Capital-Intensity Hypothesis and Factor Price Equalization Theorem: Intriguing Relationship?

By Hiroshi Ohta, SIPEC, Aoyama Gakuin University
Prepared for RSIJ Annual Conference at Tokushima University, October 14, 2013

ABSTRACT: Predicated upon a 2x2x2 model of general equilibrium, this paper shows conditions under which theso-called factor price equalization theorem may not hold. Consider a developing nation with a strong propensity tosave and accumulate capital on one hand and a developed country that can afford specializing in consumption on theother hand. If the consumption good sector is more capital-intensive than the investment sector, then free trade canlower the developing nation’s wage rate while increasing the developed nation’s wage rate. The assumed conditions
are all empirically plausible, and the outcome sounds surprising, and deserves theoretical scrutiny.
Key Words:
2x2x2 general equilibrium, capital intensity, factor price equalization, trade, capital-rich versus -poor countries

Introduction
According to the factor price equalization theorem real factor prices such as relative wage rates tend to equalize
among nations after trade. But this well-known theorem may not hold, if the less well-known hypothesis on certain
particular technological conditions holds true. The latter, called the capital-intensity hypothesis (R.M. Solow, 1956;
Y. Shinkai, 1960), assumes two sectors of production, and states that stable economic growth requires, though notnecessarily, the consumption good sector to be more capital intensive than the investment good sector. Thiscondition has recently been confirmed empirically by H. Takahashi, et. al. (2012) utilizing data from the IO tables of
the OECD countries, and H. Kawano, et. al. (2009, 2012) using a simple applied general equilibrium model.
The present paper challenges the factor price equalization theorem on the basis of an empiricallyplausible condition, albeit less well known, on capital intensities in consumption and investment sectors. We showthat if a rich developed nation endowed with large capital and a poor developing country with small capital trade
their products, then their real factor prices may tend to diverge rather than converge.
The crucial conditions required for such a startling outcome other than the capital-intensity hypothesis
aforementioned are that the developing nation being capital-poor has a strong taste for investment goods, and thedeveloped capital-rich nation specializes in consumption. The poor stay hungry, and the rich remain foolish.
BackgroundA paper by H. Ohta (2004) has revealed certain moral scientific nature of the well-known Stolper-Samuelson (SS)theorem and the related less well-known proposal to ‘bribe’ for free trade. Ohta assumed two identical countries,
identically endowed with homogeneous labor and capital (two factors) but with internationally different tastes.
Pursuant to these basic assumptions he presented a simple account of the SS theory in a simple formal 2x2x20general equilibrium framework. The H.O. (2005) presented his heretical ideas on different tastes in addition to theorthodox HO ideas on different endowments. It in turn inspired H.O. to ponder on R. Jones and R. Ruffin (2005,2007) discussing their “technology transfer paradox” along with some other seeming paradoxes in trade theory in
H.O. (2006), elaborated further with Hiro Nakagawa (APJAE, 2008). The more recent H. O. (2009, 11, 12) are
getting more directly focused on a bitter maxim of free trade that the Stolper-Samuelson theorem revealed.
With these prior inquiries in the background the present paper goes beyond the static model of generalequilibrium of production and exchange by incorporating in it capital formation and economic growth. It shows
conditions under which how factor prices, product prices, and sectoral factor intensities may be related to the stagesof economic growth. If the bitter maxim of free trade is an inconvenient truth despite the factor price equalization
theorem, the bitterness may intensify all the more if and when the rosy theorem is reversed.
In what follows Section 1 reviews the basic 2x2 general equilibrium model of two clone nations with nogrowth, nor trade. The model for a representative clone country is presented using just one CD parameter torepresent both technology and taste. Section 2 in turn presents a 2x2x2 by 1 CD parameter model of two contrasting
nations with asymmetric national tastes before and after trade to examine what impact free trade may have on factor
prices and commodity prices. We confirm the orthodox outcomes of free trade and the related factor-priceequalization theorem. Section 3 then introduces growth, asymmetric growth in particular, to see what happens tofactor prices between the trading nations. The question is if factor price equalization will be achieved. The answer is,
no. Moreover, factor prices that may be equal before trade will necessarily diverge after trade. Section 4 concludes.
Section 1. The 2x2 GE model of clone economies before asymmetric growth and trade
Consider an autarky economy endowed with fixed amounts of two factors: labor and capital. These factors are used
to produce consumption goods and investment goods under constant returns to scale, of the Cobb-Douglas type:
The Consumption Good Sector: C = KCα LC(1−α)
The Investment Good Sector: I = KI(1−α)LIα
where C stands for consumption, I for investment, α ‘output elasticity’ of capital in the C sector and also ‘output
elasticity’ of labor in the sector I. This is a deliberate contrivance to differentiate the two sectors’ methods ofproduction by a single parameter α.
Now in order are the following optimization problems and equilibrium conditions.
Consumptive Optimum:
U1 (I, C)/p = U2(I, C)(1)

where U1 is MU of the first good, I, U2 MU of the second good C, and p the relative price of investment goods I in
terms of consumption goods C assumed as the numeraire.
Individual Budget Constraints:
w = pIL + CL; r = pIK + CK,(2), (3)Productive Optimum:1

w/r = f2/f1= g2/g1,(4), (5)where w is wage rate, r rent on capital, f1(or g1) is the C (or I) sector’s MP of capital, f2 (=f(kI,1)−kIf1(kI,1))=φ(kI) is
the I sector’s MP of labor, which is a function of capital/labor ratio or capital intensity in the I sector, and g2(=g(kC,1)–kCg1(kC, 1))=γ(kC) is MP of labor in the C sector, as a function of C sector’s capital/labor ratio.
Factor Market Equilibrium:
KI + KC = 1; LI + LC = 1(6), (7)

Product Market Equilibrium:
I = f(KI, LI); C = g(KC, LC)(8), (9)Walras’ Law:
pI + C = w + r(10)

Equi-MRTS (of L for K) for Optimal Resource Allocation
The following optimization conditions are to be observed along a contract curve, which is concave when α > 1/2.
(−dKI/dLI) = ((1−α)/α)(KI/LI) = (−dKC/dLC) = ( α/(1−α))(KC/LC)
Combined with the conditions of a given endowment of L and K, this yields.

KC =

1
1
1-" 2
( -1)() +1
LC"(K = L = 1)

ECA

0I

!

C0

KC0

C0

CC

PPF

I0

K

0C

LC0

EIA

L

I0

Figure 1. Given α, LI determines KI, hence (LI, KI) as a one-to-one correspondence with (C, I), and w/r(=MRTS)
Figure 1 illustrates an autarkic equilibrium, labeled ECCA, for country C, and ECIA for Country I on the conflictcurve CC on the left and the production possibility frontier on the right. Here Country C produce and consume moreconsumption goods than the I do. The C workers, before trade, are poorer than the capitalists. Not only are the C
workers poorer than the C capitalists, but also they are poorer than the I workers. The I workers, by comparison, arericher than not only their domestic capitalists, but also the C workers abroad.
These observations are to be made on Figure 2 below by comparing a square point with a triangularpoint along the conflict curve on the left. Wage rate given by factor allocation identified at the triangular point must
be strictly greater than unity and hence greater than at the square point, which is less than unity. Related to this2

particular wage rate of unity is a circular point at which MRTS is required to be unity. Not only wage rate is unity at
this point, but also so is product price identified at a circular point on the production possibility frontier on the right.

0I

KIA

CCA

ECA

EIA

IA

C

IIA

!

0C LIAIIA

Figure 2. Alternative resource allocations and related autarky equilibria

Underlying the diagram above is a formal 2x2x2 general equilibrium model of production, trade, and consumption.
Section 2. 2x2x2! GE of clone economies with hetero-national tastes before/after trade(Omitted)
Section 3. Impact of trade under asymmetric growth and tastes upon factor prices
Despite the bitter outcomes of trade both nationals C and I each as a nation are richer after trade, and the C canconsume more consumption goods while the I can invest more than they could before trade. This implies that while
the factor endowment in C remains the same regardless of trade the capital endowment in I tends to increase overtime, all the more after trade. As the I grows to be a developed economy, it can afford turning itself into a country
specializing in consumption. The ex-I is now a developed country DC. The ex-C realizing no growth, bycomparison, may learn from the ex-I to turn itself to a new developing country DI.
What then would happen to factor prices in these countries after trade? Would they equalize between
DC and DI? To answer this question we restate formally the following basic assumptions/conditions of our analysis.
A1) Two clone countries exist with divergent national tastes: the one addicted to consumption, called
DC, and the other to saving and investment, called DI.
A2) Each country is endowed with two factors of production, capital K and labor L, to produce twokinds of goods, one for consumption and the other for investment: C and I.
A3) In both countries labor is owned by a given number of workers L, and capital by a given number ofcapitalists K, in the same number, say, L=K=1. However, as ex-I grows capital accumulated K* becomes
distinctively higher than in the ex-C so that K* > K=1 in the long run or asymmetric growth, say, K*x-CI = 4 (forcontrasting purposes only)
A4) Each country has two sectors of production: one for consumption. another one investment goods.
A5) The C sector is more capital-intensive than the I sector. (This requires α > 1/2.)
A6) The developing country is investment-oriented: producing investment goods extensively with little

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consumption goods, therefore living on thin air. The poor stay hungry.
A7) The developed country is more consumption- than investment-oriented. They specialize inconsumption taking advantage of rich capital while producing little investment goods.
These specifications of the Assumptions above are needed only for the sake of yielding the initialconditions for autarky equilibrium for the two nations’ CPs and FPs before trade as illustrated by Figure 5 below.

FP (CP, k4)

MRTS (kI)ω = w/p

MRTS (kC)

FP (CP, k1)

ω =1

p=pI/pC
pC >1 pI < 1

kI (= k=1)

kC (= k=4)

Figure 5. Sectoral ki’s, and aggregate ks that yield equal FPs (=1), yet distinct CPs
The differences in CPs in two countries provide apparent incentives for them to trade. The question thenis what happens to prices CPs and FPs. Remember that given k = 1 for two clone countries trade tends to equalize
them all, as illustrated by Figure 4 supra, along which the high prices must fall and the low prices rise after trade.
But in the long run FPs could diverge insofar as capital accumulation proceeds in Country I where theworkers may tend to resist opening their market in fear of their real wage declines at least in the short run when k is
fixed. But over time while the stagnant C’s factor endowment remains unchanged the I’s capital must increase.
Moreover, free trade will accelerate their capital accumulation, thereby increasing their wages over time to such an
extent that their real wage will exceed the stagnant C’s when capital accumulation proceeds beyond k = 4. The
growing I’s real wages will also increase with increasing output of I to be exported by the stagnant C to the growing
I, but will peak out when it reaches ω (=w/r) 2 in the figure. Thus, real wages of a growing economy can surpass
those in the stagnant economy, all the more with trade, accelerating capital accumulation.
Further accumulation beyond k = 4 keeps price of investment goods increasing as the ex-I can now
specialize in consumption on their production possibilities frontier that has kept shifting outward with increasing
slopes at the C axis. They are now on PPFDC and capable of offering the entire amount of the C in exchange for the
maximum I the ex-C can produce along PPFDI that intersects with the I axis. The maximum wage ωDImax that the DI
can earn by producing the investment goods at EDIP* is now strictly lower than the minimum ωDCCmin that the DC
earn by specializing in consumption now at the left corner of PPFDC with kDI > 4. See Figure 6.

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C
EDCP*:

ω DCmin
PPFDC: kDC > 4

ω increases

PPFDI: kDI= 1

ω DCmin > ω DI max when kDC large enough.

ω diminishes

Trade can cause wages to diverge.

I
EDIP*: ω DImax
Figure 6-1. Trade can cause factor prices to diverge
FP (CP, k=4)

FP (CP, k=1)

MRTSC (kC)

MRTSI(kI )

ω=2

EDIP*

EDCP*

p = pI/pC
pDI> p*>pDC

kI=1/2

kDI=1

kI >1

kC=2

kDC=4

kC > 4

Figure 6-2. Trade can cause factor price diversions under asymmetric tastes and growth

Underlying Figure 6 are assumptions A1) through A7). Figure 6-1, upper panel, depicts two contrasting production
possibility frontiers PPFDI and PPFDC of the ex-developing (now rich) DC and the ex-consumption-addicted (nowhungry) country DI, respectively. Note initially that asymmetric autarky equilibria are represented by EDIP* andEDCP* and related equilibrium commodity prices of pDC > p DI.
Then the two arrows show how trade may yield resource reallocations and related changes in thecommodity bundles produced and, of particular interest to the present inquiry, the factor prices between the
countries under consideration. The rich DC that now prefers consumption to investment nevertheless had better
produce less consumption goods and more investment goods for export while the poor DI that now prefers
investment to consumption may produce more consumption goods for export. Thus they move along their own
PPCs until when pDC=p DI is established. In this process toward equilibrium what happens to factor prices within the
trading countries is that they must necessarily diverge as indicated by the dotted arrows in opposite directions.

5Moreover, trade lowers the poor nation’s real wages and increases the rich nationals’ wage income! An inconvenient
truth would it be? Would it then call for the SS proposal to ‘bribe’ for free trade with no apology? H.O. disagrees.
Section 4. Conclusions
A simple model of general equilibrium with trade by H. Ohta (2004, 5, 6, 11, 12) has been revisited to probe theStolper-Samuelson theorem and the related orthodoxy of the HO factor price equalization theorem. We find that free
trade may dis-equilibrate factor prices under certain particular empirically plausible conditions of asymmetricinternational tastes among nations at the outset of asymmetric growth, followed by a plausible shift in, or reversal ofnational tastes upon asymmetric growth. Taste reversal is likely to occur insofar as an ex-hungry, now rich nation
can afford specializing in consumption while a stagnant nation with little prior saving realizes that they need to
become hungry to take off for growth/development.
This conclusion, if surprising, can be confirmed by the departure point conditions, that factor prices, say,real wages, happened to be equal in a poor developing country and a rich developed country under autarky,
respectively. Within the confines of a simple 2x2x2 GE model presented in Section 3, related Figures 4 - 6, thisrequires capital stock of the developed country to be four times that of the developing country. The assumed
technological clones are represented by just one CD parameter α = 2/3. This last specification reflects our otherrelated assumption that requires the consumption sector to be more capital intensive than the investment sector.
This required condition is crucial to our present inquiry, crucial for two counts. First, our conclusion above
crucially depends upon this seemingly unrealistic assumption. Second, if this seemingly unrealistic assumption werein fact unrealistic, then our surprising conclusion must collapse. Fortunately, the seemingly unrealistic assumption isempirically supported, and hence crucial and relevant in the second sense. And our conclusion is valid.

SELECTED REFERENCES
Ohta, H. (2004) “The Stolper-Samuelson Theorem: Diagranomics Pedagogica,” presented at the Tokyo
Workshop on Αλπα and Οµεγα Topics in international trade and Finance.
Ohta, H. (2012) “The Moral Scientific Nature of the Stolper-Samuelson Proposal to ‘Bribe’ for Free
Trade,” (eds. M. Kemp, et al.) Positive and Normative Analysis in International Trade in Honour of Hiroshi Ohta,
Chapter 2, Palgrave Macmillan.
Solow, R. M. (1956) “A Contribution to the ‘Theory of Economic Growth’,” QJE, vol. 70(1).
Shinkai, Y. (1960) “On Equilibrium Growth of Capital and Labor,” International Econ Rev., vol. 1(2).
Stolper, Wolfgang and Paul Samuelson (1941) “Protection and Real Wages,” RES, vol. 1.
Takahashi, H., K. Mashiyama, and T. Sakagami (2012) “Does the Capital-intensity Matter?: Evidence
from the Post-war Japanese Economy and Other OECD Countries,” Macroeconomic Dynamics, vol. 12.
Ruffin, R. and R. Jones (2007) “International Technology Transfer: Who gains and Who loses,” RIE, 15.

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