Intermediate Microeconomics and Its Application 12th Edition by Walter Nicholson,Christopher Snyder

Exercise 19

One way economists measure total factor productivity is to use a Cobb-Douglas production function of the form q = A(t)K a L 1-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 (see Problem)b. Taking logarithms of this production function yields
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?
The production function
q = K a L b
where 0 ? a, b ? 1 is called a Cobb-Douglas production function. This function is widely used in economic research. Using the function, show the following:
a. The production function in Equation is a special case of the Cobb-Douglas.
b. If a + b = 1, a doubling of K and L will double q.
c. If a + b 1, a doubling of K and L will less than double q.
d. If a + b 1, a doubling of K and L will more than double q.
e. Using the results from part b through part d, what can you say about the returns to scale exhibited by the Cobb-Douglas function?
Equation