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Chapter 16: Problem 5

Suppose labor's share of GDP is 70 percent and capital's is 30 percent, total factor productivity is growing at an annual rate of 2 percent, the labor force is growing at a rate of 1 percent, and the capital stock is growing at a rate of 4 percent. What is the annual growth rate of real GDP?

Short Answer

Expert verified
Answer: The annual growth rate of real GDP is 3.9%.

Step by step solution

01

Write the Cobb-Douglas production function with given values

We have the following values: labor's share β = 0.7, capital's share α = 0.3, total factor productivity (A) growth rate = 0.02, labor force (L) growth rate = 0.01, and capital stock (K) growth rate = 0.04. Write the Cobb-Douglas production function as: \(Y = A * K^{0.3} * L^{0.7}\)
02

Differentiate the production function with respect to time

Take the natural logarithm of the function and differentiate with respect to time t: \(\frac{d\ln(Y)}{dt} = \frac{d\ln(A)}{dt} + 0.3 * \frac{d\ln(K)}{dt} + 0.7 * \frac{d\ln(L)}{dt}\)
03

Substitute the growth rates into the equation

Plug in the growth rates for A, K, and L: \(\frac{d\ln(Y)}{dt} = 0.02 + 0.3 * 0.04 + 0.7 * 0.01\)
04

Calculate the annual growth rate of real GDP

Simplify the equation and solve for the growth rate of GDP: \(\frac{d\ln(Y)}{dt} = 0.02 + 0.012 + 0.007\) \(\frac{d\ln(Y)}{dt} = 0.039\) The annual growth rate of real GDP is 3.9%.

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