Automobile Prices in Equilibrium Berry, Levinsohn and Pakes. Empirical analysis of demand and supply in a differentiated product market.

Transcription

1 Automobile Prices in Equilibrium Berry, Levinsohn and Pakes Empirical analysis of demand and supply in a differentiated product market. about 100 different automobile models per year each model has different observed characteristics: size, fuel efficiency, etc. unobserved characteristics: brand image, reliability, etc. Price is determined in a oligopolistic market equilibrium. 1

2 Data availability observed product characteristics: size, fuel efficiency. aggregate data: sales share, consumer data: income distribution, etc. no individual level consumer data. no cost information. 2

3 Hedonic Approach automobile: bundle of observed hedonic characteristics (size, mileage, etc) and unobserved hedonic characteristics (brand image, etc). reduce 100 models into a bundle of hedonic characteristics. Questions Derive own cross price elasticities of different models. Consumers values of different characteristics. 3

4 Policy impact: expected demand of a new product. Impact of voluntary export restraint in the U.S. automobile industry. 4

5 Model U ( ζ i, p j, x j, ξ j, θ ) : Consumer i s utility from purchasing a model ζ i : consumer characteristics. Distribution is known. (income, family size, etc) p j : price of model j. x j : observed characteristics of model j. ξ j : unobserved characteristics of model j. θ: parameters. 5

6 Consumer i chooses model j if and only if U ( ζ i, p j, x j, ξ j, θ ) U (ζ i, p r, x r, ξ r, θ) for all r = 0, 1,..., J, r j r = 0: consumer did not buy a car. A j : a set of characteristics that buys model j. A j = { ζ : U ( ζ i, p j, x j, ξ j, θ ) U (ζ i, p r, x r, ξ r, θ) ; r = 1, 2,..., J } A j : Consumers whose random utility shock ζ is such that she obtains highest utility from purchasing model j. 6

7 Each consumer buys at most one car per year. Market share of model j: fraction of consumers who buy model j ( s j pj, x j, ξ j ; θ ) = P 0 (dζ) ζ A j ( Demand: Ms j pj, x j, ξ j ; θ ) where M: total number of consumers. 7

8 Firms: Profit:π j = r F j (p r mc r ) Ms r (p, x, ξ; θ) Flat MC: ln (mc r ) = w r γ + ω r w j : observable cost characteristics firms. ω j : unobservable cost characteristics, assume E [ω r x, w] = 0 F j : models for a firm. 8

9 Bertrand Nash equilibrium of the differentiated goods economy. Given the prices of other firms p k, k / F r, F.O.C.for profit maximization is s j (p, x, ξ, θ) + k F r (p k mc k ) s k (p, x, ξ; θ) p j = 0 9

10 Moment conditions for estimation Demand Side: Suppose individual i s utility for car model j has the following specification: u ij = u ( x j, p j, θ ) + ξ j + ɛ ij x j : model specific observed characteristics. ξ j : model specific unobserved characteristics (brand image, etc), assumed to be mean zero, conditional on x, w. That is, E [ξ x, w] = 0 10

11 ɛ ij : individual specific unobserved taste component, assumed to be i.i.d. extreme value distributed. p j : price of the model. The utility of not buying the car (outside option) is set to 0. Then, the probability of consumers who buy model j can be expressed as follows, if ɛ is extremely value distributed. exp [ u ( p j, x j ; θ ) ] + ξ j s j (p, x, ξ; θ) df (ɛ) = ɛ A j 1 + J j=1 exp [u (p l, x l ; θ) + ξ] for j = 1,..., J 11

12 The share of not buying a car: s 0 (p, x, ξ; θ) = J j=1 exp [u (p l, x l ; θ) + ξ] Then, ln [ s j (p, x, ξ; θ)] ln [s0 (p, x, ξ; θ)] = u ( p j, x j ; θ ) + ξ j Now, we know that E [ξ x, w] = 0. Hence, E [ξ (x, w)] = E {E [ξ x, w]} (x, w) = 0 If we want our model parameter to be close to the true parameter value, then the parameter should satisfy the following sample analog of the above equation. 1 J J j=1 ξ j ( x j, w j ) = 0 12

13 That is, the parameters should be chosen so that the following equation is as close to zero as possible. 1 J J j=1 [ lnsj lns 0 u ( p j, x j ; θ )] ( x j, w j ) Steps to construct demand side moments: Step 1: From the data, derive the share of model J. No. of model j sold s j = Tota sample no. From the data, derive the no purchase share: s 0 = No. of no purchase Total sample no. 13

14 Step 2: Given p j,x j from the data, and given parameters θ, derive the sample moment 1 J J j=1 [ lnsj lns 0 u ( p j, s j ; θ )] ( x j.w j ) Given the parameters, you can get back ξ j by using ξ j = lns j lns 0 u ( p j, x j ; θ ) 14

15 Supply Side: Marginal Cost Equation ln ( mc j ) = wj γ + ω j w j : observable cost characteristics. ω j : unobservable cost characteristics: error term of the regression equation. γ: parameters to be estimated. 15

16 There is no data on marginal cost mc j. We recover the marginal cost from the model. That is, from the F.O.C. of the profit maximization problem of the firm. s j + (p k mc k ) s k (p, x, ξ, θ) = 0 p k F r k s j : market share of model j: calculated from the automobile purchase data. p k : price of model k: get it from the data. Also, we know that s k (p, x, ξ; θ) = exp [ u ( p j, x j ; θ ) + ξ ] 1 + J l=1 exp [u (p l, x l, ; θ) + ξ] 16

17 By taking derivative of the above equation with respect to p j, we can derive s k p j. Together, we get jk = s k p j mc = p 1 s if j, k F r, same firm s model 0 otherwise Then, we get the following linear regression equation: [ p (p, x, ξ, θ) 1 s ] j = w jγ + ω j We know that E [ω k x, w] = 0. Hence, E [w (x, w)] = E {E [ω x, w] (x, w)} = 0 17

18 If we want our model parameters to be close to the true parameter value, then the parameter should satisfy the following sample analog to the above equation. 1 J J j=1 ω j ( x j, w j ) = 0 That is, the parameters θ,γ should be chosenso that teh following equation is as close to zero as possible. 1 { [p J (p, x, ξ, θ) 1 s ] } J j w jγ ( ) x j, w j j=1 In sum, parameters θ, γ should be chosen so that the following two sets of moments are made as close to zero as possible. Another possibility of instruments: observable demand and cost characteristics of other firms, market characteristics such as number of firms, etc. 18

19 Generalized Method of Moments (GMM) Estimation. Moments of consumer choice problem: 1 J J j=1 m 1 (p, x, w, θ, γ) [ lnsj lns 0 u ( p j, x j ; θ )] ( x j, z j ) Moments from firms profit maximization 1 J J j=1 m 1 (p, x, w, θ, γ) { [p (p, x, ξ, θ) 1 s ] } j w jγ ( ) w j, z j We choose θ,γ to make those moments for consumer choice problem and profit maximization problem as close to zero as possible. 19

20 A good way to do that is to choose θ, γ to minimize the weighted sum of squares of the moments. [ ] [ ] m1 (p, x, w, θ, γ) m1 (p, x, w, θ, γ) G (θ, γ) W m 2 (p, x, w, θ, γ) m 2 (p, x, w, θ, γ) 20

21 Further Issues in Estimation Other firms (x, w) and market variable such as number of firms are used as instruments for the price, which is endogenous. Assume that u ( x j, p j, θ ) = x j β p j α. Then, the cross price elasticity is s j (x, p, θ) p l / sj p l = s l p l u(x l, p l, θ) p l = s l p l α That is, the cross price elasticity of the price of model s to a model j only depends on the model l s price and model l s share. 21

22 Yugo and Benz have the same market share. Increase in BMW price has the same effect on the market share of Yugo and Benz. Same increase in market share for Benz and Yugo, which is unreasonable. Consumers who choose Benz have a preference over a larger car (because of larger family size, higher income, etc.). They would be more interested in BMW. Introduce random coefficients. u itj = αln ( y jt p jt ) + xjt β + ξ jt + k σ k x jkt v ik + ɛ itj u it0 = αln (y 0t ) + ξ 0t + σ 0 v i0 + ɛ it0 22

23 Suppose x j1t, j = 1,...J is the size of a model. Then, for consumers who have high v i1, size of a car is very important for her utility. She is more likely to purchase cars with larger size (such as Benz) than the others. For her, Benz and BMW are in a similar group and more attractive than Yugo. For people like her, increase in BMW price will increase Benz sales more than Yugo sales. Suppose that x j2t is the mileage per gallon. Consumers who buy Benz: likely to have high v 1i (love for size). Likely to buy more Benz when BMW price increases. Likely to not react to Fiat price increase. 23

24 Consumers who buy Yugo. Likely to have high v 2i (care for fuel efficiency). Likely to buy more Yugo when Fiat price increases. Likely to not react to BMW price increase. Cross elasticities are high for similar car models and low for different car models, which is more realistic. Simulators for market share Step 1. Draw from income distribution y mt Draw v mk, k = 1,..., K from standard normal distribution. 24

25 Calculate the utility component without the utility shock. u mtj = αln (y mt p mt ) + x jt β + ξ jt + k σ k x jkt v mk u mt0 = αln (y mt ) + ξ 0t + k σ 0 v m0 Then, calculate the shares. s mj (p, x, ξ, θ) = exp [ u mtj + ξ j ] 1 + J l=0 exp [u mtl + ξ l ] 25

26 Step 2 Repeat Step 1 m = 1,..., M times and derive the average. s j (p, x, ξ; θ) = 1 M m=1 Ms mj (p, x, ξ, θ) We also need to derive ξ that makes the simulated market share s j, j = 0,..., J equal to the observed market share s d j, j = 0,..., J. Fortunately, the algorithm of finding the right ξ is known to be a contraction. Hence, even the number of parameters ξ is large, the solution algorithm is relatively easy. 26

27 Data: Product characteristics: from Automobile News Market Data Book: no. of cylinders, no. of doors, weight, engine displacement, horsepower, length, width, EPA MPG rating, front wheel drive, automatic transimission, power steering, air conditioning. 27

28 Price: list retail price for the base model (1983 $) Additional data: Gasoline price, March Current Population Survey: income distribution of consumers, Consumer reports reliability ratings. 28

29 Trends from 1971 to 1990 No. of models: increases from 72 (1974) to 150 (1988). Sales per model: decrease Price: flat until 1979, rises 50% during 80 s More fuel efficient. Increase in Japanese cars market share, European shares constant. 29

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31 Estimation Results: Logit Demand estimation with Dependent variable: lns j lns 0 Variable OLS logit IV logit Const (0.253) (0.493) HP/Weight (0.277) (0.909) Air Cond (0.073) (0.248) MP$ (0.043) (0.086) Size (0.125) (0.247) Price (0.004) (0.123) No. Inelastic Demands R sq n.a. 30

32 1494 models have elasticity less than 1. Not consistent with profit maximization. Possible correlation between price and unobserved product attributes. IV: size, No. of own and rival firm s products, etc. Only 22 products that are inelastic. 31

33 Price Elasticity: (1 s j )p j α Higher share models have higher markup and lower share models have lower markup, which is unrealistic. More expensive cars have lower markup and cheaper cars have higher markup. 32

34 Results from the Full Model Demand Parameter Std. Error Mean β s Constant (0.941) HP/weight (2.019) Airbag (0.891) Miles per $ (0.320) Size (0.610) Std. Dev σ β Constant (1.485) HP/weight (1.885) Airbag (1.695) Miles per $ (0.272) Size (0.585) ln (y p) (6.427) 33

35 Cost Parameter Std. Error Constant (0.194) ln(hp/weight) (0.056) Airbag (0.038) ln(mpg) (0.055) ln(size) (0.081) Trend (0.002) Negative MPG coefficient: popular cars were the ones with low fuel efficiency. The elasticity of demand w.r.t. MP$ declines monotonically with the car s MP$ rating. Individuals who purchase BMW, Lexus are not concerned with fuel efficiency. 34

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37 Consumers who purchase the smallest cars value more on increased acceleration. All 2217 models in the sample have elasticity higher than 1. The most elastically demanded products are those that are in the most crowded market segments, the compact and subcompacts. Cross price elasticities are large for cars with similar price characteristics. Markup: rises monotonically with price. Lowest markup cars: Mazda, Sentra and Escort. Highest markup: Lexus and BMW 35

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