DISCUSSION PAPER
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ECONOMEmIC X T H O D S FOR AGRICULTURAL SUPPLY
FERTILIZER VSE AND CROP RESPONSE
by
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Xchael J. Hartley
Ecc~~cmis
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Development Research Department i
Economics and Research Staff
World aank .
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The views presented here are those o f the author, and they should not be
interpreted as reflecting those of the World Bank
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METHODS FOR AGRICULTURAL SUPPLY UNDER UNCERTAINTY:
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FERTILIZER USE AND CROP RESPONSE
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by I
Michael J. Hartley
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, E c o n o m i s t
P u b l i c E c o n o m i c s D i v i s i o n I~
D e v e i o p m e n t R e s e a r c h D e p a r t m e n t I
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V o r l d Bank
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~orthcornin~,Journal of Mathematical Analysis and Applications. July, L
The paper develops general econometric methods for the treatmeht
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of modelling decisionkaking under conditions of uncertainty and risk. - 4
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problem proposed to the author by John Holsen, Chief Economist, South ~ s j a
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Regional Office -- the problem of modelling f e r t i l i z e r use and crop respknse
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i n the context of irrigated and/or rainfed water supply when t h e i r is unier-
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; tainty and r i s k with respect t o weather conditions over the crop-cycle ahd
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the subsequent price a t marketing -- suggested the present approach. A s a
by-product, we also derive the stochastic duality theory associated w i t h the
present primal problem.
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have included government extension service advice to farmers on cultivation
practices; the introduction and distribution of High Yielding v a r i e t i e s ( W s )
to replace T ~ d i t i o u a lLocal Varieties (TLVs) of seed, etc.; the provis$on of
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public worlcs and other complementary inputs to f e r t i l i z e r (such a s i r r i g a t i o n
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and v a t e r c o n t r o l systems, pesticides, etc.); subsidies t o the output p$ice of
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crops u t i l i z i n g f e r t i l i z e r; provision of credit to farmers t o r f e r t i l i z $ r .
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purchases via government credit institutions and/or subsidies t o i n t e r e t i
rates; etc. I
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I n the case where a l l other aspects regarding the use of f e r t j l i z e r
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are l e f t t o the voluntary decisions of farmers, assessment of the costs!and
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benefits associated with any c ~ l l e c t i o nof government programs requires ;an
explicit behavioral-model of the farmer's response to the resulting 'stllucture
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'of inceatives." i~
We shall consider these issues within the context of a s t a t i c
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neoclassical mod-: of individual farmer behavior. We shall adopt the
maintained hypothesis that tho bbjective of each farm unit is to maximize the
conditional expectation of profit with respect to the levels of all v a r a b l e 4
inputs (including f e r t i l i z e r ) , subject t o the constraint of a given technology
for a given crop, and given prices 'of a l l variable inputs a d output.
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refer to t h i s a s the Standard Neoclassical b d e l (SNM) of agrLcultura1
production sod input.>emand-ncudn the derived concepts of output supply,
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cost and profit funceons, obtained a s a consequence of duality theory. I I
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Econometric'methods appropriate for the estimation of the par+eters
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,:ontained within the production function of a SNM f o r a specific crop w i l l be
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treated in section 2. We shall extend the present state-of-the- art w i t 4
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respect to t h i s problem by exhibiting econometric and computational methbds
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behaved ,"but o t h e r w i ~ cunrestricted parametric specifications of the I
production function required under any maintained hypothesis. Our apprca I
w i l l be couched entirely i n terms of the so- called "primal problem," invo ving C
a profit equation (containing the production function) the derived syItem
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of input denand functions (obtained, say, under maximization of the
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conditional expectation of profits), rather than postulating a functional form
for either the p r o f i t function or cost function, to be estimated i n
con junction vit h the derived input demand system-the so-called "dual" to the
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primal problemL/ We believe this to be a more useful approach since i
information to guide the choice of functional forms is more readily availeble
for the production function.?! Further, once th; parameters i n the produ
function of the primal problem. have been estimated, the implied profit,
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and output supply density functions can be derived and their expectat1ons;and
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other moments evaluated.
This has not been accepted current practice- except i n s p e c i a l i ~ e d 4 .
cases- because of the apparent necessity of obtaining explicit closed-fo
analytic solutions to the system of marginal productivity equations i n p l i j d by
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a par=tic~;lar choice of a production function. Tradition has required this/ i n
o r d e r that: (a) t h e d e r i v e d functinna!frln Fr)t tho I
+?ST:+ ? ~ n n n J~ n - ~ , ? : * ~ r
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(including the resulting homogeneity conditions %- prices) is consistent Qth
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#he postulated production function, and (b) t o s&cify the s e t of "cross-
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See, e.g., Sidhu and Baanante I19791 for a recent example of the use oif a
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translog profit function.
-2/ Such information is available from the agronomic l i t e r a t u r e on cultivation
practices for each crop- often obtained from experimental t r i a l s .
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equation" restrictions, functionally relating every demand equation ~ a r + e t e r
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to the set of original parameters i n the production function. By exhibi~ting
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an algorithm, i n vhich the solution values to 3well-behaved optinizat&on
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2roble.m specified i n the m+intained hypothesis automatically s a t i s f y botb (a)
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and (b), we eliminate a major constraint on the set of "econometrically
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estimable" parametric forms for a productiou function and a s ~ o c i a t e dinp t
demanZ functions. This obviates the necessity of proceeding t o the
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"econometrics of duality" approach- since the associated profit, cost an
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supply density functions can then be derived d i r e c t l y from within the
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stochastic structure of the p r i d problem and, hence, t h e i r expectation+ and
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variances can be numericallz evaluated. I
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The econometric approach of section 2 w i l l then be extended to!
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several cases which a r e of particular relevance to the analysis of f e r t i i z e r
use and crop response. In section 3 we consider the comparison of irri ed
and rainfed crops, and t r e a t the typical situation i n which the availabiiity
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of irrigation and water-management systems is a binary, exogenously-dete
event- say, as a consequence of the differential access of farmers to pu
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f a c i l i t i e s . In the situation where water-managesent f a c i l i t i e s a r e avai able,
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w e assume that t h i s permits the farmer t o control the level of soil-mois/ure
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accessible to the crop. This not only provides the farmer wfth Rn addft4onal
control variable. but also eliminate= much of the uncertainty with r'9 p e i t t o w
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unpredictable weathe; conditions. Since water is a complementary i n k t , both
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of these affect the ;sage of and response ta f e r t i l i z e r . In contrast, firmer*
without access to such f a c i l i t i e s must t r e a t water (or soil-moisture) as la
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s t a t e variable, beyond their present control. Here, farmers maximize ex ected
profits with control over one less variable input than those wfth access t o
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i r r i g a t i o n f a c i l i t i e s . This, by appeal to ,the 'theory of the second-best'," ~
c e t c r i s paribus, w i l l reduce the expected levels of p r o f i t s and, i n pract/ice,
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also f e r t i l i z e r wage. As an i l l u s t r a t i o n of these issuer, w e consider the
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irrigated case, i n which the actual levels of water a r-e not observed- thu -
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representing a l a t e n t variable input. This is contrasted with the rainfe!
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case, i n which the amount of r a i n f a l l is reported but uncontrolled.
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By
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pooling data on both types of farmers cultivating a given crop t o e s t i m a t ~the
parameters within t h e i r common production functions, we may, i n principle,
quantitatively assess the benefits of watermanagement public works projects
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and t h e i r c r o p s p e c i f i c effect on f e r t i l i z e r use and yields. I
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Our discussioi of the tainfed case i n section 3 extends naturaliy to
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the problem of decision-making under uncertainty. Thus, i n section 4, we^
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consider a general econometric approach, requiting the additional
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spettffication of the (subjective) probability density function ,f a l l s t a
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variables to which the farmer attaches uncertainty. This 'endogenizesw t4e
sources of uncertainty for all 'Ptate variables within the scope of the model
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and, by "integrating out' their combined effect, we can obtain ML estimat s of
the original (production function) parameters of interest, i n addition to the
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parameters i n the marginal distribution of these "new" l a t e n t efidogenous
variables.
There are two principal sources of uncertainty i n the. economics 1of
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crop cultivation. The f i r s t arises i n the case'of rainfed crops. Here, input
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decisions over the crop's planting t o harvest lffe-span must invariably b
taken prior to harvest. This, i n conjunction with the fact that weather is,
apart f;;m seasonal trends, a random s t a t e of nature implies that, under dhe
maintained hypothesis, f e r t i l i z e r use and crop yields w i l l , i n general, "fY
with a l l of the parameters of the r a i n f a l l distribution, and not just {he with
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expected value of r a i n f a l l , a s typical. practice suggests. To the extedt that
there a r e systematic climatic differellces a t d i f f e r e n t locations, and
racional behavior on the part of farmers, our methods permit modeling (he
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range of cross- sectional f e r t i l i z e r use and crop responses to such
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uncertainties.
h e may a l s o t r e a t uncertainty with respect t o the price of dlutput
a t harvest-time along similar lines. Indeed, it is precisely t h i s pro
which occasioned the seminal contribution of Nerlove [I9581 ro the
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on uncertain future prices, whers farmers a r e assumed to form "adaptivel
expectations" involving the notion of a "normal" (in the sense of long-run) I
formulation^
price. Tile dynamics of supply response resulting from such is
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now a common theme within the post-Nerlovian agricultural supply l i t e r a t u r e -
see, again, Askari and Cummings (19771. The adaptivcexpectatiolla hypothesis
is, however, but one of many currently popular expectations formulation I-
Other time- series specifications include the partial- adjustment model;
autoregressive integrated moving-average (ARIMA) processes f o r prices; 1
rational and so-called "pseudo" rational price expectations models; etcl-see
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Nerlove, ~ r e t h i rand Carvalho (1979, ch: XIII]. A l l such choices, hove er,
boil down to making an assumption regarding the ~ a r a m e t r i cform of the ,
'ixpectation of 'the price a t harvest- time, based upon information availa Ie to
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farmer a t the t i m e a decision regarding variable inputs must be pad?.
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&nce, by adding a normal random error component, these candidates t a l l within
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our approach t c estimation- requiring the probability density function
future prices.
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It is also important to note that, by integrating out the effe t of
a l l l a t e n t endogenous variables, ve obtain the marginal distribution of ~~
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p r o f i t s end all obsened decision variables. Once t h i s is obtained, we &an
define the conditional density of profits, given the values of a l l d e c i s o n i
variables. I
In general, &moments of t h i s conditional p.d.f. are requi ed i n
order t o completely characterize the consequences for decision making unjer I
uncertainty. There is, however, no reason t o assume, a priori, t h a t f a d e r s
are "neutral with respect to risk." To the extent that a t t i t u d e s toward r i s k
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a r e an important common behavioral characteristic, and presuming that sudh
risks vary with respect to time and place, our approach may easily be 1
generalized to consider, instead, an objective function containing the 1
conditional expectation and -conditional variance of profits a s arguments
within a specified parametric form of the fanner's u t i l i t y function. ~ h j s
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permits a direct econometric treatment of the consequences of r i s k and
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uncertainty on f e r t i l i z e r and other input use.
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We should also stress, to the extent that: (1) the standard
treatment of decision-making under tmdertainty has been largely couched
terms of theories regarding the expected values ~
.- of the uncertain variables,
and (2) economic units a r e not -indiffeient t o risk, the standard approac
the problem w i l l therefore provide an incomplete explanation. - I
In additfop,
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since, except i n highly .apecializeJ cases, the structure of a neoclassicall
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model represents a n o n l i k a r transformation of the s t a t e variables into tbe
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decision variables (incluaing output and prof it) it follows that the enptcted
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value of (say) profit can not i n general be calculated solely from infonn tion
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bated on the expectations of the uncertain s t a t e variables--as i n the cask of
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"certainty equivalence," see Theil (19611. This also suggests that theor e s
of so-called rational expectations- see, e.g., Lucas and Sargent (19811-are
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unlikely t o provide a complete treatment of behavior under uncertaint , i
Rather, we need, i n addition, rational. variance models, rational t h i r 1- moment
model., etc. This suggests a convergence t o our-approach for dealing ivith
uncertainty- in which we seek t o parametrize only the
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probability density function of the uncertain state variables, and estimate
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its parameters based only on information available to the fanner up td the
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time that decisions are required,
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2. The Standard Seoclassical Model of F e r t i l i z e r Use I
and Response for a Given Crop: I
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In t h i s section we. present the standard neoclassicai model a f
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f e r t i l i z e r use m d output response for a given crop. Kc consider the
maintained hypothesis of (expected) prof1t maxfmization for the indivqdual
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farm unit i n vhich che prices of all vatiable inputs and output a r e gdven and
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known t o the farmer with certainty at the t i m e of decision making. We shall
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consider a situation i n which the land-use decision has already been dade, 1/
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The problem, here, is t o decide upon the optinal levels of the decisiohr
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variables-a. set of variable inputs, including fertilizer-dgiven knovlCdge of
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the production technology for the crop and given knowledge 02 a l l pric
other ecate variables which serve to disting+sh the individual farm u its, n
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In subsequent sections we shall extend t h i s p d a l to incorporate khe vbrious
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other features noted i n section 1.
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decision is also made will not be treated i n the present paper and will be
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discussed i n a sequel. -
(a) Notation 1
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The general situation requires each Earm unit to choose b 'tween t
..., 04
-1, c o p U e ahall consider the (conditional) question optiaal
f e r t i l i z e r usage given that-the fasm unit, i, has decided to allcca
portion of i z s available land, aij' to the arbitrary crop j. Let i
observation index for the farm unit, 1-l,Z,. ..,N, where N denotss t?e d z e of
a random sample of such units. Let s ( i ) denote the farm unit
denote the day and ~(1) denoto the year of observation. Thus,
permite the random sample to constitute a pure crosa-section, a pure time-
series, or, more importantly, a mixed tCmeseries/crosssection data sample.
Further, the analysis can easily be restructured f o r the case of str t i f i e d C
random samples associated with each yesr of obserration, by the intrbduction
of suitable sampling weights attached t6 each sanple observatioa. I
Foz any variable, (say) z, we shall use the shorthand notation, xi,
t o refer to an o b s e r ~ a t i o non farm unit, s(i), and, where relevant t b crop j,
the notation, x etc. Further, within year ~ ( i ) and for crop, j/i),we
i j' , .
shall let tj(i) denote :he date of harvest; tj(l) - L (1)denote t h e d a t e of
j
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planting and R (1) denote the length (in days) of the cr3p cycle. Fo any
j a
observation, 1, and crop, j, we define the following variables:- 1
F = F
K, --vector of f e r t f l t z c r i q ~ v (tC~: J Y ! ~ , ~ . ? ~ :7: ~ 5 r - e -1 .
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t9
I=
element vector of N,'P,K nutrient levels),
xO = KO --vector of other variable inputs, I
-1il j ? i ,
-1/ Different varieties of a given crop, to the extent that t h e i r prdductioa
functions d i f f e r , should be viewed as different 'crops.' The nodation
here (particularly the subscript j) is also applicable to the seduel, i n
which c r o r c h o i c e vill also be modelled.
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= area of land under cultivation,
"ij
a = L vector of fixed non-land inputs and other noatprice state
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variables,
= output,
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= price of output, ~
P i j,
vector of f e r t i l i z e r prices
O = K0 vector of other variable input prices ,
lij J
Crj = t o t a l fixed costs and
= total profit .
We now turn to the Standard Neoclassical Model.
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(b) The Deterministic Standard ~ e o c l a s s i c a lModel
A deterministic conditional neoclassical model of f e r t i ' izer usage
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and output response, given that a positive area of land has been e,llocated t o
-crop I
j, assumes that farmers maximize profits with reipect to a l l variable
--
input Levels associated with the area of land, aij. t
The technolo y associated
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with crop j, here taken as invariant over the sample period, may e summal-ized
by a production function,
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where x0 d 9 is an
-ij -= I-ij
XXIJxoO] is a K 1-vector-- of variable input levels, an
-ij - -1
M vector of parameters of interest. Expsted profit associated d i t h crop j
.
j-
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may then be defined
F' 0'
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vhere q, s [q-ij q 1 is the Kj-vector of a l l variable input p r i c e s . The I
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A, -ij
maintained hypothesis is that farmers maximize the function, @ w i t h respect
j' I
t o x for given q?aluesof aij, zij, pi, and zij. %b,f
Under the customa y
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assumption of continuous d i f f e r e n t i a b i l i t y f o r f with respect to t h i s
1
leads to the familiar marginal productivity conditions,
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o r , i n the alternative o m , exhibiting the role of relative prices,
where 31, * 1
is the vector of normalized or relative prices--
9'j
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(19781, using' the crop price, pi,, a s aukeraire. Under the conditio
implicit function theorem, a solution, , for the optimal
G v e l s exists, and, at l e a s t locally, may be calculated via
L
1 *and a,quation system,
where xf denotes a K
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The major ?roblem with the generality of t h i s theory- at b a s t in
terms of current practice i n both mathematical economics and ecdnometric
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empirical work-Is that, even f o r "well-behaved" production functionb ( i .e.,
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those satisfying the typical neoclassicai second-order conditions fok a
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mm-mum for any 9 which solves (2.3b)), the implicit function theoken is
-ij I
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only an existence-theorem, and does not necessarily permit closed-fob
analytic solutions for the K input demand functions
1 --.
i?. Indeed, onby f o r
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c e r t a i n specific choices f o r the production function, 'J' is it poss ble to
analytically solve (2.3 b) uniquely for the optimal x* functions, as, )
-j ~ e.g., i n
the Cobb-Dooglas and quadratic cases. However, given a particular w 11- i
behaved functional form for f j, and given specific values for tte vafiables,
4tj, aid, E ~&~t h e, parameters, 9 a suitable algorithm, progr
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computer, can e a s i l y calculate numerical solution values f o r the opt$mal i n p u t '
levels, x* , such that I
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-ij
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Our purpose i n the remaf~derof t h i s section w i l l be t o inkroduca r
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stochnstie version of the Standard ~ e d c l n s e i c a lModel and to e*hibit
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computational ~ e t h o d sfor the calculation of maximum likelihood est3 a t e s of
the parameters of the p;oduction function contained within the prof i k equation
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and system of input demznd equations regardless of whether o r not th4 choice
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of the production f unct fon permi
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marginal productivity equations of the form (2.4). Our approach w i l d be to
employ, instead, an i t e r a t i v e procedure, such a s the avido on-~letcheJ - P ~ ~ ~ ~ I
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a
(1963, 1964, 19681 (DFP) algorithm, to solve t'.o-.(expected) p r o f i t rn ximi-
zation problem, (2.5), for the optimal x* values, given values of t
+j
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parameters, 9 and the s t a t e variables,Sj,aij and zij, associate
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each individual sample member. This set of "inner" maximization pro lems w i l l
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then be embedded within an "outer" maximization problem, i n which t h log-
likelihood function of the parameters is maximized given the data, 1
l
z i j qij. pijs aijs aj}* -and the solution values, By i terating
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between the "inner" and "outer" maximization problems such that the ikelihood
d
of the observed sample of data under the maintained hypothesis is a onotone
increasing sequence, convergence t o the maximum likelihood estimates of the
parameters can be establishod- see Hartley 11981b] l! This simple divice-
replacement of the derived analytic solution for the optimal input dmand 9
functions by an unconstrained optimization algorithm- not only penni s ti
estimation of zgpply response model? f o r 3well-behaved choice of dhe
production function (and automatically imposes a l l cross- equation ree!t r i c t i o n s
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on the parameters of the demand system), but also permits extension a rich
array of alternative maintained' hypotheses encompassing problems of r i s k and
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uncertainty.
(c) A Stochastic Standard Neoclassical Hodel:
e I t
We now turn to a stochastic specification of the Standard
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tieoclaesical Model to implement t h i s approach We shall assume that
'f
choose input levels t o maximize the condition& expectation,
=
!pr o f i t s , n , given the vector of decision var9ableas, x
ij -ij'
-1/ I
In keceral, convergence to a local maximum of the likelihood func ion is
a l l that can be guaranteed --Hartley
see [1981b]. -
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uncertainty regarding the level of p r o f i t s , xij, t h a t w i l l o b t a i n 2 I ti
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follows that the conditional p.d.f. of xij given -i j' is given by
x
whereas the marginal p.d.f. of x -4is
1
(2. lb)
Hence, the joint p.d.f. of x and x is the (K + 1)- variate normal I1
i J -iJ 1
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density,
It should be stressed that the foregoing analysis is
upon the fact that farmers have devoted a positive land area, a to crqp
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j. We therefore define the index set of relevant observt.tions, I
containiug N such o b ~ e r v aions
r 2 The conditional
1
given the decision to cultivate crop j, is thus defined 5y
. , * 1
P
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-1/ See Zellner, Kmenta and Dr'eze (19661 for a Cobb-Douglas version of t h s
specification.
-2
1 .
Observations with a = 0 should be excluded from the conditional
estimation of 9 anadz In the sequel, however, we exhibit
vhich u t i l i z e d e i o i o b a t i o n that a particular farmer has
cultivate crop j.
bhere C is the implied symmetric covariance matrix of the joint p.d.fl
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of eij a d lij'i.e., consistent with (2.8) and (2.10)' i.e.,
II
..aloj ..a.llj
.. .
"'%lCjj
..
bKjOj :a*Kjlj..* bK jKjj
1
The problem, in general, is to provide econometric methods a d
caputational algorithms to permit the maximization of the log-likelihdod
function L*(8 C ) with respect to 0 and C without requiring a closed-form
j - j
1' --j 1
analytic solution for the function, x*, of (2.4),
-j which solves the mar
productivity equations (2.3b)' but, rather, requires only that numeric 1-4-
solu!Aons to the optimization problem (2.5).
(d) The Algorithm
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We are now in a position to state an algorithm by which the imum
n n
Likelihood (ML) estdmates, 8 and
-j y be calculate
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n 8 19: :vecr.1 t i . 1 5 )
.. ~
- m - - -
d -J I
3
denote &he R Z H l)(K +2) parameter vector, containing the) Mj-
*
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1 1+z(K1 + j
rector of structural parameters in the production function, 8 and the(I .E
e
-1 '
additional parameters in the lower triangle of the symmetric covarianc
matrix, C1 'where vecC denotes the functionally independent elements
-1
in C organized into a vector,
1 '
-4
w CI( r( \d n
--*--+---
9)
rlscec 325 5
U r( t cecum 5
0, m uo4rl&d u rlo
9) 0 a~m U cd (r d (r 4 K 0 "a"
" 2 G
(d u CJ rl Ua, LC rl 0 cd rl U 9) (r a 4 0
2 50 *I *.
9) M U !?
'
e
tii
may be used a s an estimate of the (asmptotic) covariance matrix of a
I
It is hportant to stress-that, in the context of the present 1
problem, the algorithm applies whether o r not the explicit analytic solu ions
-
t o &he"inner" maximization p r o b l a s i n ( I ) of Step (n.1.r) can be calcu ated
i
for any a(nsr). If the konner is the case, the solution values, x
-J
be calculated d i r e c t l y using the explicit expressions for the optimal da$mnd
functions, x*(nsr), of (2.4). Otherwise, we must employ an i t e r a t i v e
-3
procedure (such as the DFP algorithm), with the use of the actual decisic!m
vectors', x -iJ' a s the i n i t i a l vector t o commence the algorithm for the "i
marbization problem, (2.5) -L! I n the present case it w i l l also be no
- ..-.
-
that the solution vectors, {& 1, are independent
. l d
-iJ
the parameters, 8 Hence, i n Step (n.l.r), since x*(','I x*(n~o) for
-1 -1) --Ij
values r = M + 1, M + 2,...,R (corresponding to the perturbations of
J J J
the C elements), the calculations i n ( i ) may be by-passed for r > M dn
J J' I
other problems (see the discussion of r i s k and uncertainty i n section 4 1
belov), we shall confront situation. in ~ h i c h ~ t helements of E
e e n t r r tt/e
J
optimal decision vectors, and nere, evaluation of x*(npr) for a l l r val es
-
-11 Y
,,-- .
_ . - -
J
C - ,
Y C . L .
'2
In other contexts we ahall confront cases i n which the log-
- -
l *
likelihood function, L*(8 ,C f, and/or the conditional expectation of I
I .I
. 1-J 1-
I
I
i
-1/ I f the actual input vector, sl,is not "too far*' from the optimal ve tor,
, then we should obtain -t h e global maximum for (2.5) even i.n cases
r e f j permits multiple local maxima as solutions to (2.3b).
I
p r o f i t s , E[n ( x ] of (2.6), themselves, can only be evaluated by 5
i j -ij
numerical aethods. Our algorithm applies, mutatis mutandis, t o thesd
situations. Further, i n problems involving decisiolrrnaking under
w e may even postulate a more general objective function--e.g., a
-
function of 3 specific parametric form i n which both the conditional
expectation and the conditional variance (risk) of profits (or
order conditional moments) enter as arguments. There can all be treaked
within the confines of ouz approach for general awell-behavad" choice$ of
functional forms. ~ l i u s t r a t i o n sof such cases w i l l be given i n subsebuant
.
sections
(e) Stochastic Duality Theory:
We shall cortlplete t h i s section with a discussioa of the
theory of the p r o f i t , oueput supply and cost density functions for th
representative micro farm unit. However, rather than postulating func;tional
-
forms for the (expected) profit and/or cost functions, and deriving t4e input
I
d.emand and output supply functions corresponding to them under perfec l y I
competitive profit-maximization o r cost-minimization, we shall,
methods for any s e t of s t a t e variables, i j ~ j ~ P iand j qij.
We begin with the marginal p.d.f. for proflts ~learld,frcm
%if
8
the joint p.d.f. of x and x given i n ( 2 . 1 1 ~ ) weohave
~
i j +j
with marginal expectation,
and marginal variance,
.,
In general) even though h is a notmal p.d. f in the absence of clo:?
expressions for E[sr ] and V[x ] e~aluationof (2.23),
, (2.24a) and ( .24b),
iJ I
1) f
w i l l require the use of numerical integration (see, e.g., Haber [1970)/ using I
the estimates, 8 and C This w i l l provide point estimates of the de
-j 1
expectation and variance of p r o f i t s for any choice of the s t a t e variab es, t I
!
a i j ~ ~ i j . P i j ad Ldjm
This provides an implicit representation of the p r o f i t densi y
I
I
function h and expectation function, E[x 1, treated previously i n the I I
x, i J
i
context of a deterministic SIN. Indeed, a s Lau [I9781 has shown, the se of
the Legendre transformation, t o obtain a closed Pcrm expression for th{ form
of the profit function corresponding to a s i v e n prodcction function un er d
p r o f i t maximization, La, i n practice, applicable t o only z iimited set of
choices for the ~ r i g i n a lproduction function.
. I f such analytic expressions
*
I
are infeasible, a flexible functional form (say, the trans-log) is o f t e
selected- f o t tye -rqff.Y z u n c i 4 , 7 ~ ,an? ~ C ~ ~ T - ?~- CC IO ~Z: ~ ~ > - _ > L O : I::'.I ? ' -
-9
resulting input demand functions- see, e.g., Sidhu and Baanante
-
approac& however, obviates t h e necessity for such
e
can numerically evaluate (2.2.3) , (2.24 a) and (2.24 b) for well-behavkd
choice of the production function, fj, i n (2.1).
- 22-
I
We next consider the output supply, yi j, defined by (2.1) Cud note
&
that provided the conditions of the implicit function theorem are s a i s f i e d
..,K
(i.e., suppose af /ax +0 for some k-1,. ), then we are assur
j i j k J
existence of an inverse function of the form with k=l (say):
where x' [xijl zij2], with Jacobi.ur,
-13
Thus, the marginal p.d.f. of yij is given by
and can be evaluated by numerical methods. It follows that the
of the quantity of crop j eapplied is given by the function of
~
I
v l t h variance,
vhich require further numerical integration for evaluation as Q ljandid.
-?-
Finally, we consider the stochastic version of the ex e c t e cost
function. L e t Cij=GjE~~+ ci be the t o t a l cost and ~:onsidert h
;l
transf ormatlon,
Since the conditional p.d.f. of n given x haa already been def
- i j -ij
(2.11), w e have
Hence the marginal p.d.f for Cij is
and the general analysis follows similar l i n e s for evaluation of E
A A
any choice of the stat= variable values 8.t 0 and C etc.
-3 1'
i
In short, &exploiting the distribution theory (and ass ciated
numerical integrations) i n the above primal problem xe have abtaindd the
I
marginal p.d.f.'s cf all duality concepts uhich are internally con
I
the stochastic version of the primal SNM-:hereby obviating the nadd for the -
--
so-called profit and cost- function econometric approaches- .see, e.a/., ~ a u
4
[I9781 acd McFadden [1978], unless we a r e i n the peculiar s i t u a t i o of having
prior informscion regarding the specification of these l a t t e r cor;cefpts,
inotead of the production function, i t s e l f .
-
'3 1
I
- 3. I r r i ~ a t 2 dversus Pninfed Crops:
-
I
I
I
I!
- r
The f i r s t extension of the Standard IIeoclas'.ical Model t o be
considezed is the problem of modelling f e r t i l i z e r use and response
crop i n one cf two situations: Either
J
+?
(1) the farmer has access t o i r r i g a t i o n (and water manageme t) systems t o
control water inputs, but the amounts of water availabl to the crop
are not recorded, or
(2) the farmer has no control over water inputs ( i n the abs nce of t
i r r i g a t i o n .ad water management systerns) and, instead, r e l i e s upon
,
*
the actual r a i n f a l l , which is recorded.
C '
The problem is how to estimate the parameters i n the production unction and
determine the demand for f e r t i l i z e r , etc., i n the case of data f
crop, j, f o r farmers i n e i t h e r of these circumstances.
I
To simplify our analysis we shall assume that the presence or
I
absenre of i r r i g a t i o n and water Panagement systems is exogenous1 jr
I
determined. Thus we map defineathe binary variable,
--
- i 1 i f irrigaticn and water management system
a r e available to unit i on crop j
0 othervise.
1
Suppose we now mite the production function f o r the particular top j as
4
= f ( X ' 0 1,
Y i j j -ijBuijBa: jB%jl-j
.#
;-.. . . -
e -e.13L,J a , ~ - i a o - ~ z c r~bz;;,.G ;atc. ,npuL. Iil case { L ) i n which
'9
Z
6' = 1, the variable wis is an unobserved decision variable i n he contra1 of
i j
C
the farmer, and, hence, i. to be treated as an additional variab e i n u t ;thus ,
augmenting the X j-vector, 3j. On the other hand, i n case (2), +ere
= 0, the variable, w , plays the role of an observed s t a t e ariable, and
iJ
hence augments the Lj-vector, z,j. We ccnsider each case i n tur
(a) Case (1): Irrigated Crops
I n case (1) we have the model consisting of:
c,ij=5 1
where denotes the normalized price of water (yos
%,ij
i f uater is a free public good) and, for any 9 the values of x and
-f ' -ij
w('Ir are the joint solutions to the problem of -imi~ing the cond
.il
expectation of profit,
.
with respect to both xt and w Let denote the marginal p.d.f. of the
j- il -u* j
errors i n the complete set of input demand equations, defined by
P
I E
acd l e t g denote the
El:,j
defined as
(1)
gclu, j irij -j
(3-5)
( 1 2- 1 - * - (1))
-j 1 zj
5"
-
I
-
-
-
-
"K+ Z -
d v-- 5
LI 0 cU w r'l r'l r* t! a t.~ 3 u
Q) m at+.
.r)
3 cu d d g 5 5
Ip 3* cI u 0 rn
.m ld 9)
*
dm C, w4 d.r) C,
3* I 3 n
L) .r)
w r'l a
*
h -4 d +In n
3* Ids" 3&>
a rl �
! 2 o ld a "r 0 wd 1 5
u4 rd KT
.r)
r ..
u g a z w 0 d g rn
a
ti4 .? n
3a 3"
.r) 4 .r) 4 .r) -.r) V r'l P.
.'s
2
ld r( d.0 0 rna 9) bl I Wdrn a a
P) a a! a at a U
8 513 H U 3-$ 4 dm rd cl m 0 P w
at 0
.r) tt .r)
4 1
u b. cU !i 0, U
n 5 3" w r'l a0
n .r) -l
sz a d 4 a 49-
N 9)cdWd 0 ld rlma
9) a UUd 3
+&--
S
U 0 4 3 5
0)
P) r( 4 0
a
I n general, the 'incomplete' data sample (due to a l a t e n t dbpendent
variable, wi ) means that certain of '.he parameters i n 9 and ~j( l ) ,w 11 be C
j -j
inestimable vithout further information. Though it may not be possib e tc i.
decide analytically exactly which parameters t h i s w i l l affect, t h i s d es not A
-of e
preclude use of our algorithm. Feasible i n i t i a l values f o r all th
complete data parameters, 9(') and c ( ~ ) ( ~ )w,i l l still have t o be cho
-3 j
i t e r a t i o n may proceed. A l l parameters, , i n i t e r e t i o n n such t
=j
(L*(n*r) - is zero, ~ 1 . 2 , . , are clearly inestimable, a n d thus may
..
j 1
be deleted from ( i ) of Step (n.1.r). For the remainder of the giaramet/ers,
functional dependencies may still exist- in which case the log- likeli ood h
surface will have a "ridge" at the global maximum, represenring the 4 rameter
sibspace within which all parameter points are observationally equivallent.
1/
The algorithm then w i l l simply converge to one such solution on the rddge,
-
Second, we may employ a Moore-Panrose generalized inverse o$ H(1)
I j
(given i n (2.20) and (2.22)) i f it is of i n i e r e s t to determine, f o r a 1
I
particular functional form of f J i n (3.1), exactly how many
independent parameters a r e present. This w i l l be revealed
-rank of H(') and can be calculsted by means of the Singular Value
L j
Decomposition algorithm--see Goluh r19681.
7'-"r d , ' -
"0nvn! Zho ' z ~ ; l z ; : j zi.23 .. ;.-J 3 , ,:5 1-3 ~e-;dLfi LC L~ie
-3
decision variables, thooagh kcown t o be of i=poetsnco, do no; happen to be
-
t
*
present in the data sample, it may be empting to, instead, apply the ode1 of
m
section 2, treating the x vector as the complete set of decision var able.
-il f
h
As is well-known, t h i s r e s u l t s i n a specification error in employing t e
-1/ For a more complete discussion of identification problems caused b
unobserved dependent variable i n such contexts, see Hartley [1981a
I .
I
production function, (2.1), when i n fact (3.1) is know to hold. $us
I
I
1 1 I
the and estimates w i l l not be consistent. Rather, to correct for such a
-1 1 1
,
the specification error, one should endure the additional computati
I
I
by estimating the 'observationally relevant" parameters i n i,of (31.1) and
I
Fourth, i f wij is also observed in case (1)--as, e.g., wh n farm-
t
level soil- moisture measurements are available t o construct an ind
mount of water accessible to the area, we a r e back to the correct
model i n section 2, but using the density, h(1) q
j .=ij~zij~~ij
(b) Case (2): Rainfed Crops
We now consider a model in which wij is a state variable, and hence
4
- beyond the control of the farm unit; but where 'wi is observed, a s n index of
l
I
data on r a i n f a l l over the cropcycle. In t h i s situation, provided ji, -is
know t o the farmer at the tine the amer must decide upon h i s optdmal
f e r t i l i z e r and ofher variable input levels, xP and xu then we
-ij 4 3 '
i
with "complete data" (as i n section 2), but with a structure given y:
vij, now enters as an argument i n the input demand functions,
normalized price, %,ij. Further, while the production
crop j are identical--and given by equation (3.1) i n both cases--the demand
functions f o r f e r t i l i z e r and other inputs, x('I* -j versus xj-(2)*, w i l l d i,f f e r
depending upon whether or not i r r i g a t i o n is present.
Thus, i n case (2) we proceed a s i n section 2.c, but with an
-
additional s t a t e variable, vi Here,
defines the conditional expectation of profit, as opposed to x(I)* of (3.3) in
i J
case (1). As before, i f
and
then
h(2) (2) "'"I - - (2)*
-g6;1, j(mj Xi, kij - zij
(2)*) . I 0 . u )
~ , ~ , j ( * i j s ~ i9j 8u,ij(zij-zij
- )- I
Thus, the joint p.d.f. of E ( ~ a) d u ( ~ )is given by
+j
- -
( 2 ) ( ~ ( 2 ) (2)) N ( o , z ~ ~ ) )ri2)
;
=j i j *xij ,--*
4
and the log- likelihood function f o r cage (2) is given by
w
I
.
=*(2) (2) I (2)
J ( i j * " iErj ( 1 - 6 i j ) - b g X , x , j i j B i j
-
and the marginal p.d.f. of n Is h(2) (xi,) = ... (2
iJ h j ( ~ ~ ~ ~ f ~ , ) d
x s j
(c) Pooled Samples:
I
In cases i n which the data sample, x i , ,pi , , a , , bll,
contains farm units which employ crop j under both irrigated and raiqfed
-
condition..we may po-1 the complete sample to improve the precision (of our
I
estimbtes, obtained by treating the two subsamples within czses (1)
'~
separately. The improved precision occurs due to the common paramete s, 8 ic -Y
i n each of the log- likelihoods, (3.8) and (3.14), since the
function, (3.1) applies t o each case.
In t h i s case the p o l e d log- likelihood is simply
Apart from improving the efficiency of the estimation of 8 , pooling
-j
"salvage" certain parameters i n 8, of case (1) which would have othe
P1
i 'I
inestimable due t o the f a i l u r e to report wi Thus, provided the Sam le
j
I contains both irrigated and rainf ed farmers cultivating a given crop, the
-
i1
reported data for r a i n f a l l i n the l a t t e r may be pooled with data for rrigated
.
. I
farmers t o identify the parameters even though w is not observed. ndeed,
Ij
a
1
- T C - 11 7 ~
~ 1 ~ ~ ! ~ ~ : : p ~~f
2r3?-7L2rri ;deri: i : i - 3 > i 1 - 1 : y .
'9 .
- 4. F e r t i l i z e r Usage and Response under Uncertainty:
*
e Y
0
The fact that the length of the production period
1 harvest is of significaat duration for most crops creates various typ s of
i
1
additional problems involving decision-making under uncertainty. Inp t
decisions with respect to f e r t i l i z e r use must often be made w e l l i n a vance of
I
I
- 31 -
I
4
the harvest time. As a result, variations i n the pattern of r a i n f a l l nd
other weather conditions, vhich occur a s random events i n the interven ng i
time-span, create uncertainty with respect to both the level of output and the
price a t vhich the crop map be sold, and, thus, influence decisions reg)arding
f e z t i l i z e r use. W e shall now i l l u s t r a t e econometric methods to handle (such
types of decision-makin~unde~uncertaiatpsituations.
(a) Case (3): Rainfed Crops with Uncertain Rainfall 1
I
1
I
To i l l u s t r a t e our approach to t h i s c l a s s of problems, we shal
I
reconsider the problem of deciding upon the optimal levels of f e r t i l i z e (and
I
other variable inputs) for the case of rainfed cro-ps. In case (2) of sgction
3, we assumed that rainfall, vij was the only source of orater for crops i n the
I
absence of i r r i g a t i o n and water-management systems. EIowever,- the.major
prablem v i t h the use of the model, (3.10a) and (3.10b), and associated 1
coaditional log- likelihood, L*(2) (0 ,Z(2) ) of (3.15), is that the valu
1 . - j
1
the s t a t e variable, wl denoting the r a i n f a l l index over the crop cycl
j'
cannot be assumed to be imom to the decision-maker at the t i m e of fertfllizer
I
application, etc.
, e
Suppose that r a i n f a l l conditions i n the t i m e period under
1
C"v,":~"7: !3T; ,-2 +z_C> _ - -- . -
-h i - l "
* , - , d f i T ,- p a -" ' J C i- - .
J - -
7 : ' ' I
i . J
't
-
trend and a sizeable random white-noise component with location- specific - '
parameters. Also, sup&se that, by utilizillg a sufficiently lengthy serkes of
w I
prior historical data on nonthly o r even daily r a i n f a l l , i n relation to ihe
I
I
-expected time--span of the crop, j, the probability distribution,
( ), can be estimated for an appropriate rainfall index, vij.
~v,ij(wij;-iJ
Here, denotes a vector of parameters associated v i t h the t i m e , t j ( i )
Jij
place, P ( i ) of observation i on crop j> Finally, suppose that the lamer
I
has knowledge of the historical rainfall patterns a t his location and, indeed, I
I
enploys % ij(wij;&j) F 0
t o guide h i s decisions on zij and sij.
8
In a particular example, agronomic/meteorological knowledge
- .
used to define a suitable rainfall- index, w il' associated with crop j
location L*(i), a s w e l l a s a simple stochastic model f o r the r a i n f a l l
*
*P
bution a t P ( i ) f o r the typical tlmrspan (from planting, -
t i ) ( ) t o
e
hamest, tj(i)) associated with crop j. For present purposes we assum that
such a p.d.f., i
%,ij ("ijiAij ), has been defined and that its paranet rs can
be estimated via MI, methods from a .extraneoixs sample of r a i n f a l l data f o r
1 I
each location, L*(i), and crop j.
L e t aj denote the vector of a l l s t a t e variables observed a t t h e
I
t i m e of decision-making,
I
In case (2) of section 3 we described a neoclassical model f c r muimira(tion of
the conditional expectation of profits, given the set of decision
the set of state variables, %j,
lij' -
and the r a i n f a l l index, w
~ ~ for -4.8n (21,d ~ [ n ~ ~W ilj , alj ~1 , ~ir, however,
,
)*
knowledge of w1j and may be denoted by ni (2 s
j w i of
.
$ffect by taking the marginal expectation of x
ij=
present situation,
~
-
1/ In practice, 15 farm level data are not collected, we would have to rely
upon data from neighboring weather reporting stations.
I
I
I
I
Thus, i n comparing (2)* of (3.11) with n of (4.2). we see that th
iJ
Q
consequence of uncertainty is to replace the latent variablle, wij, by the
parameters, I$ i n its p.d.f., and to change the form of the o jective
-
-ij' % , i d S
function to one requiring numerical itegration for its evaluation. 1 .
Let x(3)* be the solurioa to the problem,
-iJ
which, by the implicit function theorem, vill represent the optimal declision- .
vector aa a function of the form,
(3>* *(3>*
zij (4.4)
-ij (2i,;2j&&,)*
Comparison with x(2)* of (3.10b), the optimal decisfon vector when
-1 j
known, exhibits again the replacement of. w
i Jby the parameter vector,
and the difference i n th6 functional form of the demand functions--x
8
versus x(3)* m
-J
Given x ( ~ ) * ,.the observed decision vector, zij, can now be
-i J
=.
represented by the model,
(3)* (3) .(3)* (3)
= X
"ij -iJ +4j -j ( 4 j s j*Aij) +
'2 4~ * ( 4 . 5 )
-(2)
v ~ e r eu ( ~ )follovs the p.d .f., of ( 3 . ~ a ) , i.e.,
-ij gu,j'
Thus, the marginal p.d.f. of x is simply
*j
+- I
and comparison with (3.U) reveals that uncertainty with respect to
chamsea the mean of x from 3(2)* to x(3)*, while preseming the ran
-ij -il
covariance matrix, z!~)*, as i n (3.2a) of case (2).
Consider next the conditional p.d..f. of profits, zij, given that
is determined through the model (4.5 j in ignorance, ex-ante, of the ex
3 j I -
post value of the rainfall index, wij, which subsequently obtains as 4 randon
. I
In the case where the resulting w value is
i j
..- ..
,at harvest- tine), then, conditional upon both x- -and
the p.d.f. of case (2), ~
vhere g(2) is defined in (3.2 b); n(2)* (w ) is defined by. (3.11) , and is a
cI:, l i j ij
function of the actual value, wij; and x(3)* is defined by (4.4), and depends 4
-if
I
conditional joint p.d.f. of kjand E ~ ~ ,
is
whereas the unconditional p.d.f. is defined by
~
* s
alo aa
5 5 d
Q-4
Ur, r'l n
W WWW0 Mdd ea4 3 guu
M cd "4ci a
cd d cd td
#4 cd a td PI 5 d cd
I
degenerate point-mass distribution a t w13 ' whence n(3)* of (4.2) c o ~ v e r ~ e s
i J 1
The algorithm of. section 2(c) applies, mutatis mutandis, do this 1
I
I ,'
case. The major difference is the use of nunarical integration to evaluate I
I
I
the objective function, (3)* of (4.2j, associated with each Inciivid/ual, which ,
I
I
is then maximized with respect to x i n (I) of Step (n.1.r). I
4 3 I
It should also be evident that this approach readily gene
the treament of other sources of uncertain? with respect to any o
variables la s An obvious extedsiorr would be to treat prices of
4 3
pij, a t harvest t i m e as uncertain a t the Lime of planting and devel
to generate the p.d.f. of pij based on historical data on prices and other
. 1/
I _determinants up to the t i m e of planting-
(b) Risk and Uncertaintp:
~*
We now generalize our discussion to the case where, in the presence
I
of uncertainty, farmers are not assumed to be neutral to risk.
- - We dpfine r i s k
~
I
as the conditional variance of profits, given the decision variables1,x and
+IS
the vector of s t a t e variables, s of (&.I), where wlj is now an el6 ent
4 3 I
A
of z Suppose, upon reordering, that ge partition s as
-ij- -1 j
*
where s ( l ) is h o r n with eerfalnty a t the t i m e 05 decision making, but
-il
uncertainty surrounds the elements of a(2 . (2
...--- Let g (2) ( E ;~l j )~d e k t e the
-ij 9
-9 j i
-11 Such uncertainties can be eliminated i f farmers contract
t (I) - k t time,
E i s for delivery of their crop' a t time, t j ( i ) , a t the currently
pJevailin$ futures market prices. ~
I
I .
i 4 11'
g
u I-ln
f;l s
u m -a
4 4 3 I
LC A KT cd 0 ku cd *
Q Q)4 d 4a w al LCLC-
r, 14 -m
3 ili
Ill w s Y B
w
rrj
PO 5
LI 0 rw
4'),
Thus x(4)* depends on the state variables known wi:h certainty, s the
-ij 1
parameters, 8 i n the production function; the parameters, P , defining th
-j '
covariance structure of &e errors; the parameters, l,,a s w e l l as other
J
possible determinants, Ln the distribution of the s t a t e
which uncertainty is attached; and, f i n a l l y , the parameters,%, i n the u t i l i w
function, defining the trade-off between the expectation and r i s k of p r o f i t s ,
Hence, i n the case where s ( ~ )is l a t e n t we ha-~bthe mod&*
-il
I
I
I
= x(4)* + -= X
U.,=., 4 s ! . r (4 2Obir
l i j -ij Eij *~ 1
-A, -j j j j ' l j 9 +
where
bij)= N(o,q) (4.21a)
j
~ u , j ( ~iui ij
- -
) = N ( d t l -
% ' jb - -j
d ~*-l ) ,
j --j a (4.21b)
j
+i -j
-A".
Thus, the joint p.d.f. of r and x is given by
ij -ij
with log- likelihood function,
I f , on the other hand, s ( ~ )is obaerved
-ij
(4.20a) is replaced by
I
I
p . / d . f .
where fi is defined by (2.4) ~ i i hthe notation change, and the is
j
i
to be inserted into (4.23).
I
Thus, apart from these changes, the algorithm cf section 2 uiay be
I
-
usad to estimate the parameter vector, a I vecZ2' 02'],
where ljis
y a s s d
-1 [ti-J
n
to have been estimated from aa extrarieous sample--providing a gendral a p p r d
for the econometrPc treatment of risk i n the presence of uncettai ty.
5. Conclusions:
I
This paper has outlined a general approach to the
agricultural productgon, input demand and output sugply i n the coltext of--
and uncerrainty. As a by-product, we have.also indicated.(section 2e) how tge
,
4,
econometrics of the dual concepts of profit and cost functions ma be treatd
in the context of a stochastic model. Our purpose has been to utfllize the
issues of f e r t i l i z e r use, irrigation systems and crop response as a vehicle m
I
illustrate these ideas. It should, however, be clear that the approach is
. I
rn applicable to a vast Array of other agricultural (and non-agriculkral)
I
problems. I
Implenentation of these methods on the Bangladesh [I981 data has.
]i
I
a t present, to await clearance by the Internationgl Pertilizer ~evklopment
Ic I
Center and the Government of Bangladesh. It is hgpedBhowever. t h ~ in a
t
I
subsequent paper we w i l l be able to report on such erpirlcal resul
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