SITUATION: PRICE COMPETITION WITH DIFFERENTIATED PRODUCTS
Our first example in this chapter was the case of shops selling the same goods. In that simple setting, consumers cared only about price, so if one store charged a lower price, they would all buy from it. In most markets, however, companies offer distinct products. You can buy Coke only from Coca-Cola, a Big Mac just from McDonalds, and a Dell computer from Dell alone. Of course, you can buy similar products from other companies, such as a Pepsi, a Whopper, and an HP computer. Nevertheless, even if a Big Mac costs more than a Whopper, some people will choose to buy a Big Mac because they pre- fer its flavor and shape. Similarly, there are those who prefer the Whopper and would buy it even if it were more expensive. These are markets in which prod- ucts are said to be differentiated.
So let us explore price competition between Dell and HP in the PC market.* Taking account of the properties just mentioned, assume that the demand for Dell computers (i.e., how many units it sells) is of the following form:
Let’s think about the properties of this formula. The higher the price of a Dell PC, the fewer of them are sold. This makes sense, as some people decide not
DDell(PDell, PHP) � 100 � 2PDell � PHP.
*To model the PC market properly, one would need to include all manufacturers, as well as relevant sub- stitutes, such as the Mac (not to be confused with the Big Mac). To keep things simple, we’ll focus on just two suppliers in that market.
6.3 Solving for Nash Equilibria with Calculus (Optional) 161
to buy a PC or switch from buying a Dell to an HP. Furthermore, the higher the price of an HP, the more PCs Dell sells, since some prospective HP buyers choose instead to buy a Dell.
The revenue that Dell earns is equal to the number of units it sells multi- plied by the price it charges:
Finally, assume the cost of manufacturing and distributing a Dell PC is 10 per unit. Dell then makes a profit of for each unit it sells, so its total profit (and payoff) is
[6.2]
We define HP’s total profit analogously, but assume that HP has a higher cost of 30:
[6.3]
The strategic form of the game is as follows: There are two players—Dell and HP—and a strategy is a price. Let the common strategy set be the in- terval [0,100]. The payoff functions are as described in Equations (6.2) and (6.3).
To find the Nash equilibria, we’ll derive each firm’s best-reply function. Take the case of Dell. FIGURE 6.11 plots Dell’s payoff function in relation to Dell’s price if Note that Dell’s best reply is 45, as that is the price thatPHP � 60.
VHP(PDell, PHP) � (PHP � 30)(100 � 2PHP � PDell).
VDell(PDell, PHP) � (PDell � 10)(100 � 2PDell � PHP).
(PDell � 10)
PDell � DDell(PDell, PHP) � PDell � (100 � 2PDell � PHP).
FIGURE 6.11 Dell’s Payoff Function when HP Prices at 60
Pa yo
ff
0 25 26
2,450
2,288 2,250
1,728
1,650
5445 55
V Dell(pDell, 60)
pDell
162 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
maximizes Dell’s payoff (which reaches a level of 2,450). Note also that the slope of the payoff function is zero at Contrast this scenario with Dell’s charging a price of, say, 25, where the slope of the payoff function is pos- itive. This means that Dell’s payoff is increasing. For example, raising from 25 to 26 raises Dell’s payoff from 1,650 to 1,728. Thus, the best reply can- not be anywhere that the payoff function is increasing. Nor can it be optimal to choose a price where the slope is negative. For example, lowering the price from 55 to 54 raises the payoff from 2,250 to 2,288. Thus, the best reply is where the slope is zero, so that a higher payoff cannot be had by either rais- ing or lowering the price.
What we know thus far is that Dell’s best reply is the price at which the slope of the payoff function is zero. Since the slope of a function is simply its first derivative, the payoff-maximizing price is that price which makes the first derivative equal to zero:
Solving this equation for yields Dell’s best-reply function:
Voilà! Just as a check, substitute 60 for the price of an HP PC, and you’ll find that the answer shown in Figure 6.11.
FIGURE 6.12 plots Dell’s best-reply function against the price charged by HP. Note that it increases as a function of HP’s price: The higher the HP price, the greater is the number of consumers that want to buy Dell and stronger demand
PDell � 45,
PDell � 120 � PHP
4 or BRDell � 30 � .25PHP.
PDell
� 120 � 4PDell � PHP � 0.
0VDell(PDell, PHP)
0PDell � 100 � 2PDell � PHP � 2PDell � 20 � 0
PDell
PDell � 45.
30
45
0 60 pHP
pD el
l
30 � .25pHP
FIGURE 6.12 Dell’s Best-Reply Function Plotted Against the Price Charged by HP
6.3 Solving for Nash Equilibria with Calculus (Optional) 163
for its product allows Dell to charge a higher price. Dell is able to extract more money out of consumers because their alternative—buying an HP computer— is not as attractive when is higher.
Performing the same series of steps, we derive the best-reply function for HP:
Add a dash of algebra to this stew of best-reply functions, and we can con- coct a meal of Nash equilibrium. A price pair is a Nash equilib- rium when both computer manufacturers are simultaneously choosing best replies:
[6.4]
[6.5]
There are two equations and two unknowns. We want to find the pair of prices that satisfies both equations. The pair of prices we seek is depicted in FIGURE 6.13 as the intersection of the two best-reply functions. At that price pair, each com- pany’s price is a best reply.
Algebraically, we can solve the simultaneous equations (6.4) and (6.5) as follows. Substitute the right-hand side of (6.5) for in (6.4):
P̂Dell � 30 � .25(40 � .25P̂Dell).
P̂HP
P̂HP � 40 � .25P̂Dell.
P̂Dell � 30 � .25P̂HP,
(P̂Dell, P̂HP)
BRHP � 40 � .25PDell.
PHP � 160 � PDell
4
� 160 � 4PHP � PDell � 0
0VHP(PDell, PHP)
0PHP � 100 � 2PHP � PDell � 2PHP � 60 � 0
PHP
42.67
0 50.67 pHP
pD el
l
40 � .25pDell
30 � .25pHP
FIGURE 6.13 The Nash Equilibrium Is Where the Two Best-Reply Functions Intersect
164 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
Now we have one equation in one unknown. Next, perform a few algebraic manipulations:
Rounding up, the equilibrium price of a Dell com- puter is 42.67. Substituting this price into HP’s best-reply function gives us the price for HP:
There is, then, a unique Nash equilibrium— since there is only one solution to the pair of
equations (6.4) and (6.5)—and it has Dell pricing at 42.67 and HP pricing at 50.67. Newton and Leibniz, take a bow.