<p>Your basic assumptions seem to be
$$
p(q_a) = AVC(q_a)
$$ 
and as the difference between price and average variable cost is decreasing before $q_a$ and increasing after it you also assume
$$
\frac{d \ \left(p(q) - AVC(q)\right)}{d \ q} &lt; 0.
$$
(More on the necessity of this assumption later.)<br>
 We can rewrite this second assumption to
$$
\frac{d \ p(q)}{d \ q} - \frac{d \ \frac{VC(q)}{q}}{d \ q} = \frac{d \ p(q)}{d \ q} - \frac{MC(q) \cdot q - VC(q)}{q^2} &lt; 0.
$$
Multiplying by $q&gt;0$ yields
$$
\frac{d \ p(q)}{d \ q} \cdot q - MC(q) - AVC(q) &lt; 0.
$$
Adding $p(q) + AVC(q)$ we get
$$
\frac{d \ p(q)}{d \ q} \cdot q + p(q) - MC(q) &lt; p(q) - AVC(q).
$$
The left hand side is now $MR(q) - MC(q)$. It follows from our two initial assumptions that
$$
\forall q &gt; q_a: \ p(q) &lt; AVC(q).
$$ 
Then for such values of $q$
$$
MR(q) - MC(q) &lt; p(q) - AVC(q) &lt; 0.
$$
Because of this there can be no $q_b &gt; q_a$ such that $MR(q_b) = MC(q_b)$.</p>

<hr>

<p>On the necessity of the second assumption: <br>
Seems to me that without this assumption the goal function $p(q) \cdot q - C(y)$ would not be concave and hence $MR(q) = MC(q)$ would not be a sufficient condition for optimum.</p>

CC BY-SA 3.0
2016-01-07T11:23:09.993Z
2016-01-07T13:18:05.663Z
2016-01-07T13:18:05.663Z
Giskard
<p>Is the following graph possible?</p>

<p><a href="https://i.sstatic.net/aJLQX.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/aJLQX.jpg" alt="enter image description here"></a></p>

<p>I've attempted to sketch the TC/TR curves for it but they don't seem to be able to satisfy the following two properties at the same time:</p>

<p>1) Slope of Total Cost (TC) being less than the slope of Total Revenue (TR) until q=b, that is until quantity is at the zero marginal profit level.</p>

<p>2) TC being lesser than TR at first but going on to be greater than it after q=a, that is after quantity is at the zero economic profit level.</p>

CC BY-SA 3.0
2016-01-03T02:47:32.970Z
Doesn't your graph imply both conditions (except perhaps TC being greater than TR close to zero, because revenue is $0$ when $q=0$, which FC $\not = 0$)? Slope of TC is MC, and slope of TR is MR, so 1) is true, and TC = AC *q, and price = D, so $2)$ is satisfied...
majmun
CC BY-SA 3.0
2016-01-03T07:07:39.900Z
This question could be greatly improved by defining $a$ and $b$ in the text part of the question because search engines cannot search pictures and future users will be unable to find relevant info in questions such as these.
Giskard
CC BY-SA 3.0
2016-01-07T10:58:42.270Z
Do you assume $F = 0$? Or do you mean $AVC$ instead of $AC$?
Giskard
<microeconomics><cost>
Is the following graph possible?
HeyThereGur
# Is the following graph possible?

(Asked by: username_0 on 2016-01-02)

Is the following graph possible?

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

I've attempted to sketch the TC/TR curves for it but they don't seem to be able to satisfy the following two properties at the same time:

1) Slope of Total Cost (TC) being less than the slope of Total Revenue (TR) until q=b, that is until quantity is at the zero marginal profit level.

2) TC being lesser than TR at first but going on to be greater than it after q=a, that is after quantity is at the zero economic profit level.

**Tags:** microeconomics, cost

> **Comments:**
> **username_1**: Doesn't your graph imply both conditions (except perhaps TC being greater than TR close to zero, because revenue is $0$ when $q=0$, which FC $\not = 0$)? Slope of TC is MC, and slope of TR is MR, so 1) is true, and TC = AC *q, and price = D, so $2)$ is satisfied...  
> **username_2**: This question could be greatly improved by defining $a$ and $b$ in the text part of the question because search engines cannot search pictures and future users will be unable to find relevant info in questions such as these.  
> **username_2**: Do you assume $F = 0$? Or do you mean $AVC$ instead of $AC$?  

---

## 1 answers

### Answer by username_2 on 2016-01-07. Score: 1

Your basic assumptions seem to be
$$
p(q\_a) = AVC(q\_a)
$$
and as the difference between price and average variable cost is decreasing before $q\_a$ and increasing after it you also assume
$$
\frac{d \ \left(p(q) - AVC(q)\right)}{d \ q} < 0.
$$
(More on the necessity of this assumption later.)  
We can rewrite this second assumption to
$$
\frac{d \ p(q)}{d \ q} - \frac{d \ \frac{VC(q)}{q}}{d \ q} = \frac{d \ p(q)}{d \ q} - \frac{MC(q) \cdot q - VC(q)}{q^2} < 0.
$$
Multiplying by $q>0$ yields
$$
\frac{d \ p(q)}{d \ q} \cdot q - MC(q) - AVC(q) < 0.
$$
Adding $p(q) + AVC(q)$ we get
$$
\frac{d \ p(q)}{d \ q} \cdot q + p(q) - MC(q) < p(q) - AVC(q).
$$
The left hand side is now $MR(q) - MC(q)$. It follows from our two initial assumptions that
$$
\forall q > q\_a: \ p(q) < AVC(q).
$$
Then for such values of $q$
$$
MR(q) - MC(q) < p(q) - AVC(q) < 0.
$$
Because of this there can be no $q\_b > q\_a$ such that $MR(q\_b) = MC(q\_b)$.

---

On the necessity of the second assumption:   
Seems to me that without this assumption the goal function $p(q) \cdot q - C(y)$ would not be concave and hence $MR(q) = MC(q)$ would not be a sufficient condition for optimum.


---

