One-way economists measure total factor productivity is to use a Cobb-Douglas production function of the form q = A (t) KaL1-a, where A (t) is a term representing technical change and a is a positive fraction representing the relative importance of capital input.
a. Describe why this production function exhibits constant returns to scale.
b. Taking logarithms of this production function yields
ln q = ln A (t) + alnK + (1 – a) ln L
One useful property of logarithms is that the change in the log of X is approximately equal to the percentage change in X itself. Explain how this would allow you to calculate annual changes in the technical change factor from knowledge of changes in q, K, and L and of the parameter a.
c. Use the results from part b to calculate an expression for the annual change in labor productivity (q/L) as a function of changes in A (t) and in the capital-labor ratio (K/L). Under what conditions would changes in labor productivity be a good measure of changes in total factor productivity? When would the two measures differ greatly?