<p>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:</p>

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

<p>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 $p_2&lt;q_2$, then the budget set is </p>

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

<p>If $q_2&lt;p_2$, then the budget set is </p>

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

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

CC BY-SA 3.0
2017-09-14T03:16:55.540Z
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
CC BY-SA 3.0
2017-09-14T07:20:27.717Z
Yes, it certainly makes some sense after I drew it, however, how can I prove it algebraically, rather than graphically?
elbarto
<microeconomics><consumer-theory>
Non-linear budget constraints written as union/intersection of linear budget constraints
elbarto
# Non-linear budget constraints written as union/intersection of linear budget constraints

(Asked by: username_0 on 2017-09-13)

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

$$\{x \in \mathbb{R}^2: p\_1x\_1+p\_2x\_2 \le w\} \cap \{x \in \mathbb{R}^2: p\_1x\_1+q\_2x\_2 \le w+(q\_2-p\_2)M\}$$

If $q\_2

$$\{x \in \mathbb{R}^2: p\_1x\_1+p\_2x\_2 \le w\} \cup \{x \in \mathbb{R}^2: p\_1x\_1+q\_2x\_2 \le w+(q\_2-p\_2)M\}$$

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

**Tags:** microeconomics, consumer-theory

> **Comments:**
> **username_1**: 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.  
> **username_0**: Yes, it certainly makes some sense after I drew it, however, how can I prove it algebraically, rather than graphically?  

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## 0 answers

