# Decreasing and increasing returns to scale question

Hi, I have deduced that this function exhibit increasing returns to scale but I am not sure how to verify part d. My answer doesn't show that there is decreasing returns to scale but I can't be sure d is wrong. I am unsure between choosing c and d.

Thanks

• Hint: How did you arrive to the conclusion that the function exhibits increasing returns to scale? Does the level of K + L matter for your conclusion? – user20105 May 4 '19 at 14:37
• @user20105 I deduced increasing returns to scale by doing f(tK, tL) which gives me t^1.2 f(K,L) since t^1.2 is greater than t, I concluded that the returns to scale is increasing. – 6.19 May 4 '19 at 14:46
• I am guessing K+L matters because it is a part of the production equation but returns to scale normally relates to their power so I don't really understand what K+L>1 is indicating. – 6.19 May 4 '19 at 14:48
• Since $f(\lambda K, \lambda L) \Rightarrow \lambda^{1.2}f(K,L)$, say you double both your inputs. Increasing returns to scale tells you that your output will be more than doubled, which is precisely what $\lambda^{1.2}$ represents. The level of $K + L$ does not affect this conclusion. – user20105 May 4 '19 at 14:54
• I have added this as an answer but please note that it is always better when asking questions to show your own work and to type in the questions instead of pasting an image! – user20105 May 4 '19 at 15:18

## 1 Answer

From your production function: $$f(K,L)= 1.8(K^{0.6} + L^{0.6})^{2} \Rightarrow f(K,L) = 1.8(K^{1.2} + 2(KL)^{1.2} + L^{1.2})$$

You are able to see that $$f(\lambda K, \lambda L) \Rightarrow \lambda^{1.2} f(K,L)$$. Say you double both your inputs. Increasing returns to scale tells you that your output will be more than doubled, which is precisely what $$\lambda^{1.2}$$ represents. The level of $$K$$ and $$L$$ does not affect this conclusion. The point where the level of $$K$$ and $$L$$ play a "role" is that for levels such that $$K + L >1$$ each unit of input results in more than one unit of output (i.e.$$f(K,L) > 1$$) (again note that we are talking about levels here and not changes, which would regard the returns to scale). You can easily see this by plugging in different values into your formula. But, again, this is not to do with the returns to scale of the function so it doesn't pertain the question you are asked.

• Should it not be $2K^{0.6}L^{0.6}$ instead of $2(KL)^{1.2}$ ? – Bertrand Jan 29 at 17:09