## Optimal Replacement Problem

Optimal Replacement Problem

In a stationary environment, prices of used goods are constant over time and depend only on the age of the good. Consumer h arrives at the beginning of the current period with a good of age t � {0, … , T � 1}. The state variable for the consumer is the age of her current good. In the current period, this good yields utility flow xth. At the end of the period, the consumer sees the realization of the transaction cost and then decides whether to sell the good and purchase a replacement or keep the good for another period. Since resale is always preferredl ! 1,t to scrappage.

In equilibrium, all consumers will follow a stationary decision rule: they will sell their goods either when they draw a zero transaction cost or when the good reaches a threshold age, whichever happens first. After resale, each consumer type will update to her optimal vintage, and the holding cycle will start again. Let be the discountedV(h; t) present value of consumer h with the good of age t at the beginning of the current period2 and let be the discount factor. The Bellmanb ! 1

2 Technically speaking, the value function V and the optimal decision rules will depend on the price vector as well, but this dependence is suppressed in the notation for convenience.

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equation describing this optimal replacement problem reads as

V(h; t) p x h � ba max max [V(h; S) � p ] � p , V(h; t � 1){ }t S t�1 S

�b(1 � a) max max [V(h; S) � p ] � p (1 � l ),{ S t�1 t�1 S

V(h; t � 1) . (3)}

The first term is the value of service provided by the good during the current period, the second term is the expected value of the resale opportunity with a zero transaction cost, and the third term is the ex- pected value of the resale opportunity with a positive transaction cost. Let

S p arg max [V(h; S) � p ] (4)h S S

be the optimal vintage for consumer h. Since the problem is stationary, consumer h will purchase the good of age Sh every time she replaces her durable. I shall call Sh the buying point for consumer h. Since for every t

max [V(h; S) � p ] ≥ V(h; t � 1) � p ,S t�1 S

any consumer who draws a zero transaction cost resells her good and returns to her buying point. A consumer who draws a positive transaction cost replaces her good when the gain from resale exceeds the transaction cost:

max [V(h; S) � p ] � [V(h; t � 1) � p ] 1 l p .S t�1 t�1 t�1 S

Let

t p min {t : V(h; S ) � p � [V(h; S � t) � p ] 1 l p }. (5)h h S h S �t S �t S �th h h h

Scrapping is free, so any good will be replaced before age T. Expression (5) says that consumer h will sell her good at age even if sheS � th h draws a positive transaction cost. I shall therefore use the term selling point for and the term holding time for th. The intervalS � t [S , S �h h h h

will be called the holding interval.t � 1]h If consumer h follows a decision rule with the buying point S and the

holding time t, we can compute her lifetime utility by substituting the

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decision rule into the Bellman equation (3). This yields the following expressions for the value function on the holding interval:

V(h; S � i � 1) p x h � ab[V(h; S) � p � p ]S�i�1 S S�i

� (1 � a)bV(h; S � i), i p 1, … , t � 1;

V(h; S � t � 1) p x h � ab[V(h; S) � p � p ]S�t�1 S S�t

� (1 � a)b[V(h; S) � p � p � l p ].S S�t S�t S�t

Using the notation

g p (1 � a)b, ab p b � g

and making recursive substitution, we can find the expression for the optimal value function at the buying point:

V(h; S) � p p max U(h, S, t), (6)S t

where

U(h, S, t) p t ti�1 i�1 th� x g � p � (b � g)� p g � g p (1 � l )ip1 ip1S�i�1 S S�i S�t S�t

. (7) t(1 � b)[(1 � g )/(1 � g)]

In this expression, equals the consumer’s expected lifetimeU(h, S, t) utility measured at the buying point. The optimal decision rule must yield the maximum expected utility:

(S , t ) p arg max U(h, S, t) (8)h h S,t

subject to

0 ≤ S ≤ T � 1, 1 ≤ t ≤ T � S.

If initially the consumer has a good of age thent � [S , S � t � 1],0 h h h by construction she will find it optimal to follow the decision rule (Sh, th). However, since we have not determined the optimal value function outside the holding interval, we do not know how the consumer will behave if her initial state is not in 3 Nevertheless, (8)[S , S � t � 1].h h h will be sufficient for all consumers to abide by the equilibrium behavior,

3 The optimal policy may not be an (S, s) rule because, depending on the values of xt and lt for different t, the value function may not be concave. If, e.g., a consumer is given a good whose age is greater than her selling point, she may not want to return to the buying point immediately, but may instead choose to keep holding the good.

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since in equilibrium the ages of goods owned by consumers will always belong to their respective holding intervals.

I conclude the discussion of decision rules by proving that they are monotonic in h. It turns out that no matter what the prices are, higher types prefer to buy younger goods and resell them earlier. Formally, the buying point Sh and the selling point are step functions that mapS � th h [0, into The following proposition says that these steph ] {0, … , T }.max functions are nonincreasing functions of h.

Proposition 1. Monotonicity of decision rules.—Let be the so-(S , t )h h lution to the optimal replacement problem. Then Sh and areS � th h nonincreasing functions of h for almost all prices.4

Proof. See the Appendix. The essential assumption on uh(x, c) that is required for proposition

1 is that it is linear in c.5 The result in the proposition will be important in characterizing the equilibrium.

I shall now turn to describing the distribution of durable goods across consumer types that gives rise to the steady-state equilibrium.

B. Steady-State Holdings Distribution

In the steady state, the distribution of durable goods across consumer types must stay the same every period and replicate itself indefinitely. This holdings distribution will generate constant purchases and constant resales for every vintage, which will make up steady-state supply and demand.

Let be the number of consumers of type whof(h, t) h � [0, h ]max hold the goods of age at the beginning of the currentt p 0, … , T � 1 period. Since trade happens at the end of the period, no one holds goods of age T at the beginning of the period. In the steady state,

must be the same every period. The optimal decision rulesf(h, t) (S ,h impose a certain law of motion on the holdings distributiont ) f(h,h In particular, consumers do not own goods whose ages are outsidet).

of their holding interval:

f(h, t) p 0, t ! S or t 1 S � t � 1. (9)h h h

4 Except for price vectors for which all three functions Sh, and th have a discon-S � t ,h h tinuity at the same point However, the subset of such price vectors hasĥ � [h , h ].min max measure zero.

5 Linearity in h and x is not essential because the utility function can be transformed by an appropriate choice of units of h and x.

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Inside the holding interval, the transitions of consumer h across states can be summarized by the following matrix:t # th h

a 1 � a 0 … 0⎛ ⎞ a 0 1 � a … 0

P p … … 0 … … .h a 0 … … 1 � a⎜ ⎟ 1 0 … 0 0⎝ ⎠

The first row of Ph corresponds to state Sh and the last row corresponds to state In every state, the consumer draws a zero transactionS � t � 1.h h cost with probability a and returns to her buying point Sh next period. With probability the consumer moves to the state in which her1 � a good is one period older. When the consumer is one period away from her selling point, she will return to the buying point for sure. For each consumer type, the steady-state holdings distribution is the stationary distribution of her transition matrix Ph:

0 t ! S or t 1 S � t � 1h h h f(h, t) p (10)t�Sha(1 � a){ n(h) S ≤ t ≤ S � t � 1.h h hth1 � (1 � a)

We can now compute steady-state flows of goods as functions of con- sumer type6 and age of the good.

Type h consumers demand goods only at the buying point Sh. There- fore, their contribution to demand for goods is simply

0 t ( Shq (h, t) p (11)d {f(h, S ) t p S .h h Similarly, type h consumers supply used goods at every point of their holding interval, because whenever the transaction cost is zero, they sell their current good. Accordingly, the supply function for type h consum- ers whose decision rule is can be written as(S , t )h h

0 t ! S � 1 or t 1 S � th h h q (h, t) p af(h, t � 1) S � 1 ≤ t ≤ S � t � 1 (12)s h h h{f(h, t � 1) t p S � t .h h

Using the expressions (9)–(12) and integrating over consumer types with the same decision rule, we can determine the steady-state supply and demand for every vintage. The steady-state supply and demand depend on the price vector through the decision rulesTp p (p ) (S ,t tp0 h

6 More precisely, consumer type and which are themselves functions of h.(S , t ),h h

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All the elements are now in place to define the steady-statet ).h equilibrium.

C. Equilibrium

Definition. Steady-state equilibrium consists of the price vector p p the steady-state holdings distribution the mar-(p , p , … , p , 0), f(h, t),0 1 T�1

ginal consumer and the optimal decision ruleh , (S , t ), h � [h ,min h h min such that the following conditions hold:h ]max

1. The steady-state holdings distribution f(h, t) is given by (10) for every participating consumer hmax] and f(h, for nonpar-h � [h , t) p 0min ticipating consumers h � [0, h ).min

2. Prices for all useful goods are positive:

p 1 0, t p 0, … , T � 1,t

p p 0,T

and supply equals demand for any used good that is not scrapped: h hmax max

Q (t) p q (h, t)dh p q (h, t)dh p Q (t),s � s � d d h hmin min

t p 1, … , T � 1. (13)

3. Consumers choose the decision rule that maximizes their lifetime utility:

(S , t ) p arg max U(h, S, t).h h (S,t)

4. The marginal consumer7 is indifferent between buying a used good and taking a useless good for free:

pT�1 h p .min xT�1

For the applications, it makes sense to restrict attention to steady- state equilibria with positive prices. Positive equilibrium prices of all useful vintages can be guaranteed if new goods price p0 is large enough relative to the values of T and xt.

Although the equilibrium can be computed only numerically, it is possible to characterize it in an important way.

Proposition 2. In any steady-state equilibrium with positive prices,

7 Nonnegativity of utility is equivalent to The proof is inU(h, S , t ) ≥ 0 h ≥ p /x .h h T�1 T�1 the Appendix.