1 Differentiated Products: Motivation

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Lucinda Garrison
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1 1 Differentiated Products: Motivation Let us generalise the problem of differentiated products. Let there now be N firms producing one differentiated product each. If we start with the usual demand function and quasi functional analysis, we will get q 1 = f 1 (p 1, p 2,..., p N, Y 1 ) q 2 = f 2 (p 1, p 2,..., p N, Y 2 ) : q N = f N (p 1, p 2,..., p N, Y 2 ) There are then N N own and cross price elasticities which are too many parameters to estimate (fewer if we make further assumptions on cross price effects being symmetric but still too many). There have been attempts in the literature to get around these. But the most influential of these approaches is through modelling the more basic problem using the utility function. 2 Bresnahan (1981,1987) The background that we seek to model is very similar to other papers. The American automobile market is characterized by oligopoly and product differentiation. The aim is to empirically explore the difference of prices from marginal cost. We will first seek to draw an easier model, extending the procedures we used for Sheppard and Leslie (which are of much later vintage). But before we do so, let us state as before, that firms choose types of products that they market. But year to year, they set prices, unable to change, their product type. This paper will look at only the second part of the game: the interplay between product types, market prices and quantities. Assumption 1: There is a continuum of consumers. Each consumers purchases 1 unit or none. Assumption 2: Heterogeneity in tastes are modelled as a random variable v, which is distributed uniformly with density δ on 0, V max ]. v is the constant marginal rate of substitution between buying a car and other goods. Each consumer has a v i. Assumption 3: Each consumer i has a utility function U(x ij, Y i, v i ) = Y i + v i x j p j if some car is bought = V (Y i, v i ), otherwise Here x j is the quality of car j. Note that Consumers buy different cars not because of difference in income but because of difference in tastes. So someone who really loves cars (high v), will be more likely to buy a higher quality car. To see this note that, if a consumer i buys a car, then she will select the product j if U(x ij, Y i, v i ) U(x ik, Y i, v i ) for all k 1

2 or, in other words, the product that maximises v i x (.) p (.).(Recall in Leslie, we modelled this differently) As before, for any two goods, 1 and 2, with quality x 1 and x 2 respectively, let us calculate who is the indifferent consumer. She is one for whom: v i x 1 p 1 = v i x 2 p 2 let us denote this consumer by v jh. So v 12 = p 2 p 1 x 2 x 1 Now consider the situation where we arrange quality of goods by indices 1, 2, 3,... with x 1 being the worst quality good. Now suppose we look at the decision between goods 2 and 3.Then: v 23 = p 3 p 2 x 3 x 2 Then the demand function for good 2 is then p3 p 2 q 2 = δ v 23 v 12 ] = δ p ] 2 p 1 x 3 x 2 x 2 x 1 Notice that the own price demand derivative is: ] q = δ + p 2 x 3 x 2 x 2 x 1 The cross price derivative is: q 2 p 3 = δ 1 x 3 x 2 The smaller the quality difference between products (that is the difference in the xs), the close the own price derivate is to cross price derivates. Next we model the choice to buy any car. Comparing buying and not buying a car: 1. There is p (.) less to spend on other goods if you buy a car 2. Consumption of good x (.) adds vx (.) to utility if you buy a car ] 3. V Y, v] = max vγ l E l, vγ u E u ] where 0 < γ l < x i < γ u for all i.here γ l and γ u reflect the quality of the unknown lowest quality good or highest quality good. E l and E u are their shadow prices (Note this is another way to take care of how to model those who are not buying). The E u, E l, γ l and γ u are functions of income, tastes and other 2

3 prices. Therefore those who buy some car are those whose value v lies between vml and vmu where vml = P E l x γ l and vmu = P E u γ u x. Note that we will have to estimate E l, E u, γ l and γ u. Therefore the demand system is: p2 p 1 q 1 = δ p ] 1 E l x 2 x 1 x 1 γ 1 pj+1 p j q j = δ p ] j p j 1 x j+1 x j x j x j 1 Eu p N q N = δ p ] N p N 1 γ u x N x N x N 1 Two points about the demand system: 1. Income does not enter (2) 2. Income enters through (1) and (3). Corresponds to the fact that the most newness of a car is the most income elastic characteristic. 2.1 Firm Problem So far we have assumed that one firm produces one product. But this is not necessary. Let a firm k produce a vector of products given by X k, quantities of which are given by Q k. Let the indices of the products produced by the firm be given by J k. The only restriction is that the same product cannot be produced by both firms. Let the marginal cost of each x j be given by mc(x j ) = µ exp(x j ) Therefore the firm s shor term problem is maxπ k (p j mc(x j )).q j (P, X) p j j J k where we abstract from fixed costs components (which may affect the number of products being produced). Here P is the vector of prices of all products. To solve for p j, let us define an indictor variable C ijk which takes the value 1 if i, j J k, 0 otherwise. The first order condition for a typical product j is given by: 0 = q j +(p j mc(x j )). q j +C jj+1k (p j+1 mc(x j+1 )). q j+1 +C jj 1k (p j 1 mc(x j 1 )). q j 1 p j p j p j 3 (1) (2) (3)

4 We have to write out the first order conditions for all products: which forms a system of equations (Read Bresnahan to see more details). In the end we can write this is in a compact manner pγ = where we have to calculate the equilibrium prices by and the equilibrium quantities are given by p (X, θ) = Γ 1 (4) Q (X, θ) = Q(p, θ) (5) where θ = (E u, E l, γ u, γ l, δ, µ). The conditions for existence of equilibrium are worked out in various papers of Bresnahan. 2.2 Econometric Model: p i = p i + ε pi q i = q i + ε qi But the problem is that the quantities are all functions of x that is unknown and so are the prices. The trick in Bresnahan is to project the different product types into the space of their observable characteristics. x j (z j, β) = β 0 + t z tj β j + ε xj (6) where z tj are a list of variables that are attributes of product j. These are characteristics that are proxies for product quality (length of car, weight, engine horse power, nymber of cylinders, gallons per litre. The square root is arbitrary but intended to capture decreasing returns to each individual attribute. Assume all three error distributions are normal. The Likelihood function is then given by L 3 = φ(p P )φ (Q Q ) φ (X X ) Two technical problems: First X is unobservable. Second x j = x j+1 makes the problem discontinuous in X. So another option is to maximize the expected likelihood: L 2 = L 3 dx It can be shown that this is continuous in parameters. But this does not have any closed form. An alternate method is to use an EM algorith where in 4

5 E Step: Set X i iteration. = E X P, Q, θ i 1, β i 1, ] i 1. where i stands for the ith M Step: Treat X i as data and maximize L 3. Dempster et. al (1977) showed this procedure converges to the maximimum of the L 2. L 3 is less computationally burdensome than calculating intergrals in L 2. 3 Problem: In the model above, there is only one characteristic, the "quality" that varies across consumers. That greatly restricts the substitution patterns. Products have non zero cross price elasticities only with two products that are adjacent to it in the ranking of quality (prices). Consider, for example, the possibility that the price ranking contains, in order, a Rs 7.5 lakh Ford Echo Sport, a Rs 7 lakh Swift Dzire and Rs 8 lakh Honda City. In this case the model will guarantee that the echo sport will be substitute for Swift Dzire but not Honda City. This would need us to consider products that are differentiated in multiple dimensions (sports, non sports; big, small). 5

Bresnahan, JIE 87: Competition and Collusion in the American Automobile Industry: 1955 Price War

Bresnahan, JIE 87: Competition and Collusion in the American Automobile Industry: 1955 Price War Spring 009 Main question: In 1955 quantities of autos sold were higher while prices were lower, relative