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APPENDIX A2.B

3.5 INTEGRATING RISK INTO A PMP MODEL: A NEW METHODOLOGICAL PROPOSAL

75 where W is the initial wealth,0 V is the variance-covariance matrix of the gross margin per 1 unit of activity of the first phase and the other variables have the same meaning as previously declared. The idea is that V is a national or regional matrix, while the farm makes his choice 1 based on an individual variance-covariance matrix which is not known. Petsakos and Rozakis proposed to use the dual information of the first phase to adjust the matrix V and get an 1 individual farm matrix V which calibrates the model according to equation (3.42). 2

2 2 2 2 1 1

2 3 2 3

0 0 0 0

( ) ' ' ( ) ' '

( ) ' ( ) '

( ( ) ' ) ( ( ) ' ) ( ( ) ' ) ( ( ) ' )

E E

E E

W E W E W E W E

   

− = − +

 + +   + + 

   

V x g x V x V x g x V x

g g λ

g x g x g x g x

ɶ ɶ

ɶ ɶ

ɶ ɶ ɶ ɶ

(3.42) where V is the only unknown variables. 2

The third step consists in a calibrated model which perfectly reproduces the base year observations without the calibration constraints

2

0 2

0

( ) '

max ( ) ln( ( ) ' ) 1 '

2 ( ) '

EU W E E

W E

 

= + −  + 

x g x g x V x

g x ɶ ɶ

ɶ (3.43)

subject to Axb (3.44)

x≥0 (3.45)

The work of Petsakos and Rozakis represents a remarkable proposal to include risk into a PMP framework, without applying the E-V approach. Their proposal avoids the direct estimation of the absolute risk aversion coefficient which can be calculated by applying the Arrow-Pratt rule. In addition their model is based on DARA preferences. The weaknesses of their study concerns what they assume that the misspecification of the initial variance-covariance matrix of the unitary gross margin is the only reason why the starting model does not reproduce the observed activity level. Besides the risk, the cost function is still linear.

3.5 INTEGRATING

RISK INTO A PMP MODEL: A NEW

76 mathematical and economic justification and takes advantage of the recent developments in both PMP and risk modelling literature.

We have explored this research area since we think the combination of PMP with risk modelling will be a new research frontier in the analysis of farmers’ behaviour. Given the importance of accounting for risk in agricultural production and the powerful calibration ability of PMP, the integration of the two elements represents a relevant development in farm modelling. The proposed model estimates both the farm non-linear cost function and the farmer’s absolute risk aversion coefficient.

This new methodological proposal relies upon the model of Arfini and Donati (2008, 2011), that merges the first linear phase of PMP with the second non-linear phase by using the dual relationships of a farmer’s expected utility maximisation problem. The model estimates simultaneously the differential marginal cost and the shadow price of resources which usually belong to the first PMP phase, as well as the farm non-linear cost function and the farmer’s coefficient of absolute risk aversion. The model specification is the following:

, , ,

min 1 ' ' ' '( ) ' ( ) '

2 E

α + + + + +α −

u y λ u u y b c x λ x ε x Vx p xɶ (3.46)

subject to cVx+A y' + ≥λ E( ) (3.47)

c+ =λ Qx+u (3.48)

y≥0,λ≥0,α ≥0 (3.49)

The meaning of the symbols is the same as before. The objective function minimises the square of the individual farm deviations, 1

2u u' , from the common cost function and the difference between the primal and the dual objective function of a farmer’s expected utility maximisation problem. This difference at the optimum should be equal to zero. The constraint (3.47) represents the dual constraint which indicates the economic equilibrium condition stating that the marginal cost must be larger or equal to the marginal revenue. The constraint (3.48) indicates the relationship between the marginal cost of the first phase of the standard PMP and the marginal cost of the farm non linear cost function. It is worth to notice that the observed activity levels are directly introduced in the model without defining any calibration constraints, which raised several critiques to the standard three-phase PMP approach. Solving the model either by GME or by LS leads to the simultaneous estimation of the shadow prices

77 of resources, y, the shadow prices of activities, λ, the quadratic matrix of the cost function, Q, the individual farm deviations from the cost function, u, and the farmer’s absolute risk aversion coefficient α. Since the model (3.46) - (3.49) is based on a farmer’s expected utility maximisation problem following the E-V approach, the coefficient of farmer’s absolute risk aversion, α , is farm specific and it is independent of the level of wealth (thus, it corresponds to CARA risk preferences).

Since the model (3.46) - (3.49) is a mathematical programming model with inequality constraints and sign restricted variables, a set of KKT conditions provides the solution of the model. In order to derive the KKT conditions, we write the Lagrange function of the model:

1 ' ' ' '( ) ' ( ) ' '( ( ) ' ) '( )

L=2u ux Vx+y b λ x ε+ + +c xE p x wɶ + E pɶ − −c αVx A y λ− − +v c λ Qx u+ − − (3.50) where w'and v'represent the Lagrange multipliers associated to each constraint. From the Lagrange function we can derive the KKT conditions, where the sign is dictated by the direction of the optimisation and by the sign of the variables, and their associated complementary slackness conditions:

dL 0

d =u v- =

u (3.51a) 'dL '( ) 0

d = - = u u u v

u (3.51b)

dL 0

d = −b Aw

y (3.52a) 'dL '( ) 0

d = − =

y y b Aw

y (3.52b)

dL 0

d = + − + ≥x ε w v

λ (3.53a) 'dL '( ) 0

d = + − + =

λ λ x ε w v

λ (3.53b)

' ' 0

dL

dα =x Vxw Vx (3.54a) ( ' ' ) 0

dL αd α

α = x Vx w Vx = (3.54b)

( ) ' 0

dL E

d = p − −c αVx A y λ− − ≤ w

ɶ (3.55a)

'dL '( ( ) ' ) 0

d = E − −α − − =

w w p c Vx A y λ

w

ɶ (3.55b)

78 dL 0

d = + −c λ Qx u− =

v (3.56a) 'dL '( ) 0

d = + − − = v v c λ Qx u

v (3.56b)

KKT condition (3.51a) indicates that the dual value, v, associated to the marginal cost function equation is equal to the farm deviation from the cost function,u; since the model tries to keep uas small as possible,vshould result in a small positive or negative number too.

w is the dual value of the economic equilibrium constraint (3.47) and it can be interpreted as the shadow output quantity, thus w = x. Substituting v=uand w =x in (3.52a) and (3.53a), we can recognize in these two conditions the resource constraints and the calibration constraints respectively. Hence, the model (3.46)-(3.49) implicitly represents the constraints of a first phase model of the standard PMP and as a consequence the model calibrates to the base year activity level without making the first phase explicit. This prevents from the critiques raised against the standard PMP approach. The other KKT conditions represent a tautology (condition 3.54a) and the constraints of the model (conditions 3.55a and 3.56a).

The estimated variables of the model (3.46) - (3.49) are then used to construct a non-linear model which includes both the estimated farm quadratic cost function and the estimated risk term (equations 3.57 -3.59). The model calibrates the endogenous variable levels to the base year without the calibration constraints and it can been used in simulation analysis:

1 1

max ( ) ( ) ' ' ' '

2 2

EU πɶ =E p xɶ − x Qx u x− − αx Vx (3.57)

subject to Axb (3.58)

x≥0 (3.59)

where xis the vector of endogenous activity levels, Q , uand α have been estimated previously by equations (3.46)-(3.49) and E p( ) ', V , A and bare exogenous parameters.

Equation (3.57) is the farmer expected utility to be maximised which is equal to the expected revenue minus the estimated farm non linear cost function and the risk premium. Equation (3.58) is the usual resource constraint.

The new methodological proposal for the incorporation of risk in a PMP framework represents an innovative approach compared to the previous studies in this challenging

79 research area. Our model differs from the work of Paris and Arfini (2000) as we estimate endogenously the farmer’s coefficient of absolute risk aversion and we do not rely upon the standard three-step PMP. Although the endogenous variables in our model are the same variables estimated in the model of Severini and Cortignani (2011), our proposal differs from their model in the viewpoint adopted to solve the problem. While Severini and Cortignani skipped the first phase of the PMP and they estimated directly the optimality conditions of the desired model, we merged the first linear phase with the second non-linear phase by using the dual relationships of an expected utility maximisation problem. Finally, our approach differs from the proposal of Petsakos and Rozakis (2011) as it applies the E-V approach and it estimates both the non-linear cost function and the non-linear risk term.

The advantages of our proposed estimation approach compared to the direct estimation of the optimality conditions of the desired model (Heckelei and Wolff, 2003; Severini and Cortignani, 2011) consists in the possibility of using additional information such as the variable accounting cost per unit of activity available in our dataset and the difference between the primal and the dual objective function of a farmer expected utility maximisation problem. In addition, our model specifies only the dual constraint on marginal costs while the estimation of the optimality conditions involves the specification of both the dual constraint and the resource constraint. However, one of the two constraints is redundant in an optimisation model and it may affect the parameters’ estimation. Finally, the estimation carried out by Severini and Cortignani, according to the procedure proposed by Heckelei and Wolff (2003) introduces the deviations in the output level, while our deviations concern the individual departure from the common cost function.