Durable Goods Monopoly with Varying Demand

Transcription

1 Durable Goods Monopoly with Varying Demand Simon Board Department of Economics, University of Toronto June 5, 2006 Simon Board, Back to school sales Motivation New influx of demand reduce prices in September. But causes people to delay purchase in August. How much should reduce price? Pricing with varying demand What happens if new demand falls over time? What happens if new demand is uncertain? Objective: Derive optimal pricing strategy for durable goods monopolist facing fluctuating demand from new cohorts. Simon Board,

2 Durable Goods Monopoly No new entry of consumers (Stokey, 1979) Consumers enter market in period 1. Firm choose prices {p 1,...,p T }. Agents choose when to buy. Solution: charge static monopoly price forever. Identical entry each period (Conlisk et al, 1984) Solution: charge static monopoly price forever. Simon Board, Varying Demand What if new demand varies over time? Theory of dynamic pricing. Scope for intertemporal price discrimination. Technique Method to solve dynamic mechanism design problems. Simple marginal revenue interpretation. Fast rises and slow falls Demand growing = price increases quickly. Demand dying = price decreases slowly. Application: propagation of demand cycles. Prices exceed the average demand price. The lowest price is at last period of the slump. Simon Board,

3 The Model Time is discrete, t {1,...,T}, where allow T =. Consumers and firm s information represented by filtered space (Ω,F,{F t },Q). Common time t discount rate, δ t (ǫ,1 ǫ), is F t adapted. Total discount factor t := t i=1 δ s. Consider consumer with value [,] If buy at time t and price p t get ( p t ) t. If do not purchase get zero. Each period consumers of measure f t () enter market Distribution function F t (), survival function F t (). Total measure F t (). New demand, f t (), is F t adapted. The Model 5 Payoffs Consumer s Problem Consider consumer (,t) with value who enters at time t. Given sequence of F t adapted prices {p 1,...,p T }. Choose purchasing time τ(, t) to maximise expected utility. u t () = E[( p τ ) τ ] Firm s Problem Assume marginal cost is zero. Choose F t adapted prices {p t } to maximise expected profit [ T ] Π = E τ (,t)p τ (,t)df t Notable assumptions: No resale. Firm commits to prices. t=1 where τ (,t) maximises the consumer s utility, u t (). The Model 6

4 Consumer Surplus and Welfare Purchase time optimal so use envelope theorem, [ ] u t () = E τ (x,t)dx+u(,t) using Milgrom Segal (2002) since space of stopping times complex. Consumer surplus from generation t, [ ] u t ()df t = E τ (,t)f t ()d Welfare from generation t, [ ] W t = E τ (,t)df t Costs are zero so the welfare is maximised by setting p t = 0. Solution Technique 7 Firm s Problem Define marginal revenue with respect to price as m t () := f t () F t () Expected profit is welfare minus consumer surplus, [ T ] Π = E τ (,t)m t ()d t=1 Profit is discounted sum of marginal revenues. Marginal revenue sticks to each agent (, t). The firm s problem is to chooses prices {p 1,...,p T } to maximise Π subject to τ (,t) maximising u t (). Solution Technique 8

5 Consumer s Problem and Cutoffs Lemma 1. The earliest purchasing rule, τ (,t), obeys: [existence] τ (,t) exists. [ monotonicity] τ (,t) is decreasing in. [non discrimination] τ (,t L ) t H = τ (,t L ) = τ (,t H ), for t H t L. Characterise τ (,t) by F t adapted cutoffs Back out prices from cutoffs: t := inf{ : τ (,t) = t} ( t p t ) t = max τ t+1 E [( t p τ ) τ] Solution Technique 9 General Solution Definition. Cumulative marginal revenue equals M 1 () := m 1 () and M t () := m t ()+min{m t 1 (),0}. Assumption (MON). M t () is quasi increasing ( t). Theorem 1. Under (MON) the optimal cutoffs are t = M 1 t (0). Period t = 1 Sell to agent iff m 1 () 0 Period t = 2 Form cumulative MR, M 2 () = m 2 ()+min{m t 1 (),0} Sell to agent iff M 2 () 0 Cutoffs are determined by past demand. Prices are determined by future cutoffs. Optimal Pricing 10

6 (1) Monotone Deterministic Demand Suppose demand deterministic. Proposition 2a. Suppose demand is increasing, m 1 t+1 (0) m 1 t (0). Then t = m 1 t (0) and prices are p t = m 1 t (0). Proposition 2b. Suppose demand is decreasing, m 1 t+1 (0) m 1 t (0). Then t = m 1 t (0) and prices are p t = T E s=t Myopic price: p M t := m 1 t (0). [( s ) ] s+1 m 1 s t (0) F t t Average Demand price: p A t := m 1 T (0) Applications 11 (2) Deterministic Cycles Suppose demand follows K repetitions of {f 1 (),...,f T ()} Proposition 4. For k 2, cycles are stationary. Proposition 5. For k 2, optimal prices always lie above the average demand price, m 1 T (0). Price discrimination bad for all customers. Proposition 6. For k 2, if cycles are simple the price is lowest at the end of the slump. Applications 12

7 (3) IID Demand Demand drawn from {m i ()} with prob {q i }. Average marginal revenue m A () = i q im i (). Average demand price p A := [m A ] 1 (0). Proposition 7. The SLLN implies lim t t p A and lim t p t pa a.s.. Stochastic equivalent of Proposition 5 (i.e. with deterministic cycles, prices exceed the average demand price). Applications 13 Summary Derived optimal pricing strategy for durable goods monopolist facing varying demand. Award good to agents with positive cumulative MR. Prices rise quickly and fall slowly. Asymmetry pushes prices upwards. The End 14