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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

- 2 -

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,

\

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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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