@elguapador,

I'll continue here, for some reason it doesn't allow me to edit.

The MRTS in a complementary inputs function makes no sense. Economically, it tells you how much of the input K you must increase in you give up 1 unit of L (or how much K you must give up to use one more unit of L). Think about the bike example now: If you have 3 tyres and give up one, you don't need to add more capital to produce one single bike, which can't be said mathematically: that's why it doesn't exist.

Now, let's answer to the question: indicate whether each marginal product is diminishing

1)

MPL = K

MPK = L

I remember I had a similar question on my book, and after doing the exercise I checked the correct answers. What my book says about it is:

MPL Is constant when the Labour used changes

MPK Is constant when the Capital used changes

Basically, according to my book if you are asked if they are constant, you must consider how the MPx changes to the change of the x (L or K) input used.

2)

MPL = 1

MPK = 3

Of course they're both cnonstant.

3) Not defined.

RETURS TO SCALE:

The returns to scale tell you, if you increase the amount of both the inputs by a certain amount λ, how much the quantity produced will increase. Let's call the increase in the quantity produced ϕ.

What you need to do, is multiplying bot L and K by λ.

-----------------From the theory I know that:---------------------

1) Perfect substitutes production functions have constant return to scale

2) Perfect complements production functions (or Leontief function) have constant return to scale

3) Cobb-Douglas Production functions can have Constant, deminishing or increasing returns to scale.

The Cobb Douglas function must be like that:

AL^α K^β, with A, α, β > 0

If α+β = 1 Then you'll have constant returns to scale

If α+β > 1 Then you'll have increasing returns to scale

If α+β < 1 Then you'll have decreasing returns to scake

Let's check it out.

1) Q1= L3K

Q2 = (λL)3(λK)

= λL3λK

Here we have: A=1 , α = 1, β = 1, α+β= 1+1=2 -->We expect to have increasing returns to scale

= λ²(L3K)

Q1 = L3K

Q2 = λ²Q1.

Which means that by increasing both L and K to λL and λK, we now produce λ² units of good.

2) Q1 = L+3K

Q2 = (λL) + 3(λK)

= λ (L+3K)

Q1 = L+3K

Q2 = λQ1. For instance, this means that by using double Labour and double capital you produce the double of what you produced before.

3)

Q1= Min {L^1/3+K^1/3}

Q2 = Min {(λL)^1/3 + (λK)^1/3}

Q2 = Min {λ^3/3 (L^1/3+K^1/3)}

Q2 = Min {λ (L^1/3+K^1/3)}

Q1 = λQ2

You need to raise λ to the third power in order to bring it out of the brackets