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:
- How does "price effect" relate to the concept of "elasticity"?
- what is the difference between the interpretation coefficient $b_2$ and $c_2$?
