# Profit Maximisation if MC is still falling after intersecting MR

really basic one here, I am just in the process of re-covering old ground.

I understand that for any profit maximising firm FOC: MC=MR SOC: MR'< MC'

But suppose MC has a minimum beyond the point at which MR=0. So MR still intersects MC from above but MC is still falling (does this still satisfy SOC?). In that case would the max difference between TR and TC be greatest at some Q higher than where MC=MR?

Also I am looking for sources/examples of where economic assumptions are really taken apart in an applied setting. A lot of the Industrial Organisation I am looking at sticks to the same neat assumptions.

• Sorry typo, yes you're right, amending. Nov 18 '15 at 10:44
• Consider making this into two separate questions: 1. MR and MC, tricky functions and 2. Economic assumptions in applied settings. These are really quite different and merit separate answers. Nov 18 '15 at 11:00

Since the OP "recovers old-ground", let's turn the flashlight on some dark spots of this ground.

It is almost "automatic" to think that "marginal revenue" is the price at which the additional unit of output will be sold 9so that it is "the revenue that the additional unit brings in"), and that "marginal cost" is what this additional unit will cost. We should remember that this only holds when the firm is a price-taker, and its choices on output level does not affect the price of output, nor the prices to be paid for input factors. For if

$$TR(q) = pq \implies MR = p$$

but if

$$TR(q) = p(q)\cdot q \implies MR = p'(q)q + p < p$$

since $p'(q) <0$

The same goes for Total Cost, where denoting by $\mathbb w$ the prices commanded by input factors

$$TC(q) = C(\mathbb w, q) \implies MC = \frac {\partial C}{\partial q}$$

but if

$$TC(q) = C(\mathbb w(q), q) \implies MC = \frac {\partial C}{\partial \mathbb w}\cdot \frac {\partial \mathbb w}{\partial q} + \frac {\partial C}{\partial q}$$ (I have avoided matrix-differentiation notation). Note that the effect of increased quantity on input prices is not clear-cut: in the usual monopsonistic setting, it will tend to decrease the unit cost of the inputs -but if the currently supplied quantity of input has reached a ceiling, it may push prices up (although that is considered more of a short-term effect).

The usual setting, and the one indicated by the OP, is to assume price-taking behavior in the markets for inputs (in which case the non-constancy of marginal cost stems purely from technological factors), while assuming some monopolistic power in the product market, i.e. a downward sloping demand curve.

So it is more general to think "marginal revenue is the increase in total revenue if we increase quantity by one unit", and "marginal cost is the increase in cost if we increase quantity by one unit", disassociating the concepts from the revenues and the costs of the "additional unit" itself.

Having cleared this, the OP asks

But suppose MC has a minimum beyond the point at which MR=0. So MR still intersects MC from above but MC is still falling (does this still satisfy SOC?)

This is something like

which looks pretty standard. As to whether Second-order conditions hold, in this post one can find a mnemonic about slopes, since the SOC is phrased in terms of derivatives of the curves shown in the above graph.

In that case would the max difference between TR and TC be greatest at some Q higher than where MC=MR?

Since at $MR=MC$ the marginal revenue curve is assumed to cross the marginal cost curve from above, it follows that for higher quantities, marginal cost will be higher than marginal revenue. Given the previous suggestion as to how it is best to think about Marginal Revenue and Marginal Cost, I believe the answer to the question is easy.

Sorry I have been visualising the corresponding TR, TC curves wrong / mixed up with another topic. Where the MC minimum is is of no consequence to the problem, I see that now.

I don't have an issue with the marginal conditions, as the above answer points out they hold irrespective.