ÒÄÆÉÖÌÄ. ÐÄÒÉÏÃÖËÉ ÀÌÏÝÀÍÉÓ x = f(t, x), x(0) = x(1) ÀÌÏáÓÍÀÃÏÁÉÓ ÊÀÒÂÀÃ ÝÍÏÁÉËÉ

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2 x = f(t, x x( = x(1 1 1 x(1 = dh(s x(s x( = dh(s x(s h ÒÄÆÉÖÌÄ. ÐÄÒÉÏÃÖËÉ ÀÌÏÝÀÍÉÓ x = f(t, x, x( = x(1 ÀÌÏáÓÍÀÃÏÁÉÓ ÊÀÒÂÀÃ ÝÍÏÁÉËÉ 1 ÊÒÀÓÍÏÓÄËÓÊÉÓ ÃÀ ÂÖÓÔÀÅÓÏÍ-ÛÌÉÔÉÓ ÛÄÃÄÂÄÁÉ ÂÀÍÆÏÂÀÃÄÁÖËÉ ÉØÍÀ x(1 = dh(s x(s ÃÀ x( = 1 dh(s x(s ÓÀáÉÓ ÀÒÀËÏÊÀËÖÒÉ ÓÀÓÀÆÙÅÒÏ ÐÉÒÏÁÄÁÉÓ ÛÄÌÈáÅÄÅÀÛÉ (h ÍÀÌÃÅÉËÉ ÀÒÀÊËÄÁÀÃÉ ÖÍØÝÉÀÀ, ÒÏÌËÄÁÉÝ ÌÏÉÝÀÅÄÍ ÐÄÒÉÏÃÖË ÓÀÓÀÆÙÅÒÏ ÐÉÒÏÁÄÁÓ, ÒÏÂÏÒÝ ÊÄÒÞÏ ÛÄÌÈáÅÄÅÄÁÓ. ÊÏÍÔÒÌÀÂÀËÉÈÄÁÉÓ ÓÀÛÖÀËÄÁÉÈ ÛÄÃÀÒÄÁÖËÉÀ ÐÄÒÉÏÃÖËÉ ÃÀ ÀÒÀ- ËÏÊÀËÖÒÉ ÓÀÓÀÆÙÅÒÏ ÐÉÒÏÁÄÁÉÓ ÛÄÌÈáÅÄÅÄÁÉ, ÒÏÌËÄÁÉÝ ÀÅËÄÍÄÍ ÐÄÒÉÏÃÖËÉ ÛÄÌÈáÅÄÅÉÓ ÓÐÄÝÉÀËÖÒ áàóéàèó. ÀÍÀËÏÂÉÖÒÉ ÊÏÍÔÒÌÀÂÀËÉÈÄÁÉ ÀÂÒÄÈÅÄ ÂÅÉÜÅÄÍÄÁÄÍ, ÒÏÌ ÌÄÏÒÄ ÒÉÂÉÓ ÓÉÓÔÄÌÄÁÉÓ ÛÄÌÈáÅÄÅÀÛÉ ÂÀÒÊÅÄÖËÉ ÀÒÀËÏÊÀËÖÒÉ ÓÀÓÀÆÙÅÒÏ ÐÉÒÏÁÄÁÉÈ ÆÏÂÉÄÒÈÉ ÖÔÏËÏÁÉÓ ÍÉÛÍÉÓ ÛÄÁÒÖÍÄÁÀ ÀÒ ÛÄÉÞËÄÁÀ.

3 R n B R R n R f : [, 1] R n R n x = f(t, x, x( = x(1. R > u f(t, u (t, u [, 1] B R, u f(t, u (t, u [, 1] B R, x([, 1] B R τ = 1 t e C([, 1], R n λ R \ {} x = λx + e(t, x( = x(1 d dt R n f : [, 1] H H f(t, t [, 1] R > u f(t, u < (t, u [, 1] B R C R n Φ i C 1 (R n, R (i = 1,..., r C = {u R n : Φ i (u (i = 1,..., r} Φ i (u Φ i (u = u C Φi (u f(t, u (t, u [, 1] C i α(u, Φi (u f(t, u (t, u [, 1] C i α(u, α(u := {i {1,..., r} : Φ i (u = } x([, 1] C C = B R r = 1 Φ 1 (u = 1 ( u R C R n R n u C ν(u R n \ {} ν(u u > C {v R n : ν(u v u < } ν(u C u ν : C R n \ {} C ν ν(u C u C R n ν C ν(u f(t, u > (t, u [, 1] C,

4 ν(u f(t, u < (t, u [, 1] C, x([, 1] C C = B R C C C u C i α(u Φ i (u C u 1 x = f(t, x, x(1 = dh(s x(s, x = f(t, x, x( = 1 dh(s x(s h : [, 1] R 1 dh(s = 1. x( = x(1 x( x(1 x [, 1] h( < h(α α (, 1 C R n ν C ν(u f(t, u (t, u [, 1] C, x x([, 1] C h(α < h(1 α (, 1 C R n ν C ν(u f(t, u (t, u [, 1] C, x x([, 1] C C = B R h( < h(α α (, 1 R > u f(t, u (t, u [, 1] B R, x x([, 1] B R

5 h(α < h(1 α (, 1 R > u f(t, u (t, u [, 1] B R, x x([, 1] B R x = λx + e(t, x(1 = 1 [ ( 1 ] x + x(, x ( = 1 [ ( 1 ] x + x(1, e C([, 1], R n λ R \ {} d dt f λ e C([, 1], C z = λz + e(t, z : [, 1] C x 1 = Rz, x = Iz, f 1 (t, x = R(λz + e(t, f (t, x = I(λz + e(t, u f(t, u t [, 1] u 3 n x, f(t, x x = R R > x = g(t, x, x, x( =, x (1 = dh(s x (s, 1 g : [, 1] R n R n R n h : [, 1] R 1 dh(s = 1 h( < h(α α (, 1 C R n ν C ν(v g(t, u, v (t, u, v [, 1] C C, x x([, 1] C x ([, 1] C

6 C = B R h( < h(α α (, 1 R > v g(t, u, v (t, u, v [, 1] B R B R, x x([, 1] B R x ([, 1] B R x = g(t, x, x, x( =, x ( = dh(s x (s, 1 h(α < h(1 α (, 1 g h(α < h(1 α (, 1 C R n ν C ν(v g(t, u, v (t, u, v [, 1] C C, x x([, 1] C x ([, 1] C x = g(t, x, x, x( =, x ( = x (1 R > v g(t, u, v (t, u, v [, 1] B R B R, v g(t, u, v (t, u, v [, 1] B R B R. z (t = λz(t, z(1 = 1 [ ( 1 z( + z, ] λ C z : [, 1] C s =, h(s = 1/ s (, 1/] 1 s (1/, 1]. λ tc,1,k = k(πi λ tc,,k = 4 + (k + 1(πi (k Z

7 λ C e λ = eλ/. µ := e λ/ µ µ 1 µ 1 =, µ tc,1 = 1 µ tc, = 1 eλ/ = µ tc,1 = 1 λ = kπi (k Z λ tc,1,k = k(πi (k Z. e λ/ = µ tc, = 1 λ = + πi + kπi = + (k + 1πi (k Z λ tc,,k = 4 + (k + 1(πi (k Z. z (t = λz(t, z( = 1 [ ( 1 ] z + z(1, λ C z : [, 1] C s [, 1/, h(s = 1/ s [1/, 1, 1 s = 1. λ ic,1,k = k(πi λ ic,,k = 4 + (k + 1(πi (k Z λ C µ := e λ/ µ 1 = 1 eλ/ + 1 eλ. 1 µ + 1 µ 1 = µ ic,1 = 1 µ ic, = λ ic,1,k = k(πi (k Z λ ic,,k = 4 + (k + 1(πi (k Z. z = λz, z( = z(1 λ p,k = k(πi (k Z Rz = 4 Rz = 4

8 λ ( e ( Lz := z z = e(t, z( = 1 ( 1 z + 1 z(1 Mz := z + z = e(t, z( = 1 z ( z(1 z = L 1 e z = M 1 e e C([, 1], C L 1 M 1 C([, 1], C z λz = e(t, z(1 = 1 z( + 1 ( 1 z z + λz = e(t, z( = 1 z ( 1 z = (λ 1L 1 z + L 1 e, z = (λ + 1M 1 z + M 1 e, + 1 z(1 λ tc,, = 4 + (4k + πi e : [, 1] C z (t = ( 4 + πiz(t + e(t, z(1 = 1 z( + 1 ( 1 z z(t = x 1 (t + ix (t e(t = e 1 (t + ie (t x 1(t = ( 4x 1 (t πx (t + e 1 (t, x (t = πx 1 (t ( 4x (t + e (t, x 1 (1 = 1 x 1( + 1 ( 1 x 1, x (1 = 1 x ( + 1 ( 1 x. f(t, u := ( ( 4u 1 πu + e 1 (t, πu 1 ( 4u + e (t. u f(t, u = u 1 [ ( 4u1 πu + e 1 (t ] + u [ πu1 ( 4u + e (t ] = ( 4(u 1 + u + u 1 e 1 (t + u e (t ( 4 u + e(t u <,

9 u R R f R > u f(t, u (t, u [, 1] B R, λ ic,, = 4 + πi e : [, 1] C z (t = ( 4 + πiz(t + e(t, z(1 = 1 z( + 1 ( 1 z z(t = x 1 (t + ix (t e(t = e 1 (t + ie (t x 1(t = ( 4x 1 (t πx (t + e 1 (t, x (t = πx 1 (t + ( 4x (t + e (t, x 1 ( = 1 ( 1 x x 1(1, x ( = 1 ( 1 x + 1 x (1. f(t, u := ( ( 4u 1 πu + e 1 (t, πu 1 + ( 4u + e (t. u f(t, u = u 1 [( 4u 1 πu + e 1 (t] + u [ πu1 + ( 4u + e (t ] = ( 4(u 1 + u + u 1 e 1 (t + u e (t ( 4 u e(t u >, u R R f R > u f(t, u (t, u [, 1] B R, f n = 1 n n = 3, x 3 = ( 4x (x 1 + x, x 3 (1 = 1 [ ( 1 ] x 3 ( + x 3 x 3 = ( 4x (x 1 + x, x 3 ( = 1 [ ( 1 ] x 3 + x 3 (1 4 n = n = 3 n n = 1

10 z = πiz + e πit, z( = z(1 z (e πit z = 1, [, 1] z = x 1 + ix x 1 = πx + (πt, x = πx 1 + (πt, x 1 ( = x 1 (1, x ( = x (1, f 1 (t, x 1, x = πx + (πt, f (t, x 1, x = πx 1 + (πt, x = (x 1, x, f(t, x = ( f 1 (t, x 1, x, f (t, x 1, x, x, f(t, x = πx x 1 + (πtx 1 + πx 1 x + (πtx = (πtx 1 + (πtx. x = R[ (πθ, (πθ] B R (θ [, 1] x, f(t, x = R [ (πt (πθ + (πt (πθ ] = R [π(t θ] (t, θ [, 1], t [, 1] x, f(t, x B R n z (t = λz (t, z( =, z (1 = 1 z ( + 1 z ( 1, λ C x : [, 1] C z λz λz λ bc,j,k (j = 1, k Z 4 w(t = z (t z( = z(t = w (t = λw(t, w(1 = 1 w( + 1 w ( 1 t w(s ds λ e,

11 z λz = e(t, z( =, z (1 = 1 z ( + 1 z ( 1 w = z w λw = e(t, w(1 = 1 [ ( 1 w( + w, ] λ bc,, = 4 + πi e : [, 1] C z (t = ( 4 + πiz (t + e(t, z( =, z (1 = 1 z ( + 1 z ( 1 z(t = x 1 (t + ix (t e(t = e 1 (t + ie (t x 1(t = ( 4x 1(t πx (t + e 1 (t, x (t = πx 1(t ( 4x (t + e (t, x 1 ( =, x 1(1 = 1 x 1( + 1 ( 1 x 1, x ( =, x (1 = 1 x ( + 1 ( 1 x. g(t, v := ( ( 4v 1 (t πv (t + e 1 (t, πv 1 (t ( 4v (t + e (t. v, g(t, v < v R R g R > v, g(t, u, v (t, u, v [, 1] B R B R,

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Inference. Jesús Fernández-Villaverde University of Pennsylvania

Inference. Jesús Fernández-Villaverde University of Pennsylvania Inference Jesús Fernández-Villaverde University of Pennsylvania 1 A Model with Sticky Price and Sticky Wage Household j [0, 1] maximizes utility function: X E 0 β t t=0 G t ³ C j t 1 1 σ 1 1 σ ³ N j t