Class 6: The Pricing Problem: MR = MC

Textbook reading: Kreps, Chapter 6, “Marginal Cost Equals Marginal Revenue.” Read Sections 6.1 and 6.2, the Freedonian Steel example in 6.3, and Section 6.5 on incremental versus allocated costs. Kreps uses calculus throughout; wherever he takes a derivative, we will use a table of small steps instead, and every calculation in this class can be done by subtraction. Skip the stadium example, the “Multivariable Optimization” subsection, Section 6.4, and the subsection “What’s Wrong with This Picture.”

Key Ideas

The pricing problem

Part I was a firm dealing with rivals. Part II is a firm dealing with its customers: how should a firm that can choose its own price choose it?
The firm faces a demand function $D(p)$, the number of units it can sell at price $p$. Demand slopes down, so the firm cannot choose price and quantity separately: choosing a quantity $x$ fixes the price $P(x)$ at which exactly $x$ units sell, and vice versa.
Profit is total revenue minus total cost, $\pi(x) = TR(x) - TC(x)$, where $TR(x) = P(x)\cdot x$ is what the firm collects from selling $x$ units and $TC(x)$ is what producing them costs. The firm chooses $x$ to make $\pi(x)$ as large as possible.

Marginal cost and marginal revenue

Marginal cost ($MC$) is the amount total cost rises when the firm produces one more unit. Marginal cost changes with output. A brewery running one shift can add a batch cheaply: the vats and the crew are already there. Once it is running around the clock, the next batch means overtime or a new vat, and marginal cost jumps. Software runs the other way: the first copy costs years of engineering, the millionth copy costs nearly nothing.
Marginal revenue ($MR$) is the amount total revenue rises when the firm sells one more unit. If the firm must cut its price to sell that unit, the cut applies to every unit it was already selling, so marginal revenue is less than the price. An airline that lowers fares to fill three more seats collects less from the hundred passengers who would have paid full fare.
We work with tables in blocks of output. The marginal values are the differences between adjacent columns, divided by the block size.

Why profit is highest where MR = MC

Selling one more unit adds $MR$ to revenue and $MC$ to cost, so it changes profit by $MR - MC$.
If $MR > MC$, the unit raises profit: produce it. If $MR < MC$, the unit lowers profit: don’t.
$MR$ falls as output grows and $MC$ eventually rises, so starting from zero, profit climbs while $MR > MC$ and falls once $MR < MC$. The peak is at the output where $MR = MC$. In a table of blocks, this is the last block whose $MR$ still covers its $MC$.

Problems of the Day

Q16: The Bakery

A bakery can sell loaves according to the demand schedule below. Its total cost of baking is also given. Everything is in dollars, output in blocks of 10 loaves.

Loaves	0	10	20	30	40	50	60
Price per loaf to sell this many	50	45	40	35	30	25	20
Total revenue	0	450	800	1050	1200	1250	1200
Total cost	10	85	210	385	610	885	1210

Explain: how was total revenue computed?
Add a row labeled Marginal Revenue: the extra revenue per loaf for each block of ten, by subtracting total revenue by the previous total revenue.
Add a row labeled Marginal Cost: the extra cost per loaf for each block of ten, by subtracting total cost by the previous total cost.
Find the block at which extra revenue stops covering extra cost. How many loaves should the bakery sell, and at what price?
Check your answer to part D by computing profit at every column.
The bakery’s owner says: “The price is $35 but the last ten loaves only added $25 each to revenue. Where did the other $10 a loaf go?” Answer her, using the table.

Q17: Chicolini Steel

In the mythical land of Freedonia, Chicolini Steel is the only steelmaker and is protected by tariffs no import can get past. Domestic revenue and total cost, in millions of dollars, for output in blocks of 10,000 tons:

Tons	60,000	70,000	80,000	90,000	100,000	110,000
Price needed to sell them at home ($/ton)	760	720	680	640	600	560
Domestic revenue ($M)	45.6	50.4	54.4	57.6	60.0	61.6
Total cost ($M)	25.6	28.9	32.4	36.1	40.0	44.1

Add two rows: the extra domestic revenue per ton and the extra cost per ton of each block of 10,000 tons. How much should Chicolini sell at home, and at what price? Check your answer by computing profit at every column.
The world price of steel is $375 a ton, and Chicolini could sell as much as it liked abroad at that price. Its executives say exporting is absurd: why take $375 when you can get $680 at home? Look at the next block of 10,000 tons beyond your answer to (a). What does it cost to make, and what would it earn if exported? Should the block after that be exported too?
Write down the firm’s best plan: total production, tons at home, tons exported, and the extra profit compared with the executives’ plan. Then explain to the executives why a price far below $680 can be worth taking abroad when it would not be worth taking at home.

Q18: Two Bad Arguments

Each argument below is made by a smart, experienced businessperson. Each is wrong for the same reason. Find the flaw in each, then say what the reason is.

(a) The gadget line. A firm’s income statement, in thousands:

	Widgets	Gidgets	Gadgets	Total
Sales revenue	120	160	70	350
Variable costs	70	90	55	215
Allocated overhead	40	40	40	120
Net contribution	10	30	(25)	15

“Gadgets lose $25,000 a year. Drop them and profit rises to $40,000.” The overhead is rent and management salaries that do not change if gadgets go. What happens to profit if the line is dropped?

(b) Google’s advice. You sell a waterproof phone case online for $20. It costs you $10 to make and ship, so each sale is worth $10 of profit. You pay Google to show your ad whenever someone searches for a keyword you have bought, and you can spend as much or as little on each keyword as you like. Google’s help page tells you to measure each keyword by its return on investment:

\[\text{ROI} = \frac{\text{profit from the sales the keyword produced} - \text{spending on the keyword}}{\text{spending on the keyword}}\]

and then to move budget from keywords with a low ROI to keywords with a high ROI.

You have bought two keywords, “phone case” and “waterproof phone case,” at $2,000 each, and you have $1,000 more to spend. Your data say:

Keyword	Sales at $2,000	Sales at $3,000
“phone case”	1,000	1,100
“waterproof phone case”	500	900

Compute each keyword’s ROI at $2,000. Google’s rule says the $1,000 goes to the keyword with the higher ROI. Instead, compute how much profit the extra $1,000 produces on each keyword. Where should it go, and what did Google’s rule get wrong?

Answers

Q16:
- Block 0→10 10→20 20→30 30→40 40→50 50→60
  
  Extra revenue per loaf 45 35 25 15 5 −5
  
  Extra cost per loaf 7.5 12.5 17.5 22.5 27.5 32.5
  
  Extra revenue stops covering extra cost at the 30→40 block. Sell 30 loaves at $35. Profit by column: −10, 365, 590, 665, 590, 365, −10. ✓
- 1. To sell loaves 21 through 30, the price had to fall from $40 to $35. The $5 cut applies to the 20 loaves that were already selling: 20 × $5 = $100, or $10 for each of the ten new loaves. $35 − $10 = $25. The missing $10 went to the existing customers. This is why extra revenue per loaf falls faster than the price does.
Q17:
- 1. Per ton, for each block of 10,000:
  Block 60→70 70→80 80→90 90→100 100→110
  
  Extra domestic revenue ($/ton) 480 400 320 240 160
  
  Extra cost ($/ton) 330 350 370 390 410
  
  The 70→80 block adds $400 a ton and costs $350: make it. The 80→90 block adds $320 and costs $370: don’t. Sell 80,000 tons at home at $680. Profit by column ($M): 20.0, 21.5, 22.0, 21.5, 20.0, 17.5. ✓
- 1. The 80→90 block costs $370 a ton to make and earns $375 a ton exported, so make it and export it. The 90→100 block costs $390 a ton, more than $375, so stop there.
- 1. Produce 90,000 tons, sell 80,000 at home at $680, export 10,000 at $375. Extra profit: 10,000 × (375 − 370) = $50,000. Selling the extra block at home would have earned only $320 a ton, because the price on all 80,000 tons already sold would have to fall to $640. Exporting it earns the full $375 a ton, because the export sale does not touch the domestic price. $375 is far below $680, but the relevant comparison is with the $370 the block costs to make, not with the price the other tons fetch.
Q18:
- 1. Dropping gadgets loses $70,000 of revenue and saves $55,000 of variable cost. The $120,000 of overhead is still paid. Profit falls by $15,000, from $15,000 to zero. Allocated overhead is not a cost of the gadget line; it would be paid with or without it.
- 1. ROI at $2,000: “phone case” = (10,000 − 2,000)/2,000 = 4; “waterproof phone case” = (5,000 − 2,000)/2,000 = 1.5. The extra $1,000 on “phone case” brings 100 sales, $1,000 of profit, so it exactly pays for itself. The extra $1,000 on “waterproof phone case” brings 400 sales, $4,000 of profit, net +$3,000. The next $1,000 goes to “waterproof phone case,” the keyword with the lower ROI. ROI is an average over the whole budget; the decision is about the next dollar.
- The common flaw: both arguments use a number describing the whole (overhead per line, return per dollar) to answer a question about a change. Only what the change adds to revenue and to cost matters.