<p>I guess the discount holds for the 100 first units of food even if she buys more than 100. Therefore the new budget line is
\begin{equation*}
\left\{
   \begin{array}{ll}
   &amp; 200 = 0.2 F + 2C \text{ if } F &lt; 100 \\
   &amp; 200 = 0.2 \times 100 + F-100+2C \text{ otherwise }
   \end{array}
\right.
\end{equation*}
which is continuous.</p>

CC BY-SA 3.0
2016-02-16T09:03:24.210Z
2016-02-16T09:03:24.210Z
Oliv
<p>It's been awhile since I've taken introductory microeconomics. I read <a href="https://books.google.com/books?id=xeg8AAAAIAAJ&amp;pg=PA46&amp;lpg=PA46&amp;dq=should+budget+lines+be+continuous&amp;source=bl&amp;ots=vhO9gmehqA&amp;sig=kIXEcrjw99zsCW5CddRnEi5LzFM&amp;hl=en&amp;sa=X&amp;ved=0ahUKEwjbmLWy7fvKAhWC3KYKHQpAAEoQ6AEIGTAA#v=onepage&amp;q=should%20budget%20lines%20be%20continuous&amp;f=false" rel="nofollow noreferrer">budget lines can be discontinuous</a> awhile ago but am suspicious as to such a concept being discussed in intro micro classes particularly calling them budget 'lines'.</p>
<hr />
<p>The problem:</p>
<blockquote>
<p>Say Jill likes food and clothing. Jill has 200 to spend on food and clothing. One unit of food costs \$1 while one unit of clothing costs $2.</p>
</blockquote>
<p>Budget line is 200=F+2C</p>
<blockquote>
<p>Say Jill receives an 80% discount on food but it applies up to 100 units.</p>
</blockquote>
<p>New budget line is 200=0.2F+2C for 0-100 then same as old for 100 onward?</p>
<p>So it's discontinuous? So it's not really a budget 'line' right (though piecewise, it's a line, I guess).</p>

CC BY-SA 3.0
2016-02-16T08:59:21.077Z
What you described seems to be a kink in the line (not a discontinuity), unless you mean that if she does not receive any discount at all in case she buys more than 100 units.
HRSE
CC BY-SA 4.0
2021-09-14T04:21:15.933Z
@HRSE so piecewise differentiable but wholly continuous?
BCLC
<microeconomics><optimization>
Can budget lines be discontinuous?
BCLC
# Can budget lines be discontinuous?

(Asked by: username_0 on 2016-02-16)

It's been awhile since I've taken introductory microeconomics. I read [budget lines can be discontinuous](https://books.google.com/books?id=xeg8AAAAIAAJ&pg=PA46&lpg=PA46&dq=should+budget+lines+be+continuous&source=bl&ots=vhO9gmehqA&sig=kIXEcrjw99zsCW5CddRnEi5LzFM&hl=en&sa=X&ved=0ahUKEwjbmLWy7fvKAhWC3KYKHQpAAEoQ6AEIGTAA#v=onepage&q=should%20budget%20lines%20be%20continuous&f=false) awhile ago but am suspicious as to such a concept being discussed in intro micro classes particularly calling them budget 'lines'.

---

The problem:

> Say Jill likes food and clothing. Jill has 200 to spend on food and clothing. One unit of food costs \$1 while one unit of clothing costs $2.

Budget line is 200=F+2C

> Say Jill receives an 80% discount on food but it applies up to 100 units.

New budget line is 200=0.2F+2C for 0-100 then same as old for 100 onward?

So it's discontinuous? So it's not really a budget 'line' right (though piecewise, it's a line, I guess).

**Tags:** microeconomics, optimization

> **Comments:**
> **username_1**: What you described seems to be a kink in the line (not a discontinuity), unless you mean that if she does not receive any discount at all in case she buys more than 100 units.  
> **username_0**: @username_1 so piecewise differentiable but wholly continuous?  

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## 1 answers

### Answer by username_2 on 2016-02-16. Score: 2 | Accepted answer

I guess the discount holds for the 100 first units of food even if she buys more than 100. Therefore the new budget line is
\begin{equation\*}
\left\{
\begin{array}{ll}
& 200 = 0.2 F + 2C \text{ if } F < 100 \\
& 200 = 0.2 \times 100 + F-100+2C \text{ otherwise }
\end{array}
\right.
\end{equation\*}
which is continuous.


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