© 2011  Steven Tschantz
Logit demand model
Steven Tschantz
3/1/11
Problem
Determine the impact of a merger between two of a number of competing firms in a differentiated product market where
consumers choose between products. Find an analytically and computationally tractable choice model of demand that can be
used for more than just two products, that has a reasonably small number of free parameters, can be calibrated to limited
demand data, and can be used for calculating Nash equilibrium, profit maximizing, pricing for firms selling the products.
Model
We imagine that firms are profit maximizing before the merger, use current price and quantity data and any available demand
elasticity estimates to calibrate a demand model and estimate firms' marginal costs. Then, assuming the firms will be profit
maximizing after the merger, we can determine price increases for the merged firm and the responses of the other firms in Nash
equilibrium. To make any of this practical, we need a reasonable computational model of demand. In particular, if the model
has too many parameters, then it will be difficult to get enough data to adequately calibrate the model. We will have to make
some arbitrary simplifying assumptions in any case, and there is some sense in which the best idea is to take a model derived
from certain basic assumptions. What we need in merger analysis is some idea of the magnitude of the effects of a merger, not
a precise prediction. We rarely have enough data or time to make a detailed industry projection. If enough data is available to
contradict the basic assumptions of the model, then a more elaborate model can be formulated.
The specification of a parsimonious demand model (one with relatively few parameters) is a practical question, not a theoreti
cal one. We have given a general choice model of demand; it is simply(?) a matter of defining the joint distribution of values
for products over the population of all potential consumers. The choice probabilities of products, as functions of the prices of
products, are given by certain (multidimensional) integrals, which can at least be numerically integrated. Derivatives of choice
probabilities, as derivatives of the integrals defining the choice probabilities, can be expressed in turn as more complicated
integrals, so elasticities and first order equilibrium conditions can all be written down. Computation of profit maximizing
prices can be computed by numerical root finding where the functions to be evaluated are computed by numerical integration.
At least, in principle.
A general enough demand model would allow us to calibrate the model to arbitrary quantities at given prices and own and
cross price elasticities. If there are
n
products, the elasticity matrix has
n
2
entries. However, rarely is there enough good data
to estimate a full elasticity matrix. Such an estimation would require prices of the products to vary sufficiently and vary
independently of each other enough to determine how demand shifts between products in response to changes in each price
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 Spring '11
 Schantz
 Math, Supply And Demand, Cumulative distribution function, CDF, p1

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