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Answer:
If the Cobb Douglas production funtion is
This function is homogeneous of degree 3: To understand that, we first must know that a function f(K,L) is homogeneous of degree "m" if . Intuitively, this means that, when you increase your productive factors (in this case, we are talking about a production function), by a factor "
", your output increases by
. Depending on the value of m, the function will exhibit increasing returns to scale (m>1), decreasing returns to scale (m<1) or returns to scale equal to 1 (when m=1).
- In this case,
. Applying distributive power's property, we get
.
- Because of power property, we can associate terms and get
(remember that
.
- Finally,
. In this case the function is homogeneous of degree 3 because when multiplying K and L by
, the function as a whole is multiplied by
.
Euler's Theorem: this theorem states that, if a function is homogeneous of degree "m", the following must hold: .
- To prove it, we should then calculate the partial derivative of Q with respect to L and K respectively, and apply the previous definition to see if the statement holds.
- Applying Euler's Theorem then means
should be equal to
(remember that in this case, m=3, see previous exercise).
- Solving
- Then the Euler's Theorem is verified!