Calculating Elasticity From Regression Equations with Different Functional Forms
In this video we're gonna look at calculating elasticity from regression results with different functional forms. So some background what is the elasticity it's a percentage change in one variable say Y divided by the percentage change in another variable to call that X using calculus notation. The percentage change in. Y is represented by this dy. Here is just a change in Y divided by Y DX is just change in X divided by X if we were to multiply the denominator and numerator by the reciprocal or X divided by DX. You have the classic formula for elasticity so example one we have a regular linear regression dependent variable is y independent variable ax so we want to calculate the elasticity from this estimated regression a and B our coefficients a is a constant B is the slope coefficient on X. So first thing we do to build this elasticity we're gonna get dy / DX just a derivative of the equation with respect to good respect to ax it's just B next we're going to multiply both sides by X divided by Y. That'll put us into this elasticity format so doing that. We have the estimated elasticity of Y with respect to variable ax and in a regression context. It's common to use the mean values in your data set for ax and y you could choose any values but it's common to use the mean values for x and y to evaluate this elasticity so here is a numerical example. We estimate a linear regression. Q's Quine demanded P is the price we just saw in the last screen based on this functional form the elasticity will be the coefficient x variable ax / variable. Y in this case variable ax is the price and variable. Y the dependent variable is Q. So what is B it's minus 4 and once again we can adjust evaluate the P and Q terms at their mean values. Here's a different different functional form a double log form that dependent variable is the natural logs. The independent variable is also natural logs. We wanted to solve for the elasticity so the first thing. I'm going to do is take the. I'm going to.
Just totally differentiate this equation. The derivative of the natural log of Y is just 1 over Y and we're taking the total differential so have dy here and the derivative of the natural log of X is just 1 over ax but that's being multiplied by B so the B is in the numerator and again it's a total differential so we're going to multiply that through by the change in X or DX let's divide everything through now by DX so doing that our left hand side now looks like this and now we want to multiply everything through by ax. And if we do that you'll notice the left-hand side is our typical elasticity expression and given this functional form the elasticity will just equal B so applying that to some numerical results. We have a double log functional form. Estimating a demand equation based on what we saw. The elasticity of demand here is just going to be minus 2 point 5 5 moving on to our last functional form example 3 a semi-log form where the dependent variables and natural logs. But that's it so once again. Let's take the total differential of this equation so 1 over. Y times a change in y equals the derivative of B ax is just be multiplied by the change in X let's divide everything through by the change in X or DX. So doing that. We're left with this result. Now let's multiply everything through by. Y so the derivative of Y with respect to X given this functional form is just B times y now let's put it in elasticity format by multiplying both sides by X divided by Y. So you'll notice. The left hand. Side is now in our standard elasticity format and again we multiplied both sides through by X divided by Y so that's X divided by Y here notice the. Y's cancel and you're left just with B times X. So that's the elasticity given this functional form it's just the coefficient B times ax and here once again we can use the mean value for ax to evaluate this elasticity so a numerical example. You're the semi-log estimated demand equation. Looks like this using our result from the last slide we've got the parameter B down in front and that's just going to be multiplied by the independent variable which is P and we can evaluate P at its mean value or any other value if we choose so I hope you found this video helpful.
Just totally differentiate this equation. The derivative of the natural log of Y is just 1 over Y and we're taking the total differential so have dy here and the derivative of the natural log of X is just 1 over ax but that's being multiplied by B so the B is in the numerator and again it's a total differential so we're going to multiply that through by the change in X or DX let's divide everything through now by DX so doing that our left hand side now looks like this and now we want to multiply everything through by ax. And if we do that you'll notice the left-hand side is our typical elasticity expression and given this functional form the elasticity will just equal B so applying that to some numerical results. We have a double log functional form. Estimating a demand equation based on what we saw. The elasticity of demand here is just going to be minus 2 point 5 5 moving on to our last functional form example 3 a semi-log form where the dependent variables and natural logs. But that's it so once again. Let's take the total differential of this equation so 1 over. Y times a change in y equals the derivative of B ax is just be multiplied by the change in X let's divide everything through by the change in X or DX. So doing that. We're left with this result. Now let's multiply everything through by. Y so the derivative of Y with respect to X given this functional form is just B times y now let's put it in elasticity format by multiplying both sides by X divided by Y. So you'll notice. The left hand. Side is now in our standard elasticity format and again we multiplied both sides through by X divided by Y so that's X divided by Y here notice the. Y's cancel and you're left just with B times X. So that's the elasticity given this functional form it's just the coefficient B times ax and here once again we can use the mean value for ax to evaluate this elasticity so a numerical example. You're the semi-log estimated demand equation. Looks like this using our result from the last slide we've got the parameter B down in front and that's just going to be multiplied by the independent variable which is P and we can evaluate P at its mean value or any other value if we choose so I hope you found this video helpful.