Numerical solution of a conspicuous consumption model with constant control delay

Abstract

We derive optimal pricing strategies for conspicuous consumption products in periods of recession. To that end, we formulate and investigate a two-stage economic optimal control problem that takes uncertainty of the recession period length and delay effects of the pricing strategy into account.This non-standard optimal control problem is difficult to solve analytically, and solutions depend on the variable model parameters. Therefore, we use a numerical result-driven approach. We propose a structure-exploiting direct method for optimal control to solve this challenging optimization problem. In particular, we discretize the uncertainties in the model formulation by using scenario trees and target the control delays by introduction of slack control functions.Numerical results illustrate the validity of our approach and show the impact of uncertainties and delay effects on optimal economic strategies. During the recession, delayed optimal prices are higher than the non-delayed ones. In the normal economic period, however, this effect is reversed and optimal prices with a delayed impact are smaller compared to the non-delayed case.

Pricing conspicuous consumption products in recession periods with uncertain strength

“…Objective function U 1 (22a) In addition, we choose the initial reputation and cash to be A 1 0 ¼ 20:0; B 1 0 ¼ 10:0; A 2 0 ¼ 40:0; B 2 0 ¼ 50:0; ð30cÞ each of which corresponds to economic starting points, where the firm can cope with an intermediate (A 0 1 ,B 0 1 ) or severe (A 0 2 ,B 0 2 ) recession for a certain time period, but which does not hold enough capital reserves to survive a continuing recession (Caulkins et al 2011;Huschto et al 2011). Finally, the random variable a for the linearization and the sigmapoint approaches is characterized by its mean value " a ¼ 0:836 ð31Þ…”

“…In Huschto et al (2011) we extended the aforementioned results by establishing a new numerical methodology based on a scenario tree formulation and the introduction of a delayed effect of the current price on the reputation. This implies that price changes have a delayed effect on the consumer's reaction as they have to get accustomed to the new situation.…”

“…While the duration of the economic crisis has been considered as a random variable by Caulkins et al (2010Caulkins et al ( , 2011 and Huschto et al (2011), its strength was assumed to be known. In fact, it has been deduced that one has to distinguish three major types of recession, depending on the rate of the demand reduction within the recession period: mild, intermediate, and severe crises.…”

“…] j ¼ 2:0; c ¼ 5:0; C ¼ 7:5; d ¼ 0:05; m ¼ 3:0; b ¼ 0:5; r ¼ 0:1; k ¼ 0:5; ð30aÞand the assumptions(Huschto et al 2011) …”

“…For a detailed description of the pricing strategies depending on the economic stages, the initial cash and reputation, time delays, and the three major types of recession, compareCaulkins et al (2011),Huschto et al (2011).…”