<p><em>Short-run</em> implies that some decision variable cannot be free set, it is fixed for a time (in the short-run). In the long-run all variables may be freely set. Let us denote the fixed variable by <span class="math-container">$x$</span>. This can take different values, so there is a family of short run cost functions is <span class="math-container">$SC(x,q)$</span>.</p>
<p>Given a value <span class="math-container">$q$</span> let us denote the cost minimizing value of <span class="math-container">$x$</span> by <span class="math-container">$x^*$</span>, i.e.
<span class="math-container">$$
x^* = \arg\min_x SC(x,q). 
$$</span>
Cost minimization implies that in the long-run this will be the chosen value of <span class="math-container">$x$</span>, thus
<span class="math-container">$$
C(q) = SC(x^*,q) = \min_x SC(x,q).
$$</span>
Dividing by <span class="math-container">$q$</span> we get
<span class="math-container">$$
AC(q) = SAC(x^*,q) = \min_x SAC(x,q).
$$</span>
The above holds for any <span class="math-container">$q$</span>.</p>

CC BY-SA 4.0
2021-05-08T15:47:47.200Z
Thanks for the answer, but your final conclusion doesn't agree with other parts of the graph right? At other places, the SAC's minima is not tangent to the AC.
reasonStore
CC BY-SA 4.0
2021-05-08T16:36:21.007Z
I am not sure what you mean, as I don't think I wrote anything about this, but note that the graph displays minima with respect to $q$, not $x$. (Your graph denotes variable $x$ by $k$). That is the lowest point of curve $SAC(k_0)$ on the graph has height $\min_q SAC(k_0,q)$ and not $\min_{k_0} SAC(k_0,q)$.
Giskard
CC BY-SA 4.0
2021-05-08T13:27:06.183Z
2021-05-08T13:27:06.183Z
Giskard
<p>@Giskard and @tdm have given you perfectly valid responses. I am not convinced that your question is perfectly clear.</p>
<p>Are you asking why only at <span class="math-container">$q_1$</span> (in your graph) an SRAC and AC are tangent at the min of an SRAC?  If so, that's a result of the underlying assumptions about the production functions that lead to your graph.</p>
<p>In short, you have economies of scale (i.e., AC has a negative slope), a minimum efficient scale at <span class="math-container">$q_1$</span> (i.e., AC's slope is 0 at a single point), and diseconomies of scale (i.e., AC has a positive slope). SRACs have slope 0 at their minimum, while AC has slope 0 only at <span class="math-container">$q_1$</span> (at its minimum efficient scale). Hence, only at <span class="math-container">$q_1$</span> can the min of AC be tangent with the min of an SRAC.</p>
<p>Now had you made different assumptions about the production function the situation would be different. AC is the lower envelope of your SRACs. So, for example, if your minimum efficient scale was a range (instead of a point), or if your production function had constant returns to scale, then you would have infinite number of tangent points at min SRAC and min AC.</p>
<p>See:</p>
<ol>
<li><p><a href="https://www.economics.utoronto.ca/osborne/2x3/tutorial/C2LSFRM.HTM" rel="nofollow noreferrer">https://www.economics.utoronto.ca/osborne/2x3/tutorial/C2LSFRM.HTM</a></p>
</li>
<li><p>Varian (1992): Microeconomic Analysis p.p.71-72.</p>
</li>
<li><p><a href="https://uh.edu/%7Eghong/fina3334/chap_07.doc" rel="nofollow noreferrer">https://uh.edu/~ghong/fina3334/chap_07.doc</a></p>
</li>
</ol>
<p>Edit: fixed username</p>

CC BY-SA 4.0
2021-05-08T20:40:25.010Z
2021-05-08T20:40:25.010Z
Evangelos Con
<p>This result comes from the &quot;zero-profit&quot; condition used to define long run equilibrium in this model.</p>
<p>Recall the fact that in the &quot;firm&quot; part of the model, the marginal cost curve intersects with the average total cost curve at the minimum of the average total cost curve. Since marginal cost is the individual firm's supply curve, it follows that when price equals the minimum of total cost, the firm produces the quantity at the crossing point and has zero profit. A higher price will give the firm positive profit since P &gt; ATC, so Revenues &gt; Costs. A lower price will give the firm negative profit since P &lt; ATC, so Revenues &lt; Costs.</p>
<p>Long run equilibrium is defined by the number of firms being at the level that gives each firm zero profit. Thus the long run supply curve must be at the zero-profit price level, i.e. the minimum of average total cost.</p>

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2021-05-12T07:52:07.863Z
(-1) This is not true. The result holds even without assuming prices or perfect competition.
Giskard
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2021-05-12T10:26:54.103Z
Oh!  That was not clear from the poster's original question, and the post was tagged "competitive equilibrium" so I interpreted it in that context.  Could you point to the part of the question that tells you that this information?
Jonathan
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2021-05-12T10:27:35.187Z
"*Could you point to the part of the question that tells you that this information?*" What do you mean?
Giskard
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2021-05-12T10:28:10.343Z
You are right about competitive equilibrium being an irrelevant tag, I removed it.
Giskard
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2021-05-12T10:29:00.960Z
OK.  Is the setting you are considering basically a single firm where capital is flexible in the long run but fixed in the short run?
Jonathan
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2021-05-12T10:29:41.653Z
Single firm or industry, it does not matter, capital or any other input(s), it does not matter, "*flexible in the long run but fixed in the short run*" yes.
Giskard
CC BY-SA 4.0
2021-05-12T10:38:13.050Z
OK.  In that case I agree with your answer above.  I think you are overstating the case a little bit when you say "it does not matter" because I can imagine (noncompetitive) settings where the firm's long run choice of k will not be the solution to the cost minimization problem (E.g. a natural monopoly where demand < q_1).  But assuming cost minimization, I agree with your answer.
Jonathan
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2021-05-12T10:39:01.373Z
Why would a natural monopoly (or any firm) not minimize costs, why is this an unusual assumption?
Giskard
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2021-05-12T10:39:46.960Z
The OP also asks something about the SRAC curve being tangent to the LRAC at the LRAC point, but other SRAC curves not being tangent to the LRAC. I think this follows from the envelope theorem applied to the cost minimization problem, you might edit your answer to add that in!
Jonathan
CC BY-SA 4.0
2021-05-12T10:41:20.913Z
Woah, woah, woah, so now you take cost minimization for granted? And didn't you say before this follows from the long-run equilibrium? What about these things?
Giskard
CC BY-SA 4.0
2021-05-12T10:41:49.920Z
I won't edit my answer, I feel it gives an exact reasoning. Feel free to write an answer based on the envelope theorem, if it is valid I will vote for it!
Giskard
CC BY-SA 4.0
2021-05-12T10:45:37.973Z
I am not going to argue with you about this, you corrected me about the setting and so I accepted that you are correct and changed my position in response.  Your answer above uses cost minimization, and I agree with it. Sorry if that was unclear.
Jonathan
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2021-05-12T10:46:08.043Z
Comp. eq. assumes cost minimization, so let us just go with that.
Giskard
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2021-05-12T01:42:11.593Z
2021-05-12T01:42:11.593Z
Jonathan
<p>At the lowest point of long run average cost curve AC, the SAC is also at its minimum and that is not the case with any other SAC curve bounded by the envelope. Why is that?</p>
<p><a href="https://i.sstatic.net/sNKnP.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/sNKnP.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2021-05-08T09:27:44.660Z
Does this answer your question? [Why is the short run average cost curve not a tangent to the long run average cost curve at the lowest point on the short run average cost curve?](https://economics.stackexchange.com/questions/14900/why-is-the-short-run-average-cost-curve-not-a-tangent-to-the-long-run-average-co)
tdm
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2021-05-08T10:30:46.843Z
NO @tdm ! It's completely different.
reasonStore
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2021-05-08T10:31:21.057Z
@tdm I know that they are tangnet, but they are tangent at their minimas at the minima of LRAC
reasonStore
CC BY-SA 4.0
2021-05-08T17:36:17.127Z
At the other points, say $q_0$, the $SMC(k_0)$ is below the $SAC(k_0)$. This means the $SAC(k_0)$ is decreasing at that point. So the tangent point can not be at the minimum.
tdm
<microeconomics><cost><cost-functions>
Why min AC = min SRAC at the minima of AC curve?
reasonStore
# Why min AC = min SRAC at the minima of AC curve?

(Asked by: username_0 on 2021-05-08)

At the lowest point of long run average cost curve AC, the SAC is also at its minimum and that is not the case with any other SAC curve bounded by the envelope. Why is that?

[![enter image description here](https://i.sstatic.net/sNKnP.png)](https://i.sstatic.net/sNKnP.png)

**Tags:** microeconomics, cost, cost-functions
> **Comments:**
> **username_1**: Does this answer your question? [Why is the short run average cost curve not a tangent to the long run average cost curve at the lowest point on the short run average cost curve?](https://economics.stackexchange.com/questions/14900/why-is-the-short-run-average-cost-curve-not-a-tangent-to-the-long-run-average-co)  
> **username_0**: NO @tdm ! It's completely different.  
> **username_0**: @tdm I know that they are tangnet, but they are tangent at their minimas at the minima of LRAC  
> **username_1**: At the other points, say $q_0$, the $SMC(k_0)$ is below the $SAC(k_0)$. This means the $SAC(k_0)$ is decreasing at that point. So the tangent point can not be at the minimum.  

---

## 3 answers

### Answer by username_2 on 2021-05-08. Score: 3

*Short-run* implies that some decision variable cannot be free set, it is fixed for a time (in the short-run). In the long-run all variables may be freely set. Let us denote the fixed variable by $x$. This can take different values, so there is a family of short run cost functions is $SC(x,q)$.

Given a value $q$ let us denote the cost minimizing value of $x$ by $x^\*$, i.e.
$$
x^\* = \arg\min\_x SC(x,q).
$$
Cost minimization implies that in the long-run this will be the chosen value of $x$, thus
$$
C(q) = SC(x^\*,q) = \min\_x SC(x,q).
$$
Dividing by $q$ we get
$$
AC(q) = SAC(x^\*,q) = \min\_x SAC(x,q).
$$
The above holds for any $q$.

> **Comments:**
> **username_0**: Thanks for the answer, but your final conclusion doesn't agree with other parts of the graph right? At other places, the SAC's minima is not tangent to the AC.  
> **username_2**: I am not sure what you mean, as I don't think I wrote anything about this, but note that the graph displays minima with respect to $q$, not $x$. (Your graph denotes variable $x$ by $k$). That is the lowest point of curve $SAC(k_0)$ on the graph has height $\min_q SAC(k_0,q)$ and not $\min_{k_0} SAC(k_0,q)$.  

---

### Answer by username_3 on 2021-05-08. Score: 1

@Giskard and @tdm have given you perfectly valid responses. I am not convinced that your question is perfectly clear.

Are you asking why only at $q\_1$ (in your graph) an SRAC and AC are tangent at the min of an SRAC? If so, that's a result of the underlying assumptions about the production functions that lead to your graph.

In short, you have economies of scale (i.e., AC has a negative slope), a minimum efficient scale at $q\_1$ (i.e., AC's slope is 0 at a single point), and diseconomies of scale (i.e., AC has a positive slope). SRACs have slope 0 at their minimum, while AC has slope 0 only at $q\_1$ (at its minimum efficient scale). Hence, only at $q\_1$ can the min of AC be tangent with the min of an SRAC.

Now had you made different assumptions about the production function the situation would be different. AC is the lower envelope of your SRACs. So, for example, if your minimum efficient scale was a range (instead of a point), or if your production function had constant returns to scale, then you would have infinite number of tangent points at min SRAC and min AC.

See:

1. <https://www.economics.utoronto.ca/osborne/2x3/tutorial/C2LSFRM.HTM>
2. Varian (1992): Microeconomic Analysis p.p.71-72.
3. [https://uh.edu/~ghong/fina3334/chap\_07.doc](https://uh.edu/%7Eghong/fina3334/chap_07.doc)

Edit: fixed username


---

### Answer by username_4 on 2021-05-12. Score: -1

This result comes from the "zero-profit" condition used to define long run equilibrium in this model.

Recall the fact that in the "firm" part of the model, the marginal cost curve intersects with the average total cost curve at the minimum of the average total cost curve. Since marginal cost is the individual firm's supply curve, it follows that when price equals the minimum of total cost, the firm produces the quantity at the crossing point and has zero profit. A higher price will give the firm positive profit since P > ATC, so Revenues > Costs. A lower price will give the firm negative profit since P < ATC, so Revenues < Costs.

Long run equilibrium is defined by the number of firms being at the level that gives each firm zero profit. Thus the long run supply curve must be at the zero-profit price level, i.e. the minimum of average total cost.

> **Comments:**
> **username_2**: (-1) This is not true. The result holds even without assuming prices or perfect competition.  
> **username_4**: Oh!  That was not clear from the poster's original question, and the post was tagged "competitive equilibrium" so I interpreted it in that context.  Could you point to the part of the question that tells you that this information?  
> **username_2**: "*Could you point to the part of the question that tells you that this information?*" What do you mean?  
> **username_2**: You are right about competitive equilibrium being an irrelevant tag, I removed it.  
> **username_4**: OK.  Is the setting you are considering basically a single firm where capital is flexible in the long run but fixed in the short run?  
> **username_2**: Single firm or industry, it does not matter, capital or any other input(s), it does not matter, "*flexible in the long run but fixed in the short run*" yes.  
> **username_4**: OK.  In that case I agree with your answer above.  I think you are overstating the case a little bit when you say "it does not matter" because I can imagine (noncompetitive) settings where the firm's long run choice of k will not be the solution to the cost minimization problem (E.g. a natural monopoly where demand < q_1).  But assuming cost minimization, I agree with your answer.  
> **username_2**: Why would a natural monopoly (or any firm) not minimize costs, why is this an unusual assumption?  
> **username_4**: The OP also asks something about the SRAC curve being tangent to the LRAC at the LRAC point, but other SRAC curves not being tangent to the LRAC. I think this follows from the envelope theorem applied to the cost minimization problem, you might edit your answer to add that in!  
> **username_2**: Woah, woah, woah, so now you take cost minimization for granted? And didn't you say before this follows from the long-run equilibrium? What about these things?  
> **username_2**: I won't edit my answer, I feel it gives an exact reasoning. Feel free to write an answer based on the envelope theorem, if it is valid I will vote for it!  
> **username_4**: I am not going to argue with you about this, you corrected me about the setting and so I accepted that you are correct and changed my position in response.  Your answer above uses cost minimization, and I agree with it. Sorry if that was unclear.  
> **username_2**: Comp. eq. assumes cost minimization, so let us just go with that.  

---

