions
become same. We assume that the input cost of renewable resources depends
on the total amount of renewable resources in each region. And we also assume
the cost increase for scarcity of renewable resources.
CRN = CRS = CR ≡ CR (R )(19)CR′ (R) < 0(Where R ≡ R N +R S )Next, we consider the line h=0. it needs the condition of input equal
reproduction of renewable resources for zero change of the amount of renewable
resources.
K
Kr(K N ) N + r(KS ) S = a(R)(20)
c(K N )c(KS )
From the assumption neutral technical growth(c:v:r=constant), we have a
expression for the line h=0 common to both regions.
c(21)
K N = −KS + a Rr
The dynamic process of this case is presented in FIGURE 4. In this case, thedecrease of renewable resources in one region causes decrease the both regions
profit. The result is an ever increasing divergence between the regions, which
end only when the renewable resource are exhausted.
Ⅴ . Conclusion
The two regions model shows that necessary condition for evendevelopment is high elasticity of manufactured good price. And the case of noresource trade has higher possibility for even development compare with thecase of trading in resource.

The result of the case of even development leads to the conclusion that
infinite accumulation of capital is impossible on condition that finite growth ofresources. Therefore, the sustainable growth is to attain steady state economy inwhich fixed amount of resources is thrown into production of a manufactured
good and fixed amount of manufactured good are produced in every period.

Appendix
Local stability of equilibrium
1. The case of no renewable resources trade
We have the system of dynamic model with in this paper as follows.
KÝNKÝS≡ gN (K N ,K S , RN ) ,≡ gS (K S , KN , RS )
(A1)
KN
KS
RÝN
RÝS
≡ h(K N , RN ),≡ h(KS , RS )RN
RSThe Jacobian matrix of this system is as follows.

(A2)

∂gN
∂K N
∂gS
∂K
J = ∂h N
N
∂K N
0

∂gN
∂KS
∂gS
∂KS
0

∂hS
∂KS

∂gN
∂RN
0

∂hN
∂RN
0

0

∂gS
∂RS
0

∂hS
∂RS

The following formula is realized at the equilibrium by the symmetry of two
regions and neutral technical growth.
∂ gN ∂ gS∂ gN ∂ gS∂ hN ∂ hS∂ gN ∂ gS
=
,
=
,
=
,
=
(A3)
∂K N ∂K S∂K S ∂KN∂K N ∂K S∂RN ∂RSThe following formula is obtained by calculating the Jacobian using (A3).
 ∂g ∂h∂h ∂g J = N ⋅ N − N ⋅ N  > O
 ∂K N ∂RN ∂KN ∂RN 
2(A4)

The divergence of the Jacobian matrix (A2) is as follows.
(A5)

 ∂g

D = 2 N + a ∂K N( ∂ h N / ∂R N =a)

Therefore, Jacobian always takes positive value. Then, the value of thedivergence must be negative for local stable equilibrium. Now we havefollowing condition as stability of equilibrium.

(A6)

∂ gN
< −a
∂K N

There were two effects to profit rate: external effect and price effect. When theprice effect is relatively strong, the following condition is satisfied: ∂ g N / ∂K N <O. Therefore, we can confirm that the more manufactured good priceelasticity of resources is high, the more stability of equilibrium increases.

2. The tradable case of renewable resources with no cost
We have the system of dynamic model with in this paper as follows.
KÝNKÝS≡ gN (K N ,K S , R ) ,≡ gS (K S , KN , R )
(A7)
KN
KS
RÝ
≡ h(K N ,K S , R )( Where R ≡ R N +R S )R
The stability condition are obtained in the same way of case 1 as follows:
(A8)

−2⋅

 ∂g
∂h ∂gN∂g ⋅> a − N − N ∂K N ∂RN
 ∂K S ∂KN 

2⋅

∂ gN
+a<0
∂ KN

(J>0)

(D<0)

Therefore the condition of negative divergence is higher manufactured goodprice elasticity of renewable resources. And there is the condition of positive
Jacobian for stable equilibrium. If J<0 equilibrium is a saddle point.

Reference
Krugman, Paul, “Trade, Accumulation, and Uneven Development” Journal of
Development Economics, vol.8, no.2, Apr. 1981, pp.149-161.