Trade, accumulation and uneven development.
HALVOR MEHLUM
January 2009
Paul Krugman Trade Accumulation and Uneven Development , (First seminar question at the end)
Consider the two regions North and South of equal size in labour. The manufacturing production requires
the input both of labour and capital. The technology is Leontief i.e. fixed coefficients. The capital/output ratio
and the labour/output ratio are characterized by external effects leading to increasing returns to scale. There
is full employment. Profits determine investments (as in a classical Ricardian growth model).
In the model below I follow Krugman, but set L̄ = 1 and specify the c function and the v function. You will
see that as I specify the same functional form for c and v, things become simple, as fractions v (K) /c (K) is
one irrespective of K. In addition I specify the wage level in North and south as wS and wN
LN = LS = 1
(1)
cN = 1 − KN /5,
cS = 1 − KS /5
vN = 1 − KN /5, vS = 1 − KS /5
KS
KN
, MS =
MN =
cN
cS
AN = LN − vN MN , AS = LS − vS Ms
(2)
(3)
(4)
(5)
From the equations above (in particular the simplification v (K) = c (K)) it follows that in manufacturing the
use of capital is equal to the use of labour. As the total labour supply is equal to unity in each region. the
capital accumulation cannot go beyond unity. Hence Kmax = 1.
The demand for manufactures is determined by the wage income (investments are built using agriculture
good). Under the assumption about constant expenditure shares (µ being spent on manufacturing) it follows
that
PM (MN + MS ) = µ (LN wN + LS wS )
(6)
Remember that LN = LS are assumed to be equal to one. Note also that, as one unit of labour produces one
unit of agriculture goods, the wage will be one as long as we are in an equilibrium where there are workers
producing agriculture.
Investments are carried out as long as there is profits. The profit per unit of capital (the rate of return) is
K̇N
PM − v N w N
= ρN =
,
KN
cN
K̇S
PM − vS wS
= ρS =
KS
cS
(7)
If profit rates turn negative, capital will shrink.
Long run equilibria
Krugmans analysis is quite sloppy. He does not recognize the fact that as long as wages departs from unity,
because of short labour supply when K = Kmax , demand for manufactures will go up and with it the price PM .
Hence, as soon as one region bumps into the Kmax constraint and wages are bid up PM will shoot up and the
profit rate in the region, where wages are still equal to unity, will go up. In the lecture I got most of it right
but I was wrong when I claimed that workers in the lagging region are better of in all long run equilibria with
trade than in the corresponding equilibria without trade. It all depends on where the long run equilibrium is
and on the assumption about capital movement.
The interior equilibrium (no trade)
Without trade we get an interior equilibrium where MN = MS , where wN = wS = 1, and where ρS = ρN = 0.
From (6) and the technology (c and v) above it follows that
PM MS = µ ⇐⇒ PM =
µ
µcS
=
MS
KS
(8)
(North is symmetric). From (7) and the technology (c and v) above it follows that
0=
It then follows that
0=
µcS
KS
cS
PM
−1
cS
− 1 ⇐⇒ KS = µ
(9)
(10)
Intuition: When there is no profits, the sales value in both sector corresponds to the wage bill in each sector.
When v = c and w = 1, the wage bill is equal to KS . Since all workers spend µ of their income on food, a
fraction µ of the workers has to be employed producing food.
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Now, what is the price in the interior equilibrium? Using the equations above and the technology specification
for c:
∗
PM
= cS = 1 − µ/5
(11)
µ low and trade
When µ is low, demand for manufacturing is never sufficient for one region to specialize in manufacturing.
Wages will be kept at unity in both regions. Consider the long run equilibrium where north produces all manuf
goods. Then KS = MS = 0, ρN = 0, and ρS ≤ 0 while wN = wS = 1. Doing the same exercise as above we
then get
2µcS
(12)
PM MN = 2µ ⇐⇒ PM =
KS
0=
PM
−1
cN
(13)
∗
KN = 2µ and PM = cS = 1 − 2µ/5 < PM
= 1 − µ/5
(14)
Given our (non substantial) simplification the use of capital in one region is exactly the double of the use in
each region without trade. Due to the efficiency improvement, prices are lower to the benefit of all. World
production of agriculture is the same as in the no trade long run equilibrium.
Krugmans model has as a result that trade is better for all than no trade. Starting out in a long run
equilibrium without trade, shipping all the capital to one region will benefit all.
µ high
Now, if µ is higher we may have a long run equilibrium with north completely specialized in manuf and
south producing some manuf. In such a long run equilibrium wN ≥ wS = 1, KN = 1 and ρN = ρS = 0. (The
high wN pushes ρN all the way down to zero). In this equilibrium
PM (MN + MS ) = µ (wN + 1)
0=
PM − vN wN
,
cN
0=
From the last it follows that
PM = cS and wN =
(15)
PM − v S
cS
(16)
cS
>1
cN
(17)
The price is determined by productivity in south and the wage gap between north and south is determined by
north’s productivity advantage.
Since KN = 1 we know that cN = 4/5 and that MN = 5/4. Inserting in the price equation yields
4/5cS (5/4 + KS /cS ) = µ (4/5 + cS )
(18)
⇓
4/5cS (5/4 + (5 − 5cS ) /cS ) = µ (4/5 + cS )
(19)
⇓
20 − 4µ
cS =
5µ + 15
hence
PM =
and KS =
9µ − 5
µ+3
(20)
20 − 4µ
20 − 4µ
∗
> PM
=
5µ + 15
20µ
(21)
5−µ
3+µ
(22)
wN =
higher with trade than without. For the workers in the lagging region the long run equilibrium without trade
is the best of the two. This is counter to what I said at the lecture. (For other formulations of v and c I still
suspect that trade can be better than no trade for all and I’m working on an example) . The workers in north
earns
cS
wN =
>1
(23)
cN
therefore they can buy more food than in the interior equilibrium. Their purchasing power in terms of manufacturing goods is
wN
1
5
1
µ
5
=
= > ∗ =
=
(24)
PM
cN
4
PM
1 − µ/5
5/µ − 1
which means that the workers in north also can by more manufacturing goods than in the interior. This is more
than sufficient in order to conclude that the workers are unambiguously better of than in the interior.
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Transition
There are numerous ways transition may happen. They all have their specific features. I will here only
consider transition from the point where KN = 1 and KS = 0 and towards a long run equilibrium as the one
above where KN = 1 and KS = 9µ−5
µ+3 > 0.
In the lecture we considered a case where we came to the point KN = 1 and KS = 0 as a result of
international movement of capital from the least profitable to the most profitable region. We then considered
wage dynamics given that capital was not free to move the other way (from north to south). When KN = 1
(cN = 4/5 and that MN = 5/4) and ρN = 0 and KS = 0 we get that
PM 5/4 = µ (wN + 1)
0=
PM − vN wN
⇒ wN = 5/4PM
cN
(25)
(26)
We see immediately that the purchasing power in terms of manuf goods is equal to the one that prevail in long
run equilibrium. Moreover we get
µ
4µ
PM =
and wN =
(27)
5 (1 − µ)
1−µ
We see that when µ is close to 1 wN is extremely high. The reason is that manuf is in extremely high demand
while only 1/2 of the worlds population is producing it. Therefore, if µ is high, an increase in KS will lower
wN until the new equilibrium is reached. Thus workers in north will prefer that the transition stops right here:
With KS = 0 and KN = 1. They will prevent the profit margin, that accrues to the workers in north themselves,
to be undermined by higher supply of manufactures and lower supply of food from the south.
Questions:
1. Check whether all of the above is correct.
2. Work out the consequences for prices and wages as KS increase along the KN = Kmax = 1 boundary.
(work out the algebra and draw a figure.)
3. Work out the consequences for prices and wages if KN decreases marginally from KN = 1 (while KS = 0).
4. Work out the consequences for prices and wages as KS increase from zero along the KN = Kmax = 1
boundary in the case where capital is free to move from north to south and from south to north (i.e.
ρS = ρN ≥ 0).
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