Estimating Demand Systems for Differentiated Products

ESTIMATING DEMAND SYSTEMS FOR DIFFERENTIATED PRODUCTS

Vertical product differentiation: Consumers have the same ordinal preferences, but not the same cardinal preferences. – e.g. all consumers prefer better fuel efficiency, but their willingness to pay will differ.

 Products: Bundles of characteristics  Preferences: Defined on characteristics

 Each consumer chooses bundle that maximizes utility. Consumers have different relative (marginal) preferences for different characteristics  different choices by different consumers (assume consumes at most 1 unit of the good)  Aggregate Demand = sum over individual demands. Depends on entire distribution of consumer preferences

1 Utility of individual i for product j:

Ui,j = U(xj, pj, vi; ) j = 0,1,2….,J j=0= “outside” good (with p and q exogenous) vi are factors that determine consumer i preferences  determines the impact of those preferences on utility. A vector of parameters. xj vector of product characteristics (these do not vary over consumers) pj price of good j

Estimation from Product Level Data: The Simple Cases

2 1. The Vertical Model Bresnahan Automobile Paper (1987):  The first paper to set up a full utility, cost and Nash Equilibrium pricing model for differentiated products, and as such, serves as a useful tool for examining structural estimation of demand and cost systems.  Estimates competitive conduct for a differentiated products industry to  try to explain why automobile prices fell and quantity rose in 1955 (relative to 1954 and 1956), although 1955 was an economy boom year  Was this due to a breakdown in collusion (Rotemberg and Saloner, 1986)? Or a temporary shift to the right in the supply curve?  Bresnahan estimates the vertical model

3 Vertical model: each consumer makes a discrete choice among the set of alternatives  Alternatives = set of new cars  Outside alternative = buy used car/take bus/walk  (i) price for new car does not go to infinity (ii) AD for new cars can change due to e.g. some policy changes Consumers are differentiated by their taste for

quality i. (Everybody agrees on which products have more quality) Consumer utility:

U(xj, yi, i, pj) = ixj + yi – pj

Consumer Price car j income Consumer Quality of car j taste for quality

max Assume i ~ U(0,  ) Total per unit mass  of consumers Order goods by increasing p (must also be increasing by quality - good with a higher price and lower quality than another good will not be purchased)

4 Graph of utility (2 cars) Uj car 2: slope x2 Uj car 1: slope x1

yi – pj

p2>p1 so lower intercept for car 2

Purchase car 1 Purchase car 2 i 2

To characterise demand (assume 3 cars), we want to determine the cut-off points

Take the cut-off between car 1 and 2, 2. At this point, the consumer is indifferent between purchasing car 1 and 2

At 2: U1(2) = U2(2)

 2x1 + yi – p1 = 2x2 + yi – p2

 2x1 – p1 = 2x2 – p2

p2  p1 Solving, we get  2  x2  x1

5 p3  p2 The cut-off for 2 is similarly  3  x3  x2 Total per unit mass of consumers = 

 p2  p1  D1(p1,p2,p3) =    x2  x1 

 p3  p2 p2  p1  D2(p1,p2,p3) =     x3  x2 x2  x1 

 max p3  p2  D3(p1,p2,p3) =     x3  x2 

These demand conditions are necessary conditions

What if car 2 is more expensive than 3? In this case, 3 > 2. Given this, no consumers will choose 2, and we get a two-good problem. A good with a higher price and lower quality than another good will not be purchased. This can’t happen in equilibrium, but is a practical concern

6 From the demand functions:  Cross-price elasticity of car 3 for 1 = 0  Cross-price elasticity of car 1 for 3= 0  Only goods located closest to consumer are substitutable

We can readily derive the demand derivatives as the following: dq2  1 1       dp2  x3  x2 x2  x1  dq2  1      dp1  x2  x1  dq2  1      dp3  x3  x2 

 good j only competes with goods j–1 and j+1 in the vertical model

7 Specifying the Cost System: (only MC is relevant in determining p and q)

They assume: ln MC(xj) =  + xj

How to find xj? Assume quality is related to zj characteristics of the car e.g. size, horse power, fuel efficiency…. x j   z j

How to compute the equilibrium of the model?  Given utility and cost system, solve for vector of p and q  To do this, first write down a vector of FOC’s (these are all linear)

8 First assume single product firms: Single product FOC’s

 j ( p1,...... , pJ )  [ p j  MC j ]DJ ( p1,...., pJ )

 j dD dD  p j  D(P)  MC j  0 p j dp j dp j dD  p j  MC j   D(P) dp j

 1 1   p j1  p j p j  p j1   p j  MC j         0  x j1  x j x j  x j1   x j1  x j x j  x j1 

This then forms a system of 2J linear equations, where the variables are D1….DJ and p1….pJ , and the equations are J FOC’s and J demand equations These equations encapsulate supplier and consumer behaviour, and hence are necessary to determine the equilibrium

9 Multiproduct FOC’s

 j ( p1,...... , pJ )  [ p j  MC j ]DJ ( p1,...., pJ )

Now suppose products j and k are owned by the same firm. Then max sum of their profits: ~ and  jk   j  k ~ D  jk j Dk  p j  MC j   D j  pk  MCk   0 p j p j p j

“ extra term” internalises the effect of pj on product k’s profit  higher pj – MCj It is large if high degree of substitution between the products

From the FOC of typical product j, pj – MCj increases if the same firm produces neighbouring products, or if colluding with neighbouring firms.

10 Need to specify and simultaneously solve FOCs for all products: Can write this as a vector of FOCs:

T dD  p  MC  .x Own  D  0  dp  where ownjk = 1 if j and k owned by the same firm ownjk = 0 otherwise e.g. products 1 & 2 owned by same firm, and 3 1 1 0   by different firm: Own  1 1 0   0 0 1

1 0 0   Single product case: Own  I  0 1 0   0 0 1

1 1 1   Perfectly Collusive: Own  1 1 1   1 1 1

11 J FOC’s and J demand equations are necessary to determine the equilibrium These equations are linear (so equilibrium can be computed as a matrix inverse making it less computationally demanding than if conditions were non-linear)

Having fully specified the model, we need to estimate parameters:  = (, , , max, )

MC Utility parameter parameters quality of outside good from Characteristi c Regression

How to estimate the parameters? Using a structural approach, where we actually solve for industry equilibrium in order to derive the likelihood for any parameter vector Specifically, we want to solve for

L [p1,…pJ, q1….qJ | z1…zJ, ]

12 There is no closed form solution for L, so instead we do the following: ^ 1) pick candidate 

^ 2) solve for x1…..xJ( )

^ 3) solve for the equilibrium, P and Q, given  There are 2 J variables (J = number of cars) Need 2 J equations (J demand levels and J FOCs for each product) 4) Compute the likelihood for this parameter vector. 5) search in parameter space and find  that maximises the likelihood

In the base model, there are no unobservable components to either demand or costs. Thus, if we know the parameter vector , we could perfectly predict equilibrium. So can’t define a likelihood.

13 Bresnahan assumes the “true” Q and P are not observable to him, and instead he observes them with error

obs ^ ^ qj ( ) = qj( ) + qj qj ~ N(0,q) iid obs ^ ^ pj ( ) = pj( ) + pj pj ~ N(0,p) iid

What if prices in the data do not match the ordering of qualities x, for any parameter value ^  ? Prices observed with error – thus, must conclude that actual prices are recorded with (unobservable) measurement error. This is the only error in the defined model.

Bresnhan evaluates estimates for the Bertrand collusive, and both Bertrand single product and multi-product competitive models. He concludes that, as conjectured, collusive pricing better fits the data in 1954 and 1956, but a Nash in prices assumption fits the 1955 data better. Problems with Estimates from the Vertical Model:

14  Cross-price elasticities only with neighbouring goods, with neighbours defined by prices.  Own-price elasticities. Often not smaller for high priced goods, even though this is intuitive (higher income purchasers care less about price) and we think true in most data sets.

Vertical model is the most simple specification of demand used in estimating demand for differentiated products. Pioneering work by Bresnahan. Generalisations of the Demand Function can yield more reasonable properties and thus allow for richer estimation of demand systems Comparing the basic demand models:

1. Vertical Model

Uij = xj + pj

15 Observed price characteristics Bresnhan (1987) assumes that products compete only with the product located on either side of it in product space And that all characteristics of the product are observed (only error is due to measurement error)

2. Logit: Mean quality plus idiosyncratic consumer tastes for a product

Uij = xj + pj + j + ij or re-writing,

Uij = j + ij (j describes the mean quality of product j)

where j is unobservable (to the econometrician) demand characteristics and ij is iid (over both products and individual) This is a random co-efficient model, where each consumer consumes 1 unit of the good yielding the highest utility (including the outside good) Berry (1994). Develops an approach to estimating differentiated demand systems to (i)

16 allow for more flexible substitution patterns (substitution patterns are symmetric over all product types) using the Logit model, and (ii) to correct for price endogeneity (since we don’t observe all product characteristics, price is endogenous) To estimate the logit model, Berry proposes an IV estimator. Thus, we don’t actually have to solve for the equilibrium to estimate the demand parameters (as in Bresnahan). Aggregating over all consumers, we can define product level demand

j = xj + pj + j Which Berry (1994) shows can be written as the following linear demand equation:

j = lnsj – lns0 = xj + pj + j

unobservable Product j outside price characteristics good share of entire observable market characteristics Product j share of entire market (outside + inside goods) Thus, we have a simple IV regression of difference in the log of market shares on (xj,pj)

17 To estimate price cost mark-ups, we need to consider the cost function. Two approaches:

1) Without fully specifying a cost function, one can ‘back out’ out Marginal Cost (as in Mariuzzo, Walsh and Whelan, 2003). Using observed p and the estimated demand parameters (thus, not fully structural model), you ‘back out’ MC from system of J FOCs (such that MCj = price + mark up). Note that MC in this case is estimated with error (since demand parameters are estimated with error).

2) Simultaneously estimate Demand and Supply (as proposed in Berry, 1994).

DEMAND: j = lnsj – lns0 = xj + pj + j

SUPPLY: given MCj = ’xj + Qj + j , Instrument Demand using cost shifters not in demand Instrument Supply using demand shifters not in cost Jointly estimate demand and costs using GMM, where the set of instruments need to be jointly orthogonal to the errors

18 Problems with Estimates from Logit Models 1. Own-price derivative only depends on market shares. Two goods with the same share must have the same mark-up in a single-product firm Nash equilibrium in prices 2. The Independence of Irrelevant Alternatives property of the Logit Model. This indicates that two goods with the same shares have the same cross-price elasticities with any other good.

3. Nested Logit: This is just a simple extension of the Logit case, to allow for the fact that you have various segments or groups in the market. It allows for correlations in error terms for products within a group, where groups are exogenously specified.

uij = j + ig+ (1-  )ij 0 <  < 1. where j = xj - pj + j Higher values of  indicate greater substitutability of products within groups

19 4. BLP (Berry, Levinsohn, Pakes):

Random co-efficients model

Uij = xji + ipj + j + ij Thus, the  co-efficients vary across individuals (different people have different tastes for different characteristics) Random coefficients are designed to capture variations in the substitution patterns. Although more realistic than the Logit model, the estimation procedure is not so straightforward. Requires (i) simulation to derive demand and (ii) numerical methods to solve out J FOC’s

20 5. An Alternative Discrete Choice Models Discrete Choice models can allow for heterogeneity of consumers. Instead of estimating demand using discrete choice models, one could use a Representative Consumer Choice Model – these allow the representative consumer to buy more than one product in varying amounts. Examples include distance metric model (Pinkse, Slade & Brett, 2002), or the multi-stage budgeting model (Hausman, Leonard & Zona, 1994).

Heterogenous Consumer models however are generally preferred as: 1. They use more of the available information 2. “ Differences” among consuming units (often households) includes differences caused by such things as “demographics”. So allowing for consumer heterogeneity is more realistic