In this Section I examine three natural experiments that allow inferring the impact of an increase in the average of some characteristic that occurs without any change in its variance. I use these to examine whether and how the agents’ treatment changes in order to estimate the cross-partial derivative in (3). The first experiment—a change in height in a population—occurred naturally over a period of several decades. The other two—differences in average beauty across small groups of people—were generated by apparently random mixing of the individuals to form those groups. Unlike the previous Section, where the examples gave consistent answers to the question, here the results are more mixed.

4.1 The increasing height of Dutch men, 1981–2010

Two striking facts stand out about international differences in the distributions of human heights over time: 1) By the middle of the 20th century American males were the tallest in the world; 2) In the early 21st century Dutch males are the world’s tallest; and American men have fallen (actually, gained only very slightly in height) far behind their counterparts in most northern European countries4. To examine how this striking change altered the relation between earnings and height in the labor market we use two Dutch data sets that allow us to study this question. The focus on the Netherlands is due to the availability of data and to the unusually large and rapid shift in the distribution of heights that occurred there5.

The Dutch data are from 1981–82, 1995–96, and 2006–10 from two sources: 1) The POLS (Permanent Onderzoek Leefsituatie), a household survey from which data are available beginning in 1981, which continued to collect earnings data through the mid-1990s and which also included anthropometric data. Here I use the first two years of the POLS, 1981–82, which like later years only presented data on height in five-centimeter categories. I also merge the 1995 and 1996 POLS data to enlarge the sample for the middle of this nearly thirty-year time period; and 2) The DNB Household Survey, a panel study begun in 1993 and continuing through today. I use data from the 1995 wave, to match the POLS data for that period, and independent observations from the most recent waves of the DNB panel, 2006–10. The samples are restricted to men who are at least 150 cm tall and who are between the ages of 25 and 59 inclusive6.

Some evidence on the startling change in Dutch men’s heights is provided in Table 4, which presents summary statistics for men 25–59 from the POLS and the DNB for each of the three periods, 1981–82, 1995–96 and 2006–10. Over this period the average heights of men in this age group increased by a highly statistically significant 4.3 cm (1.7 inches), an increase in the median height to what would have been about the 75th percentile of Dutch men’s heights in 1981–82. It is also worth noting that the means for the two different samples in the mid-1990s are very similar (although the DNB estimate is statistically significantly above the POLS estimate).

Table 4 Descriptive statistics, Dutch household data, men's height (in centimeters), 1981-82 -- 2006-2010 a Full size table

The variances are nearly the same in both POLS samples; they are also identical, but lower, in both DNB samples. It is thus fairly likely that that the variance in men’s heights did not increase over this period. Assuming that all the information is correct (and the fact that average heights in both 1990s samples are quite similar is encouraging), comparing the 1981–82 POLS to the 2006–10 DNB we can conclude that the assumption of a constant variance is reasonable7.

To examine the earnings-height relationship I estimate log-(annual) earnings regressions, controlling for vectors of indicators of educational attainment, a quadratic in age, and marital status, with the results presented in Table 5. The estimates for both samples from the mid-1990s are nearly identical. Also, they are remarkably similar to those for 1981-82—there was very little change in the earnings-height relationship in the POLS data between the early 1980s and the mid-1990s. As a comparison of the two set of results using the DNB data shows, however, the strong relationship observed in the mid-1990s had become smaller by 2010 (although not significantly so due to the larger standard error on the estimate for 2006–10). The earnings-height elasticity dropped from 0.85 to 0.30 in these data over this period. With a linear specification the evidence suggests that the variance-preserving increase in men’s heights in the Netherlands sharply reduced the impact of height on earnings.

Table 5 Regression estimates, Dutch household data, men’s height, 1981–82 – 2006–10, (dependent variable is ln(annual earnings)) a Full size table

Is the linear specification correct, however? Because information on height was provided in five-centimeter ranges in the POLS data, there are insufficient support points in those data to present meaningful kernel densities and estimates. There are sufficient support points in the DNB data, and Figure 5 shows the kernel density of the height measure in the 1995 DNB sample. Comparing it to Figure 6, which presents the kernel density for these data for 2006–10, the densities are shaped fairly similarly, with the 2006–10 density shifted rightward, as is also suggested by a comparison of the means and standard deviations in Table 4. The kernel estimates of the log-earnings-height relationship for the mid-1990s, shown in Figure 7, do not look at all like those shown in Figure 8 for 2006–10. In the recent period the relationship rises up to just short of 180 cm—below the mean height, falling thereafter over most of the density in the rest of the range. In the mid-1990s the relationship was increasing much further up the distribution of height.

Figure 5 Kernel of height density, DNB 1995. Full size image

Figure 6 Kernel of height density, DNB 2006–2010. Full size image

Figure 7 Kernel estimation of the effect of height on ln(earnings), DNB 1995. Full size image

Figure 8 Kernel estimation of the effect of height on ln(earnings), DNB 2006–10. Full size image

Without any further considerations the results suggest that an increase in the mean of the distribution of this characteristic reduced the cross-partial derivative in (3). It is, however, worth noting that the earnings-height relationship weakened in the most recent sample because it turned flat or even slightly negative at a height slightly above a level that in the early 1980s would have been just above the mean. One explanation for the change might be that in 2006–10 we are observing a market that had not yet adjusted to the long-run equilibrium. It seems likely that employers, presumably those making decisions that led to an earnings-height relationship in 2006–10, were older (and thus shorter) than the average Dutch male ages 25–59. If they did not perceive, or at least did not react to differences in height among potential workers who are substantially taller than they, we would observe an earnings-height relationship shaped exactly like that shown in the kernel estimation shown in Figure 8.

To examine this possibility I separated the 2006–10 DNB sample into half sub-samples of workers age 25–42 and ages 43–59 and re-estimated the equation for which results were shown in the last column of Table 5. Of course, the men in the sub-sample of older workers are shorter than those in the younger sub-sample (181.05 cm vs. 184.28 cm), and, indeed, the variance is higher in the younger sub-sample. The crucial thing to note is that the parameter estimate of the impact of height on log-earnings in the older sub-sample is 0.0049, nearly identical to the estimates shown in Table 5 for the mid-1990s for all men ages 25–598. Among workers in the younger sub-sample the estimated impact of height on log-earnings is −0.0039 in 2006–10. Apparently the market only rewarded the height of older workers, those whose distribution of heights matched more closely that of the people who were likely to have been their employers than did the right-shifted distribution of the heights of younger workers.

4.2 Varying average beauty in the Dutch game show and the AEA elections

The winner’s decision about whom to “send away” in the Dutch game show and the results of the four-person elections for office in the American Economic Association analyzed in Sections 3.2 and 3.3 provide natural experiments for inferring how a one-unit increase in a characteristic, in this case beauty, alters an outcome when the characteristic’s mean changes. Columns (1) and (3) of Table 6 indicate that there is substantial variation in the average looks of “losing” contestants across games and of candidates across elections. In the game show the range of average looks is nearly five times the standard deviation of the averages, and the range of average looks in the AEA elections is also roughly that relatively large.

Table 6 Descriptive statistics and conditional logit estimates, Dutch game show and AEA elections (dependent variable is “sent away” or “elected”) * Full size table

I assume that the average looks of participants in an individual game are independent of the outcomes. Indeed, one might imagine that, if anything, the television producers who considered the issue would attempt to keep the averages as close as possible in order to maintain audience interest in the game. That consideration suggests that any inter-game differences in the mean of looks are unplanned. The average looks of candidates in a particular AEA election are almost certainly exogenous: It is difficult to believe that the AEA’s Nominating Committee, which selects the candidates, chooses sets of especially good- or bad-looking candidates to match up in a particular election in order to affect the outcome or generate additional voter interest in the process.

I expand the equations whose estimation underlay the results shown in Tables 2 and 3 to include interactions of the average looks of the candidates in the game or election with the individual participant’s or candidate’s looks. The focus is on the interaction term—does the impact of a given difference in the looks of candidates change when the average looks in the game or election change?9 The same controls as before are included, so that only the addition of the interaction terms distinguishes the results of estimating these equations from the results reported in the second columns of Tables 2 and 3.

As the estimates in Columns (2) and (4) of Table 6 show, the impacts of differences in beauty on the probability of being “sent away” are somewhat smaller when the average beauty in a game increases. The interaction term is, however, statistically insignificant (t=0.92). The impact of individual beauty on the probability of election in the AEA is slightly larger when the average candidate is better looking. The change is, however, also quite insignificant statistically (t=0.51) and substantively very minor. The appropriate inference from these results is that, if all the participants’ or candidates’ beauty increases by the same amount, the impacts of differences in their looks remain unaltered. A change in the mean beauty among the choices facing agents does not change the impact of differences in beauty among those choices.