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THE PERSISTENCE OF MILK SUPPLY IN WASHINGTON STATE:
AN ANALYSIS OF SUPPLY RESPONSIVENESS TO PRICE
AND TECHNOLOGY EFFECTS
by
Don Blayney and Ron C. Mittelhammer*
*Don Blayney is with CED. ERS. USDA and Ron C. Mittelhanuner is a Professor in the Department of Agricultural Economics. Washington State University. Pullman, Washington. The views expressed are those of the authors and do not necessarily reflect the policies of the USDA or the views of other USDA personnel.
, .'
0,'
The Persistence of Milk Supply in Washington State:
Analysis of Supply Responsiveness to Price
and Technology Effects
Abstract
Persistent supply/demand imbalances in the Washington State dairy industry are a chronic problem. Econometric analysis of highly productive Washington State dairymen using a dual profit function approach with a technology parameterization indicates that technological advances primarily account for persistent supply expansion in the State.
The Persistence of Milk Supply in Washington State: An Analysis of Supply Responsiveness to Price
and Technology Effects
Introduction
Dairymen in the West have achieved large productivity gains in recent
years which have tended to exacerbate what many dairy analysts and govern-
ment policymakers consider to be a chronic oversupply of raw milk in the
United Siates. Washington State dairies in particular are highly produc-
tive, having averaged 16,892 pounds of milk per cow in 1985, a figure which
was well above the national average of 13,031 pounds per cow. The increas-
ing productivity of milk cows, coupled with an increasing aggregate herd
size resulted in steadily increasing aggregate milk production in Washing-
ton State that reached 3.75 billion pounds in 1985. Increasing milk
production persisted even in the face of market conditions and government
policies that would seemingly motivate reductions in milk supply.
In this paper, the responsiveness of Washington's milk supply to price
changes and technological advancement are econometrically analyzed in an
attempt to provide insight into the rationale for continued and persistent
supply expansion in the state. The analysis is based on an aggregate dual
profit function approach that utilizes a flexible generalized Box-Cox
representation of the profit function which incorporates a technology term
representing both neutral and biased technological effects.
Model of Output Supply and Input Demand
Several reasons have been advanced for the high productivity of
Washington's milk producers. First of all, they have been characterized as
effective and progressive managers. A significant portion of the producers
actively participate in Dairy Herd Improvement Association programs which
foster good record-keeping of both costs and production, and include
regular component testing of farm milk. Profit-maximizing behavior is dlso
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promoted by a well-established University extension program in dairying.
Second, on-going technological advancements in feeding and milking systems,
herd health care, and genetic potential have been readily adopted by
dairymen in the state. Other factors, indigenous to the region, that have
contributed to a favorable environment fostering high-producing cows
include a favorable climate for dairying on the west side of the state,
where the vast majority of dairying takes place, and the development of the
Columbia Basin irrigation project, which created a source of high quality
forages, especially alfalfa hay, which are instrumental to feeding regimes
for high-producing cows.
Given the high level of managerial expertise and the demonstrated
willingness to adopt technological innovation exhibited by Washington
dairymen, a dual profit function approach incorporating a technology
parameterization was thought to be a useful paradigm within which to model
dairymen behavior. Regarding the application of the dual profit function
to an industry aggregate, recent theoretical contributions by Blayney and
Mittelhammer, and Blayney provide a benchmark rationale.
The generalized Box-Cox specification of the aggregate profit func-
tion, with technology term, was patterned after the application of the
functional form in an aggregate cost function analysis by Berndt and
Khaled. The specification of the profit function was
1
(1) -1 2A TI (P,W,t) = (A (Z'AZ)) exp{t(c + y'R)}
where the arguments of TI(') are P, the price of milk, W, the vector of
input prices for six input categories; cows, concentrates, hay, silage,
labor, and capital, and t is the technology variable represented by time.
The vectors Z and R are defined as Z = (PA, ... , and R =
(lnP, lnW1
, ... , lnW6
) , . Linear homogeneity of TI in prices is maintained
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7 , under the restriction that ~r, = 0 (r
J. are the elements of r).
j=1 J
The parameters to be estimated are A. the functional form parameter.
~. the measure of neutral technology effects, r, the vector of nonneutral
technology effects, and A, a Ox]) symmetric matrix parameterizing price
relationships. The parameter A may be tested to determine if a common
functional form is applicable. In particular, if A = O. the trans log
functional form results, a value of .5 would imply the generalized Leontief
form, and 1.0 indicates the square root quadratic form.
The system of equations obtained by applying Rotelling's lemma to (1)
is as follows, where Q represents aggregate milk supply, and X. represents 1
input demand for the ith input type:
+ t(L+r'R)(,-I(Z'AZ))(1/2A)-I,-I( W2A - 1+ WA- 1pA e A A a i +1 ,i+l i a 1 ,i+l i
A-I A + ~ a, '+IW, W.)] for i = 1 •...• 6.
j:Fi 1,] J 1
The system in (2), with disturbances added, offers a variety of problems
for estimation. First note the equations are highly nonlinear in the
parameters, so that a nonlinear estimation technique is required. Second-
ly. the lead term in each equation contains a term identical to (1), the
aggregate profit function. Thus, all 37 parameters of the system appear in
each equation. For estimation only 20 annual observations on prices and
quantities, spanning the years 1966-1985, were available. Data were
obtained from several sources, including Agricultural Statistics, Economic
Indicators of the Farm Sector: Production and Efficiency Statistics, 1984,
\
- 3 -
Survey of Current Business, DHIA Annual Summary, and Washington State
Cooperative Extension Service budget studies. Prices were defined as the
average return per hundred weight (cwt) for milk, the delivered cost of hay
and silage per ton, the average value of concentrates fed per cwt, the
hired farm labor wage rate per hour, and for cows and capital, a price
formula involving both purchase and salvage values was used. Quantities
include annual aggregate milk output in Washington, aggregate herd size, ·
the quantities of hay, silage and concentrates fed, the hours of labor
required, and an index of capital services purchased. The units of mea
surement of the quantity variables and their average values are displayed
in Table 1. For further details on variable definitions, see Blayney.
If it were possible, nonlinear three stage least squares (N3SLS) would
have been the preferred technique for estimating the system of equations
(2), since the right-hand-side of (2) contains endogenous prices and it is
reasonable to suspect that contemporaneous disturbances across the equa
tions would not be independent. However, given that each equation of the
system contains 37 parameters, while the data set contained only 20 obser
vations, a straightforward nonlinear system approach to estimation was not
possible. In particular, the usual N2SLS estimate of the contemporaneous
covariance matrix of the disturbances cannot be calculated, and in any case
the Jacobian matrix of partial derivatives of the estimated residual vector
with respect to the parameters of the model used in Amemiya' s N3SLS ap
proach, which is implemented in SAS, is not of full column rank, and so the
procedure would fail.
A two-step procedure was devised to provide estimates of the model
parameters. In step one, a set of 20 aggregate profit values calculated
from Washington State dairy herd budgets was used as a proxy to substitute
for the profit term (1) which can be algebraically isolated in two posi-
tions for each of the equations in (2). With this substitution, the
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remaining number of parameters to be estimated in each equation numbered
15. 1 Treating the profit proxy as an additional endogenous variable, the
revised set of equations was estimated using a nonlinear instrumental
variables estimation procedure to generate an estimate of the contemporane
ous covariance matrix of the equation disturbances. The estimated contem
poraneous covariance matrix was then imposed using a nonlinear seemingly
unrelated . regression approach for estimating the original system of equa
tions (2).
Hypothesis tests of parameter values were conducted using a general
hypothesis testing procedure for nonlinear models suggested by Gallant and
Jorgenson. Regarding whether one of the common functional forms would be
appropriate for representing the profit function, the hypotheses that A =
0, .5, and 1.0 were each rejected at the .10 level, suggesting that neither
the translog, generalized Leontief, nor the generalized square root quad
ratic forms could be adopted. Also, the dimensionality of the parameter
ization of the model was reduced using a zero-vector hypothesis for a
subset of the model parameters which was not rejected at the .10 level.
The final estimate of the equation system (2) involved 28 parameters and
goodness of fit statistics for this model are displayed in Table 1.
Judging by the standard measures of fit, the performance of the estimated
system appears to be reasonably good.
Selected Profit Model Implications
In the estimated model of output supply and input demand neither price
elasticities nor the rates of change in output supply and input demand due
to technological advancement are constant. As a general guide to the
magni tude and signs of elasticities and technological rates of change,
Table 2 displays these response measures evaluated at the mean level of the
data. Highlighted in the table are the price elasticities of output supply
(first row), the own-price elasticities of milk supply and input demand
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(diagonal), and the elasticities of output supply and input demand with
respect to milk price (first column). The results in Table 2 suggest that
output supply and input demands are price inelastic. The input cross-price
elasticities are, by and large, consistent with the theoretical work of
Bear and Rader who suggest that inputs will generally be gross complements
when firms pursue profit maximizing objectives. The rates of technological
change suggest a rather large 5.557. output growth rate due to technological
advancement, and technological advance is seen to be input-using, as
evidenced by the positive rates of input adjustment due to technological
change. In relative terms, technological advance is evidently biased most
heavily toward feed input usage, especially concentrates, and notably,
biased least favorably toward labor usage.
In order to provide an indication of the effects of technological
change and price changes on Washington state milk production over time,
year-to-year changes in milk production were decomposed into three effects
using the estimated system of supply and demand functions. The decomposi-
tion is displayed in Table 3. The first effect, labeled "Constant Input
Technology Increment" , represents the increment to output that is
attainable if input usage is held constant while technology is advanced by
one unit in any given year. To clarify the interpretation of this
incremental measure, it can be shown that the homogeneity of degree zero in
prices exhibited by the set of supply and demand equations
(3) : fX. (p,W,t), i:l, ... , 6 1.
induces a functional dependence between Q and Xl' ••• , X6
' which is parame
terized by t, as Q : Q(Xl
, ••• , X6 ' t). This functional dependence is the
dual representation of the "economically relevant production function,"
where Q : Q(Xl
, ••• , X6
' t) represents the level of production that is
associated with technology level t and the input level "(Xl' ••• , X6
) that
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is profit-maximizing for some level of . 2 pr1ces. Letting Xl' . .. , represent the input level utilized in year t, the "Constant Input
Technology Increment" is calculated as Q(Xl
, ... , X6
, t+l) - Q(Xl
, ... , X6
,
t), and represents a shift in the production surface at input level
... , X6 due to technological advancement. 3
The second effect is referred to as the "Constant Price Input
Adjustment." If technology were advanced by one unit to the value t+l, the
level of input usage applied prior to the technological advance would
generally not be profit maximizing if applied under the new technology and
old prices (prices existing in t). Holding prices constant at the level
existing in year t, and advancing technology to t+ 1, the difference in
output resulting from utilizing the profit maximizing level of inputs
versus the level of inputs used in year t is represented by the "Constant
Price Input Adjustment." Symbolically, the effect is represented by
fQ (Pt,Wt,t+l) - Q (X l , ... ,X6 , t+l), where Xl , ... ,X6 is the level of inputs
applied in year t. The sum of the first two effects can be interpreted as
the total effect of technological advances at constant prices.
The third effect, labeled "Input Adjustment to Price Change", measures
the difference in output that would result under technology level t+l from
using profit-maximizing input levels for price levels existing in year t+l
versus year t. The adjustment reflects the effect of changing prices on
the profit-maximizing level of output, given that technology has been
incremented by one unit. In symbols, the adjustment is represented by
Adding the total . of the three
effects to milk production in year t yields milk production in year t+l.
The results in Table 3 suggest that improving technology in dairying
has played a major role in advancing the level of milk supply in the state.
The average yearly percentage increase in milk supply due to technological
advance was 4.94%. Of this average percentage increase, 1.52% could be
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viewed in terms of upward shifts in the aggregate production surface, while
the remaining 3.42% could be interpreted as the effect of optimally adjust-
ing input levels at constant prices to take advantage of profit-maximizing
opportunities made available by technological advance. Adjustments to
accommodate changing output and input prices generally mitigated the effect
of technological advance. On average, adjustments to changing prices
accounted for a 2.32% reduction in milk production, but this reduction did
not fully offset the positive effect on production induced by technological
advance. The net effect of both technological advance and price adjust-
ments was an average yearly 2.62% increase in milk production in the state.
Of particular interest in Table 3 are the results for the final two
years. The period from 1983-1985 was characterized by output and input
prices that were notably less favorable to dairying than at the beginning
of the 1980' s. During this period, milk price in the state declined by
6.7%, hay prices rose 1.3%, silage costs increased by 2.9%, labor costs
were up by 13.5%, and the cost of capital services increased by 7.03%.
Mitigating the unfavorable price changes somewhat was a 5.46% reduction in
concentrate prices. While the unfavorable price changes induced a
significant output-reducing price effect, as indicated in Table 3, the
effect of technological advances more than offset the reduction, resulting
in continued supply expansion in both 1984 and 1985.
We conclude that advancing technology, including improving managerial
skill, feeding and milking systems, herd health care and herd genetic
potential, has b~en a dominant force in expanding milk supply in Washington
state. Apart from direct government intervention to reduce the aggregate
h d . 4. er s lze, 1 t appears that only quite drastic reductions in milk price
and/ or substantial increases in the prices of inputs could reverse the
expansionary effect on milk supply induced by advancing technology in the
state's dairy sector.
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Footnotes
1 After the profit term substitution, 16 parameters remain, including
7 parameters from the A matrix, the A parameter, the. parameter, and the
seven parameters 11, ... ,17' The imposition of the homogeneity
7 restriction l:
j=l 1. = 0 reduces the parameter dimension to 15.
J
2 It is evident from the definition of Q = Q(Xl
, ... ,X6,t) that levels
of inputs that are not profit maximizing for any level of prices are not in
the domain of the function.
3 As one might suspect from the highly nonlinear nature of the output
supply and input demand functions in (2), the function Q = Q(Xl
, ... ,X6,t)
could not be explicitly solved for in algebraic terms. A numerical proce-
dure using nonlinear equation solution techniques was programmed on the
IBM-PC model 50 in the GAUSS programming language to solve for the Q that
was consistent with an input level X1"",X
6 at a given level of technology
in the system of equations (3), thereby numerically solving for the func-
tion Q = Q(X1
, ... ,X6,t).
4 The initial effect of the dairy termination program in Washington
State (which began after the final year included in the profit function
analysis) was a reduction of approximately 14% in the state I s aggregate
dairy herd. However, producers not enrolled in the program quickly expand-
ed herd sizes and offset most of the state I s loss in milk production
capacity within a year. Milk production between 1986 and 1987 increased by
roughly 4.3%, which exceeds the average yearly growth rate of 2.5% calcu-
lated from the · profit function analysis, and substantially exceeds the
growth rates exhibited in either of the final two years of the profit
function analysis.
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Table 3. Year to Year Milk Production Decomposition Using Profit Function Analysis 1
Year
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
Milk Production
22.073
22.456
22.649
22.778
22.679
23.235
23.566
23.936
23.287
24.944
26.953
27.056
28.549
27.727
27.847
29.065
31. 409
33.884
34.845
2 + Constant Input +
Technology Increment
.277 (1.257)
.314 0.396)
.349 (1.539)
.376 0.652)
.387 0.704)
.417 (1.795)
.434 (1.843)
.388 (1.620)
.324 (1.393)
.382 0.530)
.467 0.734)
.458 (1.693)
.531 (1.859)
.418 (1.508)
.352 (1.262)
.350 (1. 205)
.419 (1.335)
.474 (1. 399)
.430 (1.233)
Constant Price3
Input Adjustment
.378 (1. 712)
.387 (1.722)
.412 (1.817)
.456 (2.001)
.512 (2.258)
.542 (2.334)
.608 (2.580)
.619 (2.585)
.730 (3.133)
.775 (3.107)
.809 (3.000)
.968 (3.576)
1. 000 (3.503)
1.230 (4.436)
1.370 (4.920)
1.507 (5.186)
1.699 (5.411)
1.897 (5.599)
2.118 (6.078)
+ Input Adjustment4
to Price Change
- .272 (-1.234)
- .507 (-2.258)
- .631 (-2.787)
- .931 (-4.088)
- .343 (-1.512)
- .629 (-2.706)
- .673 (-2.854)
-1.655 (-6.916)
.603 ( 2.590)
.853 ( 3.419)
-1.173 (-4.351)
.067 ( .247)
-2.353 (-8.240)
-1.528 (-5.512)
- .503 (-1.808)
.487 ( 1.674)
.357 ( 1. 135)
-1.410 (-4.162)
-1.650 (-4.736)
1 Quanti ties of milk are measured in hundreds of millions of pounds. Percentage increments or adjustments are given in parentheses.
2 Represents the increment to output attainable if input usage were held constant at the level used in year t, but technology were advanced to the level existing in year t+1.
3 Represents the difference between the level of output that is profit-maximizing at year t price levels and year t-+:1 technology, and the level of output attainable using year t input levels and year t+1 technology.
4 Using year t+1 technology, this represents the difference between profit maximizing output levels at price levels existing in year t+1 and year t.
Table 2. Matrix of Own-Price and Cross-Price Elasticity Estimates and Technological Rates of Change, a Computed from the Reduced Model
Price Elasticities
Quantity I MILK COWS CONCENTRATES HAY SILAGE
MILK .8932 1~.1128 -.1946 -.1124 -.0675
COWS 1.1446 ~ - .1134 ............. -.2350 -.1962 .0002
CONCENTRATES .9554 -.1137 -.0841
HAY .9653 -.1661
SILAGE 1.3039 .0003 -.3306
LABOR 2.1611 -.2676 -.2294 -.2901
CAPITAL .6844 -.1532 -.1027 .1375 -.0134
aEvaluated at the means of the data points
LABOR CAPITAL I
-.3174 -.0885
-.3988 -.2009
-.1655 -.0652
-.3660 .1526
-.0333
Technology Rates
Q X. or 2 Q x .
1
.0555
.0571
.0803
.0409
.07L,5
.0278
.0330
Table 1. Goodness-of-Fit and Theil's Statistics of Forecast Errors for the Reduced Model
Endogenous Average Unit Mean Mean i. Aggregate Actual of Mean Mean i. Absolute Absolute Theil's Variable Measure Error Error Error Error Correlation U
MILK 25.8975 100 mil. .8380 4.377 1. 503 6.075 .978 .067 lbs.
COWS 19.3300 10,000 -.2427 -1. 466 .915 4.718 .917 .055 head
CONC 9.1683 100,000,000 -.4031 -5.949 .827 9.392 .971 .099 lbs
HAY 9.6385 100,000,000 -.3750 -3.276 .913 9.516 .539 .105 lbs
SILAGE 15.7647 100,000,000 -.0705 -0.182 .685 4.539 .977 .051 lbs
LABOR 9.5672 1 mil. hrs. -.3867 -2.037 .846 8.753 .975 .096
CAPITAL 12.1165 index a -.3585 -2.018 .961 7.750 .726 .097
aAnnualized expenditure index on capital items, excluding land and buildings, adjusted for price level to 1977 base year, expressed relative to 1977 expenditure levels with 1977 = 10.
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