Elasticity Panel Data

Question

I have a dataset like this:

Fileds

1. Profit x Product: It is the variable that I can change, and is the independent variable.
2. %Sales: Number of sales/Customer
3. %Sales With subscription: Number of Sales with subscription/ Sales

Question:

1. I would like to make a model that captures the relation between %Sales and %Sales with Subscription, such that

$$ Y_2 + Y_1 = \alpha_1 + \alpha_2 X_1 + \epsilon $$

• $Y_1$ = %Sales with Subscription

• $Y_2$ = %Sales

• $X_1$ = Profit x Product

What kind of model I can use?

2. After obtaining the model, I would like to calculate the elasticity of the Profit x Product by group, like this.

How can make this analysis of elasticity?

I would like to make this analysis in R.

Answer 1

What kind of model I can use?

You can use a log-log model, to estimate elasticities: $$ \log(Y) = \beta_0 + \beta_1 \log(X) + \dots + \varepsilon $$ Just set $Y$ to your desired outcome variable (e.g. %Sales) and $X$ to Profit x Product. The coefficient $\beta_i$ then gives the (ceteris paribus) percentage change in $Y$ given a $1\%$ change in $X$.

Individual elasticities per group

If you want to estimate individual elasticities for each group you can add an interaction effect: $$ \log(Y) = \beta_0 + \beta_1 \log(X)\times group + \dots + \varepsilon $$ where $group$ is a dummy variable indicating the group membership.

R implementation

You can estimate this model via:

lm(log(Y) ~ log(X):group,
   data = data)