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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.

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  • $\begingroup$ Where are you reading this from? Maybe a little context will be useful $\endgroup$ – caverac Dec 29 '18 at 4:20
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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...

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  • $\begingroup$ It would be helpful to cite a specific source for Diewert/Chambers in your answer for the user above. $\endgroup$ – Kitsune Cavalry Dec 29 '18 at 15:37
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    $\begingroup$ See for instance Chambers, Robert G., 1988, Applied Production Analysis, Cambridge University Press and Diewert, E., 1982, "Duality approaches to microeconomic theory", in Handbook of Mathematical Economics, Volume 2. $\endgroup$ – Bertrand Dec 29 '18 at 16:31
  • $\begingroup$ Looks good. Put it in your answer! $\endgroup$ – Kitsune Cavalry Dec 29 '18 at 16:34
  • $\begingroup$ I've gotten this proof from one of the exercises in Frank Cowell's textbook, Microeconomics: Principles and Analysis. I've tried doing it with the Cobb-Douglas and found that the ratio was independent of w. You find that the conditional input demand for input 1 for example is Kq(w1w2)^(1-α), where K is a constant. So if you take the ratio, everything cancels off apart from q/q'! $\endgroup$ – Robin Liao Dec 30 '18 at 14:48
  • $\begingroup$ Thanks for the detailed reply, but doesn't your proof with the Cobb-Douglas case support the idea that the ratio is independent of w? Since you have x1 = (αλq)/w1. So taking the ratio of the x1 at different quantities, e.g. q', we end up with q/q' as I did, which is an expression that is clearly independent of w1. $\endgroup$ – Robin Liao Jan 1 at 22:35
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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

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

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