On a Theme by Williamson: A Stochastic Model for the Evolution of Global Labor Markets since 1830

doi:10.4236/me.2011.24065

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Modern Economy, 2011, 2, 584-588

doi:10.4236/me.2011.24065 Published Online September 2011 (http://www.SciRP.org/journal/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

E-mail: [email protected]

Received June 14, 20 11; revised July 22, 2011; accepted August 1, 2011

Abstract

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

well.

F. HASHEMI585

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

space:



usf



 



 

where

denotes probability density, denotes the

mean of the stationary equilibrium distribution,

de-

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

parameter



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



were

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-

tively.

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.

F. HASHEMI

586

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-

dictions.

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

F. HASHEMI587

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.

5. References

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Appendix 2



The expression representing the time-development of the

distribution is:



=e1











,=ee

fstNa



















==1e e

uEf uu









222 2

=e 1e





 





where and is the normalization constant.