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When specifying a production function for regression, it is well known that one of the features of using a log-log model is that the estimated coefficients are the output elasticities w.r.t. their respective independent variables.

My question is does it then follow that if one regresses log(production) on log(price), the coefficient on log(price) will be the demand elasticity?

That is, if we specify the production model as follows:

$$\ln q=\alpha_0 + \alpha_p \ln p$$

(where q is quantity of output and p is output price)

and then differentiate w.r.t. $\ln p$

$$\frac{d \ln q}{d \ln p}=\alpha_p$$

then isn't this the demand elasticity?

If it is, then does the omission of other important variables in the production function bias the elasticity (if firms are not homogenous in those variables)?

Thanks

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2 Answers 2

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Unfortunately, the problem you will run into here is one of endogeneity. The observed p and q are market equilibrium prices and quantities, not "samples" of points along a demand curve. While occasionally you may get lucky, usually if you naively regress quantity on price, you will end up with a positive coefficient. According to economic theory, this doesn't make sense (unless you've actually stumbled upon a Giffen good), but what one is typically observes are different levels of demand, and prices adjusting accordingly.

Therefore, to estimate accurate demand coefficients on price, it is typical to either estimate a system of equations (much as one solves a simple supply-demand problem in intro-micro), or to use instrumental variables. In either case, for identification, it requires some variables that affect only the supply side of the equation.

And finally, if you've solved all that such that you can overcome the endogeneity issues with your regression, remember that while taking logarithms is often a convenient way to do regressions, it implies a fundamental distinction about the model and the error terms, and if that doesn't fit the actual process generating the data, the "elasticity" you get will be the right units, but won't be the right number.

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  • $\begingroup$ Thanks. I am in fact regressing a system of two equations, but they are cost and revenue. Secondly, although I cannot interpret $\alpha_p$ as demand elasticity, it seems I should at least be able to interpret it as the firm's output elasticity of price. Would you agree? $\endgroup$
    – ben
    Mar 9, 2012 at 16:35
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    $\begingroup$ It all depends on what varies in your data. Consider the case of linear supply and demand. You observe the equilibrium price and quantity. Now, suppose the supply curve shifts. You observe another point. Connect the two observations, and you have traced the demand curve. What if instead, the demand curve had shifted? Connect the two observations and you have the supply curve. If both curves shift by an equal amount, you will have a horizontal line that is neither the supply curve, nor the demand curve. $\endgroup$
    – prototoast
    Mar 9, 2012 at 18:04
  • $\begingroup$ Wow... Thanks! Do you have any links with examples of a process of estimating the demand elasticity? $\endgroup$
    – moldovean
    Mar 22, 2012 at 19:58
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I don't think you can interpret $\alpha$ as the price elasticity of demand unless you are willing to assume that supply is perfectly price inelastic. Take a look at some really good IO lecture notes if you want to proceed this way. You will need to instrument the price variable using supply side variables that do not effect demand directly. This will give you variation in supply that leaves demand curve unchanged. Input prices might good candidates.

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  • $\begingroup$ How about if you're working with insurance data where the rates vary not as a function of time but an algorithm and then modelling "conversion" of quotes as the demand function. Would you be able to interpret elasticity from this kind of model? Because the time window is much more narrow than aggregate data, I feel supply shifts would be less of a concern. $\endgroup$ Jun 17, 2021 at 17:12
  • $\begingroup$ @NickCorona I think it depends on what’s driving the variation in rates and how transparent the algo and its inputs are to you. I can easily cook up a toy example where this goes awry, but the devil is in the details here. $\endgroup$
    – dimitriy
    Jun 17, 2021 at 17:22
  • $\begingroup$ The variation would be driven by the rating algorithm. So, for example we'd have a generalized additive model that calculates what are called 'pure premiums' (their cost to the insurer) for each customer and we'd have access to all the features used. $\endgroup$ Jun 18, 2021 at 13:32

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