n n 1 2 n+1 2

i N j j A j D j U [0,θ j (1 e j )] θ j (0, 1] e j [0, 1] LD j L<1 α j {0, 1} α =(α j ) N j=1 =(e j ) N j=1 Π(α, ) = j {A j ((1 D j )+α j LD j )} C( )

C E [U(Π)] = E [Π] ρv ar(π) D E [D j e j ] = 1 2 θ j(1 e j ) V ar [D j e j ] = 1 12 θ2 j (1 e j ) 2 E [Π (α, )] = N j=1 A ( j 1 1 (1 α 2 jl)θ j (1 e j ) ) C( ) V ar [Π (α, )] = 1 12 N j=1 A2 j(1 α j L) 2 θ 2 j (1 e j ) 2 α, N j=1 ( [A j 1 1 ) 2 (1 α jl)θ j (1 e j ) ρ 1 ] 12 A2 j(1 α j L) 2 θj 2 (1 e j ) 2 C( ) N j=1 α j =1 α j {0, 1} e j [0, 1] (α) = ρ 1 12 N j=1 [ A j (1 1 ) 2 (1 α jl)θ j (1 e j ) N A 2 j(1 α j L) 2 θj 2 (1 e j ) 2 ] C( ) j=1

[ ] C (1 e j ) = W j 1+ρW j e j 3 W j = A j w j w j = 1 2 (1 α jl)θ j j c(e j )=ψ j e j ψ j C( ) = N A j ψ j e j. j=1 α ψ j ρ A j w j 0 ψ j w j + 2ρA 3 jwj 2 ê j (α j,ψ j, ρ, A j,w j )= 1 3 ψ j w j 2 ρa j w wj 2 j <ψ j <w j + 2ρA 3 jwj 2 1 ψ j w j ê θ j ê ψ j > 0 < 0 ê ρ > 0 ê w j ê A j > 0 > 0 ρ W 2 j 6 < 0 ψ ψ ψ ψ

ê (optimal effort) Insured Not insured w 1 w 0 ŵ 1 ŵ 0 ψ (cost of effort) ê w 1 w 0 w j w 1 = 1 2 θ j(1 L) w 0 = 1 2 θ j ŵ 1 = w 1 + 2 3 ρa j(w 1 ) 2 ŵ 0 = w 0 + 2 3 ρa j(w 0 ) 2 w 1 <ψ<ŵ 0 ψ F ψ j = 3W j+2ρwj 2 3A j p j = Prob(ê j > 0) = Prob(ψ j < ψ j )=F( ψ j ) p j θ j > 0 p j > 0 p j ρ w j > 0 > 0 ê j =1 p j A j θ j w j ρ A j ψ ψ (w 1, ŵ 0 ))

ê j (α j,ψ j, ρ, A j,w j ) j u j (α j,ψ j, ρ, A j,w j )=A j 1 2 A jw j (1 ê(α j,ψ j, ρ, A j,w j )) ρ 12 A2 jw 2 j (1 ê(α j,ψ j, ρ, A j,w j )) 2 A j ψ j ê(α j,ψ j, ρ, A j,w j ) α V (α) = N u j (α j,ψ j, ρ, A j,w j ). j=1 α V (α) α j {0, 1} N j=1 α j =1 α j =1 A j {}}{ 1 2 θ j(1 ê j (0,ψ j, ρ, A j,w j )) A j 1 2 θ j (1 ê j (0,ψ j, ρ, A j,w j )) j {1,..,N}.

P P = LA j E [D j θ j,ψ j,α j =1]= 1 2 LA jθ j (1 ê(θ j,ψ j,α j =1)) C C = ψ j A j (ê(θ j,ψ j,α j =0) ê(θ j,ψ j,α j =1)) ψ

= + AG AGD D D = AGD AG = AGD AGD+AG(1 D) = +

D ij = β 0 + β 1 α ij + β 2 C ij + β 3 C ij α ij + β 4 A ij + λ i + ɛ ij α C A θ ij ψ ij A j θ A j θ j (1 ê(θ j,ψ j,α j =0)) β 2 (A 1,θ 1,ψ 1 ) β 1 β 1 +β 3

(A 1,θ 1,ψ 1 ) β 2 E [D 1 (A 1,θ 1,ψ 1 ),α 1 =0] E [D 1 (A 1,θ 1,ψ 1 ),α 1 =0] = 1 2 θ 1(1 ê(θ 1,ψ 1,α 1 =0)) 1 }{{} 2 θ 1(1 ê(θ 1,ψ 1,α 1 =0)) }{{} β 3 β 3 = (E [D 1 (A 1,θ 1,ψ 1 ),α 1 =1] E [D 1 (A 1,θ 1,ψ 1 ),α 1 =0]) (E [D 1 (A 1,θ 1,ψ 1 ),α 1 =1] E [D 1 (A 1,θ 1,ψ 1 ),α 1 =0]) {}}{ 1 2 θ 1 [(1 ê(θ 1,ψ 1,α 1 =1)) (1 ê(θ 1,ψ 1,α 1 =0))] 1 2 θ 1 [(1 ê(θ 1,ψ 1,α 1 =1)) (1 ê(θ 1,ψ 1,α 1 =0))] }{{} = 1 2 θ 1 {}}{ [ê(θ 1,ψ 1,α 1 =0) ê(θ 1,ψ 1,α 1 =1)] 1 2 θ 1 [ê(θ 1,ψ 1,α 1 =0) ê(θ 1,ψ 1,α 1 =1)] }{{} ψ j ψ j [ê(θ j,ψ j,α j =1) ê(θ j,ψ j,α j =0)] D U [0,θ(1 e)] θ e E [D θ, e] = 1 2θ(1 e)

β 3 D ij = β 0 + β 1α ij + β 2C ij + β 3A ij + λ i + ɛ ij. β 1 β 2 β 2 = β 2 + 1β 2 3 β 2

θ j θ j ψ j

θ

= α 0 + α 1 ij + α 2 ij i + α 3 ij i + λ i +ɛ }{{} ij α 3 α 3 > 0

D ij = β 0 + β 1 C ij + β 2 X ij + β 3 X ij 1( )+λ i + η ij. β 2 β 3

1A 2 jθ j A j θ j j v j = u j (α j =1) u j (α j =0)=cA j θ j c = 1 2 L θ = 1 N N j=1 θ j v j = ca j θ + caj (θ j θ) E [D j θ j ]= 1 2 A jθ j θj obs û j = A j Ê [ ] D j θj obs Λ(C ij )=α 0 + α 1 A ij Ê [D X, I =1]+α 2 A ij Ê [D I =1]+ɛ ij C ij =1 i j α 1 > 0

ê I j j ê 0 j j {}}{ vj 1 = u j (α j =1) u j (α j =0)= 2 A jθ j (1 ê 0 j)l ρ [ (1 L) 2 1 ] A 2 12 jθj 2 (1 ê 0 j) 2 + ρ 12 A2 jθj 2 (1 L) [ 2 (1 ê 0 j) 2 (1 ê 1 j) 2] + 1 2 A jθ j (ê 0 j ê 1 j)l }{{} }{{} + A j ψ j (ê 0 j ê 1 j) }{{} v b = 1 2 A jθ j (1 ê 0 j)l v m = 1 2 A jθ j (ê 0 j ê 1 j)l ˆv b = A ij Ê [D X, I =0] ˆv m = A ij (Ê [D X, I =1] Ê [D X, I =0]). v b v m Λ(C ij )=α 0 + α 1ˆv b + α 2ˆv m + α 3 A ij + α 4ˆv b + α 5ˆv m + ɛ ij = α 0 + α 1 A ij Ê [D X, I =0] + α 2 A ij (Ê [D X, I =1] Ê [D X, I =0]) + α 3 A ij Ê [D I =1]+α 4 Ê [D X, I =0] + α 5 (Ê [D X, I =1] Ê [D X, I =0])+ɛ ij. α 1 > 0 1θ 2 j(1 ê 0 j)=e [D I =0] α 2 > 0 ψ>0 α 2 > 0

Ê [D X, I =0] Ê [D X, I =1] D ij = β 0 + β 1 X ij + β 2 X ij 1( )+η ij X θ(1 ê 0 ) α 3 A j ψ j (ê 0 j ê 1 j)

4.7 4.4 2.6 2.6 1.8 1.7 6.8 (1.4) (2.4) (1.1) (2.0) (1.0) (1.8) (3.9) 1.3 1.1 0.3 0.4 1.9 1.9 (1.2) (1.9) (1.0) (1.6) (0.9) (1.4) 2.3 2.3 0.6 0.6 1.6 1.6 2.7 (2.2) (2.2) (1.8) (1.8) (1.6) (1.6) (5.6) 0.5 0.2 0.2 (4.0) (3.3) (2.9) +

2.0 1.8 1.9 1.6 (0.9) (0.9) (0.9) (0.9) 1.0 0.8 (0.6) (0.6) 0.6 0.4 (2.8) (2.8) 1.0 1.8 (1.6) (1.6) 1.7 0.7 (1.9) (1.9) 2.0 1.9 1.5 1.4 (1.5) (1.6) (1.5) (1.6) + +

4.7 4.8 5.7 (1.4) (1.4) (2.2) 2.5 2.2 (1.3) (1.3) 1.9 0.2 2.0 0.5 2.8 (0.9) (1.6) (0.9) (1.0) (1.2) 2.4 (1.9) 1.1 1.2 2.0 (0.9) (1.0) (1.2) 1.4 1.0 2.3 (0.9) (1.2) (1.2)

1.9 2.1 9.3 17.7 1.8 1.8 0.02 0.02 0.8 2.4 (0.9) (1.2) (17.4) (24.3) (1.2) (1.6) (0.010) (0.01) (0.8) (1.0) 0.2 0.2 3.4 5.4 0.7 0.5 0.6 2.0 (1.1) (1.5) (33.6) (55.1) (1.9) (3.0) (0.7) (1.1) 2.6 3.2 12.1 23.8 2.2 2.3 0.02 0.02 1.0 2.6 (1.1) (1.5) (20.1) (24.6) (1.5) (2.0) (0.010) (0.01) (1.1) (1.4)

1.93 2.15 0.22 0.15 0.43 0.30 (0.86) (1.18) (0.13) (0.17) (0.25) (0.33) 0.23 0.21 0.10 0.19 0.20 0.37 (1.08) (1.54) (0.22) (0.33) (0.44) (0.65) 2.64 3.16 0.46 0.39 0.90 0.77 (1.14) (1.55) (0.15) (0.17) (0.30) (0.34) log(fertilizer + fertilizer 2 + 1)

5.5 3.8 (1.4) (1.5) 2.3 3.6 (2.2) (2.4)

4.0 6.9 (2.1) (2.1) 5.2 4.3 (4.2) (4.2) 5.5 0.5 (5.2) (4.9) 7.9 1.0 (4.5) (4.2) 13.1 3.4 (5.7) (6.0) 0.2 4.8 (2.7) (2.5) 6.2 7.0 (4.6) (4.6) 8.6 9.1 (6.0) (5.8) 0.1 3.1 (5.4) (4.9) 7.1 2.5 (7.0) (7.2) 21.5 26.3 (2.9) (2.6)

β β Ê(D X, I =1) 1.12 0.025 (0.044) (0.0078) Ê(D X, I =0) 1.13 0.026 (0.052) (0.0090) (Ê(D X, I =1) Ê(D X, I =0)) 1.10 0.023 (0.061) (0.012) Ê(D X, I =1) 1.01 0.0045 (0.024) (0.0052) Ê(D X, I =0) 1.01 0.0066 (0.028) (0.0062) (Ê(D X, I =1) Ê(D X, I =0)) 0.99 0.0018 (0.032) (0.0080) Ê(D I =1) 0.96 0.95 0.0073 0.0084 (0.035) (0.040) (0.0081) (0.0089)

31.7 (26.3) 21.2 (26.8) 25.6 (25.4) 12.0 (21.5) 6.0 (13.5) 9.4 (19.9) 22.9 (52.9) 11.8 (36.9) +

1.72 1.57 0.15 1.71 1.57 0.14 (0.09) (0.14) 2.88 2.84 0.04 2.88 2.82 0.06 (0.69) (0.61) 10.20 10.45 0.26 10.11 10.40 0.29 (0.41) (0.39) 53.82 52.93 0.90 53.76 53.17 0.59 (0.37) (0.59) 0.17 0.16 0.00 0.16 0.15 0.01 (0.91) (0.68) 0.33 0.33 0.00 0.34 0.34 0.01 (0.84) (0.73) 0.59 0.60 0.02 0.59 0.60 0.01 (0.39) (0.62) 0.23 0.25 0.02 0.20 0.22 0.02 (0.36) (0.51) 0.02 0.02 0.04 0.01 0.04 0.03 (0.40) (0.59) 0.19 0.21 0.02 0.20 0.22 0.02 (0.23) (0.42) 0.15 0.14 0.01 0.16 0.15 0.01 (0.63) (0.60) 0.04 0.04 0.00 0.05 0.05 0.00 (1.00) (0.87)

Farmers in Random Insurance Group Insured plot Control plot 1st Choice Plots Non-1st Choice Plots Adverse Selection Moral Hazard

Farmers in Random Insurance Group Insured Plots Control Plots 1st Choice Plots (a) Adverse Selection (c) Moral Hazard on 1st choice (b) Selection On Baseline Risk Non-1st Choice Plots (d) Moral Hazard on non-1st choice

Density 0.005.01.015 Full harvest per hectare (all seasons) 0 100 200 300 400 Potential harvest per hectare

E [Π(α, )] = E [ N ] j=1 (A j(1 D j )+α j A j LD j ) C( ) = N j=1 A j(1 (1 α j L)E [D j e j ]) C( ) = N j=1 A j(1 (1 α j L) θ j(1 e j ) 2 ) C( ) V ar(π(α, )) [ N = V ar j=1 A j ] N j=1 A j(1 α j L)D j = N j=1 A2 j(1 α j L) 2 V ar(d j ) = N j=1 A2 j(1 α j L) 2 θ2 j (1 e j) 2 12 [ ] C (1 αj L)θ j = A j ρa j (1 α j L) 2 θ 2 2(1 e j ) j e j 2 12 [ ] 1 = A j (1 α j L)θ j 2 + ρa 1 e j j(1 α j L)θ j 6 [ ] 2(1 e j ) = W j 1+ρW j 3

W j = 1 2 A j(1 α j L)θ j ρw j 2(1 e j ) 3 [ ] 2(1 e j ) A j ψ j = W j 1+ρW j 3 = A jψ j W j 1 1 e j = 3A jψ j 2ρW 2 j e j =1 3A jψ j 2ρW 2 j 3 2ρW j + 3 2ρW j e j =1 3 ψ j w j 2 ρa j wj 2 e j (0, 1) e j > 0 ψ j <w j + 2 3 ρa2 jwj 2 e j < 1 w j <ψ j. e = 3A j < 0 ψ j 2ρWj 2 e ρ = 3(A jψ j W j ) > 0 4ρ 2 Wj 2 e θ j = = e W j W j θ j [ ( 2) 3A jψ j 2ρWj 2 3 ] ( 1 2ρWj 2 2 A j(1 α j L)) = 6A jψ j 3W j ( 1 2ρWj 3 2 A j(1 α j L)) > 0 ψ F ψ = 3W j+2ρw 2 j 3A j p = Prob(ê >

0) = Prob(ψ < ψ) =F ( ψ) p w = F w = F 3+4ρW ( ψ) 3A > 0 p θ = F 3+4ρW 1 ( ψ) A(1 αl) > 0 3A 2 p ρ = F 2 2W ( ψ) 3A > 0 p A = F ( ψ) 1 6 ρθ2 (1 αl) 2 > 0 ψ F ˆψ = w j q = Prob(ê =1)=Prob(ψ < ˆψ) =F ( ˆψ) q w = F ( ˆψ) > 0 q θ = F ( ˆψ) 1 A(1 αl) > 0 2 q ρ =0 q A =0 Λ(α j,ψ j, ρ, A j,w j )= N j=1 1 2 A jw j (1 ê(α j,ψ j, ρ, A j,w j )) + ρ 12 A2 jw 2 j (1 ê(α j,ψ j, ρ, A j,w j )) 2 + A j ψ j (ê(α j,ψ j, ρ, A j,w j )) Π(α, ψ j,,ρ,, = N j=1 Λ(α, ψ j,,ρ,, ) ˆα = α Λ(α, ψ j,,ρ,, )

ê(0,ψ j, ρ, A j,w j ) j ê j 0 ê(0,ψ j, ρ, A j,w j ) Λ(α, ψ j, ρ,, ) = N { 1 2 A jw j (1 ê j 0)+ ρ 12 A2 jwj 2 (1 ê j 0) 2 j=1 } + 1 2 A jψ j ê j 0 N N λ(a j θ j, (1 α j L)) A j ψ j ê j 0 j=1 j=1 λ λ(y, x) = 1 4 yx+ ρ 48 y2 x 2 N j=1 A jψ j ê j 0 λ h l A h θ h (1 ê h 0) >A l θ l (1 ê l 0) h l Λ((α h =1,α h =0)) Λ((α l =1,α l =0)) = λ(a h θ h (1 ê h 0), (1 L)) + λ(a l θ l (1 ê l 0), 1) λ(a l θ l (1 ê l 0), 1 L) λ(a h θ h (1 ê h 0), 1) = λ(a h θ h (1 ê h 0), (1 L)) λ(a l θ l (1 ê l 0), 1 L)+λ(A l θ l (1 ê l 0), 1) λ(a h θ h (1 ê h 0), 1) M(A h θ h (1 ê h 0)) h M(A h θ h (1 ê h 0) < 0 λ(aθ(1 ê 0 ), (1 αl)) = 1 Aθ(1 ê 0 ) 4 (1 αl)+ ρ 24 Aθ(1 ê 0)(1 αl) 2 > 0 λ(aθ, (1 αl)) = 1 (1 αl) 4 Aθ(1 ê 0)+ ρ 24 (Aθ(1 ê 0)) 2 (1 αl) > 0 λ(aθ, (1 αl)) = 1 Aθ (1 αl) 4 + ρ 12 (Aθ(1 ê 0))(1 αl) > 0

M(A l θ l (1 ê 0 )) = 0 M(A h θ h (1 ê h 0)) = M(A h θ h (1 ê h 0)) M(A l θ l (1 ê l 0)) Ah θ h (1 ê h 0 ) A l θ l (1 ê l 0 ) Ah θ h (1 ê h 0 ) A l θ l (1 ê l 0 ) Ah θ h (1 ê h 0 ) < 0 A l θ l (1 ê l 0 ) Ah θ h (1 ê h 0 ) 1 A l θ l (1 ê l 0 ) M(s) Aθ(1 ê 0 ) ds ( (s, 1 L) Aθ(1 ê 0 ) ( (s, 1 L) Aθ(1 ê 0 ) 1 L λ(s, m) dmds } s m {{} >0 ) (s, 1 L) ds Aθ(1 ê 0 ) (s, 1 L) Aθ(1 ê 0 ) ) ds Aθ(1 ê 0 ) 1 2 θ(1 ê 0)