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Estimation in markets with unobservedprice barriers: an application to theCalifornia retail milk marketRobert G. Chambers a , Richard E. Just a & L. Joe Moffitt aa Department of Agricultural and Resource Economics , Universityof Maryland , Maryland, 20742, USACollege ParkPublished online: 24 May 2006.

To cite this article: Robert G. Chambers , Richard E. Just & L. Joe Moffitt (1985) Estimation inmarkets with unobserved price barriers: an application to the California retail milk market, AppliedEconomics, 17:6, 991-1002, DOI: 10.1080/00036848500000063

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Applied Economics, 1985, 17, 99 1 - 1002

Estimation in markets with unobserved price barriers: an application to the California retail milk market

R O B E R T G. C H A M B E R S , R I C H A R D E. J U S T and L. J O E M O F F I T T

Department of Agricultural and Resource Economics, L:nicersiry of Maryland, College Park, 20742. Maryland, USA

I. I N T R O D U C T I O N

Following the seminal work of Fair and Jaffee (1972). econometric methods for disequilibrium markets have been extended in recent papers by Maddala and Nelson (1974). Hartley (1976). Goldfeld and Quandt (1972). Gourieroux et al. (1980) and Ito (1980). For the most part attention has been restricted to fixed price models and endogenous price formulations characterized by insufficient equilibrium for market clearing. Recently several studies have developed procedures for dealing with markets characterized by the existence of price barriers (Maddala (1 983). Chambers et al. (1980) and Quandt (1 98 1)). In these cases. competitive market adjustment mechanisms are sometimes fully operative but are also often bounded by an accumulation point in the form of one-sided price barriers. Since price barriers are usually set administratively, there is no a priori reason to expect correspondence to market clearing or equilibrium prices. Hence, disequilibria may or may not occur in markets where competitive adjustments are constrained by the existence of price barriers. Specifically, excess supply may be anticipated in a competitive market if a minimum price is established which is higher than the market clearing price.

Because the event of equilibrium is random with regulated price bounds (occurring with a probability that is neither zero nor one), previous disequilibrium estimators are generally not applicable. That is, in cases with one-sided price barriers, the probability of market clearing is neither zero nor one. Disequilibrium econometrics assumes markets clear with probability zero while conventional econometrics models assume markets clear with probability one. Since the assumed price-quantity couples in each model are different, different econometric approaches are necessary for each case.

This paper develops and demonstrates the asymptotic properties of a limited information estimation technique that can be applied to a variety of situations where price barriers exist. The proposed estimator is particularly valuable in those instances where the existing price barriers are not observable from collected data.

After the theoretical findings are presented, the limited information estimator is applied in an

0003-6846185 503.00 + . l 2 0 1985 Chapman and Hall Ltd. 99 1

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992 Robert G . Chambers, Ricl~ard E. Just and L. Joe Mofi t t

analysis of the California retail milk market which has been characterized historically by minimum price regulation. The problems presented by examination of the California retail milk market were the original motivation for the theoretical developments included below and emphasis is thus given in the theoretical development to the peculiarities of that market - namely the unobservability of a minimum price even though the analysis can be easily extended to a maximum price or maximum and minimum price regulation scheme.

11. T H E O R E T I C A L M O D E L

Assume that demand and supply relationships are representable as

where D, is demand, P, is price prevailing in the market. X, is a vector o r predetermined variables. S, is supply and U , (i = 1 , 2) is a serially independent, random error term (all at time 1 ) .

E(Ui , ) = 0. Traditional equilibrium econometrics assumes that quantity transacted ( Q , ) in the market can

be represented as

while disequilibrium econometric models usually assume

Q, = min (D , , S,). (3)

Suppose, however. that the minimum price depends linearly on a vector of predetermined l variables:

P{ = i.'Z, + U , , (4)

where P{ is the minimum price at time I. Z, is a vector ofassociated predetermined or exogenous variables, and U , , is a random error term. For example. regulated price supports in agricultural markets are often adjusted approximately in accordance with production costs.' Substituting Equation 2 into Equation 1 and solving the resulting expressions obtains the reduced form equations corresponding to market equilibrium:

Q : = A)QX, + V l r ;

P,? = n>X, + V,,;

' Another possible specification could incorporate a rational expectations approach to price regulation. For example. specifying

Pi' =/(S:, D:) where S:and D: are expected supply and demand, respectively, suggests that the administrative agency could be trying to balance supply and demand in a longer-run context. Such a specification may be particularly appropriate for dynamic problems where supply and demand depend on expected price as well as P!.

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Esrintatiun in naurkels ~ . i i l t unubserced price barriers 993

tvhere Q* and P* correspond to the market clearing pricequantity couple. On'the other hand, if P: P / then P, (the prevailing market price) is given by Equation 4 and Q, = a , P;' + a'X, + I; ,,.l Thus, in general

Q,' if P{ P: P / i f e/ 2 P: Q , = { D, otherwise P , = { P: otherwise.

111. T H E L I M I T E D I N F O R M A T I O N A P P R O A C H

There are instances in which the minimum price is not directly observed (the California retail milk market considered later provides an example). Hence, the sample cannot be generally divided a priori into the periods of equilibrium and periods of disrquilibrium. In this section, a limited information approach to estimating the parameters of the demand and supply model detailed above is outlined. The approach uses limited information in the sense that i t only relies on the unconditional marginal density for observed transactions and not on the joint density for quantity transacted and observed market price.

The conditional density for Q,, given that Q, = Q:, can be written as

Note that the subscript of f in each case throughout this paper denotes the variable for which i t is the probability density function. This expression can also be retvritten as

where 4, = Vz, - U,, and B, = y'Z, + X;np. Expression 7 in turn is equivalent to

Using a similar argument establishes that the conditional density of Q,. given Q, = D,, is

Thus, the unconditional density of Q,, i.e. the sum of the conditional densities weishted by their probability of occurrence, can be written as

If the vector of parameters is denoted as 8, then the log-likelihood function for the parameter

'This model assumes that producers know P { . When P{ 3 P: the quantity transacted equals market demand since thecommodity is in excess supply. Excess supply is calculable by evaluating S, -D, at P / (not P:). Observable excess supply could provide additional information for estimation.

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994 Robert G. Chambers, Richard E. Just and L. Joe Mofitt

vector can be written as

IV. A S Y M P T O T I C P R O P E R T I E S OF T H E L I M I T E D I N F O R M A T I O N M A X I M U M L I K E L I H O O D E S T I M A T O R

To consider maximization of the likelihood function in Equation 10 requires some distri- butional assumptions about the error structure underlying the demand and supply relationships.

Assumption 1. The structural error terms, U,(i = 1,2,3), are serially independent, normally distributed with zero mean and a contemporaneous covariance structure given by

which induces the covariance matrix for Viz (i = 1, 2).

Remark 1. Under Assumption 1. the limited information likelihood functions, as in many previous disequilibrium and switching regression-studies, is unbounded over the parameter space. This necessitates a further but relatively harmless restriction on the parameter space.

Assumption 2. The parameter space O is a compact subset of E" which does not contain the region U: = 0 (i = 1.2). The true parameter vector (00) is an interior point in O.

Assumption 3. The regressors X , (i = 1.2, . . . , n) are each i.i.d. bounded random variables with plim(l/T)X;U, = 0.

Assumption 4. The elements of 8 are functionally independent with or without the involvement of the regressors. The vectors a and B are defined such that demand and supply each have at least one predetermined variable that is specific to it. There is at least one observation from the equilibrium regime.

Remark 2. Assumptions 1 to 4 are quite similar to those introduced by Hartley and Mallela in their analysis of the asymptotic properties of the standard disequilibrium econometric model. Assumptions 2 and 3 are necessitated by the unboundedness of the likelihood function. Assumption 2 further ensures that both demand and supply always possess a random component, while Assumption 4 insures identifiability of the parameters. Also, note that for the case where the market clears, the conditions of Assumption 4 are sufficient to satisfy the usual rank and order conditions for identifiability in simultaneous equation models.

Under these assumptions, the following result can be established. ,. Theorem. If U, is the true parameter point, there exists a unique, consistent estimator U

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Esrimorion ir t murkrls bcirh trnobserred price barriers

which corresponds to a solution of the likelihood equations given by

Furthermore. ,/T(O^ -0,) is asymptotically normally distributed with mean zero and covariance matrix given by - ,M - ' (U,) where M = E[S2 In L, (O)/L'U SO']. Proof. The proof is similar to that of Kiefer in his analysis of the switching regression model

and relies on verifying the multivariate Cramer conditions for aiymptotic normality and consistency (see Chanda, 1954). The conditions are:

(a) For the likelihood function given by = f(Yj, U) wheref(X U) is a probability density function for Y with parameter vector 8, and the 1; are independent observations on Y, the derivatives

S l n j S21n f and -- Z3 In/ i , j , k = l , 2 . . . . , n

?Ui ' SOi 28, ' ?Ui ?Uj ?Bk

exist for all O,E O and almost all X (b) For almost all Y and all 8 E O

where H is such that J Hi,,( Y) f ( Y. U) d Y C j B m, and F,( Y) and F,,( Y) are bounded for - 4

all i, j, k. (C) For all B E O the matrix,

Slnf dlnf ' = [;a[x] [=]fdY

is positive definite.

Since a complete verification of these conditions would require a prohibitive amount of space, a sketch of the proof is offered (a more detailed proof is available upon request from the authors). Condition (a) is easily verified by direct computation of the derivatives involved (general expressions for these derivatives are presented in the Appendix). For the normal distribution, all will exist. Condition (c) in turn is guaranteed by Assumptions 3 and 4. Condition (b), however, is much more difficult to demonstrate and in fact requires bounding all the derivatives involved for extreme values of Q, and P, (when jointly dependent). Direct, though laborious, computation reveals that the largest of the first derivatives is on the order of M2-exp(- M)', M = P. Q; the largest of the second derivatives is on the order of M4.exp ( - M)'; and the largest of the third derivatives is on the order of M6 eexp (-M)'. For very large values of M, each of these will be bounded since there will always exist an arbitrarily large, though finite, W such that W 2 M6.exp(- M)'.

Remark 3. The above theorem demonstrates that, given an appropriate compact parameter

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996 Robert G. Chambers, Richard E. Just and L. Joe Mofitt

space, maximum likelihood estimation will yield consistent and asymptotically normal estimates of the structural parameters. In general, however, the conditions of the theorem also provide for consistent estimation of the structural parameters via indirect maximum likelihood methods. That is, the likelihood function also could be maximized in terms of (nband n',); under the identifiability assumption, consistent estimates of /ll and jl' could then be obtained by algebraic manipulation.

V. A L T E R N A T I V E M O D E L S A N D E S T I M A T I O N P R O C E D U R E S

To this point the analysis has centred on the case where sample separation is not known and only limited information procedures are considered. This is because the empirical analysis that concludes this paper only relies on these developments. However, other models of bounded price variation have been developed (see Chambers et al. (1980) for a more detailed discussion).

For the model above, a straightforward extension of previous arguments establishes that the full information log-likelihood function is

L:(u) = f: ~n [l: ID,.+,. P! (Q,. P,. 41)ddt + /c?:. P:. ~ ( Q t v PI- o I ) ~ ~ I ] .

1'1 L

For reasons similar to those established for L,(O), this likelihood function will be unbounded over the parameter space under the assumption of normal errors. However, the above proofcan* be extended in a straightforward manner to this case as well.

Thus far, it has been implicitly assumed that information does not exist which would allow the researcher to separate the sample into equilibrium and disequilibrium observations (i.e. P{ is not directly observed). In many instances, however. Pi/ is nonstochastic and both Pi/ and P, will be directly observab1e;and if P;' < P,, then the market must beclearing. Otherwise, the market will not necessarily clear. For this situation. the appropriate limited information log-likelihood function is

L2(U) = X 1. [f:zfD,. (Q,. v:.)d v,, b. ,,,, (Q,. v,,,d v,. 1€7,

where A, = P{-X;np, j', = { I ; D, # S1), and y2 = {I; DI = SI}. This likelihood function is bounded over the parameter space if normality is assumed. Hence, maximization should present no theoretical problems and, under appropriate regularity conditions, the maximum likelihood estimator is consistent and asymptotically normal.

Full information maximum likelihood estimation of this model with sample separation has been considered in recent work by Maddala (1983) and Chambers er al. (1980). In essence, estimation boils down to the maximization of a Tobit-like full information likelihood function with tractable regularity problems. The Heckman sample selection procedure can then be adopted to attain a consistent limited information estimator of the model above when sample separation is available (see Chambers et al. or Maddala). On a practical level, therefore, either Equation 12 or the Heckman (1976) procedure can be used to generate a first round consistent estimator to be used as the startingvalue for the linearized full information maximum likelihood

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Eslin~cltion in markets ill^ unobserved price barriers 997

approach. One Newton step will then be sutlicient to achieve results asymptotically equivalent to the full information maximum likelihood estimator (Rothenberg and Leenders, 1964).

Finally, consider the case where because of, say, data problems as described in the introduction, P, is not directly observable. In this situation, full information methods based on observed price are not likely to be available. Hence, the limited information approach suggested above will be particularly attractive. A straightforward extension of previous arguments indicates that the appropriate likelihood function is

Unfortunately, this likelihood function is unbounded over the parameter space.

l VI. AN A P P L I C A T I O N O F T H E M I N I M U M P R I C E M O D E L l I N T H E F L U I D M I L K MARKET

With approximately 95", of all milk sold in the United States meeting fluid3 use standards 1 priced under state andlor federal marketing orders. fluid milk markets are subject to more

governmental regulation than most other markets in the United States. This type and degree of regulation is seldom considered in simultaneous-equation econometric analyses of demand

I and supply in milk markets (Wilson and Thompson, 1967; Pa to , 1973). although the potential 1 importance of the issue is often acknowledged. In particular, the California retail milk market

has been regulated perhaps morethan any other. The principal features of this milk stabilization plan are briefly described below before presenting estimates of a limited information, inobservable minimum price model of this market.

California wholesale and retail milk trade has been regulated since the 1930s.' The stabilization plan included regulation of both wholesale and retail milk prices according to milk quality standards and allocation of production rights or quota to producers as well. Although the provisions of this complex marketing programme have evolved since initiated, the programme determined the minimum price for fluid dairy products at the retail level prior to 1977.

The minimum retail price of fluid milk in California was established administratively several times a year after detailed periodiccost analyses and public hearings. In addition, the minimum price was determined separately for each region in the state. Thus, milk prices could vary by region with the minimum price being the effective price in some regions but lying below the market-determined equilibrium price in other regions.

Given data limitations, this latter feature of the marketing programme contributes

/ substantially to the econometric problems associated with estimating demand and supply in this market. Sample separation of observed quantity into demand and equilibrium regimes is

'Fluid milk is produced under more rigid sanitary standards than milk produced for manufacturing uses. 'An historical account of milk regulation in California is contained in California Department of Food and Agriculture (1974).

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998 Robert G. Chambers, Richurd E. Just and L. Joe Mofitt

precluded since it is generally impossible to say whether the administered minimum price acted as a constraint in the market when one possesses only much more highly aggregated (e.g. statewide) prices. However, it is possible to conceptualize an overall minimum price which is related stochastically to overall production cost (Equation 4) since by law the minimum retail price (as described above) was set after periodic cost of production analysis.

The final model for demand and supply in the California retail milk market estimated using the limited information estimation technique developed above is:

P f = -0.128Pc; and

(0.176) .

a: = 3 a: = 0.559 a: = 0.985

(0.384) (0.438) where

D = fluid milk demanded (tens of millions of gallons) P, = fluid milk price deflated by the consumer price index (dollars per-gallon) C = cereal price deflated by the consumer price index (tens of cents per 12 ounces) X = school lunch dummy variable (one if school is in session; zero, otherwise) S = fluid milk supplied (tens of millions of gallons) P, = fluid milk production cost deflated by the consumer price index.

To estimate this model, 30 monthly observations from mid-1974 through 1976 were used. All price data were obtained from the US Bureau of Labor Statistics. The Californiaconsumer price index was taken from the California Department of Finance Sales and production quantities were obtained from the California Crop and Livestock Reporting Service. Asymptotic standard error estimates (reported in parentheses) were estimated from the Hessian of the log-likelihood function. To accommodate the unboundedness of the likelihood function, a search procedure was undertaken on the variance in the demand equation; i.e. a: was held constant at alternative values while optimization was carried out with respect to the other parameters. The reported estimates reflect convergence at the 0.01 level on components of the gradient for these parameters.

By way of comparison with results reported by other investigators. the own-price elasticity estimates implied by the coefficient estimates do not appear unreasonable in magnitude. Both demand and supply were estimated to be highly price inelastic with elasticities of 0.107 and 0.085. respectively (evaluated at the mean). While the estimated demand elasticity is quite consistent with previous work, estimated supply response is somewhat more inelastic than in other studies.

Estimated coeficients on explanatory variables in the demand equation suggest the following. First, the positive coefficient attached to the school-in-session indicator variable suggests, not

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Estinlarion in markets with unobserved price barriers 999

, surprisingly, that the school lunch programme in California stimulates fluid milk demand. Additionally, a complementary relationship between cereal and milk is suggested by the positive estimated coelficient on cereal price. Cost of production. as expected. was found to have a

~ negative impact on fluid milk supply. The price of manufacturing-grade milk represented here I by the price of butter, was also found to have a negative impact on quantity supplied to the fluid l milk market. And this makes good economic sense given the structure of the California dairy

industry where most milk produced (even that ultimately devoted to manufacturing-grade uses)

I is capable of meeting fluid milk standards. Hence, increases in the prices of products using manufacturing grade milk will divert some fluid milk from the retail milk market into the production of commodities like butter and cottage cheese. Any changes in the margin between the fluid milk price and the manufacturing-grade price should encourage processors to take more of the retail grade fluid milk, especially when the minimum retail price is in effect and there is excess supply in the fluid milk market.

Theestimated coefficient on production cost in the minimum-price equation is of the opposite sign from what was originally expected given the minimum-price adjustment mechanism operative during the sample period. At first glance, one might, therefore, suspect that minimum prices were not adjusted as prescribed by California legislation but in fact were raised even in the face of cost decreases thus ensuring increased profits to fluid milk producers and processors. While such an interpretation cannot be ruled out conclusively, there are some other plausible explanations for the sign of the estimated coefficient. First, the lack of statistical precision suggests that sampling variation may have been inadequate to provide a sound enough basis for accurate point estimation of this parameter in such a complicated nonlinear estimation procedure. Moreover, the lack of statistical precision, in fact, raises the possibility that the true parameter is positive since the asymptotic confidence interval at reasonable levels of confidence includes positive numbers. Perhaps more obvious, however, is the very real possibility that this minimum price specification is just too simple to describe the actual price-setting process that takes place periodically. Any process that involves periodic cost analyses and public hearings may be so complex as to escape accurate modelling without a host of information on pertinent socioeconomic variables (such as milk lobbying expenditures) which were simply not available when the study was conducted. Thus, our results in this portion of the model must remain preliminary.

VII. C O N C L U S I O N S

Estimation in markets subject to one-sided and unobserved price regulation has been investigated in this paper. A limited information maximum likelihood approach has been suggested and the asymptotic properties of this estimator have been examined. Under fairly

1 weak regularity conditions, the limited-information, maximum likelihood estimator is con- sistent and asymptotically normal. Finally, an application of the model to estimate demand and

1 supply in the California retail milk market was presented. It seems that models such as the unobserved, price barrier case developed in this paper should

find many applications in economic analysis because of the increasing tendency of governments to regulate prices on an intermittent basis in recent decades. In particular, the nature of this price

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1000 Robert G. Chambers, Richard E. Just and L. Joe Mofitt

regulation; i.e. regulation based on hard-to-obtain or confidential firm-specific factors, suggests that the model developed here is of much more practical importance than alternative price barrier formulations.

APPENDIX

In this appendix, expressions for the first three derivatives of the log-likelihood function in Equation 10 are derived. These expressions are the basis for establishing that under the stated assumptions the likelihood function satisfies the multivariate Cramer conditions.

Note that

Cln h,,(O) - 1 ?hl, --- (20,. / I , , ce, *

c"ln hl,(0) - 1 ?'h,, 1 ?hlf ?hll _ - - - - - i-Oi CO, h , , CO, COj h:, COi (20,

and C n h I ( 0 ) 1 Z3h,

=- 1 h h , 1 ?'hl, ?h,, ------ -- ? ? h,, 80, ?Oj?dk h:, COi ?dj ?Bk h:, cOj?Ok (1.0,'

1 S2hl, ?hl, 2 ?hl, ?h,, Ch,, ---- + ---- h;, ? O j t O k CO, h;, COi a, ?Ok *

Now under Assumption 1 , further computation reveals that -- ?hl f CZl CfD, C = ~ ; f ~ Q.

7 = f D , . 4 ( E l ) ~ +F @(zl ) - fQ; .d (Ez) ?+ [ l -@(E2)] .A COi . CO, coi t Ui (1.4

where fD, = [Znu:]-'. ' exp { - (20:)-' (Q, - a,P, - z'x,)'}

l ! ' {- [, a:.; +@:U: 1- l . ( - (21 -P , I2 - Q, -

a1 -B1

(~ l -Pl )y 'ZI+(a ' -B?X,+Q,-z lP , -z 'X, = l ( =

[ ( z l - /3])'u: + 0: ] l i Z

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I Eslimation in markets will1 unobserved price barriers l

and r x

@(S) = J [ 2 n ] 1 1 2 exp[-f C'] dS - ,%

l Furthermore.

and ;'h,, ( 8 ) d 3 t l S2P1 i7E, d2E1 ?Pl SZP1 ?Pl

-----p---

S O , S O , ~ O ~ = aei 28, sg as, 20, ztl, 20,21), ?Bi SO, i.6, ?dj

l R E F E R E N C E S '

California Bureau of Milk Stabilization. Standard Milk Produoion Costs. Sacnmento. Various issues. California Crop and Livestock Reporting Service, California Dairy Industry Statistics, Sacramento, various

issues. California Department of Finance. Calijornia Economic Indicators. Sacramento, various issues. California Department of Food and Agriculture (1974) The Calijornia ;Lfilk hfarketing Program, A Special

Report to the Senate Committee on Agriculture and Water Resources, Sacramento. Chambers. R. G.. Just, R. E. and Moffitt, L. J. (1980) Estimation Methods for Markets Subject to

Threshold Price Regulation, Working Paper, Department of Agricultural and Resource Economics, University of California.

Chanda, K. D. (1954) A note on the consistency and maxima of the roots of likelihood equations, Biometrika 4 1, 54-6 1.

Fair, R. C. and Jaffee. D. M. (1975) Methods of estimation for markets in disequilibrium, Econometrica, 40, 1 497-5 14. Goldfeld, S. E. and Quandt, R. E. (1972) Estimation in a disequilibrium model and the value of

information, Journal of Econometrics, 3, 32548. Gourieroux. D.. Laffont, J. J. and Monfort, A. (1980) Disequilibrium econometrics in simultaneous

equation systems. Econometrica. 48, 75-96. Hartley, M. J. (1976) The estimation of markets in disequilibrium: the fixed supply case, International

Economic Review, 17, 687-99. Heckman. J. J. (1976) The common structure of statistical models of truncation, sample selection, and

limited dependent variables, Annals of Economic and Social Measurement, 5, 475-91. Ito, T. (1980) Methods of estimation for multi-market disequilibrium models. Econometrica. 48.97-126.

i tvladdala, G. S. (1983) Limited-dependent and Qualitaticr Variables in Econometrics, Cambridge University Press. New York.

Maddala, G. S. and Nelson, F. (1974) Maximum likelihood methods for models of markets and disequilibrium, Econometrica, 41, 101 3-30.

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1002 Robert G. Chambers, Richard E. Just and L. Joe Mofitt

Prato, A. A. (1973) Milk demand. supply, and price relationships. 1950-1968, American Journal of Agricultural Economics. 55, 21 7-22.

Quandt, R. E. (1980) Equilibrium and disequilibrium: Transitional Models, Research Memorandum 269, Princeton University.

Rothenberg, T. 1. and Leenders, C. T. (1964) Efficient estimation of simultaneous equation systems, Econometrica, 32, 57-76.

U.S. Bureau of Labor Statistics, Estimated Retail Food Prices by Cities, US Government Printing Office, Washington, DC, various issues.

Wilson, R. R. and Thompson, R. G. (1967) Demand, supply, and price relationships for the dairy sector, post-World War I1 period, Journal of Farm Economics, 49. 360-71.

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