Comparing Growth to Equality

Tyler Cowen explains why the rate of economic growth is important, emphasizing that:

"had America grown one percentage point less per year, between 1870 and 1990, the America of 1990 would be no richer than the Mexico of 1990."

Abiola wholeheartedly agrees, and declares that he is glad to see that also Yglesias does so. And indeed, the question should really be who does not?

However, when comparing befits for different populations at different times, you are implicitly assuming that you can aggregate individual's gains into a population gain. Then let's do so in order to see to which extent equality should be weighed into the growth equation. It seems to me that a logarithmic utility function is a good tool for this, at least it was in this paper on taxes and leisure quoted here by Arnold Kling. Experiencing such utility adds equally much of it for every doubling of the consumption. With geometrical income growth you hence get arithmetic utility growth.

So - let's assume that the Land-of-the-Free, F, steadily leaves a share, p, of its population behind at constant consumption, while annually multiplying the consumption for the rest by a factor 1+Gf.

The Welfare State, W, on the other hand advances consumption for all its citizens each year by a smaller factor 1+Gw.

Now, for simplicity, let's further assume that all people in both W and F, initially upholds the same consumption c0, and hence the same utility u0. Each and every year, W hence adds Gw*u0 to the total utility experienced (assuming Gw small enough to make ln(1+Gw) close enough to Gw) by the population from its consumption while F adds (1-p)Gf*u0. The utility experienced by people in the Land-of-the-Free hence outgrows the Welfare-State if, its economic growth times one minus its share of people left behind, is larger that the Welfare-State's growth. For instance, growing at a rate of 5% while leaving 10% of the population behind would advance the population's welfare at the same rate as if growth were 4.5% and equally distributed. It should be noted for completeness that the total growth-rate for F is initially 1+Gf/(1+p), but that it converges (as the share consumed by the poor goes to zero) to 1+Gf.

I'm not an economist, at least I am not a macro-economist, but let me tentatively suggest an approximate growth rate for the Land-of-the-Free of something like 3.5% and that a 1/7th of its population is left behind. This would let F add 3% to its utility each year and with a 2.5% growth in W, F adds 0.5% of the initial utility more to its total utility than does W. In 200 years, F's utility added would have exceeded that added by W by the whole amount of the initial utility. In steady state, the welfare in the the Land-of-the-Free will with these numbers eventually converge to an amount that is 50% higher than that of the Welfare-State's.

An impressing win for the Land-of-the-Free - but hardly a sudden Welfare-State collapse!

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