Math Spring 2017 Mathematical Models in Economics

2017 - Steven Tschantz Math 3660 - Spring 2017 Mathematical Models in Economics Steven Tschantz 2/7/17 Profit maximizing firms Logit demand model 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

2 05.1-LogitDemandModel.nb 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 theoretical 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 alone. But prices frequently vary up and down together, because of competition between firms, in response to changing conditions and changing demand. It is simple enough to calibrate a linear or constant elasticity demand to n quantities and n 2 elasticities at given prices, if we had such estimates. Instead, focusing on a choice model of demand constrains how demand shifts between products. What one product loses in demand because of an increase in price, may partially add to the demand for competing products, the rest of the lost demand choosing not to purchase any product. We would not expect to see more added demand for competing products than is lost by the product with increased price. However further simplifications are necessary. Independent values - normal distributions It is hard to specify and evaluate multivariate distributions where the variables are not independent. In any case, we would not want to have to calibrate a model requiring perhaps n 2 independent covariance parameters. Instead, consider models where the consumer values of the different products vary independently over the population of consumers. For example, suppose there are n 3 products, selling for prices p 1, p 2, and p 3, where consumer values are independent random variables V i, given by cumulative distribution functions F i (x), or probability density functions f i (x). Then the joint probability density function for values is f(x 1, x 2, x 3 ) f 1 (x 1 ) f 2 (x 2 ) f 3 (x 3 ), and the joint distribution function is F(v 1, v 2, v 3 ) Prob(V 1 v 1, V 2 v 2, and V 3 v 3 ) v 1 v 2 v 3 - - - f(x 1, x 2, x 3 ) dx 3 dx 2 dx 1 v 1 v - f 1 (x 1 ) dx 2 v 1 - f 2 (x 2 ) dx 3 2 - f 3 (x 3 ) dx 3 F 1 (v 1 ) F 2 (v 2 ) F 3 (v 3 ) The choice probability for, say, product 1 is then

05.1-LogitDemandModel.nb 3 π 1 (p 1, p 2, p 3 ) Prob(V 1 - p 1 > 0, V 2 - p 2, and V 3 - p 3 ) x 1 -p 1 +p 2 - x 1 -p 1 +p 3 p1 - f(x 1, x 2, x 3 ) dx 3 dx 2 dx 1 x p1 f 1 (x 1 ) 1 -p 1 +p 2 x - f 2 (x 2 ) dx 2 1 -p 1 +p 3 - f 3 (x 3 ) dx 3 dx 1 p1 f 1 (x 1 ) F 2 (x 1 - p 1 + p 2 ) F 3 (x 1 - p 1 + p 3 ) dx 1 0 f 1 (u 1 + p 1 ) F 2 (u 1 + p 2 ) F 3 (u 1 + p 3 ) du 1 For example, with V i independent normally distributed variables with means μ i and standard deviations σ i, we would compute In[1]: Clear[choiceprob1, mu1, mu2, mu3, sigma1, sigma2, sigma3] In[2]: choiceprob1[p1_, p2_, p3_] Integrate[PDF[NormalDistribution[mu1, sigma1], u1 + p1] * CDF[NormalDistribution[mu2, sigma2], u1 + p2] * CDF[NormalDistribution[mu3, sigma3], u1 + p3], {u1, 0, Infinity}] (-mu1+p1+u1)2 e- 2 sigma12 Erfc mu2-p2-u1 mu3-p3-u1 Erfc 2 sigma2 2 sigma3 Out[2] 0 4 2 π sigma1 du1 And though Mathematica will try very hard, there is no reasonable formula for this integral. Instead, we would have to do a numerical integration, which since it is one-dimensional would not be too bad. Assign some values for the means and standard deviations for example. In[3]: Clear[choiceprob1, mu1, mu2, mu3, sigma1, sigma2, sigma3] In[4]: mu1 20.; mu2 15.; mu3 25.; sigma1 5.; sigma2 2.; sigma3 7.; In[10]: choiceprob1[p1_, p2_, p3_] : NIntegrate[PDF[NormalDistribution[mu1, sigma1], u1 + p1] * CDF[NormalDistribution[mu2, sigma2], u1 + p2] * CDF[NormalDistribution[mu3, sigma3], u1 + p3], {u1, 0, Infinity}]; In[11]: choiceprob1[20., 15., 25.] Out[11] 0.314441

4 05.1-LogitDemandModel.nb In[12]: Plot[choiceprob1[p1, 15., 25.], {p1, 0., 30.}] 1.0 0.8 Out[12] 0.6 0.4 0.2 5 10 15 20 25 30 In[13]: ContourPlot[choiceprob1[p1, p2, 25.], {p1, 0., 30.}, {p2, 0., 20.}] 20 15 Out[13] 10 5 0 0 5 10 15 20 25 30 Computing the elasticity of demand requires that we evaluate the derivative of the choice probability integral. But since this integral is numerically defined, we need to compute this numerically as well. dπ 1 (p 1,p 2,p 3 ) dp 1 0 f 1 (u1 + p 1 ) F 2 (u 1 + p 2 ) F 3 (u 1 + p 3 ) du 1 In[14]: dpdf1[x_] D[PDF[NormalDistribution[mu1, sigma1], x], x] Out[14] -0.00319154 e -0.02 (-20.+x)2 (-20. + x) In[15]: dchoiceprob1dp1[p1_, p2_, p3_] : NIntegrate[dpdf1[u1 + p1] * CDF[NormalDistribution[mu2, sigma2], u1 + p2] * CDF[NormalDistribution[mu3, sigma3], u1 + p3], {u1, 0, Infinity}];

05.1-LogitDemandModel.nb 5 In[16]: dchoiceprob1dp1[20., 15., 25.] Out[16] - 0.0613408 In[17]: choiceprob1[20.01, 15., 25.] - choiceprob1[20., 15., 25.] 0.01 Out[17] - 0.0613196 In[18]: elast11 dchoiceprob1dp1[20., 15., 25.] * 20. choiceprob1[20., 15., 25.] Out[18] - 3.90158 The total demands are given by multiplying the choice probabilities by the size of the population of consumers. This model requires 2 n + 1 parameters, the means and standard deviations of the n products and the population. It requires one-dimensional numerical integration to evaluate choice probabilities and derivatives. While practical we can do better. The outside good We make one modification to the assumption that consumer values for products are independent. The n products are considered the inside goods, meaning the products that we explicitly consider, evaluating their price response to a merger. The consumer may choose not to purchase any of the inside goods, deciding not to purchase, or perhaps choosing a product not modeled, a product whose price is considered constant and not responding to a merger and which we do not observe. Either of these cases is considered the outside good. We have imagined that the values of inside goods are measured relative to a fixed value for the outside good. But we could as well imagine that the outside good, say indexed by i 0, also has a value V 0 independent as well of the other V i, say taking a price for the outside good of p 0 0. The consumer chooses one of the inside goods or the outside good, whichever gives the maximum of V i - p i even if this value is negative, choosing the least bad choice in essence, since we include for i 0 an explicit value for the outside good. Then the probability of choosing any given product is a function of the prices and the differences V i - V 0, and since V 1 - V 0 and V 2 - V 0 include the same component V 0, these relative values will no longer be independent, unless V 0 is a constant. As an aside, in programming I usually take a list of n + 1 values representing the n inside goods plus outside good, and since lists start indexing at 1 not 0, it often proves convenient to take the last element of the list as the outside good. Mathematica takes negative indices as counting from the end of the list backwards, so the -1 index is the last element of the list. The choice probability integral for 3 products plus the outside good becomes π 1 (p 1, p 2, p 3 ) Prob(V 1 - p 1 > V 0 - p 0, V 2 - p 2, and V 3 - p 3 ) x 1 -p 1 +p 0 - x 1 -p 1 +p 2 - x 1 -p 1 +p 3 - - f(x 1, x 2, x 3 ) dx 3 dx 2 dx 0 dx 1 - f 1 (x 1 ) F 0 (x 1 - p 1 + p 0 ) F 2 (x 1 - p 1 + p 2 ) F 3 (x 1 - p 1 + p 3 ) dx 1 - f 1 (u 1 + p 1 ) F 0 (u 1 + p 0 ) F 2 (u 1 + p 2 ) F 3 (u 1 + p 3 ) du 1 To make this work out nicely, the translations F i (x + p i ) of the F i (x), should be simply related to each other. Also the products of F i (x) should have a simple formulation. The product F 0 (u 1 + p 0 ) F 2 (u 1 + p 2 ) F 3 (u 1 + p 3 ) Prob(u 1 max(v 0 - p 0, V 2 - p 2, V 3 - p 3 ))

6 05.1-LogitDemandModel.nb is the distribution function for the maximum of translations of independent random variables, so it would be nice to have a class of distributions, closed under translations and under maximums. The extreme value distribution The average of n independent samples from a distribution, under broad assumptions, tends to a normal distribution as n, the mean being the mean of the distribution and the variance decreasing as 1/n. On the other hand, the maximum of n independent samples from a distribution, under certain assumptions, tends to an extreme value distribution. In[19]: sample : Table[Random[NormalDistribution[0, 1]], {60}]; In[20]: maxsample : Max[sample]; In[21]: maxsample Out[21] 2.74928 In[22]: maxsample Out[22] 2.73949 In[23]: sortedmaxsamples Sort[Table[maxsample, {500}]] Out[23] {1.20213, 1.27576, 1.31433, 1.31816, 1.36891, 1.42945, 1.45688, 1.50796, 1.50873, 1.52684, 1.5282, 1.54455, 1.57005, 1.57526, 1.59648, 1.59888, 1.60144, 1.60305, 1.60324, 1.61703, 1.62164, 1.6243, 1.62803, 1.6341, 1.64649, 1.66866, 1.67149, 1.69116, 1.69202, 1.69687, 1.6971, 1.70529, 1.71542, 1.71623, 1.71894, 1.72709, 1.74021, 1.74076, 1.7408, 1.74915, 1.75011, 1.75072, 1.75139, 1.76118, 1.77382, 1.78231, 1.78533, 1.78962, 1.79372, 1.79485, 1.7989, 1.7995, 1.80848, 1.81062, 1.81064, 1.81336, 1.81409, 1.81737, 1.81787, 1.81922, 1.81944, 1.82442, 1.82772, 1.82873, 1.83009, 1.83191, 1.833, 1.83717, 1.83819, 1.83843, 1.84362, 1.84439, 1.84672, 1.8494, 1.86365, 1.87135, 1.87273, 1.87454, 1.87506, 1.881, 1.88772, 1.89018, 1.893, 1.89939, 1.89976, 1.90227, 1.90332, 1.90432, 1.90599, 1.91019, 1.9139, 1.91613, 1.91691, 1.91825, 1.91932, 1.92063, 1.92521, 1.92567, 1.92893, 1.93681, 1.94233, 1.94488, 1.94806, 1.95195, 1.95434, 1.95886, 1.95921, 1.96076, 1.96463, 1.96737, 1.96922, 1.96942, 1.9724, 1.98806, 1.99427, 1.99462, 1.99958, 2.00137, 2.00211, 2.00272, 2.00351, 2.00372, 2.00374, 2.00541, 2.00662, 2.00934, 2.00951, 2.01402, 2.01633, 2.01787, 2.01943, 2.02209, 2.02361, 2.02769, 2.02858, 2.03036, 2.03038, 2.03426, 2.03507, 2.03604, 2.03667, 2.03978, 2.03991, 2.04065, 2.04398, 2.04874, 2.04876, 2.04889, 2.04896, 2.0493, 2.05411, 2.05801, 2.05928, 2.06258, 2.06343, 2.06437, 2.06665, 2.0712, 2.07463, 2.0776, 2.08029, 2.08063, 2.08568, 2.0909, 2.09106, 2.09615, 2.09747, 2.09815, 2.09902, 2.09957, 2.1004, 2.10182, 2.10452, 2.1157, 2.11766, 2.12056, 2.12105, 2.12595, 2.13063, 2.13104, 2.13154, 2.14133, 2.14974, 2.15282, 2.15383, 2.15897, 2.16255, 2.16677, 2.16769, 2.16791, 2.16871, 2.16993, 2.17161, 2.17211, 2.1768, 2.17687, 2.17763, 2.18038, 2.18291, 2.18863, 2.19188, 2.20085, 2.20109, 2.20148, 2.2022, 2.20455, 2.2046, 2.21665, 2.2182, 2.21898, 2.21976, 2.2204, 2.222, 2.22322, 2.23024, 2.23278,

05.1-LogitDemandModel.nb 7 2.23295, 2.23298, 2.2349, 2.2352, 2.23615, 2.23936, 2.24127, 2.24197, 2.2428, 2.2434, 2.24653, 2.24931, 2.25007, 2.25265, 2.25548, 2.25647, 2.25871, 2.25943, 2.261, 2.26324, 2.2639, 2.26434, 2.2678, 2.26869, 2.27028, 2.27378, 2.27712, 2.27938, 2.28264, 2.28408, 2.28455, 2.28497, 2.28544, 2.28647, 2.28937, 2.29337, 2.30103, 2.30456, 2.30484, 2.30568, 2.30712, 2.30774, 2.30815, 2.31213, 2.31316, 2.31353, 2.3142, 2.31623, 2.32197, 2.32311, 2.32446, 2.32677, 2.3303, 2.33535, 2.33812, 2.33911, 2.34317, 2.3477, 2.34802, 2.3488, 2.34919, 2.35453, 2.35784, 2.35803, 2.35857, 2.36021, 2.36043, 2.36184, 2.36204, 2.36268, 2.36308, 2.36431, 2.36645, 2.37096, 2.37412, 2.37586, 2.37721, 2.38003, 2.38099, 2.38275, 2.38551, 2.38597, 2.3978, 2.40061, 2.40203, 2.40361, 2.40408, 2.40585, 2.40594, 2.40608, 2.40941, 2.40988, 2.41099, 2.4131, 2.41412, 2.41459, 2.4228, 2.42281, 2.42429, 2.42728, 2.42794, 2.42815, 2.4293, 2.43005, 2.43031, 2.43297, 2.4349, 2.43515, 2.43941, 2.44895, 2.45132, 2.45509, 2.45804, 2.46735, 2.46787, 2.46813, 2.47109, 2.47119, 2.47246, 2.47353, 2.47708, 2.47777, 2.48291, 2.48386, 2.48847, 2.49034, 2.49435, 2.49489, 2.50182, 2.50475, 2.50587, 2.5064, 2.50957, 2.51079, 2.51252, 2.51518, 2.51553, 2.51582, 2.52633, 2.53627, 2.54089, 2.54484, 2.54927, 2.55101, 2.55786, 2.56652, 2.5671, 2.5679, 2.56912, 2.57014, 2.57229, 2.57837, 2.57992, 2.58202, 2.58361, 2.59332, 2.5949, 2.59716, 2.6044, 2.60453, 2.60456, 2.6072, 2.6073, 2.60737, 2.611, 2.61821, 2.61973, 2.62161, 2.62531, 2.6383, 2.64107, 2.64312, 2.64416, 2.64507, 2.6506, 2.65319, 2.65464, 2.65659, 2.66359, 2.67056, 2.67315, 2.67498, 2.67688, 2.67846, 2.6817, 2.69016, 2.6921, 2.69218, 2.69641, 2.69747, 2.72326, 2.72644, 2.73259, 2.73389, 2.73401, 2.7352, 2.73787, 2.74027, 2.74467, 2.74889, 2.76039, 2.76251, 2.76259, 2.76399, 2.76617, 2.77291, 2.77603, 2.78246, 2.78988, 2.7911, 2.79462, 2.8134, 2.82542, 2.82822, 2.83036, 2.84689, 2.84891, 2.85178, 2.85441, 2.85886, 2.85978, 2.85986, 2.86156, 2.86358, 2.86621, 2.8665, 2.8693, 2.87027, 2.87121, 2.87123, 2.87695, 2.87756, 2.88575, 2.89835, 2.91537, 2.9168, 2.91996, 2.92803, 2.93383, 2.93987, 2.95273, 2.95434, 2.95799, 2.96586, 2.98476, 2.98478, 2.98642, 2.99098, 2.99865, 3.02624, 3.0332, 3.04174, 3.04716, 3.06472, 3.07668, 3.07965, 3.08273, 3.08542, 3.08844, 3.11512, 3.11731, 3.13253, 3.18972, 3.20038, 3.21886, 3.21953, 3.22964, 3.23194, 3.23437, 3.24591, 3.27637, 3.28218, 3.30676, 3.36583, 3.36592, 3.366, 3.39869, 3.46327, 3.50414, 3.53135, 3.73342, 3.88111, 3.95048, 4.10837} Sorting the results from taking the maximum of 60 standard normally distributed samples gives an idea of how the maxima are distributed. Here is a picture approximating the CDF for the maximum of 60 standard normally distributed samples.

8 05.1-LogitDemandModel.nb In[24]: maxsamplescdfplot ListPlot[Table[{sortedmaxsamples[[i]], i / Length[sortedmaxsamples]}, {i, 1, Length[sortedmaxsamples]}]] 1.0 0.8 Out[24] 0.6 0.4 0.2 1.5 2.0 2.5 3.0 3.5 4.0 It is harder to get a reasonable picture of the PDF. In[25]: maxsamplespdfplot ListPlot Transpose Drop[Drop[sortedmaxsamples, 10], -10], 20 Length[sortedmaxsamples] Drop[sortedmaxsamples, 20] - Drop[sortedmaxsamples, -20] 1.4 1.2 1.0 Out[25] 0.8 0.6 0.4 0.2 2.0 2.5 3.0 The density looks "spiky", and while we should expect some random clustering of results and so noise in this plot, it might also partly be due to roundoff or other imperfections in the random number generation process. This no longer looks like a normal distribution; it is no longer symmetrical, with a fatter tail on the right. In[26]:? ExtremeValueDistribution ExtremeValueDistribution[α, β] represents an extreme value distribution with location parameter α and scale parameter β. ExtremeValueDistribution[] represents an extreme value distribution with location parameter 0 and scale parameter 1. The mean and standard deviation of an extreme value distribution are not the same as the location and scale parameters, though they are simply related. Since I use α and β frequently for other purposes, I use η and λ for the location and scale parameters.

05.1-LogitDemandModel.nb 9 In[27]: Mean[ExtremeValueDistribution[eta, ]] Out[27] eta + EulerGamma In[28]: Out[28] StandardDeviation[ExtremeValueDistribution[eta, ]] π 6 To match the above distribution to the extreme value distribution approximating it we solve for the scale and location parameters giving the same mean and standard deviation. In[29]: Mean[sortedmaxsamples] Out[29] 2.32447 In[30]: StandardDeviation[sortedmaxsamples] Out[30] 0.450306 In[31]: evsoln Solve[{Mean[ExtremeValueDistribution[eta, ]] Mean[sortedmaxsamples], StandardDeviation[ExtremeValueDistribution[eta, ]] StandardDeviation[sortedmaxsamples]}, {eta, }] Out[31] {{eta 2.1218, 0.351102}} In[32]: evdist ExtremeValueDistribution[eta, ] /. evsoln[[1]] Out[32] ExtremeValueDistribution[2.1218, 0.351102] We can visually see the correspondence. In[33]: evcdfplot Plot[CDF[evdist, x], {x, 1.0, 4.5}] 1.0 0.8 Out[33] 0.6 0.4 0.2 1.5 2.0 2.5 3.0 3.5 4.0 4.5

10 05.1-LogitDemandModel.nb In[34]: Show[evCDFplot, maxsamplescdfplot] 1.0 0.8 Out[34] 0.6 0.4 0.2 1.5 2.0 2.5 3.0 3.5 4.0 4.5 In[35]: evpdfplot Plot[PDF[evdist, x], {x, 1.0, 4.5}] 1.0 0.8 Out[35] 0.6 0.4 0.2 1.5 2.0 2.5 3.0 3.5 4.0 4.5 In[36]: Show[evPDFplot, maxsamplespdfplot, PlotRange All] 1.4 1.2 1.0 Out[36] 0.8 0.6 0.4 0.2 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Here is the CDF of the extreme value distribution and some properties of this distribution. In[37]: CDF[ExtremeValueDistribution[eta, ], x] Out[37] e -e- -eta+x Translation of an extreme value distribution is again an extreme value distribution.

05.1-LogitDemandModel.nb 11 In[38]: Out[38] Simplify[CDF[ExtremeValueDistribution[eta, ], x + p] True CDF[ExtremeValueDistribution[eta - p, ], x]] The maximum of extreme value distributions with the same scale parameter is again an extreme value distribution. In[39]: Simplify CDF[ExtremeValueDistribution[eta1, ], x] * Out[39] True CDF[ExtremeValueDistribution[eta2, ], x] CDF ExtremeValueDistribution * Log Exp eta1 + Exp eta2,, x If F(x) is the CDF and f(x) is the PDF of an extreme value distribution function with location η and scale λ, then log(f(x + p)) -exp(-(x + p - η)/λ) -exp(-p/λ) exp(-(x - η)/λ) exp(p/λ) log(f(x)) and so by differentiating, f (x+p) f(x) exp(-p/λ) F(x+p) F(x) In[40]: Simplify[PDF[ExtremeValueDistribution[eta, ], x + p] / Out[40] True CDF[ExtremeValueDistribution[eta, ], x + p] Exp[- p / ] * PDF[ExtremeValueDistribution[eta, ], x] / CDF[ExtremeValueDistribution[eta, ], x]] Logit demand model We mean to evaluate the choice probability π 1 (p 1, p 2, p 3 ) - f 1 (u 1 + p 1 ) F 0 (u 1 + p 0 ) F 2 (u 1 + p 2 ) F 3 (u 1 + p 3 ) du 1 f 1 (u 1 +p 1 ) F - F 1 (u 1 +p 1 ) 0(u 1 + p 0 ) F 1 (u 1 + p 1 ) F 2 (u 1 + p 2 ) F 3 (u 1 + p 3 ) du 1 Suppose now the V i are independent extreme value distributed random variables with location parameters η i and the same scale parameter λ. Set p 0 0. Since only differences in values matter, we may as well set η 0 0. Then F i (x) exp -e (x-η i)/λ, so F i (x + p i ) is the CDF of an e.v. distribution with location η i - p i, and F max (u 1 ) Prob(u 1 max(v 0 - p 0, V 1 - p 1, V 2 - p 2, V 3 - p 3 )) F 0 (u 1 + p 0 ) F 0 (u 1 + p 1 ) F 2 (u 1 + p 2 ) F 3 (u 1 + p 3 ) is the CDF of an extreme value distributed random variable V max max(v 0, V 1, V 2, V 3 ) with location parameter η max λlog(exp((η 0 - p 0 )/λ) + exp((η 1 - p 1 )/λ) + exp((η 2 - p 2 )/λ) + exp((η 3 - p 3 )/λ)) and scale parameter λ. Let f max (u 1 ) be the corresponding PDF. Then log(f 1 (u 1 + p 1 )) -exp(-(u 1 + p 1 - η 1 )/λ) -exp((η 1 - p 1 - η max )/λ) exp(-(u 1 - η max )/λ) exp((η 1 - p 1 - η max )/λ) log(f max (u 1 )) so by differentiating we have

12 05.1-LogitDemandModel.nb f 1 (u 1 +p 1 ) exp((η F 1 (u 1 +p 1 ) 1 - p 1 - η max )/λ) f max(u 1 ) F max (u 1 ) So the choice probability integral becomes f π 1 (p 1, p 2, p 3 ) 1 (u 1 +p 1 ) F - F 1 (u 1 +p 1 ) 0(u 1 + p 0 ) F 1 (u 1 + p 1 ) F 2 (u 1 + p 2 ) F 3 (u 1 + p 3 ) du 1 f max (u 1 ) exp((η 1 - p 1 - η max )/λ) F - F max (u 1 ) max(u 1 ) du 1 exp((η 1 - p 1 - η max )/λ) exp((η 1 -p 1 )/λ) exp((η 0 -p 0 )/λ)+exp((η 1 -p 1 )/λ)+exp((η 2 -p 2 )/λ)+exp((η 3 -p 3 )/λ) exp((η 1 -p 1 )/λ) 1+exp((η 1 -p 1 )/λ)+exp((η 2 -p 2 )/λ)+exp((η 3 -p 3 )/λ) since η 0 p 0 0. The choice probabilities for the other products are of course similar. The choice probability for the outside good is then likewise exp((η 0 -p 0 )/λ) exp((η 0 -p 0 )/λ)+exp((η 1 -p 1 )/λ)+exp((η 2 -p 2 )/λ)+exp((η 3 -p 3 )/λ) 1 1+exp((η 1 -p 1 )/λ)+exp((η 2 -p 2 )/λ)+exp((η 3 -p 3 )/λ) π 0 (p 1, p 2, p 3 ) Of course, exactly similar formulas apply if there are more than 3 products. The total demands for each product are given by multiplying the choice probabilities by the total population of consumers, or a market size scale factor, M. q i (p 1, p 2, p 3 ) M exp((η i -p i )/λ) 1+exp((η 1 -p 1 )/λ)+exp((η 2 -p 2 )/λ)+exp((η 3 -p 3 )/λ) This is called the logit demand model. There are n + 2 parameters, the n location parameters η i, the scale parameter λ, and the market size M. In[41]: etamax * Log Exp eta0 - p0 + Out[41] Exp eta1 - p1 + Exp eta2 - p2 + Exp eta3 - p3 Log e eta0-p0 + e eta1-p1 + e eta2-p2 + e eta3-p3 In[42]: Simplify PDF[ExtremeValueDistribution[eta1, ], u1 + p1] * Out[42] True CDF[ExtremeValueDistribution[eta0, ], u1 + p0] * CDF[ExtremeValueDistribution[eta2, ], u1 + p2] * CDF[ExtremeValueDistribution[eta3, ], u1 + p3] Exp eta1 - p1 - etamax * PDF[ExtremeValueDistribution[etamax, ], u1] In[43]: Integrate[PDF[ExtremeValueDistribution[eta1, ], u1 + p1] * CDF[ExtremeValueDistribution[eta0, ], u1 + p0] * CDF[ExtremeValueDistribution[eta2, ], u1 + p2] * CDF[ExtremeValueDistribution[eta3, ], u1 + p3], {u1, - Infinity, Infinity}, Assumptions { > 0}] Out[43] e eta3+p0+p1+p2 + e eta2+p0+p1+p3 e eta1+p0+p2+p3 + e eta1+p0+p2+p3 + e eta0+p1+p2+p3

05.1-LogitDemandModel.nb 13 In[44]: Simplify Exp eta1 - p1 - etamax Out[44] e eta3+p0+p1+p2 + e eta2+p0+p1+p3 e eta1+p0+p2+p3 + e eta1+p0+p2+p3 + e eta0+p1+p2+p3 In[45]: Simplify Exp eta1 - p1 - etamax /. {eta1 - p1 nu1} /. {nu1 eta1 - p1} Out[45] e eta0-p0 + e eta1-p1 e eta1-p1 + e eta2-p2 + e eta3-p3 In[46]: Clear[choiceprob1]; In[47]: choiceprob1[p1_, p2_, p3_] Simplify Exp eta1 - p1 - etamax /. {eta1 - p1 nu1} /. {nu1 eta1 - p1} /. {eta0 0, p0 0} Out[47] 1 + e eta1-p1 e eta1-p1 + e eta2-p2 + e eta3-p3 Model calibration Next, we need to be able to calibrate a logit demand model to given conditions. The demands q i (p 1,..., p n ) M exp((η i -p i )/λ) 1+exp((η 1 -p 1 )/λ)+...+exp((η n -p n )/λ) are determined by the n + 2 parameters η i, λ, and M Suppose that the demands at prices p i * are q i * for i 1,..., n. The conditions q i (p i * ) q i * give n conditions on the parameters. Generally, we cannot observe the consumers that choose not to buy one of the inside goods, so we cannot know M and the choice probabilities directly. For given λ and M, it is not difficult to solve for the location parameters η i. The problem is how to estimate reasonable values of λ and M. Now we may want simply to guess at λ and M, trying a range of values for these parameters, possibly computing the characteristics of the implied demand functions to determine which combinations are plausible, and then derive the merger effects for these combinations of parameters to get a feel for the range of likely consequences of a merger. A more systematic procedure is to calibrate these parameters to corresponding demand characteristics, estimate these characteristics directly, or else try a range of plausible values for these more intuitive quantities. In any case, getting reliable information about demand beyond current prices and quantities is often problematic. What merger modeling does is give a systematic procedure for translating expert opinion about the basic characteristics of demand into a specific demand model and the consequential merger effects. The parameter M represents the total population of consumers. The quantity q 0 M - q 1 -... - q n represents the number of consumers who choose not to buy, the larger M is, the larger q 0 and the more attractive the outside good. As the prices of all inside goods increase, consumers will switch to the outside good, and the more attractive the outside good the more readily consumers will switch. To capture this sensitivity of the total demand for the inside goods with respect to the prices of the inside goods, we define an aggregate elasticity of demand. Imagine increasing all of the prices of inside goods by the same Δp. Then the total demand for inside goods increases by some Δq inside. To

14 05.1-LogitDemandModel.nb express this relationship as an elasticity we should multiply Δq inside /Δp by some average price and divide by the total demand for inside goods q iniside. The appropriate average price is to weight the p i by the quantities q i. The aggregate elasticity is then i1 ϵ aggregate lim n q i (p 1 +Δp,...,p n +Δp)-q i (p 1,...,p n ) Δp 0 Δp i1 n p i q i (p 1,...,p n ) ( i1 n q i (p 1,...,p n )) 2 To express this more simply, note that q 0 M - q inside, and imagine that we take p 0 as a variable instead of a constant equal to 0. Increasing the prices of all the inside products is equivalent to decreasing the price p 0 of the outside good; the choice of the maximum of V i - (p i + Δp) or V 0 - p 0 gives the same result as the choice of the maximum of V i - p i or V 0 - (p 0 - Δp), the only difference being the addition of Δp to each of the quantities to be compared. Thus with q 0 we have M exp((η 0 -p 0 )/λ) exp((η 0 -p 0 )/λ)+exp((η 1 -p 1 )/λ)+...+exp((η n -p n )/λ) i1 lim n q i (p 1 +Δp,...,p n +Δp)-q i (p 1,...,p n ) Δp 0 dq 0 Δp dp 0 then substituting p 0 0. We calculate dq 0 dp 0-1 λ q 0 + 1 λ q 0 2 M - q 0 λm (M - q 0) and (M - q 0 ) is the total demand for the inside goods so ϵ aggregate - q 0 p λm - p λ where p i1 n p i q i (p 1,...,p n ) n q i (p 1,...,p n ) i1 i1 1 - n M q i - π 0 p λ - p λ 1 1+exp((η 1 -p 1 )/λ)+...+exp((η n -p n )/λ) is the average price. As M increases starting from n i1 q i, ϵ aggregate decreases from 0 approaching a limiting value of - p. Within this range, assuming p and λ are given, we can solve for the M giving a λ specified aggregate elasticity of demand for the inside goods. That leaves the parameter λ to be determined. Recall that λ is proportional to the standard deviation of the values distribution of each product. The smaller λ is, the more sensitive demand will be to price differences, that is, the greater will be the price elasticity of demand of any product. For example, the own price elasticity of demand for product 1 is q1 1- M ϵ 11 - p 1 - (1-π 1) p 1. λ λ The aggregate elasticity and own price elasticity of one product can be used in combination to determine the λ and M parameters. In[48]: Out[48] Simplify D[choiceprob1[p1, p2, p3], p1] * p1 choiceprob1[p1, p2, p3] True - 1 - choiceprob1[p1, p2, p3] * p1 Summary The logit choice model for n inside products is specified by n location parameters η i (reflecting mean consumer values), a scale parameter λ (in proportion to the standard deviations of consumer values), and a total consumer population, or market size, M. The choice probability for product i is given by

05.1-LogitDemandModel.nb 15 π i (p 1,..., p n ) exp((η i -p i )/λ) 1+exp((η 1 -p 1 )/λ)+...+exp((η n -p n )/λ) and the demand for product i is then q i (p 1,..., p n ) Mπ i (p 1,..., p n ) M exp((η i -p i )/λ) 1+exp((η 1 -p 1 )/λ)+...+exp((η n -p n )/λ). The choice probability and demand for the outside good are π 0 (p 1,..., p n ) and 1 1+exp((η 1 -p 1 )/λ)+...+exp((η n -p n )/λ) q 0 (p 1,..., p n ) Mπ 0 (p 1,..., p n ) M 1+exp((η 1 -p 1 )/λ)+...+exp((η n -p n )/λ). The aggregate elasticity of demand is defined by ϵ aggregate - p λ where p i1 n p i q i (p 1,...,p n ) n q i (p 1,...,p n ) i1 i1 1 - n M q i is the average price. The own price elasticity of demand for product 1, for example, is q1 1- M ϵ 11 - p 1. λ * Given initial prices and demands p i and q * i, the location parameters η i can be found in terms of λ and M from the conditions q i (p i * ) q i *. The values of λ and M can be determined from specified estimates of * the aggregate elasticity ϵ aggregate and an own price elasticity of one product, say, ϵ * 11, so that * ϵ aggregate ϵ aggregate and ϵ 11 ϵ * 11.