Derive the cost function for a Homothetic production function

Question

I'm having trouble understanding the steps in showing that a Homothetic production function's cost function must be expressible in the form $C(w, q) = a(w)b(q)$.

Since the production function is homothetic, I know that the optimal cost minimizing input ratio given the same input costs must be exactly the same for different quantities of output i.e.

$$ \frac{H^j(w,q)}{H^i(w,q)} = \frac{H^j(w,q')}{H^i(w,q')} $$

Rearranging this we arrive at the ratio:

$$ \frac{H^i(w,q)}{H^i(w,q')} = \frac{H^j(w,q)}{H^j(w,q')} $$

This makes sense because in order to the maintain the input ratio constant, we would need both cost-minimizing inputs to increase by the same factor when going from $q$ to $q'$. However, the proof then states that "for the above to true it is clear that the ratio must be independent of w, thus, setting $q' = 1$"

$$ \frac{H^1(w,q)}{H^1(w,1)}= \frac{H^2(w,q)}{H^2(w,1)} = \cdots =\frac{H^m(w,q)}{H^m(w,1)} = b(q) $$

and so

$$ H^i(w,q) = b(q)H^i(w, 1) $$

The steps after this to get to $C(w, q) = a(w)b(q)$ is pretty understandable and straightforward for me.

However, what I don't understand is how you can clearly see that the ratio must be independent of w. Surely the ratio just states that for a given w the inputs must increase by the same factor/ratio. But how does that imply that the ratio has to be exactly the same for all w?

Edit: for those who are asking, this comes from an exercise question in Frank Cowell's textbook: Microeconomics: Principles and Analysis.

Answer 1

There must be an error in your proof. Where does your proof come from? The part "for the above to true it is clear that the ratio must be independent of $w$," cannot be True. Your ratio must be independent of $q$ but not of $w$. Take for instance the Cobb-Douglas case (which corresponds to a homothetic production function) for a counter example. See Diewert or Chambers for a proof of your result:
Chambers, Robert G., 1988, Applied Production Analysis, Cambridge University Press.
Diewert, E., 1982, "Duality approaches to microeconomic theory", in Handbook of Mathematical Economics, Volume 2.
In the Cobb Douglas case, the production function $$q=x_1^\alpha x_2^\beta$$ yields the first order conditions $$\frac{w_1}{\lambda}=\alpha\frac{q}{x_1}$$ $$\frac{w_2}{\lambda}=\beta\frac{q}{x_2},$$ so that optimal input demand satisfy (in your notations) $$\frac{x_1}{x_2}=\frac{\alpha}{\beta}\frac{w_2}{w_1} \equiv \frac{c_1(w)}{c_2(w)},$$ which is independent of $q$ and implies (show it) that $$x_1=H^1(w,q)=c_1(w)b(q),$$ $$x_2=H^2(w,q)=c_2(w)b(q).$$ It follows that the cost function $$c(w,q)=w_1H^1(w,q)+w_2H^2(w,q)=a(w)b(q),$$ with $a(w)=...$. A similar derivation holds for the more general homothetic case, because by definition of homotheticity, the production technology can then be written as $$y=f(x)=g(h(x))$$ where h is homogeneous of degree one...

Answer 2

The fact the ratio is independent of w comes from one of the properties of homothetic functions. A homothetic function by definition is a monotonic transformation of a homogenous function. Thus, for any homothetic function, a known result is that $Φ(z_1) = Φ(z_2)$ implies that $Φ(tz_1) = Φ(tz_2)$ for any input combination $z_1$ and $z_2$. Thus, for any two input combinations/ratios along the same isoquant (i.e. $Φ(z_1) = Φ(z_2)$) and therefore any price ratio (since for a homothetic function price ratio = MRTS is solely determined by input ratio) multiplying by the exact same factor $t$ gets you from the isoquant where $Φ(z_1) = Φ(z_2)$ to $Φ(tz_1) = Φ(tz_2)$.

As a result, the ratio above is independent of the price w (i.e. which point along the isoquant you are) because no matter where it is, you always have to multiply by the same t to get from q to q', which in my case is the ratio

$$ t = \frac{H^i(w,q)}{H^i(w,q')} = \frac{H^j(w,q)}{H^j(w,q')} $$