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

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?

• 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...?) Mar 1 at 5:47
• A good question is what is the reason that the same/similar functional forms are used in different cases of econometric estimation. Mar 1 at 5:50

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 $$p$$).