Horizontal Differentiation Spatial & Price Competition (H. Hotelling) http://wiwi.uni-bonn.de/shaked/ISET
Firms (sellers on the beach) choose location and price. Assume there are two firms charging p, q. What is the demand of each??
As before, customers are uniformly distributed on an interval of length 1 each buys one unit of a homogeneous good sold by various firms. Customers pay transportation cost t per unit of distance. They choose a firm to minimize the total price of the good (transportation cost + mill price).
The Game Form This defines a two stage game between the firms They first choose location (simultaneously) After this stage, their locations are revealed and they (simultaneously) choose prices.
1 2 2 1 1 2 2 Location Information set Location Location revealed Price 2 Information set 2 Payoffs
q p b c x a a + b + c = 1 Given the two locations and prices we compute he demand of each firm lines with slope t tx q p x x b c x a a + b + c = 1 A customer whose distance from the right firm is x pays a total of p + tx when he buys from the right firm.
q p x x b a c A customer buying from the right firm pays according to this function Similarly, for the left firm Each customer pays the lower of the two costs The demand of the right firm is therefore
q p x x x b a c The marginal consumer can be found by:
a c b p q x The demands are: The profits are (assuming no production costs):
The profits: Given the locations, each firm chooses its price to maximize its profit (given the price and location of the other firm) The equilibrim prices: Note that: p* + q* = 2t, q*- p* =2t(b-a)/3
The Equilibrium Profits: This increases as c diminishes i.e the right hand firm will move towards the other.
The Principle of Minimal Differentiation It seems that the two firms tend to move towards each other to increase their profits. Hotelling termed it: The Principle of Minimal Differentiation It has implications for location of firms and the product quality. Hotelling termed it: The Principle of Minimal Differentiation It has implications for location of firms and the product quality. But Hotelling’s analysis is false d’Aspremont, C, J. Gabszewicz and J.-F. Thisse On Hotelling’s Stability in Competition Econometrica 17:1145-1151 (1979) (1979)
Once two firms are sufficiently close, there can be undercutting. q p p x a c b Once two firms are sufficiently close, there can be undercutting. Undercutting means lowering the price so that the other firm is left with no customers.
q a c b x Undercutting occurs discontinuously. As long as there is no undercutting the Left firm has all the b customers. Then they are all indifferent between the two firms. And any further decrease shifts them to the Right firm
q tc x a c b The Right firm can undercut the Left one when q – tc > 0. The right firm can then charge p < q – tc and undercut the other.
Similarly, the Right firm is undercut when q x a c b Similarly, the Right firm is undercut when q + tc < p.
p π Given the locations of the two firms, and the price q of the Left firm, the profit of the Right firm as a function of its own price p is given by:
For p < q – tc the Right firm undercuts and has the whole market, π q-tc For p < q – tc the Right firm undercuts and has the whole market, π =1p .
p π q-tc q+tc For p > q + tc the Right firm is undercut and has the no market share π = 0
Recall the computation in slide 9: π q-tc q+tc In the intermediate interval, both firms co-exist with positive market shares. Recall the computation in slide 9:
p π q-tc q+tc For the intermediate interval, both firms co-exist with positive market shares. Recall the computation in slide 9: The profit of the Right firm is an inverted parabola in p.
p π q-tc q+tc For the intermediate interval, both firms co-exist with positive market shares. Recall the computation in slide 9: The profit of the Right firm is an inverted parabola in p.
p π q-tc q+tc An equilibrium p cannot be in the undercutting intervals. For then, one of the firms can gain by lowering its price.
where the firms co-exist with positive market shares. π q-tc q+tc If a pure strategy equilibrium exists, the equilibrium p must be in the interior of the intermediate interval, where the firms co-exist with positive market shares.
Equilibrium prices are at least 2t/3 > tc . From slide 10: The equilibrim prices, and the equilibrium profits are: For a sufficiently small distance (c) between the firms, each could undercut the other. Equilibrium prices are at least 2t/3 > tc . (c is small < 2/3)
From slide 10: The equilibrim prices, and the equilibrium profits are: However, in equilibrim no firm wants to deviate, by undercutting and serving the whole market: Substitute the values, to find:
As c → 0, the two inequalities become: It follows that: Contradiction !! since 0 a 1
Hence, it cannot be true that neither firm gains by. undercutting Hence, it cannot be true that neither firm gains by undercutting. (at least one firm would gain by deviating) It follows that there exists no equilibrium in pure strategies (prices) when the firms are close. There exist equilibria in mixed strategies. but as the distance between the two firms diminishes they approach the Bertrand equilibrium (zero profit). Hotelling’s Principle of Minimum Differentiation is false.
A beach with quadratic transportation costs Assume now that a customer pays the mill price plus td2 where d is his distance from the shop. q p x a c b
With quadratic transportation costs there is no discontinuity With quadratic transportation costs there is no discontinuity. By lowering price a firm gradually steals the other’s customers. x a c b
A beach with quadratic transportation costs x a c b c - x x
A beach with quadratic transportation costs Both firms wish to move away from each other to the edge of the interval Maximal DIfferentiation x x b c a
A Circular City Salop, S.: Monopolistic Competition with outside goods Bell Journal of Economics, 10, 141-156, 1979p
A Circular City Customers are uniformly distributed along the circumference of a circle. The circumference of the circle is 1. Firms selling a homogeneous product enter, pay entry fee f (fixed costs) and are symmetrically distributed along the circle. Firms compete in prices but all charge the same price (symmetric equilibrium). Customers buy one unit of the product, they pay linear transportation cost of t per unit of distance. They go to the firm in which the total cost of a unit is lowest for them. we ignore the existence question of pure equilibrium in prices. We could take the costs to be quadratic.
A Circular City Assume that n firms entered the city. 1/n 1/n 1/n
gross profit, net profit is: Π – f A Circular City Assume that n firms entered the city. Assume that all firms except one, charge q, find the optimal price for that firm. gross profit, net profit is: Π – f p q q x 1/n
x A Circular City p q q x Assume that n firms entered the city. Assume that all firms except one, charge q, find the optimal price for that firm. x In a symmetric equilibrium, the p that solves this equation is the q used by all other firms. p q q x 1/n
A Circular City The common equilibrium price is Free Entry means that firms will enter as long as their net profit is positive Hence, the number of firms that will enter in competiton is:
A Circular City – A Planner Now assume that a planner determines the number of entrants (n). They locate symmetrically around the circle. His aim is to choose n so as to minimize total costs: cost of entry + transportation costs Each consumer buys a single unit, so the price is merely a transfer within the society.
A Circular City – A Planner
Market forces lead to more variety than is optimal. In general, new products create new demand but they also steal demand from existing products. New demand is good for consumers, and does not harm the industry. Stealing demand enhances competition, may be good for consumers but is not good for the firms. The balance between the two effects may lead to too much or too little product differentiation.
Dixit, A. and J. Stiglitz: Monopolistic Competition and Optimum Product Diversity, American Economic Review, 67: 297-308, 1977
Monopolistic Competition Firms earn 0 profits (Free Entry). Yet, each firm faces a downward sloping demand * * In contrast to Perfect Competition, where a firm cannot gain by changing its price.