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

Begin by identifying the relationship given: y = A k^0.3, where y is per capita income, A is total factor productivity, and k is capital per person. The task asks for A relative to the U.S. level such that India matches U.S. per capita income after normalization.

Step 1: Normalize to the U.S. baseline. We set U.S. values to 1 for both y and k. This means k_US = 1 and A_US = 1, so y_US = A_US * k_US^0.3 = 1 * 1^0.3 = 1.

Step 2: Express India’s given condition in the normalized framework. It’s stated that capital per person in India is 6 percent of the U.S. level, so k_IND = 0.06. We want India’s per capita income to equal the U.S. level, so y_IND must also be 1 when normalized.

Step 3: Write India’s condition using the model. With y_IND = A_IND * (k_IND)^0.3, and y_IND = 1, we have 1 = A_IND * (0.06)^{0.3}.

Step 4: Solve for A_IND. Rearranging gives A_IND = (0.06)^{-0.3}.

Step 5: Compute numerically. First compute (0.06)^{-0.3} = exp(-0.3 * ln(0.06)). The natural log of 0.06 is approximately ln(0.06) ≈ -2.81341. Multiplying by -0.3 yields ≈ 0.844023. Exponentiating gives e^{0.844023} ≈ 2.326.

Step 6: Round to two decimal places as requested. The computed value is approximately 2.33.

Summary of the logic: normalization sets U.S. k and A to 1, the target is to have y_IND = 1, plug k_IND = 0.06 into y_IND = A_IND k_IND^{0.3}, and solve for A_IND, which yields (0.06)^{-0.3} ≈ 2.33.

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