<p>As stated in the comments, there is no economic connection between a demand function and a production function.</p>
<p>The only connection between the two functional forms is that both of them are log-linear. In other words, the log of the demand/production function is linear in the logs of the independent variables.</p>
<p>This type of functional form is widely used in economics as it easily allows one to retrieve the elasticities as the coefficients of this linear function.</p>
<p>This follows from the identity that for any strictly positive function <span class="math-container">$f(p)$</span> and strictly positive variable <span class="math-container">$p_i$</span>,
<span class="math-container">$$
\varepsilon^f_p = \frac{\partial f}{\partial p_i} \frac{p_i}{f(p)} = \frac{\partial \ln(f(p)}{\partial \ln(p_i)}
$$</span>
So log linear functions are the ones where the elasticities can easily be read of by the coefficients. It's also the functional form for which the elasticities are constant (do not vary with the value of <span class="math-container">$p$</span>).</p>

CC BY-SA 4.0
2024-03-01T07:34:41.387Z
2024-03-01T07:34:41.387Z
tdm
<p>This may seem like a silly question... Well... I guess, because it really is... But I have just realized that the intuition behind both principles may not be mutually exclusive.</p>
<p>Consider we would like to estimate a cross-price elasticity of demand such as:</p>
<p><span class="math-container">$$\ln(x_i) = \beta_0 + \beta_{i,i} \ln (P_i) + \beta_{i,j} \sum_{j \neq i} \ln(P_j) + \epsilon$$</span></p>
<p>The marginal effects then correspond exactly to cross-price elasticity:</p>
<p><span class="math-container">$$\frac{d \ln(x_i)}{d \ln(P_j)} = \frac{dx/x}{dP_j /P_j}$$</span></p>
<p>Interestingly, this is also similar to estimation of cobb-douglas production function, defined as:</p>
<p><span class="math-container">$$ y = z_1^{\beta_1} \cdot z_2^{\beta_2} \cdot \ldots \cdot z_m^{\beta_m} \cdot \epsilon$$</span></p>
<p>Which could be logarithmized as:</p>
<p><span class="math-container">$$ \ln(y) = \beta_1 \ln(z_1) + \beta_2 \ln(z_2) + \dots + \beta_m \ln(z_m) + \ln(\epsilon) $$</span></p>
<p>This would imply that in the first model, we could view the demand as a result of production function (if <span class="math-container">$\beta_0$</span> is equal to 0):</p>
<p><span class="math-container">$$x_i = P_i^{\beta_{i,i}} \cdot \prod_{j \neq i} P_j^{\beta_{i,j}} \cdot e^{\epsilon}$$</span></p>
<ul>
<li><p><strong>Is this intuition correct?</strong></p>
</li>
<li><p><strong>Can we use it to something?</strong></p>
</li>
<li><p><strong>How would we interpret the elasticity of substitution of cross-price elasticity?</strong></p>
</li>
</ul>

CC BY-SA 4.0
2024-03-01T05:47:53.380Z
IMO there is no economic connection here, this is just superficial mathematical similarity. My really silly example: I have 2 hands with 5 fingers on each, $2\cdot5$. A year contains 52 weeks with 7 days in each, $52\cdot7$. Notice that the $\cdot$ binary operator appears in both! Can we use this for something? How would we interpret my fingers as days? (When I clip my nails short, are those like short winter days...?)
Giskard
CC BY-SA 4.0
2024-03-01T05:50:48.793Z
A good question is what is the reason that the same/similar functional forms are used in different cases of econometric estimation.
Giskard
<microeconomics><elasticity><cobb-douglas><cross-price-elasticity>
Is cross-price elasticity a Cobb-Douglas production function?
Athaeneus
# Is cross-price elasticity a Cobb-Douglas production function?

(Asked by: username_0 on 2024-02-29)

This may seem like a silly question... Well... I guess, because it really is... But I have just realized that the intuition behind both principles may not be mutually exclusive.

Consider we would like to estimate a cross-price elasticity of demand such as:

$$\ln(x\_i) = \beta\_0 + \beta\_{i,i} \ln (P\_i) + \beta\_{i,j} \sum\_{j \neq i} \ln(P\_j) + \epsilon$$

The marginal effects then correspond exactly to cross-price elasticity:

$$\frac{d \ln(x\_i)}{d \ln(P\_j)} = \frac{dx/x}{dP\_j /P\_j}$$

Interestingly, this is also similar to estimation of cobb-douglas production function, defined as:

$$ y = z\_1^{\beta\_1} \cdot z\_2^{\beta\_2} \cdot \ldots \cdot z\_m^{\beta\_m} \cdot \epsilon$$

Which could be logarithmized as:

$$ \ln(y) = \beta\_1 \ln(z\_1) + \beta\_2 \ln(z\_2) + \dots + \beta\_m \ln(z\_m) + \ln(\epsilon) $$

This would imply that in the first model, we could view the demand as a result of production function (if $\beta\_0$ is equal to 0):

$$x\_i = P\_i^{\beta\_{i,i}} \cdot \prod\_{j \neq i} P\_j^{\beta\_{i,j}} \cdot e^{\epsilon}$$

* **Is this intuition correct?**
* **Can we use it to something?**
* **How would we interpret the elasticity of substitution of cross-price elasticity?**

**Tags:** microeconomics, elasticity, cobb-douglas, cross-price-elasticity
> **Comments:**
> **username_1**: IMO there is no economic connection here, this is just superficial mathematical similarity. My really silly example: I have 2 hands with 5 fingers on each, $2\cdot5$. A year contains 52 weeks with 7 days in each, $52\cdot7$. Notice that the $\cdot$ binary operator appears in both! Can we use this for something? How would we interpret my fingers as days? (When I clip my nails short, are those like short winter days...?)  
> **username_1**: A good question is what is the reason that the same/similar functional forms are used in different cases of econometric estimation.  

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## 1 answers

### Answer by username_2 on 2024-03-01. Score: 4 | Accepted answer

As stated in the comments, there is no economic connection between a demand function and a production function.

The only connection between the two functional forms is that both of them are log-linear. In other words, the log of the demand/production function is linear in the logs of the independent variables.

This type of functional form is widely used in economics as it easily allows one to retrieve the elasticities as the coefficients of this linear function.

This follows from the identity that for any strictly positive function $f(p)$ and strictly positive variable $p\_i$,
$$
\varepsilon^f\_p = \frac{\partial f}{\partial p\_i} \frac{p\_i}{f(p)} = \frac{\partial \ln(f(p)}{\partial \ln(p\_i)}
$$
So log linear functions are the ones where the elasticities can easily be read of by the coefficients. It's also the functional form for which the elasticities are constant (do not vary with the value of $p$).


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