<p>The fact the ratio is independent of <strong>w</strong> 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 <span class="math-container">$Φ(z_1) = Φ(z_2)$</span> implies that <span class="math-container">$Φ(tz_1) = Φ(tz_2)$</span> for any input combination <span class="math-container">$z_1$</span> and <span class="math-container">$z_2$</span>. Thus, for any two input combinations/ratios along the same isoquant (i.e. <span class="math-container">$Φ(z_1) = Φ(z_2)$</span>) and <strong>therefore any price ratio</strong> (since for a homothetic function price ratio = MRTS is solely determined by input ratio) multiplying by the exact same factor <span class="math-container">$t$</span> gets you from the isoquant where <span class="math-container">$Φ(z_1) = Φ(z_2)$</span> to <span class="math-container">$Φ(tz_1) = Φ(tz_2)$</span>.</p>

<p>As a result, the ratio above is independent of the price <strong>w</strong> (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 </p>

<p><span class="math-container">$$
t = \frac{H^i(w,q)}{H^i(w,q')} = \frac{H^j(w,q)}{H^j(w,q')}
$$</span></p>

CC BY-SA 4.0
2019-01-02T10:53:00.467Z
2019-01-02T10:53:00.467Z
Robin Liao
<p>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 <span class="math-container">$w$</span>," cannot be True.  Your ratio must be independent of <span class="math-container">$q$</span> but not of <span class="math-container">$w$</span>.
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:<br>
Chambers, Robert G., 1988, Applied Production Analysis, Cambridge University Press.<br>
Diewert, E., 1982, "Duality approaches to microeconomic theory", in Handbook of Mathematical Economics, Volume 2.<br>
In the Cobb Douglas case, the production function <span class="math-container">$$q=x_1^\alpha x_2^\beta$$</span> yields the first order conditions <span class="math-container">$$\frac{w_1}{\lambda}=\alpha\frac{q}{x_1}$$</span>
<span class="math-container">$$\frac{w_2}{\lambda}=\beta\frac{q}{x_2},$$</span> so that optimal input demand satisfy (in your notations) <span class="math-container">$$\frac{x_1}{x_2}=\frac{\alpha}{\beta}\frac{w_2}{w_1} \equiv \frac{c_1(w)}{c_2(w)},$$</span> which is independent of <span class="math-container">$q$</span> and implies (show it) that <span class="math-container">$$x_1=H^1(w,q)=c_1(w)b(q),$$</span> <span class="math-container">$$x_2=H^2(w,q)=c_2(w)b(q).$$</span> It follows that the cost function <span class="math-container">$$c(w,q)=w_1H^1(w,q)+w_2H^2(w,q)=a(w)b(q),$$</span> with <span class="math-container">$a(w)=...$</span>. A similar derivation holds for the more general homothetic case, because by definition of homotheticity, the production technology can then be written as <span class="math-container">$$y=f(x)=g(h(x))$$</span> where h is homogeneous of degree one...</p>

CC BY-SA 4.0
2018-12-29T15:37:53.833Z
It would be helpful to cite a specific source for Diewert/Chambers in your answer for the user above.
Kitsune Cavalry
CC BY-SA 4.0
2018-12-29T16:31:20.263Z
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.
Bertrand
CC BY-SA 4.0
2018-12-29T16:34:45.033Z
Looks good. Put it in your answer!
Kitsune Cavalry
CC BY-SA 4.0
2018-12-30T14:48:30.300Z
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'!
Robin Liao
CC BY-SA 4.0
2019-01-01T22:35:52.853Z
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.
Robin Liao
CC BY-SA 4.0
2019-01-04T20:48:46.017Z
Ok, the proof can also be written this way round.
Bertrand
CC BY-SA 4.0
2018-12-29T12:10:25.150Z
2018-12-30T15:40:17.550Z
2018-12-30T15:40:17.550Z
Bertrand
<p>I'm having trouble understanding the steps in showing that a Homothetic production function's cost function must be expressible in the form <span class="math-container">$C(w, q) = a(w)b(q)$</span>.</p>

<p>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. </p>

<p><span class="math-container">$$
\frac{H^j(w,q)}{H^i(w,q)} = \frac{H^j(w,q')}{H^i(w,q')}
$$</span></p>

<p>Rearranging this we arrive at the ratio:</p>

<p><span class="math-container">$$
\frac{H^i(w,q)}{H^i(w,q')} = \frac{H^j(w,q)}{H^j(w,q')}
$$</span></p>

<p>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 <span class="math-container">$q$</span> to <span class="math-container">$q'$</span>. However, the proof then states that "for the above to true <strong>it is clear that the ratio must be independent of w</strong>, thus, setting <span class="math-container">$q' = 1$</span>"</p>

<p><span class="math-container">$$
\frac{H^1(w,q)}{H^1(w,1)}= \frac{H^2(w,q)}{H^2(w,1)} = \cdots =\frac{H^m(w,q)}{H^m(w,1)} = b(q)
$$</span></p>

<p>and so</p>

<p><span class="math-container">$$
H^i(w,q) = b(q)H^i(w, 1)
$$</span></p>

<p>The steps after this to get to <span class="math-container">$C(w, q) = a(w)b(q)$</span> is pretty understandable and straightforward for me.</p>

<p>However, what I don't understand is how you can clearly see that the ratio must be independent of <strong>w</strong>. Surely the ratio just states that for a given <strong>w</strong> 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 <strong>w</strong>? </p>

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

CC BY-SA 4.0
2018-12-29T04:20:03.087Z
Where are you reading this from? Maybe a little context will be useful
caverac
<microeconomics><production-function><cost-functions>
Derive the cost function for a Homothetic production function
Robin Liao
# Derive the cost function for a Homothetic production function

(Asked by: username_0 on 2018-12-27)

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.

**Tags:** microeconomics, production-function, cost-functions
> **Comments:**
> **username_1**: Where are you reading this from? Maybe a little context will be useful  

---

## 2 answers

### Answer by username_0 on 2019-01-02. Score: 0 | Accepted answer

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

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


---

### Answer by username_2 on 2018-12-29. Score: 0

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...

> **Comments:**
> **username_3**: It would be helpful to cite a specific source for Diewert/Chambers in your answer for the user above.  
> **username_2**: 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.  
> **username_3**: Looks good. Put it in your answer!  
> **username_0**: 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'!  
> **username_0**: 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.  
> **username_2**: Ok, the proof can also be written this way round.  

---

