# Are there any situations where the elasticity version of the Slutsky equation can only be used compared with the regular Slutsky equation?

Regular Slutsky Equation: $$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x} - \frac{\partial x_M}{\partial m} x _M$$ Elasticity Slutsky Equation: \begin{align*}\varepsilon_{x,p_x}^M &= \varepsilon_{x,p_x}^H - \varepsilon_{x,m}^M s_x \\&=- s_y \sigma- \varepsilon_{x,m}^M s_x \end{align*} Note: $\sigma$ is the elasticity of substitution. Informal proof $\varepsilon_{x,p_x}^H = -s_y \sigma$.

I have always preferred to use the regular Slutsky equation. But as I was preparing for my midterm last week, I came across the following problem:

Margarine is an inferior good, yet consumption of margarine has increased over the last several decades as has real per capita income. Can you explain the apparent contradiction?

My study buddy and I struggled and struggled and eventually decided the elasticity version of the Slutsky equation was a better way to think about the problem. It was the first example I encountered where the regular Slutsky equation didn't seem sufficient and that I really needed the elasticity version.

My Question:

Are there any (other) situations where the elasticity version of the Slutsky equation can only be used compared with the regular Slutsky equation?

• Is the elasticity-version required here? I'm not so sure! We know the direction of the income effect (as we're told it's an inferior good) so it's easy to argue the only explanation is that the substitution effect has dominated? May 11, 2015 at 8:51
• I have no idea tbh. If you or anyone can disprove or rebut what I said, awesome. May 11, 2015 at 8:54
• The two relationships are equivalent. We can easily go from one to the other, so the answer is no - there isn't any situation where we can't. May 12, 2015 at 9:03