<p>A counterexample: <br>
Consider the production function <span class="math-container">$f(x_1,x_2) = x_1 + x_2$</span>, and assume input prices of 1. The long-run cost function is <span class="math-container">$C(y) = y$</span>. Assume that in the short run <span class="math-container">$x_2 = 5$</span>. Then the short-run cost function is <span class="math-container">$C_s(y) = \max(5,y)$</span>.
LRAC is 1 for all output levels, SRAC is <span class="math-container">$5/y$</span> for <span class="math-container">$y&lt;5$</span> and 1 for all other output levels. LRAC and SRAC coincide at all points <span class="math-container">$y\geq 5$</span>.</p>

CC BY-SA 4.0
2022-10-19T05:05:05.320Z
2022-10-19T05:05:05.320Z
Giskard
<p>Why does the LRAC (long-run average cost) curve intersect with the SRAC (short-run average cost) curve at exactly one point? I understand why there's at least one intersection (it's because, given an output quantity, we can find the optimal value of capital <span class="math-container">$k$</span> that minimizes the short-run total cost, and this is by definition the optimal LRTC for the given <span class="math-container">$y$</span>. Since <span class="math-container">$y$</span> is common to both, the point of intersection of the LRTC and SRTC is the point of intersection of the LRAC and the SRAC). I also know that <span class="math-container">$SRAC \text{ at } y \geq LRAC \text{ at } y$</span>. But I don't know why the tangency is at only one point.</p>

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2022-10-18T16:12:55.673Z
Hi! 1. Are you sure you mean to write "intersect" and not something like "share a point of tangency"? 2. Can you please support your claim with a reference? It does not seem to be generally true; but a reference would help by providing context (and show that you did research before posting).
Giskard
CC BY-SA 4.0
2022-10-18T19:00:56.910Z
@Giskard 1. Yes, it's a point of tangency as I mentioned in the last line (or by, "one point of intersect" and "SRAC $\geq$ LRAC", which together point to a tangency point). 2. [Varian, p. 406 (last paragraph)](http://candrafajriananda.lecture.ub.ac.id/files/2017/09/e-books-MICRO-INTERMDEDIATE-ed9-VARIAN.pdf). Another link: [Cost curve, Wikipedia](https://tinyurl.com/4ud82sjj). If you can please show me the counter-example, it would be even more enlightening (in the sense, I would know which assumptions I am making incorrectly).
Rick_Morty
CC BY-SA 4.0
2022-10-19T05:06:48.290Z
Your claim that there is always only one point of tangency is supported by this sentence from Varian: "*f the short-run cost is always greater than the long-run cost and they
are equal at one level of output*"? Notice that it starts with the word "if". Not sure where on the Wikipedia page you want me to look, I am not going to read the whole thing.
Giskard
CC BY-SA 4.0
2022-10-22T08:06:53.567Z
@Giskard It's a highlighted link. If you're using Chrome, it should direct you to the exact sentence. Otherwise, please see the second point [here](https://en.wikipedia.org/wiki/Cost_curve#Relationship_between_short-run_and_long-run_cost_curves).
Rick_Morty
<microeconomics><cost-functions>
Why does the LRAC (long-run average cost) curve intersect with the SRAC (short-run average cost) curve at exactly one point?
Rick_Morty
# Why does the LRAC (long-run average cost) curve intersect with the SRAC (short-run average cost) curve at exactly one point?

(Asked by: username_0 on 2022-10-18)

Why does the LRAC (long-run average cost) curve intersect with the SRAC (short-run average cost) curve at exactly one point? I understand why there's at least one intersection (it's because, given an output quantity, we can find the optimal value of capital $k$ that minimizes the short-run total cost, and this is by definition the optimal LRTC for the given $y$. Since $y$ is common to both, the point of intersection of the LRTC and SRTC is the point of intersection of the LRAC and the SRAC). I also know that $SRAC \text{ at } y \geq LRAC \text{ at } y$. But I don't know why the tangency is at only one point.

**Tags:** microeconomics, cost-functions
> **Comments:**
> **username_1**: Hi! 1. Are you sure you mean to write "intersect" and not something like "share a point of tangency"? 2. Can you please support your claim with a reference? It does not seem to be generally true; but a reference would help by providing context (and show that you did research before posting).  
> **username_0**: @Giskard 1. Yes, it's a point of tangency as I mentioned in the last line (or by, "one point of intersect" and "SRAC $\geq$ LRAC", which together point to a tangency point). 2. [Varian, p. 406 (last paragraph)](http://candrafajriananda.lecture.ub.ac.id/files/2017/09/e-books-MICRO-INTERMDEDIATE-ed9-VARIAN.pdf). Another link: [Cost curve, Wikipedia](https://tinyurl.com/4ud82sjj). If you can please show me the counter-example, it would be even more enlightening (in the sense, I would know which assumptions I am making incorrectly).  
> **username_1**: Your claim that there is always only one point of tangency is supported by this sentence from Varian: "*f the short-run cost is always greater than the long-run cost and they are equal at one level of output*"? Notice that it starts with the word "if". Not sure where on the Wikipedia page you want me to look, I am not going to read the whole thing.  
> **username_0**: @Giskard It's a highlighted link. If you're using Chrome, it should direct you to the exact sentence. Otherwise, please see the second point [here](https://en.wikipedia.org/wiki/Cost_curve#Relationship_between_short-run_and_long-run_cost_curves).  

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## 1 answers

### Answer by username_1 on 2022-10-19. Score: 1 | Accepted answer

A counterexample:   
Consider the production function $f(x\_1,x\_2) = x\_1 + x\_2$, and assume input prices of 1. The long-run cost function is $C(y) = y$. Assume that in the short run $x\_2 = 5$. Then the short-run cost function is $C\_s(y) = \max(5,y)$.
LRAC is 1 for all output levels, SRAC is $5/y$ for $y<5$ and 1 for all other output levels. LRAC and SRAC coincide at all points $y\geq 5$.


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