# Need some OLS model help

I need some help with calculating a OLS regression model.

The model is $sales = b_{0} + quantity^{b_{1}}$, and the objective is to find the own-price elasticity calculated at the means, $\frac{\delta \, quantity}{\delta \, price} \times \frac{price}{quantity}$.

Where $sales = price \times quantity$.

So, as far as I understand, I have to model a OLS with quantity as the dependent variable, and price as the independent variable, log transformed in some fashion to get the $b_{1}$ estimator down. But any way I try to manipulate the formula, I get a very ugly expression, and I'm not sure how I would implement it in R. Anyone got some advice?

• Are you trying to estimate a demand curve (i.e. how consumer demand for the product varies as a function of price)? Or are you trying to estimate a supply curve (i.e. how change price lead firms to produce more or less)? Commented Oct 4, 2016 at 21:20
• Your model supposes there is some floor on sales equal to $b_0$, regardless of quantity. Is that really plausible? And to find the price elasticity wouldn't you want to model quantity vs price rather than sales vs quantity?
– whuber
Commented Oct 4, 2016 at 21:30

This is more "thinking out loud" than any kind of firm guidance, but it might perhaps be of some help.

let $y=\text{price}$, and $x=\text{quantity}$. Then your model says something like

$y \cdot x = \beta_0+x^\beta_1 +\eta$

but nothing was stated about $\eta$. I assume that for a multiplicative-type variable likes sales the variance would be roughly proportional to the mean (and your mention of logs encourages me in that line of thinking). In that case:

Dividing through by $x$

$y = \beta_0/x+x^{\beta_1-1}+ \eta/x = \beta_0 x_0+ x^{\beta_1^*} + \epsilon$

Now if on this scale you can regard the variance as nearly constant, you could fit the equation via nonlinear least squares. Alternatively you could do some form of variance modelling, re-weighting via IRLS, or variance-adjustment if necessary; I'm not sure what would be seen as suitable for what you're doing (but it may not be needed if the variance specification it broadly reasonable).

now $\frac{\partial y}{\partial x} = -\beta_0/x^2+ (\beta_1-1) x^{\beta_1-2}$

and so the reciprocal of own price elasticity would be

$\frac{x}{y} (-\beta_0/x^2+ (\beta_1-1) x^{\beta_1-2})$

$=\frac{1}{y} (-\beta_0/x+ (\beta_1-1) x^{\beta_1-1})$

and hence I think the elasticity you want would be

$\frac{y}{-\beta_0/x+ (\beta_1-1) x^{\beta_1-1}}$

or

$\frac{y\times x}{-\beta_0 + (\beta_1-1) x^{\beta_1}}$

... which you can evaluate at the mean or whatever as needed.