# Substitution effect when price of both goods change by the same percentage

I am trying to grasp the concept of income- and substitution effects.

The way, I've understood it, the decomposition bundle is found, at the original indifference curve, where the slope equals that of the new budget line.

Does this mean that if the price of both goods (assuming a simple model of 2 goods) change by the same percentage, the substitution effect is zero? As the slope of the budget line will remain unchanged.

This seems to intuitively make sense to me, but I would greatly appreciate a 100% confirmation, that this is correct.

Thank you!

Consider the budget line with 2 goods: $$$$p_xx+p_yy=m \quad\text{or}\quad y=\frac{m}{p_y}-\frac{p_x}{p_y}x,$$$$ where the relative price is $$\frac{p_x}{p_y}$$.
Suppose now both goods become $$10\%$$ more expensive (e.g. due to a $$10\%$$ tax on them). Then the budget line becomes $$$$1.1p_xx+1.1p_yy=m \quad\Leftrightarrow\quad p_xx+p_yy=\frac{m}{1.1} \quad\text{or}\quad y=\frac{m}{1.1p_y}-\frac{1.1p_x}{1.1p_y}x=\frac{m}{1.1p_y}-\frac{p_x}{p_y}x$$$$ where the relative price is still $$\frac{p_x}{p_y}$$. From the equations, we see that a uniform percentage change in prices produces a pure income effect ($$m$$ drops to $$m/1.1$$), but no substitution effect, since the relative price remain the same after the change.