A stochastic dynamic pricing and advertising model under risk aversion

Abstract

This article analyzes a dynamic pricing and advertising model for the sale of perishable products under constant absolute risk aversion. We consider a time-dependent version of Gallego and van Ryzin’s dynamic pricing model with exponential demand and include isoelastic advertising effects as well as marginal unit costs. We derive closed-form expressions of the optimal risk-averse pricing and advertising policies of the value function and of the certainty equivalent. The formulas provide insight into the (complex) interplay between risk-sensitive pricing and advertising decisions. Moreover, to evaluate the optimally controlled sales process over time we propose efficient simulation techniques. These are used to analyze the characteristics of different degrees of risk aversion, particularly the concentration of the profit distribution and the impact on the expected evolution of price and advertising rates.

Appendix

Model description

For any given admissible Markov policy (p, w), cf. the section ‘Model description’, the jump intensity in state (t, n, r) is given by λ _n,r (t):=λ(t, p _n,r (t), w _n,r (t)). For all t, n and r let W(t, n, r):=u(r)=−e^−γ⋅r. Using the Itô formula for jump processes and taking expectations we obtain

where A^(p,w) is the generator that characterizes the dynamic of the three-dimensional Markovian process (t, X _t , R _t )_0⩽t⩽T; (0, X₀, R₀)=(0, N, 0) is the starting state. Applied to a function F(t, n, r)≔g(t)⋅f(n)⋅h(r) (where g(t) and h(r) are differentiable functions with compact support), for given controls (p, w) A has the form, 0⩽t⩽T, 0⩽n⩽N, ,

cf. Øksendal, Sulem (2005), Sec. 1. We want to show that the solution of the value function, cf. Theorem 3.1, is of the following separable form

where the function is independent of r and differentiable in t for all n. Applying the generator A to (A.3), we obtain from (A.2)

If is chosen such that it solves the non-linear difference-differential equation, n=1, …, N, 0⩽t<T,

with the boundary conditions

the policy (p^*, w^*)=(p _n ^*(t), w _n ^*(t)) determined by (A.4), cf. Theorem 3.1, maximizes the right-hand side of (A.1), and we obtain

Hence, for all policies (p, w)=(p _n,r (t), w _n,r (t)) we have

Thus, is the value function J^*(t, n, r). Moreover, using the relation , it follows from (A.4) – (A.5) that J^*(t, n, r) satisfies the non-linear difference-differential equation, n=1, …, N, 0⩽t<T, ,

with the boundary conditions, ,

The equation (A.6) is called the Hamilton-Jacobi-Bellman equation that is associated with the utility maximizing problem described in the section ‘Model description’.

Analytical solution

Proof of Proposition 3.1

• In our special case with exponential demand, we compute the CARA version of the dynamic Dorfman–Steiner identity as follows. Following the definition of the sales rate, the ratio of optimal advertising rate and optimal rate of sales is,

  

   □

Proof of Theorem 3.1

• Plugging the solution of the value function in the optimality condition for price, that is, , cf. (7), we obtain, d≔ɛ/(v⋅(1−δ)),

  

  The explicit formula for optimal advertising rates can be derived by plugging the formula for optimal prices derived above into the optimality condition for optimal advertising rates, cf. (7):

  

  The optimal rate of sales λ _n ^*(t) (cf. Theorem 3.1) can be computed via p _n ^*(t) and w _n ^*(t) as follows:

  

  Simple algebra reveals that the function solves equation (8). Note that the optimal controls are associated with

  

  cf. (7), and thus the conditions and guarantee that p _n ^*(t)⩾0 and w _n ^*(t)⩾0. Consequently, the optimal controls are admissible. □

Expected evolution of optimal prices

Proof of Theorem 4.1

• For sufficiently small h>0, the probability of the sale of one article between the time points t and t+h is approximately λ^*(t)⋅h. At time t, with n items left, we consider the following inequality comparing the actual price p _n (t)=c(t)/v+ln(1+v·γ/ɛ)/(v·γ)+(1−δ)/ɛ·ln(K _n (t)/K_n−1(t)) and the expected future price in t+h:

  
  
  
  

  For h→0, this inequality holds if is satisfied; or equivalently

  
  

  If the unit cost function c(t) is differentiable and not decreasing at time t, the last inequality holds if

  
  
  
  

  Since K is concave in the inventory level n, we have x≔K _n (t)⋅K_n−2(t)/K_n−1(t)²∈(0, 1), and for all 1⩽n⩽N and 0⩽t<T, the last inequality can be expressed as 1−x<−ln x, ∀x∈(0, 1). For x=1, both sides of the inequality are the same, and ∀x∈(0, 1) the right-hand side is decreasing faster in x than the left-hand side, that is, |−1|<|−1/x|. Consequently, ∀x∈(0, 1), the graph of 1−x is strictly dominated by −ln x, and the assertion follows. □

Evaluation and further properties

Proof of computation of simulated revenue and advertising expenditure

For any given (or simulated) sales times, from the definition of the accumulated revenue minus production costs from initial time 0 up to time t, we directly obtain the following formula:

The cumulative advertising expenditure until time t is given by the integrated trajectory of realized (optimal) advertising rates, characterized by given (simulated) sales times τ _j . The integration of optimal advertising rates over time from 0 up to t can be set together by the partition of time intervals with the same inventory level:

Because of the structure of the optimal advertising rates, all single integrals can be explicitly solved, and we obtain a simple sum for the advertising expenditures until time t.