Estimating Demand for Spatially Differentiated Firms with Unobserved Quantities and Limited Price Data

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1 Estimating Demand for Spatially Differentiated Firms with Unobserved Quantities and Limited Price Data Charles C. Moul May 2014 Abstract: I extend the Salop circle treatment to construct a model in which consumers first make discrete choices among symmetric sellers and then choose a continuous amount to purchase. The reduced-form pricing equation of this model identifies several pertinent structural parameters. Monte Carlo simulations indicate that 1) demand shifters and rotators can be distinguished (thus dominating the descriptive regression), 2) cost parameters are identified but imprecise, and 3) the most likely specification error conservatively biases estimates toward zero. Keywords: spatial differentiation, demand estimation, monopolistic competition JEL: D43, L13 A key contribution of economics is leveraging theory to glean insights from imperfect datasets. In this paper, I lay out a model that enables the analyst to infer various demand and cost parameters from a single firm s data on prices and product characteristics. The central idea is that a consumer follows a two-stage decision plan, first choosing from among symmetric and spatially differentiated sellers and then, conditional on that choice, choosing a purchase-quantity. By using the assumptions of utility maximization, symmetric sellers, and profit-maximizing pricing, I show how this model s estimation improves markedly on simple descriptive regressions in terms of both fit and insight, specifically in distinguishing demand shifters and rotators. I begin with Salop s (1979) circular-city extension of Hotelling s (1929) linear city. 1 While Salop emphasized the importance of an outside option in the consumer s choice set, I assume that all consumers make a purchase each period. They do, however, have discretion over the amount of the good that they purchase. This matches the spirit of elastic demands found in Smithies (1941), Hanneman (1984), and Dubin and McFadden (1984). I consequently solve my model of consumer behavior backward, first finding the consumer s utility-maximizing purchase quantity at each seller and then having the consumer choose the seller that yields the highest indirect utility. Symmetrically spaced firms face these implied demands, recognizing how prices affect both how many consumers patronize the firm and how much each consumer buys. These Economics Department, Farmer School of Business, Miami University, Oxford, OH Ph: moulcc@miamioh.edu. 1 The oftentimes unrealistic circle assumption avoids the corner difficulties of a finite line segment and was especially benign in the application that prompted this model, namely Australian bookmakers scattered around a racecourse (Moul and Keller, 2014). 1

2 combined forces yield a cubic first order condition to the firm s profit-maximization problem. I prove that, under reasonable parametric assumptions, this cubic has a single real solution. My estimation algorithm first guesses at the identified parameters, and these values imply a unique real predicted price for each observation. I then search over the parameter-space to minimize the sum of squared errors between the (log) predicted and observed prices. When the model assumes that (log) predicted and observed prices differ because of measurement error, Monte Carlo simulations show that the true parameter values are recovered with reasonable precision. This is true even when a variable has conflicting impacts on price through shift and rotation mechanisms. When (log) predicted and observed prices differ because of product characteristics that are unobserved by the econometrician, estimated parameters are biased toward zero and so provide conservative measures of the relevant impacts. Precision declines as marginal costs become strictly positive and declines further when marginal costs must be estimated, but demand estimates are still generally statistically distinct from zero. I conclude with F-tests that pit the fit of the atheoretical descriptive regression against that of the reduced form for both prices simulated from the model and simple hedonic prices. 1. Model I assume that consumers have quasilinear utility, specifically that, when purchasing from firm k, consumer i derives utility (1) where q i denotes amount purchased, d ik distance from consumer i to firm k, and y i the amount of numeraire good consumed. Assume that,, are strictly positive. Given the budget constraint, the utility maximization problem becomes (i subscripts suppressed). max (2) : 2 0 (3) This generates linear demand and inverse demand 2. With this demand function, I construct consumer i s indirect utility function for firm k: (4) The consumer has now calculated how much to purchase from each firm, and the question becomes which firm will yield the highest indirect utility. Given the circular city model and the assumption that firms in equilibrium do not price so that a consumer will leapfrog a firm, each firm faces a marginal consumer on either side. From centre looking outward, denote a given firm as m, with firm a to the left and firm z to the right, and denote the leftward (rightward) marginal consumer as x (y). For illustration, assume a unitcircumference circle of uniform consumer depth with three symmetrically located firms at,, and (2:00, 6:00, and 10:00 on a clockface). The leftward marginal consumer for firm m at is defined by (5) which reduces to (6) Location y to the right is analogously (7) 2

3 Firm m faces the demand function (8) (9) Assuming that 0, firm m maximizes profits: max (10) : 0 (11) Broadening this to an N-firm model and imposing symmetry yields 0 (12) Substituting in and simplifying (eventually) yields (13) Letting, this becomes (14) The cubic equation 0 has a unique real root and two complex roots if and only if Δ Plugging in the structural parameters for each of the s (eventually) reduces to (15) If is not always negative, continuity requires that it must at some point equal zero, so I consider the quadratic (16) If a solution exists, it satisfies (17) This complex solution, however, contradicts my assumption that and thus a real number. Therefore, for any 0 and 0, 0 and my cubic equation has a unique real price solution. The value of this reduced-form approach is its potential to gain insight about the underlying structural parameters, specifically in distinguishing demand shifters and rotators. From the above, one sees that, for each product,,, and are all identified. Much of the tractability of this approach stems from the fact that many structural parameters are aggregated in. Letting the population of consumers be M rather than normalized to 1, Ω. The relevant restriction for estimation is that the number of consumers per firm is constant, i.e., the standard long-run equilibrium condition under a shared technology and no impact of entry on factor prices. I will therefore specify Ω as a constant and reserve the demand-rotators for. 2. Monte Carlo simulations I now explore how estimation that uses the above cubic equation performs. I set 250 observations of a firm s price and product offering (denoted as k) for 100 simulations. Product characteristics, are drawn from a standard normal distribution and incorporated into the 2 See, e.g., Irving (2004), Integers, polynomials, and rings, pp

4 above parameters as Ψ and Ω. These specifications allow inverse demand to be shifted by X (an intercept and two regressors) and rotated by Z (one regressor). 3 While one ideally has prior beliefs about which characteristics shift demand rather than rotate it, in practice these roles are rarely delineated. I therefore consider two cases: X and Z have no overlapping regressors, and the first X regressor repeats in Z. To these product characteristics I add three cases of marginal cost: 0, 0 in which is known and 0 in which is unknown and must be estimated. I use Matlab s solve command applied to the cubic equation to find the price solutions for selected parameters and,. Disturbances are introduced in one of two ways. The model first assumes measurement error in the observed (log) price: ln ln, letting ~0,1. I also allow for the possibility that the disturbance is the mean valuation of unobserved characteristics. In this case, Ψ and ln ln. I choose values of to generate plausible descriptive regression fits (e.g., 0.4). All qualitative results are robust on this margin. I employ Matlab code and search for parameters using fminsearch. Solving for each observation s cubic equation s real root at every parameter guess is computationally intensive. Convergence time therefore increases proportionally with sample size. On a 2.6GHz machine with 4GB RAM, each benchmark estimation (data generated by model, initial parameter guess at true values) took about one hour. The inference of marginal cost is frequently problematic in industrial organization, and so I begin with the simplest case of 0. Table 1 displays this first set of descriptive and reduced form results. The first two columns contain the measurement-error cases when X and Z do and do not overlap, and the last two columns show the analogous cases in which the disturbance is an omitted variable. Goodness of fit measures in all cases use residuals based on the difference between observed and predicted prices and are therefore comparable between descriptive and reduced form. In both cases in which the simulated data were generated by the model, the estimation routine successfully recovers the true parameters with great precision. Fits using the reducedform equation are substantially better than those of the descriptive, a result confirmed by formal F-tests. In the case when the demand rotator Z is distinct from the demand shifters X, the reduced-form estimates also improve on the descriptive estimates. Specifically, the reduced form identifies that consumers don t dislike product characteristic Z as they do with X 2, but rather the greater elasticity prompts the firms with market power to lower mark-ups. 4 This same insight appears differently when X 1 serves as both a demand shifter and rotator. The descriptive regression suggests that X 1 is unimportant in consumer demand when it is simply aggregating the two countervailing forces. While the estimation algorithm leans heavily on the measurement error interpretation of the disturbance, the idea that the econometrician may fail to observe all relevant product characteristics is perhaps more compelling. Estimation when the simulated data were generated under this scenario indicates that most coefficients are conservatively biased toward zero. This bias does not appear to affect the rotator parameter, which remains consistent and loses little if any precision. 3 Demand s horizontal intercept is therefore unaffected by changes in Z. 4 As shown in Bresnahan (1982), demand rotation is a fundamental source of identifying market power. 4

5 The above results are useful when firms face no variable costs, but positive marginal costs are far more frequent. Table 2 displays analogous Monte Carlo estimates under the cases in which strictly positive marginal cost is respectively a known and an unknown constant. In both cases, precision substantially worsens. Using the simulated data generated by the model, standard errors when marginal costs are a known constant are about 50% larger than when marginal cost is zero. When marginal costs must be estimated, standard errors rise further, with those on the marginal costs typically so large as to prevent the rejection that such costs are zero. Nevertheless, point estimates are generally close to their true values, and demand estimates can usually be distinguished from zero. Larger samples (and the faster computing to accommodate them) would presumably address this issue. When disturbances are the mean impact of unobserved characteristics rather than measurement error, the conservative bias of most parameters that was noted above is still apparent. I conclude by applying this technique to data that are generated by a much different process. Specifically, I examine prices that are generated by exp in which I chose the variance of the disturbance to generate comparable measures of fit to the previous Monte Carlos. Such prices would loosely correspond to those that would arise in a competitive market. In the vast majority of cases (92%), F-statistics using the R 2 s were either unworkable (because the reduced-form estimation failed to converge or its fit was less than that of the descriptive estimation) or could not reject the hypothesis of no benefit with 95% confidence. 5 This approach therefore offers a ready technique that may markedly improve on simple atheoretical analysis, even with highly incomplete data. References Bresnahan, T. F., The oligopoly solution concept is identified. Economics Letters 10: Dubin, J. A. and D. L. McFadden, An econometric analysis of residential electric appliance holdings and consumption. Econometrica 52(2): Hanneman, W. M., Discrete/Continuous models of consumer demand. Econometrica 52(3): Hotelling, H., Stability in competition. Economic Journal 39: Irving, R. (2004). Integers, polynomials, and rings. Springer: New York. Moul, C. C. and J. M. G. Keller, Time to unbridle U.S. thoroughbred racetracks? Lessons from Australian bookies. Review of Industrial Organization 44(3): The parameter increasing to positive infinity was the typical cause of non-convergence. 5

6 Salop, S. C., Monopolistic competition with outside goods. Bell Journal of Economics 10(1): Smithies, A., Optimum location in spatial competition. Journal of Political Economy 49(3):

7 Table 1: Monte Carlo Simulations when marginal costs are zero * 100 simulations of n = 250 * Specifications: = exp(x Z), b = exp( Z) (I) (II) (III) (IV) Z Z X 1 Z X 1 Disturbance M.E. M.E. O.V. O.V E(p) std(p) Descriptive regressions: p = 0 + X x (+ Z z ) + u E(Estimate) E(Estimate) E(Estimate) E(Estimate) (0.030) (0.028) (0.023) (0.021) X (0.030) (0.028) (0.023) (0.021) X (0.031) (0.028) (0.023) (0.021) Z (0.031) (0.023) E(R 2 ) Reduced form regressions Truth E(Estimate) E(Estimate) E(Estimate) E(Estimate) (0.028) (0.030) (0.032) (0.034) (0.027) (0.065) (0.030) (0.069) (0.027) (0.028) (0.030) (0.031) (0.024) (0.066) (0.027) (0.068) (0.593) (0.786) (0.496) (0.591) E(R 2 ) E(F), vs. descriptive Mean point estimates with mean standard errors in parentheses Simulated data disturbance either Measurement Error (M.E.) or Omitted Variable (O.V.) 99 th percentile of F distribution is 6.63 for (I) and (III), 4.61 for (II) and (IV)

8 Table 2: Monte Carlo Simulations when marginal costs are c = 0.5 * 100 simulations of n = 250 * Specifications: = exp(x Z), b = exp( Z) (I) (II) (III) (IV) Z X 1 X 1 X 1 X 1 Disturbance M.E. M.E. O.V. O.V c Known Unknown Known Unknown E(p) std(p) Descriptive regressions: p = 0 + X x + u E(Estimate) E(Estimate) (0.034) (0.023) X (0.034) (0.023) X (0.034) (0.024) E(R 2 ) Reduced form regressions Truth E(Estimate) E(Estimate) E(Estimate) E(Estimate) (0.034) (0.183) (0.031) (0.235) (0.110) (0.171) (0.094) (0.142) (0.036) (0.087) (0.032) (0.075) (0.109) (0.154) (0.093) (0.128) (1.323) (5.095) (0.757) (8.796) c (0.372) (0.531) E(R 2 ) E(F), vs. descriptive Mean point estimates with mean standard errors in parentheses Simulated data disturbance either Measurement Error (M.E.) or Omitted Variable (O.V.) 99 th percentile of F distribution is 6.63 for (I) and (III), 4.61 for (II) and (IV)