Neoclassical econometrics : non-negativity constraints

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The paper considers the problem of determining the parameters in equality-constrained neoclassical economic models in which the decision variables are also constrained to be non-negative. Such problems frequently arise in models of household and farmer behavior. The household maximizes utility subject to income and/or time constraints and the fact that its demand for various goods and services and its supply to various segments of the lab market are non-negative. The profit maximizing farmer allocates a given amount of land over a set of possible crops, where the land allocation, input demand, and output supply associated with each crop are non-negative. The paper formulates a canonical form of this model and discusses the nature of the inverse (neoclassical econometric) problem. The traditional econometric approach is generalized to a multi-decision situation and the computational difficulties and inherent paradigmatic limitations are discussed. A simple alternative deterministic neoclassical econometric approach, avoiding such problems, is proposed and a simple algorithm is discussed. Finally, extensions of the model and topics for future research are considered.
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DISCUSSION PAPER DRD137 ~rEOCl...\SSICALECONOMETRICS: NON-NEGATIVITY CONSTRAINTS by Michael J. Hartley October 1985 . \ Development Research Department Economics and Research Staff World Bank The World Bank does not accept responsibility for the v1ews expressed herein which are those of the author(s) and should not be attrlbuted to the World Bank or to its affiliated organizations. The findings,~interpretations) and conc~~sions are the results of research supported by the Bank; they do not necesiarily represent official policy of the Bank. The;Pesignations employed, the ~esentation of material, and any maps used in this-document are solely for the convenience of the reader and do not imply the expression of any opinion whatsoever on the part of the World Bank or its affiliates conc(.rning the legal status of any co~ntry, territory, city, area, or of its authorities, or concerning the delimitations of its boundaries, or national affiliation. NEOCLASSICAL ECONOMETRICS: NON-NEGATIVITY CONSTRAINTS by MichaP..1. J .. H4Tthy* Development Research Department World Bank** October, 1985 * .If This paper is based on portions of Hartley (1981a, 1981b, 1983i). The author is indebted co Arne Drud for helpfuL discussions. ** The WorLd BaRk does not accept responsibiLity for the author's ~ie~s. These shouLd noc be attributed to the WorLd Bank or les affiliated organizations. The findings, interpretations and concLusions are the results of research supporceJ ~y the Bank; but do not necessariLy represent its officiaL poLicy. Abstract The paper considers the problem of determining the parameters in equality-constrained neoclassical economic models in which the decision variables are also constrained to be non-negative. Such problems frequently arise in models of household and farmer behavior. The household maximizes utility subject to income and/or time constraints and the fact that its de~ for various goods and services and its supply to various segments of the lab market are non-negative. The profit maximizing farmer allocates a given amount of land over a set of possible crops, where the' land allocation, inp demand and output supply associated with each .:;;:op are. T,ll)n ;--:.a~ ltive. The paper formulates a canonical form of this model and discusses the nature of the inverse (neoclassical econometric) problem. The traditional (regressi{}t based) econometric approach is generalized to a multi-decision situation and ~he computational difficulties and inherent paradigmatic limitations are discussed. A simple alternative deterministic neoclassical econometric approach, ~voiding such problems, is proposed and a simple algorithm is discussed. Finally, extensions of the model and topics for future research are considered .! .. .. 11 Table of Contents I. Introduction ................................................... 1 II. The Standard Neoclassical Model with ................ ...... 3 Non-Negativity Constraints A. The Maintained Hypothesis (Rationality) 3 .................................. B. Two Simpl~ Illustrations 4 1. The Representative Household 4 2. The Competitive Farm 5 C. Nonlinear/Dynamic Programming ........................... 7 D. The Inverse Problem ~~ 8 III. The Traditional (Regression-Based) Approach ................... 9 A. General Considerations ................................. 9 B. The Representative Household 12 l. The Two-Good Case 12 2. The Three-Good Case .................. 15 c. Generalizat ion ........................................... 19 IV. A Neoclassical Econometric Approach 20 A. A Deterministic Model ................................... 21 8. A Simple Algorithm .. ................................... 22 C. ,Di seus s ion................................................. . 2S v. Conclusions, Extensions and Future Research 27 A. Conclusions . ,. .................................. -. ...... 27 B. Extensions and Future Research 29 1/ I would ~ike to thank ~iriam Bailey for her excellent ~ord-processlng. I. Introduction Neoclassical micro-economic theory is based upon the maintained hypothesis that individual decision-making units households, firm~, etc. -- maximize a specified objp.ctive function or goal, g, with respect to a decision or policy vector, ~, given the value of a state vector, !, and, 1n a general equilibrium model, subject to the fact that their decisions are not independent of those of their neighbors. In many such applications, the decision vectors are, by definition, non-negative. ~or example, in household expendi ture ;lurveys wi th even modet 8.t:a disaggregat ion, thl'! ;A:~N(rseigator wi 11 typicalty encounter numerous instances of individual units spen.thng ~ income on one or more categories of goods and serV1ces over the duration of the study. Similar situations obtain, for example, in the allocation of land, input-demand and output-supply over all crops in surveys of agricultur:l producers. We shall utilize these two examples in developing general econometric methods for such allocation problems involving non-negativity constraints on the domain of the decision-variables. Such problems have come to be termed multivariate Tobit-type models (following the s~minal work of Tobin (1958 . ' or, more generally, Limited Dependent Vari~ble (LDV) econome~ric models (see, e.g., Amemiya (1984) or :iaddala (1983)). These m.odels ha.ve their origins tn ! the linear/nonlinear regression model,combined with the problem of estimating the parameters of a It~cnsoredtl probabi!;ity density' function (p.d.f.) in statistical theory. S'..:r::1 methods will be termed the traditional (regresslon- based) approach. In section [I we shall begin by formulating the maintained (neoclassical) hypothesis governing the behavior of the individual economic u~it. We then illust ta by exhibiting two standard examples -- the utility- maximizing household, sUbject to an income constraint; and the profit- maximizing farmer, subject to the constraints of a given amount of land and a given technology associated with each possible crop or 1and use. We then turn to a brief revie~'of available nonlinear and dynamic programming algorithms to compute the optimal decision-values in the situation where the parameters are known. ~inally, we formulate the nature of the inverse problem confronting t~e investigator. In the next section, we consider application of the traditional (regression-based) approach to these problems. The essence of this stochastic model involves introduction of a vector of (normal) random errors, added to the optimal (unconstrained) decision functions. If infeasible, t~ese must then be mapped back onto the boundary of the feasible decision space. We formulate the I'closest boundary value rule to define a unique mapping. Apart from the implausibility of the underlying paradigm, implementation of this approach quickly breaks down in problems of moderate dimension, as the order of multiple numerical integrations required to estimate the true parameters soon exceed~ present computer capabilities. We illustrate these difficulties 1n terms of our leading example. In section IV we intrOduce an alternative neoclassical econometric app~oach to this class of problems. We consider a deterministic model, and formulate the inverse problem. A simple algorithm is proposed to calibrate the unknown par:meters. Some concludi~~ ~emarks and a discussion of possible extensions and future research are reser~ed for section V. II. The Standard Neoclassical ~odel A. The ~aintained Hypothesis We assume (here) that individual economic units have cumplete knowledge of the Sxl state vector, !; the Pxl parameter vector, ~; the objective function or goal, g; and the Cxl equality-constraint vector function, c. Each unit (hen chooses a Dxl decision vector, ~, to maximize the objective function, subject to the C D functionally-independent equality constraints, (2.2) and the 0 non-negativity constraints, d 0 (2.3) We assume that g 1S quasi-convex 1n d Ear all values In some ~eglon. 0 0 DO, including the region, did 0 Ear a suitably largeconst~nt 0 0 .! containing the non-negative o=thant, R~+; !I thdt the vector, !, belong9 to the state spac~, So; and that the parameter vector, ~, lS restricted to a com~act parameter space, Further we assume that the functions, g and ~, are twice continuously differentiable in ~ and 0 21 9 over DOx Soy' 0 B. Two Simole Illustrations 1. The Representative Household: We consider (here) only the income-allocation aspects of the 31 representative househol.d.- Each economic unit mAximizes the utility function, (2.4) subject to a budget constraint, (2)' (3) s d - s = 0 , (2.5) II ~or example, if g denotes a utility function, and we wish to model cases where non-n~gativi~y const~aints on ~ may be binding, then g must be defined ~ve~ ne3acive values of ce~tain elemencs of~. This would appea~ to c~le out~fo~ p~eser.t pu~?oses such specifications as the di~ect t~ans LfJg ucility. fur.cc:''J:'. (Ch~istenson, Jo~genson, Lau ~1 J7::', involving oni.y t.he Loga~it~s ot e~ements in ~ -- see seccion IV. 21 Some of these clajs:'cal assumptions, howeve~, aLe not ~equi~ed 1n ce~tac~ of the methods lee below. 3/ Gene~alization of the model co ir.co~?o~ace, say, time-allocation, involving a non-ne~aeive labo~ fo~ce pa~eicipation/labo~ suppLy decision by household ;nemben -- ;:e~o, if noei posieive, if labo~ is supplied -- cs st~aightfo~wa~d, gee Ha~eley (L985b). and the restriction that all demands are non-negative, d 0 (2.6) where 11 g is any well-behaved utility function, d = demand vector for goods and services, s (1) = household characteristics vector, s(2) = price vef;tor fot goods and serVices (also E) , g(3) = income (also y). Thus, here we have the state vector, (1)' (2)'(3) , s _ [.! .! s 1 (2.7) 2. The Competitive ~arm: We assume the farmer operates a holding of a given cultivable area, on which the land must be allocated among K possible crops -- each of which has a known techmllogy. Each unic maximizes profi.ts, g = (2.8 ) subject to the K constraints of a given technology Eor each crop, o , (2.9) 11 Alternative notation LS incroduced for subsequent purposes. with k =1, ,K; the land-allocation restriction, k I (2.10) k=l and the 0 overall non-negativity constraints, (Z.ll) g(1):;- (d(l (1) (1) with vee k ' s _ vee(sk ), and where del) = output of crop k k g~Z) = variable input vector for crop k, d(3) = land-allocation vector over K crops, s~l) = output price f0r crop k (2) s = variable input price vector, (3) s = fixed input prlce vector (4) !k = fixed input v~ctor for crop k, 3(5) = holding/farmer characteristics vector, (6) s = area of holdin~ 11 1n a vector.:-:- 1/ A dynamic multi-pertod '/erSLon of chis ~odel. applicable co mixed hoLdin~~ involving both annuals ~nd perennials and utilizing a ~arkovlan land- rotation system, is ~iven in Bellman and Hartley (l985). C. Nonlineat /Dynamic Pro~;:ammin.3. In the event that an Itinterior solutionlt obtains to the canonical form of the problem under the maintained hypothesis, then classical methods may be employed to f lnd the so l' .- ' utton, * 11 ~ However, in many micro-economi~ applications, one or more of the inequality constraints will be binding; and, for a given value of 9 and given Itwell-behavedll specifications of the functions, g and ~,optimal feasible values for d* can only be computed as the solution to a nonlinear and/or dynamic programming problem. This is a long-standing problem -- see, e.g., Hadley (1964) for a reView of the early literature. Various nonlinear. programming algorithms may be employed, and no single method, at present, appears dominant in ~ll applications. The reduced gradient method (Lasdon, Waren, Jain and Ratner (1978); the NAg Libraries software package, SALQDR;~I and the SOL/NPSOL Fortran package, involving solving a sequence of quadratic programming problems (Gill, Murray and Wright (1981); Gill, Murray, Saunders and Wrlght (1983), etc.) are widely recommended. See also Drud (1985). Finally, as we subsequently note, many of the allocation-cype proble~s in mathematical economics with non-negativity constraints can be formulated as dynamic programming problems (see, e.g., Bellman (1957, 1963' 11 See, e.g., Hartley (l985b) for a parallel discussion of the case where inequality constraints in (2.3) are removed or are unnecessary (Christenson, Jor~er.son. Lau (1975)). ? I ~I AvaiLable from NAg Libraries, 1250 Grace Court, Downers Grove, [llinois. USA. and Bellman and Dreyfus (l962.1/ Such methods, in avoiding any use of the cla~sical methods of the calculus, thus permit a richer array of potentialLy more realistic maintained hypvr1leses -- a point repeatedly stressed by Bellman. This matter assumed even greater significance as we generalize the nature of the domain constraints, since the state of ccmputer technology continues t~ escalate by leaps and bounds. D. The Inverse Problem The problem confronting the investigator, however. is the inverse of that of the decision-maker under the maintained hypothesis. Suppose waare given a suitably-sized sample/set of data,~/ fed. ,s. ) :i=l, . ,N; t=l, .T} 2.l2) , -It -1.t on t~e actual decisions. ~it' taken by unit i in period t (with N,T ~ 1), wh'~ndecision-makers are confronted wi th the state of nature , s . Then -It the problem for the investigator is ~o determine the unknown parameter values, a , under parti.cular specifications of g and ~, j~hich best approximate ! .. 1/ Dynamic pr~gramming is a mathematical theory of multi-stage 'er::'ision processes, and of ~rocesses which may be viewe~ as multi-stage decision processes. One of the most important of these is an allocation process. 2/ Our choice of notation ?er~its time-series, cross-section and panel data samples. ,Also, ' ,e permit lagged dements of the decisi.on-/ector, di.r:-L' 1it-2' etc:. to be included i.n Sit' 1/ the observed data, {d. ,so } .- -u: -It III. The TraditionaL (Regression-Based) Approach A. GeneraL Considerations In terms of its historical antecedents within the statisticaL Literature, the prnbLem of estimation of the parameters of a (non-negative) censored p.d.f. apparently w~s considered first by Pearson and Lee (L908), with later important contributions by nsher (1931), HaLd (1949), and Cohen (1949, L957) -- generaLly in the context of a univariate normal distribution. In the econometric Literature, Tobin's (l958) seminal paper, extended the analysis to a censored normaL regression modeL (motivated by the probLem of modeLi~g the demand for a consumer durabLe good); and Amemiya's (1973) influential discus~ion of the asymptotic statisticaL properties in such (independent, but not identicaLly distributed) cases, spawned considerable interest among economists in so-caLl~d Lu~ ~odels, and their generalization co multi-decision economic problems
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