Monopoly firm's block pricing strategy making max. profit

Question

I have question about max. profit condition for monopolist block pricing.

I learned that to maximize profit in block pricing, given a linear demand curve function and constant MC, calculate $Q_2(Q_1)$, and $P_S(Q_1)$ then find $Q_1$ maximizing profit and then find $P_1$ and $Q_2$ the $P_2$.

This is what I read from the textbook (Microeconomic, Besanko).

But I heard somewhere that when in following case,

with right-downward sloping $MC$ and $AC$,

Profit maximizing first block price and quantity is $P_1$ and $Q_1$ where $AC=MR$ and second block price and quantity should be $P_2$ and $Q_2$ where $AC=D$

I've thought about this for 1 day and still couldn't get why maximizing condition is like that.

Answer 1

Given linear inverse demand $P=a-bQ$, if the monopolist wants to sell in two blocks, then the prices are $$ \begin{cases} P_1=a-bQ_1& \text{if }Q\le Q_1\\ P_2=a-bQ_2& \text{if }Q\in(Q_1,Q_2]. \end{cases} $$ Thus, the profit maximization problem is $$ \max_{Q_1,Q_2}\; (a-bQ_1)Q_1+(a-bQ_2)(Q_2-Q_1)-TC(Q_2) $$ subject to $Q_1\le Q_2$. The FOCs are $$ \begin{aligned} a-2bQ_1-a+bQ_2&=0 \qquad\qquad(1)\\ a-2bQ_2-MC(Q_2)&=0\qquad\qquad(2) \end{aligned} $$ Since marginal cost is constant, i.e. $MC(Q_2)=c$, we get from $(2)$ $$Q_2=\frac{a-c}{2b}.$$ Substituting this into $(1)$, we get $$Q_1=\frac{a-c}{4b}.$$ The prices are then $$P_1=\frac{3a+c}{4}\qquad P_2=\frac{a+c}{2}.$$ These are the profit maximizing block prices (assuming zero fixed cost).

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The graph you show to illustrate the second approach is lacking some relevant information. In particular, marginal cost is not constant and there may be fixed costs as well. Moreover, having $P_2=AC(Q_2)$ implies that the monopolist earns zero profit on the sales in the second block (note that $P_2$ is the average revenue in this block), which doesn't make sense.

Answer 2

The reason why the profit maximising condition is like that is because the monopoly is Price Discriminating between those who are willing to pay at $P_1$ and $P_2$ for $Q_1$ and $Q_2$ respectively.

This increases the area of producer surplus to an area greater than what it would be by charging $P_1$ and $P_2$ alone.

Answer 3

A monopoly's pricing and production decisions are calculated this way: Equate the marginal cost and marginal revenue functions. This will give you the production quantity. Now, you plug in this quantity in the demand function to get the price. I don't think, MR is equated with average cost.