Diminishing returns to labour and returns to scale.

Definitions:
for x>1:
xF(L,K)>F(xL,xK) means increasing returns to scale.
then = and < for constant and decreasing, right?

What's the exact definition for diminishing returns to labour? Is it when MPL is decreasing? So when the derivative of MPL with respect to L is negative, we have diminishng returns to labour? Which is the same as when the second derivative of F(L) with respect to L is negative, we have diminishing returns to labour?

What if we had a production function that was a something like F(K,L) = K/L * Min(L,K)?
So it's constant returns to labour for L<K, then as soon as there's a DISCRETE jump from L<k to L>K, surely there's diminishing returns? Then, from there on, although each labour added actually decreases output, the decrease is less and less, so we actually have INCREASING returns to labour?

So, okay, what if we had a weird production function like F(L) = 1/L. This means that capital does not exist.
Clearly for x>1: xF(L) > F(xL).

But the gradient of the curve is getting less and less negative, so do we have a situation of DECREASING returns to scale and ALSO INCREASING returns to labour, even though labour is the only input?

First off, a quick read of this particular subsection may help: http://en.wikipedia.org/wiki/Marginal_product_of_labor#The_Law_of_Diminishing_Marginal_Returns

Key thing to take away: diminishing marginal returns to labour means MPL is falling, diminishing returns to labour means MPL is NEGATIVE (which is sometimes called negative returns to labour).



Yup, just think of it like "I doubled all the inputs, but I got more than double my output, zOmg"



Exact definition of diminish returns to labour is that MPL is negative.

Exact definition of diminishing marginal returns to labour is that MPL is positive and MPL is decreasing. So that first partial derivative of F is positive, second partial derivative of F is negative (equivalently, first partial derivative of MPL is negative).

And usually (i.e. when dealing with neoclassical theories) we assume that production functions and utility functions are such that dF/dL > 0 for all L and d^2F/dL^2 < 0 for all L.



As you've basically done, it's best here to always split the production function:

Case 1: L < K, then F(K,L) = K/L * L = K, so F is constant, so MPL = 0. What does this mean? It means hiring another worker does nothing for you! This is trivially constant returns to labour I think.

Case 2: L = K, then F(K, L) = K. As you'll see from below, the point L = K isn't smooth, i.e. the derivative doesn't exist at that point, so you can't really talk about MPL here.

Case 3: L > K, then F(K, L) = K/L * K = K^2/L. First thing to ask your self is if there's really a jump? Well, no, there isn't: check the point L = K, then F(K, L) = K^2/K = K exactly as case 2! So that this isn't even a 3 case problem, it's a 2 case problem as case 2 is just included into case 3.

Now the best thing to do is to sketch F holding K constant (so F on the y axis, L on the x axis). This is fairly easy, horizontal line at F = K and then draw K^2/L (like a 1/x curve) cutting the horizontal line exactly where L = K.

So now to actually answer you (I'm not sure why I've gone off on a rant since I'm sure you've figured all of this out already!),

For L >= K $\text{MPL} = \frac{\partial F}{\partial L} = \frac{-K^2}{L^2} < 0$ and $\frac{\partial^2 F}{\partial L^2} = \frac{2K^2}{L^3} > 0$ for all L and K > 0

Since MPL is always negative for L >= K, we say that we have diminishing returns to labour (as per the definition in bold above), so it only really makes sense to talk about increasing marginal returns when MPL is positive. Usually, MPL is positive because think about what negative MPL means: hiring more people REDUCES your output, which is a bit crazy, so it's unlikely to come up in any micro question that requires actual thinking (I know Prof. Young throws it into some multiple choice questions though, but that's more of a maths test really, don't expect a section C question with negative MPL).



Urrr, take x = 2. Then F(xL) = F(2L) = 1/(2L) < 2/L because 1/2 < 2, so F(xL) < xF(L) which gives us increasing returns to scale.



Again, this is negative returns to labour, so no point thinking of decreasing/increasing marginal returns.

Any tips for tackling EC102? Where exactly were you 2 weeks before the exam revision wise?
Also, did you spend more time revising EC102 compared to other subjects?

Thanks

It's not a fair comparison because we had 3 weeks after Easter to prepare for EC102 (since LT was only 10 weeks last year, as it usually is).

I remember I only started revising macro the weekend before summer term started, but I'd been through all but 1 of the micro quizzes once or twice, which meant that I'd read the relevant chapters from the micro book and attempted the questions + understood the solutions (not memorised though). I was more relaxed about macro because I'd worked hard on it during term time and had read most things in the book at least once (except the annoying stuff on policy/Phillips curve /AD-AS) before Easter.

Summer term started and I went to every single revision lecture, I found them extremely helpful, even the repeats.

Most of my revision in those 3 weeks of summer term involved doing, then redoing, then redoing quizzes/problem sets and basically memorising the solutions off by heart. The solutions are extremely good - they act as notes because he always gives you a lot of theory on top of the solution and so they're more practical than lecture slides/the book. I can't tell you the number of times I read the AD-AS solutions (which are like 19 pages or something as well!). Literally just memorise them the best you can.

I can tell you that even a week before the exam, I was getting very basic things wrong and finding it hard to remember the difference between stuff like the PCC and Engel curves at times. But practise makes perfect (not that I'm perfect lol). If I'm honest, I think a 2:1 would be a better indicator of my ability at economics, I just got lucky with section B questions which were pretty mathematical last year, but then I think I understood certain concepts much better than most people - my economic intuition is weaker than the LSE average I reckon.

I definitely spent way more time on EC102 than anything else - the other stuff I thought was repetitive (i.e. you can pretty much predict exactly what's in the MA100 paper, all they really do is change the numbers and phrasing around a bit).

*Edit*

Arrgghhhh, been meaning to do some macro work since 12 AM and now it's 8:30 AM somehow

I don't know if it's just me, but didn't I get increasing/decreasing returns to scale mixed up in my OP? xF(L,K) > F(xl,xk) is decreasing isn't it? I doubled all input and got less than if I just double outputs.

With regards to these definitions though, I am seriously sick of it. Wikiedia MUST be different to the way it's taught at LSE then.
See this link to the first couple questions, particularly 2):
http://moodle.lse.ac.uk/file.php/745/Problem_Set_2_Microeconomics_2007-09_Solutions.pdf

q2 says &quot;consider these six production functions in box 6. Which of these never has diminishing returns to labour?&quot;

The first production function F(L,K) = k^0.5 * L^0.5
the solution says MPL is decreasing (slope of the curve is getting flatter) hence it does exhibit diminishing returns.

The 2nd production function = k^2 + L^2.
Solution say MPL increases as labour does, hence increasing returns.

3rd = min(K,L).
Solution says constant, decreasing, then zero returns to labour.

4th= 4k+ 2L
Solution says MPL is constant, so constant returns to labour.

5th = k^0.6 * L^0.5
says decreasing.

6th is pretty interesting: = (k/L) * min(K,L)
it says that they're initially zero for L&lt;K, and then INCREASING because MPL slope is negative, but getting less negative. Then if you look back at the 4th production function, it talks about discrete changes causing it to have diminishing returns. Does this not have a discrete diminish return to labour as well?

Basically, we make no distinction between returns and marginal returns I think?

I even emailed this to Ian Long: &quot;
Is it that diminishing returns to labour/capital means as labour/capital is added to a fixed factor of production, MPL/MPK falls?

So diminishing returns to labour is when the derivative of MPL with respect to labour is negative? And since MPL is the derivative of output with respect to labour, diminishing returns to labour is when the second derivative of output with respect to labour is negative (when the gradient of the output function is decreasing)?&quot;

And he replied saying that I'm correct.

thanks for any help guys. I know I'm probably being really anal about this but it's just confusing me.