Set-up
As in the baseline model, in each period one new consumer arrives. Since consumers can now choose to wait, more than one consumer might “accumulate” in the market, such
12Note that ifµis not too (insensibly) high, this price is lower than consumer valuation v. Refer to the discussion of the comparative statics ofµin Section 5.3 for more details.
13Only under symmetric aggregate capacities even lower prices are possible (see Section 5.1).
14One could also consider a third, “intermediate” case, where consumers are not informed about ca-pacities, but know the remaining time, allowing them to be still fully forward-looking. This case would require more complex analysis involving consumers having beliefs about capacities and possibly firms using prices as capacity signals.
that at the deadline at most as many consumers as total periods can be in the market.
Letc∈N+be the number of consumers in the market at a certain time. Again, I assume that in each period only one trade can occur, i.e. at most one of possibly several waiting consumers can buy from only one firm. The remaining consumers are “rationed” and have to wait for (at least) one more period.
Then the timing in a period is such that first a new consumer arrives into the market with probability one, while all consumers who did not buy in the previous period are in the market as well. Second, firms simultaneously post prices and a lottery draws out of all consumers in the market one lucky consumer.15 The probability to be the lucky consumer is equal to 1/c, while the probability to be rationed is equal to (c−1)/c. Third, the lucky consumer chooses to buy from an active firm or to further wait in the market.
Possible trade between the lucky consumer and a firm takes place and the period ends.
All consumers who did not buy will remain in the market for the next period.
State Transition
The state is now given by the vector ω= (t,x, c), i.e. it additionally features the number of consumerscin the market, and is common knowledge among all firms and consumers.
Depending on the choice of the lucky consumer, the state transition function from ω= (t,x, c) into a next state ω0 = (t0,x0, c0) is
ω0|(ω, j) =
(t−1,x, c+ 1) if j = 0, (t−1,x−ej, c) if j ∈J(ω),
(16)
where ej is a 1×nvector with all zeros but the jth entry equal to 1. If the consumer waits (j = 0), x capacities remain and one additional new consumer arrives, such that there will be c+ 1 consumers in the next period t−1. If the consumer chooses firm j ∈ J(ω) with xj ≥ 1, then this firm j’s capacity is reduced by 1, while (possibly) unserved consumers as well as the one newly arriving consumer, i.e. c−1 + 1 = c, will be in the market in the next period. Note that time cannot decrease past its deadline boundary t= 0.
Consumer Utility and Demand
In the world of forward-looking consumers waiting gives the consumer utility ofδcWc(ω), whereWc(ω) is the consumer’s expected valuation of waiting which is introduced in the
15Note that the order of the consumer lottery and firms’ price setting is irrelevant.
4. Forward-looking Consumers 99 next paragraph. Let δc ∈ [0, δ] be the discount rate for all consumers, which could be smaller than the firms’ discount rate δ.16 I will interpret δc as consumers’ degree of patience, i.e. the degree to which they are forward-looking (see Section 5.4 for a discussion). Note that for δc = 0 the model becomes identical to the baseline model of myopic consumers. The lucky consumer chooses from J(ω) ∪ {0}, where choices j ∈J(ω) exhibit the products of the active firms, while j = 0 represents the choice of waiting. Utility of the lucky consumer when choosing an active firm’s product iis again Ui =v−pi+µ i.Waiting gives the consumer utility of17
U0 =µ δcWc(ω) +µ 0. (17)
All choice options are again associated with independent Extreme Value Type-I dis-tributed taste shocksj. Then the multinomial logit demand system yields the probabil-ities that the consumer chooses a product i∈J(ω) or waits (0),
Di(p(ω),ω) =
expv−pµi(ω) exp [δcWc(ω)] + P
J(ω)
expv−pµj(ω)
, (18)
D0(p(ω),ω) = exp [δcWc(ω)]
exp [δcWc(ω)] + P
J(ω)
expv−pµj(ω)
, (19)
while inactive firms obtain zero demand, i.e. Dj(p(ω),ω) = 0 if xj = 0.
Consumers’ Problem
In each period the lucky consumer chooses between buying from a firm or waiting for (at least) one more period in the market. Thereby she considers the following trade-off.
On the one hand, she could purchase at current prices from one of the active firms with certainty. On the other hand, she could wait, expecting that firms become less confident about selling their goods and hence will further decrease prices. However, waiting involves the risk of being rationed in the next period(s), amplified by the arrival of new consumers.
When the consumer chooses where to buy or whether to wait, she anticipates all possible future prices of all sub-games, which determine her expected valuation of waiting.
16Note thatδ can be normalized to one.
17Note that I multiply the observable part of the outside option utility withµ, too, such that the ratio to the shock 0 remains constant, as in the baseline model, where the observable part of the outside option is zero. This is merely a normalization and hence does not affect the results qualitatively.
Given the current state ω and considering future rationing risks, let the transition probability matrix of the lucky consumer be Φ(ω), a matrix of three dimensions,T×X× C, with entries in [0,1].18 This matrix shall state for each entry, i.e. for each possible state ω0 = (t0,x0, c0) ∈ {{0, .., T} × {0, ..,X} × {1, .., C}}, the probability that in that stateω0 the currently lucky consumer will be for the first time again the lucky consumer, given that she decides to wait in the current state ω.19 In Appendix A.2 I provide the details of this transition matrix Φ(ω).20 In order to solve for the equilibrium it is important to note that the consumer transition matrix Φ(ω) only depends on the current state ω. All future sub-games’ prices, demand decisions and rationing probabilities are fully anticipated in expectation by consumers as they only depend on the current state.
Next, let Ω(ω) be a matrix of three dimensions, T ×X ×C, where the entries are equal to the lucky consumer’s value function in all possible states, as expected after choosing to wait in ω. All entries of Ω(ω) at the deadline, i.e. for t0 = 0, shall be equal to zero as all goods will have perished at the deadline. Each other entry (t0,x0, c0) ∈ {{0, .., T} × {0, ..,X} × {1, .., C}}is equal to the consumer’s value function in that state, Ω(ω)[t0,x0,c0] = Vc(t0,x0, c0). (20) The value function of the lucky consumer in a state ω is equal to her consumer surplus as derived for the logit model e.g. in the textbook by Train (2009) and reads21
Vc(ω) = ln
The expected valuation of a consumer upon selecting to wait shall be then given by Wc(ω) = X
18C=c+tis the maximum number of consumers that can accumulate until the end.
19For example, consider a state (t= 2, x1 = 1, c= 1). If the only (and hence lucky) consumer waits, her transition probability matrix consists of zeros for all possible states, with one exception: in state (t= 1, x1= 1, c= 2) she will be for the first time again the lucky consumer with probability 1/2, which is the probability that she and not the newly arrived consumer will be drawn to be lucky. If the new consumer is drawn, which happens with probability 1/2, the older consumer does not get to be the lucky consumer again and reaches the deadline, obtaining the outside option of no purchase.
20Note that unlucky consumers do not become active until they are drawn by the lottery, hence their transition function is irrelevant to the solution of the problem until then.
21Note that this term is the log of the denominator of a choice probability and therefore is often called
’the log-sum term’, although this equivalence “has no economic meaning” (Train, 2009).
4. Forward-looking Consumers 101 i.e. the sum of all elements obtained from element-wise multiplying the transition prob-ability matrix with the matrix of value functions. Note that in the last selling period t = 1 there are no continuation values, hence waiting grants the outside option valua-tion Wc(ω) = 0 whenever t= 1, completing the recursion. Intuitively, Wc(ω) gives the expected equilibrium-path- and rationing-risk-probability-weighted valuation of waiting.
Firms’ Problem
In each state ω = (t,x, c), all active firms i simultaneously post their prices pi(ω) to maximize their value function Vi(p(ω),ω). The value function is basically identical to the one in the baseline model as in equation (6). The only difference is that consumers here have the choice of waiting rather than exiting the market. Since in each period only one consumer is selected to choose, the firms’ problem remains equal: firms compete against each other for one trade per period, thereby considering all possible continuation values. However, consumers’ demand as in equations (18) and (19) now also depends on their valuation of waiting (equation 22), which firms’ need to take into account.