Ok fuck it I'm bored. If you don't care about this then you'd probably be better off not reading any further.
Start with a production function.
Y = K^a * N^1-a where Y = output; K = capital; N = Labor or Population; ^ indicates exponent; and 0 < a < 1.
Because we are examining growth, we are more concerned with per-capita output rather than just output.
Divide each side by N to get per-capita output.
Y/N = (K^a * N^1-a / N = K^a * N^-a = (K/N)^a
This is generally written as y = k^a, where y = Y/N and k = K/N
Look what happens to output per laborer as the capital/labor ratio changes. You can see that output per laborer is positively affected by the capital/labor ratio. But we want to see how growth in output per laborer is affected by growth in the capital/labor ratio. In order to do this you will need to know some of the simple properties of logarithms and how to differentiate logarithms.
So we have y = k^a, after differentiating with respect to T you get Δy/y = aΔk/k
This says that growth in output per laborer is positively related to growth in the capital labor ratio. So now we need to find out what is happening to growth in the capital/labor ratio (Δk/k).
Basically we need to do the same thing we just did for y. The result:
Δk/k = ΔK/K - ΔN/N
As you can see growth in the capital labor ratio, which is postively related to growth in per-capita income, is positively related to growth in capital and negatively related to growth in labor or population.
The point is that growth in per-capita income, which is the best measure of a nation's standard of living, is positively related to the growth in the capital/labor ratio. This ratio increases as capital grows, and decreases as labor or population grows. When the government runs a deficit, capital investment decreases so ΔK/K gets smaller. Because the population is always increasing, ΔN/N is increasing. This means Δk/k is decreasing which means growth in per-capita income is decreasing.
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