That depends on the slope of the line and how far we are above the 0 utility threshold for a life worth living. I found this book with a table (table B21 on p. 264) of estimated historical GDP going back to the year 0, and there have been less than six doublings in that time. Current countries with per capita GDP at the same levels as pre-1900 Western Europe (and even year 0 Western Europe) are included in some of those analyses that found the log-linear fit, and their self-reported well-being is in line with the regression that fits the rest of the world. The log-linear relationship might break down if we go far enough into the past (or future), to places that are poor enough (or rich enough), or to societies that are different enough so that GDP won’t be a good measure of their material quality of life, but the data I’ve seen suggest that the relationship is more robust than I would’ve expected.
The Stevenson & Wolfers paper that I linked uses self-report measures of welfar including happiness, satisfaction with life, and amount of smiling, and finds similar log-linear relationships with income on all of them (though with different slopes and intercepts), which suggests that this relationship will apply to whichever definition of utility we use.
That depends on the slope of the line and how far we are above the 0 utility threshold for a life worth living. I found this book with a table (table B21 on p. 264) of estimated historical GDP going back to the year 0, and there have been less than six doublings in that time. Current countries with per capita GDP at the same levels as pre-1900 Western Europe (and even year 0 Western Europe) are included in some of those analyses that found the log-linear fit, and their self-reported well-being is in line with the regression that fits the rest of the world. The log-linear relationship might break down if we go far enough into the past (or future), to places that are poor enough (or rich enough), or to societies that are different enough so that GDP won’t be a good measure of their material quality of life, but the data I’ve seen suggest that the relationship is more robust than I would’ve expected.
The Stevenson & Wolfers paper that I linked uses self-report measures of welfar including happiness, satisfaction with life, and amount of smiling, and finds similar log-linear relationships with income on all of them (though with different slopes and intercepts), which suggests that this relationship will apply to whichever definition of utility we use.