Bio-based and Applied Economics 7(3): 191-215, 2018 ISSN 2280-6180 (print) © Firenze University Press ISSN 2280-6172 (online) www.fupress.com/bae Full Research Article DOI: 10.13128/bae-7675 Positive Mathematical Programming and Risk Analysis Quirino Paris Department of Agricultural and Resource Economics, University of California, Davis Date of submission: 2018 3rd, July; accepted 2019 20th, March Abstract. In 1956, Freund introduced the analysis of agricultural price risk in a mathematical programming framework. His discussion admitted only constant abso- lute risk aversion. This paper generalizes the treatment of risk preference in a math- ematical programming approach along the lines suggested by Meyer (1987) who demonstrated the equivalence of expected utility of wealth and a function of mean and standard deviation of wealth for a wide class of probability distributions that dif- fer only by location and scale. This paper extends the definition of calibration under Positive Mathematical Programming (PMP) by considering limiting input prices along with the traditional decision variables. Furthermore, it shows how to formu- late an analytical specification for the estimation of the risk preference parameters and calibrates the model to the base data within small deviations. The PMP approach under generalized risk allows also the estimation of output supply elasticities and the response analysis of decoupled farm subsidies that recently has interested policy mak- ers. The approach is applied to a sample of farms that do not produce all the sample commodities. Keywords. Risk analysis, positive mathematical programming, model calibration, chance constraint, policy analysis. JEL. C6. 1. Introduction This paper accomplishes several objectives: 1. It presents a procedure to estimate generalized risk preferences in combination with Positive Mathematical Programming (PMP). 2. It obtains a unique calibrating solution of a PMP model even with a sample of farms that produce zero levels of some crops. 3. It estimates a complete cost function that can be used in a calibrating model for poli- cy analysis. 4. It shows that Phase I and Phase II of the classical PMP procedure give identical and unique results. 5. It shows how to incorporate exogenously given supply elasticities. 192 Quirino Paris 6. It extends the meaning of calibration in PMP by minimizing the distance of optimal solutions from observed output levels and limiting input prices. In this way, it dis- penses from the necessity of a user-determined parameter that was originally intro- duced to guarantee a positive shadow price of binding constraints. The treatment of agricultural price risk in a mathematical programming framework has dealt mainly with either an exponential utility function and constant absolute risk aversion (CARA) or the minimization of total absolute deviation (MOTAD) of income. The first approach, originally proposed by Freund (1956), appealed to the expected util- ity (EU) hypothesis and assumed that random prices were normally distributed. These assumptions lead to a mean-variance specification of the certainty equivalent (CE) defined as total expected revenue minus a risk premium. Such a premium corresponds to half the variance of revenue multiplied by a constant absolute risk aversion coefficient. The MOTAD approach was proposed by Hazell (1971) who justified its introduction with the difficult access – at that time – to a quadratic programming computer software necessary to solve a mean-variance model. According to Hazell (1971, p. 56), the MOTAD specifica- tion “has an important advantage over the mean-variance criterion in that it leads to a lin- ear programming model in deriving the efficient mean-absolute deviation farm plans.” The MOTAD model approximates a mean-standard deviation (MS) criterion but it says noth- ing about the economic agent’s risk preference with regard to either decreasing (constant, increasing) absolute or relative risk aversion. Recently, Cortignani and Severini (2012), Arata et al. (2017) and Paris (2018) have combined PMP with a CARA specification of risk preferences. It is difficult, however, to accept the idea that farmers risk behavior does not account for changes in wealth as the CARA approach stipulates. Petsakos and Rozakis (2015) have presented a combina- tion of the traditional PMP specification with a decreasing absolute risk aversion (DARA) parameter. The present paper combines a more encompassing specification of PMP (cali- bration of output quantities and limiting input prices) with generalized risk preferences where the behavior of the risk-avert farmer can vary over all theoretically possible prefer- ences (CARA, DARA, IARA, constant, decreasing and increasing relative risk aversion). The paper deals with market price risk leaving the treatment of production risk for further research. The mean-standard deviation approach has a long history [Fisher (1906), Hicks (1933), Tintner (1941), Markowitz (1952), Tobin (1958)]. Meyer (1987) presented a rec- onciliation between the EU and the MS approaches that may be fruitfully applied in a positive mathematical programming (PMP) analysis of economic behavior under risk. The main objective of Meyer was to find consistency conditions between the EU and the MS approaches in such a way that an agent who ranks the available alternatives according to the value of some function defined over the first two moments of the random payoff would rank those alternatives in the same way by means of the expected value of some utility function defined over the same payoffs. It turns out that the location and scale condition is the crucial link to establish the consistency between the EU and the MS approaches. We reproduce here Meyer’s argument (1987, p. 423): “Assume a choice set in which all random variables Yi (with finite means and vari- ances) differ from one another only by location and scale parameters. Let X be the ran- dom variable obtained from one of the Yi using the normalizing transformation Xi = 193Positive Mathematical Programming and Risk Analysis (Yi-μi)/σi where μi and σi are the mean and standard deviation of Yi. All Yi, no matter which was selected to define X, are equal in distribution to μi+σiX. Hence, the expected utility from Yi for any agent with utility function u( ) can be written as EU(Yi )= u(µi +σ ix)dF(x)≡V( a b ∫ µi ,σ i ) (1) where a and b define the interval containing the support of the normalized random varia- ble X.” “… under the location and scale condition, various popular and interesting hypoth- eses concerning absolute and relative risk-aversion measures in the EU setting can be translated into equivalent properties concerning V(μi,σi).” Given the assumptions made by Meyer about first and second derivatives, V(μ,σ) is a concave function of μ and σ. Concav- ity is established when second derivatives Vμμ and Vσσ are non-positive and VμμVσσ-Vµσ 2 ≥0. The structure of absolute risk (AR) is measured by the slope of the indifference curves in the (μ,σ) space that is represented as AR(µ,σ )= −Vσ (µ,σ ) Vµ(µ,σ ) (2) where Vμ(μ,σ) and Vσ(μ,σ) are first partial derivatives of the V(μ,σ) function. Some proper- ties of this risk measure are: 1. Risk aversion is associated with AR(μ,σ)>0, risk neutrality with AR(μ,σ)=0 and risk propensity with AR(μ,σ)<0. 2. If u(μ+σx) displays decreasing (constant, increasing) absolute risk aversion for all μ+σx, then ∂AR(µ,σ ) ∂µ <(=,>) 0 for all μ and σ>0. 3. If u(μ+σx) displays increasing (constant, decreasing) relative risk aversion for all μ+σx, then ∂AR(tµ,tσ ) ∂t >(=,<) 0 for t>0. Saha (1997) proposed a two-parameter MS utility function that conforms to Meyer’s specification: V(μ,σ)= μθ-σγ (3) and assumed that θ>0. According to this MS utility function, the absolute risk measure (AR) is specified as AR(µ,σ )= −Vσ (µ,σ ) Vµ(µ,σ ) = γ θ µ(1−θ )σ (γ −1) . (4) 194 Quirino Paris Hence, risk aversion, risk neutrality and risk propensity are specified by γ>0, γ=0 and γ<0, respectively. As economic agents do not, in general, operate directly upon expected wealth and its standard deviation but, rather, upon a string of decision variables such as output and input levels, it is important to analyze the behavior of the absolute risk meas- ure (AR) under risk aversion and risk propensity. The justification for this requirement is due to the fact that knowledge of parameters θ and γ is obtained only by empirical esti- mation of economic relations involving entrepreneur’s decisions. The sign of these param- eters, therefore, is an empirical question. For γ>0, (risk aversion), decreasing, constant and increasing absolute risk aversion is defined by ∂AR(µ,σ ) ∂µ = (1−θ )γ θ µ−θσ (γ −1) < (=,>)0 (5) and, therefore, by θ>1, θ=1, θ<1, respectively. For γ>0, (risk propensity), decreasing, con- stant and increasing absolute risk propensity is defined by θ<1, θ=1, θ>1, respectively. For γ>0, (risk aversion), decreasing, constant and increasing relative risk aversion is defined by ∂AR(tµ,tσ ) ∂t t=1 = (γ −θ )AR < (=,>)0 (6) and, therefore, by θ>γ, θ=γ, θ<γ respectively. For γ<0, (risk propensity), neither decreas- ing nor constant relative risk propensity are applicable because the combination of param- eters’ signs produces always a positive derivative. Increasing relative risk propensity is defined by any value of θ>0. The meaning of decreasing absolute risk aversion relates to an economic agent who experiences a wealth increase and chooses to augment his investment – measured in abso- lute terms – in the risky asset. Decreasing relative risk aversion relates to an economic agent who experiences a wealth increase and chooses to increase the share of his invest- ment in the risky asset. It is possible, therefore, for an economic agent to behave according to a decreasing absolute risk aversion framework and an increasing relative risk aversion scenario if the absolute amount of increase in the risky asset is not sufficient to increase also the share of that asset. In any given sample of economic agents’ performances, there- fore, the prevailing combination of risk preference is an empirical question. The risk anal- ysis of Meyer (1987) admits all possible combinations of risk behavior (risk aversion and risk propensity). Saha (1997) listed the risk aversion combinations for the MS utility func- tion specified in relation (3) when γ>0. Table 1, for example, admits absolute risk aversion behavior that may be decreasing, when θ>1 and γ>0, in association with either increasing relative risk aversion when γ>θ>0 or decreasing relative risk aversion when θ>γ. Decreas- ing, constant and increasing absolute risk aversion are denoted by DARA, CARA and IARA, respectively. Decreasing, constant and increasing relative risk aversion are denoted by DRRA, CRRA and IRRA, respectively. 195Positive Mathematical Programming and Risk Analysis Table 1. Possible risk preferences under risk aversion (θ>0, γ>0) DRRA CRRA IRRA DARA θ>1, θ>γ θ>1, θ=γ θ>1, θ<γ CARA θ=1, θ>γ θ=1, θ=γ θ=1, θ<γ IARA θ<1, θ>γ θ<1, θ=γ θ<1, θ<γ Table 2. Possible risk preferences under risk propensity (θ>0, γ<0) DRRP CRRP IRRP DARP θ<1, NA θ<1, NA θ<1, YES CARP θ=1, NA θ=1, NA θ=1, YES IARP θ>1, NA θ>1, NA θ>1, YES “NA” stands for “Not Applicable” because the combi- nation of parameters’ signs produces always a posi- tive value of the derivative (6). When θ>0 and γ<0, risk propensity is active and the behavior of the risk measure AR, under the given MS utility, assumes the specification reported in Table 2. Decreasing, constant and increasing absolute risk propensity are denoted by DARP, CARP and IARP, respectively. Decreasing, constant and increasing relative risk propensity are denoted by DRRP, CRRP and IRRP, respectively. The V(μ,σ)=μθ-σγ function is concave with respect to μ and σ when θ<1 and γ>1. The same function V[μ(x),σ (x)]= μ (x)θ-σ (x)γ, however, exhibits a flexible behavior with respect to entrepreneur’s decisions, x. This behavior depends on the relative values of parameters θ and γ. In other words, the upper contour sets of V[μ(x),σ (x)]= μ (x)θ-σ (x)γ are convex for a wide range of values of parameters θ and γ. A few examples illustrate the function’s graph and the associated upper contour sets in the appendix. The rest of the paper is organized as follows. Section 2 discusses a PMP model that combines a generalized risk analysis with an extension of calibration constraints involving observed prices of limiting inputs. This extension integrates the traditional PMP specifi- cation of calibration constraints dealing only with observed levels of realized outputs. In particular, the extension provides a unique estimate of the optimal decision variables and avoids the user-determined perturbation parameters introduced by Howitt (1995a, 1995b) to guarantee that the dual variables of binding structural constraints will assume positive values. Section 3 discusses a chance-constrained relation that anchors the θ and γ param- eters to the decision quantities and, therefore, provides an independent relation for their estimation. Section 4 assembles a Phase-I estimation model of the novel PMP approach. Section 5 defines and estimates a complete cost function involving output quantities and limiting input prices. The derivatives of the cost function are used in calibrating models that are suitable for policy analysis. Section 6 discusses how to obtain endogenous (to a farm sample) output supply elasticities. This section matches exogenous (to the farm sam- 196 Quirino Paris ple) supply elasticities (available through econometric estimation, for example) with the endogenous supply elasticities. Section 7 states that optimal decision variables are identi- cal whether estimated as solution of the Phase I model or solution of Phase I and Phase II models combined. Section 8 defines two alternative calibrating equilibrium models which reproduce calibrating solutions that are identical to those ones obtained in section 4. Sec- tion 9 presents the empirical results of the more elaborate PMP and risky model applied to a sample of 14 farms when not all farms produce all commodities. Conclusions follow. 2. Generalized Risk Preference in a PMP Framework A Positive Mathematical Programming approach has been adopted frequently to ana- lyze agricultural policy scenarios ever since Howitt proposed the methodology (1995a, 1995b). In this section, we extend the PMP methodology to deal with generalized risk preference and risky market output prices. Furthermore, we extend the PMP methodol- ogy to deal with calibration constraints involving observed prices of limiting inputs, say land. This extension modifies the traditional specification of calibration constraints and the notion of calibrating solution, as explained further on. Suppose N farmers produce J crops using I limiting inputs and a linear technology. Let us assume that, for each farmer, the (J×1) vector of crops’ market prices is a ran- dom variable !p with mean E( !p) and variance-covariance matrix ∑p. A (J×1) vector c of accounting unit costs is also known. The (I×1) vector b indicates farmer’s availability of limiting resources. The matrix A of dimensions (I×J,I 0 . And let V be a nonsingular diagonal matrix of dimensions (I×I) with positive diagonal terms bi/yobs,i>0. The purpose of matri- ces W and V is twofold. First, to render homogeneous the units of measurement of all terms in the objective function of models defined below. Second, to weigh the deviations h and u according to the scale of the corresponding expected price and input size, respec- tively. Using a least-squares approach for the estimation of deviations h and u, it turns out that, by the self-duality of least squares (LS), λ=Wh and ψ=Vu, where ψ is the vector of Lagrange multipliers associated with constraints (9): see Paris (2015). To show this result, consider the following weighted LS problem minLS=h´Wh/2+u´Vu/2 subject to x=xobs+h dual variable λ y=yobs+u dual variable ψ. The corresponding Lagrange function and first-order-necessary conditions with respect to h and u are L=h´Wh/2+u´Vu/2+λ´(x-xobs-h)+ψ´(y-yobs-u) ∂L ∂h =Wh−λ = 0λ 198 Quirino Paris ∂L ∂u =Vu−ψ = 0 with the result that λ=Wh and ψ=Vu as asserted. A crucial issue concerns parameters θ and γ. On the one hand, an economic entrepre- neur wishes to maximize her utility of random wealth while minimizing the disutility of its risk. On the other hand, it is a fact that high levels of current income (a component of wealth) are associated with high risk of losses. Another fact is that this entrepreneur has already made her choice and executed a production plan, xobs, in the face of output price risk. It is also likely that she does not know (or that she is not even aware of) parameters θ and γ. The challenge, therefore, is to infer – from her decisions – the values of parameters θ and γ that could explain the behavior of this entrepreneur in a rational fashion. 3. A Chance-Constrained Relation for θ and γ Charnes and Cooper (1959) proposed a very interesting approach to deal with risky prospects based upon the notion of chance-constrained programming. This idea is par- ticularly useful within the context of this paper because it establishes an independent link between the θ and γ parameters, on one side, and the entrepreneur’s decisions, x, on the other side. Consider the following scenario. With some probability, a farmer may survive unfavorable events such as total revenue being less than total cost. In terms of the chance- constrained methodology this risky scenario is expressed by the following probabilistic proposition: Prob{ ! ′p x ≤ ′y Ax + (c + λλ ′) x} ≤1− β (10) where the probability that uncertain (random) total revenue ′!p x be less than or equal to certain total cost y´Ax+(c+λ)´x should be smaller than or equal to 1-β. Intuitively, for how many years could a farmer survive while operating in the red? As an example, say once every ten years. In this case, the estimated probability equals to 1-β=1/10=0.10. The y´Ax term is total cost associated with fixed limiting inputs (y´Ax= y´x). The (c+λ)´x term is total variable cost associated directly with output levels. To derive a deterministic equivalent of relation (10) it is convenient to standardize the random variable ′!p x by subtracting its expected value E( !p) ´x and dividing it by the cor- responding standard deviation (x´∑px)1/2: Prob ! ′p x − E( !p ′) x ( ′x Σ px) 1/2 ≤ ′y Ax + (c + λλ ′)) x − E( !p ′) x ( ′x Σ px) 1/2 ⎛ ⎝⎜ ⎞ ⎠⎟ ≤1− β Prob τ ≤ ′y Ax + (c + λλ ′)) x − E( !p ′) x ( ′x Σ px) 1/2 ⎛ ⎝⎜ ⎞ ⎠⎟ ≤1− β Prob[E( !p ′) x +τ ( ′x Σ px) 1/2 ≤ ′y Ax + (c + λλ ′)) x]≤1− β. (11) ψ 199Positive Mathematical Programming and Risk Analysis By assuming that τ is a standard normal random variable and choosing a value, say τ = τ , that corresponds to probability 1-β, the deterministic equivalent of relation (11) assumes the specification E( !p ′) x +τ ( ′x Σ px) 1/2 ≤ ′y Ax + ′c x + ′λλ x (12) To establish the relation between the τ parameter and the MS coefficients θ and γ the dual complementary slackness condition of constraint (8) is subtracted from the deter- ministic equivalent (12) (recall that λ=Wh): E( !p ′) x +τ ( ′x Σ px)1/2 ≤ ′y Ax + ′c x + ′h Wx −θ[w + (E( !p)− c ′) x]θ−1(E( !p)− c ′) x = − ′y Ax − ′h Wx −γ ( ′x ΣΣ px)γ /2 . (13) With simplification, relation (13) corresponds to E( !p ′) x − ′c x +τ ( ′x Σ px) 1/2 −θ[w + (E( !p)− c ′) x]θ−1(E( !p)− c ′) x + γ ( ′x ΣΣ px) γ /2 ≤ 0 (14) Relation (14) establishes a simultaneous and independent link between the risk parameters θ, γ and the decision variables x, once the value of τ is selected by the researcher. As an example, if the survival probability is determined to be 1-β=0.10, the one tail value of the standard normal random variable is τ =-1.285. 4. Phase I PMP Model – Estimation of Calibrating Primal and Dual Solutions The components of Phase I PMP model are ready to be assembled. For estimation purposes, deviations h and u will be minimized in a weighted least-squares objective function subject to relevant primal and dual constraints, their associated complementary slackness conditions and relation (14). This task leads to the following Phase I model minLS=h´Wh/2+u´Vu/2 (15) subject to Ax≤b+Vu (16) θ[w + (E( p)− c ′) x](θ−1)[E( p)− c]≤ ′A y +Wh ++ γ ( ′x Σ px) (γ /2−1)Σ px (17) x=xobs+h (18) y=yobs+u (19) y´(b+Vu-Ax)=0 (20) Σ Σ ΣΣ 200 Quirino Paris ′x { ′A y +Wh ++ γ ( ′x Σ px) (γ /2−1)Σ px −θ[w + (E( p)− c ′) x](θ−1)[E( p)− c]} = 0 (21) E( !p ′) x − ′c x +τ ( ′x Σ px) 1/2 −θ[w + (E( !p)− c ′) x]θ−1(E( !p)− c ′) x + γ ( ′x ΣΣ px) γ /2 = 0 (22) with x≥0,y≥0,θ>0,γ,h and u free. With the specification of the calibration constraints as in relations (18) and (19), the notion of a PMP calibrating solution differs from the traditional concept according to which the optimal calibrating solution is equal to the observed output levels, that is, x* ≅ xobs , as the perturbation results in a very small (user-determined) positive number. With the methodology proposed in this paper, a calibrating solution (x̂, ŷ) will not, in general, be exactly equal to the corresponding vectors of the observed production plan and input prices (xobs,yobs). The objective of model (15)-(22), therefore, is to minimize the deviations h and u in the amount allowed by the technological and risky environment fac- ing farmers. Constraints (16) represent the structural (technological) relations of input demand being less-than-or-equal to the effective input supply. Constraints (17) represent the dual relations with marginal utility of the production plan being less-than-or-equal to its mar- ginal cost. Here marginal cost has two parts: the marginal cost due to limiting and vari- able inputs, A´y+Wh, and the marginal cost of output price risk, γ(x´∑px)(γ/2-1)∑px. Con- straints (18) and (19) are the calibration relations. Constraints (20) and (21) are comple- mentary slackness conditions of constraints (16) and (17). Constraint (22) results from the chance-constrained specification (10). Because constraints (16)-(22) represent primal and dual relations and their complementary slackness conditions, any feasible solution of rela- tions (16)-(22) constitutes an admissible economic equilibrium that is consistent with the behavior of decision making under price risk. Furthermore, the calibrating solution (x̂, ŷ) is unique because the least-squares solution of (ĥ, û) is also unique. 5. Phase II PMP Model – Estimation of the Cost Function Phase II of the PMP methodology deals with the estimation of a cost function that embodies all the technological and behavioral information revealed in Phase I. Typically, a marginal cost function expresses a portion of the dual constraints in a Phase I PMP mod- el. In the absence of risk, PMP marginal cost is defined as A´y+Wh+c, where A´y stands for the marginal cost due to limiting inputs and Wh+c for the effective marginal cost due to variable outputs. In the risky price case, marginal cost is given by the right-hand-side of relation (17) where all the elements are measured in utility units. It is crucial to obtain a dollar expression of marginal cost, as in the familiar relation MC ≥ E( p) . To achieve this result, the elements of relation (17) will be divided by the term θ[w + (E( p)− c ′) x](θ−1) to write MC ≥ E( p) (23) c + 1 θ [w + (E( p)− c ′) x](1−θ )[ ′A y +Wh]++ γ θ [w + (E( p)− c ′) x](1−θ )( ′x Σ px) (γ /2−1)Σ px ≥ E( p) Σ Σ Σ Σ Σ Σ Σ 201Positive Mathematical Programming and Risk Analysis In relation (23), all the terms are measured in dollars. The marginal cost due to limit- ing and variable inputs is given by c + 1 θ [w + (E( p)− c ′) x](1−θ )[ ′A y +Wh]⎧ ⎨ ⎩ ⎫ ⎬ ⎭ . The marginal cost due to risky output prices is given by γ θ [w + (E( p)− c ′) x](1−θ )( ′x Σ px) (γ /2−1)Σ px ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ . The cost function selected to synthesize the technological and behavioral relations of Phase I is expressed as a modified Leontief cost function such as C(x,y) = ( ′f x)( ′g y)+ ( ′g y)( ′x Qx) / 2 + ( ′f x)[(y1/2 ′) Gy1/2 ] (24) A cost function is non-decreasing in output quantities and input prices. It is lin- early homogeneous and concave in input prices, y. The (I×I) matrix G has elements Gi,ii=Gii,i≥0,i≠ii,i,ii=1,…,I. The diagonal elements Gi,i can take on either positive or nega- tive values. The (J×J) matrix Q is symmetric positive semidefinite. The components of vec- tors f and g are free to take on any value as long as f´x>0 and g´y>0. The reason for intro- ducing a term like (f´x)(g´y) is to add flexibility to the cost function. The marginal cost function associated with cost function (24) is given by MCx = ∂C ∂x = f( ′g y)+ ( ′g y)Qx + f[(y1/2 ′) Gy1/2 ] (25) The derivative of the cost function with respect to input prices corresponds to Shephard’s lemma that produces the demand function for inputs: ∂C ∂y = ( ′f x)g + g( ′x Qx) / 2 + ( ′f x)[Δ(y−1/2 ′) Gy1/2 ]= Ax (26) where ∆(y-1/2) represents a diagonal matrix with elements yi -1/2 on the main diagonal. With knowledge of the solution components resulting from the Phase I model (15)- (22), x̂, ŷ, ĥ, û,θ̂ ,γ̂ , a Phase II model’s goal is to estimate the parameters of the cost func- tion, f,g,Q,G. This task is accomplished by means of the following specification minAux=d´d/2+r´r/2 (27) subject to f( ′g ŷ)+ ( ′g ŷ)Qx̂ + f[(ŷ1/2 ′) Gŷ1/2 ]= (28) 202 Quirino Paris c + 1 θ̂ [w + (E( !p)− c ′) x̂](1−θ̂ )[ ′A ŷ +Wĥ]++ γ̂ θ̂ [w + (E( !p)− c ′) x̂](1−θ̂ )( ′x̂ Σ px̂) (γ̂ /2−1)Σ px̂ + d ( ′f x̂)g + g( ′x̂ Qx̂) / 2 + ( ′f x̂)[Δ(ŷ−1/2 ′) Gŷ1/2 ]= Ax̂ + r (29) Q=LDL´ (30) QQ-1=I (31) with ′f x̂ > 0, ′g ŷ > 0,D ≥ 0 , f and g free. The GAMS software requires an objective func- tion. The vector variables d,r perform the role of slack variables in the estimation of the marginal cost function and Shephard’s lemma, respectively. The objective function (27) is a typical least-squares specification. Relation (28) rep- resents the marginal cost function. Relation (29) is Shephard’s lemma. Relation (30) is the Cholesky factorization of the Q matrix with D as a diagonal matrix with nonnegative ele- ments on the main diagonal and L is a unit lower triangular matrix. The Cholesky factori- zation guarantees symmetry and positive semidefiniteness of the Q matrix. Relation (31) defines the inverse of the Q matrix and, thus, guarantees the positive definiteness of that matrix. This constraint assumes relevance for computing the supply elasticities of the vari- ous outputs. Any feasible solution of model (27)-(31) is an admissible cost function for representing the economic agent’s decisions under price risk. 6. PMP and Output-Supply Elasticities It may be of interest to estimate price supply elasticities for the various commodity outputs involved in a PMP-MS approach. The supply function for outputs is derivable from relation (25) by equating it to the expected market output prices, E( p) , and invert- ing the marginal cost function: x = −Q−1f −Q−1f[(y1/2 )Gy1/2 ] / ( ′g y)+ [1 / ( ′g y)]Q−1E( p) (32) that leads to the supply elasticity matrix Ξ = Δ[E( p)] ∂x ∂E( p) Δ[(x−1)]= Δ[E( p)]Q−1Δ[(x−1)] / ( ′g y) (33) where matrices Δ[E( p)] and Δ[x−1] are diagonal with elements E( pj ) and x j −1 on the main diagonals, respectively. Relation (33) includes all the own- and cross-price elasticities for all the output commodities admitted in the model. PMP has been applied frequently to analyze farmers’ behavior to changes in agricul- tural policies. A typical empirical setting is to map out several areas in a region (or state) and to assemble a representative farm for each area (or to treat each area as a large farm). When supply elasticities are exogenously available (say the own-price elasticities of crops) at the regional (or state) level (via econometric estimation or other means), a connection of Σ Σ 203Positive Mathematical Programming and Risk Analysis all area models can be specified by establishing a weighted sum of all the areas endogenous own-price elasticities and the given regional elasticities. The weights are the share of each area’s expected revenue over the total expected revenue of the region. The advantage of using exogenously supply elasticities has been asserted by Mérel and Bucharam (2010) and Petsakos and Rozakis (2015) in order to account for second-order conditions’ information. Let us suppose that exogenous own-price elasticities of supply are available at the regional level for all the J crops, say η j , j = 1,..., J . Then, the relation among these exog- enous own-price elasticities and the corresponding areas’ endogenous elasticities can be established as a weighted sum such as η j = wnj n=1 N ∑ ηnj where the weights are the areas’ expected revenue shares in the region (state) wnj = E( pnj )xnj E( ptj )xtjt=1 N∑ (34) ηnj = E( !pnj )Q jjxnj −1 / ( ′gnyn ) (35) where Qjj is the jth element on the main diagonal in the inverse of the Q matrix. The Phase II model that executes the estimation of the cost function parameters and the disaggregated (endogenous) output supply elasticities for a region (state) that is divid- ed into N areas takes on the following specification: minAux = ′dndn / 2 n=1 N ∑ + ′rnrn / 2 n=1 N ∑ (36) subject to fn ( ′gnŷn )+ ( ′gnŷn )Qx̂n + fn[(ŷn 1/2 ′) Gŷn 1/2 ]= (37) cn + 1 θ̂n [wn + (E( !pn )− cn ′) x̂n ](1−θ̂n )[ ′Anŷn +Wnĥn ] ++ γ̂ n θ̂n [wn + (E( !pn )− cn ′) x̂n ](1−θ̂n )( ′x̂nΣ px̂n )(γ̂ n /2−1)Σ px̂n + dn ≥ E( !pn ) ( ′fnx̂n )gn + gn ( ˆ ′xnQx̂n ) / 2 + ( ′fnx̂n )[Δ(ŷn −1/2 ′) Gŷn ]= Anx̂n + rn (38) Q=LDL´ positive semidefiniteness (39) QQ-1=I positive definiteness (40) Σ Σ 204 Quirino Paris Ξn = Δ[E( !pn )]Q −1Δ[(xn −1)] / ( ′gnyn ) endogenous own- and cross-price elasticities (41) wnj = E( pnj )x̂nj E( ptj )x̂tjt=1 N∑ expected revenue weights (42) ηnj = E( !pnj )Q jj x̂nj −1 / ( ′gnŷn ) own-price elasticities (43) η j = wnj n=1 N ∑ ηnj disaggregation of exogenous elasticities (44) with Dn≥0,gn and fn free and ′fnx̂n > 0 , ′gnŷn > 0 . The GAMS software requires an objective function. The objective function Aux mini- mizes the pseudo slack variables, rn and dn, of the primal and dual constraints. 7. Phase I Versus Phase I-II Estimates of the Calibrating Solution A strand of the PMP literature has discussed the issue of whether the Phase I esti- mates of decision variables and input shadow prices, x,y, are consistent with the corre- sponding Phase II estimates where the cost function parameters are estimated simul- taneously with them. The short answer is positive because the amount of information is the same in the two Phases. With the limitations of a two-dimensional diagram, Figure 1 illustrates the issue. In Phase I, total cost is a linear function of the decision variables while in Phase II total cost is a nonlinear function of the same variables. Hence, the cali- brating optimal solution, x*, is the same in the two Phases. In the context of this paper, Phase I model is stated as a LS specification of relations (15) through (22). This model results in a unique Least-Squares solution of deviations h and u and, therefore, of the decision variables x̂, ŷ . The Phase II model that estimates Figure 1. Phase I and Phase II estimates of decision variables x and input shadow prices y. 205Positive Mathematical Programming and Risk Analysis simultaneously the cost function parameters and the optimal decision variables is stated as the LS specification in Phase I combined with constraints (28) through (31) (where the “ ⋅̂ ” symbol is removed from the decision variables). The original information is identical in the two models and, therefore, the LS methodology guarantees the unique and identical solution for the two sets of estimates. 8. Phase III PMP Model – Calibrating Models With the parameter estimates of the cost function, f̂n , ĝn ,Q̂,Ĝ , derived from either Phase II model (27)-(31) or model (36)-(44), it is possible to set up a calibrating equilib- rium model to be used for policy analysis. Such a model takes on the following economic equilibrium specification minCSC=y´zp+x´zd=0 (45) subject to ( ′f̂ x)ĝ + ĝ( ′x Q̂x) / 2 + ( ′f̂ x)[Δ(y−1/2 ′) Ĝy1/2 ]+ z p = b +Vû (46) f̂( ′ĝ y)+ ( ′ĝ y)Q̂x + f̂[(y1/2 ′) Ĝy1/2 ]= E( p)+ ẑd (47) with x≥0,y≥0,zp≥0,zd≥0. The objective function represents the complementary slackness conditions (CSC) of constraints (46) and (47) with an optimal value of zero. The varia- bles zp and zd are surplus variables of the primal and the dual constraints, respectively. The solution of model (45)-(47) calibrates precisely the solution obtained from the Phase I model (15)-(22), that is, x̂LS = x̂CSC and ŷLS = ŷCSC . This remarkable result is due simply to the fact that all the information of the Phase I model has been transferred to the cost function. Note that the matrix of fixed technical coefficients A does not appear in either constraint (46) or (47). The calibrating model, then, can be used to trace the production and revenue response to changes in the expected output prices, subsidies and the supply of limiting inputs in a more flexible technical framework. An alternative calibrating equilibrium model is suitable for dealing with a crucial aspect of a risky policy scenario. Wealth is the anchoring measure of risk preference of an economic agent. As illustrated above, wealth is composed of accumulated income (or exogenous income) and net revenue derived from the current production cycle as in [w + (E( p)− c ′) x] where w measures the amount of exogenous income. Agricultural poli- cies in many countries deal with subsidies to farmers for cultivating (or not cultivating) crops. These subsidies may or may not be coupled to the level of crop production. Subsi- dies that are decoupled from the crop production decisions of farmers constitute exoge- nous income and end up in the term of wealth that becomes an important target of policy makers. The w term, then, must appear in the calibrating model to allow the representa- tion of decoupled subsidies as in the following specification minCSC=y´zp+x´zd=0 (48) 206 Quirino Paris subject to ( ′f̂ x)ĝ + ĝ( ′x Q̂x) / 2 + ( ′f̂ x)[Δ(y−1/2 )Ĝy1/2 ]+ z p = b +Vû (49) c + 1 θ̂ [w + (E( !p)− c ′) x](1−θ̂ )[ ′A y +Wĥ] + γ̂ θ̂ [w + (E( !p)− c ′) x](1−θ̂ )( ′x Σ px)(γ̂ /2−1)Σ px = E( !p)+ zd (50) with x≥0,y≥0,zp≥0,zd≥0. Also the solution of model (48)-(50) calibrates precisely the solu- tion obtained from the Phase I model (15)-(22), that is, x̂LS = x̂CSC and ŷLS = ŷCSC . 9. Empirical Implementation of PMP-MS With Supply Elasticities The PMP-MS approach described in previous sections was applied to a sample of N = 14 representative farms of the Emilia-Romagna region of Italy. There are four crops: sugar beets, soft wheat, corn and barley. There is only one limiting input: land. Empirical real- ity compels a further consideration of the above methodology in order to deal with farm samples where not all farms produce all commodities. It turns out that very little must be changed for obtaining a calibrating solution in the presence of missing commodity levels, their prices and the corresponding technical coefficients. Using the GAMS software, it is sufficient to condition the various constraints of Phase I, Phase II and Phase III models by the nonzero observations of the output levels. To exemplify, the available farm sample displays the following Table 3 of observed crop levels while Table 4 presents the variance- covariance matrix of the market output prices. Table 3. Observed output levels, xobs, with non produced commodities. Farm Sugar Beets Soft Wheat Corn Barley 1 1133.4240 0 341.3693 18.2398 2 3103.7830 841.7445 0 59.8025 3 0 450.7937 881.9748 0 4 3488.3540 821.3934 1493.3320 51.1247 5 959.1102 468.2848 0 28.2406 6 942.2039 801.1288 1283.5910 152.5810 7 1600.7310 0 899.4739 66.9718 8 0 1212.8550 1237.5840 98.0497 9 1050.5370 332.3773 0 63.6696 10 3473.6780 952.5199 774.7402 0 11 0 765.1689 501.9673 59.5366 12 3276.1450 1100.1680 0 177.9740 13 877.0970 380.9171 564.6091 76.2122 14 1430.9460 0 1309.3920 0 Σ Σ 207Positive Mathematical Programming and Risk Analysis Other missing information deals with prices and unit accounting costs associated with the zero-levels of crops. Furthermore, the technical coefficients of farms not pro- ducing the observed crops also equal to zero. Hence, we can state that, for n=1,…,N, the number of farms, and j=1,…,J, the number of crops, if xnj obs = 0 , also pnj=0, cnj=0 and Anij=0. Furthermore, suppose that only one input, land, is involved in this farm sample. Let us assume also that the land price is observed for all farms. The procedure to deal with this type of sample data consists in conditioning the relevant constraints on the posi- tive values of the output levels. In GAMS, this procedure requires a conditional statement using the $ sign option. Table 4. Variance-covariance matrix of the market output prices. Sugar Beets Soft Wheat Corn Barley Sugar Beets 0.0024719 -0.0164391 -0.0117184 -0.0121996 Soft Wheat -0.0164391 0.2386034 0.1821288 0.2049011 Corn -0.0117184 0.1821288 0.1530464 0.1610119 Barley -0.0121996 0.2049011 0.1610119 0.1830829 Tables 5 and 6 present the estimated output levels and input prices ( x̂, ŷ ). They also exhibit the percent deviation of the solution ( x̂, ŷ ) of model (15)-(22) from the corre- sponding targets (xobs,yobs). It is of interest to report that the same identical solution was obtained in three different ways. All the estimations were performed with the GAMS soft- ware. The first round of estimates were obtained by solving model (15)-(22) one farm at a time. The second round of estimates were obtained by solving model (15)-(22) using the entire sample of observations. This means that the objective function was specified as minLS = hnWnhn n=1 N ∑ + unVnun n=1 N ∑ subject to constraints (16)-(22) specified for each single farm observation. The third round of estimates of the optimal decision variables were obtained by solving model (36)- (44) with the “ ⋅̂ ” symbol removed from the variables. Table 7 presents the estimates of the parameters θ and γ of the MS utility function. The sample is composed of relatively homogeneous farms. Hence, the limited numeri- cal range of variation of the MS utility parameters is not a surprise. Within that range, however, a wide variety of risk preferences is detected. Seven farmers exhibit decreasing absolute risk aversion accompanied by increasing relative risk aversion. This result match- es a statement of Tsiang (1972, p. 357): “…the most commonly observed pattern of behav- ior toward risk of a risk-averter individual is probably decreasing absolute risk-aversion coupled with increasing relative risk-aversion when his wealth increases…” Two farmers exhibit increasing absolute risk aversion associated with decreasing relative risk aversion. Four farmers exhibit decreasing absolute risk propensity and increasing relative risk pro- pensity. It should be noted that the negative gamma coefficients of these four farmers are 208 Quirino Paris rather small, suggesting that risk neutrality may – probably – be a better risk-preference representation of these farmers. The approach does not allow for a statistical testing of this conjecture. Finally, one farmer exhibits increasing absolute and relative risk aversion. This Table 5. Estimated LS solution, x̂ , and percent deviation from the observed levels, xobs with zero lev- els for some crops and some farms. Farm Optimal Decisions x̂ Percent deviation from xobs Sugar Beets Soft Wheat Corn Barley Sugar Beets Soft Wheat Corn Barley 1 1133.851 0 341.622 18.156 0.0377 0 0.0741 -0.4587 2 3104.392 861.829 0 52.923 0.0196 0.0098 0 0.2021 3 0 450.794 881.975 0 0 -0.0000 0.0000 0 4 3488.400 821.340 1493.477 51.165 0.0013 -0.0065 0.0097 0.0791 5 959.234 468.140 0 28.308 0.0129 -0.0310 0 0.2399 6 942.488 801.394 1283.947 152.923 0.0301 0.0331 0.0278 0.2238 7 1601.381 0 899.724 67.104 0.0406 0 0.0278 0.1975 8 0 1213.157 1237.937 98.080 0 0.0249 0.0285 0.0307 9 1051.373 332.592 0 63.767 0.0796 0.0645 0 0.1528 10 3474.183 952.606 774.966 0 0.0145 0.0085 0.0291 0 11 0 765.267 502.186 59.659 0 0.0128 0.0436 0.2052 12 3276.657 1100.245 0 178.324 0.0156 0.0070 0 0.1964 13 877.324 380.970 564.926 76.467 0.0258 0.0138 0.0561 0.3347 14 1431.231 0 1309.653 0 0.0199 0 0.0199 0 Table 6. Deviation of ŷ from yobs. Farm Observed Land Prices yobs Estimated Land Prices ŷ Percent Deviation 1 4.42 4.4213 0.0287 2 4.38 4.3810 0.0219 3 6.98 6.9800 0.0000 4 5.73 5.7302 0.0036 5 4.40 4.3995 -0.0111 6 1.86 1.8609 0.0458 7 3.65 3.6517 0.0454 8 3.36 3.3609 0.0266 9 2.75 2.7521 0.0780 10 4.28 4.2807 0.0158 11 3.28 3.2810 0.0318 12 1.93 1.9305 0.0281 13 2.32 2.3213 0.0579 14 4.03 4.0308 0.0199 209Positive Mathematical Programming and Risk Analysis empirical result is a clear illustration of the flexible structure of risk preferences as stated by the theoretical analysis. The corresponding meaning of the various acronyms is derived from Table 1 and Table 2. The estimated parameters of the cost function are reported in Tables 8 and 9. In this numerical example, the G matrix contains only one parameter whose value is Gi,i=- 11.39904. Regional, exogenous own-price supply elasticities were available in the magnitude of 0.6 for sugar beets, 0.5 for soft wheat, 0.7 for corn and 0.4 for barley. The endogenous own-price elasticities of all farms were aggregated to be consistent with the regional exog- enous elasticities according to relation (44). Table 10 presents the farms’ own-price supply elasticities used in the aggregation relation. 10. Conclusion This paper accomplished several objectives. First, it extended the treatment of risk in a mathematical programming framework to include any combination of risk prefer- ences represented by absolute risk aversion (or absolute risk propensity) and relative risk aversion (or relative risk propensity). Second, it modified the traditional PMP approach to deal with calibration constraints regarding observed output levels and observed input prices by eliminating the user-determined perturbation parameter. The combination of these two approaches provides suitable models for agricultural policy analysis that take into consideration farmers’ risk preferences associated with the randomness of output prices. Third, this paper integrated the use of exogenous supply elasticities observed for, say, an entire region with the endogenous elasticities derived from the supply functions of the sample farms. This objective is achieved by specifying a complete and flexible total cost function that fulfills all the theoretical requirements. Fourth, it resolves in a positive Table 7. Estimates of θ and γ. Farm Parameter θ Parameter γ Risk Preference 1 1.0131215 1.1397862 DARA, IRRA 2 1.0050568 1.0766995 DARA, IRRA 3 1.1313873 1.2841485 DARA, IRRA 4 0.9836798 0.9273945 IARA, DRRA 5 0.9578977 -0.1867746 DARP, IRRP 6 0.9645178 -0.1465580 DARP, IRRP 7 1.0183502 1.1310367 DARA, IRRA 8 1.0562629 1.1969911 DARA, IRRA 9 1.0277583 1.1992494 DARA, IRRA 10 1.0043433 1.0570120 DARA, IRRA 11 0.9503372 -0.1567640 DARP, IRRP 12 0.9986044 1.0263446 IARA, IRRA 13 0.9577406 -0.1663556 DARP, IRRP 14 0.9797649 0.8443536 IARA, DRRA 210 Quirino Paris way the dispute debated in the PMP literature whether Phase I calibrating estimates are consistent with Phase II estimates. Fifth, a calibrating model resulting from the PMP-MS framework described here allows for the analysis of policy scenarios dealing with farm subsidies that are decoupled from the current crop production. Consider the parameter Table 8. Intercepts f̂ and ĝ of the marginal cost and input demand functions. Farm f̂ ĝ ′f̂ x̂ ′ĝ ŷ Sugar Beets Soft Wheat Corn Barley 1 0.00949 0 0.00666 -0.00320 0.00465 12.923 0.02055 2 0.00364 0.03154 0 -0.06940 0.00149 34.378 0.00654 3 0 -0.00290 0.00374 0 0.00129 1.965 0.00901 4 0.00734 -0.00284 0.00902 -0.06489 0.00132 33.448 0.00756 5 -0.00307 0.02349 0 -0.05730 0.00202 6.426 0.00888 6 -0.02082 0.08018 0.08193 0.26459 0.00658 190.289 0.01224 7 0.00473 0 0.02687 0.05681 0.00271 35.568 0.00992 8 0 0.08121 -0.00668 -0.05092 0.00220 85.254 0.00738 9 0.00408 0.04408 0 0.07081 0.00610 23.462 0.01679 10 0.00905 0.03772 -0.02931 0 0.00164 44.673 0.00703 11 0 0.08005 -0.04152 -0.05365 0.00300 37.213 0.00985 12 0.00395 0.11439 0 0.06448 0.00329 150.291 0.00635 13 0.00041 0.05078 0.03585 0.19131 0.00950 54.584 0.02205 14 -0.00159 0 0.03287 0 0.00192 40.770 0.00773 Table 9. Estimated matrices Q̂ and D̂ . Matrix Q̂ Sugar Beets Soft Wheat Corn Barley Sugar Beets 0.0408842 -0.0269584 -0.0084418 -0.0405819 Soft Wheat -0.0269584 0.8509183 -0.1850005 -0.5925655 Corn -0.0084418 -0.1850005 0.3698877 -0.0130721 Barley -0.0405819 -0.5925655 -0.0130721 7.7008830 Matrix D̂ Sugar Beets Soft Wheat Corn Barley Sugar Beets 0.0408842 Soft Wheat 0.8331423 Corn 0.324558 Barley 7.1182454 211Positive Mathematical Programming and Risk Analysis w in the measure of wealth that may represent exogenous income subsidy. With a Freund approach to risk based upon a constant absolute risk aversion utility function, the wealth parameter disappears from the programming model. On the contrary, one version of the calibrating equilibrium model presented in this paper allows for the analysis of decoupled farm subsidies that are more frequently the target of policy makers. This general model has been tested on different farm samples with satisfactory results including a data sample where not all farms produce all the commodities. 11. References Arata, L., Donati, M., Sckokai, P. and Arfini, F. (2017). Incorporating Risk in a Positive Mathematical Programming Framework: a Dual Approach. The Australian Journal of Agricultural and Resource Economics 61: 265-284. Brooke, A., Kendrick, D., Meeraus, A. (1988). GAMS, A User’s Guide. The Scientific Press, Redwood City, California. Cortignani, R., Severini, S. (2012). Modeling Farmer Participation to a Revenue Insurance Scheme by Means of Positive Mathematical Programming. Agricultural Economics – Czech 58: 324:331. Fisher, I. (1906). The Nature of Capital and Income. MacMillan Company, London. Freund, R.J. (1956). The Introduction of Risk into a Programming Model. Econometrica 24: 253-263. Hazell, P.B.R. (1971). A Linear Alternative to Quadratic and Semivariance Programming for Farm Planning Under Uncertainty. American Journal of Agricultural Economics 53: 53-62. Table 10. Disaggregation/aggregation of the regional, exogenous own-supply elasticities with zero observations of some output levels. Farm Exogenous Sugar Beets: 0.6 Exogenous Soft Wheat: 0.5 Exogenous Corn: 0.7 Exogenous Barley: 0.4 1 0.4422 0 0.9144 0.7529 2 0.6093 0.6344 0 0.8320 3 0 0.8010 0.7415 0 4 0.3126 0.5757 0.6101 0.8427 5 1.1497 0.6980 0 1.1181 6 0.9824 0.3464 0.4243 0.1741 7 0.5677 0 0.7124 0.4221 8 0 0.3910 0.6719 0.4075 9 0.5837 0.5928 0 0.2624 10 0.3482 0.4793 1.1446 0 11 0 0.4196 1.2469 0.4779 12 0.6557 0.4610 0 0.2876 13 0.5061 0.3453 0.4804 0.1667 14 1.0249 0 0.6590 0 212 Quirino Paris Hicks, J.R. (1933). The Application of Mathematical Methods in the Theory of Risk. Lec- ture presented at the meeting of the Econometric Society in Leyden, September- October 1933, summarized by J. Marschak. Econometrica 1934(2): 187-203. Howitt, R.E. (1995a). A Calibration Method for Agricultural Economic Production Mod- els. Journal of Agricultural Economics 46: 147-159. Howitt, R.E. (1995b). Positive Mathematical Programming. American Journal of Agricul- tural Economics 77: 329-342. Markowitz, H. (1952). Portfolio Selection. Journal of Finance 7: 77-91. Mérel, P. and Bucharam, S. (2010). Exact Calibration of Programming Models of Agricul- tural Supply Against Exogenously Supply Elasticities. European Review of Agricultur- al Economics 37: 395-418. Meyer, J. (1987). Two-Moment Decision Models and Expected Utility Maximization. American Economic Review 77: 421-430. Paris, Q. (2015). The Dual of the Least-Squares Method, Open Journal of Statistics 5:658- 664. DOI: 10.4236/ojs.2015.57067  Paris, Q. (2018). Estimation of CARA Preferences and Positive Mathematical Program- ming. Open Journal of Statistics 8: 1-13. DOI: 10.4236/ojs.2018.81001 Petsakos, A. and Rozakis, S. (2015). Calibration of Agricultural Risk Programming Mod- els. European Journal of Operational Research 242: 536-545. Saha, A. (1997). Risk Preference Estimation in the Nonlinear Mean Standard Deviation Approach. Economic Inquiry 35: 770-782. Tintner, G. (1941). The Theory of Choice Under Subjective Risk and Uncertainty. Econo- metrica 9: 298-304. Tobin, J. (1958). Liquidity Preference as Behavior Toward Risk. Review of Economic Studies 67: 65-86. Tsiang, S.C. (1972). The Rationale of the Mean-Standard Deviation Analysis, Skewness Preferences, and the Demand for Money. American Economic Review 62: 354-371. 213Positive Mathematical Programming and Risk Analysis Appendix The function V(μ,σ)=μθ-σγ is concave in μ and σ when the corresponding Hessian matrix is negative definite. This event occurs when θ<1 and γ>1. When the mean and standard deviation of wealth, μ and σ, are expressed in terms of decision variables, x, μ(x) and σ(x), the resulting function assumes a flexible structure whose concavity depends on different values of parameters θ and γ. This appendix illustrates the possible shapes of the MS utility function (as a function of decision variables) by means of simple graphs and the associated upper contour sets that are conditional upon the magnitude of the θ and γ parameters. The value of θ and γ are chosen to reflect the estimates of Table 7. The MS utility function is simplified to show two decision variables, x1 and x2. The expected prices are chosen as E( !p1) = 4 and E( !p2 ) = 6 with standard deviation σp1=0.5, σp2=0.7 and σp1p2=0.1. With these stipulations, all the figures’ functional forms and the upper contour sets exhibit the following specification V[µ(x),σ (x)] = µ(x)θ −σ (x)γ = [E( !p1)x1 + E( !p2 )x2 ]θ − [σ p1 2 x1 2 +σ p2 2 x2 2 + 2σ p1p2 x1x2 ]γ /2 = [4x1 + 6x2 ]θ − [0.52 x1 2 + 0.72 x2 2 + 0.2x1x2 ]γ /2 In all the figures, the upper contour sets appear to be convex even though the contour levels appear rather flat in some figures. The convexity of the upper contour sets is a cru- cial reason for obtaining an optimal solution. The flatness of the contour levels may make it more laborious for the algorithm to converge to an optimal solution. The figures were drawn using Mathematica. 1500 2000 2500 3000 3500 4000 4500 5000 2000 2500 3000 3500 4000 4500 5000 θ=1.1, γ=1.36 DARA, IRRA 214 Quirino Paris 1500 2000 2500 3000 3500 2000 2200 2400 2600 2800 3000 3200 3400 θ=1.05, γ=1.20 DARA, IRRA 1500 2000 2500 3000 3500 4000 4500 5000 2000 2500 3000 3500 4000 4500 5000 θ=0.98, γ=0.92 IARA, DRRA 215Positive Mathematical Programming and Risk Analysis 1500 2000 2500 3000 3500 4000 4500 5000 2000 2500 3000 3500 4000 4500 5000 θ=0.99, γ=1.03 IARA, IRRA 2000 3000 4000 5000 6000 2000 3000 4000 5000 6000 θ=0.95, γ=-0.15 DARP, IRRP Positive Mathematical Programming and Risk Analysis Quirino Paris The hedonic contents of italian super premium extra-virgin olive oils Luca Cacchiarelli1,*, Anna Carbone2, Tiziana Laureti1, Alessandro Sorrentino1 Corporate R&D and the performance of food-processing firms: Evidence from Europe, Japan and North America Heinrich Hockmann1, Pedro Andres Garzon Delvaux2,*, Peter Voigt3, Pavel Ciaian2, Sergio Gomez y Paloma2 Can menu labeling affect away-from-home-dietary choices? Elena Castellari1,*, Stéphan Marette2, Daniele Moro3, Paolo Sckokai1 A preliminary test on risk and ambiguity attitudes, and time preferences in decisions under uncertainty: towards a better explanation of participation in crop insurance schemes Attilio Coletta1, Elisa Giampietri2, Fabio Gaetano Santeramo3,*, Simone Severini1, Samuele Trestini2