Modern Economy, 2011, 2, 584-588
doi:10.4236/me.2011.24065 Published Online September 2011 (
Copyright © 2011 SciRes. ME
On a Theme by Williamson: A Stochastic Model for the
Evolution of Global Labor Markets since 1830
Fariba Hashemi
Swiss Federal Institute of Technology, Lausanne, Switzerland
Received June 14, 20 11; revised July 22, 2011; accepted August 1, 2011
The cross-sectional distribution of wages has so far been neglected compared to the study of income differ-
ences across countries over time. We propose a stochastic model, building on the theory of diffusion pro-
cesses, in order to describe the evolution of global labor markets since 1830. The model is applied to em-
pirical data collected by Williamson, in order to describe the level and the variation of wages for 15 countries.
The empirical application validates the proposed method.
Keywords: Evolution of Global Labor Markets, Spatial Dynamics, Stochastic Analysis
1. Introduction
The economics literature is rich with studies which ex-
amine wage differentials across countries over time. The
dynamics in the cross-sectional distribution of wages
have been relatively neglected however. The main ob jec-
tive of this paper is to examine the disequilibrium pro-
cess of wage adjustment in face of an exogenous shock.
We hypothesize that the growth distribution of wages
can be generated by a single stochastic process that
builds upon the theory of diffusion. A drift-diffusion
model is proposed which describes the dynamics of
cross-sectional distribution of real wages. The dynamics
of the model are governed by the speed of diffusion of
knowledge, the mobility o f capital and lab or, and decline
in transportation costs which come along with integration
and globalization. An empirical application of the pro-
posed model to the level and variation of real wages
using data collected by Williamson for 1 5 co untries from
1830 - 1988 tests the validity of th e metho d.
2. Theoretical Framework
In a classic study of global labor markets, Jeffrey Willi-
amson [1] explores the debate over the economic con-
vergence of currently industrialized nations and high-
lights a number of shortcomings which include limita-
tions in data availability. Williamson offers a new data
base consisting of purchasing-power-parity-adjusted real
wage rates for unskilled labor. The new data base has
several advantages over the standard GNP estimates, as
factor prices generally, and real wages specifically, are
likely more suitable measures for the analysis of econo-
mic performance and standards of living. This difference
becomes significant und er condition s of incomplete co m-
modity price equalization. Moreover, labor participation
rates differ across space and over time in an environment
of migration and differential rates of population growth.
Both these considerations hold true with greater strength
the farther back in history we look.
This paper explores the dynamics in the evolving dis-
tribution of real wages across Williamson’s data. Willi-
amson describes his data via four ‘regimes’. We explore
whether there is an equilibrium distribution of wages in
general and to what extent it is ‘regime’ dependent. Our
picture of world development is one where some econo-
mic forces push in the direction of convergence whilst
other forces are divergent. Globalization typically rein-
forces the convergent trend through the flow of capital
towards capital poor economies and through trade-
induced factor price equalization. At the same time, a
decline in transportation costs, and faster population
growth lead to divergence [2-5]. Con sistent with this ob-
servation, a model is proposed to describe the fluctu-
ations over time in the density of cross-sectional distri-
bution of real wages. It is hypothesized that these flows
follow simple stochastic laws that can be described with
five param e t er s 1.
1Our study is in the spirit of probabilistic models which study city and
firm sizes [6-8]. Our suggestion is that these models could provide
interesting insights, if applied to the spatial dynamics of real wages as
Consider a region consisting of a constant number of
countries with different levels of wages. The set of
wages forms a distribution which evolves over time.
Fierce competition in labor markets generates some sta-
tionary equilibrium distribution of real wages with a cer-
tain mean and variance, towards which the ensemble of
countries considered tend. The equilibrium is a result of
tension between counteracting forces of convergence and
divergence. Convergence is a result of adjustment of
capital-labor ratios to common steady-state levels,
starting from different initial values [9,10]. Call this the
drift spread, driven by diminishing returns to capital. A
counteracting diffusion spread is at work, driven by
bottlenecks in the flow of labor and capital and by
random effects, which cause a spread of wages from high
density towards lower density. Diffusion of knowledge
and learning [11-16] is limited by the presence of
obstacles in the form of trade barriers and the like.
Consistent with the above, the following drift-diffu-
sion model is proposed to express the wage adjustment
process with noise, describing diffusion of shocks across
 
denotes probability density, denotes the
mean of the stationary equilibrium distribution,
notes wages,
the wage adjustment rate, and
a di-
ffusion parameter2.
3. Empirical Analysis
The empirical analysis uses Williamson’s data [1] which
consists of purchasing power parity adjusted real wage
rates for unskilled labo r recorded from 1830 - 1988. The
data is for the following 15 countries3: Argentina, Aus-
tralia, Belgium, Canada, Denmark, France, Great Britain,
Germany, Ireland, Italy, Netherlands, Norway, Spain,
Sweden and USA.
The evidence presented by Williamson suggests that
there have been four distinct global labor market ‘regi-
mes’ since 1830. In this paper, we adopt Williamson’s
four regimes: (1) 1830-1869, (2) 1870-1913, (3) 1914-
1945, and (4) 1946-1988. The first is associated with
early industrialization in Belgium, Denmark, France,
Great Britain, Germany, Ireland, Italy, Netherlands, No r-
way, Spain and Sweden, settlement in Australia, Argen-
tina, Canada and the United States, international migra-
tions, high transport costs on commodity trade, and ba-
rriers to trade. The second covers the age of industria-
lization and free international migration, the Victorian
boom amidst an age of imperialism, and a general world
boom under free trade and the gold standard. The third
covers the two World Wars and the interwar period when
world commodity and factor markets break down. The
fourth is the po s t W orl d Wa r II period.
The evolution of the distribution of real wages for the
15 countries across the four time periods has been
investigated. Table 1 reports the descriptive of the four
phases (regi mes).
It can be observed that in general, real wages rose
sharply, especially in regime 4. At the same time, the
distribution of real wages became more heterogeneous.
Table 1 reports a lower mean wage (M = 58.28) as com-
pared with the reported mean wage of Phase 2 (M =
90.95), Phase 3 (M = 124.38), and Phase 4 (M = 280 .49) .
However, although Phase 1 displays a lower mean, it dis-
plays less fluctuations, suggesting a relatively stable
labor market. In what follows, we test the reliance of
these developments on five parameters: the initial mean
wage 0, the initial standard deviation 0
, the mean
wage at stationary equilibrium , the velocity of con-
vergence to stationary equilibrium
, and the diffusion
3.1. Estimation
The model has been fitted to the log real wage distri-
bution of the four populations as a function of time,
using the non-linear least-squares estimation using a
two-step procedure. First, the values for 0, and uu
were estimated using the first moment of the distribution.
In the second step, the values for
and 0
computed using the second moment. Tables 2-5 report
estimates for the five model parameters, along with the
standard errors and t-values for the four phases respec-
Table 1. Descriptive of the four Phases.
N MinimumMaximum Mean Std. Deviation
Phase 14048.38 67.58 58.28 4.02
Phase 24464.93 126.67 90.95 15.96
Phase 32591.93 141.47 124.38 15.99
Phase 443144.23 417.53 280.49 97.60
Table 2. Phase 1 Parameter estimates.
Parameter Value Std Error t-value
λ 1.24 1.02 0.02
u 1.27 0.03 3.43
u0 1.15 0.03 5.31
σ0 1.07 0.04 2.31
ε 0.57 0.05 4.53
2For an elaboration of this model see [17-21] and the Appendix.
3Williamson [1] reports data for 11 countries. We are grateful to Wil-
liamson for data on the remaining countries, which were collected and
received post-publication.
Copyright © 2011 SciRes. ME
Table 3. Phase 2 Parameter estimates.
Parameter Value Std Error t-value
λ 1.64 1.06 0.59
u 1.56 0.03 0.85
u0 1.30 0.03 15.52
σ0 1.16 0.04 15.12
ε 0.63 0.05 3.12
Table 4. Phase 3 Parameter estimates.
Parameter Value Std Error t-value
λ 1.78 0.76 0.05
u 1.92 0.03 3.34
u0 1.97 0.03 7.93
σ0 1.65 0.03 5.23
ε 1.03 0.04 3.37
Table 5. Phase 4 Parameter estimates.
Parameter Value Std Error t-value
λ 2.09 2.08 0.08
u 2.32 1.04 6.00
u0 2.13 0.04 7.10
σ0 1.95 0.04 4.06
ε 1.25 0.04 5.67
It can be observed that the parameter estimates con-
form to the real data presented in the descriptive analysis.
Phase 1 displays the smallest estimate for wages as
compared to Phase 2, Phase 3, and Phase 4. Moreover,
Phase 1 displays the lowest standard error as compared
to all other Phases, suggesting that the predictive model
developed in this paper, captures most of what is hap-
pening in the real data.
To ascertain this, one may look at the evolution of
actual versus predicted distributions. Figures 1-4 grap-
hically illustrate the evolution of the distribution of real
wages over time, superimposed on histograms which
describe the time evolution of the distribution of wages
in the data across the four Phases. The solid curves in
these figures illustrate the distribution of real wages as
predicted by the model, and the dotted curves illustrate
the distribution of real wages in the data. The vertical
axes in these figures denote frequency, and the horizontal
axes denote real wages in logarithms. These figures illu-
strate that the neat pattern which we see in the fitted log
normals is being pulled out of a set of histograms whose
shape is irregular. It can be inferred from these illu-
strations that the predicted values generated by the model
are reliable in characterizing the actual data.
The following observations can be made concerning
our results:
Figure 1. Phase 1 (selected years 1830 and 1869).
Figure 2. Phase 2 (selected years 1870 and 1913).
1. The mean and variance of all four distributions are
clearly evolving, corresponding to our theoretical pre-
2. The value for the wage adjustment rate
positive for all four sub-periods and varies from sample
to sample as expected.
3. The value for the diffusion parameter
is small
and positive for all four sub-periods, conforming to our
theoretical predictions. The diffusive is: 2=.
lim t
The results predict that if we start with a normal
Copyright © 2011 SciRes. ME
Figure 3. Phase 3 (selected years 1914 and 1945).
Figure 4. Phase 4 (selected years 1946 and 1988).
distribution and let the mode l drive the distribution, the
distribution variance will tend toward a constant
and concentrated around a mean which is largest in
order of importance for Phase 4, Phase 3, Phase 1 and
Phase 2.
4. Conclusions
An empirical application of the proposed model to the
dynamics in four subperiods following the classification
of Williamson [1] illustrates the applicability of our
method. The empirical analysis confirms the results by
Williamson and offers some new insights. Like Willi-
amson, we find a significant variance in the rate of con-
vergence since the mid-19th century, suggesting that the
world economic environment mattered a great deal, and
that the forces driving convergence are likely to have had
very different quantitative significance within different
epochs. Our results are consistent with historical obs-
ervation that dramatic convergence took place from 1870
to 1913 as international trade boomed and capital flows,
as well as international migrations rose. Long-run con-
vergence slowed down and eventually ceased during and
between the World Wars while world commodity trade
and capital markets collapsed and international mig-
rations slowed due to quotas and a Great Depression.
Convergence resumed after World War II while inter-
national trade and flows of capital and labor picked up.
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Appendix 2
The expression representing the time-development of the
distribution is:
==1e e
uEf uu
222 2
=e 1e
where and is the normalization constant.