# Non-linear budget constraints written as union/intersection of linear budget constraints

Let $w$ denote wage. Let the price of each unit of food be $p_1$, the price of the first $M$ units of electricity be $p_2$ and the price of each unit of electricity over $M$ units is $q_2$. $M$ is fixed. My question is regarding how to express the (non-linear) budget constraint set $B_{p_1, p_2, q_2, w}$. I know one way is as follows:

$$B_{p_1, p_2, q_2, w}=\begin{cases}p_1x_1 + p_2x_2 \le w & \text{ if } x_2 \le M \\ p_1x_1 + p_2M + q_2(x_2-M) \le w & \text{ if } x_2>M \end{cases}$$

However, I am told that the budget set above can also be equivalently expressed as the union or intersection of linear budget sets. Namely, if $p_2<q_2$, then the budget set is

$$\{x \in \mathbb{R}^2: p_1x_1+p_2x_2 \le w\} \cap \{x \in \mathbb{R}^2: p_1x_1+q_2x_2 \le w+(q_2-p_2)M\}$$

If $q_2<p_2$, then the budget set is

$$\{x \in \mathbb{R}^2: p_1x_1+p_2x_2 \le w\} \cup \{x \in \mathbb{R}^2: p_1x_1+q_2x_2 \le w+(q_2-p_2)M\}$$

How is the union/intersection expression derived? I can not see the connection between my expression and the above one at all.

• Drawing the two different budget sets might help you understand what is happening. If electricity is getting more expensive after a certain point, then the budget set is just like if the price is the same, but it "bends in" once $x_2 > M$. You can think of the analogous case for if electricity becomes cheaper after a certain amount of it is bought. – Kitsune Cavalry Sep 14 '17 at 3:16
• Yes, it certainly makes some sense after I drew it, however, how can I prove it algebraically, rather than graphically? – elbarto Sep 14 '17 at 7:20