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387hw4 - δ 1 β − δ 2 β = 0(7 and the complementary...

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Economics 387. Banking and Financial Intermediation. Spring 2002. Department of Economics, University of Texas Instructor: Dean Corbae, BRB 3.118, (o) 512-475-8530 email: [email protected], web: www.eco.utexas.edu/°corbae Homework #4 - Due 3/6/02 Consider the system of equations that solve the 2 state version of Atkeson and Lucas (1992) in Problem P with log preferences on p. 440. φ ( α ) = min { r i ,h i } 2 i =1 2 X i =1 µ i exp( r i ) subject to 2 X i =1 µ i [exp( r i ) α exp( h i )] = 0 (1) 2 X i =1 µ i [(1 β ) θ i r i + βh i ] = 0 (2) (1 β ) θ i r i + βh i (1 β ) θ i r j + βh j , i, j { 1 , 2 } , i 6 = j (3) If we assign the multipliers λ, ξ, and δ i with the constraints (1)-(3, i ), then the f.o.c. are given by: (1 λ ) µ 1 exp( r 1 ) ξ (1 β ) µ 1 θ 1 δ 1 (1 β ) θ 1 + δ 2 (1 β ) θ 2 = 0 (4) (1 λ ) µ 2 exp( r 2 ) ξ (1 β ) µ 2 θ 2 + δ 1 (1 β ) θ 1 δ 2 (1 β ) θ 2 = 0 (5) λαµ 1 exp( h 1 ) ξβµ 1 δ 1 β + δ 2 β = 0 (6) λαµ 2 exp( h 2 ) ξβµ 2
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Unformatted text preview: + δ 1 β − δ 2 β = 0 (7) and the complementary slackness conditions are: δ 1 [(1 − β ) θ 1 r 1 + βh 1 − (1 − β ) θ 1 r 2 − βh 2 ] = 0 (8) δ 2 [(1 − β ) θ 2 r 2 + βh 2 − (1 − β ) θ 2 r 1 − βh 1 ] = 0 (9) as well as the for a & xed point α = φ ( α ) : µ 1 exp( r 1 ) + µ 2 exp( r 2 ) − α = 0 (10) Thus we have 9 equations (1),(2),(4)-(10) to solve for the the endogenous variables ( r 1 , r 2 , h 1 , h 2 , λ, ξ, δ 1 , δ 2 , α ) . Use GAUSS±s nonlinear systems equa-tion solver (NLSYS) to solve for these variables where the parameters are given by θ 1 = 1 . 5 , θ 2 = 0 . 5 , µ 1 = µ 2 = 0 . 5 , β = . 8 . 1...
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