Questions
When our Isoquant map exhibits constant MRTS along a ray from the origin making. Why do we make specific reference to.
- Constant returns to scale
- Homothetic Functions
I'm asking because it seems to me that any homogenous function is satisfactory i.e. according to my text, for a homogenous function of degree $d>0$ $MRTS$ will depend only on the ratio of it's input. E.g. Looking at our standard cobb-duglas we get $MRTS_{l,k} = \frac{βk}{αl}$ This seems to depend only on the ratio of $k$ & $l$, for whatever α & β chosen, which depending on choice can give DRS, CRS, or ICS.
Context
My book "intermediate Microeconomic Extensions" - Walter Nicholson (A great book btw which i really love). Is emphasising the role of homogenous function of degree one (CRS), as giving us Isoquant maps where MRTS are constant along a ray.
I'm confused why CRS and not just homogeneity greater than 0 are being emphasised here. For example i don't get the significance of the following:
Because A CRS production function is homothetic, the RTS depends only on the ratio of $k$ to $l4 not on the scale of production.
Why doesn't it just read. 'Because the function is homogenous to a degree greater than 0, RTS depends only on the ratio of $k$ to $l$ not on the scale of production'
My understanding is as follows:
- Any functions homogenous to degree $d > 0$ will be Homothetic.
- Any function $f$ that satisfies the following is homothetic.
- It can be written as $f(x,y) = q(r(x,y))$
- Where $r$ is a function that is homogenous of degree 1
- $q$ is strictly increasing.
- This makes sense, and i can prove it.
- The ratio of the partial derivatives e.g. the MRTS, of any homogenous function can be shown to depend only on the ratio of inputs and not their absolute value. I'm happy with the proof of this.
Question recap: For our constant MRTS along a ray from the origin why do we care about:
- CRS
- Homothetic functions
- I think Im generally missing the added value of a homothetic function vs homogenous function.