Worksheet and Points to ponder:

Formalization: Solow ==> Growth Accounting

If Y = f (K, L, TFP) is reduced to a specific function

Yt = TFPtKtαLt1-α

then log Y = log TFP + α log K + (1-α) log L

and if we convert to per laborer terms

log Y/L = log Y – log L then (subtracting from both sides)

log Y/L = log TFP + α log K + (1-α) log L – log L

= log TFP + α log K – α log L

= log TFP + α log K/L

Now the total growth rate for something multiplicative is the sum of the individual growth rates. If there are exponents, it becomes a weighted average. You can see that by taking the derivative of the logs, as d(log X) = dx/X or (when multiplied by 100) the rate of change in percent.

Rule of 70/72

[it’s an approximation, so pick the one that gives the easiest arithmetic]

log 2 = 0.69 so if the growth rate is x (single-digit percentage), we can use the Math 101 “fact” that [cf. the infinite series expansion] log (1 + x) ≈ x. So n log (1 + x) is growth rate x compounded for n years. If we write it in percentage terms, then nx = 70 or 72 is a doubling, or 70/x [72/x] is years to double.

So here are thought experiments. How long does it take to double in the following. Does it make much difference if we set α = 0.4 or a bit lower / higher at 0.3 (70% of income accrues to labor) or 0.5?

Practice calculations

Plug the number in (use α = 0.4 and (1-α) = 0.6) to calculation (i) growth and (ii) growth per capita. How many years to double?

**Premodern**

pop growth 1%

K growth 1%

TFP growth 0.5%

**Health improvements that improve LF productivity a bit but also increase population growth**

pop growth 2%

K growth 1%

TFP growth 1%

**China**

pop growth -0.5%

K growth 6%

TFP growth 2%

**Japan**

pop growth -1.5%

K growth 1%

TFP growth 0.5%

From investment to growth.

If a country has K/Y capital-output ratio of 1.5, then with a depreciation rate of 8%, total depreciation comes to 12% of GDP. So a savings rate of 16% gives net investment of 4% or a growth of the capital stock of 4%. Savings of 20% gives net investment of 8% of GDP or growth of 8% of K. China has savings-cum-investment of 40% of GDP. which gives net investment of 28% and capital stock growth of 28%. But α is about 0.4.