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Suppose I have two different models to predict the sold quantity (Q) of apples. For simplicity, price (P) is the only explainatory variable in the model.

I assume that the following holds true for all models. I define here Y as the dependent variable and $X_i$ with $i=1...n$ as the explainatory variable.

$Y = a_1 + a_2X_1 + ... + a_nX_n + e$

Except for the intercept, I can interpretate the coefficients $a_2....a_n$ as $a_i = \frac{\partial Y}{\partial X_i}$. In words, the change in the dependent variable given a change in the explainatory variable, given that the other variables are kept constant. The latter implies the ceteris paribus condition.

Model 1

$log(Q) = b_1 + b_2*log(P)$

The interpretation of $b_2$ under the above explaination of the partial derivative will result in $\frac{\frac{\mathrm{d} Q}{Q}}{\frac{\mathrm{d} P}{P}}$, which is the expression of the elasticity of demand. I excluded the derivation for brievity.

Model 2

The second model is

$Q = c_1 + c_2*P$

For the interpretation of $c_2$, which is also a partial derivative. I use an example: If the price of apples increases by a euro, the sold quantity will be $c_2 * 1$ more/less, dependent on the sign of the coefficient. In my course $c_2$ is called the "price effect"

Question

The bottleneck is relating the "price effect" to the concept of "elasticity". I think I understand what elasticity means, oversimplified: If the price changes with x%, the quantity sold changes y%. Although coefficent $c_2$ is not an elasticity, I can still formulate a simillar phrase. In other words, what is the difference between the interpretation coefficient $b_2$ and $c_2$.

I hope I provided enough context. In short:

  1. How does "price effect" relate to the concept of "elasticity"?
  2. what is the difference between the interpretation coefficient $b_2$ and $c_2$?
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  • $\begingroup$ Your two models are different models; in one, the relationship between price and quantity is (modelled as) linear, in the other, exponential. This means that the effect of price changes on quantity will not be the same in the two models. You give the definition of the two coefficients; not sure what else there is to write? Could you perhaps explain what kind of information you are looking for, what type of answer you are expecting? $\endgroup$
    – Giskard
    Jan 25, 2023 at 14:29
  • $\begingroup$ @Giskard Thanks for your reponse, I agree that it is a bit vague what I'm specifically looking for. I see mathematcally that the effect changes on quantity are not the same, however I find it hard to make economic sense from it. The ideal answer would contain an overview of the differences and similarities between "elasticity" and "price effect" from an economical standpoint. $\endgroup$
    – Tim
    Jan 25, 2023 at 15:17
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    $\begingroup$ From an "economical standpoint", both of these things are numbers, but they do not measure the same thing. They measure the two different things that you describe they measure; i.e.: an elasticity of $b_2$% means that if the price changes with 1%, the quantity sold changes $b_2$% (approx.), while a "price effect" of $c_2$ means that if the price of the good increases by 1, the sold quantity will be higher by $c_2$. $\endgroup$
    – Giskard
    Jan 25, 2023 at 18:50

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