used cars as a depreciating asset

USED CARS AS A DEPRECIATING ASSET

SUSAN ROSE ACKERMAN' University of Pennsylvania

Previous econometric studies have shown that the prices of used cars decline with age at a constant exponential rate [ 21 [ 31 [41[ 11 1. This paper presents new data confirming the pattern observed by others and makes a first attempt to explore some of the underlying causes of the rapid fall in prices. A model is developed which decomposes the services generated by an automobile into a function containing a number of discrete attributes. It is then possible-with the aid of Consumers Union data-to assess the importance of several basic factors at work in the used car market.

I . THE BASIC MODEL

Following a fundamental tenet of capital theory, we shall assume that the price of an automobile of a given age, K , can be expressed as the dis- counted present value of its remaining services.'

P ( K ) = $ S ( ~ ) e + ( x - ~ ) d ~ (1)

where :

D

K

K = present age of car D = age of scrappage x = age r = discount rate (assumed constant)

S(x)= value of services provided by a car of age x P(K) = price of a car of age K

In order to obtain an expression amenable to estimation, we differen- tiate (1) with respect t o K and obtain:

(2) P ' ( K ) = - S ( K ) + rP(K)

Equation (2) can be used t o estimate the service function. By trans- posing terms, (2) states that the value of the services provided by a car of age K is equal to the sum of the rate of change in price with age (the loss in value) plus r times the automobile's price (the opportunity cost of holding the car). This model is a completely accurate description of

*This paper is adapted from the author's Ph.D. dissertation [ 11. Two anonymous referees made

1 . This formulation of the basic model assumes that the services provided by an old automobile many helpful suggestions.

are independent of model year, an assumption that will be modified in the empirical work.

463

464 WESTERN ECONOMIC JOURNAL

reality only if purchasers expect that no capital gains or losses will affect the value of their assct. Since actual data on market prices paid for used cars include both depreciation and ex post capital gains and losses, however, our estimates of the service function will be biased by this omission. An attempt is made to remedy this defect at the end of the paper by developing a model that explicitly considers capital gains and losses as well as thc value of automotive services.

The discrete form of ( l ) , which will be used in Section C, is:

P ( K ) = 5 s ( ~ ) / ( I +r)X-K x = K

(3)

Let: A P ( K ) = P(K + 1 ) - P(K) Substituting for P ( K ) gives:

(4) A P ( K ) = - S ( K ) + [ r / ( l + r ) lP(K + 1 )

or.

II. A SIMPLE SERVICE FUNCTION

We shall begin our attempt t o estimate the service function by assuming a very simple form in which services fall in value at a constant exponential rate with age. Thus

(5) S(x) = hP(0)edX where :

h = constant P ( 0 ) = price of car when new

a = constant rate of exponential decay of car services

Substituting (5) in the expression for used car prices, ( l ) , we obtain

( 6 )

Thus the price of age K cars relative t o the price of new cars is:

D P ( K ) = $ hP(0)e-ax e-'(x- K)dX

K

ACKERMAN: USED CARS 465

If D4 00, this expression can be substantially simplified since

lim erK- (a +r)D= erK[lim e-la+rlD] = 0

Thus we have

( 8) P(ldIPf0) = Be-aK

where B = h f fa + r) .

For estimation purposes (8) can be expressed in terms of vintage, v , and calendar date, t , where P(K) = P f v , t ) and K = t - v. By converting (8) to logs and adding an error term, linear regression techniques may be used to estimate B and a.' In the empirical work presented here, separate estimates of B and a were made for each of six different automobile models, m, produced between the years 1954 and 1963. The price of one- year-old cars, P f v , v + I), was used in the denominator instead of P f O l = P(v ,v ) since data on new car prices are not gathered in the same manner as information on older automobiles and a bias may be introduced by combining the two price ~ e r i e s . ~ The six estimating equations are thus of the form:

(9)

where:

In [Pm(v, t)] - In [P"(v, v + I ) ] = In B, - a,(t - v - 1 ) $. u,

rn = Chevrolet -6 cylinders Ford -6 cylinders, Plymouth-6 cylinders, Chevrolet-8 cylinders, Ford-8 cylinders, Ply- mouth-8 cylinders

2. Since this depreciation function is strictly correct only if the age of scrappage approaches infinity, some error is present because an automobile's life is finite (about 95% of any vintage has been scrapped after ten years. See Friedman (6, p. 191). However, the fact that the simple function in (9) fits the data quite well (see Table 1) indicates that the date of scrappage is relatively unim- portant in explaining the fall in prices with age.

3. Price data are for fourdoor sedans for the years 1956 to 1965 from the July price books of the National Automobile Dealers Association (NADA) for the Eastern Region. The author ex- amined several other series on used car prices and concluded that the NADA information was the most reliable. Automobiles from one to seven years old are included in this study since NADA price data are not available for older automobiles. 1954 was chosen as the starting vintage to avoid the extreme shortage conditions and price controls prevalent from the beginning of world War I1 through the end of the Korean War.


v = 1954, ..., 1963 t = v + 2 , . . . , v + 7 if v < 1960 t = v + 2 , ..., 1966ifv231960

P"(v, t ) = used car retail price in year t for cars of make rn and vintage v from the July price books of the National Automobile Dealers Association for the Eastern Region. The prices are those of the cheapest standard four-door sedan with six cylinders and the most expensive eight. All prices are divided by the CPI.

The calculated values of B,, a,, and E2 and the Standard Error of Estimate (SEE,) are provided in Table 1 for each model studied. The t values are given in parentheses below the coefficients of In B , and a,.

Table 2 presents the results for six-cylinder automobiles in a different form, in which the price of a car of any given age is expressed as a percentage of its price when one year old. The left side of the table records the average prices observed in the sample, while the right-hand side expresses the prices estimated through equation (9).

The tables demonstrate that a very simple geometric service function reflects the decline in price with age quite well. The estimates of a , are highly significant for all models. R2's are uniformly high, and the esti-

Table 1 -Estimates of Simple Exponential Decay Model for Six Automotive Makes Over the Years 1956-1966

Make m

Chev - 6

Ford - 6

Ply-6

Chev - 8

Ford - 8

Ply - 8

Degrees of Freedom

48

48

48

47

48

47

In B , Bm

.05 1 1.05

.006 1.01

.016 1.02

.129 1.14

.080 1.08

.086 1.09

(1.38)

(.I51

(.33)

(2.87)

(1.95)

(2.53)

'rn SEEM*

+.279 .94 .123

+.294 .93 .134

+.342 .93 .155

+.288 .94 .I20

+.297 .93 .132

+.332 .97 .I04

(2.79)

(2.67)

(2.53)

(2.61)

(2.72)

(3.68)

*Standard error of estimate of In (Pm(v,t)) - In (Pm(v, v f I ) ) . This term is approximately equal to average percentage error in ~ ~ ( v , t ) / ~ ~ ( v , v + I ) .


Table 2

Means of Actual Observations Means of Actual Observations Make Age Age --

2 3 4 5 6 1 2 3 4 5 6 1

Chevrolet - 6 .77 .61 .48 .37 .25 .20 .79 .60 .46 .34 .26 .20 Ford - 6 .75 .57 .42 .30 .23 .18 .75 .56 .42 .3 1 .23 .17 Plymouth- 6 .72 .52 .38 .26 . I 6 .15 .72 .52 .37 .26 .18 .13

mates of relative prices are very close to actual average relative prices. Moreover, the analysis isolates important intermake differences in the rate of price decline. Plymouths, for example, lose value at a faster rate than Che~role t s .~ However, while the simple model estimated here is complete in the sense that vintage prices include all the attributes of a car that users consider valuable, it does not explain which characteristics of the different makes of car account for the fall in value. It is to this more difficult problem that we now turn.

Ill. A MORE COMPLEX SERVICE FUNCTION

To explain the factors which account for a fall in the value of automotive services, several important changes were made in the simple model discussed previously.’ First, we substituted a hedonic price index (Griliches [ 7]), A(v, m), for the actual new car prices used in the simple model.6 This permits us to treat each car as if it were a bundle of services and purges new car retail prices of random components. As in our simple model, however, we assumed that the price index declined with age at a constant exponential rate, since no independent data exist to support a more so- phisticated treatment.

Second, we included a term R f v , t , m) which was based upon auto repair data provided by Consumers Union. The Consumers Union yearly survey of members asks car owners to indicate which of a dozen possible diffi-

4. Three previous studies have also obtained good fits by assuming that the relative price of old cars falls at a constant exponential rate. Two of the studies, W. B. Bennett 121 and M. Boiteux [3] , were performed without formal discussion of the model underlying their work. While Frank Wykoff (11) developed a model similar to that used here, he did not attempt to discover which automative attributes caused the fall in the value of services.

5 . Although their purposes and models vary, the studies most closely related to the present work are those by Cagan [4], Hall [S] and Ramm [ 9 ] . Both Cagan and Hall try to discover the importance of quality change in new cars and trucks by examining data about these vehicles when they are traded in the used car market. Ramm [ 9 ] attempts to estimate the value of automotive services as a function of age and measurable automotive characteristics.

6. In the actual estimating equations A(v,m) is substituted not for new car prices, but for the prices of one-yearald cars.


culties “have given you considerable trouble or required major repairs since you have had the car.”’ The list of repairs includes both major repairs, like transmission overhauls, and more minor difficulties, like muffler replace- ments.8 R(v, t , rn) measures the average repair records calculated by CU for each make, vintage and age by multiplying the frequency of each kind of breakdown by a number between ten and one hundred representing the repair’s relative costliness in dollars.’

This term must be interpreted with great care. It is not a measure of relative repair costs over any single annual period, but instead represents the relative repair costs accruing since the car was new. In the empirical estimates which follow, it is one of our main objectives to determine the extent to which R(v, t ,m) may be used as a measure of consumer percep tions of the average quality of each make, vintage, and age of automobile. If R(v, t ,m) is a measure of average automotive quality, we would expect it to be negatively correlated with the value of services expected over the period t to t + I. This is so because the repair index represents the average historical record for a given kind of car and rational car purchasers should expect ceteris paribus that a model whose past record has been poor will continue to generate above average repair costs. Of course, for any indi- vidual car, relatively high spending for repairs in the past will suggest a relatively high level of services in the future. Nevertheless, so far as all cars of a given model are concerned, the fact that they have generated a high level of repair costs should only depress the value of services expected in the future.

Adding a trend term to S(v,t,rn) to account for any overall rise or fall in the value of services with calendar date and assuming an additive form for the service function, we obtain:

7. This question was asked of owners who had bought cars both new and used. In 1966, however, when the responses from both groups of owners were separated, the number who had bought their automobiles used was a very small proportion of the total even for the older vintages.

8. The car owners queried by Consumers Union are a very special group that is hardly repre- sentative of all automobile owners. If subscribers to Consumers Reports are more careful with their automobiles than the average owner, using their repair records will bias repair costs downwards.

9. The weighted sum of these repairs for any particular make and vintage is used as the repair term for the automobile. This average ranges from 14.4 for one-year-old Darts in 1964 to 168.9 for five-year*ld Ford eights in 1963. The variable’s average value is about 40 for one-yearald cars and about 120 for five-year-olds. Unfortunately the repairs included in the CU list have varied markedly over time, and thus it was only possible to obtain compatible data for the years 1963 through 1966. The data include cars from one to five years of age which, given the 12 makes studied, produce close to 60 observations for each year. The actual annual number of observations is less than 60 times 4 or 240 because of the introduction of new automobile models over the period studied.


where: v = model year t = calendar date

rn = make c = year previous to start of price series

R(v,t,rn) = repair costs for car of model year v , and make rn, at

A(v,rn) = hedonic price index for cars of make rn and model year

b = rate of exponential decay of A(v ,m) (to be estimated)

time t

v calculated from (12).

Substituting (10) for S(v,t,rn) in (4') permits one to use actual used car prices to estimate the service function.

( 1 1) -AP(v,t,rn) + (rf(1 + r ) ) P(v , t + I , m) = so + sl ( t - c ) + s2A(v,rn)e-b(t-v) + s3R(v, t ,m) + uVtm

To explore further the relationship between repair records and automotive services, we then expanded our model to consider the previously ignored problem of capital gains and losses. The change in the price of an automobile is now assumed to equal the sum of services used minus (r/(l + r))P(v, t + I , rn) plus capital gains or losses over period, A K , or:

(12) -AP(v,t,rn) = S(v,t,rn) - (r / ( l + r ) )P(v , t + 1 , rn) + AK(v,t,rn)

Thus:

To gain an insight into the new term AK(v,t,rn), we considered the way in which the change in repair records, A R (v , t ,m) = R (v, t + 1, rn) - R (v,t,rn), affected both services and capital gains and losses. In this case, we expect that the change in the level of repair records over a given period affects expectations of future repair costs and, hence, causes capital gains or losses in automobile prices. The term A R will also have an impact upon the value of services generated over the period t to t + I , with a high A R reducing the value of services generated." The relative importance of the impacts on capital changes and S(v,t,rn) will be indicated in our empirical work by the sign and size of the coefficient on A R , with a positive coefficient indicating that capital effects are dominant.

1. Estimating A(v,rn). Before attempting to estimate equations (1 1) or

10. The term S(v,t,rn) in our model represents net services generated over the period r to t + 1. In other words, expenditures on maintenance and repair are subtracted from gross services provided by use in order to obtain the term S(v,t ,m).


( 127, it is first necessary to construct a hedonic price index for new cars. Using a methodology similar to that of Fisher, Griliches, and Kaysen (151 hereafter FGK), we regressed log,, of the new car list prices of all makes and vintages considered on a linear combination of measurable automotive characteristics." The estimating equation using 12 makes and 8 vintages is shown below. The choice of makes and vintages was determined by the availability of Consumers Union data on repair costs and thus includes more makes and fewer vintages than the data analyzed in the previous section of the paper. The t values are given in parentheses below the coefficients.

R2=.97 df = 77

where : A(v,m) = new car list price including federal tax and handling and

Efv,m) = I if car has eight cylinders, = 0 if car has six cylinders W(v,m) = weight in thousands of pounds L(v,m) = length in hundreds of inches H(v,m) = brake horsepower in hundreds C(v,m) = I if car is a compact, = 0 if not

transportation charges divided by CPI.

v = 1958 . . . , 1965 (model year) m = Chevy I1 - 6, Corvair, Chevrolet - 6, Chevrolet - 8, Dodge

Dart, Falcon, Ford Galaxie 500 - 8, Oldsmobile, Valiant, Pontiac, Rambler Classic - 6 (all four-door sedans); Volks- wagen (two-door sedan).

As the high R2 indicates, our analysis has isolated a set of measurable attributes whose weighted average is closely correlated with actual new

1 1 . Such a form assumes straight line indifference curves between any two attributes and implies that equal absolute changes in the level of an attribute have an equal percentage impact on A(v,m) no matter what amount of the attribute is already possessed. Although surely an over-simplifica- tion, this assumption, which was also used by FGK, produces estimates that fit the actual data very closely.


car list prices. Moreover, horsepower is the only attribute whose coefficient is not highly significant.'* A possible explanation for this fact is that the dummy for number of cylinders, which is very collinear ( -86 ) with horsepower, reflects engine characteristics more accurately.

These results take on an increased interest when they are compared with those obtained by FGK. The only salient distinction between the methodology used here and that pursued in the earlier study lies in the fact that equation (13) pools data across car vintages while FGK derive separate weighting equations for each year considered in their analysis. Never- theless, the coefficients on weight and length are within the range of the FGK values, and the R2 is higher than that of the vintage by vintage estimates obtained in the earlier work. This last fact is of special significance since one might expect that combining vintages as we have done would reduce the R2 because of the omission of variables reflecting fluctuations in demand conditions. The fact that (1 3) explains ninety-seven percent of the variance in new car prices over the eight-year period studied, however, implies that list prices are determined with little or no reference to market conditions. Automobile dealers are instead likely to absorb any fluctuations in demand through shifts in the size of discounts and trade-in allowances.

2. The Service Function. The service function was first estimated by regressing -AP(v , t ,m) on S ( v , t , m ) - ( r / ( l + r ) )P(v , t + I , m ) , using estimates of A(v ,m) and R f v , t , m ) . Since the service function contains an exponential term, s2A (v ,m) e-blt-vj, an iterative procedure was followed to obtain the value of b leading to the highest R2. The maximum occurred at b = . 2 O . I 3 The resulting equation is given in (14), with (r / ( l + r ) ) P (v , t + I , m) put on the left-hand side to permit a clearer focus on the service function. The t values are in parentheses below the coefficients. l4

12. The positive but small coefficient on the dummy for compact cars implies that this particular measure of styling raises a new car's value a small amount over other cars that are in other measurable respects identical.

13. This result is consistent with the rate of depreciation found by other scholars using models different in many respects from the one presented here. Cagan 14, p. 227) and Wykoff [ 11, p. 172, Table 71 obtained overall exponential depreciation rates of .25 and .20 respectively for similar makes and vintages of automobiles. Hall [ 8 , p. 2571 obtained a rate of .17 in his study of half-ton pickup trucks. Ramm [ 9 ] obtains annual depreciation rates that range from .17 to 5 8 depending upon calendar year and age. The highest rates are for five- and six-yeardd automobiles.

14. The I values in (14) are overestimates of the true values because they are conditioned upon the value of b chosen.


( 14) S(v,t,ml= .15P(v, t + I , m) - AP(v,t,m) (4.78)

= 131.58 - 13.90(t-1962) - . 76R(v,t,m) + . 2 8 4 A ( ~ , m ) e - . ~ ~ ( ~ - ” ) (3.84) (3.50) (3.62) ( 1 2.45)

E 2 = . 6 8 ; d f = 2 1 0 t (calendar date) = 1963, . . . , 1966 v = t - 1 , ..., t - 5

m = same as (13)

The service function was also estimated directly by obtaining independent estimates of r and regressing (r/(l + r)) P(v , t + 1 , m) - AP(v,t,m) on S(v,t,m). The interest rates are estimates of new car finance rates em- ployed by Ramm [9].15 Ramm’s estimates, based on the lending rates of four automobile finance companies, are approximately 12.5% for the four years 1963 to 1966 while the rate that arises from equation (14) is 17.7%. This latter rate, although close to the interest charge on many charge accounts, seems high as a measure of automobile finance costs. Forcing r to Ramm’s estimates yields results that are similar to (14) but indicate a somewhat slower rate of exponential decay for the hedonic price index, A , and a smaller negative coefficient on the repair term. In both (14) and (15) the coefficients on R(v,t,m) and A(v,m)e-bft-v) are significant at 98%.

(15) S(v,t,m) = (r / ( l + r))P(v, t + 1 , m ) - AP(v,t,m) = 69.19 - 8.30(t-l962) - 0.43R(v,t.m) +.20A(v,mle- J 7 ( t - v )

(2.10) (2.08) (2.49) ( 13.2 1 )

E2=.66;df=211

3. Adding cizpital Gains and Losses. Moving to the second version of our complex model, equation (12’), we attempt to assess the extent to which the change in an automobile’s repair record, AR(v,t,ml, explains capital gains or losses in a car’s value.

15. Ramm’s interest rate series is an extrapolation from estimates of new automobile finance rates made by Shay [ 101 for the years 1924 to 1962. Ramm uses the Federal Reserve Board’s series on small business loans as the extrapolation series ( [9] p. 154). In the estimates in fn. 16 Shay’s and Ramm’s series are combined. Since many automobiles are purchased for cash, John Kendricks in a comment on Ramm’s article ( [9] p. 160) criticizes Ramm’s exclusive reliance on borrowing rates. Kendricks suggests that a weighted average of consumer borrowing and lending rates would be more appropriate. Although this argument has merit, Kendricks’ suggestion has not been followed in this paper because of the lack of data on the proportion of used car purchases financed by credit and the difficulty of choosing a consumer lending rate as well as the fact that the particular interest rate chosen does not seriously affect the empirical results in the text. An experiment using arbitrary interest rates between Seven and nine percent produced results that differed little from (15).


Following basically the same approach as in ( 1 9 , we regress AR(v,t,m), A(v,m)e-b(t-v) and a trend term on -AP(v,t,m) + (r/(l + r ) ) P(v , t + I, m) and obtain:

(16) S(v,t,m) + AK(v,t,ml = -AP(v,t,m) + ( r / ( l + r))P(v, t + I , ml = -32.77 + 23.75(t - 1962) + l.SIAR(v,t,m) + 0.2lA(v,m)e-. 23(t - V )

( I . 04) (3.44) (2.40) (14.72) R2 =.62; d f = I28

t = 1963-1965 v = t - I , . . . , t - 4

m = same as (13)

The coefficient on AR in (16) is positive and significant at over 9576, indicating the importance of the rate at which repairs accrue upon the level of capitalgains and losses. The size of the coefficient indicates, moreover, that if AR has no effect on S(v,t,m), then an increase of one in AR causes a capital loss in the value of an automobile of $1.8 1 . The capital effect is even greater if AR also negatively affects S(v,t,m) as seems prob- able. In other words, an unexpectedly high rate of repair for a given make, vintage, and age of automobile lowers automobile purchasers expectations about the future patterns of services after t +I and hence lowers dramati- cally the price, P(v, t + I , m), they are willing to pay.16

16. The results in (14). (15) and (16) appear inconsistent with the results presented inTable 1 for the simple exponential decay model. In particular, the proxy for new car prices, A (v.m), decays at an exponential rate between .17 and .23 while the rate of decay in relative prices is between .28 and .34 in Table 1. However, when the simple service function is estimated in linear form using the data for v = 1954, ..., 1963 and pooling the six automobile models, we obtain:

S(v,t,m) = ( r / ( l+r) )P(v , t + l , m ) - AP(v,t,m) = -77.67+.40P(v, ~ + l , m ) e - . ~ ~ ( ~ - ~ ) (4.18) (20.44)

R2 = .65 df = 228

v = 1954 - 1963 t = v + 2 , ..., v + 7 i f v < 1 9 6 0 t = v + 2 , ..., 1966 if ~ 2 1 9 6 0

m = same as (9) This equation is not identical to (5) since the constant term has not been forced to zero and is, in fact, significant and negative. Furthermore, the use of P(v, v + 1 , m) instead of P(v,v,m) as in ( 5 ) will affect the calculated value of the coefficient of this term but not the estimate of b, the rate o f exponential decay. If we assume that P(v, v + l , m) = P(v,v,m)e-b, then, for b = . I 7 , we can write the second term as: . 3 4 P ( v , ~ , m ) e - . ~ ~ ( ' - ~ ) .

The rate of decay is equivalent to the rate in (15) and is much lower than the estimates using the log-linear form of (9). The f l is comparable to those in (15) and (16). This result does not, however, imply that repair records are irrelevant since vintages, makes anddates are not comparable between the simple model and (14), (15) and (16). The explanation for the difference in decay rates is the flatness of the function estimated in this footnote over the rate . I 7 to .30. Although the constant term becomes positive, the coefficient on P(v, v + 1, m)e-b(r-v) only falls shghtly to +.38 and the R2 drops only three points to .62.

414 WESTERN ECONOM IC JOURNAL

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used cars as a depreciating asset

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