Consider a simple quasi-linear utility function of the form $U(x,y)=x +ln(y)$. For this problem, assume that you have “enough” income, so that the optimal consumption bundle is where: $x,y >> 0$.

I was wondering how to calculate income and substitution effects. I've started by using Lagrange function to derive that $y=\frac{P_x}{P_y}$, (price of $x$ over the price of $y$) and $x=\frac{M}{P_x}-1$ ($M$ is income). What do I do now to find these two effects?


First of all: In general, for preferences that are quasi-linear in $x(y)$ it holds true that:

Overall effect = Substitution effect, if in the initial situation both goods or if only good $y(x)$ are consumed.

Overall effect = Income effect, if in the initial situation only good $x(y)$ is consumed.

Substitution effect= $x( p(x)', m') - x( p(x), m)$

Income effect= $x(p(x)', m) - x( p(x)', m') $

If you need a detailed step for step calculation, feel free to ask for it!

Have a good day!

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