MMElogit - 2011 Steven Tschantz Logit demand model Steven...

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© 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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MMElogit - 2011 Steven Tschantz Logit demand model Steven...

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