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.

Given %ΔX the degree of returns to scale γ is defined by the equation:

%ΔY = γ %ΔX

if there is no change in technology.

γ = the degree of returns to scale

DON'T CONFUSE SMALL GREEK LETTER γ WITH Y OR y!

The other key equation that you need is 

γ = AC/MC

where 

AC = average cost

MC = marginal cost

Your task in this exercise is to deduce the missing value in each row. One out of AC, MC, %ΔX and %ΔY is missing in each row of the questions. It is there in the answers. The degree of returns to scale γ is never explicitly stated in the row. But since each row assumes no change in technology, 

 γ = AC/MC =  %ΔY / %ΔX

Here is the way to show that  γ = AC/MC. Let C stand for total cost. (Sometimes we use C to stand for "consumption" but not here.) Then if Ω (the capital Greek letter Omega) is a price index for all the factors put together,   

MC = Ω ΔX / ΔY

AC = Ω X/Y

(The changes ΔX and ΔY in the definition of marginal cost stipulate that technology is not changing.) Since the Ω cancels,  

MC/AC = (ΔX/ΔY) / (X/Y)  =  (ΔX/X) / (ΔY/Y) ≈  %ΔX / %ΔY = 1/γ

AC/MC = (X/Y)  / (ΔX/ΔY)   =  (ΔY/Y) / (ΔX/X)  ≈   %ΔY / %ΔX =  γ

The only reason there is an approximation sign is that γ might not be constant. For small changes %ΔX and %ΔY, the degree of returns to scale γ will be approximately constant over the relevant range. For big changes it is some kind of complicated average γ over the relevant range that matters.

Why might γ not be constant? Here is an important case. Suppose there is a fixed cost FC and a constant marginal cost MC, then average cost AC will be declining from a very high level (essentially infinite for tiny amounts of output) toward MC. As the amount of output goes to infinity, the average cost AC converges to the constant marginal cost MC. This makes it clear that γ can't be constant in this case. 

You can either print out this one-sheet pdf file with the questions on one side and the answers on the other side, or you can look at the questions and later on the answers below in this post. Remember that these calculations all assume no change in technology.