Ann Oper Res (2010) 176: 369–378DOI 10.1007/s10479-008-0472-5

Estimation of production risk and risk preferencefunction: a nonparametric approach

Subal C. Kumbhakar · Efthymios G. Tsionas

Published online: 15 November 2008© Springer Science+Business Media, LLC 2008

Abstract While estimating parametric production models with risk, one faces two mainproblems. The first problem is associated with the choice of functional forms on the meanproduction function and the risk (variance) function. The second problem is associatedwith the specification of the risk preference function. In a parametric model the researcherchooses some ad hoc functional form on all these. It is obvious that the estimated (i) tech-nology (mean production function), (ii) risk and (iii) risk preference functions are affectedby the choice of functional form. In this paper we consider an estimation framework thatavoids assuming parametric functions on all three. In particular, this paper deals with non-parametric estimation of the technology, risk and risk preferences of producers when theyface uncertainty in production. Uncertainty is modeled in the context of production theorywhere producers’ maximize expected utility of anticipated profit. A multi-stage nonparamet-ric estimation procedure is used to estimate the production function, the output risk functionand the risk preference function. No distributional assumption is made on the random termrepresenting production uncertainty. No functional form is assumed on the underlying util-ity function. Rice farming data from Philippines are used for an empirical application of theproposed model. Rice farmers are, in general, found to be risk averse; labor is risk decreas-ing while fertilizer, land and materials are risk increasing. The mean risk premium is about3% of mean profit.

Keywords Production risk · Risk preference function · Risk premium · Kernel method

S.C. Kumbhakar (�)Department of Economics, State University of New York, Binghamton, NY 13902, USAe-mail: kkar@binghamton.edu

E.G. TsionasDepartment of Economics, Athens University of Economics and Business, 76 Patission Street,104 34 Athens, Greecee-mail: tsionas@aueb.gr

370 Ann Oper Res (2010) 176: 369–378

1 Introduction

Risk in production theory is mostly analyzed under (i) output price uncertainty and (ii)production uncertainty (commonly known as production risk). Production uncertainty isquite popular in applied work because it is often explained by inputs. That is, input quantitiesnot only determine the volume of output produced but some of these inputs also affectvariability of output (often labeled as production risk). For example, fertilizer might be riskaugmenting while labor might decrease output risk. Here we address the implications ofproduction risk in a framework where producers maximize expected utility of anticipatedprofit. In particular, we examine input allocation decision in the presence of production risk.

Although the theoretical work on risk in the production literature is quite extensive,there are relatively fewer empirical studies devoted to analyzing different sources of riskon production and input allocation. Most of these studies either looked at output price un-certainty (Appelbaum and Ullah 1997; Kumbhakar 2002; Sandmo 1971; Chambers 1983)or production risk along the Just and Pope (1978) framework (Tveterås 1999, 2000; Ascheand Tveterås 1999; Kumbhakar and Tveterås 2003, and many others). To examine produc-ers’ behavior under risk some parametric forms of the utility function, production func-tion and output risk function along with specific distributional assumptions on the errorterm representing risk are considered in the existing literature (Love and Buccola 1991;Saha et al. 1994). Thus, the risk studies in the production literature have some or all ofthese features built in, viz., (i) parametric forms of the production and risk function, (ii)parametric form of the utility function, (iii) distributional assumption(s) on the error term(s)representing production risk and/or output price uncertainty.

This paper estimates nonparametrically the production function, the risk function andrisk preference function associated with production risk. The main advantage of the non-parametric approach used here is that the results are robust to functional form used. In arecent survey Yatchew (1998) argues that economic theory rarely, if ever, specifies precisefunctional forms for production or risk functions. Consequently, its implications are not,strictly speaking, testable when arbitrary parametric functional forms are specified. To theextent that the production or risk functions are misspecified estimates of risk preferencefunctions may be biased.1 By using nonparametric technique it is possible to estimate therisk preference function that do not depend on specific functional form of the underlyingutility function, the production and output risk functions. Similarly, estimates of producer-specific risk premium are obtained without making any assumptions on the functional formof the underlying utility function.

The rest of the paper is organized as follows. The production risk model is presentedin Sect. 2. Nonparametric estimation of the production, risk and risk preference functionsare considered in Sect. 3. Section 4 discusses results from the application of our model toPhilippines rice farming data. Section 5 concludes the paper with a brief summary of results.

2 A production risk model

Assume that the production technology can be represented by the Just-Pope (1978) form,viz.

y = f (X) + g(X)ε, ε ∼ (0,1) (1)

1For an excellent review of nonparametric methods in econometrics, see Pagan and Ullah (1999).

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where y is output, X is a vector of variable inputs and f (X) is the mean output function. Weassume that E(ε|X) = 0 and E(ε2|X) = 1. Since output variance is represented by g2(X),the g(X) function is labeled as the output risk function. In this framework an input j is saidto be risk increasing (decreasing) if gj (X) > (<)0.

We consider the case where output and input markets are competitive and their prices areknown with certainty. The production is, however, uncertain. Assume that producers maxi-mize expected utility of anticipated profit E(U(πe)) to determine optimal input quantities,which in turn determines output supply. Define anticipated profit2 πe as

πe = py − rX = pf (X) − rX + pg(X)(ε) ≡ μπ + pg(X)ε (2)

where μπ = E(πe) = pf (X) − rX,p being the output price and r the input price vector.The first-order conditions (FOCs) of expected utility of anticipated profit E(U(πe)) max-

imization can be written as

E(U ′(πe){pfj (X) − rj + pgj (X)ε}) = 0 (3)

where U ′(πe) is the marginal utility of anticipated profit, fj (X) and gj (X) are partial deriv-atives of f (X) and g(X) functions with respect to input Xj , respectively.

Rewrite the above FOCs as

fj (X) = rj /p − gj (X)θ(.) (4)

where

θ(r,p,X.) ≡ E(U ′(πe)ε)

E(U ′(πe))(5)

The θ(.) term in the first-order conditions (4) is the risk preference function associated withproduction risk. If producers are risk averse then θ(.) < 0 (i.e., an increase in ε increasesπe which in turn reduces U ′(πe) since U ′′(πe) < 0 (utility function being concave)). Simi-larly, θ(.) is positive if producers are risk lovers and is zero for risk neutral producers. It isgenerally a function of X,p, and r , since in general, U(πe) is a function of p, r , and X.

The risk preference function θ(.) can be related to the Arrow-Pratt measure of risk aver-sion. To show this we write πe = μπ + p.g(X).ε and take a Talyor series expansion ofU ′(πe) at μπ . This enables us to rewrite θ(.) as

θ(r,p,X.) ≡ E(U ′(πe)ε)

E(U ′(πe))= E[(U ′(μπ) + U ′′(μπ).p.g(X).ε + · · ·)ε]

E(U ′(μπ) + U ′′(μπ).p.g(X).ε + · · ·) = U ′′(μπ).p.g(X)

U ′(μπ)

= −AR(μπ).p.g(X) (5′)

where AR(μπ) = −U ′′(μπ)/U ′(μπ) is the Arrow-Pratt measure of risk aversion.The risk preference function θ(.) plays an important role in input use decisions. This can

easily be seen by expressing the FOCs in (4) as

fj (X) = rj

pφj (6)

2We call it anticipated (instead of actual) profit because πe in (2) is random.

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where φj (.) ≡ 1 − (p/rj )gj (x)θ(.). If gj (X) > 0, then for risk averse producers φj > 1which means that producers do not equate marginal product (expected) of an input to theobserved price of that input. Since the value of (expected) marginal product of the input Xj

exceeds its price (pfj (X) > rj ), a risk averse producer will use the input less relative to arisk neutral producer (θ = 0). Alternatively, the risk averse producer internalizes cost of therisky input by raising its virtual (shadow) price to rjφj > rj . Similarly, if producer A is morerisk averse than an otherwise identical producer B (i.e., φj for A is greater than φj for B),producer A will use less of input Xj than producer B, ceteris paribus.

3 Econometric model and estimation

Estimating risk preference function θ requires deriving algebraic expressions for θ involvingunknown parameters and data. This is not always possible without making some assump-tions about the functional form of the underlying utility function and distributional assump-tions on the error terms. Furthermore, one needs to make assumptions on the functionalforms of f (X) and g(X). All these problems can be avoided if one uses nonparametrictechniques to estimate f (X) and g(X) as well as the risk preference function.

3.1 Estimation of f,g functions and their partial derivatives

Suppose Xj ∈ Rd (j = 1, . . . ,m) is a vector of explanatory variables (that include bothvariable and quasi-fixed inputs), and yj denotes output (the dependent variable). We assumethat there is a production function of the form

y = f (X) + v (7)

where f : Rd → R is an unspecified functional form, and v is an error term. Our objectiveis to obtain estimates of f (X) and g(X) as general possible. So we do not consider separa-ble specifications that are popular when dimensionality reductions are desired. We use themultivariate kernel method to obtain an estimate of f (X) at a particular point X ∈ Rd asfollows. First, we estimate the density of X (p̃(X)) as:

p̃(X) = (Nh)−1N∑

i=1

Kh(X − Xi) = (Nh)−1N∑

i=1

d∏

j=1

K(Zj − Zji) (8)

where Kh(w) = exp(− 12h2 (w −w)′�̃−1

X (w −w)) is the d-dimensional normal kernel, h > 0is the bandwidth parameter, K(w) = exp(− 1

2w2) is the standard univariate normal kernel,�̃X is the sample covariance matrix of Xi (i = 1, . . . , d),

Zi = A(Xi − X̄)/λ,

A�̃XA = Id,

X̄ = N−1N∑

i=1

Xi,

and λ is a smoothing parameter. The optimal choices for h and λ are

h = λd |�̃X|1/2,

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λ =(

4

(2d + 1)N

)d+4

.

The unknown function is then estimated as

f̃ (X) = (Nh)−1N∑

i=1

Whi(X)yi (9)

where

Whi(X) ≡ Kh(X − Xi)/p̃(X)

(see Hardle 1990, pp. 33–34). The estimates are adjusted near the boundary using the pro-cedures discussed in Rice (1984), Hardle (1990, pp. 130–132), and Pagan and Ullah (1999,Chap. 3).

First derivatives of f (X) is obtained from

∂f̃ (X)

∂X= (Nh)−1

N∑

i=1

∂Whi(X)yi/∂X.

More specifically,

∂f̃ (X)/∂Xj = −(Nh)−1

[N∑

i=1

GjiKh(X − Xi)yi − f̃ (X)

N∑

i=1

GjiKh(X − Xi)

]/p̃(X)

(10)where

Gji = λ−2d∑

k=1

σ̃jk

X (Xk − Xki)

and

�̃X = [σ̃ jk

X , j, k = 1, . . . , d].Given the estimate of f̃ (Xi) one can obtain the residuals ei from ei = yi − f̃ (Xi). Anestimate of the variance can then be obtained from

σ̃ 2(X) = (Nh)−1N∑

i=1

Whi(X)e2i (11)

(see Hardle 1990, p. 100; Pagan and Ullah 1999, pp. 214–215). Since g(X) = σ̃ (X), esti-mates of the g(X) function and its gradient ∂g(X)/∂X can be obtained. Alternatively, theg(X) can be obtained from a nonparametric regression of |ei | on Xi in a second step. Thegradient of g(X) could then be obtained by a procedure similar to the one used to obtain thegradient of f (X) in (10).

3.2 Estimation of risk preference functions and risk premium

To estimate the risk preference function θ ≡ θ(X, r/p) we rewrite the relationship in (4) as

Dj ≡ f̃j (X) − rj /p

−g̃j (X)≡ ζ(X, r/p) + ηj , j = 1, . . . ,m (12)

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where [η1, . . . , ηm]′ ∼ (0,�η) and m ≤ d is the number of variable inputs. Since Dj iscomputed using the estimated values of fj (X) and gj (X)—the error term ηj captures thediscrepancy between the true and estimated values of θ ≡ θ(X, r/p). Our objective is toestimate the ζ(X, r/p) function that will be the estimator of the risk preference functionθ(.).

To get a better understanding of the importance of risk preference and the degree of riskaversion among firms, researchers often compute risk premium (RP) defined as the amountof money that would make a producer indifferent between uncertain profit πe and certainprofit E(πe) − RP. Taking a first-order approximation of U{E(πe) − RP} = U(μπ − RP)

around RP = 0, and a second-order approximation of U(πe) around μπ , Antle (1987) andChavas and Holt (1996) have shown that

RP = 0.5AR(μπ)Var(πe), (13)

where AR is the Arrow-Pratt measure of absolute risk aversion (i.e., AR =−U ′′(μπ)/U ′(μπ)). RP is the risk premium which is the implicit cost of private risk taking.

Using the definition in (13), the formula for risk premium can be expressed as:

RP = −1

2θ(.)pg(X). (14)

4 An application to Philippines rice farming

The model presented in the preceding sections is applied to Philippines rice farming data.The sample consists of 43 rice farms from four villages in the province of Tarlac in Philip-pines. The period covered is from 1990–1997. Thus total number of observations is 344.Details on the data can be found in Villano (2004).3 The output variable is tones of paddy.The inputs are: hectares of land (area) used in rice planting, man-days of hired and familylabor, kilogram of fertilizer, materials (quantity index of all other inputs).

First we report the estimated elasticities of the mean output function (f (X)) with respectto area, labor, fertilizer and materials (listed as other). These are popularly known as inputelasticities. We plot the empirical distribution of these elasticities in Fig. 1.4 The mean valuesof these elasticities are: 0.49,0.31,0.25, and 0.02, respectively. It can be seen that none ofthe distributions (except for fertilizer) is symmetric. In fact they are all skewed to the right. Itcan be clearly seen that these elasticites vary quite a lot and hence a parametric form (suchas the Cobb-Douglas) that restricts these elasticities to be the same for all farms in everyyear is inappropriate.

Since the input elasticities are related to returns to scale (RTS is the sum of input elastic-ities), it is natural to report RTS measures. The histogram of RTS is reported in Fig. 2. It canbe seen from the figure that majority of the farms are operating near unitary returns to scale.In other words, very few farms can benefit from expanding (contracting) their operations.

3Data used in this paper is downloaded from the website, www.uq.edu.au/economics/cepa/crob2005.4These elasticities are positive for all farms in every year. This shows that there are no violations of theproperties of the underlying production technology (viz., positive marginal product). Violations often occurwhen one uses a flexible parametric production function such as the translog. In nonparametric models itis, however, possible to eliminate negative marginal products by restricting them with their lowest allowablebound (zero), see Pagan and Ullah (1999, pp. 175–176).

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Fig. 1 Distributions of input (f (x)) elasticities

Fig. 2 Distribution of returns toscale

In rice production, risk plays an important part. Consequently, it is important to knowwhich input(s) is (are) risk increasing (decreasing). For this we estimate the partial deriva-tives of production risk, the g(X) function. These elasticities can also be regarded as partialvariance elasticities, and are useful to determine whether an input is risk increasing (de-creasing) from the sign on them (positive elasticity indicated risk increasing). Based onthe estimates of these elasticities we find that labor is, in general, risk reducing. All otherinputs are found to be risk increasing. In Fig. 3 we report them for area, labor, fertilizerand materials. The mean values of these elasticities for area, labor, fertilizer and materials:0.59,−0.16,0.24, and 0.06, respectively. It is clear from Fig. 3 that these partial risk elastic-ities vary across farms. Thus the assumption of a CD function for the g(X) function (which

376 Ann Oper Res (2010) 176: 369–378

Fig. 3 Distributions of risk (g(x)) elasticities

Fig. 4 Distribution of totalvariance elasticity

makes these elasticities constant for all farms) is inappropriate. In Fig. 4 we report sumof these partial risk elasticities which are also known as total variance elasticity (Tveterås1999). For most of the farms total variance elasticities are quite close, although the partialrisk elasticities are different. This means that one might not get the true picture unless thepartial risk elasticities are examined.

Elasticities of the mean output and risk functions for each input are derived from theestimates of the f (X) and the g(X) functions and their partial derivatives. We use the es-timated values of f (X) and g(X) and their partial derivatives to obtain estimates of therisk preference function θ(.) and estimates of risk premium (RP) in the second step. SinceRP gives a direct and more readily interpretable result, reporting of RP is often preferred.Given that the RP measure is dependent on units of measurement, a relative measure of RP

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Fig. 5 Distributions of relativerisk premium

(defined as RRP = RP/μπ) is reported. Relative risk premium (RRP) is independent of theunits of measurement. RRP also takes farm heterogeneity into account by expressing RP inpercentage terms.

The frequency distribution of RRP is reported in Fig. 5. The distribution is skewed to theright. The mean value of RRP is 0.03. RP shows how much a risk averse farm is willing topay to insure against uncertain profit due to production risk. The RRP, on the other hand,shows what percent of mean profit a risk averse farm is willing to pay as insurance. Theabove results show that on average a farm is willing to pay about 3% of the mean profit asan insurance against possible profit loss due to production risk.

5 Summary and conclusions

This paper deals with nonparametric estimation of production risk. We consider an approachin which producers maximize expected utility of anticipated profit to solve input allocationproblem. In contrast to the risk studies in the production literature that are based on built-in features such as (i) parametric forms of the production and risk function, (ii) parametricform of the utility function, (iii) distributional assumption(s) on the error term(s) represent-ing output risk, our nonparametric approach avoid all these restrictive features. We estimatethe production function, the risk function (output risk), and risk preference function non-parametrically and avoid making any functional form assumption on them. Furthermore,we do not make any distributional assumption on the error term representing productionrisk.

We use the Philippines rice farming data as an application of our model. Results showthat input elasticities are far from constant across farms. This finding shows that a Cobb-Douglas functional form is inappropriate for modeling the mean production function. Wealso find that labor is risk reducing, while other inputs (area, fertilizer and materials) arerisk increasing. Finally our results show that the rice farmers risk averse. We examinedfarm-specific values of risk premium (as a percent of profit) to study the cost of private riskbearing. These risk premiums are positive, but vary among farms, and over time. The meanvalue of relative risk premium is found to be 0.03—thereby meaning that, on average, farmsare willing to pay about 3% of their mean profit as insurance to protect against profit lossesdue to production risk.

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