Oxford University Multinomial Logit Model Economics Questions


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1. Consider a version of the multinomial logit model of demand for differentiated products. Let consumer
i’s utility from good j (produced by firm j) be
Wij = x;ß-ap; + &j + € ij
= 8j + €ij
where each ej has the “type-I extreme value distribution” discussed in class. As usual, there is also an
outside good (“good 0”) giving utility (normalizing do = 0)
U?0 = €10.
j = 1,…, J.
a. Show that market shares depend only on the values of d; for j = 1,…, J.
b. Derive an expression for the price elasticity of demand for good j and for the cross-price elasticity of
demand for good j with respect to the price pk of good k.
c. Show that in the multinomial logit model the identities of the top three competitors faced by firm k
(i.e., the firms that would attract the largest share of k’s consumers if pk were to rise) can be determined
from the market shares alone (no price data needed!).
d. Offer an argument that this feature of the mulitnomial logit model, while convenient, may not be
desirable when attempting to use data prices and quantities to asure demand elasticities.
2. Consider a multinomial logit model of differentiated products demand. In each market t, there is a
continuum of consumers i and two products (plus the outside good). Consumer i’s conditional indirect
utility for good j is given by
Uijt = ?jt – ?pjt + Eijt,
where it is a measure of good j’s quality, pjt is its price, and each Eijt is an independent draw from a type-1
extreme value distribution. Each good j in a given market is produced by a different firm with constant
marginal cost Cjt. In each market, firms compete under full information by simultaneously choosing prices.
For parts a-d below, focus on a single market (so you may drop the subscript t).
a. Provide expressions for the following
(i) the market share of good j
(ii) the own-price elasticity of demand for good j
(iii) the cross-price elasticity of demand for good j with respect to pk (k = j)
b. Derive the first-order conditions characterizing each Nash equilibrium price p, in terms of cj, 8; and
c. Holding p2 fixed, how does firm 1’s optimal price vary with its quality A?? Holding p? fixed, how does
firm 2’s optimal p2 vary with A??
d. Now consider the case of many markets. Suppose that the firm producing good 1 is a national firm,
competing in all markets, with the same marginal cost regardless of the market in which sales take place.
“Firm 2” (the producer of good 2), however, is a different firm in each market. Based on your previous
answers, give at least two reasons that firm 1’s market share and equilibrium price may vary across markets.
e. Suppose you are able to observe firm 1’s price pit and market share sit in each market. Argue that
the first order conditions for firm 1 and the variation discussed on part e allow you to either (i) determine
the value of the parameter a, or (ii) determine that the assumptions of the model cannot all be true.
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