# Is the following graph possible?

Is the following graph possible?

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.

• 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... Jan 3, 2016 at 2:47
• 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. Jan 3, 2016 at 7:07
• Do you assume $F = 0$? Or do you mean $AVC$ instead of $AC$? Jan 7, 2016 at 10:58

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 $MR(q_b) = MC(q_b)$.
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.