CES-Frchet modeling of farmer choices Christophe Gouel INRA - PowerPoint PPT Presentation

CES-Fréchet modeling of farmer choices Christophe Gouel INRA November 8, 2018

Model setup • Risk-neutral farmer/landowner facing the choice of allocating its land endowment x 1 to crops. • Crops are indexed by k = l ∈ K ≡ { 1 , . . . , K } • Crop production requires 2 bundles of inputs: 1 R inputs that are partial substitutes (e.g., land, fertilizers and water), with land indexed r = 1, with substitution elasticity 0 < σ k < 1. 2 Unspecified for now and corresponds to non-land value-added and is non substitutable to the first bundle. • Land is heterogeneous and composed of a continuum of parcels indexed by ω defined over [ 0 , 1 ] .

Production function �� � r = 2 A r , k ( x r , k ( ω )) ( σ k − 1 ) /σ k � σ k / ( σ k − 1 ) A 1 , k ( ω ) ( x 1 , k ( ω )) ( σ k − 1 ) /σ k + � R Q k ( ω ) = min , N k ( ω ) /ν k , • A 1 , k ( ω ) ≥ 0 a parameter governing land productivity • A r , k ≥ 0 with r � = 1 are productivity shifters for the inputs affecting yields • x r , k ( ω ) is input demand, • N k ( ω ) is the value added demand, • ν k > 0 is a productivity shifter for value added.

Price indexes From the Leontief structure: p k = P X k + w ν k , where P X k is the price of the first input bundle and w is the wage. From CES standard algebra, � 1 / ( 1 − σ k ) � R � ( A 1 , k ( ω )) σ k − 1 ( π 1 , k ( ω )) 1 − σ k + P X A σ k − 1 π 1 − σ k k = . r , k r , k r = 2 Then, we can express the land rents per hectare as: � 1 / ( 1 − σ k ) � R � 1 − σ k − � � A σ k − 1 π 1 − σ k P X π 1 , k ( ω ) = A 1 , k ( ω ) , k r , k r , k r = 2 � 1 / ( 1 − σ k ) � R � ( p k − w ν k ) 1 − σ ku − A σ k − 1 π 1 − σ k = A 1 , k ( ω ) . r , k r , k r = 2 � �� � = r k

Fréchet assumption A 1 , k ( ω ) are i.i.d. from a Fréchet distribution with shape θ > 1 and scale γ A 1 , k > 0: � � − θ � � a Pr ( A 1 , k ( ω ) ≤ a ) = exp − ∀ a ∈ R > 0 . γ A 1 , k • γ ≡ (Γ ( 1 − 1 /θ )) − 1 a scaling parameter introduced so that A 1 , k is the unconditional productivity of land, A 1 , k = E [ A 1 , k ( ω )] , the productivity if all the land was planted with crop k . • θ measures the dispersion of yields around their average A 1 , k : a higher θ indicates more homogeneity and a lower θ more heterogeneity. It follows that π 1 , k ( ω ) is distributed Fréchet with parameters θ and γ r k A 1 , k

Crop choices The probability that crop k is the most profitable on parcel ω is defined by � � λ k = Pr π 1 , k ( ω ) ∈ arg max π 1 , l ( ω ) , l ∈K = also the share of land allocated to k . π θ 1 , k λ k = , � l ∈K π θ 1 , l where π 1 , k ≡ r k A 1 , k denotes the unconditional land rents if all the land is planted with crop k .

Crop production From CES and Fréchet algebra: � r k � σ k � � Q k = x 1 λ k E A 1 , k ( ω ) | π 1 , k ( ω ) ∈ arg max π 1 , l ( ω ) , P X l ∈K k � r k � σ k = x 1 A 1 , k λ ( θ − 1 ) /θ . k P X k

Input demand � � x r , k = E x r , k ( ω ) | π 1 , k ( ω ) ∈ arg max π 1 , l ( ω ) , l ∈ K which gives � � � π r , k � − σ k A σ k x r , k = E r , k Q k ( ω ) | π 1 , k ( ω ) ∈ arg max π 1 , l ( ω ) , P X l ∈ K k � π r , k � − σ k � � = A σ k E Q k ( ω ) | π 1 , k ( ω ) ∈ arg max π 1 , l ( ω ) , r , k P X l ∈ K k � π r , k � − σ k = A σ k Q k . r , k P X k

Parcel-level production function 0.8 0.6 Production 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Input level

In exact hat algebra � � θ ˆ A 1 , k ˆ r k ˆ λ k : ˆ λ k = � θ , � � ˆ l ∈K λ l A 1 , l ˆ r l � � − σ k π r , k ˆ ˆ x r , k : ˆ ˆ x r , k = Q k , for r ≥ 2 ˆ P X k � � σ k r k ˆ Q k : ˆ ˆ Q k = ˆ A 1 , k ˆ λ ( θ − 1 ) /θ , k ˆ P X k � � ˆ k ˆ P X k : α X P X 1 − α X k = ˆ p k − w k , ˆ k � 1 / ( 1 − σ k ) � R � r k : ˆ P X α X r 1 − σ k α X π 1 − σ k ˆ k = 1 , k ˆ + r , k ˆ , r , k k r = 2

3 extensions • Multiple fields • Non zero production at zero input • More flexible acreage elasticities (not here).

Multiple fields • There are f ∈ 1 , . . . , F fields that are heterogeneous in their productivity. Fields can be defined on a grid or on land classes (GAEZ). • There are no transport costs between fields, so that they all face the same prices, and labor productivity shifters ν k are the same.

New equations = Q f k � �� � � ( θ − 1 ) /θ � r f � σ k F � � x f 1 A f λ f k Q k : Q k = , 1 , k k P X k f = 1 � � θ A f 1 , k r f k λ f k : λ f k = � θ , � � A f 1 , l r f l ∈K l � � 1 / ( 1 − σ k ) R � σ k − 1 π 1 − σ k � 1 − σ k + � � � r f k : P X r f A f k = , k r , k r , k r = 2 P X k : p k = P X k + w ν k , � π r , k � σ k F � � σ k Q f � A f x r , k : x r , k = k . r , k P X k f = 1

Elasticities F Q f ∂ ln Q k 1 � � �� � � � 1 − α X , f k 1 − λ f = ( θ − 1 ) + σ k . k 1 , k k α X , f ∂ ln p k Q k α X f = 1 1 , k

Non zero production at zero input Each crop can be produced using two technology: a CES technology and a no-input technology (except value added). Let’s use ˜ x for the variable x under the CES technology and ˇ x for the no-input one. Let’s assume that when produced under these two technologies, the same crop has different productivity distribution with the following cumulative distribution:  1 − ρ k    � � − θ/ ( 1 − ρ k )  � � − θ/ ( 1 − ρ k )    a k ˜ a k ˇ � F ( a ) = exp − + ,   γ ˇ γ ˜ A 1 , k A 1 , k     k ∈ K where ρ k parameterizes the correlation between the two technology.

New equations � ˜ � � σ k � r k Q k = x 1 λ ( θ − 1 ) /θ λ ( θ − 1 + ρ k ) /θ λ ( θ − 1 + ρ k ) /θ ˜ ˜ + ˇ A 1 , k ˇ A 1 , k , (1) k P X k k k where 1 = ˜ λ k + ˇ λ k , (2) � � θ/ ( 1 − ρ k ) ˜ A 1 , k ˜ r k ˜ λ k = � θ/ ( 1 − ρ k ) , (3) � ˇ � � θ/ ( 1 − ρ k ) ˜ A 1 , k ˜ r k + A 1 , k ˇ r k �� � θ/ ( 1 − ρ k ) � 1 − ρ k � ˇ � θ/ ( 1 − ρ k ) ˜ A 1 , k ˜ r k + A 1 , k ˇ r k λ k = � θ/ ( 1 − ρ l ) � 1 − ρ l , (4) �� � ˇ � θ/ ( 1 − ρ l ) � ˜ A 1 , l ˜ + A 1 , l ˇ r l r l l ∈ K r k = p k − w ˇ ˇ ν k (5)

Data • Share of land revenues or Acreage share (for λ f k ) • Acreage or supply elasticities ( θ ) • σ k : yield elasticities or fertilizer response function. • α X k share of land and other inputs in production costs. • α X , f r , k share of each input in the bundle or in a biophysical approach the input levels. • Q f k / Q k or A f 1 , k • Others