<p>The rigorous definition of demand says that it is a correspondence (i.e., set-valued mapping), and need not be a function. In your example, the demand for perfect substitutes is a correspondence. If you want to restrict it to be a function, then yes, you have to pick some value for the equality case (but you need to always pick the same value for the same function, e.g., one demand function is piecewise linear with <span class="math-container">$x_1(p,m) = 0$</span> if <span class="math-container">$p_1/p_2 \geq a/b$</span>; <span class="math-container">$x_1(p,m) = m/p_1$</span> if <span class="math-container">$p_1/p_2 &lt; a/b$</span>.</p>

CC BY-SA 4.0
2024-07-26T01:45:03.233Z
2024-07-26T01:45:03.233Z
DiZ
<p>For perfect substitutes preferences <span class="math-container">$U(x_{1}, x_{2}) = ax_{1} + bx_{2}$</span> and income <span class="math-container">$m$</span>, the demand curve for <span class="math-container">$x_{1}$</span> is:</p>
<p><span class="math-container">\begin{equation}
x_{1} = \begin{cases}
0 &amp; \text{if } \frac{p_{1}}{p_{2}} &gt; \frac{a}{b} \\
{[0, \frac{m}{p_{1}}}] &amp; \text{if } \frac{p_{1}}{p_{2}} = \frac{a}{b} \\
\frac{m}{p_{1}} &amp; \text{if } \frac{p_{1}}{p_{2}} &lt; \frac{a}{b}
\end{cases}
\end{equation}</span></p>
<p>For the equality case (<span class="math-container">$\frac{p_{1}}{p_{2}} = \frac{a}{b}$</span>), shouldn't we choose one endpoint of <span class="math-container">$[0, \frac{m}{p_{1}}]$</span> so that the function <span class="math-container">$x_{1}(p_{1})$</span> is well-defined and passes the vertical line test? Is it arbitrary which endpoint we use?</p>

<demand><perfect-substitutes>
Discontinuity in perfect substitutes demand function
Macaulay
# Discontinuity in perfect substitutes demand function

(Asked by: username_0 on 2024-07-25)

For perfect substitutes preferences $U(x\_{1}, x\_{2}) = ax\_{1} + bx\_{2}$ and income $m$, the demand curve for $x\_{1}$ is:

\begin{equation}
x\_{1} = \begin{cases}
0 & \text{if } \frac{p\_{1}}{p\_{2}} > \frac{a}{b} \\
{[0, \frac{m}{p\_{1}}}] & \text{if } \frac{p\_{1}}{p\_{2}} = \frac{a}{b} \\
\frac{m}{p\_{1}} & \text{if } \frac{p\_{1}}{p\_{2}} < \frac{a}{b}
\end{cases}
\end{equation}

For the equality case ($\frac{p\_{1}}{p\_{2}} = \frac{a}{b}$), shouldn't we choose one endpoint of $[0, \frac{m}{p\_{1}}]$ so that the function $x\_{1}(p\_{1})$ is well-defined and passes the vertical line test? Is it arbitrary which endpoint we use?

**Tags:** demand, perfect-substitutes


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## 1 answers

### Answer by username_1 on 2024-07-26. Score: 0

The rigorous definition of demand says that it is a correspondence (i.e., set-valued mapping), and need not be a function. In your example, the demand for perfect substitutes is a correspondence. If you want to restrict it to be a function, then yes, you have to pick some value for the equality case (but you need to always pick the same value for the same function, e.g., one demand function is piecewise linear with $x\_1(p,m) = 0$ if $p\_1/p\_2 \geq a/b$; $x\_1(p,m) = m/p\_1$ if $p\_1/p\_2 < a/b$.


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