\documentstyle[12pt,epsf] {article} \parskip 10pt \reversemarginpar \topmargin -.4in \oddsidemargin .25in \textheight 8.7in \textwidth 6in \begin{document} \parindent 24pt \parskip 10pt \setcounter{section}{13} \setcounter{page}{386} \section*{ 13 Pricing } \noindent June 28, 1993.April 18, 1999 \subsection{Quantities as Strategies: Cournot Equilibrium Revisited} %13.1 \noindent Chapter 13 is about how firms with market power set prices. Section 13.1 generalizes the Cournot Game of section 3.5, in which two firms choose the quantities they sell, while section 13.2 sets out the Bertrand model of firms choosing prices. Both the Bertrand and Cournot models are then expanded to allow for differentiated products. Section 13.3 goes back to the origins of product differentiation, and develops two Hotelling location models. Section 13.4 shows how to do comparative statics in games, using the differentiated Bertrand model as an example and supermodularity and the implicit function theorem as tools. Section 13.5 shows that even if a firm is a monopolist, if it sells a durable good it suffers competition from its future self. \bigskip \noindent {\bf Cournot Behavior with General Cost and Demand Functions} \noindent In the next few sections, sellers compete against each other while moving simultaneously. We will start by generalizing The Cournot Game of section 3.5 from linear demand and zero costs to a wider class of functions. The two players are firms Apex and Brydox, and their strategies are their choices of the quantities $q_a$ and $q_b$. The payoffs are based on the total cost functions, $c(q_a)$ and $c(q_b)$, and the demand function, $p(q)$, where $q=q_a + q_b$. This specification says that only the sum of the outputs affects the price. The implication is that the firms produce an identical product, because whether it is Apex or Brydox that produces an extra unit, the effect on the price is the same. Let us take the point of view of Apex. In the Cournot-Nash analysis, Apex chooses its output of $q_a$ for a given level of $q_b$ as if its choice did not affect $q_b$. From its point of view, $q_a$ is a function of $q_b$, but $q_b$ is exogenous. Apex sees the effect of its output on price as \begin{equation} \label{e13.1} \frac{ \partial p}{\partial q_a}= \frac{dp}{ d q} \frac{\partial q}{ \partial q_a}= \frac{dp}{ d q}. \end{equation} Apex's payoff function is \begin{equation} \label{e13.2} \pi_a = p(q) q_a - c(q_a). \end{equation} To find Apex' reaction function, we differentiate with respect to its strategy to obtain \begin{equation} \label{e13.3} \frac{d\pi_a}{dq_a} = p + \frac{dp}{dq} q_a - \frac{dc}{d q_a} = 0, \end{equation} which implies \begin{equation} \label{e13.4} q_a = \frac{\frac{dc}{dq_a} - p}{\frac{dp}{dq}}, \end{equation} or, simplifying the notation, \begin{equation} \label{e13.5} q_a = \frac{c' - p}{p'}. \end{equation} If particular functional forms for $p(q)$ and $c(q_a)$ are available, equation (\ref{e13.5}) can be solved to find $q_a$ as a function of $q_b$. More generally, to find the change in Apex' best response for an exogenous change in Brydox's output, differentiate (\ref{e13.5}) with respect to $q_b$, remembering that $q_b$ exerts not only a direct effect, but possibly an indirect effect on $q_a$. \begin{equation} \label{e13.6} \frac{d q_a}{d q_b} = \frac{(p-c')(p'' + p'' \frac{d q_a}{d q_b})}{p'^2} + \frac{ c'' \frac{d q_a}{d q_b} - p' - p' \frac{d q_a}{d q_b}}{p'}. \end{equation} Equation (\ref{e13.6}) can be solved for $\frac{d q_a}{d q_b}$ to obtain the slope of the reaction function, \begin{equation} \label{e13.7} \frac{d q_a}{d q_b} = \frac{(p-c')p''- p'^2} {2p'^2 - c''p' - (p- c')p''} \end{equation} If both costs and demand are linear, as in section 3.5, then $c''=0$ and $p''=0$, so equation (\ref{e13.7}) becomes \begin{equation} \label{e13.8} \frac{d q_a}{d q_b} = - \frac{p'^2 }{2p'^2 } = -\frac{1}{2}. \end{equation} The general model faces two problems that did not arise in the linear model: nonuniqueness and nonexistence. If demand is concave and costs are convex, which implies that $p'' < 0$ and $c''> 0$, then all is well as far as existence goes. Since price is greater than marginal cost ($p >c'$), equation (\ref{e13.7}) tells us that the reaction functions are downward sloping, because $2p'^2 - c''p' - (p-c')p''$ is positive and both $(p-c')p''$ and $-p'^2$ are negative. If the reaction curves are downward sloping, they cross and an equilibrium exists, as was shown in figure 3.1 for the linear case represented by equation (\ref{e13.8}). We usually do assume that costs are at least weakly convex, since that is the result of diminishing or constant returns, but there is no reason to believe that demand is either concave or convex. If the demand curves are not linear, the contorted reaction functions of equation (\ref{e13.7}) might give rise to multiple Cournot equilibria as in figure 13.1. \begin{center} {\bf Figure 13.1} Multiple Cournot-Nash Equilibria \end{center} \epsfysize=3in \epsffile{/Users/erasmuse/AAANewChapters/Figures/f_12.1.eps} If demand is convex or costs are concave, so $p'' > 0$ or $c''<0$, the reaction functions can be upward sloping, in which case they might never cross and no equilibrium would exist. The problem can also be seen from Apex' payoff function, equation (\ref{e13.2}). If $p(q)$ is convex, the payoff function might not be concave, in which case standard maximization techniques break down. The problems of the general Cournot model teach a lesson to modellers: sometimes simple assumptions such as linearity generate atypical results. \bigskip \noindent {\bf Many Oligopolists} \noindent Let us return to the simpler game in which production costs are zero and demand is linear. For concreteness, we will use the particular inverse demand function \begin{equation} \label{e13.9} p(q) = 120 - q. \end{equation} Using (\ref{e13.9}), the payoff function (\ref{e13.2}) becomes \begin{equation} \label{e13.10} \pi_a = 120q_a - q_a^2 - q_b q_a. \end{equation} In section 3.5, firms picked outputs of 40 apiece given demand function (\ref{e13.9}). This generated a price of 40. With $n$ firms instead of two, the demand function is \begin{equation} \label{e13.11} p \left(\sum_{i=1}^n q_i \right) = 120 - \sum_{i=1}^n q_i, \end{equation} and firm $j$'s payoff function is \begin{equation} \label{e13.12} \pi_j = 120q_j - q_j^2 - q_j\sum_{i \neq j} q_i. \end{equation} Differentiating $j$'s payoff function with respect to $q_j$ yields \begin{equation} \label{e13.13} \frac{d\pi_j}{dq_j} = 120 - 2q_j - \sum_{i \neq j} q_i= 0. \end{equation} The first step in finding the equilibrium is to guess that it is symmetric, so that $q_j = q_i,( i = 1,\ldots, n)$. This is an educated guess, since every player faces a first-order condition like (\ref{e13.13}). By symmetry, equation (\ref{e13.13}) becomes $120 - (n+1)q_j = 0$, so that \begin{equation}\label {e13.14} q_j = \frac{120}{n+1}. \end{equation} Consider several different values for $n$. If $n=1$, then $q_j = 60$, the monopoly optimum; and if $n=2$ then $q_j = 40$, the Cournot output found in section 3.5. If $n = 5$, $q_j = 20$; and as $n$ rises, individual output shrinks to zero. Moreover, the total output of $ nq_j =\frac{120n}{n+1}$ gradually approaches 120, the competitive output, and the market price falls to zero, the marginal cost of production. As the number of firms increases, profits fall. \bigskip \noindent {\bf Conjectural Variation} \noindent Conjectural variation, an equilibrium concept different in flavor from any that has yet appeared in this book, is a way to quantify the degree of cooperation between oligopolists. Let us continue to specify the strategies as quantities. In a Nash equilibrium, no player wants to deviate, and his beliefs about how the other players would behave are confirmed whatever nodes are reached. Under conjectural variation, a player believes, for reasons outside the model, that if he deviated, the other players would deviate in specified ways. This should seem quite an unnatural idea to anyone who has read this far in the book, since it violates the basic assumptions of Bayesian games and it is rather hazy about what is happening in this simultaneous-move game. The idea may be clearer in an example. Returning to the two-player model, we can use equation (\ref{e13.3}) to write Apex' self-perceived first order condition as \begin{equation}\label{e13.15} \frac{d\pi_a}{dq_a} = p + \left( \frac{dp}{dq} \right) \left( \frac{dq }{d q_a} \right) q_a - \frac{dc}{dq_a} = 0. \end{equation} The difference between the first-order-conditions (\ref{e13.3}) and (\ref{e13.15}) is that (\ref{e13.15}) contains \begin{equation}\label{e13.16} \frac{dq }{d q_a} = 1 + \frac{d q_b }{d q_a}. \end{equation} Equation (\ref{e13.16}) says that the expected effect on industry output of an increase in $q_a$ by one unit has two components: a direct increase of one unit, and an indirect increase from Brydox increasing his output in response. The first-order condition (\ref{e13.15}) must be qualified by ``self-perceived'' because Apex might be mistaken in his beliefs about Brydox' response. The belief implicit in Nash equilibrium, that Apex' deviation is not followed by a response from Brydox, is the only belief that supports an equilibrium in which one player or the other is not mistaken. But if consistency of beliefs is not required, other beliefs are possible that lead to different behavior. \noindent {\it Firm $i$'s {\bf conjectural variation} is the rate $\frac{d q_{-i} }{d q_i}$ at which he conjectures that the output of other firms would change if $i$'s own output changed.} \noindent {\it CV} = 0\\ In a Cournot-Nash equilibrium, Apex believes that if he deviated by producing more, Brydox would not deviate, so the conjectural variation equals 0. \noindent {\it CV} = $-1$\\ If Apex believes that an increase in his output is matched by a decrease in Brydox' output, so the total industry output is left unchanged, the conjectural variation is $-1$. If both firms use this conjectural variation, the industry output is the competitive level; firms ignore the effect of their output in depressing the price. Of course, if both firms use a negative value, their beliefs are inconsistent. \noindent {\it CV} = 1\\ If Apex believes that Brydox would exactly match his output changes, the conjectural variation is 1. With two firms, with identical cost curves, industry output is at the cartel level, though an $n$-player game would need $CV = n-1$ to achieve that level. In Stackelberg equilibrium (section 3.5), the conjectural variation of the Stackelberg follower is between 0 and 1, and takes the value given by a reaction function like equation (\ref{e13.7}). In the world oil market, fringe producers like Britain face the OPEC cartel. If Britain's conjectural variation equals $-1$, Britain believes that producing more would make OPEC cut back an equal amount; if 0, that OPEC would ignore Britain; if 0.5, that OPEC would follow with a smaller increase; if 1, that OPEC would match every increase; and if 10, that OPEC would respond by flooding the market. Setting up equations with the appropriate value for the conjectural variations of all the players, we could solve for the equilibrium output. The idea is useful for organizing different models of duopoly and it is simple enough to be empirically estimated. Even without knowing the correct theory, an estimate could be made of how much OPEC actually does respond to Britain. \subsection{Prices as Strategies: Bertrand Equilibrium} %13.2 \noindent The Bertrand (1883) duopoly model seems to be only slightly different from the Cournot model, but it reaches radically different conclusions. The Bertrand solution is nothing more than a Nash equilibrium in prices rather than quantities. We will use the same two-player, zero-cost, linear-demand world as before, but now the strategy spaces will be the prices, not the quantities. We will also use the same demand function, equation (\ref{e13.9}), which implies that if $p$ is the lowest price, $q = 120 - p$. In the Cournot model, firms chose quantities but allowed the market price to vary freely; in the Bertrand model, they choose prices and sell as much as they can. The strategies for Apex and Brydox are $p_a$ and $p_b$. The payoff function for Apex (and analogously for Brydox) is \begin{tabular}{ll} $ \;\;\;\;\;\;\;\; \pi_a =$ & $\left\{ \begin{tabular}{ll} $ p_a (120 - p_a)$ &if $p_a < p_b$ \\ $ \frac{p_a(120 - p_a)}{2}$& if $p_a = p_b $ \\ 0 & if $ p_a > p_b$ \\ \end{tabular} \right.$ \end{tabular} \noindent The Bertrand game has a unique Nash equilibrium: $p_a = p_b = 0$. No other pair of prices could be an equilibrium, because one firm could capture the entire market by slightly undercutting the other's price. The only pair of prices where undercutting is not a temptation is (0,0). Duopoly profits are not just less than monopoly profits, they are zero. Like the surprising outcome of Prisoner's Dilemma, the Bertrand equilibrium is less surprising once one thinks about the model's limitations. What it shows is that duopoly profits do not arise just because there are two firms. Profits arise from something else, such as multiple periods, incomplete information, or differentiated products. Both the Bertrand and Cournot models are in common use. The Bertrand model can be awkward mathematically because of the discontinuous jump from a market share of 0 to 100 percent after a slight price cut. The Cournot model is useful as a simple model that avoids this problem and which predicts that the price will fall gradually as more firms enter the market. There are also ways to modify the Bertrand model to obtain intermediate prices and gradual effects of entry, and we will proceed to look at some of these modifications. \bigskip \noindent {\bf Capacity Constraints: The Edgeworth Paradox} \noindent Let us start by altering the Bertrand model by constraining each firm to sell no more than $K = 70$ units. The industry capacity of 140 exceeds the competitive output, but do profits continue to be zero? When capacities are limited we require additional assumptions because of the new possibility that a firm with a lower price might attract more customers than it can supply. We need to specify a {\bf rationing rule} telling which customers are served at the low price and which must buy from the high-price firm. The rationing rule is unimportant to the payoff of the low-price firm, but crucial to the high-price firm. One possible rule is \noindent {\bf Intensity rationing.} { \it The customers that value the product most buy from the firm with the lower price.} The inverse demand function from equation (\ref{e13.9}) is $p = 120- q$, and under intensity rationing, the $K$ customers with the strongest demand buy from the low-price firm. Suppose that Brydox is the low-price firm, charging a price of 30 so that 90 consumers wish to buy from it. The residual demand facing Apex is then \begin{equation}\label{e13.18} q_a = 120 - p_a - K. \end{equation} The demand curve is shown in figure 13.2a. \begin{center} {\bf Figure 13.2 Rationing Rules} (a) intensity rationing if K=70; (b) proportional rationing \end{center} \epsfysize=3in \epsffile{/Users/erasmuse/AAANewChapters/Figures/f_12.2.eps} \noindent Under intensity rationing, the payoff functions are, given that $K=70$, \begin{equation}\label{e13.19} \begin{array}{ll} \pi_a = & \left\{ \begin{array}{llr} p_a \cdot Min \{ 120 - p_a, 70 \} &{\rm if} \;p_a < p_b & (a) \\ \frac{p_a(120 - p_a)}{2} & {\rm if }\; p_a = p_b & (b) \\ 0 & {\rm if }\; p_a > p_b, p_b \geq 50 & (c)\\ p_a (120 - p_a- 70) & {\rm if}\; p_a > p_b, p_b < 50 & (d) \\ \end{array} \right. \end{array} \end{equation} The appropriate rationing rule depends on what is being modelled. Intensity rationing is appropriate if buyers with more intense demand make greater efforts to obtain low prices. If the intense buyers are wealthy people who are unwilling to wait in line, the least intense buyers might end up at the low-price firm which is the case of {\bf inverse-intensity rationing}. An intermediate rule is proportional rationing, under which every type of consumer is equally likely to be able to buy at the low price. \noindent {\bf Proportional rationing.} {\it Each customer has the same probability of being able to buy from the low-price firm.} Under proportional rationing, if $K= 70$ and 90 customers wanted to buy from Brydox, 2/9 ($=\frac{q(p_b)-K}{q(p_b)}$) of each type of customer will be forced to buy from Apex (for example, 2/9 of the type willing to pay 120). The residual demand curve facing Apex, shown in figure 13.2b and equation (\ref{e13.20}), intercepts the price axis at 120, but slopes down at a rate three times as fast as market demand because there are only 2/9 as many remaining customers of each type. \begin{equation}\label{e13.20} q_a = (120 - p_a) \left(\frac{120- p_b - K}{120 - p_b} \right) \end{equation} The capacity constraint has a very important effect: (0,0) is no longer a Nash equilibrium in prices. Consider Apex' best response when Brydox charges a price of zero. If Apex raises his price above zero, he retains most of his customers (because Brydox is already producing at capacity), but his profits rise from zero to some positive number, regardless of the rationing rule. In any equilibrium, both players must charge prices within some small amount $\epsilon$ of each other, or the one with the lower price would deviate by raising his price. But if the prices are equal, then both players have unused capacity, and each has an incentive to undercut the other. No pure-strategy equilibrium exists under either rationing rule. This is known as the {\bf Edgeworth paradox}, after Edgeworth (1897). Suppose that demand is linear, with the highest reservation price being $P=100$ and the maximum market quantity $Q=100$ at $P=0$. Suppose also that there are two firms, Apex and Brydox, each having a constant marginal cost of 0 up to capacity of $Q=80$ and infinity thereafter. We will assume intensity rationing of buyers. Note that industry capacity of 160 exceeds market demand of 100 if price equals marginal cost. Note also that the monopoly price is 50, which with quantity of 50 yields industry profit of 2500. But what will be the equilibrium? Prices of $(P_a=0, P_b=0)$ are not an equilibrium. Apex's profit would be zero in that strategy combination. If Apex increased its price to 5, what would happen? Brydox would immediately sell $Q=80$, and to the most intense 80 percent of buyers. Apex would be left with all the buyers between $P=20$ and $P= 5$ on the demand curve, for $Q_a=15$ and profit of $\pi_a= (5) (15) = 75$. So deviation by Apex is profitable. (Of course, $P=5$ is not necessarily the most profitable deviation-- but we do not need to check that; I looked for an {\it easy} deviation.) Equal prices of $(P_a, P_b)$ with $P_a = P_b>0$ are not an equilibrium. Even if the price is close to 0, Apex would sell at most 50 units as its half of the market, which is less than its capacity of 80. Apex could deviate to just below $P_b$ and have a discontinuous jump in sales for an increase in profit, just as in the basic Bertrand game. \newpage Unequal prices of $(P_a, P_b)$ are not an equilibrium. Without loss of generality, suppose $P_a > P_b$. So long as $P_b$ is less than the monopoly price of 50, Brydox would deviate to a new price even close to but not exceeding $P_a$. And this is not {\it just} the open-set problem. Once Brydox is close enough to Apex, Apex would deviate by jumping to a price just below Brydox. If capacities are large enough, the Edgeworth Paradox disappears. Consider capacities of 150 per firm, for example. The argument made above for why equal prices of 0 is not an equilibrium fails, because if Apex were to deviate to a positive price, Brydox would be fully capable of serving the entire market, leaving Apex with no customers. If capacities are small enough, the Edgeworth Paradox also disappears, but so does the Bertrand Paradox. Suppose each firm has a capacity of 20. They each will choose to sell at a price of 60, in which case they will each sell 20 units, their entire capacities. Apex will have a payoff of 1200. If Apex deviates to a lower price, it will not sell any more, so that would be unprofitable. If Apex deviates to a higher price, it will sell fewer, and since the monopoly price is 50, its profit will be lower; note that a price of 61 and a quantity of 19 yields profits of 1159, for example. \footnote{Inverse intensity rationing might change this result. Think about it.} We could have expanded the model to explain why the firms have small capacities by adding a prior move in which they choose capacity subject to a cost per unit of capacity, foreseeing what will happen later in the game. A mixed strategy equilibrium does exist, calculated using intensity rationing by Levitan \& Shubik (1972) and analyzed in Dasgupta \& Maskin (1986b). Expected profits are positive, because the firms charge positive prices. Under proportional rationing, as under intensity rationing, profits are positive in equilibrium, but the high-price firm does better with proportional rationing. The high-price firm would do best with {\bf inverse-intensity rationing}, under which the customers with the least intense demand are served at the low-price firm, leaving the ones willing to pay more at the mercy of the high-price firm. Even if capacity were made endogenous, the outcome would be inefficient, either because firms would charge prices higher than marginal cost (if their capacity were low), or they would invest in excess capacity (even though they price at marginal cost). \bigskip \noindent {\bf Product Differentiation} \noindent The Bertrand model without capacity constraints generates zero profits because only slight price discounts are needed to bid away customers. The assumption behind this is that the two firms sell identical goods, so if Apex' price is slightly higher than Brydox' all the customers go to Brydox. If customers have brand loyalty or poor price information, the equilibrium is different and the demand curves facing Apex and Brydox might be \begin{equation} \label{e13.21} q_a = 24 - 2p_a + p_b \end{equation} and \begin{equation} \label{e13.22} q_b = 24 - 2p_b + p_a. \end{equation} The greater the difference in the coefficients on prices in demand curves like these, the less substitutable are the products. As with standard demand curves like (\ref{e13.9}), we have made implicit assumptions about the extreme points of (20) and (21). These equations only apply if the quantities demanded turn out to be nonnegative, and we might also want to restrict them to prices below some ceiling, since otherwise the demand facing one firm becomes infinite as the other's price rises to infinity. With those restrictions, the payoffs are \begin{equation} \label{e13.22} \pi_a = p_a (24 - 2p_a + p_b) \end{equation} and \begin{equation} \label{e13.23} \pi_b = p_b (24 - 2p_b + p_a). \end{equation} Maximizing Apex' payoff, we obtain the first-order condition \begin{equation} \label{e13.24} \frac{ d\pi_a}{d p_a} = 24 - 4p_a + p_b = 0, \end{equation} and the reaction function \begin{equation} \label{e13.25} p_a = 6 + p_b/4. \end{equation} Since Brydox has a parallel first-order condition, the equilibrium occurs where $p_a = p_b = 8.$ The quantity each firm produces is 16, which is below the 24 each would produce at prices of zero. Figure 13.3 shows that the reaction functions intersect. Apex' demand curve has the elasticity \begin{equation} \label{e13.26} \left( \frac{\partial q_a}{\partial p_a} \right) \cdot \left( \frac{p_a}{q_a} \right) = - 2 \left( \frac{p_a}{q_a} \right), \end{equation} which is finite even when $p_a = p_b$, unlike the case of the standard Bertrand model. \begin{center} {\bf Figure 13.3} Bertrand Reaction Functions with Differentiated Products \end{center} \epsfysize=3in \epsffile{/Users/erasmuse/AAANewChapters/Figures/f_12.3.eps} \bigskip \noindent {\bf Cournot Equilibrium with Differentiated Products} \noindent We can also work out the Cournot equilibrium for demand functions (20) and (21), but product differentiation does not affect it by much. Start by expressing the price in terms of quantities alone, obtaining \begin{equation} \label{e13.28} p_a = 12 - \frac{1}{2}q_a +\frac{1}{2} p_b \end{equation} and \begin{equation} \label{e13.29} p_b = 12 - \frac{1}{2}q_b + \frac{1}{2}p_a. \end{equation} After substituting from (28) into (27) and solving for $p_a$, we obtain \begin{equation} \label{e13.30} p_a = 24 - 2q_a/3 - q_b/3. \end{equation} The first-order condition for Apex' maximization problem is \begin{equation} \label{e13.30} \frac{d \pi_a}{dq_a} = 24 - 4q_a/3 - q_b/3 = 0, \end{equation} which gives rise to the reaction function \begin{equation} \label{e13.31} q_a = 18 - q_b/4. \end{equation} We can guess that $q_a = q_b$. It follows from (31) that $q_a = 14.4$ and the market price is 9.6. On checking, you would find this to indeed be a Nash equilibrium. \subsection{Location Models }%13.3 \noindent In section 13.2 we analyzed the Bertrand model with differentiated products using demand functions whose arguments were the prices of both firms. Such a model is suspect because it is not based on primitive assumptions. In particular, the demand functions might not be generated by maximizing any possible utility function. A demand curve with a constant elasticity less than one, for example, is impossible because as the price goes to zero, the amount spent on the commodity goes to infinity. Also, demand (20) and (21) were restricted to prices below a certain level, and it would be good to be able to justify that restriction. Location models construct demand functions like (20) and (21) from primitive assumptions. In location models, a differentiated product's characteristics are points in a space. If cars differ only in their mileage, the space is a one-dimensional line. If acceleration is also important, the space is a two-dimensional plane. An easy way to think about this approach is to consider the location where a product is sold. The product ``gasoline sold at the corner of Wilshire and Westwood,'' is different from ``gasoline sold at the corner of Wilshire and Fourth.'' Depending on where consumers live, they have different preferences over the two, but, if prices diverge enough, they will be willing to switch from one gas station to the other. Location models form a literature in themselves. We will look at the first two models analyzed in the classic article of Hotelling (1929), a model of price choice and a model of location choice. Figure 13.4 shows what is common to both. Two firms are located at points $x_a$ and $x_b$ along a line running from zero to one, with a constant density of consumers throughout. In the Hotelling Pricing Game, firms choose prices for given locations. In the Hotelling Location Game, prices are fixed and the firms choose the locations. \begin{center} {\bf Figure 13.4:} Location Models \end{center} \epsfysize=3in \epsffile{/Users/erasmuse/AAANewChapters/Figures/f_12.4.eps} \begin{center} {\bf The Hotelling Pricing Game}\\ (Hotelling [1929]) \end{center} {\bf Players}\\ Sellers Apex and Brydox, located at $x_a$ and $x_b,$ where $x_a < x_b$, and a continuum of buyers indexed by location $x \in [0,1]$. \noindent {\bf Order of Play }\\ (1) The sellers simultaneously choose prices $p_a$ and $p_b$.\\ (2) Each buyer chooses a seller. \noindent {\bf Payoffs}\\ Demand is uniformly distributed on the interval [0,1] with a density equal to one (think of each consumer as buying one unit). Production costs are zero. Each consumer always buys, so his problem is to minimize the sum of the price plus the linear transport cost, which is $\theta$ per unit distance travelled. \begin{equation} \label{e13.33} \pi_{buyer \;at \;x} = -Min\{ \theta |x_a -x| + p_a, \; \theta |x_b - x| + p_b \}. \end{equation} \begin{tabular}{ll} $ \pi_a = \left\{ \begin{tabular}{llr} 0 & if $p_a - p_b > \theta (x_b - x_a)$ & (33a)\\ & (Brydox captures entire market) & \\ & & \\ $p_a$ & if $p_b - p_a > \theta (x_b - x_a)$ & (33b)\\ & (Apex captures entire market) & \\ & & \\ $p_a ( \frac{1}{2\theta} \left[ (p_b - p_a) + \theta(x_a + x_b) \right] )$ & otherwise (market is divided)& (33c) \\ \end{tabular} \right. $ \end{tabular} \noindent Brydox has analogous payoffs. The payoffs result from buyer behavior. A buyer's utility depends on the price he pays and the distance he travels. Price aside, Apex is most attractive to the customer at $x=0$ (``Customer 0'') and least attractive to the customer at $x = 1 $ (``Customer 1''). Customer 0 will buy from Apex so long as \begin{equation}\label{e13.34} \theta x_a + p_a < \theta x_b + p_b, \end{equation} which implies that \begin{equation} \label{e13.35} p_a - p_b < \theta (x_b - x_a), \end{equation} which yields payoff (33a). Customer 1 will buy from Brydox if \begin{equation}\label{e13.36} \theta(1- x_a) +p_a > \theta (1-x_b) + p_b, \end{equation} which implies that \begin{equation}\label{e13.37} p_b - p_a < \theta (x_b - x_a), \end{equation} which yields payoff (33b). Very likely, inequalities (35) and (36) are both satisfied, in which case Customer 0 goes to Apex and Customer 1 goes to Brydox. This is the case represented by payoff (33c), and the next task is to find the location of Customer $x^*$, defined as the customer who is at the boundary between the two markets, indifferent between Apex and Brydox. First, notice that if Apex attracts Customer $x_b$, he also attracts all $x > x_b$, because beyond $x_b$ the customers' distances from both sellers increase at the same rate. So we know that if there is an indifferent consumer he is between $x_a$ and $x_b$. Knowing this, (32) tells us that \begin{equation}\label{e13.38} \theta(x^*- x_a) + p_a = \theta (x_b -x^*) + p_b, \end{equation} so that \begin{equation}\label{e13.39} p_b - p_a = \theta (2x^*- x_a - x_b ), \end{equation} and \begin{equation}\label{e13.40} x^* = \frac{1}{2\theta} \left[ (p_b - p_a) + \theta(x_a + x_b) \right]. \end{equation} Keep in mind that equation (40) is valid only if there really does exist a consumer who is indifferent-- if such a consumer does not exist, equation (40) will generates a number for $x^*$, but that number is meaningless. Since Apex keeps all the customers between 0 and $x^*$, equation (40) is the demand function facing Apex so long as he does not set his price so far above Brydox's that he loses even Customer 0. The demand facing Brydox equals $(1 - x^*)$. Note that if $p_b = p_a$, then from (40), $x^* = \frac{x_a + x_b}{2}$, independent of $\theta$, which is just what we would expect. Demand is linear in the prices of both firms, and looks similar to demand curves (20) and (21), which were used in section 13.2 for the Bertrand game with differentiated products. \begin{large} \newpage \noindent HOTELLING PRICING GAME INTERIOR EQUILIBRIUM Now that we have found the demand functions, the Nash equilibrium can be calculated in the same way as in section 13.2, by setting up the profit functions for each firm, differentiating with respect to the price of each, and solving the two first-order conditions for the two prices. If there exists an equilibrium in which the firms are willing to pick prices to satisfy inequalities (35) and (37), then it is \begin{equation}\label{e13.41} p_a = \frac{(2 + x_a + x_b)\theta}{3}, \;\;p_b = \frac{(4 - x_a - x_b)\theta}{3}. \end{equation} From (41) one can see that Apex charges a higher price if a large $x_a$ gives it more safe customers or a large $x_b$ makes the number of contestable customers greater. The simplest case is when $x_a = 0$ and $x_b =1$, when (41) tells us that both firms charge a price equal to $\theta$. Profits are positive and increasing in the transportation cost. We cannot rest satisfied with the neat equilibrium of equation (41), however, because the assumption that there exists an equilibrium in which the firms choose prices so as to split the market on each side of some boundary consumer $x^*$ is often violated. Hotelling did not notice this, and fell into common trap of game theory situations. Economists are used to models in which the calculus approach gives an answer that is both the local optimum and the global optimum. In games like this one, however, the local optimum is not global, because of the discontinuity in the objective function. Vickrey (1964) and D'Aspremont, Gabszewicz, \& Thisse (1979) have shown that if $x_a$ and $x_b$ are close together, no pure-strategy equilibrium exists, for reasons similar to why none exists in the Bertrand model with capacity constraints. If both firms charge non-random prices, neither would deviate to a slightly different price, but one might deviate to a much lower price that would capture every single customer. But if both firms charged that low price, each would deviate by raising his price slightly. It turns out that if Apex and Brydox are located symmetrically around the center of the interval, then, if $x_a \geq 0.25$ and $x_b \leq 0.75$, no pure- strategy equilibrium exists.\ \newpage $$ p_a = \frac{(2 + x_a + x_b)\theta}{3}, \;\;p_b = \frac{(4 - x_a - x_b)\theta}{3}. $$ $$ x^* = \frac{1}{2\theta} \left[ (p_b - p_a) + \theta(x_a + x_b) \right]. $$ Case 1: Try $x_a = 0, x_b = .7$ and $\theta =.5$. Then equation (41) says $ p_a= (2+0+.7).5/3 =0.45 $ and $p_b= (4-0-.7).5/3 = 0.55$. Equation (40) says that $ x^* = \frac{1}{2*0.5} \left[ (0.55-0.45) + 0.5(0+.7)\right]= .45$. That works out just fine. Case 2: Try $x_a = .9, x_b = .9$ and $\theta =.5$. Then equation (41) says $ p_a= (2+.9+.9).5/3 \approx .63$ and $p_b= (4-.9-.9).5/3 \approx .37$. But that means Firm B would capture the entire market! This result is nonsense, because its derivation relied on the assumption that $x_a < x_b$, which is false. Case 3. Try $x_a = .7, x_b = .9$ and $\theta =.5$. Then equation (41) says $ p_a= (2+.7+.9).5/3 =.6$ and $p_b= (4-.7-.9).5/3 =.4$. But what about the split-up of the consumers? Equation (40) says that $ x^* = \frac{1}{2*.5} \left[ (.4-.6) + .5(.7+.9) \right]= .6$. But that is less than $x_a$, violating our implicit assumption that the players split the market! Equation (40) is based on the premise that there does exist some indifferent consumer, and when that is a false premise, equation (40) will still spit out a value of $x^*$, but it will not mean anything. And, in fact, consumer $x=.6$ is not really indifferent between Apex and Brydox. He could buy from Apex at a total cost of .6 + .1(.5) = .65 or from Brydox, at a total cost of .4 + .3 (.5) = .55. There are no consumers who strictly prefer Apex, in fact. Even Apex's `home' consumer at $x= .7$ would have a total cost of buying from Brydox of $.4 + .5 (.9-.7) =0.5$, and would prefer Brydox. Similarly, the consumer at $x=0$ would have a total cost of buying from Brydox of $.4 + .5 (.9-0) = .85$, compared to a total cost of buying from Apex of $.6 + .5(.7-0)= .95$, and would prefer Brydox. \newpage The problem in both Cases 2 and 3 is that the firm with the higher price would do better to deviate with a discontinuous price cut to just below the other firm's price. Equation (41) was derived by calculus, with the implicit assumption that a local profit maximum was also a global profit maximum, or, put differently, that if no small change could raise a firm's payoff, then it had found the optimal strategy. Sometimes a big change will increase a player's payoff even though a small change would not. Perhaps this is what they mean in business by the importance of ``nonlinear thinking'' or ``thinking out of the envelope''. The everyday manager or scientist as described by Joseph Schumpeter and Thomas Kuhn concentrates on analyzing incremental changes and only the entrepreneur or genius breaks through with a discontinuously new idea, the profit source or paradigm shift.\footnote{See Schumpeter, Joseph (1911/1934) {\it Theory of Economic Development}, translated from the German 3rd Edition by Redvers Opie, Cambridge, Mass: Harvard University Press, 1934. and Kuhn, Thomas (1970) {\it The Structure of Scientific Revolutions}, Chicago: University of Chicago Press, 1970.} Hotelling should have done some numerical examples. And he should have thought about the comparative statics carefully. Equation (41) implies that Apex should choose a higher price if both $x_a$ and $x_b$ increase, but it is odd that if the firms are locating closer together, say at .9 and .91, that Apex should be able to charge a higher price, rather than suffering from more intense competition. This kind of odd result is a typical clue that the result has a logical flaw somewhere. Until the modeller can figure out an intuitive reason for his odd result, he should suspect an error. \newpage So what is the equilibrium in Case 3? (In Case 2 it is simple: $p_a=p_b=0$.) First note that in Case 3, $p_a=p_b=0$ is not an equilibrium. Suppose Apex deviated to $p_a=0.05$. If there exists an indifferent consumer , Equation (40) tells us where he is, and in this case Equation (40) yields $ x^* = \frac{1}{2*.5} \left[ (.0-.05) + .5(.7+.9)\right]= 0.75.$. Consumer $x = .75$ has a cost of buying from Apex of $.05+.5(.75-.7) = .075$, and a cost of buying from Brydox of $0+ 0.5 (0.9-0.75) = .075$, so he is truly indifferent, an indifferent consumer does exist, and Equation (40) is valid. Since Apex will now sell to all consumers in the interval $[0,.0.75]$ at the positive price of $p_a =.05$, Apex's deviation is profitable, and $p_a=p_b=0$ is not an equilibrium. In starting to look for the mixed strategy equilibrium, think about the support of the mixing distribution. In the equilibrium, Apex will mix using density $f_a(p)$ on the support $[L_a, U_a]$. Notice that it cannot be that $L_a=0$, because at that price, Apex would make zero profit, and at higher prices it can earn positive expected profit. Thus, the lower bound for mixing is strictly greater than marginal cost. How about the upper bound? $U_a = 10$, equal to the reservation price, is a good guess. I think it is not worth going into the calculations further, tho, because this is a particularly tricky mixed strategy equilibrium to calculate. That is because the payoff to a particular price $p_a$ takes one of two forms. First, if $p_b$ is low enough, Apex is shut out of the market and earns zero. Second, if $p_b$ is moderately greater, we would calculate an indifferent consumer $x^*(p_a,p_b)$ and find Apex's profit. Third, if $p_b$ is much greater, Apex gets the entire market. The expected profit from Apex charging the pure strategy of $p_a$ thus has three components, depending on the part of the density $f_b(p_b)$ that results in each possibility. This will get computationally intricate. \newpage \begin{center} {\bf The Hotelling Location Game}\\ (Hotelling [1929]) \end{center} {\bf Players}\\ $n$ Sellers. \noindent {\bf Order of Play }\\ The sellers simultaneously choose locations $x_i \in [0,1].$ \noindent {\bf Payoffs}\\ Consumers are distributed along the interval [0,1] with a uniform density equal to one. The price equals one, and production costs are zero. The sellers are ordered by their location so $x_1 \leq x_2 \leq \ldots \leq x_n$, $x_0 \equiv 0$ and $x_{n+1} \equiv 1.$ Seller $i$ attracts half the customers from the gaps on each side of him, so that his payoff is \begin{equation}\label{e13.42} \pi_1 = x_1 + \frac{x_2 - x_1}{2}, \end{equation} \begin{equation}\label{e13.43} \pi_n = \frac{x_n - x_{n-1}}{2} + 1 - x_n, \end{equation} or, for $i = 2, \ldots n-1$, \begin{equation}\label{e13.44} \pi_i = \frac{x_i - x_{i-1}}{2} + \frac{x_{i+1} - x_i}{2}. \end{equation} \bigskip With {\bf one seller}, the location does not matter in this model, since the customers are captive. If price were a choice variable and demand were elastic, we would expect the monopolist to locate at $x=0.5$. With {\bf two sellers}, both firms locate at $x= 0.5$, regardless of whether or not demand is elastic. This is a stable Nash equilibrium, as can be seen by inspecting figure 13.4 and imagining best responses to each other's location. The best response is always to locate $\varepsilon$ closer to the center of the interval than one's rival. When both firms do this, they end up splitting the market since both of them end up exactly at the center. \newpage With {\bf three sellers} the model does not have a Nash equilibrium in pure strategies. Consider any strategy profile in which each player locates at a separate point. Such a strategy profile is not an equilibrium, because the two players nearest the ends would edge in to squeeze the middle player's market share. But if a strategy profile has any two players at the same point, the third player would be able to acquire a share of at least $(0.5 -\epsilon)$ by moving next to them; and if the third player's share is that large, one of the doubled-up players would deviate by jumping to his other side and capturing his entire market share. The only equilibrium is in mixed strategies.Suppose all three players use the same mixing density, with $m(x)$ the probability density for location $x$, and positive density on the support $[g,h]$. Firm 2 has location $x$ with density $m(x)$, and Firm 3's location is greater than that with probability $1-M(x)$, so the density for Firm 2 having location $x$ and it being smaller is $m(x)[1-M(x)]$. The density for either Firm 2 or Firm 3 choosing $x$ and it being smaller than the other firm's location is then $2m(x)[1-M(x)]$. Firm 2 has location $x$ with density $m(x)$, and Firm 3's location is less than that with probability $ M(x)$, so the density for Firm 2 having location $x$ and it being smaller is $m(x) M(x)$. The density for either Firm 2 or Firm 3 choosing $x$ and it being smaller than the other firm's location is then $2m(x)M(x)$. If Player 1 chooses $x=g$, then his expected payoff is $$ \pi_1(x_1=g) = g + \int_g^h 2m(x)[1-M(x)] \left( \frac{ x-g }{2} \right) dx, $$ where $g$ is the safe set of customers to his left, $2m(x)[1-M(x)]$ is the density for $x$ being the next biggest firm location, and $\frac{ x-g }{2}$ is Firm 1's share of the customers between his own location of $g$ and the next biggest location of $x$. If Player 1 chooses $x=h$, then his expected payoff is, similarly, $$ \pi_1(x_1=h) = h + \int_g^h 2m(x) M(x) \left( \frac{ h-x }{2} \right) dx $$ In a mixed strategy equilibrium, Player 1's payoffs from these two pure strategies must be equal, and they are also equal to his payoff from a location of 0.5, which we can plausibly guess is in the support of his mixing distribution. Going on from this point, the algebra and calculus start to become fierce. Shaked (1982) has computed the symmetric mixing probability density $m(x)$ to be \begin{equation}\label{e13.45} m(x)= \left\{ \begin{array}{ll} 2 &{\rm if}\;\; \frac{1}{4} \leq x \leq \frac{3}{4} \\ 0 &{\rm otherwise}.\\ \end{array} \right. \end{equation} I do not know how Shaked came to his answer, but I would tackle the problem by guessing that $M(x)$ was a uniform distribution and seeing if it worked, which was perhaps his method too. \newpage Strangely enough, three is a special number. With {\bf more than three sellers}, an equilibrium in pure strategies does exist (Eaton \& Lipsey [1975]). Dasgupta \& Maskin (1986b), as amended by Simon (1987), have also shown that an equilibrium, possibly in mixed strategies, exists for any number of players $n$ in a space of any dimension $m$. Since prices are inflexible, the competitive market does not achieve efficiency. A benevolent social planner or a monopolist who could charge higher prices if he located his outlets closer to more consumers would choose different locations than competing firms. In particular, when two competing firms both locate in the center of the line, consumers are no better off than if there were just one firm. The average distance of a consumer from a seller would be minimized by setting $x_1 = 0.25$ and $x_2 = 0.75$, the locations that would be chosen either by the social planner or the monopolist. The Hotelling Location Model, however, is very well suited to politics. Often there is just one dimension of importance in political races, and voters will vote for the candidate closest to their own position, so there is no analog to price. The Hotelling Location Model predicts that the two candidates will both choose the same position, right on top of the median voter. This seems descriptively realistic; it accords with the common complaint that all politicians are pretty much the same. One way to modify the model is by looking at two-dimensional location. It turns out that this is difficult to analyze and generally has mixed-strategy solutions, even with just two firms or politicians. \newpage \subsection{Comparative Statics and Supermodular Games} %13.4 section. Comparative statics is the analysis of what happens to endogenous variables in a model when the exogenous variable change. This is a central part of economics. When wages rises, for example, we wish to know how the price of steel will change in response. Game theory presents special problems for comparative statics, because when a parameter changes, not only does Smith's equilibrium strategy change in response, but Jones' strategy changes as a result of Smith's change as well. A small change in the parameter might produce a large change in the equilibrium because of feedback between the different players' strategies. Let us use a differentiated Bertrand game as an example. Suppose there are $N$ firms, and for firm $n$ the demand curve is \begin{equation} \label{e13.46} Q_n = Max \{ \alpha - \beta_n p_n + \sum_{m \neq n} \gamma_m p_m, 0\}, \end{equation} with $\alpha \in (0, \infty)$ , $\beta_n \in (0, \infty)$, and $\gamma_n \in (0, \infty)$ for all $n$. Assume that the effect of $p_n$ on firm $n$'s sales is larger than the effect of the other firms' prices, so that \begin{equation} \label{e.47} \beta_n > \sum_{m \neq n} \gamma_m. \end{equation} Let firm $n$ have constant marginal cost $\kappa c_n$, where $\kappa \in \{1,2\}$ and $c_n \in (0, \infty)$, and let us assume that firm $n$'s costs are low enough that it does operate in equilibrium. The shift variable $\kappa$ represents the effect of the political regime on costs. The payoff function for firm $n$ is \begin{equation} \label{e13.48} \pi_n = (p_n - \kappa c_n)(\alpha - \beta_n p_n + \sum_{m \neq n}\gamma_m p_m). \end{equation} Firms choose prices simultaneously. Does this game have an equilibrium? Does it have several equilibria? What happens to the equilibrium price if a parameter such as $c_n$ or $\kappa$ changes? These are difficult questions because if $c_n$ increases, the immediate effect is to change firm $n$'s price, but the other firms will react to the price change, which in turn will affect $n$'s price. Moreover, this is not a symmetric game--- the costs and demand curves differ from firm to firm, which could make algebraic solution of the Nash equilibrium quite messy. It is not even clear whether the equilibrium is unique. Two approaches to comparative statics can be used here: the implicit function theorem, and supermodularity. We will look at each in turn. \noindent {\bf The Implicit Function Theorem} The implicit-function theorem says that if $f(x,y) = 0$, then \begin{equation} \label{e13.49} \frac{ \partial x} { \partial y } = - \left( \frac{ \frac{\partial f}{ \partial y} } { \frac{\partial f}{ \partial x} } \right). \end{equation} This is especially useful if $x$ is a choice variable and $y$ a parameter, because then the first-order condition takes the form $f(x, y) =0$, and the second-order condition determines the sign of $\frac{\partial f}{ \partial x}$. One only has to make certain that the solution is an interior solution, so the first- and second-order conditions are valid. In the differentiated Bertrand game, equilibrium prices will lie inside the interval ($c_n, \overline {p}$) for some large number $\overline {p}$, because a price of $c_n$ would yield zero profits, rather than the positive profits of a slightly higher price, and $\overline{p}$ can be chosen to yield zero quantity demanded and hence zero profits. The equilibrium or equilibria are, therefore, interior solutions, in which case in this well-behaved problem they satisfy the first-order condition, \begin{equation} \label{e.50} \frac{\partial \pi_n }{ \partial p_n } = \alpha - 2\beta_n p_n+ \sum_{m\neq n} \gamma_m p_m + \kappa c_n \beta_n = 0, \end{equation} and the second-order condition, \begin{equation} \label{e.51} \frac{\partial^2 \pi_n }{ \partial p_n^2 } = -2 \beta_n < 0. \end{equation} We can apply the implicit function theorem by letting $\frac{\partial \pi_n(p_n, c_n) }{ \partial p_n } = 0$ from equation (50) be our $f(x,y) = 0$ and applying equation (49). Then \begin{equation} \label{e.52} \begin{array}{ll} \frac{ \partial p_n} { \partial c_n } &= - \left( \frac{ \frac{\partial^2 \pi_n}{ \partial p_n \partial c_n} } { \frac{\partial^2 \pi_n}{ \partial p_n^2} } \right) \\ & \\ & = - \left( \frac{ \kappa \beta_n } { - 2 \beta_n } \right)\\ & \\ & = 0. \end{array} \end{equation} Thus, an increase in $n$'s individual cost parameter increases its price at a rate of $ frac {/kappa}{2}$. Keep in mind, however, that the implicit-function theorem only tells about infinitesimal changes, not finite changes. If $c_n$ increases enough, the nature of the equilibrium changes drastically, because firm $n$ goes out of business. We cannot go on to discover the effect of changing $\kappa$ on $p_n$, because $\kappa$ is a discrete variable, and the implicit-function theorem only applies to continuous variables. The implicit-function theorem is nonetheless very useful when it does apply. This is a simple example, but the approach can be used even when the functions involved are very complicated. In complicated cases, knowing that the second-order condition holds allows the modeller to avoid having to determine the sign of the denominator if all that interests him is the sign of the relationship between the two variables. \noindent {\bf Supermodularity} The second approach uses the idea of the supermodular game, an idea related to that of strategic complements. Suppose that there are $N$ players in a game, subscripted by $m$ and $n$, and that player $n$ has a strategy consisting of $k_n$ elements, subscripted by $i$ and $j$, so his strategy is the vector $ x_n = (x_{n1}, \ldots, x_{nk_n})$. Let his strategy set be $S_n$ and his payoff function be $\pi_n(x_n, x_{-n}; \tau)$, where $\tau$ represents a fixed parameter. We say that the game is a {\bf smooth supermodular game} if the following four conditions are satisfied: (A1$'$) The strategy set is an interval in $\boldmath{R^{kn}}$: \begin{equation} \label{e13.53} S_n = [\underline{x_n}, \overline{x_n}]. \end{equation} (A2$'$) $\pi_n$ is twice continuously differentiable on $S_n$. (A3$'$) (supermodularity) Increasing one component of player $n$'s strategy does not decrease the net marginal benefit of any other component: for all $n$, and all $i$ and $j$ such that $ 1 \leq i < j \leq k_n$, \begin{equation} \label{e13.54} \frac{\partial^2 \pi_n}{ \partial x_{ni} \partial x_{nj}} \geq 0 . \end{equation} (A4$'$) (increasing differences in one's own and other strategies) Increasing one component of $n$'s strategy does not decrease the net marginal benefit of increasing any component of player $m$'s strategy: for all $n \neq m$, and all $i$ and $j$ such that $1 \leq i \leq k_n$ and $1 \leq j \leq k_m$, \begin{equation} \label{e13.55} \frac{\partial^2 \pi_n}{ \partial x_{ni} \partial x_{mj}} \geq 0 . \end{equation} In addition, we will be able to talk about the comparative statics of smooth supermodular games if a fifth condition is satisfied, the increasing differences condition, (A5$'$). (A5$'$) (increasing differences in one's own strategies and parameters) Increasing parameter $c$ does not decrease the net marginal benefit to player $n$ of any component of his ownstrategy: for all $n$, and all $i$ such that $ 1 \leq i \leq k_n$, \begin{equation} \label{e13.56} \frac{\partial^2 \pi_n}{ \partial x_{ni} \partial \tau } \geq 0 . \end{equation} The heart of supermodularity is in assumptions $(A3')$ and $(A4')$. Assumption $(A3')$ says that the components of player $n$'s strategies are all {\bf complementary inputs}; when one component increases, it is worth increasing the other components too. This means that even if a strategy is a complicated one, one can still arrive at qualitative results about the strategy, because all the components of the optimal strategy will move in the same direction together. Assumption (A4$'$) says that the strategies of players $m$ and $n$ are {\bf strategic complements}; when player $m$ increases a component of his strategy, player $n$ will want to do so also. When the strategies of the players reinforce each other in this way, the feedback between them is less tangled than if they undermined each other. I have put primes on the assumptions because they are the special cases, for smooth games, of the general definition of supermodular games (see list in Appendix B). Smooth games use differentiable functions, but the supermodularity theorems apply more generally. One condition that is relevant here is condition (A5): (A5) $\pi_n$ has increasing differences in $x_n$ and $\tau$, for fixed $x_{-n}$; for all $x_n \geq x_n'$, the difference $\pi_n( x_n, x_{-n},\tau ) - \pi_n( x_n', x_{-n}, \tau ) $ is nondecreasing with respect to $\tau$. \bigskip \noindent Is the differentiated Bertrand game supermodular? The strategy set can be restricted to [$c_n$, $\overline{p}$] for player $n$, so (A1$'$) is satisfied. $\pi_n$ is twice continuously differentiable on the interval[$c_n, \overline{p}$], so (A2$'$) is satisfied. A player's strategy has just one component, $p_n$, so (A3$'$) is immediately satisfied. The following inequality is true, \begin{equation} \label{e13.57} \frac {\partial^2 \pi_n} {\partial p_n \partial p_m} = \gamma_m >0, \end{equation} so (A4$'$) is satisfied. And it is also true that \begin{equation} \label{e13.58} \frac {\partial^2 \pi_n} {\partial p_n \partial c_n} = \kappa \beta_n > 0, \end{equation} so (A5$'$) is satisfied for $c_n$. From equation (50), $\frac{\partial \pi_n}{\partial p_n}$ is increasing in $\kappa$, so $\pi_n(p_n, p_{-n}, \kappa ) - \pi_n( p_n', p_{-n}, \kappa) $ is nondecreasing in $\kappa$ for $p_n> p_n'$, and (A5) is satisfied for $\kappa$. Thus, all the assumptions are satisfied. This being the case, a number of theorems can be applied. Two of them are Theorems 13.1 and 13.2. {\bf Theorem 13.1} // {\it If the game is supermodular, there exists a largest and a smallest Nash equilibrium in pure strategies.} {\bf Theorem 13.2}// {\it If the game is supermodular and assumption (A5) or (A5$'$) is satisfied, then the largest and smallest equilibrium are nondecreasing functions of the parameter $\tau$.} Applying Theorems 13.1 and 13.2 yields the following results for the differentiated Bertrand game: (1) There exists a largest and a smallest Nash equilibrium in pure strategies (Theorem 13.1). (2) The largest and smallest equilibrium prices for firm $n$ are nondecreasing functions of the cost parameters $c_n$ and $\kappa$ (Theorem 13.2). Note that supermodularity has yielded comparative statics on $\kappa$, unlike the implicit function theorem. It yields weaker comparative statics on $c_n$, however, because it just finds the effect of $c_n$ on $p_n^*$ to be nondecreasing, rather than telling us its value. Theorem 13.2 is also useful in proving that the equilibrium here is, in fact, unique--- the largest and smallest equilibrium are one and the same. Since \begin{equation} \label{e13.59} \frac{\partial^2 \pi_n }{ \partial p_n p_m } = \gamma_m, \end{equation} it will be true that \begin{equation} \label{e13.60} - \left( \frac{\partial^2 \pi_n }{ \partial p_n^2 } \right) > \sum_{m \neq n} \frac{\partial^2 \pi_n }{ \partial p_n p_m } . \end{equation} Condition (60) is what is commonly called a {\bf dominant-diagonal condition}. It says that direct effects on profits are more important than all the indirect effects, so if one expresses the second derivatives in matrix form, the main diagonal would have the largest elements. For a three-firm case that matrix would be \begin{equation} \label{e13.61} \left[ \begin{array}{lll} \frac{\partial^2 \pi_1 }{ \partial p_1^2 } & \frac{\partial^2 \pi_1 }{ \partial p_1 \partial p_2 } & \frac{\partial^2 \pi_1 }{ \partial p_1 \partial p_3 }\\ \frac{\partial^2 \pi_2 }{ \partial p_2 \partial p_1 } & \frac{\partial^2 \pi_2 }{ \partial p_2^2 } & \frac{\partial^2 \pi_2 }{ \partial p_2 \partial p_3 }\\ \frac{\partial^2 \pi_3 }{ \partial p_3 \partial p_1 } & \frac{\partial^2 \pi_3 }{ \partial p_3 \partial p_2 } & \frac{\partial^2 \pi_3 }{ \partial p_3^2 }\\ \end{array} \right] \end{equation} Suppose there were two equilibrium price profiles, $p$ and $\hat{p}$. Theorem 13.1 says that the largest and smallest equilibria can be ranked, so for every strategy in the strategy profile, it would be true that $\hat{p} \geq p$. But because the first-order condition applies at both equilibria, we know that \begin{equation} \label{e13.62} \frac{\partial \pi_n(p) }{ \partial p_n } - \frac{\partial \pi_n(\hat{p}) }{ \partial p_n } = 0. \end{equation} One can rewrite equation (62) differently. Starting at equilibrium $p$ and moving to $\hat{p}$, the first derivative would change as all the components of $p$ changed. If we use $t$ to index the slow changes in the components, we can write these changes as \begin{equation} \label{e13.63} \int_0^1 \left\{ \left( (\hat{p_n} - p_n) \cdot \frac{\partial^2 \pi_n [t \hat{p } + (1-t) p ] }{ \partial p_n^2 } \right) + \left( \sum_{m \neq n} (\hat{p_m} - p_m) \cdot \frac{\partial^2 \pi_n [t \hat{p } + (1-t) p ] }{ \partial p_n \partial p_m } \right) \right\} dt. \end{equation} Expression (63) equals expression (62). But from equation (60), expression (63) must be negative, and equation (62) equals zero. This is a contradiction, so there cannot really be two different equilibria. The biggest and smallest equilibria are one and the same, and the equilibrium is unique. \subsection{Durable Monopoly} % sctn 13.5. \noindent Introductory economics courses are quite vague on the issue of the time period over which transactions take place. When a diagram shows the supply and demand for widgets, the $x$-axis is labelled ``widgets,'' not ``widgets per week'' or ``widgets per year.'' Also, the diagram splits off one time period from future time periods, using the implicit assumption that supply and demand in one period is unaffected by events of future periods. One problem with this on the demand side is that the purchase of a good which lasts for more than one use is an investment; although the price is paid now, the utility from the good continues into the future. If Smith buys a house, he is buying not just the right to live in the house tomorrow, but the right to live in it for many years to come, or even to live in it for a few years and then sell the remaining years to someone else. The continuing utility he receives from this durable good is called its {\bf service flow}. Even though he may not intend to rent out the house, it is an investment decision for him because it trades off present expenditure for future utility. Since even a shirt produces a service flow over more than an instant of time, the durability of goods presents difficult definitional problems for national income accounts. Houses are counted as part of national investment (and an estimate of their service flow as part of services consumption), automobiles as durable goods consumption, and shirts as nondurable goods consumption, but all are to some extent durable investments. In microeconomic theory, ``durable monopoly'' refers not to monopolies that last a long time, but to monopolies that sell durable goods. These present a curious problem. When a monopolist sells something like a refrigerator to a consumer, that consumer drops out of the market until the refrigerator wears out. The demand curve is, therefore, changing over time as a result of the monopolist's choice of price, which means that the modeller should not make his decisions in one period and ignore future periods. Demand is not {\bf time separable}, because a rise in price at time $t_1$ affects the quantity demanded at time $t_2$. The durable monopolist has a special problem because in a sense he does have a competitor--- himself in the later periods. If he were to set a high price in the first period, thereby removing high-demand buyers from the market, he would be tempted to set a lower price in the next period to take advantage of the remaining consumers. But if it were known he would lower the price, the high-demand buyers would not buy at a high price in the first period. The threat of the future low price forces the monopolist to keep his current price low. To formalize this situation, let the seller have a monopoly on a durable good which lasts two periods. He must set a price for each period, and the buyer must decide what quantity to buy in each period. Because this one buyer is meant to represent the entire market demand, the moves are ordered so that he has no market power, as in the principal-agent models in section 7.3. Alternatively, the buyer can be viewed as representing a continuum of consumers (see Coase [1972] and Bulow [1982]). In this interpretation, instead of ``the buyer'' buying $q_1$ in the first period, $q_1$ of the buyers each buy one unit in the first period. \begin{center} {\bf Durable Monopoly} \end{center} {\bf Players}\\ A buyer and a seller. \noindent {\bf Order of Play }\\ (1) The seller picks the first-period price, $p_1$.\\ (2) The buyer buys quantity $q_1$ and consumes service flow $q_1$.\\ (3) The seller picks the second-period price, $p_2$.\\ (4) The buyer buys additional quantity $q_2$ and consumes service flow $(q_1+q_2)$. \noindent {\bf Payoffs}\\ Production cost is zero and there is no discounting. The seller's payoff is his revenue, and the buyer's payoff is the sum across periods of his benefits from consumption minus his expenditure. His benefits arise from his being willing to pay as much as \begin{equation} \label{e13.64} B(q_t) = 60 - \frac{q_t}{2} \end{equation} for the marginal unit service flow consumed in period $t$, as shown in figure 13.5. The payoffs are therefore \begin{equation} \label{e13.65} \begin{array}{lllr} \pi_{seller} & = & q_1 p_1 + q_2p_2 & \\ \end{array} \end{equation} and \begin{equation}\label{e13.66} \begin{array}{lllr} \pi_{buyer} & =& [consumer \;surplus_1] + [consumer \;surplus_2]\\ & & & \\ & =& [total \;benefit_1 - expenditure_1]+ [total \;benefit_2 - expenditure_2] &\\ & & & \\ & =& \left[\frac{(60-B(q_1))q_1}{2} + B(q_1)q_1 - p_1q_1 \right] &\\ & & & \\ & & + \left[\frac{60-B(q_1 + q_2)}{2} \left( q_1 + q_2 \right) + B(q_1+q_2)(q_1+q_2) - p_2q_2 \right] & \end{array} \end{equation} Thinking about durable monopoly is hard because we are used to one-period models in which the demand curve, which relates the price to the quantity demanded, is identical to the marginal-benefit curve, which relates the marginal benefit to the quantity consumed. Here, the two curves are different. The marginal benefit curve is the same each period, since it is part of the rules of the game, relating consumption to utility. The demand curve will change over time and depends on the equilibrium strategies, depending as it does on the number of periods left in which to consume the good's services, expected future prices, and the quantity already owned. Marginal benefit is a given for the buyer; quantity demanded is his strategy. The buyer's total benefit in period 1 is the dollar value of his utility from his purchase of $q_1$, which equals the amount he would have been willing to pay to rent $q_1$. This is composed of the two areas shown in figure 13.5a, the upper triangle of area $ \left(\frac{1}{2} \right)\left( q_1 + q_2 \right) \left( 60-B(q_1 + q_2) \right) $ and the lower rectangle of area $(q_1+q_2)B(q_1+q_2)$. From this must be subtracted his expenditure in period 1, $p_1q_1$, to obtain what we might call his consumer surplus in the first period. Note that $p_1 q_1$ will not be the lower rectange, unless by some strange accident, and the ``consumer surplus'' might easily be negative, since the expenditure in period 1 will also yield utility in period 2 because the good is durable. \begin{center} {\bf Figure 13.5 Buyer's Marginal Benefit per Period in the Game of Durable Monopoly } \end{center} \epsfysize=3in \epsffile{/Users/erasmuse/AAANewChapters/Figures/f_12.5.eps} To find the equilibrium price path one cannot simply differentiate the seller's utility with respect to $p_1$ and $p_2$, because that would violate the sequential rationality of the seller and the rational response of the buyer. Instead, one must look for a subgame perfect equilibrium, which means starting in the second period and discovering how much the buyer would purchase given his first-period purchase of $q_1$, and what second-period price the seller would charge given the buyer's second-period demand function. In the first period, the marginal unit consumed was the $q_1$-th. In the second period, it will be the $(q_1+q_2)$-th. The residual demand curve after the first period's purchases is shown in figure 13.5a. It is a demand curve very much like the demand curve resulting from intensity rationing in the capacity-constrained Bertrand game of section 13.2, as shown in figure 13.2a. The most intense portion of the buyer's demand, up to $q_1$ units, has already been satisfied, and what is left begins with a marginal benefit of $B(q_1)$, and falls at the same slope as the original marginal benefit curve. The equation for the residual demand is therefore, using equation (66), \begin{equation}\label{e13.67} p_2 = B(q_1) - \frac{q_2}{2}= 60 - \frac{(q_1 )}{2}- \frac{(q_2 )}{2}. \end{equation} Solving for the monopoly quantity, $q_2^*$, the seller maximizes $q_2p_2$, solving the problem \begin{equation}\label{e13.68} \stackrel{Maximize}{q_2} q_2\left(60 - \frac{q_1 + q_2 }{2} \right), \end{equation} which generates the first-order condition \begin{equation}\label{e13.56} 60 - q_2 - q_1/2 = 0, \end{equation} so that \begin{equation}\label{e13.70} q_2^* = 60 - q_1/2. \end{equation} From equations (64) and (70), it can be seen that $ p_2^* = 30 -q_1/4$. We must now find $q_1^*$. In period one, the buyer looks ahead to the possibility of buying in period two at a lower price. Buying in the first period has two benefits: consumption of the service flow in the first period and consumption of the service flow in the second period. The price he would pay for a unit in period one cannot exceed the marginal benefit from the first-period service flow in period one plus the foreseen value of $p_2$, which from (70) is $30-q_1/4$. If the seller chooses to sell $q_1$ in the first period, therefore, he can do so at the price \begin{equation}\label{e13.71} \begin{array}{ll} p_1 (q_1)&= B(q_1) + p_2 \\ & = ( 60 - q_1/2) +( 30 - q_1/4),\\ &= 90 - \frac{3}{4}q_1.\\ \end{array} \end{equation} Knowing that in the second period he will choose $q_2$ according to (70), the seller combines (70) with (71) to give the maximand in the problem of choosing $q_1$ to maximize profit over the two periods, which is \begin{equation}\label{e13.72} \begin{array}{ll} \left(p_1 q_1 + p_2q_2 \right) & = (90 - \frac{3}{4}q_1)q_1 + ( 30 - q_1/4) ( 60 - q_1/2)\\ & = 1800 + 60q_1 - \frac{5}{8}q_1^2, \end{array} \end{equation} which has the first-order condition \begin{equation}\label{e13.73} 60 - \frac{5}{4} q_1 = 0, \end{equation} so that \begin{equation}\label{e13.74} q_1^* = 48 \end{equation} and, making use of (71), $p_1^* = 54$. It follows from (70) that $q_2^*=36$ and $p_2 = 18.$ The seller's profits over the two periods are $\pi_s = 3,240$ ($ = 54(48) + 18(36))$. The purpose of these calculations is to compare the situation with three other market structures: a competitive market, a monopolist who rents instead of selling, and a monopolist who commits to selling only in the first period. A {\it competitive market} bids down the price to the marginal cost of zero. Then, $p_1 = 0$ and $q_1 = 120$ from (66), and profits equal zero. If the monopolist {\it rents } instead of selling, then equation (66) is like an ordinary demand equation, because the monopolist is effectively selling the good's services separately each period. He could rent a quantity of 60 each period at a rental fee of $30$ and his profits would sum to $\pi_s = 3,600$. That is higher than 3,240, so profits are higher from renting than from selling outright. The problem with selling outright is that the first-period price cannot be very high or the buyer knows that the seller will be tempted to lower the price once the buyer has bought in the first period. Renting avoids this problem. If the monopolist can {\it commit to not producing in the second period}, he will do just as well as the monopolist who rents, since he can sell a quantity of 60 at a price of 60, the sum of the rents for the two periods. An example is the artist who breaks the plates for his engravings after a production run of announced size. We must also assume that the artist can convince the market that he has broken the plates. People joke that the best way an artist can increase the value of his work is by dying, and that, too, fits the model. If the modeller ignored sequential rationality and simply looked for the Nash equilibrium that maximized the payoff of the seller by his choice of $p_1$ and $p_2$, he would come to the commitment result. An example of such an equilibrium is ($p_1=60$, $p_2=200$, {\it Buyer purchases according to $q_1 = 120-p_1$, and $q_2=0$}). This is Nash because neither player has incentive to deviate given the other's strategy, but it fails to be subgame perfect, because the seller should realize that if he deviates and chooses a lower price once the second period is reached, the buyer will respond by deviating from $q_2=0$ and will buy more units. With more than two periods, the difficulties of the durable-goods monopolist become even more striking. In an infinite-period model without discounting, if the marginal cost of production is zero, the equilibrium price for outright sale instead of renting is constant--- at zero! Think about this in the context of a model with many buyers. Early consumers foresee that the monopolist has an incentive to cut the price after they buy, in order to sell to the remaining consumers who value the product less. In fact, the monopolist would continue to cut the price and sell more and more units to consumers with weaker and weaker demand until the price fell to marginal cost. Without discounting, even the high-valuation consumers refuse to buy at a high price, because they know they could wait until the price falls to zero. And this is not a trick of infinity: a large number of periods generates a price close to zero. We can also use the durable monopoly model to think about the durability of the product. If the seller can develop a product so flimsy that it only lasts one period, that is equivalent to renting. A consumer is willing to pay the same price to own a one-hoss shay that he knows will break down in one year as he would pay to rent it for a year. Low durability leads to the same output and profits as renting, which explains why a firm with market power might produce goods that wear out quickly. The explanation is not that the monopolist can use his market power to inflict lower quality on consumers--- after all, the price he receives is lower too--- but that the lower durability makes it credible to high-valuation buyers that the seller expects their business in the future and will not lower his price. \begin{small} \bigskip \noindent {\bf Notes} \noindent {\bf N13.1} {\bf Quantities as Strategies: the Cournot Equilibrium Revisited} \begin{itemize} \item Articles on the existence of a pure-strategy equilibrium in the Cournot model include Novshek (1985) and Roberts \& Sonnenschein (1976). \item {\bf Merger in a Cournot Model.} A problem with the Cournot model is that a firm's best policy is often to split up into separate firms. Apex gets half the industry profits in a duopoly game. If Apex split into firms $Apex_1$ and $Apex_2$, it would get two thirds of the profit in the Cournot triopoly game, even though industry profit falls. $\;\;\;$ This point was made by Salant, Switzer, \& Reynolds (1983) and is the subject of problem 13.2. It is interesting that nobody noted this earlier, given the intense interest in Cournot models. The insight comes from approaching the problem from asking whether a player could improve his lot if his strategy space were expanded in reasonable ways. \item An ingenious look at how the number of firms in a market affects the price is Bresnahan \& Reiss (1991), which looks empirically at a number of very small markets with one, two, three or more competing firms. They find a big decline in the price from one to two firms, a smaller decline from two to three, and not much change thereafter. Exemplifying theory, as discussed in the Introduction to this book, lends itself to explaining particular cases, but it is much less useful for making generalizations across industries. Empirical work associated with exemplifying theory tends to consist of historical anecdote rather than the linear regressions to which economics has become accustomed. Generalization and econometrics are still often useful in industrial organization, however, as Bresnahan \& Reiss (1991) shows. The most ambitious attempt to connect general data with the modern theory of industrial organization is Sutton's 1991 book, {\it Sunk Costs and Market Structure}, which is an extraordinarily well-balanced mix of theory, history, and numerical data. \item The idea of conjectural variation is attributed to Bowley (1924) and is discussed in Jacquemin (1985) and Varian (1992, p. 302). \item Do not confuse a conjectural variation of $-1$ with perfect competition, even though both may lead to the efficient output. In perfect competition, individuals do not believe that they affect the rest of the market, but if $CV = -1$, a firm believes that other firms will cut back when it produces more. Perfect competition is more like a game with players so small relative to the market that even though $CV = 0$, as in Nash equilibrium, each player correctly believes that his actions have a trivial effect on the market price. \end{itemize} \bigskip \noindent {\bf N13.2} {\bf Prices as Strategies: the Bertrand Equilibrium} %10.3 \begin{itemize} \item Intensity rationing has also been called {\bf efficient rationing}. Sometimes, however, as in section 13.2, this rationing rule is inefficient. Some low-intensity customers left facing the high price decide not to buy the product even though their benefit is greater than its marginal cost. The reason intensity rationing has been thought to be efficient is that it is efficient if the rationed-out customers are unable to buy at any price. \item OPEC has tried both price and quantity controls (``OPEC, Seeking Flexibility, May Choose Not to Set Oil Prices, but to Fix Output,'' {\it Wall Street Journal}, October 8, 1987, p. 2,29; ``Saudi King Fahd is Urged by Aides To Link Oil Prices to Spot Markets,'' {\it Wall Street Journal}, October 7, 1987, p. 2). Weitzman (1974) is the classic reference on price vs. quantity control by regulators, although he does not use the context of oligopoly. The decision rests partly on enforceability, and OPEC has also hired accounting firms to monitor prices (``Dutch Accountants Take On a Formidable Task: Ferreting Out `Cheaters' in the Ranks of OPEC,'' {\it Wall Street Journal}, February 26, 1985, p. 39). \item Kreps \& Scheinkman (1983) show how capacity choice and Bertrand pricing can lead to a Cournot outcome. Two firms face downward-sloping market demand. In the first stage of the game, they simultaneously choose capacities, and in the second stage they simultaneously choose prices (possibly by mixed strategies). If a firm cannot satisfy the demand facing it in the second stage (because of the capacity limit), it uses intensity rationing. (The results depend on this.) The unique subgame perfect equilibrium is for each firm to choose the Cournot capacity and price. \item Haltiwanger \& Waldman (unpub) have suggested a dichotomy applicable to many different games between players who are {\bf responders}, choosing their actions flexibly, and those who are {\bf nonresponders}, who are inflexible. A player might be a nonresponder because he is irrational, because he moves first, or simply because his strategy set is small. The categories are used in a second dichotomy, between games exhibiting {\bf synergism}, in which responders choose to do whatever the majority do (upward sloping reaction curves), and games exhibiting {\bf congestion}, in which responders want to join the minority (downward sloping reaction curves). Under synergism, the equilibrium is more like what it would be if all the players were nonresponders; under congestion, the responders have more influence. Haltiwanger \& Waldman apply the dichotomies to network externalities, efficiency wages, and reputation. $\;\;\;$ If the reaction functions of two firms are upward sloping, it has been said that the actions are {\bf strategic complements}, and if they are downward sloping, {\bf strategic substitutes} (Bulow, Geanakoplos, \& Klemperer [1985]). Gal-Or (1985) notes that if reaction curves slope down (as in Cournot) there is a first-mover advantage, while if they slope upwards (as in differentiated Bertrand) there is a second-mover advantage. \item Section 13.3 shows how to generate demand curves (20) and (21) using a location model, but they can also be generated directly by a quadratic utility function. Dixit (1979) states with respect to three goods 0, 1, and 2, the utility function \begin{equation} \label{e13.75} U = q_0 + \alpha_1 q_1 + \alpha_2 q_2 - \frac{1}{2} \left(\beta_1 q_1^2 + 2 \gamma q_1 q_2 + \beta_2 q_2^2 \right) \end{equation} (where the constants $\alpha_1,\alpha_2, \beta_1$, and $\beta_2$ are positive and $\gamma^2 \leq \beta_1 \beta_2$) generates the inverse demand functions \begin{equation} \label{e13.76} p_1 = \alpha_1 - \beta_1 q_1 - \gamma q_2 \end{equation} and \begin{equation} \label{e13.77} p_2 = \alpha_2 - \beta_2 q_2 - \gamma q_1. \end{equation} \end{itemize} \bigskip \noindent {\bf N13.3} {\bf Location Models} \begin{itemize} \item For a booklength treatment of location models, see Greenhut \& Ohta (1975). \item Vickrey notes the possible absence of a pure-strategy equilibrium in Hotelling's model in pp. 323-24 of his 1964 book {\it Microstatics}, but does not go on to consider the mixed-strategy equilibrium that can be found in D'Aspremont et al. (1979). \item Location models and switching cost models are attempts to go beyond the notion of a market price. Antitrust cases are good sources for descriptions of the complexities of pricing in particular markets. See, for example, Sultan's 1974 book on electrical equipment in the 1950's, or antitrust opinions such as {\it US v. Addyston Pipe \& Steel Co. et al.}, 85 Fed 271. \item It is important in location models whether the positions of the players on the line are moveable. See, for example, Lane (1980). \item The location games in this chapter model use a one-dimensional space with end points, i.e., a line segment. Another kind of one-dimensional space is a circle (not to be confused with a disk). The difference is that no point on a circle is distinctive, so no consumer preference can be called extreme. It is, if you like, Peoria versus Berkeley. The circle might be used for modelling convenience or because it fits a situation: e.g., airline flights spread over the 24 hours of the day. With two players, the Hotelling location game on a circle has a continuum of pure-strategy equilibria that are one of two types: both players locating at the same spot, versus players separated from each other by 180$^\circ$. The three-player model also has a continuum of pure-strategy equilibria, each player separated from another by 120$^\circ$, in contrast to the nonexistence of a pure-strategy equilibrium when the game is played on a line segment. \item Characteristics such as the color of cars could be modelled as location, but only on a player-by-player basis, because they have no natural ordering. While Smith's ranking of (red=1, yellow=2, blue=10) could be depicted on a line, if Brown's ranking is (red=1, blue=5, yellow=6) we cannot use the same line for him. In the text, the characteristic was something like physical location, about which people may have different preferences but agree on what positions are close to what other positions. \end{itemize} \bigskip \noindent {\bf N13.5} {\bf Durable Monopoly} \begin{itemize} \item The proposition that price falls to marginal cost in a durable monopoly with no discounting and infinite time is called the ``Coase Conjecture,'' after Coase (1972). It is really a proposition and not a conjecture, but alliteration was too strong to resist. \item Gaskins (1974) has written a well-known article on the problem of the durable monopolist who foresees that he will be creating his own future competition in the future because his product can recycled, using the context of the aluminum market. \item Leasing by a durable monopoly was the main issue in the antitrust case {\it US v. United Shoe Machinery Corporation}, 110 F. Supp. 295 (1953), but not because it increased monopoly profits. The complaint was rather that long-term leasing impeded entry by new sellers of shoe machinery, a curious idea when the proposed alternative was outright sale. More likely, leasing was used as a form of financing for the machinery customers; by leasing, they did not need to borrow as they would have to do if it was a matter of financing a purchase. See Wiley {\it et al.} (1990). \item Another way out of the durable monopolist's problem is to give best-price guarantees to customers, promising to refund part of the purchase price if any future customer gets a lower price. Perversely, this hurts consumers, because it stops the seller from being tempted to lower his price. The ``most-favored-nation'' contract, which is the analogous contract in markets with several sellers, is analyzed by Holt \& Scheffman (1987), for example, who demonstrate how it can maintain high prices, and Png \& D. Hirshleifer (1987), who show how it can be used to price discriminate between different types of buyers. \item The durable monopoly model should remind you of bargaining under incomplete information. Both situations can be modelled using two periods, and in both situations the problem for the seller is that he is tempted to offer a low price in the second period after having offered a high price in the first period. In the durable monopoly model this would happen if the high-valuation sellers bought in the first period and thus were absent from consideration by the second period. In the bargaining model this would happen if the seller rejected the first-period offer and could conclude that he must have a low-valuation and act accordingly in the second period. With a rational buyer, neither of these things can happen, and the models' complications arise from the attempt of the seller to get around the problem. In the durable-monopoly model this would happen if the high-valuation buyers bought in the first period and thus were absent from consideration by the second period. In the bargaining model this would happen if the buyer rejected the first-period offer and the seller could conclude that he must have a low valuation and act accordingly in the second period. For further discussion, see the survey by Kennan \& R. Wilson (1993). \end{itemize} {\bf Problems} \bigskip {\bf 13.1: Differentiated Bertrand with Advertising.}\\ Two firms that produce substitutes are competing with demand curves \begin{equation} \label{e13.78} q_1= 10 - \alpha p_1 + \beta p_2 \end{equation} and \begin{equation} \label{e13.79} q_2= 10 - \alpha p_2 + \beta p_1. \end{equation} Marginal cost is constant at $c=3$. A player's strategy is his price. Assume that $ \alpha > \beta/2.$ \hspace*{16pt}(13.1a) What is the reaction function for Firm 1? Draw the reaction curves for both firms. \hspace*{16pt} (13.1b) What is the equilibrium? What is the equilibrium quantity for Firm 1? \hspace*{16pt} (13.1c) Show how Firm 2's reaction function changes when $\beta$ increases. What happens to the reaction curves in the diagram? \hspace*{16pt} (13.1d) Suppose that an advertising campaign could increase the value of $\beta$ by one, and that this would increase the profits of each firm by more than the cost of the campaign. What does this mean? If either firm could pay for this campaign, what game would result between them? \bigskip {\bf 13.2: Cournot Mergers. }\footnote{ See Salant, Switzer, \& Reynolds (1983).}// There are three identical firms in an industry with demand given by $P = 1-Q$, where $Q = q_1+q_2+q_3$. The marginal cost is zero. \hspace*{16pt}(13.2a) Compute the Cournot equilibrium price and quantities. \hspace*{16pt} (13.2b) How do you know that there are no asymmetric Cournot equilibria, in which one firm produces a different amount than the others? \hspace*{16pt} (13.2c) Show that if two of the firms merge, their shareholders are worse off. {\bf 13.3: Differentiated Bertrand.}\\ Two firms that produce substitutes have the demand curves \begin{equation} \label{e13.80} q_1= 1 - \alpha p_1 + \beta (p_2-p_1) \end{equation} and \begin{equation} \label{e13.81} q_2= 1 - \alpha p_2 + \beta (p_1-p_2), \end{equation} where $\alpha > \beta$. Marginal cost is constant at $c$, where $c < 1/\alpha$. A player's strategy is his price. \hspace*{16pt} (13.3a) What are the equations for the reaction curves $p_1(p_2)$ and $p_2(p_1)$? Draw them. \hspace*{16pt} (13.3b) What is the pure-strategy equilibrium for this game? \hspace*{16pt}(13.3c) What happens to prices if $\alpha$, $\beta$, or $c$ increase? \hspace*{16pt}(13.3d) What happens to each firm's price if $\alpha$ increases, but only Firm 2 realizes it (and Firm 2 knows that Firm 1 is uninformed)? Would Firm 2 reveal the change to Firm 1? \bigskip \noindent {\bf Problem 13.4: Asymmetric Cournot Duopoly.}// Apex has variable costs of $q_a^2$ and a fixed cost of 1000, while Brydox has variables costs of $2q_b^2$ and no fixed cost. Demand is $p = 115 - q_a - q_b$. \hspace*{16pt}(13.4a) What is the equation for Apex' Cournot reaction function? \hspace*{16pt}(13.4b) What is the equation for Brydox' Cournot reaction function? \hspace*{16pt}(13.4c) What are the outputs and profits in the Cournot equilibrium? \end{small} \end{document}