La función de producción de trigo en la Región Pampeana (Argentina)

1

the doses of N and P in [kg.ha ] applied as fertilizer,

x is a row vector of powers of x and z is the vector of powers of the organic matter (MO) content z in [%],

(j) (j)

fj(x ,z) are polynomials of degree p, where x is x but excluding the terms with xj,

g(x)̃ = g(x1, x2, x x1 2) is a linear function of N , P and the f f

interaction N × P ,f f

ln(R) = θ1 ln(w1) +…+ θkln(wk) + ... + θr ln(wr) is a climate index, where w1,...,wr is the soil moisture or the rainfall in the k-th critical period,

vm are agronomic practices and soil features, δh are geographical or varietal dummies.

2

εiis a random variable distributed εi~N(0,σ).

De Janvry concludes that in the case of wheat:

(j) 2 (j) 2

! f1(x ,z) = α1+α2 z and f2(x ,z) = α3+α4 z are the polynomials that best fit the experimental information;

! g(x1, x2, x x1 2) = 0, as none of the regression coefficients appear significant to explain yield;

! rainfall at seeding, stem elongation and tasseling are all relevant variables to the climate index ln(R);

! agronomic practices, soil features and varietal differences are negligible for sub-regional functions.

Building on these findings De Janvry arrived to

ln(yi) = c + Σj

≥

1fj(zi) ln(xij) + α ln(zi) + θ1 ln(wi1) +…+ θ3 2

ln(wi3) + εi with εi~N(0,σ). (1)

Agronomists identified technological functions through the design of experiments under controlled conditions, and proposed empirical expressions to explain the technical relationship between output and inputs following goodness of fit criteria. Álvarez (2007, pp. 91-119) presents a comprehensive In economic studies inputs are related to product through the

production function. This function is a technological relationship based on the current state of knowledge about the production process, and its representation is a mathematical expression that relates inputs with outputs in a range that is economically rational. The estimation of production functions was addressed by both economists and agronomists. While in the economic approach, the emphasis is on economic rationality, in the agronomic approach the estimation is based on goodness of fit criteria.

Neoclassical economists have proposed a set of conditions that guarantee the “economic rationale” of a technological relationship (Chambers 1994, p. 9), conditions which, ultimately, rely on assumptions about how they think the real world works. From these conditions they suggested several mathematical expressions (e.g. Leontief, Cobb-Douglas, CES, trans-log, etc.) to represent the production functions, although applied research frequently failed to fit these expressions to real data (Shumway 1995). A notable exception is the work of De Janvry (1972a), who in the early '70s proposed a general technological function for wheat and corn production in the

1

Pampas region . The general expression of De Janvry, later known as “generalized power production function” (De Janvry 1972b) is:

(j)

ln(yi) = ln(A) + Σj

≥

1fj(xi ,zi) ln(xij) + g(x̃i) + α ln(zi) + ln(Ri) +

Σm

≥

1γm ln(vim) + Σh

≥

1Dhδih + εi

where,

1 y is the yield in [kg.ha ], ln(A) is a scale constant,

x jis the dose of the j-th nutrient; in particular x1 = N and f x2 = P are f

The wheat production function in

the Pampean Region (Argentina)

La función de producción de trigo en la Región Pampeana (Argentina)

FRANK, L.

Depto. Métodos Cuantitativos Facultad de Agronomía (UBA)

[email protected]

El trabajo propone una nueva función tecnológica para relacionar la producción de trigo con dos nutrientes, N y P, y dos variables ambientales, precipitación y temperatura. Además, incluye el sistema de labranza y el tipo de suelo como factores de cambio técnico y cambio de escala, respectivamente. La función ajusta adecuadamente datos experimentales a la vez que satisface los supuestos neoclásicos sobre funciones de producción.

Palabras clave:

Trigo, función de producción

Resumen

The paper proposes a new technological function that links wheat production to two nutrients, N and P, and two environmental variables, rainfall and temperature. Additionally, it includes tillage system and soil type as factors of technical change and scalar change, respectively. The estimated function both fits the experimental data and satisfies the neoclassical assumptions about production functions.

Key words:

Wheat, production function.

Abstract

Introduction

1

review of functions (12 in total for the Pampas region) linking wheat yield and available nitrogen (N ) in conditions of non-d

limiting phosphorus (P), as well as “response” functions to

2

phosphorus under conditions of non-limiting N . Some of these d

functions also include environmental variables such as rainfall (R), MO, soil texture, etc. By comparison of all the functions, Álvarez (2007) concludes that:

! The response to nitrogen fertilization is determined by the availability of water for cultivation; this would also explain the gradient observed in yields in the Pampas region in direction East-West. However, none of the functions considers a N×R interaction term.

! The response to nitrogen fertilization is determined by the availability of water for cultivation; this would also explain the gradient observed in yields in the Pampas region in direction East-West. However, none of the functions considers a N×R interaction term.

! Interaction between nitrogen and phosphorus fertilization is

not always significant. Álvarez (2009b) reviewed this issue later, concluding that there is no interaction between the two nutrients.

! The response threshold to phosphorus fertilization is around 15 ppm of extractable P (0-20 cm depth) under non-limiting N.

! The timing of application of nitrogen fertilizer has no effect on crop yield at least when applied before stem elongation (cfr. Fischer et al. 1993).

! The preceding crop appears not to be a relevant variable to explain yield. This last conclusion does not appear explicitly but arises from a thorough reading of the paper.

A closer review of the papers quoted by Álvarez (2007) show discrepancies among and within them which are listed below:

! The technological function is not unique throughout the region either in the mathematical expression or in the parameters. The review reveals five possible functional forms (i.e. linear, linear and plateau, quadratic, log-linear and log-log) to relate wheat and fertilizer, provided that all other inputs are not limiting. However, none of the quoted authors attempted to clarify which one is the “true” technological function or why the functional form should vary with the environment.

! There is no consensus about the influence of zero tillage (SD) on yield; unlike Bono and Álvarez (2006), other studies e.g. Álvarez (2003), García and Fabrizi (1998) and García et al. (1998) find that SD does not outperform conventional tillage (LC). If the latter were the case, it is not clear why the two technologies (with different costs) co-exist.

!

expression of the technological relationship, e.g. Sánchez and Ascheri (2006) suggest that the relationship between yield and N is quadratic, but later they state a linear-plateau d

relationship between relative yield and N . The same d

happens in Calviño et al. (2002).

! Most authors estimate the parameters omitting potentially relevant variables (e.g. rainfall, pesticides), leading to (possibly) biased estimates.

! The specification of the error term is inconsistent with the way the data are collected (e.g. from different environments, different sample sizes, etc.) leading to inefficient estimates of the parameters; this point may explain why some variables (e.g. SD) sometimes appear not significant in regression analysis.

! Some functions contradict the economic theory. For example, Galantini et al. (2005) find a linear relationship

1

between yield and N (in the range of 0 to 250 kg ha ), d

suggesting constant marginal returns to fertilization.

Among the functions reviewed by Álvarez the most complete and the one we shall consider as reference is the proposed by Bono and Álvarez (2006) for the Semiarid and Sub-humid

Pampas. The function is as follows

yi = β0 + f1(xi) + f2(zi) + β5δLC + f3(wi) + β8δi,v1 + β9δi,v2 + εi with

2

εi~N(0,σ), (2) where

1 y is the yield in [kg.ha ],

-1 x is a row vector of powers of x and x is N in [kg.ha ],d

z is a row vector of powers of z and z is organic nitrogen in [%],

f1(x) and f2(z) are polynomials of second degree of the form α1x +

2

α2x,

δLC is a dummy variable indicating tillage practice (δLC = 1 if LC andδLC = 0 if SD),

w is moisture at seeding in [mm],

δv1 is a dummy indicating soil depth (δv1 = 0 if less than 60 cm, and δv1 = 1 otherwise),

δv2 is a dummy indicating soil texture (δv2 = 0 if more than 70% of sand, and δv2 = 1 otherwise).

This function is conceptually important because: (i) it considers falls in wheat yield beyond an optimal level of N-NO , (ii) 3

defines as input the sum of soil and fertilizer N-NO , (iii) the N 3 org

appears as a relevant factor to explain yield, (iv) moisture at seeding also shows an optimum, (v) P does not appear as a limiting factor, and (vi) type of tillage, soil depth and texture appear as simple changes of scale which expand production in a fixed proportion. In a later paper, but focused on the whole Pampas region, Álvarez (2009a) considered the same functional form but with some adaptation in the variables' definition.

Some papers appear inconsistent on the mathematical

2

In agronomic literature “response” is the yield difference between the fertilized crop and the control.

Objectives

In view of the many functions proposed by the agronomists to explain wheat yield from its main nutrients and other agronomic and environmental variables, the disagreement between authors on the relevance of each variable and the discrepancies listed above, we intend to:

! find a functional form compatible with the current theoretical and experimental knowledge on technological functions;

! validate the new function with experimental data used in previous studies;

!

some guidelines in technological function building.

To achieve these objectives we depart from the economic theory on production functions and the experimental findings mentioned above. Then, we relate inputs and output through a flexible functional form (see Chambers, 1994, p. 158-192) and compare the elasticities (or semi-elasticities) which arise from our relationship to those reported earlier. We note that the main contribution of the paper is the mathematical expression of the technological functional itself rather than the numerical results.

1

x2 is the level of N in [kg.ha ] on a logarithmic scale; N = N + d d s

N , where N are the N-NO in the soil (0-60 cm) and N the N f s 3 f

added by fertilization.

x3 = (zz01)○ln(x2) δz≤z0 is the difference between the

1

extractable P [kg.ha ] in the first 20 cm of the profile and the

1

critical level z0 = 15 ppm (≈ 40 kg.ha ), multiplied by ln(x2)δz≤

, where δ≤ is a Kronecker delta that equals 1 if z ≤z or 0

z0 z z0 0

otherwise. The critical level z0 is given by the empirical literature, called P in A.2. By “0 ○” we refer to the Hadamard product.

1 1

x4 = δh=SD2 +(1δe=LC)2 is a variable indicative of the type of tillage: δh=SD equals 1 if SD or 0 otherwise, and δe=LC equals 1 if the tillage is LC or 0 otherwise.

1 x5 is the mean annual rainfall in [mm.year ].

* * 1

x6 = (x5 5x 1)○δx5≥x5* where x5 = 950 mm.year and x6 is a

*

variable set to (x5 5x 1) if the annual rainfall exceeds the critical

*

level x5 or zero otherwise.

x7 is the mean annual temperature in [°C],

x8 = δl=E/A is a dummy variable for entisols or aridisols in the area.

x9 = δr=V is a dummy for vertisols.

is the error term, which includes all the variables that for some reason were omitted from the function. Each is distributed i

2 2 2 1 2

N(0,σ) where σ = σ m , and σ is the error variance in the r-th

i i i r rs r

experimental location and mrs is the s-th sample size at the r-th location.

In appendix I we explain the use of dummy variables to express broken linear functions in a linear-in-the-parameters fashion.

The data

From the graphs and tables reproduced by Álvarez (2007, p. 91-119), we computed the average yield of wheat with 50, 100, 150,

1

200 and 300 [kg.ha ] of N (N-NO down to 0-60 cm depth plus N d 3

from fertilization). Therefore,

! we averaged the mi yields at the doses mentioned above for

all experimental networks reviewed by Álvarez (2007); in this way we got 54 observations yi;

! on two occasions we averaged the two closest values to the required dose due to lack of registration at the exact desired level;

! in four cases we computed yi≈yi* Δyi, where yi* is the average yield at the lowest non-limiting level of N of the closest study in space and time, and Δyi is the response to phosphorus fertilization; we used this estimate whenever the original work did not clarify the level of N .d

1

! in three cases we computed yi≈ỹi + (ΔyiΔxi)| hxih, where ỹi is

1

the average yield of the experiment, (ΔyiΔxi) is the “agronomic efficiency” of the phosphorus fertilizer at level h

of extractable P and xih is the extractable P.

In appendix II we show the yields yi together with explanatory

4

notes on their computation . These were grouped into four sub-regions (i.e. Rolling Pampa, SE Buenos Aires, SW Buenos Aires and Sub-humid and Semi-arid Pampa) and sorted temporally. With regard to the explanatory variables we registered:

○ The model

We depart from an adapted version of the generalized power function proposed by De Janvry (1972b), inspired in the Argentinean productive context. The assumptions are the following:

A.1 N is the main nutrient of wheat and is the only input that limits the production at any level of the other nutrients. All other nutrients, except P, are available in nature in unlimited quantities. The N provided as fertilizer is indistinguishable from that present in the soil.

A.2 The elasticity of output with respect to the N is constant at d

levels of P > P but decreases linearly with phosphorus deficit. In 0

other words, P modulates the elasticity of N in a linear-plateau d

fashion. Aside from this the elasticity of N is constant at any level of other factors and at any level of N itself, and is crop specific.

A.3 Rainfall increases yield monotonically until a certain threshold at which it ceases to be limiting and becomes potentially harmful. We assume that the N × R interaction is negligible.

A.4 Temperature (T) increases yield monotonically throughout all the temperature range of the region. We assume that the R × T interaction is irrelevant for the accuracy of our function.

A.5 SD is a technical change (not a new technology) in the neoclassical sense (Chambers, 1994, p. 203-224) which expands production by a factor of φSD, regardless the level of other variables. This factor can be decomposed into a set of minor multiplicative factors that contribute to yield in small proportions. Formally, φSD = φ1φ2…φp and each φSD > 1 but not necessarily each φj > 1. For the moment we only know two of these partial factors: the greater retention of soil moisture and the replacement of traditional genotypes by superior genotypes

3

more sensitive to water deficit .

These assumptions rely on the conclusions in Álvarez (2006, 2008) mentioned above, except for A.2 and A.5 which are conjectures of the author subject to verification. Based on them, we propose a technological function of two components, one productive and another environmental, which we call f(xi2, …,

xi4) and g(xi5, …, xi9), respectively. The expression for the i-th observation is:

2

ln(yi) = (c0+c1) + f(xi2, z, xi4) + g(xi5, xi7, x , xi8 i9) + with i iN(0,σi) where,

1 1

f(xi2, z, xi4) = α2 ln(xi2) + α3 (zz0) ln(xi2) δz≤z0 + α4 [δh=SD2 +(1δe=LC)2 ] and

*

g(xi5, xi7, x , xi8 i9) = α5xi5 + α6 (x xi5 5) δx5≥x5* + α7xi7 + α8xi8 +

α9xi9. (3)

The meaning of each variable (written in vector form) is:

1 y is wheat yield in [kg.ha ].

αjfor all j = {1,…,k} are fixed parameters which we interpret as crop specific (see discussion below).

x1 = 1 is the constant associated with the scaling parameter (c0+c1); c0 is the intercept associated with f(xi2, z, xi4) and c1 with

g(xi5, …, xi9).

Methods

3

In fact, Calviño et al. (2002) point out that French varieties show higher “agronomic efficiency” than traditional varieties.

4

Since we are working with graphic information, the records show some 1

!

just one case we equated absorbed N to N because, d

according to the authors, their relationship was close to 1.

! tillage, conventional (LC) or zero tillage (SD) as reported by each author;

! mean annual temperature in [ºC], according to cartography of SNIH (2001); we averaged the isotherms of each sub-region, except in a few cases where the precise location of the experiment coincided with one of the isotherms;

! mean annual rainfall in each sub-region, according to cartography of SNIH (2001); here too we averaged isohyets in the same sub-region.

! soil order according to the Soil Taxonomy System (USDA-NRCS, 1999), from cartography of SAyDS (2003); the orders present in the area under study are: entisols and

5

aridisols, mollisols and vertisols .

This dataset is just a rough approximation to the underlying variables. However, we believe the data are reasonably accurate to explain average yields over nearly three decades in a region as vast as the Pampa. Besides, we note that practically all the available information on input-output relationships comes from experiments on fertilization, so that the conclusions will be valid only for crops reproducing such experimental conditions e.g. lots free of weeds, insects and fungi.

Parameter estimation

The econometric model associated with (3) is

2

y = X11 + X22 + with N(0,σΩ), (4)

where y is a vector of yields of dimension n1, X = [X1,X2] is a full column rank matrix of known constants and size nk, = [1',2']' is a

2

vector of k1 fixed but unknown parameters, N(0,σΩ) is an unobservable vector of n1 independent although not identically

2

distributed random variables, and σΩ is a symmetric positive definite matrix. The model is heteroskedastic (Ω ≠ I) by

2

construction as the variance of each “observation” ln(yi) is σi and

2 2 1

has cov(εi,εi') = 0. Then, each element of Ω is ωij = σr(σmrs) for

2 2

all i = j, where σr and mrs were defined earlier and σ is a scalar such that tr(Ω) = n. For all i≠j, ωij = 0. Obviously, in (3) and (4) we omit a set of potentially relevant variables but unavailable to the author. We shall assume that those variables (let's call them

X3) are orthogonal to X1 and X2. So, X1 is the matrix of the k1 = 4 productive variables, X2 is the matrix of k2 = 5 environmental variables, and and are the corresponding vectors of 1 2

parameters. Overall, we have n = 63 observations and k = 9 independent variables.

The first stage of analysis involved the computation of the

2

condition number κ(X) and the partial Rj(j) coefficients. The condition number is the ratio between the highest and lowest eigenvalue of X̃'X̃ where X̃is X re-scaled to unity but not centered. Belsley et al. (1980, p. 100-104 quoted by Judge et al. 1985 p. 902), claim that condition numbers 5 ≤κ(X) ≤ 10 reveal low to moderate multicollinearity, whereas 30 ≤ κ(X) ≤ 100

2

show moderate to strong linear dependencies. The partial Rj(j)

coefficient is the coefficient of determination between each xj

regressed on X(j) for all j≠1 and it is used to compute the variance

2

inflation factor VIF = 1/(1Rj(j)). A common rule of thumb is that if VIF > 5 then multicollinearity is high. That being the case, it is

1 1

N [kg.ha ] and P deficit in soil, both expressed in [kg.ha ]; in d known that the diagonal elements of var(bGLS) appear artificially

increased and therefore the t tests become conservative.

In the second stage, we estimate the model parameters by “feasible” generalized least squares (FGLS). Recall that this is a two steps estimator in which we first compute the parameters by ordinary least squares (OLS) and then we construct a diagonal

2 *

matrix sΩ with the residuals e = yXbOLS squared. We then plug

2 *

sΩ in the generalized estimator, so that

*1 1 *1

bFGLS = (X'Ω X)X'Ω y.

The GLS estimator is an optimal estimator in the class of linear unbiased estimators or b.l.u.e. (Greene, 2006, p. 50), where optimal stands for minimum variance. The FGLS estimator is a b.l.u.e. only asymptotically, that is, when n→∞. For small samples, as in our case, Frank (2010) suggests using White's (Greene 2006, p. 158-164) LS estimator, where bLS = bOLS and

2 1 1

var(bLS) = σ(X'X)X'ΩX(X'X) . Recall that var(bGLS) =

2 1 1

σ(X'ΩX) .

In a third stage we test (linear) hypotheses on the parameters, namely R = r. Therefore we use the test statistic

2 1

λ =(Rbr)'(RΣR') (Rbr)/q

where R is a full row rank matrix of dimension qk, r is a vector of

2

dimension q1, and Σ is either var(bLS) or var(bGLS). If is

2

normally distributed this statistic is distributed exactly λ~χ(q) and

* 2 * d 2

it may be proved that if n is big enough, λ(sΩ)→ χ(q). The hypotheses we wish to test are (a) whether the yield reaches a plateau after the 950 mm rainfall threshold, and (b) if the elasticity of N goes to 0 as Pd →0, both in (3). More formally, we want to test

(a) H : 0 α5 + α6 = 0

1

(b) H : 0 α3 α2z0 = 0.

The last stage of the analysis is to check the distribution of the residuals with the Jarque-Bera normality test. All calculations were programmed into the matrix language of the free software Euler Math Toolbox v10.1 (Grothmann, 2010). The scripts are available upon request.

The elasticities

Recall that the elasticity of output with respect to the j-th factor is defined

1

ξ = (∂f(x)/∂xj) x fj (x) , for all f(x) > 0,

while the semi-elasticity is

1

ξ = (∂f(x)/∂xi) f(x) , for all f(x) > 0.

The elasticities from (2) (showed in table 3) were computed at levels of 200 mm of moisture at sowing, 0.1% of organic N,

100-1

150 kg.ha of N as N-NO , soils with more than 60 cm deep and 3

more than 70% sandy, grown under zero tillage. For ease of interpretation, the deficit of P was introduced in absolute value as if it were a dose of 27 kg (roughly 10 ppm) of P in a P-limiting situation. In the case of tillage and soil type, the (semi) elasticities should be considered only as comparative measures because they are discrete variables and the elasticities computed this way have no clear experimental meaning. We omit any comparison with the elasticities of De Janvry in view of the time past since the paper was published.

5

1

With regard to the hypothesis α5 + α6 = 0 and α3 α2z0 = 0, the test

2

statistic λ was λ≈ 0, lower than χ(0.05,2) = 5.99, leading to failure of rejection of the null hypothesis. This means that yield reaches a plateau after rainfall exceeds 950 mm (other variables held constant), and that the elasticity of N goes to 0 as P d → 0.

In table 3 we show the elasticities and semi-elasticities (in absolute value) that arise from tables 1 and 2 and the papers reviewed by Álvarez to which we refer to as “recent literature”. It is noteworthy that our elasticities are in general lower than those reported in the literature.

Rainfall (R), is the main factor behind yield. To appreciate its relevance, notice that according to (3) ξN≈ 0.25, whileξR≈

4 1

(6.5×10 )×800 = 0.52 in, for example, an 800 mm.year rainfall scenario. If R were distributed uniformly throughout the year, its semi-elasticity during the growing season (5 out of 12 months)

4

would be (12/5)b5≈ 15.6×10 , not so far from the semi-elasticity attributed to moisture at sowing computed using (2) and shown in table 3

The condition number of X was κ(X) = 172.8, revealing a high level of linear dependence between the explanatory variables. Specifically, the VIF associated with environmental variables were all greater than 5, indication an imperfect linear relationship between them. The computation of κ(X) removing one column at a time showed that x8 (entisol/aridisol) is the variable most linearly dependent to the rest of the dataset, as the

(8)

condition number of X reduced to 89.8 when x8 was removed. However, in view of the theoretical relevance of all the variables in the model we decided to continue the analysis without eliminating any of them.

Table 1 shows the FGLS estimates and corresponding deviations. Table 2 presents the LS estimates and the estimated deviations computed using White's expression. The

2 2 2

corresponding adjusted R coefficients are RGLS = 0.5142 and RLS

= 0.5195 both lying in the usual range of 0.4-0.6 for these kind of models (see Álvarez, 2007). Simple inspection of the tables shows that all bj have the expected sign and are all “significant,” except temperature whose P(|T|>t) > 0.025 under GLS estimation. In preliminary estimations we included classification variables for decade and the source of information, but these appeared not significant and were later removed. We did not reject normality of the FGLS residuals with a type I error of 0.05.

Results

Table 1: Regression coefficients obtained by feasible GLS Table 2: Regression coefficients obtained by feasible LS Associated variable bj s(bj) t stat. P(|T|>t)

x1 = intercept 70.307 0.2114 332.583

--x2 = ln( Nd) 0.2411 0.0151 159.613

--x3 = P deficit´ln(x2) δz?z0 0.0015 0.0003 56.436

--x4 = zero tillage 0.2112 0.0247 85.672

--x5 = rainfall 0.0006 0.0001 51.678

--x6 = rainfall >950 mm -0.0025 0.0006 -41.846 0.0001

x7 = average temperature -0.0292 0.0149 -19.548 0.0279

x8 = entisol/aridisol -0.2799 0.0300 -93.343

--x9 = v ertisol 0.1811 0.0842 21.515 0.0180

Associated variable

x1 = intercept

x2 = ln( Nd)

x3 = P deficit´ln(x2) δz?z0 x4 = zero tillage

x5 = rainfall

x6 = rainfall>950 mm

x7 = average temperature

x8 = entisol/aridisol

x9 = v ertisol

bj

69.500

0.2617

0.0016

0.2061

0.0007

-0.0021

-0.0338

-0.2695

0.1110

s(bj)

0.1111

0.0081

0.0001

0.0115

0.0001

0.0003

0.0063

0.0136

0.0341

t stat.

625.685

322.334

200.322

178.858

107.327

-81.043

-53.462

-19.760

32.586

P(|T|>t)

--0.0010

Table 3: Elasticities of productive and environmental factors

Variable Estimator Recent literature Function (3)

Elasticity

Nd (z > z0)

P (z−z0=-27, x2= 102)

Zero tillage

Semi- elasticity

Rainfall (200 moist., 800 mm rain)

Temperature

Entisol/Aridisol

Vertisol

b2 + b3 |z−z0| δz?z0

b3 |z−z0| ln(x2) δz?z0

b4 [δSD+(1−δLC)]2− 1

b5 + b6 δx5?x5*

− b7

− b8

b9

0.16 - 0.42

0.17 - 0.38

0.25

20×10−4

--0.10 - 0.11

--0.24

0.19

0.21

6×10−4

2.9×10−2

0.27

0.18

0.26

0.21

0.21

7×10−4

3.3×10−2

0.28

Consider the following profit function

b(x,z) = p f(x,z) – w'x – c,

where p is the wheat price, f(x,z) is the technological function, x

is a vector of inputs, z is a vector of other (uncontrollable) factors, w is a vector of input prices and c is the sum of all fixed costs, all exogenous to b(x,z). For the j-th input, the maximum profit is achieved at

∂

b(x,z)/

∂

xj = p

∂

f(x,z)/

∂

xj – wj = 0.

Particularly, when the j-the input is N, the demand of N in (1) is f

given by

1 2 2

xj° = pwj(αj+ αj+1z) y, for all αj+1 < 0 and |αj|

≥

|αj+1 |z, (5)

while in (2) the optimal level of N isd

1 1

xj° = wj (2pβj+1) – βj(2βj+1) , for all βj+1 < 0, βj

≥

0 and βj

≥

1

p wj. (6)

*

Unlike (1) here yield increases up to a certain point, say xj, and

6

then declines, presumably because of toxicity . The second term

*

in (6) is the maximum yield xj, while the first term is the distance

*

x °j xj which is always negative under the constraint βj+1 < 0. Then, it's easy to see that the maximum profit is always attained

*

at levels of N below d xj leading to a strictly monotonous technological function along the feasible economic interval.

1

Besides, in (6) xj° depends only on the price ratio p wj, regardless

7

the levels the other variables .The optimal level of N is given in d

(3) by

1

xj ° = pwj [αj + αj+1 (zz0) δz

≤

z0] y, for all αj > 0, αj+1

≥

0, |αj|

≥

αj+1 |zz0|. (7)

In (1) both N and P are strictly essential inputs in the f f

neoclassical sense since limx(j)

→

0+f(x,z) = 0, xj being the dose of N or P . This means that when N or P approach to zero, the yield f f f f

also approaches zero, which is clearly false. In the function the marginal productivities of N and P appear modulated by the MO f f

content which De Janvry considers an indicator of the “recent use of the land.” It is unclear, however, whether the “recent use of the land” refers to the preceding crop or just to the number of years of continuous cropping. The function also rules out any soil feature, although De Janvry suggested in his original framework the use of the soil permanent wilting point as a proxy to soil texture. De Janvry's results seem to be contradictory regarding the uniqueness of the technological function throughout the region, as his theoretical framework considers crop specific parameters, but he estimates separate functions for each sub-region, implicitly admitting the possibility that the

8

parameters vary among environments .

Function (2) is also controversial about the essentiality of N and d

the role of N . Regarding N , this nutrient appears to be a org d

limiting but non-essential input, while P is neither limiting nor essential in this function. Both assertions are, however, not very realistic because N is essential for growth and also P is limiting d

according to documented empirical evidence for the Semiarid and Sub-humid Pampa (Barberis 1987). The misspecification could be easily overcome by adding to the function some constraints on the other factors xj' so as to make yield zero whenever xj approaches zero, but in this case the constraints also need to be imposed in the estimator. Regarding the role of N , it org

is unclear whether this factor is (a) a proxy to the lot's history or (b) a second source of N . If (a) were the case, the introduction of d

N into the production function does not contradict any of org

neoclassical assumptions for the input-output relationship since uncontrollable factors are not inputs. But if (b) were the case, then function (2) would be considering the existence of two inputs, N and N ' (where N ' is a function of N ) somehow d d d org

distinguishable, each with a different marginal productivity, a situation difficult to understand. Apart from this, function (2) has the merit of being the first to suggest an optimum level of moisture and to include soil texture to explain yield.

Unlike in (1) and (2), in (3) N is strictly essential. Besides, d xj 1

depends not only on the price ratio pwj but also on P, which makes it more realistic. Here we define P as a “secondary” input in the sense that it modulates the influence (below a certain threshold) of N but is not essential in the neoclassical sense. d

Aside from P, no other factor modulates the marginal productivity of N . The MO does not play a role in (3) unlike (1) d

and (2). It is worth noting that in (3) the parameters are crop specific, as in Álvarez (2009a) but in contrast to other authors quoted by Álvarez (2007) who suggest environmental dependency. To the knowledge of the author the point has never been discussed in the empirical literature. Regarding R, our findings are in line with (2) about the presence of an optimum point, although we show that an exponential-plateau relationship is also possible to represent the yield-rainfall relationship. We did not split R in critical periods as suggested in (1) (i.e. seeding, stem elongation and tasseling) under the belief that the whole dataset would result highly collinear. However, splitting R seems an appealing alternative for controlled experiments. Interestingly, there is no research performed on the interactions among R, water retention capacity of soil and SD.

As mentioned above our elasticities are in general lower than reported previously. However, it seems suspicious that the sum of the midrange elasticities of N, P and rainfall (on a symmetric distribution bases), i.e. 0.29, 0.275 and 0.534, respectively, are greater than 1 (see elasticities of recent literature in table 3). This may be due to the omission of relevant variables in the underlying econometric models proposed in the recent literature.

Discussion

6 * 1

According to the parameter estimates of Bono and Álvarez (2006) xj

≈

221.3 kg.ha . The response to moisture at seeding is also quadratic with a *

maximum at xj'

≈

464.6 mm which is above field capacity.

7 1

Interestingly, according to Bono and Alvarez's function, the demand for N-NO is zero when p w = 3 j βj1

≈

9. De Janvry (1972a) said that fertilization in 1

Argentina was incipient at a time when p wj

≈

8. 8

the De Janvry's (1972) classical paper; and (b) erroneous identification of N as a non-essential input, as well as ambiguity d

in the meaning of N in Bono and Alvarez (2006, 2008). org

Moreover, our function is unequivocal about its uniqueness and the crop dependency of the parameters, consistent with Alvarez (2009a). The function fits well to the experimental data used by competing models.

The paper proposes a new technological function that relates wheat yield with its main determinants. The function is based on both assumptions consistent with the neoclassical theory of production functions and experimental findings. The new function overcomes the flaws identified in previously proposed functions. Among them we highlight (a) misidentification of N f

and P as essential inputs and ambiguity in the meaning of MO, in f

Conclusion

This paper was written during the author's internship at the University of Manitoba supported by IICA Canada and the Emerging Leaders in the Americas Program (ELAP). The author wishes to thank the valuable comments of Roberto Álvarez (Facultad de Agronomía, UBA), Barry Coyle (Department of Agribusiness & Agricultural Economics, University of Manitoba), Rodolfo Frank (Academia Nacional de Agronomía y Veterinaria) on the manuscript and two anonymous referees.

Aknowledgements

13. FRANK, L. 2010. Constrained Estimation with Distorted Data by the Least-Squares Criterion. 2010 Proceedings of the American Statistical Association, Business and Economic Statistics Section, Alexandria, VA: American Statistical A s s o c i a t i o n : 1 9 2 6 - 1 9 3 2 . A v a i l a b l e a t : https://www.amstat.org.

14. GALANTINI, J. LANDRISCINI, M. FERNÁNDEZ, R. MINOLDO, G. CACCHIARELLI, J. and J.O. IGLESIAS 2005. Trigo: Fertilización con Nitrógeno y azufre en el sur y sudoeste Bonaerense. International Plant Nutrition Institute. Available at: http://www.ipni.net.

15. GARCÍA, F. and K. FABRIZI 1998. Fertilización de trigo y maíz bajo siembra directa en el sudeste de Buenos Aires. Bol. Téc., 150. INTA Balcarce.

16. GARCÍA, F. FABRIZI, K. BERARDO, A. and F. JUSTEL 1998. Fertilización de trigo en el Sudeste Bonaerense: respuesta, fuentes y momentos de aplicación. XVI Congreso Argentino de Ciencia del Suelo: pp. 109-110.

17. GREENE, W. 2006. Econometric Analysis. Prentice Hall.

18. GROTHMANN, R. 2010. Euler Math Toolbox. Downloadable from: http://eumat.sourceforge.net/.

19. JUDGE, G. GRIFFITHS, W. CARTER HILL, R. LÜTKEPOHL, H. and T. CHAO LEE 1985. The Theory and Practice of Econometrics. Wiley Series in Probability and Statistics.

20. SÁNCHEZ, M. and L. ASCHERI 2006. Respuesta a la fertilización nitrogenada en trigo CREA Monte Buey-Inriville: Campañas 2004/05 y 2005/06. Informaciones Agronómicas del Cono Sur, 30: 16-19.

21. S Ay D S ( S E C R E TA R Í A D E A M B I E N T E Y DESARROLLO SUSTENTABLE DE LA NACIÓN)

2003. Mapa de Suelos – Sistema Soil Taxonomy. Available at: http://medioambiente.gov.ar.

22. SNIH (SISTEMA NACIONAL DE INFORMACIÓN HÍDRICA) 2001. Mapas de precipitaciones y temperatura m e d i a a n u a l ( 1 9 6 5 - 1 9 8 2 ) . A v a i l a b l e a t : http://www.hidricosargentina.gov.ar.

23. SHUMWAY, R. 1995. Recent Duality Contributions in Production Economics. Journal of Agricultural and Resource Economics, 20 (1): 178-194.

24. USDA-NCSR, 1999. Soil Taxonomy, A Basic System of Soil Classification for Making and Interpreting Soil Surveys. Second Edition. Available at: .

References

1. ÁLVAREZ, R. STEINBACH, H. ÁLVAREZ, C. and S. GRIGERA, 2003. Recomendaciones para la fertilización nitrogenada de trigo y maíz en la Pampa Ondulada. Informaciones Agronómicas del Cono Sur, 18: 14-19.

2. ÁLVAREZ, R. 2007. Capítulo 7: Fertilización de trigo. In: ÁLVAREZ, R. (ed.), Fertilización de cultivos de grano y pasturas. Diagnóstico y recomendación en la región Pampeana. Ed. Facultad de Agronomía UBA, pp. 91-119.

3. ÁLVAREZ, R. 2009a. Predicting average regional yield and production of wheat in the Argentine Pampas by an artificial neural network approach. European Journal of Agronomy, 30: 70-77.

4. ÁLVAREZ, R. 2009b. Aditividad en la respuesta de los cultivos extensivos a la fertilización con distintos nutrientes en la Región Pampeana. Informaciones Agronómicas del Cono Sur, 43: 8-11.

5. BARBERIS, L.A. DUARTE, G. SFEIR, A. MARBAN L. and M. VÁZQUEZ 1987. Respuesta de trigo a la fertilización fosforada en la Pampa Arenosa húmeda y su predicción. Ciencia del Suelo, 5:166-174.

6. BONO A. and R. ÁLVAREZ 2006. Rendimiento de trigo en la Región Semiárida y Subhúmeda Pampeana: un modelo predictivo de la respuesta a la fertilización nitrogenada. In: XX Congreso Argentino de la Ciencia del Suelo, CD.

7. BONO A. and R. ÁLVAREZ 2008. Rendimiento de trigo en la Región Semiárida y Subhúmeda Pampeana. Un modelo predictivo de la respuesta a la fertilización nitrogenada. Informaciones Agronómicas del Cono Sur, 41: 18-21.

8. C A LV I Ñ O , P. E C H E V E R R Í A , H . E . a n d M . REDOLATTI 2002. Diagnóstico de nitrógeno en trigo con antecesor soja bajo siembra directa en el Sudeste Bonaerense. Ciencia del Suelo, 20: 36-42.

9. CHAMBERS, R. 1994. Applied production analysis. A dual approach. Cambridge University Press.

10. DE JANVRY, A. 1972a. Optimal Levels of Fertilization under Risk: the Potential for Corn and Wheat Fertilization Under alternative price Policies in Argentina. American Journal of Agricultural Economics, 54 (1): 1-10.

11. DE JANVRY, A. 1972b. The Generalized Power Production Function. American Journal of Agricultural Economics, 54 (2): 234-237.

Consider the following broken function

y = θ0 + θ1x, if x≤x0

and

y = θ2 + θ3x, si x≥x0.

Clearly in x0, θ2 = θ0 + (θ θ1 3) x0, so that we can join both parts in a single expression in the following fashion

y = (θ0 + θ1x) δx≤x0 + [θ0 + θ1 0x + θ3 (xx0)] (1δx≤x0), which, after some manipulation, is equal to

y = (θ0 + θ1 0x) + θ3 (xx0) + (θ θ1 3) (xx0) δx≤x0. (A.1) Alternatively y = (θ0 + θ1x) (1δx≥x0) + [θ0 + θ1 0x + θ3 (xx0)] δx≥x0,

and we get the expression

y = θ0 + θ1x + (θ θ3 1) (xx0) δx≥x0, (A.2)

which is the expression used to introduce rainfall in (3). Besides, if we consider that θ3 = 0 in (A.1), we may write the linear-plateau function

y = θ2 + θ1 (xx0) δx≤x0.

Appendix I

Appendix II

Table 4: Average wheat yields and explanatory variables drawn from Álvarez (2007).

Zone/Nd P∪P0 Till. Pp. T° Soil Obs. Years Source1

[kg.ha∪1_] 50 100 150 200 250 300 _[kg.ha-1

] type [mm] [°C]

-- 34.0 48.0 -- -- 54.0 -- LC/SD 1020 16.5 M Before 2000 Álvarez et al. 2000b and 2004 25.0 35.0 42.0 46.0 -- -- -- LC 1020 16.5 M [1], [8] Before 1995

28.0 37.0 44.0 49.0 -- -- -- LC 1020 16.5 M [2], [8] Before 1995

28.5 32.0 -- 41.0 41.0 -- -- LC/SD 1020 16.5 M 1997-2001 Álvarez et al. 2003 -- 25.4 -- -- -- -- -26.6 LC 1020 16.5 M [3] 1979-1983

-- 29.7 -- -- -- -- -13.3 LC 1020 16.5 M [3] 1979-1983

34.5 39.6 40.0 42.6 -- -- -- LC/SD 995 16.5 M [4] 2000-2003 Salvagiotti et al. 2004 30.6 35.7 42.8 39.8 -- -- -- SD 1170 18 M, V [4] 2000-2001 Melchiori 2000 36.5 45.0 50.0 55.0 -- -- -- SD 825 16.5 E/A, M 2005-2006 Sánchez and Ascheri 2006 23.0 39.0 41.0 41.0 -- -- -- LC 800 14.5 M [8] 1987-1994

35.0 42.0 50.0 50.0 -- -- -- LC 800 14.5 M [8] 1987-1994 -- -- 35.9 -- -- -- -13.3 LC/SD 900 14.5 M [5] Before 1994 -- -- 34.1 -- -- -- -26.6 LC/SD 900 14.5 M [5] Before 1994 30.0 39.9 43.2 -- -- -- -- LC/SD 900 14.5 M [4] 1995-1996

37.6 41.3 44.6 45.8 -- -- -- LC/SD 900 14.5 M [4] 1995-1996

-- 60.7 -- 71.8 -- -- -- SD 900 14.5 M [4] 1999-2001 Calviño et al. 2002 -- 34.4 -- -- -- -- -- LC 700 15 M [6] 1980-1985

-- 32.8 -- -- -- -- -26.6 LC 700 15 M [6] 1980-1985 -- -- 23.8 -- -- -- -26.6 LC 700 15 E/A, M [7] 1980-1992 -- -- 27.6 -- -- -- -13.3 LC 700 15 E/A, M [7] 1980-1992

21.4 32.1 32.8 41.4 -- -- -- SD 700 15 E/A, M 2003-2004 Galantini et al. 2005, 2006 -- 28.0 -- -- -- -- -13.3 LC 825 16.5 E/A, M [3] 1983-1985 Barberis et al. 1987 -- 21.0 -- 27.0 -- 24.0 -- LC/SD 825 16.5 E/A, M 1996-2004

-- 26.0 -- 36.0 -- 33.0 -- LC/SD 825 16.5 E/A, M 1996-2004

Southwest Buenos Aires Ron and Loewy 1987, y Loewy 1990

Ron and Loewy 1990, 1995

Semiarid and Subhumid Pampa

Bono and Álvarez 2006 Southeast Buenos Aires González Montaner 1997

Berardo et al. 1994

García and Fabrizi 1998, and García et. al. 1998

Average yields [qq.ha∪1_]

Rolling Pampa

Satorre et al. 2001 with data from Calderini et al. 1995

Senigagliesi et al. 1986

[1] 50% available water at sowing [2] 80% available water at sowing

[3] Average yield minus average efficiency by P dose [4] Relative yield by maximum yield

[5] Average yield with 150 kg N from García and Fabrizi (1998) and García et al. (1998) minus response to phosphorus fertilization [6] Control yield from Ron y Loewy (1987). We assumed 60 kg of N-NO in soil.3

[7] Average yield with 150 kg N from Galantini et al. (2005, 2006) minus response to phosphorus fertilization [8] Yields obtained by simulation

E/A = Entisol o Aridisol, M = Mollisol, V = vertisol

1

Econometric model

y vector of yields,

X matrix of independent variables; X̃is X re-scaled to unity but not centered,

vector of parameters,

2

random vector with 0 mean and covariance matrix σΩ,

2 b vector of estimates of ; b has covariance matrix Σ,

R and r define a set of linear restrictions on .

The profit function

p wheat price,

f(x,z) is the technological function,

x vector of inputs,

z vector of uncontrollable factors,

w vector of input prices,

c is the sum of all fixed costs.

Throughout the text

ξ elasticity,

LC conventional tillage, MO organic matter,

N available nitrogen N = N + N ,d d s f

N are the N-NO in soil,s 3

N is the N added by fertilization,f

N organic nitrogen,org

R rainfall,

SD zero tillage, which may be decomposed in φ multiplicative factors,

T temperature.

De Janvry's model

1 y yield in [kg.ha ], ln(A) is a scale constant,

1

x j is the dose of the j-th nutrient in [kg.ha ] applied as fertilizer,

x row vector of powers of x and z is the vector of powers of the organic matter content in [%],

(j) (j)

fj(x ,z) polynomials where x is x but excluding the terms with xj,

g(x)̃ linear function of N , P and the interaction N × P ,f f f f

ln(R) climate index, where ln(R) = θ1 ln(w1) + ... + θr ln(wr),

w1,...,wr are the environmental variables and θkare fixed parameters,

vm agronomic practices and soil features, δh geographical or varietal dummies, εiis a random variable.

Álvarez's model

1 y is the yield in [kg.ha ],

-1 x row vector of powers of x and x is N in [kg.ha ],d

z row vector of powers of z and z is organic nitrogen in [%],

f1(x) and f2(z) are polynomials of second degree of the form

2

α1x + α2x,

δLC dummy variable indicating tillage practice,

w moisture at seeding in [mm], δv1 dummy indicating soil depth, δv2 dummy indicating soil texture.

The proposed model

1 y is wheat yield in [kg.ha ], αjfixed parameters,

x1 scaling parameter,

1

x2 level of N in [kg.ha ] on a logarithmic scale, d 1 x3 variable associated with P deficit in [kg.ha ],

x4 variable indicative of the type of tillage,

1 x5 is the mean annual rainfall in [mm.year ],

x6 variable associated with rainfall level,

x7 mean annual temperature in [°C],

x8 dummy variable for entisols or aridisols

x9 dummy for vertisols, error term.

Appendix III