THE HOTEL PRICING DECISION UNDER UNCERTAIN DEMAND
Robert Scott, David Quigg, Edward Sattler, and Jannett Highfill, Bradley University
This paper shows that a certainty model cannot be used to accurately predict the optimal pricing behavior of hotels which face uncertain demand. Hotels have fixed capacity (number of rooms) determined long before actual demand is realized. Depending on their capacity, some hotels will choose a price which is below the certainty price and others above. Because of the discrete nature of the quantity demanded, bed and breakfasts are more affected by an unrented room, and thus price "aggressively," that is, below the certainty price. Large hotels, on the other hand, price above the certainty price. The uncertain number of persons "in the queue" to rent a room is treated in that paper as a standard queuing problem with the quantity demanded a random variable distributed Poisson with a mean inversely related to price. The behavior of large and small hotels is derived from simulation study.
In a paper "A Hotel Capacity Utilization Model" (1995) Scott, Sattler, and Highfill study the problem of an hotel with fixed capacity which faces an uncertain demand for rooms. Briefly, that paper assumes that the hotel cannot respond to uncertain demand by inventory adjustments, nor for that matter, by using high priced resources to temporarily increase production when demand is high. The uncertain number of persons "in the queue" to rent a room is treated in that paper as a standard queuing problem with the quantity demanded a random variable distributed Poisson whose mean is inversely related to price. Expected demand is the usual (linear) demand curve, however, because of the fixed capacity, there is always some probability on a given day that some of the demanders will be unable to get a room.
Thus the risk-neutral price-setting hotel faces a "timing problem" in that the hotel must set the price for its room before the actual demand for its room is known. The primary contribution of the 1995 Scott, Sattler, and Highfill paper is to show that the hotel will always choose a price for which it expects to have excess capacity even while pursuing profit maximization. The present paper is a first attempt to further examine the pricing decision of hotels under uncertainty. Briefly, the paper considers the question of how "aggressively" a hotel facing an uncertain demand sets its prices, that is, how its optimal price under uncertainty compares to the optimal price under certainty for three classes of hotels. These are (1) hotels with very small capacities, called "bed and breakfasts," (2) hotels with somewhat larger capacities, called "small hotels," and, (3) hotels with large capacities, called "large hotels." (See Figure 1 for an illustration of these classes.)
Intuitively one might suppose that large hotels facing uncertain demand would price its rooms the most aggressively as compared to small hotels or bed and breakfasts, that is, that the optimal price under uncertainty for large hotels would be less than the certainty price, and that the difference between the certainty price and the optimal price under uncertainty would be the largest. The argument would be that a large capacity implies a high probability of excess capacity, and thus an "aggressive" pricing strategy would be needed to combat the high probability of excess capacity. The paper will show, however, that the opposite results hold.
Bed and breakfast hotels price the most aggressively in that they charge a price under uncertainty which is not only less than the certainty price but also less than the certainty marginal revenue. The explanation for this comes for the discrete nature of the problem. When capacity is one or two or three rooms, a single unrented room means that the unused capacity is 100%, or 50%, or 33.3%. Small hotels have the second most aggressive pricing strategy in that the optimal price under uncertainty is less that the certainty price but greater than the certainty marginal revenue. Large hotels price least aggressively setting an optimal price under uncertainty which is greater than both the certainty price and the certainty marginal revenue.
The literature of problems with uncertain demand is strong and growing. Two papers which have been especially influential are Epstein (1978) and Turnovsky (1973). The timing problem aspect of capacity in this paper is probably closest to that of Lippman, McCardle, Rumelt, (1991). The definition of capacity itself, however, is simply the number of rooms in the hotel. Thus the paper sidesteps many of the difficulties in defining capacity, as for example, those discussed in Corrado and Mattey (1997) and Lee and Kwon (1994).
Denote unit sales by X. (We adopt the convention from statistics of using upper case letters for variable names and lower case letters for the values a variable can take.) Denote the capacity by Z, and assume that capacity is at least one, z 1. Denote the quantity demanded by Y. Because of the capacity constraint, unit sales are equal to capacity if demand is greater than capacity. Otherwise unit sales are equal to quantity demanded. These relations are summarized below.
The demand uncertainty facing the firm is such that price, P, and expected
demand, , for its output good are linearly
related:
We will assume that Y, the quantity demanded, has the Poisson distribution;
it follows that E(Y) = and the expected
quantity demanded is, for an arbitrary price, P,
Expected unit sales can now be computed as
Comparison of (3) and (4) shows that the first summation is the same for both computations; since the second summation is of precisely those terms when y > z it follows immediately that E(Y) > E(X) for any arbitrary price.
Under the assumptions of the Scott, Sattler, and Highfill (1995) paper, (the profit maximizing firm is risk neutral and marginal cost is constant and thus can be set at zero), the first order condition
(where R is revenue) can be simplified to
and a solution P* to (6) can be shown to exist. Further, the optimal solution has the following properties
That is, the optimal price must exceed one-half the maximum (zero demand) price.
To summarize, the solution of (6) is the price that maximizes profit for the hotel with fixed capacity and uncertain demand.
Because of the theoretical complexities involved, the paper proceeds from
this point by simulation. The goal was to find the optimal price under
uncertainty for various levels of capacity. While specific values of the
parameters were specified, the method is general in the sense that any other set
of parameters could have been chosen. Recall that the model assumes a linear
expected demand function and the (discrete) Poisson distribution with mean equal
to is used to model the quantity demanded.
The Poisson is theoretically appropriate; as a single parameter model it is easy
to use for simulation, further, for large
it
is approximated by the Normal distribution. The optimal price function was found
numerically by examining each capacity size from Q = 1 to Q = 50. For each
capacity size, the profit maximizing price was found by successively bracketing
the optimal price with smaller and smaller brackets which contained the maximum
profit. When the bracket was no wider than $.01, the midpoint was selected as
the optimal price.
The Bed and Breakfast, the Small Hotel, and the Large Hotel
The simulation assumes the following expected demand function
Rearranging terms, if this were a certainty model
where the subscript "C" denotes the certainty price and marginal revenue respectively. Recall that marginal cost is assumed to be zero. The simulation yields the following results.
Bed and Breakfast
For z = 2 PC = $14.60 MRC = $14.20 P* = $13.92.
For z = 3 PC = $14.40 MRC = $13.80 P* = $13.70.
Small Hotel
For z = 10 PC = $13.00 MRC = $11.00 P* = $12.37
For z = 20 PC = $11.00 MRC = $7.00 P* = $10.72.
Large Hotel
For z = 40 PC = $7.00 MRC = $-1.00 P* = $8.17.
For z = 50 PC = $5.00 MRC = $-5.00 P* = $7.59.
For this set of parameters the bed and breakfast case obtains for capacities 1, 2, and 3; the small hotel case obtains for capacities between 4 and 25 inclusive; the large hotel case obtains for a capacity of 26 or larger.
These results are summarized in Figure 1. The horizontal axis measures both expected quantity demanded and capacity while the vertical axis measure price and marginal revenue. The line labeled D is both the expected demand curve under uncertainty and the actual demand curve in the certainty case. The MRC is the certainty marginal revenue curve. The optimal price curve gives the price when demand is uncertain. As explained above and shown by the simulation, the range labeled B&B is such that the uncertainty price is exceed by both the certainty price and marginal revenue. The range labeled Large Hotel is the opposite case where the uncertainty price is less than both the certainty price and marginal revenue. The intermediate range, labeled Small Hotel, is where the optimal price is between the certainty price and marginal revenue.
While beyond the scope of this paper, it might also be noted that hotels facing uncertain demand might benefit from acquiring capacity which they would never acquire in a certain world, because there is always a positive probability that any extra unit of capacity might be rented. In fact, if the cost of acquiring capital were zero, a hotel's capacity would even approach infinity because of the positive probability that the firm could rent the rooms. In that case, the price charged would be the limit price as defined by (7) -- $7.50 in the simulation.
Conclusion
The most basic conclusion of this paper is that a certainty model cannot be used to accurately predict the optimal pricing behavior of hotels which face uncertain demand. Depending on their capacity, some hotels will choose a price which is below the certainty price and others above. Because of the discrete nature of the quantity demanded, bed and breakfasts are more affected by an unrented room, and thus price "aggressively," that is, below the certainty price. Large hotels, on the other hand, price above the certainty price.
References
Corrado, C., & Mattey, J. (1997). "Capacity Utilization. The Journal of Economic Perspectives 11(1), 151-167.
Epstein, L. G. (1978). Production Flexibility and Behavior of Competitive Firms Under Price Uncertainty. Review of Economic Studies, 45, 251-261.
Lee, Y. J., & Kwon, J. K. (1994). Interpretation and Measurement of Capacity Utilization: the Case of Korean Manufacturing. Applied Economics, 26, 981-990.
Lippman, S. A., McCardle, K.F. & Rumelt, R.P. (1991). Heterogeneity Under Competition. Economic Inquiry, XXIX(4), 774-782.
Scott, R., Sattler, E., & Highfill, J. (1995) A Hotel Capacity Utilization Model. The Journal of Economics, XXI(2), 101-105.
Turnovsky, S. (1973). Production Flexibility, Price Uncertainty, and the Behavior of the Competitive Firm. International Economic Review, 14, 395-412.