Used Durables in a Stationary Equilibrium

Turnover of Used Durables in a Stationary Equilibrium: Are Older Goods Traded More? Author(s): Dmitriy Stolyarov Source: Journal of Political Economy, Vol. 110, No. 6 (December 2002), pp. 1390-1413 Published by: The University of Chicago Press Stable URL: Accessed: 06-06-2018 14:02 UTC

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[Journal of Political Economy, 2002, vol. 110, no. 6] � 2002 by The University of Chicago. All rights reserved. 0022-3808/2002/11006-0009$10.00

Turnover of Used Durables in a Stationary Equilibrium: Are Older Goods Traded More?

Dmitriy Stolyarov University of Michigan

This paper develops a dynamic model with transaction costs to de- termine the equilibrium resale pattern in a market for a durable good. The key result is that the probability of resale is nonmonotonic in the age of the good. Trade volume is relatively low in the very beginning and in the middle of a good’s life. This result helps explain observed variations of resale rates across vintages for the U.S. market of used cars.

I. Introduction

The distinguishing feature of a durable good is its potential for resale. Resale is very active in many markets. For example, nearly 66 percent of all cars bought in the United States in 1995 were used (U.S. De- partment of Transportion 1997), more than half of all Boeing 707 air- craft changed owners during their lifetime (Goolsbee 1998), and 68 percent of all machine tools sold in the United States in 1960 were used (Waterston 1964).

When goods are long-lived and are actively traded, the demand for new goods is shaped by consumers’ decisions when to trade. Conse- quently, patterns of resale are very important in understanding how the market operates. This paper presents a model that determines the equi- librium trade pattern in a market for a durable good. This pattern can

I am grateful to John Laitner, Stephen Salant, and Ennio Stacchetti for many helpful discussions and thank Robert Barsky, Matthew Shapiro, Lones Smith, and an anonymous referee for insightful comments.

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turnover of used durables 1391

be summarized by the relationship between the good’s age and its prob- ability of resale.

The evidence on probability of resale for autos by age shows an in- teresting regularity. It turns out that the resale pattern has a “double- hump” shape. Young vehicles have low resale rates. Probability of resale subsequently rises, peaks at around three to five years, falls, and then rises again for vehicles about 10 years old.

Previous literature did not focus on explaining resale patterns for durables. Most existing models of durable goods belong to three main categories. It the first category, there are equilibrium models without frictions, where 100 percent of all goods are traded at every moment (Rust 1985; Kursten 1991; Jovanovic 1998). In the second category, there are models with infrequent replacement but no trade (Eberly 1994; Caplin and Leahy 1999; Adda and Cooper 2000). The third category includes the models in which goods live for just two periods, so that used goods of all ages are lumped together (Bond 1983; Hendel and Lizzeri 1999; Porter and Sattler 1999).1 Clearly, one needs a different model to account for the observed resale pattern.

This paper develops a dynamic model with transaction costs in which individual optimal replacement cycles are embedded into an equilib- rium framework. Transaction costs are central to the model since they make replacement infrequent. Trade occurs because goods deteriorate with age and because consumer types differ in their marginal utility of service provided by the good. Trade allows both buyer and seller to adjust to their optimal vintage. Prices and quantities are determined simultaneously in a stationary equilibrium.

The model captures the observed resale pattern for autos. The key result behind this is that the optimal holding time is hump-shaped in consumer type: in equilibrium, high and low types hold the good for a short time, whereas middle types hold it for the longest time. Goods are not resold right away because of transaction costs, so young cars have low resale rates. When cars have depreciated sufficiently, high types sell them, generating the first peak of resale. Sellers of intermediate- age used cars are middle types who have the longest holding times. Hence they supply fewer goods per period, generating a trough in the resale rate at intermediate ages. Finally, old used cars are just a few

1 Beyond these three categories, related work includes the paper by Holmes and Schmitz (1990), who consider a stationary equilibrium in the market for durable capital goods (businesses) with transfer costs. In their model the size of the secondary market and the range and distribution of qualities traded are determined endogenously. Konishi and Sandfort (2002) prove the existence of a stationary equilibrium in a generalized version of Rust’s (1985) model with transaction costs. House and Leahy (2000) consider a model with three-period-lived goods to show how adjustment costs arise endogenously because of adverse selection.

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1392 journal of political economy

years from being scrapped, so the low types who buy them again turn over quickly. Therefore, the resale rates for old used cars are high.

Beyond this, the model produces comparative statics that are consis- tent with stylized facts about the auto market documented in the lit- erature (Hendel and Lizzeri 1999; Porter and Sattler 1999). In partic- ular, cars that deteriorate faster have more convex used price profiles and are resold more frequently and earlier in life.

II. The Model

Goods.—Durable goods provide a useful service for the first T periods of their life. The good that has age at the beginningt p 0, … , T � 1 of the period provides the flow of service equal to xt during this period. Older goods provide less service:

0 ! x ! x , t p 0, … , T � 2.t�1 t

Every good becomes useless at age T: x p 0, t ≥ T.t Consumers are infinitely lived and can use only one durable good at

a time. They differ in their marginal utility of x, denoted h � [0, distributed according to an atomless density The current-h ], n(h).max

period utility to a consumer of type h from having a good x and c units of the numeraire is

u (x, c) p xh � c. (1)h

Consumers have the same rational expectations about future prices of all vintages and maximize their lifetime utility of owning an infinite sequence of durable goods. For each good in this sequence, they choose what vintage to buy and how long to hold it.

New goods are homogeneous, and their price, p0, is exogenous. Any quantity can be supplied at this price.

Trade happens at the end of every period. The buyer pays the price pt for the good of age and the seller receives where is˜ ˜t ≥ 0, p � L , Lt t t the transaction cost. Transaction cost is a random variable that assumes either a zero value with probability or a nonzero valuea 1 0

L(t, p ) p l p , l ! 1, (2)t t t t

with probability Since many infinitesimal sellers independently1 � a. draw from the same distribution, there is no aggregate uncertaintyL̃t in the model. Any good can be scrapped at zero value. Since useless goods are free and yield zero utility, consumers whose present value of participating in the market is negative do not participate.

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turnover of used durables 1393

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