Demand Model

While several choices of demand models are possible (see Chapter 7 of Talluri and van Ryzin [138] for more details and examples), we restrict attention to one of the most popular options in the RM literature, namely the linear demand model under additive noise (for extensions to other relevant demand models, we refer the reader to Section 4.4). This model is characterized by:

dt(pt) = bt+ Atpt, (4.1a)

Dt(dt, εt) = dt+ εt, (4.1b)

where the terms have the following significance:

• dt is the planned demand, i.e., the deterministic component of the demand

function, dependent only on the price vector pt

• bt ∈Rn represents a base demand in period t

• At∈Rn×n represents a matrix of price sensitivity coefficients in period t

• Dt is the realized demand in period t

• εt is an exogenous noise.

In particular, the functional form dt is linear in the prices, and the noise affects

the demand in an additive fashion. This model is quite popular due to its simplicity, and the ease of estimation from data. As such, it has been used extensively in both theoretical, as well as experimental studies (see Talluri and van Ryzin [138] and Mas- Colell et al. [104]). Standard assumptions on the matrices At include the following:

Assumption 4. The diagonal coefficients of At are non-positive, i.e., aii ≤ 0, ∀ t ∈

T .

This assumption is a fundamental law in economics, and reflects the fact that decreasing the price of a given product makes it more attractive to the customers. Items not satisfying this requirement (known as Veblen or Giffen goods Mas-Colell et al. [104]) are usually ignored in the revenue management literature.

Assumption 5. The matrices At are strictly row-diagonally dominant, i.e., |aii| >

P

j6=i|aij| , ∀ i ∈ I.

The latter fact states that the demand for a product i is more sensitive to changes in its own price, rather than simultaneous changes in the prices of other products. Alternatively, one sometimes requires that

Assumption 6. The matrices Atare strictly column diagonally-dominant, i.e., |ajj| >

P

This would reflect that changes in the price of one product impacts the demand of that product more than the total demand of other products combined. All of the above assumptions have well grounded economic justifications (Mas-Colell et al. [104]), and have been widely adopted in the operations management literature. In can be seen by standard facts in linear algebra (see Theorem 5.6.17 in Horn and Johnson [79] and Chapter 2 in Horn and Johnson [80]) that the first assumption corroborated with any of the latter two ensure that:

1. The matrices At are invertible. This is convenient, since it allows inverting

the price-demand relation to obtain a specific price pt that would generate a

particular demand dt. In this sense, we can equivalently think of the decisions

as being the demands dt, rather than the prices.

2. The eigenvalues of At all have non-positive real parts. Since the Atmatrices are

also usually taken to be symmetric, this latter fact has the direct implication that the revenue function, r(pt)

def = p′

tdt(pt), is concave in the prices, which

ensures the existence of a unique revenue-maximizing price (see Chapter 7 of Talluri and van Ryzin [138]).

Despite these attactive theoretical properties, the model does suffer from several pitfalls. On a theoretical level, it requires bounding the range of feasible prices in order to ensure the demand is non-negative (e.g., in a single product case, we would need pt ≤ −at/bt). This also implies that the model violates another typical requirement

in the OM literature, namely that the range of the revenue function r(pt) span the

entire positive half-line (refer to Section 7.3 of Talluri and van Ryzin [138] for more details). For recent work that provides a natural extension of the linear demand model which does not suffer from some of these shortcomings, we refer the reader to Farahat and Perakis [66].

On a practical level, several empirical studies (e.g., Smith and Achabal [135]), as well as several patent filings (Woo et al. [147], Boyd et al. [44]) have found the model to under-perform other functional forms, such as exponential or power sensitivity. However, despite these shortcomings, due to widespread use of the model in both

theory and practice (see, e.g., Heching et al. [78], Bertsimas and Perakis [28], Maglaras and Meissner [101], Adida and Perakis [1], Thiele [140] and references therein), we have chosen it as the main object of study in the current chapter.

In terms of the estimation requirements, since the functional form of the demand is linear (4.1a), and the noise affects the model in an additive fashion (4.1b), one can use ordinary least-squares regression techniques (OLS) (Greene [77]) to estimate the parameters of the model. More precisely, with dependent variables yit = Dit, ∀ i ∈

I, ∀ t ∈ {2, . . . , T }, and with independent variables xit = {pit, δt} (where δt is an

indicator for period t, with t ∈ {1, . . . , T − 1}), one can compute estimates ˆAt, ˆbt, as

well as associated confidence intervals.

The key underlying assumptions supporting the use of OLS techniques (see Chap- ter 2 of Greene [77]) are the standard Gauss-Markov requirements, namely

(i) The linearity of the functional form (i.e., equation (4.1a), in our case)

(ii) The full rank assumption on the data matrix X (consisting of the pi,t, δt vari-

ables)

(iii) Exogeneity of the independent variables, i.e.,E [εi,t| xjt] = 0, ∀ i, j ∈ I, ∀ t ∈ T .

In words, the expected value of the disturbance corresponding to a particular observation should not be a function of the independent variables xit corre-

sponding to any observation (including the current one).

(iv) Homoscedasticity and nonautocorrelation, i.e., the disturbances εit should have

the same finite variance and be uncorrelated across i and t. More precisely, var[εit| X] = 0, ∀ i ∈ I, ∀ t ∈ T , cov[εitεjτ| X] = 0, ∀ i, j ∈ I, ∀ t, τ ∈ T with

(it) 6= (jτ ).

(v) Normality, i.e., that the disturbances εit follow a Gaussian distribution.

Since, in reality, several of these assumptions are violated, procedures have been designed to test for mis-specifications, and several extensions of the regression tech- niques are available for more general cases (see Greene [77] for a complete account

and more references). In Section 4.6.3, we revisit some of these issues in the specific context of estimating the linear demand model (4.1a), and we also discuss several aspects related to our own data-set.