Wages and Migration
Randy Cragun
1 Feb, 2014

# Wages and migration

## Randy Cragun

### 28 February, 2014

Note: this is a supplement to the "Directed technical change as a general theory" paper. Background for the wages discussed here is in that report. This work will likely be incorporated into the larger report eventually.

We have two goals for our wage estimates. First we will show what these data imply for historical worker migration and compare these predictions to historical trends in migration. We will find that if our model is a good representation of the environment faced by workers and innovators that there must be large geography-specific costs of migration but that these costs need not necessarily vary much over time. Second we will compare our estimates to data collected by other researchers. We will find that we do not reproduce past estimates of wages in Europe and the European offshoots for the early to mid 1800s but convergence trends in our wage data from the late 1800s through the 20th century do look similar to past work. On the other hand, our work predicts much variation in the destinations for migrants, whereas others have emphasized the dominance of the European offshoots (particularly the United States) over Europe in returns to labor over most of this period.

Suppose we are interested in the implications of our results for human migration. Although Figure 1 shows that the US had the highest lifetime wages in almost every year, the relevant wage for an immigrant is not the wage earned by natives, because his human capital would differ from theirs. Absent any costs of migration, workers would go where their human capital would earn the highest wage. Although we cannot guarantee that human capital from one country would earn the same wage as native human capital after migration (perhaps there is some country-specific skillset), the local wage per unit of human capital is our best approximation for the potential wage earned by migrants' human capital.

Figure 3 gives the expected present discounted value of entire lifetimes of wages per unit of human capital in every country and year with a few countries highlighted. Thus in a given year the height of a curve tells us the value of wages a 15–24-year-old1 would expect a unit of his human capital to earn over his entire lifetime. These data use naive expectations: workers expect that as they age they will earn the current wage for older age groups. This is in line with others' work (Williamson, 1995, and O'Rourke and Williamson, 1997, both ignore expectations of future wage growth). The discount rate used is 7%—the same one used to solve the model. Figure 4 gives the actual realized lifetime wage value. The reality of expectations is likely somewhere between these extremes in most cases.

As a comparison to the present value of the lifetime wage income, we also calculate the average wage a unit of human capital would earn in each country in each year across all age groups. The results are in Figure 8. The patterns are nearly identical to the patterns in the PDV graphs.

Some of the results are as expected. Australia was a particularly desirable destination in the late 1800s. However, other results are in contrast with past work. Possibly because of high human capital in the US, American wages (per unit of human capital) are almost invariably below wages in the UK. Further, British wages pull ahead of American wages between 1830 and 1850 and then American wages catch up during the American Civil War. Contrarily, Williamson emphasized the opposite patterns: American wages grew quickly relative to Europe up to 1850, possibly due to land abundance, and then experienced a relative decline during the American Civil War in the 1860s. I have four possible explanations that could reconcile the differences.

1. We measure physical capital and (probably less accurately) human capital, but in times and places where other inputs were important, the factor wages we estimate will be distorted. For instance, others have argued that the unusual New World wage boom in the mid-1800s was due in large part to land abundance. If some technological change improved the value of land in the early 1800s, this would raise wages in the United States (and Australia, Canada, etc.) relative to Europe. Presumably resulting immigration of low-skilled workers could lower this wage as argued by Williamson (1995) and O'Rourke and Williamson (1997), but such a conclusion would need to be reconciled with observations of strong techincal bias in the directed technical change literature (Acemoglu, 1998, 2002, 2003, 2007). If strong bias always occured in the high-skill/low-skill labor markets, then immigration would cause international wage divergence. We remind the reader that strong bias depends on the elasticity of substitution between labor inputs, which likely depends on the available technology and inputs; thus there is no particular reason for us to think that simply because we observe strong bias in one period of time that we must observe it at another time. We think this hypothesis ought to be analyzed with data on labor divided by skill level rather than by age.
2. Our wages are not for the same factors used by Williamson. He used urban, low-skilled worker wages. It might be reasonable to think of low-skilled workers as people possessing some fixed level of human capital. In that case, the wage per unit of human capital would be a good comparison. However, we have no particular reason to think that our human capital measure will reproduce the actual wage distribution within age groups. It is possible that people with different skills within the same age group earn different wages per unit of our human capital measure. However, admitting this also requires admitting that there is no reason to think that we can use the wage per unit of human capital as the wage that an immigrant would earn.
3. Our measure of the labor force size does not sufficiently account for differences in actual time worked by people in the labor force. For instance, female labor force participation could be unusually low in Jordan. Because we use the average schooling per person to calculate human capital, if women are poorly educated and have low labor force representation, this will underestimate the human capital of the average worker. This would artificially increase the estimated wage per unit of human capital. On the other hand, our assumption that the male labor force is the total labor force would overestimate the size of the labor force in Jordan relative to other countries. This would artificially decrease the estimated wage per unit of human capital in Jordan relative to these other countries. We have no way of determining the sizes of these counteracting effects or even which is bigger.
4. Our model could be a poor representation of the labor market by age.
5. Williamson's data (available here) could be poor quality.

### Comparing our data to Williamson's

Comparing our wage estimates directly to Williamson's is difficult because we do not look at the same workers that Williamson does. His wage data are generally for urban industrial (probably low-skilled) workers, whereas our work did not attempt to look at workers by skill level or the urban/rural divide. Readers interested in the wage estimates produced by a framework similar to ours but applied to skill divisions should see Jerzmanowski and Tamura (2015). However, there may be some similarities between our data and Williamson's. Particularly, if we looked in our data at the average wage per unit of human capital or the wage per unit of human capital for young workers, we might expect to see patterns like in his data.

The primary statistic of interest for Williamson was the international unweighted coefficient of variation of average urban industrial wages. For instance, he collected an average real (purchasing power parity) wage estimate for urban industrial workers in France in 1830 and then did the same for the US, Great Britain, Ireland, the Netherlands, Spain, and Sweden. The coefficient of variation for these seven data points was identified as C(7)2. C(7) includes seven countries and spans 1830–1853. C(11) includes eleven countries and spans 1854–1869. C(15) includes fifteen countries and spans the rest of the period up to 1988, but differences in price level calculations over time and a few cases of missing data mean there are breaks in the early 20th century. In fact, because of the different price level calculations at different times, there are three distinct periods for C(15) that cannot be compared in his data: 1870–1913, 1914–1939, and 1950–1988. We omit his estimates for 1940-1949, because there are missing data in that period (he simply plots a C(13) over that time) and comparisons with our decadal data do not require yearly data.

We replicate his calculations and then attempt to identify similarities or differences in our data. We calculate the same dispersion statistics as Williamson using multiple wage measures:

1. Average wage per unit of human capital
2. Average wage per worker
3. Average wage per unit of human capital for workers aged 15–24
4. Average wage per worker for workers aged 15–24
5. Average wage per unit of human capital for workers aged 25–34
6. Average wage per worker for workers aged 25–34
7. The present discounted value of realized lifetime wage earnings per unit of human capital
8. The present discounted value of expected lifetime wage earnings per unit of human capital for a naive worker who expects the current age-wage relationship to remain unchanged
Because in our model

$$w = \left(1-\alpha\right)\frac{Y}{L}$$

where w is the average wage earned by a worker, we do not include income per worker separately from wage per worker. Williamson and O'Rourke in particular make much of the difference between these two measures of welfare, but our methodology does not allow us to say anything meaningful about the difference.

We expect items 1, 3, and 4 to be the most similar to Williamson's data (available here) because these are our best estimates of the wages of the lowest skilled workers. We should not expect the levels to be at all similar (a low-skilled worker might easily have 3.4 "units" of human capital, for instance).

Figure 9 plots the coefficients of variation for international wage data from Williamson (1995). Unfortunately, the actual values that I calculated for C(7), C(11), and C(15) from Williamson's data differ from those reported in his paper (see figures 1, 3, 5, and 7 in his paper). The difference is just in levels, so it is not a concern for our analysis, but I am waiting for a response from the author. Figure 10 plots the coefficients of variation for average wage per unit of human capital in our data. We place the series on the same graph in Figure 11 for comparison. Besides the long-run trends, there is little that appears muchsimilar between the two panels. Even the timing of the secular decline in international wage dispersion is different. It begins later in our data and does not dissipate in the early twentieth century as in Williamson. The reader should note that the dispersion statistic from Williamson is quite volatile, and we sample only at decadal intervals, so the curves for our data will be much flatter regardless of the fit between the two series.

This is, of course, only one of our wage measures. Figures 12 through 19 plot our coefficients of variation for each of our wage measures against Williamson's. To prodcue these graphs, we actually paired points from each data set that were close together in time even if they were not in the same year. For instance, Williamson's C(11) begins goes from 1854 to 1869, and we graph the 1854, 1860, and 1869 points against our 1850, 1860, and 1870 values, respectively. If we did not do this, C(11) would have only one point on each graph. Oddly, the measure with the tightest correlation with the Williamson statistics is the realized PDV of lifetime wages per unit of human capital (Figure 18) even though this measure has little to do with Williamson's data in theory. Because he is interested in factor price convergence, he does not look at all at workers' expectations of future wages. Rather the correlation is higher in this case because our measure is an average over time, which spreads out short-time-scale variations in favor of the long-term trends, which are shared between our data and Williamson's. Note that the PDV of expected wages for a naive worker is not an average over time, so it will not display this same tendency.

A notable result is that our data are negatively correlated with Williamson's for C11—a result that is fairly robust accross wage specifications—and often for C(7).

### Test for systematic differences in the data

Because Williamson used yearly data and our data were interpolated to get values at ten year intervals, we might simply be sampling unusual years. If the underlying wage data are volatile, interpolation might cause large errors in our data. For instance, it may be that our values for 1850 are a better match for the actual data from 1852 because we got many countries' data from 1852 and 1840 and interpolated to 1850. Although it is still difficult to imagine that our C11 and his C11 have anything in common, we might be able to test the hypothesis that the differences are due to our sampling of years. Suppose Williamson's data (available here) are the actual accurate values and that we sample points from his data and construct dispersion statistics with the samples. If the dispersion values from our data are rarely highly correlated with those from the Williamson samples, then there is likely some systematic difference in the data.

Tables 1 through 6 present correlation coefficients between the coefficients of variation of each of our measures of worker welfare and lagged values of Williamson's coefficients of wage variation. A "lag" of 2 means that we compared our values in 1950, 1960, 1970, etc. to Williamson's values in 1952, 1962, 1972, etc. Correlations of 1 or -1 occur exclusively where there are only two years to compare. Ssome sample/lag combinations for C(11) had only one pair of points to compare, so those frames are empty in Tables 3 and 4. Although we could construct bootstrap estimates of the distributions of these correlation coefficients, it would likely be a wasted exercise, as we can see clearly that the populations from which we would sample correlations for C(7) and C(11) are composed mainly of negative values, whereas the populations from which we would sample the correlations for C(15) are composd mainly of moderately strong positive values. We are unlikely to to get any strong positive correlation between our estimates and Williamson's for the early samples.

C(7) for realized PDV of wage earnings per unit of human capital and most values for C(15) are exceptions to the typical negative correlations. As emphasized before, the apparent similarity for this one value of C(7) is likely a mere accident (at least we believe that this measure of worker wages is one of the least similar to Williamson's). However, we should explore the positive correlations in C(15).

It is possible that our C(15) values are positively correlated with Williamson's only because of centuries-long convergence while we fail to reproduce Williamson's shorter fluctuations. Suppose we look at wage per worker aged 15–24, since this variable has some of the strongest correlations and also represents primarily low-skilled workers. Figure 20 splits up the C(15) series as Williamson does: essentially pre-"Great War", inter-war, and "post-war". This split is due primarily to differences in price level calculations over time. What we see is that our data for these series perform fairly well if the standard is their correspondence to Williamson's. Thus we do observe similar wage trends to those seen by Williamson in his larger sample and later years. The correspondence is weaker in the inter-war period, but that was a perdiod of great volatility, so we again look at alternative samples from Williamson's yearly data.

Tables 7 through 12 break down C(15) into the years separated by Williamson. We see that the inter-war period is, in fact, a time when our data do not match Williamson's. Before the wars, convergence dominates both our series and Williamson's, so the correlations are positive and high. The post-war period is characterized by our per person measures moving together with Williamson's while our per unit of human capital measures move opposite. If we are to accept the validity of both series (his and ours), we would likely conclude that Williamson's urban manufacturing workers were not actually low-skilled throughout that period and that their wages increased (and possibly converged internationally) because of improvements in human capital.

### Series for individual countries

Suppose we look at individual countries. Williamson normalizes the wage in the UK to 100 in 1905 for the 1830–1913 series, in 1927 for the 1914–1945 series, and in 1975 for the 1946–1988 series. For comparison, we normalize UK wages to 100 in 1900 for our 1820–1910 series, in 1930 for our 1920–1940 series, and in 1980 for our 1950–2010 series.

CVwhboth

## Textual notes and references

### Textual notes

1. I also included the expected values of wages for older workers in Figures 5 through 7.
2. When c indexes countries and w is the low-skilled wage, the statistic of interest is $$C\left(7\right) = \frac{\mbox{SD}\left(w\right)}{\mbox{mean}\left(w\right)} = \frac{\left(\frac{1}{n}\sum_c{ \left(w_c-\frac{1}{n}\sum_c{w_c}\right)^2}\right)^\frac{1}{2}}{\frac{1}{n}\sum_c{w_c}}$$

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