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Let's say you have a Cobb-Douglas utility function, U = x^.1*y^.9
This will result in Marshallian demand functions x* = .1I / Px, and y* = .9I / Py.
If you perform a Slutsky decomposition for x* as Px changes; the income effect will be 10% of the total effect.
Can someone help me understand why the effect exactly matches the percent of income spent on the good?
The income effect is given by: $$ x^\ast \frac{\partial x}{\partial I} = \frac{0.1 I}{P_x} \frac{0.1}{P_x} = 0.1\left(0.1\frac{I}{(P_x)^2}\right). $$ The Price effect is: $$ \frac{\partial x}{\partial P_x} = -0.1 \frac{I}{(P_x)^2}.$$ The first is 10% of the second (in absolute terms).
Another way to see this, is to start from the fact that for the Cobb-Douglass, the expenditure share is constant: $$ \frac{p x(p,I)}{I} = \alpha. $$ Taking derivative with respect to $p$ gives: $$ \begin{align*} &\frac{x(p,I) + p \frac{\partial x}{\partial p}}{I} = 0,\\ \to &\frac{\partial x}{\partial p} = -\frac{x(p,I)}{p} \end{align*} $$ Taking the partial derivative with respect to $I$ gives: $$ \begin{align*} &\frac{p \frac{\partial x(p,I)}{\partial I} I - p x(p,I)}{I^2} = 0,\\ \to &\frac{\partial x(p,I)}{\partial I} = \frac{p x(p,I)}{pI} = \frac{\alpha}{p} \end{align*} $$ So the income effect is: $$ \frac{\partial x(p,I)}{\partial I} x(p,I) = \alpha \frac{x(p,I)}{p} $$ This is $\alpha$ times the absolute value of the price effect.
