PRELIMINARY DRAFT: DO NOT CITE
Mergers Among Firms That Manage Revenue:
The Curious Case of Hotels
Arthur Kalnins∗
Cornell School of Hotel Management
Luke M. Froeb†, Steven T. Tschantz‡
Vanderbilt University
May 15, 2008
Abstract
When firms compete by managing revenue, i.e., pricing to fill available capacity, mergers can have an “information-sharing” effect that allows merged hotels to better forecast demand. Mergers that reduce the
frequency of over-pricing errors increase capacity utilization, which is
consistent with our empirical finding that hotel mergers increase quantity by 3%. We conclude that mergers or “call-arounds,” (the practice
of sharing information with rivals about demand) among firms operating at or near capacity have a short-run welfare benefit and only a
potential long-run cost (if capacity is reduced).
∗
School of Hotel Administration, [email protected]
William and Margaret Oehmig Associate Professor of Management, Owen Graduate
School of Management, [email protected]
‡
Department of Mathematics, [email protected]
Keywords: Revenue Management; Game Theory; Merger; Antitrust.
JEL Classifications: C61 - Optimization Techniques; C72 - Noncooperative Games;
L41: Horizontal Anticompetitive Practices.
†
1
Introduction.
The canonical revenue management problem is that price must be set before
demand is realized. A price too high means a high probability of unused
capacity; and a price too low means a high probability of foregone revenue.
Optimal pricing minimizes the expected cost of these pricing errors. In
Operations Research, the pricing problem is typically modeled as a singleproduct firm facing a stochastic demand or arrival process, sometimes with
dynamic price adjustment.
To study mergers, however, we need a multi-product characterization of
the pricing problem. Following the economics literature, e.g., Werden and
Froeb [18], we use a multinomial choice model to characterize demand and
a Bayesian Nash equilibrium in prices to characterize “supply” (firm behavior). Each firm chooses price to maximize expected profit, given their beliefs
about rivals’ information and strategies. Under various characterization of
the revenue management problem, we identify and isolate an informationsharing effect that allows the merged firm to better match demand to available capacity. Mergers that result in fewer over-pricing errors would increase
expected capacity utilization (quantity), which is consistent with our empirical results.
We conclude that for firms producing at or near capacity, the information sharing effect can potentially offset the “unilateral” effect of a merger,
i.e., the incentive of the merged firm to raise price and reduce output as it
internalizes the effects of competition among its commonly owned products
(Werden and Froeb, 1994)[18]. For a capacity-constrained firm, however,
unilateral effects are much weaker because mergers do not much affect the
profit calculus of pricing to fill capacity. This is consistent with earlier models with deterministic demand (e.g., Dowell [3], and Froeb et al., 2003)[6]
which find that mergers have no effect if the firms are capacity constrained.
For mergers among parking lots, for example, there is not much uncertainty about demand, and deterministic models can characterize the effects
of mergers. But for mergers among Hotels, where there is tremendous uncertainty about demand, we need a richer model.
In what follows, we first illustrate the basic revenue management problem in a single-product setting and then present a formal model of mergers
between firms who compete by managing revenue and can share information about demand. In our empirical section, we compare within-market
to out-of-market mergers for a set of Texas hotels, and find that mergers
increase quantity by 3%, and reduce price by 1%. We end with a discussion
of recent merger and price-fixing cases involving firms who manage revenue.
2
We conclude that antitrust enforcement agencies should take account of the
peculiar features of revenue management when evaluating competition. In
particular, our results suggest that mergers and “call-arounds” among firms
operating at or near capacity have a short-run welfare gain and only a potential long-run harm (if the merged firm reduces capacity).
2
Single-product revenue management
In this section, we present a single-product model of revenue management
to introduce some of the notation and establish some basic optimization
principles that will be familiar to students of operations research. We use the
taxonomy of Kimes [9] and limit our inquiry to services of known duration
where price is used to match demand to capacity. Other industries in this
category are hotels, airlines, rental cars, and cruise ships.
We suppose that a firm wishes to set the price on its product to maximize
its profit. The firm needs to know the demand q as a function of the price
p, and the cost C of producing that quantity q. The profit is
Π(p) = p q(p) − C(q(p))
(1)
and the profit maximizing price, assuming differentiability over a closed
interval of possible prices, will be where either the first-order condition
0 = Π0 (p) = q(p) + (p − mc(q(p)))q 0 (p)
(2)
is satisfied, with mc(q) = C 0 (q) the marginal cost, or else at an endpoint of
the interval. Marginal cost is often simply assumed to be constant over some
range, and the demand is usually such that we expect an interior maximum
over a sufficiently large interval of prices. If Π0 is decreasing, the first-order
condition is then satisfied at one point where the profit is maximum.
Moreover, a continuous Π0 implies that the profit curve will have a
rounded peak at the maximum, so small errors ∆p about the optimum price
cost the firm, at most, on the order of (∆p)2 off the maximum profit (with
some constant, for small ∆p). In other words, the firm doesn’t have to get
particularly close to the optimum price to achieve near optimum results (see
Figure 1).
Now suppose that the firm can supply up to some maximum amount K
of the product. This sharp capacity constraint idealizes the situation where
there is a sharply increasing short-run marginal cost at some quantity.
We model capacity constraints by specifying the quantity sold to be the
minimum of K and q. If the capacity constraint does not bind, the profit
3
profit
3500
3000
2500
2000
1500
1000
500
price
60
80
100
120
140
Figure 1: A typical profit curve.
maximizing price is given by the usual first-order condition in Equation 2
(also see Figure 2). The interesting feature of this graph is that the left
hand side of the profit curve drops off more sharply than the right hand
side. When we add uncertainty to the model (bottom panel of Figure 2),
this causes the firm to price slightly higher than it would in the certainty case
because the cost of over-pricing are lower than the costs of under-pricing.
To foreshadow our main empirical result, we can think about the effects
of adding more information as analogous to the difference between expected
and deterministic demand, or between the bottom and top panels of Figure
2. The peak of deterministic profit curve is higher, indicating fewer (none)
pricing errors. Most importantly from our point of view is fewer over-pricing
errors, which raises output. But price is slightly higher in the deterministic
case.
If the constraint is binding, however, the profit maximizing price is where
q(p) = K. Note that this maximum of profit is a sharp peak. If price is
lowered by ∆p, then the firm loses K∆p off the maximum profit, while
if price is raised by ∆p, profit changes by about Π0 (p)∆p, a loss roughly
proportional to ∆p again (see Figure 3). If long-run marginal costs are
greater than short-run marginal costs, then we would expect to see capacity
increase only up to the point where long-run profits are maximized and this
will be at a lower quantity and higher price than where profits are maximized
in the short-run. Hence where there is a significant difference between longrun and short-run marginal costs and a rapidly increasing short-run marginal
cost at some capacity, firms will operate, in the short-run, essentially as if
4
profit
3500
3000
2500
2000
1500
1000
500
price
60
80
100
120
140
60
80
100
120
140
profit
3500
3000
2500
2000
1500
1000
500
price
Figure 2: Deterministic vs. Expected Profit: when the cost of over-pricing
is smaller, price higher.
they are capacity constrained, carefully adjusting price to match demand to
the available capacity.
As in Figure 2, the effects of adding more information is analogous to
the difference between expected and deterministic demand, or between the
bottom and top panels of Figure 3. The peak of deterministic profit curve is
higher, indicating fewer (none) pricing errors. And fewer over-pricing errors
raises expected output. But unlike Figure 2, price is slightly lower than in
the deterministic case.
Formally, we model uncertainty by assuming that the firm is not facing
a definite demand, but instead that, for each price that the firm could set,
there is known a distribution Q(p) of possible demand q. We assume the
firm seeks to maximize the expected profit and consider first the case where
5
profit
3500
3000
2500
2000
1500
1000
500
price
60
80
100
120
140
60
80
100
120
140
profit
3500
3000
2500
2000
1500
1000
500
price
Figure 3: Deterministic (top) vs. Expected (bottom) Profit: when the cost
of under-pricing is lower, reduce price.
6
it is not capacity constrained. If marginal cost is constant (at least over
some range of interest, so C(q) is linear), then
E(Π(p)) = E(p Q(p) − C(Q(p))) = p E(Q(p)) − C(E(Q(p)))
Thus the problem of maximizing expected profit is the same as maximizing profit with the expected demand for a given price, a result known in
economics as “certainty equivalence.” This analysis will be only approximately true if marginal costs change over the range of likely demands since
E(C(Q(p))) could be either greater or less than C(E(Q(p))) depending on
whether marginal cost is increasing or decreasing. With constant marginal
costs the same first-order condition, with expected demand in place of demand, will determine the profit maximizing price.
Finally, consider the case with both capacity constraints and uncertain
demand. Suppose that the firm sets a price and then supplies either the
demand Q(p) or the capacity K whichever is smaller. If the demand is
greater than the capacity, then some consumers who would buy fail to do so
because the available stock is exhausted. Suppose now, for simplicity, that
cost is linear, say C(q) = mc q. Now expected profit is given by
E(Π(p)) = E((p − mc) min(K, Q(p))) = (p − mc)E(min(K, Q(p)))
In the examples above, demand, or expected demand, is given by a simple
logit model
100 exp(−0.1(p − 100))
q(p) =
1 + exp(−0.1(p − 100))
Marginal cost is a constant mc = 40. The non-binding capacity constraint
is K = 85, the binding capacity constraint is K = 50. Random demand
is taken to be log-normally distributed with mean q(p) and 40% standard
deviation. The large standard deviation makes the relation between random
and deterministic demand more evident. Comparisons for smaller standard
deviations are given in Table 1. Moving from a standard deviation of 40%
to 20% would yield about a 5% increase in quantity.
3
Information Sharing Merger Model
As mentioned in the introduction, the bulk of the literature in this area
comes from the field of Operations Research, and addresses yield management by a single firm. Since the purpose of this paper is to evaluate the
welfare effects of mergers among competing firms that practice revenue management, we have to extend the literature to account for competition. To
7
Demand model
price
quantity
profit
Unconstrained
86.93
78.69
3693.
Non-binding K = 85
86.93
78.69
3693.
random demand with σ = 10%
difference
88.44
1.50
75.53
-3.17
3658.
-35.
σ = 20%
difference
89.79
2.85
71.41
-7.28
3555.
-138.
σ = 40%
difference
91.16
4.22
64.69
-14.00
3309.
-384.
100.00
50.00
3000.
random demand with σ = 10%
difference
99.21
-0.79
48.81
-1.19
2890.
-110.
σ = 20%
difference
98.55
-1.45
47.47
-2.53
2779.
-221.
σ = 40%
difference
97.60
-2.40
44.55
-5.45
2566.
-434.
Binding K = 50
Table 1: Comparison of equilibrium prices, quantities, and profits.
8
do this, we put a standard differentiated-products demand curve on top of
a random arrival process, describing the number of consumers who “arrive”
or show up. The consumers who show up then choose among competing
products and a “no purchase” option, depending on price.
A simple model of uncertain demand might assume that each firm has
in mind a distribution of possible demand, and that they set price assuming
all other firms know this same or similar information. The result of such
a model would be essentially similar to the single product model. Optimal
prices would be shaded slightly up or down depending on which represented
the potentially least costly pricing error for that firm and product. We might
then posit the combination of information of merging firms to reduce the
variance of the combined distribution of demand, the effect of which is to
reduce the amount of price shading by the firms. In the case where demand
is not capacity constrained, we would expect ordinary unilateral merger
effects to swamp the information sharing effects. When demand is capacity
constrained and we would expect no price changes in a deterministic model,
the merger effects would be confined to this reduction of price shading and
could result in either small increases or decreases in price.
Instead we consider a specific detailed model of statistical analysis by
each firm that would lead to pricing strategies in Nash equilibrium and
allow us to define precisely the effects of information sharing. While this
model may not capture all of the effects to be noted in the real world, it at
least gives a rational basis for firm decisions that we can hope to extrapolate
to a hypothetical post-merger world.
We are interested in the capacity constrained case where possible merger
effects are limited to information sharing effects. We continue assuming
firms have a perishable differentiated product, and suppose, for the sake of
simplicity, that there is a common time period over which they can sell their
products. Suppose further that firms can set prices only at the beginning of
the sales period (allowing revenue management price adjustments reduces
the exposure to demand uncertainty, thereby reducing price shading and
merger effects in the capacity constrained case). Suppose that consumers
arrive at some rate, modelled by a Poisson process, and that each makes
a decision of which product to buy or whether not to buy depending on
prices according to a logit model. Assume further that a consumer denied
his first choice because supply has been exhausted does not look then for a
second choice but gives up and makes no purchase. Thus the firms attract
a certain demand but can sell only the minimum of their capacity and their
demand, without regard to which other firms may have demand in excess of
their capacity. Since we expect firms not to price so low as to have expected
9
demand greatly exceed capacity, any overflow of demand from one product
to another would be small anyway.
Let µ be the expected number of consumers that will arrive during the
sales period, total number of consumer choices made is distributed with
PDF
e−µ µQ
ψµ (Q) =
Q!
(for nonnegative integers Q). The total number of consumers that will
√
arrive is then distributed with standard deviation µ. This is a minimum
uncertainty firms have for the total number of consumers. In some cases, a
natural unit of consumer choice suggests this natural scale of uncertainty.
In cases where there is no natural unit we may wish a different standard
deviation for the total demand, but this can be approximated by re-scaling
the Poisson distribution appropriately. We assume that we can safely ignore
the discrete nature of consumer arrivals and choices.
Take logit parameters ai and b so that the shares for each product in the
total market are expected to be
si =
exp(a + bpi )
P i
1 + j exp(aj + bpj )
We imagine each arrival makes a choice among the products with these as
choice probabilities. Hence we have the potential demand for each product
described by a Poisson distribution with mean si µ (and moreover these demands are independent). We see immediately that firms with smaller shares
must deal with greater uncertainty in demand relative to their expected de√
mand, the standard deviation over expected value being 1/ si µ. That is,
even without information sharing between firms on a merger, we would expect a merger effect since the merged firm has a smaller relative uncertainty
in total demand.
In order to model information effects of a merger, we imagine that firms
have limited information about the value of µ. We assume that the firms
have full knowledge of the logit parameters, i.e., that they know the relative
strengths of their products, but we assume the problem is that no one knows
how much total business to expect. Assume that the firms share a common
prior distribution on µ, that each effectively samples the Poisson process for
some fraction of the total time period, and from the results of the sampling
each updates the prior distribution to their own posterior distribution of µ.
When firms merge, they share their sampling results, thus refining their separate posterior distributions to an improved common posterior distribution.
10
We take the conjugate prior distribution to a Poisson distribution, a
Gamma distribution, as our prior distribution. Specifically, suppose the
prior distribution on µ is Gamma distributed parameterized by (α, β) so the
PDF is given by
β α e−βµ µ−1+α
f(α,β) (µ) =
Γ(α)
This distribution has mean α/β and variance α/β 2 . If we have samples
for fractions τk of the total time period (or fraction of the total population
whose arrivals are being modelled such as the fraction si that determines
those arrivals choosing product i, or else some combination of these), and
the numbers of successes in each of these samples is nk , then the posterior distribution
we infer is again a Gamma P
distribution with parameters
P
P
(α + k nk , β + k τk ). We might think ofP k τk as the total fraction of
the Poisson distribution observed, and the k nk the number of successes
observed in that fraction of time. The information available to each firm will
thus be summarized by two values, αi and βi , so that the posterior distribution of firm i is a Gamma distribution with parameters α + αi and β + βi .
Sharing of information then amounts to adding the αi ’s and βi ’s. We use
these sampling ideas more as a metaphor for how information gathering and
sharing might work. Whether the actual information a firm gathers which
implies their improved posterior distribution comes from sampling may not
important, but this model does give us a plausible specific form for posterior
distributions and sharing of information with which to work.
We will assume that the βi of each firm is known to all firms and that the
αi characterize the private information of the firms. The size of the samples
of each firm might be imagined as in proportion to the capacities of the firms,
for example if firms all have the same time to make their pricing decisions
but only see the fraction of those consumers that choose their product during
that time. The pricing decision of firm i is to be determined as a function
of its αi . Nash equilibrium pricing strategies are determined so that the
price charged for product of firm i given αi maximize the expected profit
of firm i over firm i’s posterior distribution on µ and over the consequent
distribution of demand for firm i’s product and the sample αj of competitors
that determines competitor prices. That is, (for single product firms and
easily generalized for multiproduct firms) given pj (αj ) the pricing functions
11
of competitors, for each αi , the price pi (αi ) is the price pi which maximizes
Z
E(profiti ) =
E(profiti |µ)f(α+αi ,β+βi ) (µ) dµ where
X
X X
...
(pi − mci )qi ·
E(profiti |µ) =
α1 ≥0
αn ≥0 qi ≥0
ψsi µ (qi ) ψβ1 µ (α1 ) . . . ψβn µ (αn )
omitting the summation over αi and ψβi µ (αi ) and where si is given as above
with pj = pj (αj ) for j 6= i. Of course solving for functions pi (αi ) is much
harder than finding ordinary Nash equilibrium.
Each firm’s strategy is defined by a function which gives, for each possible number of consumers seen during their sample time, the price that
maximizes expected profit over the resulting posterior distribution on µ.
The expected profit depends on the actual number of consumers, a Poisson
process given a value of µ, and on the prices of competitors which in turn
depend on the samples each of them receives, also depending on µ. It is thus
a matter of computing the expectation over the inferred posterior distribution on µ. If we were given the pricing functions of each competitor, then
for any given sample and corresponding posterior function, the expected
profit maximizing price is thus determined. To determine the best response
pricing function of the firm, given the competitor’s pricing functions, we
evaluate the optimal price at a number of possible samples in a range, and
then interpolate within that range.
The best response pricing function of a firm mostly depends on the pricing functions of competitor’s near their expected samples in the posterior
distribution of that firm. But the mean of the posterior distribution is in
a narrower range than the µ that would be inferrd from a sample alone,
a range closer to the mean of the prior distribution. All of this is by way
of saying that the best response pricing function of a firm over a range of
possible samples centered at that corresponding to the mean of the prior
distribution depends mainly on narrower corresponding ranges of the competitor’s pricing functions. The interpolating functions approximating optimal pricing strategies for each firm can be computed almost entirely from
the interpolating functions of the other firms with a small error estimable
by extrapolation and decreasing with more interpolation points over larger
ranges. The Nash equilibrium pricing functions are computed by finding the
best response pricing functions to other firms’ pricing functions, iterating
until each firm’s pricing function is sufficiently close to a best response to
12
the other firms’ near best response pricing functions.
Needless to say, computing the expected profit for one combination of
sample and price is rather tedious, somewhat sped up by computing the
innermost expectations symbolically. To find the optimal price for any given
sample is even slower, and these optimal prices have to be computed at a
number of sample values to allow for an interpolating function adequately
approximating the best response pricing function. This has to be done for
each firm and then repeatedly until the process converges. A simplified
linear approximation to best response pricing functions can be used initially
to quickly get reasonably good pricing functions and then these can then be
refined.
3.1
Numerical Example
Consider a simple case with two products owned by two firms. Let a1 = 2.4,
a2 = 3.0, and b = −0.2 define the logit demand. Take the total number
of consumers to be Poisson distributed with an expected value µ = 10000.
Assume however that firms have as prior distribution on µ a Gamma distribution with parameters α0 = 100 and β0 = 0.01, so they assume an expected
value of µ = 10000 but with a standard deviation of 1000, much larger than
the standard deviation of 100 of the Poisson process alone. Assume that the
firms will sample demand for fractions β1 = 0.03 and β2 = 0.05 of the total
demand. The sample the firms get will be independent Poisson distributed
with means 10000β1 = 300 and 10000β2 = 500. The standard deviation of
their posterior distributions of µ will be roughly 500 and 408 respectively, at
most half the value of the prior distribution. If they combine their samples
after a merger, the standard deviation of the resulting posterior distribution
will have a standard deviation of roughly 333, one third of the value of the
prior distribution.
Suppose that prices p1 = 10 and p2 = 12 constitute optimal unconstrained pricing facing a deterministic demand with 10000 total consumers.
Then the (short run) marginal costs would have to be mc1 = 2.36 and
mc2 = 3.34 and the quantities demanded would be q1 = 3458 and q2 = 4223.
A merger to monopoly would result in a significant unilateral merger effect
with prices rising to p1 = 14.67 and p2 = 15.66 and demand falling to
q1 = 2380 and q2 = 3560. Suppose instead that the firms have capacity
constraints of K1 = 1500 and K2 = 2500 so that after the merger firms will
still be capacity constrained. Facing the deterministic demand the firms
would charge p1 = 18.93 and p2 = 19.38 both before and after a merger to
get demand to equal capacity. For these prices to be optimal prices in a long
13
prof1
24500
24000
23500
23000
22500
18.5
19
19.5
20
p1
Figure 4: Profit for fixed demand and expected profits for firm 1
run unconstrained view we would have to have long run marginal costs of
mc1 = 13.05 and mc2 = 12.71.
Taking the smaller short run marginal costs and capacity constraints,
and assuming that firms know only their prior distribution on µ, the firms
would shade prices slightly lower to p1 = 18.67 and p2 = 19.20 (see figures 4
and 5). After a merger, still only knowing the same prior, they would prefer
to shade prices higher to p1 = 18.96 and p2 = 19.42. In the post-merger
profit calculus, the merged firm views the cost of pricing low and having
demand exceed capacity to be (slightly) greater than the cost of pricing
high and having demand fall short of capacity, while the firms on their own
before the merger faced the reverse situation with the cost of pricing too high
(significantly) greater than the cost of pricing too low. The joint distribution
of demand for the products is correlated because of the common uncertainty
of the assumed prior distribution.
Geometrically, the total profit surface as a function of prices for a deterministic demand is seen to have four facets (see figure ??). The prior
distribution on the total number of consumers gives a distribution of profit
surfaces where the point moves roughly along a diagonal of positive slope,
giving the expected total profit curve (see figure 6). The merged firm must
choose the prices to balance the risks along this diagonal whereas the individual firms balance risks to their own profits in the directions of their own
14
prof2
40000
39000
38000
37000
36000
35000
18.5
19
19.5
20
20.5
21
p2
Figure 5: Profit for fixed demand and expected profits for firm 2
prices.
If instead of depending only on a prior distribution on the parameter µ
of the total consumer demand distribution, we next consider what happens
if each firm samples the demand and forms it’s own posterior distribution
for µ. The Nash equilibrium is then defined by the functions defining price
as a function of the sample observed for each firm. We translate the observed sample to the mean of the posterior distribution implied for each
firm and plot the price functions on a common graph in figure 7. Each firm
has a lower uncertainty about the value of µ and thus the ultimate total
number of consumers, but firms cannot predict the exact pricing of their
competitor because they do not know what sample the competitor has seen,
making their share of the ultimate demand more uncertain, and so partially
offsetting the reduction in uncertainty.
We might imagine that to reduce uncertainty the firms would share their
samples. This is the same as when firms priced only on the basis of a prior
distribution, but with sampling the variance of the commonly known posterior distribution of demand is reduced and firms need shade their prices less.
After a merger, the firms share their samples and coordinate their pricing
according to the improved posterior distribution, both effects reducing the
uncertainty in demand. As a function of the mean of the common posterior distribution the merger pricing functions (p1m and p2m) for the two
15
Expected total profit
62000
21
60000
58000
20
18
p2
18.5
19
19
p1
19.5
20 18
Figure 6: Expected total profit for prior distribution
16
prices
20.5
20
19.5
19
p2
18.5
p1
18
17.5
9000
10000
11000
mean post.
12000
Figure 7: Prices as a function of the mean of posterior distributions
products are plotted with the premerger pricing functions in figure 8. The
differences between post- and pre-merger pricing functions (dp1 and dp2) is
illustrated in figure 9. Assuming the actual value of µ = 10000, the samples
of each firm are Poisson distributed with means β1 µ and β2 µ while the sample of the merged firm is Poisson distributed with mean (β1 + β2 )µ. From
these we can compute the expected price the individual firms will charge to
be p1 = 18.74 and p2 = 19.26, while the merged firm can be expected to
charge p1 = 18.90 and p2 = 19.36. These results are summarized in table 2.
3.2
Extension: what if firms can update prices?
So far we have imagined that a firm sets price once and then waits to see
what sales will be. In particular, we assume the product is perishable, so
the firm cannot inventory excess production to the next period for sales.
Since sales cannot usually be taken as all occurring at one point in time, it
may be that a firm can employ more dynamic strategies. If there is some
change in demand over time, by adjusting prices over the sales interval a
firm may be able to price discriminate between early and late buyers. We
will not address this possibility. More simply, a firm may adjust prices in
response to demand fluctuations, charging more when there is a smaller
ratio of unsold product to time remaining, and charging less when the ratio
17
prices
20.5
p2m
20
p1m
19.5
p2
19
p1
18.5
18
17.5
9000
10000
11000
mean post.
12000
Figure 8: Pre- and post-merger pricing functions
change in prices
0.2
dp1
0.175
0.15
dp2
0.125
9000
10000
11000
mean post.
12000
0.075
0.05
Figure 9: Pre- and post-merger pricing functions
18
Model/variable
product 1
product 2
Unconstrained prices
10.00
12.00
post merger prices
% price increase
14.67
46.7
15.66
30.5
capacity constraints
1500
2500
short run marginal costs
2.36
3.34
long run marginal costs
13.05
12.71
pricing to capacity
18.93
19.38
Prior distribution prices
price shading
18.67
-.26
19.20
-.18
post merger prices
price shading
18.96
+.03
19.42
+.04
Posterior distribution prices
price shading
18.74
-.19
19.26
-.12
shared sample prices
price shading
18.80
-.13
19.29
-.09
post merger prices
price shading
18.90
-.03
19.36
-.02
Table 2: Comparison of competitive equilibrium prices.
19
remaining Q.
50
Optimal price
100
90
40
30
110
20
10
120
130
2
4
6
8
10
time
Figure 10: Optimal pricing over a ten interval sales period as a function of
remaining quantity.
is larger. If the firm is capacity constrained, then the firm can “steer” total
sales back toward total capacity, compensating for demand fluctuations. In
essence, the optimum sales is again given by q(p) = K but the price must
vary to achieve this in any given case to absorb the randomness of demand.
Of course, dynamic pricing strategies may evoke dynamic buying strategies on the part of consumers. If consumers can expect discounting as the
sales time nears an end, then they can wait to get a better price. Instead,
we will imagine that consumers come along at a regular rate, buying according to some function of price at that time. That is, we imagine that
consumers time their buying decisions according to some external factor and
price adjustments are relatively infrequent and insignificant in comparison.
For example, suppose the period for sales is divided into ten intervals,
the firm setting a price for each interval. Suppose demand is randomly and
independently distributed for each interval as a function of the price for that
interval, with an expectation of one tenth the demand for our baseline case of
the last section. Take the standard deviation of demand so the total demand
over ten intervals is distributed similarly to the 40% standard deviation case
from the last section. Imagine the initial stock is limited to K = 50.
During the last interval, the firm determines the expected profit maximizing price from the distribution of demand as a function of price, depending on the quantity left to sell as in the last section. During the next to
20
remaining Q.
50
40
30
20
10
2
4
6
8
10
time
Figure 11: Remaining quantity sample paths illustrating the demand process.
the last interval, the firm maximizes the total of expected profit on sales
during the penultimate time interval plus the expected profit in the last interval determined by the quantity that will be left to sell in the last interval.
Working backwards, the firm sets an initial price to maximize profit during
the first interval plus the profit on the supply remaining for the last nine
intervals.
If quantity is large enough for the time remaining, the optimal price
will be nearly the unconstrained profit maximizing price, and there will be
remaining unsold product at the end of the sales period. If quantity is
small, then the firm charges a higher price as in the capacity constrained
case. To a good approximation the optimal price is determined by the
ratio of remaining quantity and remaining sales time. Figure 10 illustrates
the numerically computed optimal prices, interpolating between intervals to
suggest a continuous pricing strategy. If demand is greater than expected in
early intervals, the remaining quantity falls below the “glideslope” and the
firm raises price. If demand is less than expected, the remaining quantity
stays above the glideslope and the firm lowers price (to a minimum of the
unconstrained profit maximizing price). A few sample paths of remaining
quantity are illustrated in Figure 11. There will be cases where capacity is
exhausted early in the sales period and others where there is unsold product.
By adjusting prices during the sales period, the firm increases expected
21
profit, in this example the expected profit is 2641 as compared to a profit of
2566 in the corresponding case from the last section. The initial price is 98.46
compared to 97.60, closer to the deterministic demand, capacity constrained,
optimal price of 100.00, since there is less need to insure against a shortfall in
demand if prices can adjust. Price adjustments reduce the variation in total
sales so that the results are roughly equivalent to the single price capacity
constrained case with random demand having 25 or 50% smaller standard
deviation.
4
4.1
The Effects of Hotel Mergerss
Data
To conduct this analysis, we merged information from two separate data sets.
For the merger information, we used publicly available data provided by the
Texas Comptroller of Public Accounts. The data includes the owner name
and address for each hotel, business name (including any brand affiliation)
and address, and room count. The data set also includes entry and exit
dates for each owner at each property. Every time an ownership transfer of
a hotel takes place, a new “entry date” is included for the hotel for the new
owner. Based on the location of each hotel and the owner information, we
can assess whether a local merger has occurred. We discuss local mergers in
detail below.
While the Comptroller’s office data set includes a panel of quarterly
revenues for each hotel, it does not include any information about price
or number of room-nights sold. For this information, we merged the data
set with an unbalanced panel data set provided by the private firm Smith
Travel Research from 1999 through 2005; we aggregated these annually to
eliminate issues related to seasonality. Participation in the STR data-sharing
is voluntary, and expensive, so STR participants are typically larger-thanaverage and brand-affiliated hotels. Hotels self-report price and room-nights
sold to STR in exchange for receiving similar data from other properties in
their local market area. The price represents the average price per roomnight actually received by each hotel. Price and room-nights sold are the
dependent variables in all regressions. For 1757 (82%) of 2144 chain-affiliated
Texan hotels, the STR data exists for some portion of the 1999-2005 period
and can be matched to the merger information in the Hotel Tax database.
Even though we have merger information for all hotels in the state thanks
to the Comptroller’s Office Database, we restrict our analysis of mergers
to branded chain properties because a large majority of chain hotels are
22
represented in the STR data while only a few independent hotels are.
We estimated fixed-effects regression models with a first-order autoregressive term. There are two dependent variables: (1) annual portion of
room-nights occupied, and (2) average price actually received by the hotel
for a room-night during the calendar year.
The independent variables of interest are binary variables that indicate
an HHI-increasing merger has taken place at a hotel. We compute HHI with
a capacity variable measuring the number of rooms. We defined hotels as
“merging locally” with the intent of selecting an area within which customers
are likely to view two hotels as substitutes to some degree and within which
the hotels’ owner would benefit by coordinating pricing decisions. To this
end, we defined each hotel’s “local area” as a specified group of proximate
hotels (e.g., the 10 closest). To measure proximity, we obtained latitude and
longitude coordinates based on street address for all 2144 branded hotels in
Texas. Using these coordinates, we identified the 10, 20, 25, 30, 40, and 50
closest hotels for each property.
If the new owner of a hotel that changes ownership also owns other
hotels among the local area of the closest 10, 20, 25, 30, 40, or 50 hotels,
and if the ownership change raises the HHI within this local area, then we
consider all hotels belonging to that owner within this local area as having
merged locally. The first column of Table 1 shows the counts of hotels that
merged locally. We only include hotels involved in mergers for which we
have valid occupancy and price data. From the calendar year of the merger
onwards, the “Hotel Merges Locally” dummy variable is set to one for all
yearly observations associated with hotels involved in local mergers. The
remainder of Table 1 provides size information for the hotels involved in the
mergers, along with median occupancy and price data. The prices are the
average daily rates (ADRs) which are the average prices actually received
by each hotel during a calendar year.
The main benefit of the “closest hotel” approach relative to a fixed distance is that it allows the local area to vary in geographic size, to take into
account differing population and commercial densities. If we chose a fixed
mileage radius of five miles, for example, several hotels in downtown Houston would have 35 chain hotels within their local area, while many would
have none, making a “local” merger impossible.
We wish to compare the effects of local mergers from other mergers that
occur across Texas between 1999 and 2005, so we created “distant” merger
variables as well. As shown in Table 1, 889 chain hotels were involved in
statewide HHI-increasing mergers. Most of these did not increase the HHI of
any meaningful local area. From the calendar year of each merger onwards,
23
Definition of “local”
10 closest
20 closest
25 closest
30 closest
40 closest
50 closest
All of TX
Non-merging
# Hotels
Avg. # rooms
Occupancy
Price
51
79
91
99
111
135
889
868
110
120
116
117
120
120
121
98
66.35%
66.58%
66.18%
65.95%
66.26%
65.74%
65.08%
61.68%
$64.22
$66.83
$66.02
$65.80
$66.15
$66.62
$64.51
$59.00
Table 3: Descriptive statistics of hotels in Texas: 1999-2005
we set a merger dummy equal to one for the affected hotels. We call this
variable“Hotel that Merged Distantly.”
We also included three control variables in all regressions. First, we
include a binary variable for all calendar years where a hotel begins operations under a new owner, regardless whether this ownership change increases
any measure of HHI or not. Ownership changes are often accompanied by
temporary disruption of operations. Further, new hotels with a new owner
often operate only partially during their first year of existence. Second, we
include the log size of the owner in terms of number of hotels-the effects of
HHI-increasing mergers need to be distinguished from the more general case
of large owners taking over properties. Third, we include the log number of
years an owner has been operating as a measure of experience. We assume
that more experienced hoteliers will enjoy higher occupancies and possibly
higher prices as well.
4.2
Fixed-effects estimation with an AR(1) term
We assign fixed effects in two dimensions. First, we include a separate fixed
effect for every hotel. This intercept spans all the calendar- observations for
that property. By doing this, we ensure that any effects of time-invariant
hotel- or region-specific heterogeneity do not appear as merger effects. Thus,
the “Hotel that Merged Locally” variables’ coefficients will be significant
only if the post-merger performance is greater than that of the same hotel
before the merger took place. Second, we add an intercept for each calendar
year for each type of location. The STR data classifies all geographic locations as: urban, suburban, small town, highway, airport and resort. Price
24
fluctuations within these geographical areas are often not highly correlated,
so we include a separate calendar year intercept for hotels of each type. For
example, highway hotels did not experience price or occupancy decreases
after the shock of September 11, while urban and airport locations experienced substantial drops. While fixed effects are used exclusively throughout
the analysis for each property, we also considered the use of random effects
models. However, for every specification we estimated, the Hausman test
rejected (at p ¡ 0.001) the null hypothesis that the coefficients of the fixed
and random effects models are identical, indicating that the random effects
approach is inappropriate.
Tables 2a and 2b presents results from fixed-effects regressions on all
branded chain properties. Table 2a has annual occupancy as the dependent
variable and Table 2b has price as the dependent variable. The coefficients
in the occupancy regression are the portion of room-nights that are occupied
while those in the price regression are dollar amounts.
The “Hotel Merged Locally” variable has positive and significant coefficients in six of the seven columns, but the magnitudes and significance levels
decrease after the local area definition of the 25 closest hotels. A merger
within the closest 10, 20 or 25 hotels is associated with occupancy increases
of 1.8%, 1.9% and 2.0% respectively, while a merger among the closest 40
or 50 is associated with an increase of 1.2% and 1.3%. As the median price
at the merged hotels shown in Table 1 is about $66 and the median number
of rooms is about 120, an occupancy increase of 2% would yield an annual
revenue gain of about a post-merger gain of approximately $57,000 per year
per property. The last columns show diminished effects on occupancy when
the local merger area is widened to 40 or 50 properties, and regressions not
shown using wider merger areas such as the closest 75 and the closest 100
showed no significant occupancy increase at all.
The importance of the “Hotel that Merged Locally” variable is further
illustrated with two F-test comparisons with other coefficients in the same
regressions. First, we compare the coefficients with those of the “ Hotel that
Merged Distantly” variable-those mergers raising Texas’ lodging HHI overall
but not that of any local area. We find a statistically significant difference
for the local area definitions of closest 20, 25 and 30. For a local merger area
of the closest 40 hotels, the difference between the two coefficients slips out
of the range of significance, consistent with the decreased coefficient. Second
we compare the “Hotel that Merged Locally” variable’s coefficient to that of
the “Hotel within Area of Merger” variable, to assess whether all hotels in
the vicinity of a merger benefit equally post-merger, regardless of whether
their owners participated in the merger or not. The local merger effects for
25
Local definitions (closest #)
Dep. Variable: Occupancy (Q)
Local Merging Hotel
(raises local HHI)
Distant Merging Hotel
(raises state HHI, not local HHI)
Local non-merging hotel
First Year of New Owner
Log Count of Owner’s Hotels
Log Years Owner in Business
F test; Ho: Local = Distant
F test; Ho: Local = Within Area
Dep. Variable: Avg. Daily Rev. (P)
Hotel that Merged Locally
(raises local HHI)
Distant Merging Hotel
(raises state HHI, not local HHI)
Local non-merging Hotel
First Year of New Owner
Log Count of Owner’s Hotels
Log Years Owner in Business
10
20
25
30
40
50
.018+
(.011)
.003
(.002)
.001
(.006)
-.024**
(.003)
-.003
(.003)
.008**
(.003)
.019*
(.008)
.002
(.002)
-.002
(.004)
-.024**
(.003)
-.003
(.003)
.008**
(.003)
.020*
(.008)
.002
(.002)
-.002
(.004)
-.024**
(.003)
-.003
(.003)
.008**
(.003)
.016*
(.008)
.002
(.002)
.000
(.003)
-.024**
(.003)
-.003
(.003)
.008**
(.003)
.012+
(.007)
.002
(.002)
.006+
(.003)
-.024**
(.003)
-.003
(.003)
.008**
(.003)
.013+
(.007)
.003
(.002)
.014**
(.003)
-.023**
(.003)
-.003
(.003)
.008**
(.003)
2.01
1.98
3.62+
5.05*
4.78*
6.51**
2.77+
3.72+
1.37
.550
2.12
.020
10
20
25
30
40
50
-.866
(.693)
-.918**
(.107)
.537
(.391)
-.553**
(.214)
.709**
(.162)
-.467**
(.165)
-1.291*
(.540)
-.892**
(.110)
.256
(.253)
-.542*
(.214)
.725**
(.162)
-.466**
(.165)
-1.301*
(.514)
-.890**
(.110)
.271
(.227)
-.536*
(.214)
.727**
(.162)
-.463**
(.165)
-1.162*
(.496)
-.874**
(.111)
1.13**
(.216)
-.521*
(.214)
.730**
(.162)
-.473**
(.165)
-1.181*
(.472)
-.855**
(.112)
.818**
(.201)
-.517*
(.214)
.714**
(.162)
-.470**
(.165)
-1.104*
(.447)
-.851**
(.112)
.883**
(.194)
-.510*
(.214)
.729**
(.162)
-.474**
(.165)
.430
16.5**
.290
18.2**
F test; Ho: Local = Distant
.010
.510
.590
.310
F test; Ho: Local = Within Area
3.210+ 7.18** 8.38** 19.2**
**: p < 0.01; *: p < 0.05; +: p < 0.10
Table 4: Fixed-effects Regressions of Q and P on Merger Dummies
26
the closest 20, 25, and 30 properties are statistically significantly greater
than that for the hotels that were within of, but did not directly participate
in, the mergers. We note that this difference also becomes insignificant for
the larger local merger areas of the closest 40 or 50 properties.
From Table 2b, we observe that price at the “Hotel that Merged Locally”
goes down post-merger, but we note that prices go down for distant mergers
as well. The local mergers using the closest 20 and closest 30 local merger
area definitions are associated with the greatest price decreases, of about
$1.30 per room, suggesting a revenue decrease of $31000 due to the merger.
Of course, these same hotels enjoy a $57000 revenue increase due to the
increased occupancy, implying a net revenue increase of $26000 due to the
merger. Even assuming a very high variable cost per room night of $25 (the
variable cost per room night is typically said to be about $15 for a threestar property; see Kalnins, 2006), the merger still yields approximately an
additional $5000 per year in gross margin for the median hotel.
The coefficients of the control variables deserve mention. First, hotels
suffer from decreased occupancies and prices in calendar years with a new
owner. This is not surprising given the temporary disruption of operations,
for existing properties, and only partial operations, in the case of new properties. The log size of the owner in terms of number of hotels has a significant
positive effect on prices but not on occupancies. Finally, the log number of
years an owner has been operating is also significant and positive in the
quantity regression but significant and negative in the price regression.
5
Conclusions.
It is difficult to make broad generalizations about the impact of information
sharing in mergers, but we have illustrated that these effects can be simulated. If there are no binding capacity constraints, the information sharing
effects are generally dwarfed by the usual unilateral merger effects. With
binding capacity constraints, firms will price so demand is near capacity, the
costs of mispricing are roughly proportional to the pricing errors, with the
proportionality constants for high and low pricing different.
In the presence of uncertainty in the demand, optimal prices will differ
from the prices where expected demand is equal to capacity depending on
whether high or low pricing is more costly. Prices will be shaded higher if
pricing low and having demand exceed capacity is more costly, while price
will be shaded lower if pricing high and having demand fall short of capacity
is more costly. Moreover, if price adjustments are possible to accommodate
27
demand fluctuations, then the necessary price shading will likely be even
smaller.
In general, we expect mergers to reduce the uncertainty in demand, especially due to information sharing, implying smaller price shading. However
the relationship between the uncertainties in demands for the merger’s products and the nature of the information sharing will change the results. In
the capacity constrained case, prices are always near to the prices making
demand equal to capacity and price shading, either higher or lower, will be
a small effect except in extreme cases.
. . . more
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30