Capital per person in India is about 6 percent of the U.S. level. Given that per capita income y=Ak0.3, calculate the level of total factor productivity (A), relative to the U.S. level, that would be needed for India to match the U.S. level of per capita income. Round your answer to 2 decimal places.

[Hint: Normalize the numbers so that U.S. values are equal to 1 and the value for the other country as a share of the U.S. value]

Question

Capital per person in India is about 6 percent of the U.S. level. Given that per capita income y=Ak0.3, calculate the level of total factor productivity (A), relative to the U.S. level, that would be needed for India to match the U.S. level of per capita income. Round your answer to 2 decimal places.

[Hint: Normalize the numbers so that U.S. values are equal to 1 and the value for the other country as a share of the U.S. value]

BlackTom AI · Accepted Answer

The task is to determine A for India so that y = A k^0.3 matches the U.S. per-capita income after normalizing U.S. values to 1.
First, normalize the baseline: set US values to 1. This implies A_US = 1 and k_US = 1, so y_US = A_US * (k_US)^0.3 = 1 * 1^0.3 = 1.
India’s capital per person is 6% of the U.S. level, so k_India = 0.06.
India’s per-capita income is y_India = A_India * (k_India)^0.3 = A * (0.06)^{0.3}.
To match the U.S. level, set y_India = y_US = 1. Therefore, A * (0.06)^{0.3} = 1, so A = 1 / (0.06)^{0.3}.
Now compute (0.06)^{0.3}. Using logarithms: ln(0.06) ≈ -2.813, multiply by 0.3 gives ≈ -0.844, and exponentiate to get (0.06)^{0.3} ≈ e^{-0.844} ≈ 0.430.
Thus A ≈ 1 / 0.430 ≈ 2.326, which rounds to 2.33 when rounded to two decimals.
In summary, to equal the U.S. per-capita level, India would need a total factor productivity of about 2.33 relative to the U.S. baseline under the given Cobb-Douglas form with k^0.3, after normalizing the U.S. values to 1.
Note: This relies on the normalization convention described in the hint, ensuring the comparison is dimensionless and centered on the U.S. benchmark.

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