# Calculating income and substitution effects

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!