Is cross-price elasticity a Cobb-Douglas production function?

Question

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?

Answer 1

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 $p$).