Thursday, July 22, 2010

Does the law of diminishing returns apply to a sense of achievement?

The law of diminishing returns is, of course, the famous economic law that as you add additional units of one factor of production, holding other factors constant, then the additional increments of output produced tend to decline. The original story was about how much additional food could be produced by adding additional units of labour to a constant amount of land, other things being equal. It has been possible, of course, to avoid the Malthusian consequences of diminishing returns by making more use of capital equipment and using more efficient technology etc.

The law of diminishing returns was probably hardwired into my brain when I was studying economics as an undergraduate over 40 years ago. In any case, when I first saw a data set providing ratings of life satisfaction and of seven domains (standard of living, health, achieving, personal relationships, safety, community connectedness and future security) it seemed natural to expect that the law of diminishing returns would apply to each of those domains. So, for example, I expected the additional life satisfaction that accompanied an increase in rating on ‘achieving in life’ from 7 to 8 would be greater than that accompanying an increase in rating from 8 to 9. It turns out, however, that my expectations were wide of the mark - at least for the data set I was using (the Australian Unity quality of life data set, Survey 13, 2005, with useable data for 1,956 respondents). The relationships between the various domains of life satisfaction aren't actually much like the relationships between fertilizer applications and crop yields.

Some time last year I decided to try to use regression to find a simple production function (there I go again) that provided a good explanation of life satisfaction in terms of the seven domains. The estimated coefficients for factors other than ‘achieving in life’ were then used to hold the influence of these factors constant at their average values in order to examine how life satisfaction varies with changes in the ‘achieving in life’ rating.

I thought a Cobb-Douglas production function, which is probably the simplest form of production function incorporating diminishing returns, would probably be appropriate. But a simple linear production function better fitted the data. The functional form that I eventually settled on is a simple linear relationship that is anchored at the top end of the scale, so that if there is a rating of 10 on all 7 domains the predicted rating for life satisfaction must also be 10. The estimated coefficients for this restricted least squares regression were similar to those for ordinary least squares, but the restriction enables better use of available information (Adjusted R squared = 0.81 versus 0.51). The estimated coefficients were as follows (followed by standard errors in brackets):

Standard of Living: 0.309 (0.020)
Health: 0.055 (0.016)
Achieving: 0.272 (0.018)
Relationships: 0.160 (0.014)
Safety: -0.006 (0.018)
Community links: 0.076 (0.016)
Future security: 0.047 (0.018)
Intercept: 10 – 10*(.309+.055+.272+.160-.006+.076+.047) = 0.876 .

Now, we know that a linear production function is inconsistent with the law of diminishing returns. The model predicts, for example, that an increase in achieving rating from 7 to 8, will result in the same increase in life satisfaction rating (+0.272) as for an increase from 8 to 9. But this doesn’t necessarily mean that the estimated model fits the data well over the full range of variation in achieving ratings. One way to test this is to use the estimated coefficients to hold other factors are constant at their average values and to examine how remaining variation in life satisfaction is related to achieving ratings. The results are shown in the chart below.

It is evident from the chart that the linear model prediction of how life satisfaction varies with achieving tracks fairly closely the estimate of life satisfaction with variables other than achieving held constant. In other words we can be fairly confident that diminishing returns does not apply to achieving.

The chart also shows large gaps between the estimates of life satisfaction with variables other than achieving held constant and average life satisfaction levels. This reflects correlation between ratings on achieving and ratings on other variables. This could be because of causal relationships between various domains or because ratings on different domains are influenced by common factors such as personal disposition or temperament.

I’m reluctant to post the results of this little piece of research because I can’t claim any expertise in this area (and my ignorance might be fairly obvious to people who do have such expertise). But the results of this exercise seem to me to have some implications for the question that I raised in my last post about the appropriate balance between different domains such as achieving and relationships. The absence of diminishing returns to achieving does not mean that high achieving by itself is likely to give many people very high life satisfaction. That usually requires high ratings on relationships and on the other domains as well. But we shouldn’t assume that achieving and relationships are completely independent. There is higher positive correlation between relationship ratings and achieving ratings (0.4) than between relationship ratings and the ratings for any of the other domains.
Does this mean that high achievers find it easier to maintain good relationships with others? Or, does it mean that people tend to view maintaining good relations with others as an achievement?

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