SOUTHERN JOURNAL OF AGRICULTURAL ECONOMICS
DECEMBER, 1983
OLS AND FRONTIER FUNCTION ESTIMATES OF LONG-RUN
AVERAGE COST FOR TENNESSEE LIVESTOCK AUCTION MARKETS
Dan L. McLemore, Glen Whipple and Kimberly Spielman
Considerable research has been conducted to explore economies of size in the livestock auction market
industry. Since auction market cost functions are expected to conform to microeconomic theory, conclusions regarding industry economies of size are often
derived from estimated long-run average total cost
(LRATC) functions (French; Stoddard). The LRATC
function suggests the least-cost firm size, as well as the
structure of size economies for the industry. Previous
economies-of-size research has generally used the least
squares method or the economic-engineering method
to estimate LRATC functions (French; Bressler; Polishuk and Buccola; and Johnson). The frontier function
method has been suggested as an alternative to these
methods (Bressler; Miller and Nauhein; Seitz). Even
though use of a frontier function is theoretically appealing in some cases, its application in economies-ofsize research has not been widespread (Bressler;
French; Farrell and Fieldhouse; Lesser and Greene).
The purpose of this paper is to evaluate the frontier
function method of estimating an LRATC function in
contrast to the ordinary least squares (OLS) method.
Both methods will be applied to evaluate size economies in the Tennessee livestock auction market industry.
METHODOLOGY
Typically, the OLS method uses cross-section data
in a regression of average total cost against output volume to estimate an LRATC function. Recent studies
utilizing OLS have confirmed that cost economies do
exist in auction markets (Wootan and McNeely; Grinnell and Shuffett; Grimes and Cramer; Wilson and
Kuehn).
A frontier function estimate of LRATC is an envelope curve fitted to the bottom of the point scatter of
average cost plotted against volume. A frontier function may be estimated by several methods (Farrell and
Fieldhouse; Aigner and Chu; Timmer; Hazell). This
study incorporates the minimum absolute deviations
(MAD) method developed by Hazell. The MAD estimate of LRATC at a given output minimizes the sum
of the deviations of estimated average total cost from
the observed average total cost, assuming that the estimated average total cost is less than or equal to observed average total cost.
Although the frontier function has not been widely
used by researchers to estimate industry cost functions, this approach may yield a more appealing estimate of the LRATC function than the OLS method.
Observations of average cost probably represent firms
operating at various points along numerous short-run
average total cost (SRATC) curves. It is unlikely that
each firm would be operating at the tangency between
its SRATC and the industry LRATC during a given
time period. Each observation on SRATC will either
be on or above the LRATC, and it is probable that most
of the observations will be above the LRATC. Thus,
cross-section data for an industry would include firms
operating not only with a variety of plant capacities but
also at a variety of volumes for a particular plant capacity. Although many factors may influence average
total cost in the short run, the LRATC function defines
the relationship between cost and volume only. Since
annual plant volume from a cross section of firms would
reflect both short- and long-run factors, one would expect a range of average total costs to be observed at each
plant volume.
OLS estimation of LRATC produces a function
which lies as near the mean of the average total costs
for a certain size firm as allowed by the particular
functional form used. The OLS estimate is a useful
predictor of short-run average cost for a range of plant
sizes, but would overestimate LRATC at a particular
output volume. It is clear from economic theory that,
at any given volume, the firm operating at a lower
SRATC is operating nearer the LRATC. Thus, a frontier function fitted to the lower extreme of the scatter
of average cost-output volume observations would
more closely approach the theoretical notion of a
LRATC function which is envelope to the industry's
SRATC curves.
Even though the frontier function more accurately
resembles the theoretical LRATC curve, if none of the
observed firms were operating on the LRATC curve,
the estimated frontier function would overestimate the
actual LRATC curve. Even so, the overestimation bias
would likely be less than that of the OLS estimate. Data
measurement error could affect the frontier function
estimate more than the OLS estimate. The location of
the frontier function is most dependent on the observations with lower SRATC for any given volume.
Thus, errors in measurement in these few observations
may disproportionately affect the estimated function.
Dan McLemore is a Professor, Glen Whipple an Assistant Professor, and Kimberly Spielman a Graduate Research Assistant, Department of Agricultural Economics and Rural Sociology,
University of Tennessee.
79
With large measurement error, the frontier function
estimate could be below the actual LRATC. Since observations on average total cost and volume are not expected to be normally distributed about the estimated
frontier function, statistical measures of fit or significance do not exist. The sum of the absolute deviations
of the estimated average total cost from observed averge total cost provides a measure of fit for the frontier
estimate. This measure is useful for comparisons of alternative frontier estimates. The sum of the absolute
deviations does not, however, provide an absolute
measure of fit or significance, nor is it useful for comparison of the frontier function with estimates utilizing
other techniques, such as OLS.
For expositional purposes, assume an LRATC function of the form:
Yi = a +
(1)
n
. Yi
maximizes
A frontier function of form (1) may be estimated by
restricting:
E (a +
3X).
Thus, y, may be eliminated and the optimizing criterion restated:
n
(5)
maximize:
E (o
+ 3Xi),
i=
subject to the restrictions that:
(4)
ot
+ 3X,1
Yl
+ p3Xn
Yn.
This problem can be solved using linear programming.
To allow ox and 3 to assume negative as well as positive values, the intercept and Xi terms must be included twice-once with negative and once with
positive signs. The linear programming problem becomes:
n
(6)
Maximize:
O
i
n
(li) +
n
Pi E (Xi) +
i=
(7)
Subject to:
(-l)
2
+
i=l
n
,32 E (-Xi)
i=
cl (1) + o2 (-1) + Pi(XI) +
cx(l) +
C-2(-1)
(xl,,a2,
, 2 : 0)
13(-XI) ;
Y
P2(-Xn)
Yn
~
+ PI(X) +
a + 15x1
Y,'whichever
The maximizing criterion will force either atl or o 2
in equation (6) to be zero; the nonzero ot I oro 2 is an
estimate of a in equation (1). Likewise, either 13 or 12,
is nonzero, is an estimate of 3 in equation
(x +
Y2
(
Restriction (2) can be described as:
X2
•.....
~...•
ax + p3Xn
. ••
•..
Y,.
Since yi is constant for any given observation, minimizing
80
3Xi)
oL + 3Xi
X
for each observation i. (This is the same as restricting
E, > 0 for all i.) Only the most efficient of the observed firms will satisfy the equality since the remainder of firms will produce at a cost above the frontier
function. An infinite set of a ox and 3 will satisfy restriction (2). Thus, to estimate a unique frontier function,
an optimizing criterion is needed. The MAD method
minimizes the absolute value of the deviations of the
estimated average total cost from the observed average
total cost. This is the same as minimizing the linear sum
of errors (Ei) and causes the estimated function to lie
as near the center of the average cost-volume point
scatter as possible, subject to restriction (2).
Summing over observations (i) and rearranging
equation 1, the optimizing criterion may be mathematically described as:
n
n
n
(o + 3Xi).
(3) minimize: E E i = E Yii=l
i=l
i=l
(4)
+
,Xi + Ei
Yi = the actual total cost for market i
Xi = the volume (output) handled by market i
Ei = random error.
Yi
(l
i=
where:
(2)
n
APPLICATION TO THE TENNESSEE
.LIVESTOCK
AUCTION MARKET
INDUSTRY
LRATC functions for the Tennessee livestock auction market industry were estimated using both OLS
and frontier function methods. Accounting records
were obtained from the Packers and Stockyards
Administration Form 130 for 1978 and 1980.1 A total
of 101 observations were available (Table 1). Market
output was measured according to volume of livestock
handled, gauged in "Animal Marketing Units"
(AMU). An AMU has been defined by the USDA as
one cow, one calf, three hogs, four sheep or goats, ori
one horse or mule (Stoddard).
Data from both 1978 and 1980 were combined to estimate the LRATC function. Costs for 1980 were deflated to 1978 dollars by the Index of Prices Paid by
Farmers.
Economic theory suggests use of an average cost
function that would decrease at a decreasing rate over
the small firm sizes and may or may not turn up for large
sizes. Four functional forms were hypothesized as theoretically appropriate for the Tennessee auction market LRATC:
(I)
Y -x
=
(8) Maximize: an(l) +
32.
n
E
+
3X +yX
-1
n
1
+ Y1
-
1-I3 +
=1 X
2
+
-1
(9)
Subjectto: (,(l)
+
+ Pi
a 2 (-1)
+
X
-2
^
,1
.
Y2. 1
2
1
+
t2n(-1)
al(1) +
Y = oa + P
(II)
As an example, the linear programming problem for
Model IV was:
^
O2(-1) +
+
P1+
Xn
+Y2Y
^ ^ X] ^
X2
^~~~~XI
X
(III)
lnY = ex + plnX
(IV)
Y = a + pL
-lI
2 X
+ Y X2
1
-n
2
YX2n
Appropriate linear programming formulations were
used to estimate frontier functions for the other three
functional forms.
where:
Y = Total cost per animal marketing unit (AMU)
X = Number of AMU's handled per year
RESULTS
Each of these models was used in an OLS regression
of average total cost against AMU's. 2 Frontier functions were estimated using the same functional forms.
Table 1. Number and Average Volume of Livestock
Auction Markets by Size Group, Tennessee, 1978 and
1980
~VoluhanNumber
of
Volume handled
per year
(AMU)
Less than 9,000
markets
1978
1980
Average
volume
Number
of
Average
volume
handled
(AMU)
markets
handled
(AMU)
7
5,814
14
5,840
9,000 - 17,999
15
13,810
14
13,115
18,000 - 26,999
13
21,648
5
20,892
27,000 - 35,999
4
31,244
6
32,276
36,000 - 44,999
5
39,455
2
40,711
45,000 - 53,999
7
51,453
3
46,046
54,000 or more
4
75,412
2
69,702
55
46
Estimates of the Tennessee auction market industry
LRATC function are given in Table 2. The OLS estimates yielded relatively small R 2 's, which seems consistent with the notion that a large part of the variation
in observed average total cost is not related to volume
in the long run. The wide range of average total costs
for a given volume apparent in Figure 1 suggests that
large R2 's are unlikely. Model IV was considered to be
the best OLS estimator of LRATC based upon R2. 3
Model IV was also considered to be the best frontier
function estimate based upon the sum of absolute deviations (1AD). Both OLS and frontier function estimates of Model IV are shown in Figure 1.
Results from both the OLS and frontier function estimates indicate that economies of size existed in the
industry. However, the two functions differ in the range
of volumes over which economies of size appear to be
important. The OLS functions for Model IV indicates
that the average market must handle almost 60,000
AMU's to achieve 90 percent of the available econ-
I The design capacity of auction market facilities cannot be determined from the Packers and Stockyards Administration data. Thus, one cannot determine how near the firms were operating
to their short-run minimum cost volumes. The phase of the cattle cycle probably affects volume moving through livestock auction markets in Tennessee. While 1978 represented an intermediate
year, cattle numbers were near the low point of the cycle in 1980. This data may contain other flaws common to accounting data. The real costs of auction market operation may not be accurately
represented because the accountant's view of costs, reflected in the firm's records, differs from the economist's view.
2 Alternatively, each model was specified to include a binary variable to capture the effect of year (1 if 1978, 0 if 1980) on average total cost. Regression results indicated that the impact
of a year on average total costs was an statistically significant at the o = 0.10 level.
2
2
3 Adjusted R's (R) for Models I, II, III, and IV were .132, .205, .180, and .254, respectively. R2 values were derived from R2 values shown in Table 2. R2 for Model III was calculated
as indicated in Table 2, footnote b.
81
Table 2.
Long-Run Average Total Cost Functions for Tennessee Livestock Auction Markets, 1978 and 1980
Ordinary Least Squares
Model
2
Y = a + 8X +YX
(I)
(II)
Y = a +
1~X
(III)
(IV)
+ Y-
Y = a+
1-X
X
MADEstimate of Frontier Function
2
a
§
Y
5.828
-0.00008643
(0.00002590)
0.0000000008083
(0.0000000003380)
R
a
Y
8
.149
2.454
-0.00001875
7988.542
~~(1544.801)
.213
2.054
2040.385
2.993
-0.15793
(0.03486)
.188
3.299
18655.230
(4152.953)
~ 3.826
lnY = a + BlnX
a
-22806680.290
(8283490.842)
b
1.506
.269
CADC
0.0000000002201
229.31
-0.07416
2.103
231.23
233.17
590.455
8154126.868
223.39
a Numbers
in parentheses below coefficients are standard errors of estimate.
2
b The R value for Model III was computed by evaluating the estimated equation for predicted average cost in logarithms at each observed volume, obtaining the antilogs of the predicted
average costs, computing the deviations of the resulting predicted average costs from the original observed average costs, computing the sum of squares [J(y - 9)2], and using:
2
R = 1_ (y - )2
;(y- y)2.
c SAD represents the sum of the absolute deviations of the observed average costs from the frontier function.
d The sum of absolute deviations for Model III was calculated from the observed average costs and the antilogs of predicted average costs.
Table 3. Comparison of Long-Run Average Total
Cost Functions Derived from Minimum Absolute Deviation Estimation of a Frontier Function and from Ordinary Least Squares for Model 4
12
Li
,, ^
9~~~~~~~~~S
*
~~Average
.·
8.
So
'^'
4a)% .
5,000
6.118
-.312
8.4
2.547
-.302
86.0
7,500
5.381
-.312
32.3
2.327
-.158
92.9
10,000
4.936
-.286
46.8
2.244
-.099
95.6
15,000
4.441
-.234
62.9
2.179
-.051
97.6
20,000
4.175
-.196
71.5
2.153
-.033
98.4
30,000
3.896
-.147
80.6
2.132
-.018
99.1
40,000
3.751
-.117
85.3
2.123
-.012
99.4
50,000
3.663
-.097
88.2
2.118
-.009
99.5
60,000
3.604
-.083
90.1
2.115
-.007
99.6
70,000
3.561
-.072
91.5
2.113
-.006
99.7
90,000
3.503
-.058
93.4
2.111
-.004
99.7
~'
.
Ordinary
astSquares
' '
'
—
*.ar_
________10
Frontier Function
%cost
Average
Cost
%cost
economieg
total
function
economieg
realized
cost
elasticity
realized
(1)
(()
($)
Volume
•(AMU)
5
* <. .'
'
• '
~»~_ '
____________
_
OLS
OLS
Cost
function
elasticity
total
cost
($)
3
Frontier Function
10
20
30
40
50
60
MU)
Volun in Thousands (A.
70
80
90
Figure 1. Long-Run Average Total Cost Functions
for the Tennessee Livestock Auction Market Industry
Using Ordinary Least Squares and Frontier Function
Using ,r.iary
Least
SqA a And Fro0n tier F unt
Methods
for Model
41 (1978
and 1980 Data Combined)
omies of size (Table 3).4 On the OLS function, elas-
ticity of cost with respect to volume declines (sign
ignored) at a decreasing rate, but remains at - .058 for
90,000 AMU's. 5 That is, the OLS function declines
substantially up to maximum observed volumes.
The frontier function estimate for Model IV indicates that auction markets achieve 90 percent of the
available economies of size at an output of less than
7500 AMU's. Of available economies, 99 percent are
achieved at 30,000 AMU's. For the frontier function,
the elasticity of cost with respect to volume reaches
-. 051 at 15,000 AMU's, while the OLS function required 90,000 AMU's to reach an elasticity of - .058.
Elasticity of average total cost with respect to volume was determined according to:
Elasticity = dATC . Vol
dVol ATC
b Percent cost economies realized was defined as the difference between predicted average total cost (ATC) at the minimum observed volume and predicted ATC at the volume
under consideration, divided by the difference between predicted ATC at the minimum oba
served volume and predicted ATC at the asymptotic minimum of the function
ATCin. vo.- ATCv,0 i
LATCin
ATCasympt. mi
The frontier function estimate indicates that economies of sizes are practically exhausted at relatively
small levels of output.
CONCLUSIONS
The appropriateness of the OLS or the frontier function as an estimator of LRATC depends on the goals
4 Percent cost economies realized was defined as the difference between predicted average total cost (ATC) at the minimum observed volume and predicted ATC at the volume under
consideration, divided by the difference between predicted ATC at the minimum observed volume and predicted ATC at the asymptotic minimum of the function.
ATCin..vol.- ATCvol.i
LATC n vol. -ATCasympt. mn.
5 Elasticity of average total cost with respect to volume was determined according to: Elasticity = dATC .
dVol
82
Vol
ATC
of the researcher. The OLS approach estimates the expected short-run average cost conditions and yields
statistical estimates of significance. The frontier function approach provides a more theoretically appealing
estimate of LRATC since it is analogous to the envelope concept, although measurement error in a few observations may be more likely to result in misestimation
of the LRATC.
The OLS estimate indicates that auctions in Tennessee may experience substantial cost economies by increasing volume up to relatively large output levels.
Along the frontier function estimate, economies of size
are important only at relatively small output levels.
Thus, the OLS estimate indicates that economies of size
are possible over a wider range of volume in the long
run than with the frontier function.
In 1978 and 1980 combined, 50 percent of the livestock auctions in Tennessee operated at volumes less
than 18,000 AMU's. These data indicate that a large
portion of auction market firms operated at volumes that
leave substantial cost economies uncaptured, assuming the OLS function to be the appropriate estimate of
LRATC. This suggests that the frontier function estimate more accurately reflects observed firm behavior
with regard to volume and thus may be the more appropriate estimator of LRATC.
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83