Hi,
I’m confused as to how the Cobb Douglas model exhibits diminishing marginal returns to capital and labor.
The formula for marginal product of capital is alpha x income/capital, alpha being the share of GDP paid out to suppliers of capital. Doesn’t that formula imply that the marginal product of capital increases with more units of capital, instead of diminishing? Because if alpha increases (meaning that the share of GDP dedicated to capital increases), marginal product of capital would increase, according to the formula.
Also, the textbook states that if alpha is close to zero, diminishing marginal returns is high and marginal output declines quickly as capital increases, and that if alpha is close to one, diminishing marginal returns is low. This can also be seen in the Cobb Douglas formula, where if alpha is high, an increase in capital would increase output by more than if alpha was low.
All of this seems counterintuitive to diminishing marginal returns - can someone help explain how the Cobb Douglas model has diminishing marginal returns?
Help would be really appreciated - I’ve been stuck on this for ages and feel like I’m missing something!
As capital increases, α(income/capital) decreases; you’re dividing by a bigger number.
Ah I always thought that alpha was capital/income. Why is it income/capital? The text states that alpha is the share of GDP paid out to suppliers of capital - doesn’t that mean that it is capital/income (GDP)?
Also, in Example 10 of reading 14, (1 - alpha) = labour/output (GDP). So, shouldn’t alpha be equal to 1 - labour/income and therefore be capital/income?
Please let me know if I’m thinking about this wrongly.
Maybe a numerical example would make it more clear, let’s say you have A=1, L=5 and alpha = 0.5. And you want to know the impact of an increase in valuea of K on Y. Let’s assume possible values for K = 1,2 and 3
if K= 1 then Y = 5
if K =2 then Y = 7.07
if K =3 then Y = 8.66
passing from one unit of capital to 2 add 2.07 in output, while passing from 2 units of capital to 3 adds 1.59 in output, so diminishing return!
“Ah I always thought that alpha was capital/income.”
It is, but you got to distinguish between your formula for the marginal product of capital from the alpha. From your formula of the marginal product of capital, as smagician said, if capital increase, your denominator increase and you have a smaller number overall.
if you like math, you can calculate the second derivatives of Y relatives to K and see that it’s below zero, which would also imply diminishing return, but beyond the scope of the exam.