# Cobb-Douglas with Constant Returns to Scale Exercise

Here, let me start with the same beginning as to the "Returns to Scale Exercise":

Let %ΔX be an overall measure of the change in inputs. If each input changes by the same percentage, this is always equal to that percentage change in each input.

If different inputs change by different percentages, this is a weighted average of the percent changes in each different input. As long as the firm is minimizing costs, the weights will be equal to the share of costs coming from paying for each input:

• s_K = share_K = RK/(RK+WL) is the cost share of capital (the share of the cost of capital rentals in total cost.)
• s_L = share_L = WL/(RK+WL) is the cost share of labor (the share of the cost of the wages of labor in total cost.)

My post "The Shape of Production: Charles Cobb's and Paul Douglas's Boon to Economics" talks about this more in the case of constant-returns-to-scale Cobb-Douglas.

Thus, the total input measure is

%ΔX = share_K  %ΔK + share_L  %ΔL

(Just to be clear about the notation, share_K and %ΔK are multiplying each other, and so on.) When (a) the degree of returns to scale is equal to 1 ("constant returns to scale"), and (b) the measures of input and output changes include everything, then the measure of technological progress is simply

%ΔA = %ΔY - %ΔX

In words, the change in technology is equal to the percent change in output minus the percent change in inputs overall (%ΔX). Technology improvements can show up as more output for the same input OR less input for the same output. It is also a technology improvement if output increases more than inputs. Things will get more complicated when there are increasing returns to scale or other issues. But this is good for now.

The percent change in outputs minus the percent change in inputs %ΔY - %ΔX is called total factor productivity growth. It is often treated as a measure of technological change. But total factor productivity growth %ΔY - %ΔX  (= %ΔY - share_K  %ΔK + share_L  %ΔL when the only inputs are capital and labor) is an imperfect measure of technological change when their are increasing returns to scale (that is, when the degree of returns to scale γ = AC/MC > 1) or when the measures of inputs and outputs used are not comprehensive. Stay tuned for more on that later on in the semester.

These equations are exact with Platonic percentage changes and Cobb-Douglas with Constant Returns to Scale. Perhaps surprisingly, these equations are also very good approximations for other Constant-Returns-to-Scale production functions even when they are not Cobb-Douglas.

Your task is to fill in the missing Platonic percent change in each row. Here is the pdf file with the questions (the missing piece for a row) on one side and the answers on the other. And here are the questions just below, and the answers further below on this post: