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 45056. Ph: 513-529-2867.
Email: [email protected].
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
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 qi denotes amount purchased, dik distance from consumer i to firm k, and yi 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)
0
(3)
:
2
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
Firm m faces the demand function
(8)
(9)
0, firm m maximizes profits:
max
Assuming that
(10)
:
0
(11)
Broadening this to an N-firm model and imposing symmetry yields
0
Substituting in
2
(12)
and simplifying (eventually) yields
2
4
2
2
0
(13)
Letting
, this becomes
2
2
4
2
2
0
(14)
The cubic equation
0 has a unique real root and two complex
roots if and only if Δ 18
4
4
27
0.2 Plugging in the
structural parameters for each of the s (eventually) reduces to
∆
13
4
32
(15)
If ∆ is not always negative, continuity requires that it must at some point equal zero, so I
consider the quadratic
13
32
0
(16)
4
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
Ω . The
. Letting the population of consumers be M rather than normalized to 1,
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. 154-6.
3
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
and ln
case,
Ψ
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
0∀ . Table 1 displays this first set of descriptive and
I begin with the simplest case of
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 X2, but rather the
greater elasticity prompts the firms with market power to lower mark-ups.4 This same insight
appears differently when X1 serves as both a demand shifter and rotator. The descriptive
regression suggests that X1 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
4
Demand’s horizontal intercept is therefore unaffected by changes in Z.
As shown in Bresnahan (1982), demand rotation is a fundamental source of identifying market power.
4
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 0 0.2
0.2
0.4
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 R2s 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., 1982. “The oligopoly solution concept is identified.” Economics Letters 10:
87-92.
Dubin, J. A. and D. L. McFadden, 1984. “An econometric analysis of residential electric
appliance holdings and consumption.” Econometrica 52(2): 345-362.
Hanneman, W. M., 1984. “Discrete/Continuous models of consumer demand.” Econometrica
52(3): 541-561.
Hotelling, H., 1929. “Stability in competition.” Economic Journal 39: 41-57.
Irving, R. (2004). Integers, polynomials, and rings. Springer: New York.
Moul, C. C. and J. M. G. Keller, 2014. “Time to unbridle U.S. thoroughbred racetracks? Lessons
from Australian bookies.” Review of Industrial Organization 44(3): 211-239.
5
The parameter  increasing to positive infinity was the typical cause of non-convergence.
5
Salop, S. C., 1979. “Monopolistic competition with outside goods.” Bell Journal of Economics
10(1): 141-156.
Smithies, A., 1941. “Optimum location in spatial competition.” Journal of Political Economy
49(3): 423-439.
6
Table 1: Monte Carlo Simulations when marginal costs are zero
* 100 simulations of n = 250
* Specifications:  = exp(X‐Z), b = exp(‐Z)
Z
Disturbance

(I)
Z
M.E.
0.30
E(p)
std(p)
1.149
0.592
(II)
X1
M.E.
0.30
(III)
Z
O.V.
0.45
(IV)
X1
O.V.
0.45
1.161
0.505
1.046
0.458
1.057
0.385
Descriptive regressions: p =  0 + X  x (+ Z  z ) + u
0
X1
X2
Z
E(R2)
E(Estimate)
1.152
(0.030)
0.197
(0.030)
‐0.198
(0.031)
‐0.224
(0.031)
E(Estimate)
1.162
(0.028)
0.010
(0.028)
‐0.265
(0.028)
‐‐‐
E(Estimate)
1.048
(0.023)
0.152
(0.023)
‐0.151
(0.023)
‐0.203
(0.023)
E(Estimate)
1.059
(0.021)
‐0.023
(0.021)
‐0.196
(0.021)
‐‐‐
0.365
0.274
0.412
0.263
E(Estimate)
0.995
(0.030)
0.517
(0.065)
‐0.501
(0.028)
0.319
(0.066)
10.228
(0.786)
E(Estimate)
0.921
(0.032)
0.453
(0.030)
‐0.460
(0.030)
0.291
(0.027)
7.715
(0.496)
E(Estimate)
0.926
(0.034)
0.456
(0.069)
‐0.457
(0.031)
0.293
(0.068)
7.581
(0.591)
0.463
43.874
0.605
122.497
0.429
36.286
Reduced form regressions
Truth
E(Estimate)
0
1.0
0.999
(0.028)
1
0.5
0.501
(0.027)
2
‐0.5
‐0.502
(0.027)

0.3
0.298
(0.024)

10.0
10.050
(0.593)
E(R2)
E(F), vs. descriptive
0.592
139.120
Mean point estimates with mean standard errors in parentheses
Simulated data disturbance either Measurement Error (M.E.) or Omitted Variable (O.V.)
99th percentile of F‐distribution is 6.63 for (I) and (III), 4.61 for (II) and (IV)
Table 2: Monte Carlo Simulations when marginal costs are c = 0.5
* 100 simulations of n = 250
* Specifications:  = exp(X‐Z), b = exp(‐Z)
Z
Disturbance

c
E(p)
std(p)
(I)
X1
(II)
X1
(III)
X1
(IV)
X1
M.E.
0.30
Known
M.E.
0.30
Unknown
O.V.
0.45
Known
O.V.
0.45
Unknown
1.533
0.635
1.420
0.451
Descriptive regressions: p =  0 + X  x + u
0
X1
X2
E(R2)
E(Estimate)
1.535
(0.034)
0.064
(0.034)
‐0.347
(0.034)
E(Estimate)
1.422
(0.023)
0.022
(0.023)
‐0.264
(0.024)
0.306
0.341
Reduced form regressions
Truth
E(Estimate)
0
1.0
0.995
(0.034)
1
0.5
0.530
(0.110)
2
‐0.5
‐0.503
(0.036)

0.3
0.331
(0.109)

10.0
10.485
(1.323)
c
0.5
‐‐‐
E(R2)
E(F), vs. descriptive
0.417
23.612
E(Estimate)
0.981
(0.183)
0.572
(0.171)
‐0.522
(0.087)
0.366
(0.154)
11.232
(5.095)
0.497
(0.372)
E(Estimate)
0.941
(0.031)
0.457
(0.094)
‐0.447
(0.032)
0.297
(0.093)
7.339
(0.757)
‐‐‐
E(Estimate)
1.068
(0.235)
0.432
(0.142)
‐0.420
(0.075)
0.277
(0.128)
13.058
(8.796)
0.179
(0.531)
0.419
16.090
0.455
26.099
0.458
17.889
Mean point estimates with mean standard errors in parentheses
Simulated data disturbance either Measurement Error (M.E.) or Omitted Variable (O.V.)
99th percentile of F‐distribution is 6.63 for (I) and (III), 4.61 for (II) and (IV)