Cobb Douglas - returns to scale

CFA Level III
#1

This should be 'low hanging fruit, but I seem to be mixed up and would appreciate unambiguous clarity.

CD assumes constant returns to scale. Does this mean;

A. A 1%increase in either Labour AND Capital will produce the same output increase, irrespective of their starting levels I.e. from 2 to 3% increase has same increase in output as from 5 to 6%

B. An INITIAL1% increase in either Labour OR capital will produce the same output increase

I still see references to diminishing marginal returns. Does this mean;

C. A CONTINUAL 1% increase in either Labour OR capital will be subject to diminishing returns I.e going from 5 to 6% capital increase produces lower output rise than from 2 to 3%

I’d be really grateful if someone can clear this up ( I guess I didn’t fully grasp it at level II !) Thanks in advance

#2

A would be your answer. If you learn the CD from an economics viewpoint, labor and capital are raised to their respective elasticity. CFA formula basically simplifies the formula and implies constant returns to scale.

good resource:

http://docentes.fe.unl.pt/~jamador/Macro/cobb-douglas.pdf

#3

Thank you Scorcerer.

However, I’ve got a 2014 Kaplan mock answer that says “CD assumes constant returns to scale e.g. 1% change in labour OR capital from 2 to 3% has the same incremental effect on real output as a change from 4 to 5%” are Kaplan trying to make a distinction using OR rather than an AND - that’s what i am trying to distinguish between my cases A & B.

also, where does diminishing returns to scale apply t please

#4

for you to really understand this, set up a CD function and run some numbers yourself. yes, a 1% change in either variable will have the same incremental effect regardless of the starting point. however, a 1% change in K or L will not equal a 1% change in output. a 1% change in K AND L will equal a 1% change in output. plug some numbers in and you should be able to see this.

#5

Ok, thank you sorcerer . You help is really appreciated, although at this stage in my study and for the small marks available , I’m just hoping rather than read a reference paper or test underlying formulae, someone could kindly write out the conclusions please

#6

Anyone please :grin:

#7

Constant returns to scale = If both inputs increse the same proportionally, output is increased proportionally.

Cobb = A * K^(a) * L (1-a)

Where A > 1, and 0

Say A = 2, K =1, L =1, a = .4, (1-a) = .6, Output = 2.

Now lets move K up to 1.5 = Output = 2.352158. % change = .176

Now lets say L = 1.5 and K = 1.5. Ouput = 3.

(3/2) -1 = .5. We moved up L &K by .5, and output increased by 50%

L = 2, K = 2 = Output = 4.

We moved L &K up by 1 from original, and output Change is 4/2 - 1 = 100%

Increasing would mean as we adjusted L&K proportionally, the ouput is increased by more than the input.

Decreasing is the opposite.

Reaching in the back of my brain right now…sooo…I may be wrong.

From a MACRO Standpoint: a = Income(GDP) share of Capital and (1-a) = Income(GDP) share of Labor. These must sum to 1 in MACRO standpoint, they come from national accounts.

From a MICRO standpoint: a = the output elasticity (%change output given percent change input), these don’t’ have to sum to 1. In a MICRO production, firm, we can have a function of Output = A * K^B * L^C. If B+C > 1, then we get increasing returns to scale. If they are less then 1, we get decreasing returns of scale.

I believe all we need to care about for the test is that the exponents must sum to 1, that Cobb gives a constant return to scale. I don’t think there are any micro based economic stuff on L3 that I recall.

#8

Thank you Jsnazz. I believe your formula is showing:

  1. increasing L&K proportionally, the output is increased by the SAME as both the inputs e.g. L, K from 1 to 2 ie double then output doubles from 2 to 4

  2. increasing L OR K prop, the output is increased by the LESS than the single inputs change - K increase to 1.5 from 1, a 50% rise produces only a 17.5% increase in output. This seems to contradict what I see in a model Kaplan answer which says “increase in either Labour OR capital will produce the same output increase”

thanks for your attention! I’m happy to just “learn” the 3 required conclusions (also the scenario with diminishing marginal returns)!

#9

Hrmmm…is that logged? AKA they brought it down. Because you can play with the model, and the only time that would happen is when A= 5. Even if it was logged, it’s all about a and (1-a). Unless they are equal, they can’t have the same change.