## Evidence on Resale of Automobiles

Application: Used Automobiles

Proposition 2 says that consumers at the extremes of the type distri- bution turn over their goods faster than the ones in the middle. This property will play an important role in interpreting the observed holding patterns for cars. Before turning to the numerical results, I shall discuss the evidence from the U.S. market for used automobiles.

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A. Evidence on Resale of Automobiles

The evidence on resale of automobiles in the United States comes from the 1995 Nationwide Personal Transportation Survey (NPTS), a data set with more than 75,000 observations. The data presented pertain to six major automobile manufacturers in the model year range 1982–96, which covers more than 75 percent of all cars in the sample. Figure 1 shows resale rates as a function of model year, which is taken as a proxy for the vehicle’s age. The vertical axis of each plot shows the observed fraction of vehicles of a particular age purchased in used condition in 1995. The horizontal axis shows the vehicle’s age, with model year 1996 at age 0 through model year 1982 at age 14.

There are several factors that can make the resale rate vary with age. Since the resale rate is a fraction, the variations in resale rates across vintages are not due to high or low sales of new vehicles in the past. Resale rates may also vary because some vintages of cars are very popular. The “good vintage” effects are controlled for by pooling together all car models by the same manufacturer. Because the introduction of new models is usually staggered, only a fraction of observations for a partic- ular model year can conceivably come from the good vintage. Besides, the effect on quantity traded is ambiguous: a good vintage is not only something that consumers want to buy (increased demand) but also something that other consumers want to keep (reduced supply).

All makes exhibit similar regularities in resale patterns: the resale rate is very low for young cars and it peaks when vehicles are three to four years old; then the resale rate stays relatively low for several years, and then goes up again when the vehicle is about 10 years old.

The patterns in figure 1 are confirmed by tests. The tests are based on the difference in resale rates for two consecutive years. The shaded bars in figure 1 show the first year, when the resale rate drops signifi- cantly, and the first subsequent year, when the resale rate rises signifi- cantly. The hypothesis that the resale rate for cars that are more than two years old is monotonic in the vehicle’s age can be rejected at 10 percent significance or better for all makes but Nissan. The p-values for each test are reported on top of the shaded bars in figure 1.

The observations record only purchases, but not the vehicle owner- ship histories. Ideally, one would like to exclude fleet sales by rental car companies because they come from agents who hold multiple vehicles at a time and thus face a different decision problem. However, since fleet sales usually involve one-year-old cars, excluding them from the sample would likely lower the resale rate at one year and make the nonmonotonic pattern in figure 1 even more pronounced.

The evidence also shows how frequency of trade varies by vehicle make. On average, Hondas and Toyotas are traded significantly less than

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other makes considered.8 Subsection B explains why resale rates are relatively low in the middle of a car’s life and why some automobiles are traded more frequently than others.

B. Results

The computation proceeds along the steps described in Section III. Each iteration starts with computing the optimal decision rules using some fixed price vector as an input. Next, using the decision rules just ob- tained, one can solve the supply equals demand equations (13) to find the market-clearing price. This market-clearing price is used as an input for the next iteration, and so forth. Essentially, each iteration computes the value of an operator that maps price vectors into price vectors. By construction, the fixed point of this operator is the equilibrium price.

1. Choice of Parameter Values

The physical lifetime of a car is taken to be years. DeteriorationT p 15 is assumed to be exponential, with a constant rate d and as ax p 10 normalization:

tx p x (1 � d) , t p 1, … , T � 1.t 0

The transaction cost of selling a used vehicle is measured with the difference between its market price and the trade-in value. In the model, the ratio of this difference to market price is equal to l. It turns out that the ratio of transaction cost to price rises with the vehicle’s age and roughly doubles over a vehicle’s lifetime.9 Accordingly, I set

(t�1)/(T�2)l p l 2 , t p 2, … , T � 1. (14)t 1

In the model, goods of all ages are traded with a probability of at least a. Typically, resale rates are minimal (i.e., equal to a) for young vintages. We can therefore choose the value of a to match the resale rate for one- to two-year-old vehicles in the data.10 This implies a p 0.1.

The values of p0, l1, and d are chosen by fitting the equilibriumh ,max price predicted by the model11 to the normalized price series for used automobiles reported in Porter and Sattler (1999, table 4). The resulting

8 Hendel and Lizzeri (1999) find a similar result: they use the data from the 1991 Consumer Expenditure Survey to conclude that Fords are traded more often than Hondas.

9 This calculation was performed using trade-in and market values reported by Ed- munds.com (http://www.edmunds.com/used/).

10 The average resale rate for used one-year-old vehicles is 0.058 and for used two-year- old vehicles it is 0.144. Source: 1995 NPTS.

11 If one assumes that is uniform.n(h)

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Fig. 1.—Resale rates as a function of age for major car makes. Shaded bars show ages when resale rate drops or rises significantly; numbers on top of the shaded bars are p- values.

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Fig. 1.—Continued

benchmark parameter values are andp p 3.95, h p 1.9, l p 0.075,0 max 1 The unit of measurement is the stream of service from thed p 0.085.

brand new good, To be more specific, the price of a new carx p 1.0 approximately equals the present value of five years of its service to the median consumer. The real interest rate is set to 0.04,12 which implies b p 0.96.

12 Since utility is linear in the numeraire commodity, utility and wealth are equivalent,

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2. Numerical Results

The simulations for prices and quantities traded are made for different plausible assumptions about the distribution of types Whenn(h). n(h) is uniform, variations in resale rates cannot be due to different densities of consumers who prefer particular vintages. This case clearly illustrates how the variation in holding times across consumer types translates into the variations in resale rates. Take, for example, consumers at the top of the type distribution who buy new cars. New-good buyers sell their cars at age 4, 5, 6 or 7, with lower types holding cars for a longer time (lower left panel of fig. 2a). Accordingly, the resale rate falls for five- to seven-year-old cars (upper left panel of fig. 2a). Next, take the con- sumers at the bottom of the type distribution who hold the good until it falls apart. This group buys goods in the 11–14-year age interval. Accordingly, the resale rate rises with age for these vintages. Roughly speaking, the number of consumers who buy a particular vintage is in the numerator of the resale rate, and their holding time is in the de- nominator.13 The number of consumers with the same buying point differs from vintage to vintage, which also affects the resale rate. For example, there are very few consumers who buy 14-year-old cars, which pushes their resale rate slightly down.

3. Robustness

Distribution of consumer types.—Consumer type in the model most likely corresponds to some increasing function of income. Because type is determined outside the model, we must explore how the resale pattern is affected by alternative assumptions about the type distribution. Let us consider two cases: when the distribution of types is extremely skewed to the left and when it is extremely concentrated in the middle. Even these extreme14 changes in the distribution of types preserve the basic two-hump resale pattern, as shown in figure 2.

At first glance, it may seem that if there are many low-type consumers, the resale rates for old used goods should also be high. In fact, this is not the case. The intuition for this result becomes transparent when

so b p 1/(1 � r). 13 Observe that the number of consumers at the buying point is approximatelyf(h, S )h

equal to

n(h) lim f(h, S ) p .h

tar0 h 14 The distribution that is skewed to the left is a truncated normal with mean h p 0

(the bottom of the market) and standard deviation This implies that ish /4. n(h )max max almost times less than The distribution that is concentrated in the middle8e ≈ 3,000 n(0). is a truncated normal with mean (symmetric) and standard deviation , soh /2 h /8max max that is again four standard deviations from the mean.hmax