Question

Yt =AtKtα(htLt)1−α Where At is TFP in period t,Kt is the capital stock, Lt is the labor force, and ht is the average education of each worker. a) Use the growth accounting framework to find an expression for the growth rate of output. (hint, take the natural log of the expression and then a derivative with respect to time, t ).

Answer 1

The growth rate of output is calculated by taking the natural logarithm of the production function, differentiating it with respect to time, and expressing the result as a weighted sum of the growth rates of total factor productivity, capital, and labor adjusted for education.

Step-by-step explanation:

To derive the growth rate of output from the given production function Yt = AtKtα(htLt)1−α, we first take the natural logarithm of both sides of the equation.

This provides a log-linear form that is easier to differentiate with respect to time. The result allows us to express the growth rate as the sum of the growth rates of each component (TFP, capital, labor, and education), weighted by their respective output elasticities.

Here's how we approach it step-by-step:

1. Take the natural logarithm of the production function:
2. ln(Yt) = ln(At) + αln(Kt) + (1-α)ln(htLt).
3. Differentiate this log-linear form with respect to time t:
4. Δln(Yt)/Δt = (Δln(At)/Δt) + α(Δln(Kt)/Δt) + (1-α)(Δln(htLt)/Δt).
5. Recognize that the derivatives of the natural logarithms with respect to time are just the growth rates of the respective variables, denoted by g:
6. gY = gA + αgK + (1-α)(gh + gL),
7. where gY is the growth rate of output, gA is the growth rate of TFP (technological progress), gK is the growth rate of capital, gh is the growth rate of education per worker, and gL is the growth rate of labor.

Growth accounting studies show that these components contribute to economic growth, with much of the residual growth attributed to TFP, implying improvements in technology.