d log w F = 1(i = k) + d log A k b Ψe k + b Ψ α d log w

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1 B Proos Proo o Theorem 2.2. By Shephard s lemma, we now that, to a rst order, the productvty shoc A wll change the prces o any ndustry who purchases nputs, ether drectly or ndrectly, rom ndustry d log p = ( = ) + Rewrte ths n matrx orm to get F d log w F α F + Ω d log p. (36) d log p = (I Ω) ( α d log w e ) = Ψ( α d log w e ), (37) where e s the th standard bass vector. Let the household s aggregate consumpton good to be the numerare, so that the household s deal prce ndex P c s always equal to one. Then we now that Combne ths wth the prevous expresson to get Note that, b Ψ = λ and b Ψ α = Λ. Hence, d log P c = b d log p = 0. (38) b Ψe + b Ψ α d log w = 0. (39) λ + Λ d log w = 0. (40) Now, note that From ths, we now that Λ = w L P c C. (4) d log Λ = d log w + d log L d log C. (42) 76

2 Substtute ths nto the prevous expresson to get λ + Λ d log Λ Λ d log L + d log C = 0, (43) where we use the act that Λ =. Rearrange ths to get the desred result.to get an explct characterzaton or d log Λ / n terms o structural parameters o the model, wthout loss o generalty, assume that each good s produced rom a dstnct prmary actor (ths can be acheved by relabellng the nput-output matrx). Now note that λ = b + µ ω λ, (44) w = α p y µ G (w /P c ), (45) λ = p y P c C, (46) p = A C (p,..., p N, w )µ, (47) P c = p c C =. (48) We can derentate these to get to our answer. Denote the elastcty o substtuton between and or the total cost uncton o ndustry by ρ. Then we can wrte l d λ = d b + ρ d ω µ λ d λ + ω µ ω µ λ d log µ, (49) d b = b b ( ) ρ c d log p d log p, (50) d ω = ω ω ( ) d log p d log p + ω α ( ) d log p d log w, (5) d log w = ω ( ) d log w d log p + d log p + d log y d log µ, (52) ρ L ρ L 77

3 d log y = λ The proo or the case wth marups s very smlar. d λ d log p + d log P cc, (53) Proo o Proposton 2.6. Tae all dervatves wth respect to A and normalze P c =. In ths case, the system reduces to λ d λ = 0, (54) d log w = d log p + d log y (55) d log y = d log p + d log C (56) d log p = ( = ) + α d log w + ω d log p, (57) d log C = w l C d log w + π ( d log λ + d log C ). (58) C We also now that b d log p = 0 by normalzaton. Hence, combnng the second and thrd relatonshp we get that d log w = d log C, (59) n other words, all wages move together there s no pecunary externalty on the actors n response to a shoc. Next, we now the prcng equaton s d log p = (I Ω) (α d log C e ). (60) Combne ths wth b d log p = 0 to get the desred result that d log C = b (I Ω) e, (6) ollowng the act that b (I Ω) α = or, wthout marups, the sum o actor payments would add up to GDP snce there are no pure prots. Proo o Proposton 3.. Frst, observe that d log p = ( = ) + ω d log p + α d log w. (62) Hence, d log p = e (I Ω) α d log w e (I Ω) e = d log w ψ. (63) 78

4 Snce, b d log p = d log w λ = 0, ths s d log p = λ ψ. (64) Ths completely characterzes the eect on prces, and the real wage s equal to output, we are done. However, n equlbrum, the real wage s not equal to output. To get the eect on output, we need to tae nto account how the labor share o ncome changes. Denote the labor share by Λ L. Usng Theorem 2.2, we have that d log C = λ Λ L d Λ L. (65) Hence, we need to characterze d Λ L. We now that d Λ L = α µ λ ( θ ) [ d log w d log p ( = ) ] + Whch smples to Now, note that d Λ L = α µ λ ( θ ) ( ψ ( = ) ) + d λ = b ( θ 0 ) d log p + ω µ ( θ ) ( d log p d log p ( = ) ) + Substtute n prces, and solve ths lnear system n d λ to get α µ d λ. (66) α µ d λ. (67) ω µ d λ. d λ m = b ( θ 0 )( λ ψ ) + ω µ λ ( θ )( ψ ψ ) + ω µ λ (θ ) ψ m. Ths can be smpled to d λ m = b ( θ 0 ) λ ψ m b ( θ 0 )ψ m ψ + ω µ ( θ )λ ψ ψ m,, ω µ ( θ )λ ψ ψ m (68) (69) 79

5 Whch can be urther smpled to Hence, m + ω µ λ (θ )ψ m. (70) d λ m = λ m λ ( θ 0 ) b ( θ 0 )ψ m ψ + ω µ ( θ )λ ψ ψ m,, ω µ ( θ )λ ψ ψ m + λ (θ ) [ ψ m (m = ) ]. (7) = (θ 0 )cov b ( Ψ (), Ψ (m) ) + (θ )λ cov ( Ψ µ Ω (), Ψ (m) ) + (θ )λ ω ψ µ ω ψ m, ω µ (θ )λ ψ ψ m + λ (θ )(ψ m (m = )), = (θ 0 )cov b ( Ψ (), Ψ (m) ) + (θ )λ cov ( Ψ µ Ω (), Ψ (m) ). (72) α m µ m d λ m = (θ 0 )cov b ( Ψ (), Puttng ths altogether, we get d Λ L = α µ λ ( θ ) ( ψ ( = ) ) + (θ 0 )cov b ( Ψ (), α m µ m Ψ (m) ) + m m α m µ m Ψ (m) )+ To nsh the proo, note that m µ m α m Ψ m = Ψ L. Then d Λ L = α µ λ ( θ ) ( ψ ( = ) ) + (θ 0 )cov b ( Ψ (), α m µ m Ψ (m) ) + m (θ )λ cov ( Ψ µ Ω (), (θ )λ cov ( Ψ µ Ω (), m m α m µ m Ψ (m) ). (73) α m µ m Ψ (m) ). (74) (θ )λ cov ( Ψ µ Ω (), Ψ (L) ), (75) 80

6 = (θ 0 )cov b ( Ψ (), α m µ m Ψ (m) ) + m (θ )λ cov ( Ψ µ [Ω,a ] (), Ψ (L) ), (76) where note that Ψ LL =. Proo o Proposton 3.2. From Theorem 2.2, we now that d log C/ d log µ = λ Λ L d λ L. (77) For a quantty m, we have d λ m = b ( θ 0 )(d log p ) + Ω µ ( θ )λ [d log p d log p ] ( = )θ Ψ m. Note that Hence, Snce, b d log p = d log w + λ = 0, ths s (78) d log p = ( = ) + ω d log p + α d log w. (79) d log p = d log w + ψ. (80) d log p = λ + ψ. (8) Substtutng ths bac nto (78) and set m = L to get d λ L = b ( θ 0 )( ψ λ )ψ L + λ ( θ )µ Rearrange ths to get Ω [ ψ ψ ]Ψ L θ µ λ d λ L =( θ 0 )Cov b ( Ψ (), Ψ (L) ) + λ ( θ )µ Cov Ω ()( Ψ (), Ψ (L) ) µ ( θ )λ Ω Ψ L θ µ λ Ω Ψ L. Ω Ψ L. (82) Combne ths wth (77). 8

7 Proo o Proposton 3.3. By Shephard s lemma, d log p = ( = ) + Ω d log p + α d log w. (83) Invert ths system to get d log p = Ψe + Ψ d log w, (84) where Ψ = (I Ω) α s a N K matrx o networ-adusted actor ntenstes by ndustry. Now consder a actor L, we have d Λ L = b ( θ 0 )[ Ψ + Ψ d log w ]Ψ L, Smply ths to + ( θ )λ µ Ω [ Ψ + Ψ d log w + Ψ Ψ d log w ]Ψ L + (θ )λ µ Ω Ψ L. d Λ L =(θ 0 ) b Ψ Ψ L b Ψ L Ψ d log w, + (θ )λ µ Ω Ψ Ψ L Ω Ψ Ψ L ( ) + ( θ )λ µ Ω Ψ Ψ d log w Ψ L + (θ )λ µ Ω Ψ L, =(θ 0 ) b Ψ Ψ L b Ψ L Ψ d log w, + (θ )λ µ Ω Ψ Ψ L Ω Ψ L Ω Ψ + ( θ )λ µ ( ) Ω Ψ Ψ d log w Ψ L, 82

8 =(θ 0 ) b Ψ Ψ L b Ψ L Ψ d log w, + (θ )λ µ Cov Ω ()( Ψ (), Ψ (L) ) + ( θ )λ µ Ω Ψ Ψ L Ω Ψ L Ω Ψ d log w, =(θ 0 ) b Ψ Ψ L b Ψ L Ψ d log w, + (θ )λ µ Cov Ω ()( Ψ (), Ψ (L) ) + ( θ )λ µ Cov Ω ()( Ψ ( ), Ψ (L) ) d log w, =(θ 0 ) b Ψ Ψ L b Ψ L Ψ d log w, + (θ )λ µ Cov Ω ()( Ψ () Ψ ( ) d log w, Ψ (L) ) =(θ 0 ) b Ψ L Ψ Ψ d log w, + (θ )λ µ Cov Ω ()( Ψ () Ψ ( ) d log w, Ψ (L) ) =(θ 0 )Cov b ( Ψ () Ψ ( ) d log w, Ψ (L) ) + (θ 0 )( λ Λ d log w )λ L, + (θ )λ µ Cov Ω ()( Ψ () Ψ ( ) d log w, Ψ (L) ). Hence, or a productvty shoc, lettng Λ L be demand or actor L, and ndexng all actors by, we have d λ L =(θ 0 )Cov b Ψ () Ψ ( ) d log w, Ψ (L) + (θ )µ λ Cov [Ω,α ] Ψ () Ψ ( ) d log w, Ψ (L) 83

9 + (θ 0 ) λ λ d log w λ L. (85) Combne ths wth the observaton that λ L (d log w L + d log L L d log C) = d λ L. (86) Fnally, we now that d log C = λ + λ d log λ. (87) Set d L L = 0, and we have a lnear system wth F + equatons and F + unnowns where F s the total number o actors. Substtute bac nto networ ormula to get λ d λ + d log C = d log w (88) Λ L d log λ L =(θ 0 )Cov b Ψ () Ψ ( ) d log λ, Ψ (L) + (θ )µ λ Cov [Ω,α ] Ψ () Ψ ( ) d log λ, Ψ (L) + (θ 0 ) λ λ d log λ d log C λ L. (89) Use Theorem 2.2 to urther smply ths λ L d log λ L =(θ 0 )Cov b Ψ () Ψ ( ) d log λ, Ψ (L) + (θ )µ λ Cov [Ω,α ] Ψ () Ψ ( ) d log λ, Ψ (L). (90) 84

10 Proo o Proposton 3.4. From Theorem 2.2, we now that d log C/ d log µ = λ Λ L d λ L. (9) For a marup shoc, we can derentate demand or a quantty m to get d λ m = b ( θ 0 )(d log p ) + Ω µ ( θ )λ [d log p d log p ] ( = )θ Ψ m. By Shephard s lemma, (92) d log p = ( = ) + Ω d log p + α d log w. (93) Invert ths system to get d log p = Ψe + Ψ d log w, (94) where Ψ = (I Ω) α s a N K matrx o networ-adusted actor ntenstes by ndustry. Substtutng ths bac nto (92) and set m = L to get d Λ L = b ( θ 0 )[ Ψ + Ψ d log w ]Ψ L, + ( θ )λ µ Ω [ Ψ + Ψ d log w Ψ Ψ d log w ]Ψ L θ µ λ Ω Ψ L. Smply ths to d Λ L =( θ 0 ) b Ψ Ψ L + ( θ 0 ) b Ψ L Ψ d log w, + ( θ )λ µ Ω Ψ Ψ L Ω Ψ L Ψ + ( θ )λ µ Ω Ψ d log w Ψ L Ω Ψ L θ λ µ Ψ L, Ψ d log w 85

11 =( θ 0 ) ( ) Cov b ( Ψ (), Ψ (L) ) + λ Λ L ( ) + ( θ )λ µ CovΩ ()( Ψ (), Ψ (L) ) ( = )Ψ L + ( θ 0 ) Cov b Ψ ( ) d log w, Ψ (L) + Λ d log w Λ L + ( θ )λ µ Cov Ω Ψ () ( ) d log w, Ψ (L) ( = )Ψ L θ λ µ Ψ L, =( θ 0 ) Cov b( Ψ () + Ψ ( ) d log w, Ψ (L) ) + λ Λ L + ( θ )λ µ Cov Ω ()( Ψ () + Ψ ( ) d log w, Ψ (L) ) ( = )Ψ L + ( θ 0 ) λ + Λ d log w Λ L λ µ Ψ L, =( θ 0 ) Cov b( Ψ () + Ψ ( ) d log w, Ψ (L) ) + λ Λ L + ( θ )λ µ Cov Ω ()( Ψ () + Ψ ( ) d log w, Ψ (L) ) λ Ψ L. The nal lne ollows rom the act that ( λ + ) Λ d log w = b d log p = d log P c = 0. Fnally, substtute d log w = d log Λ + d log C nto the expresson above to get d Λ L =( θ 0 ) Cov b( Ψ () + Ψ ( ) (d log Λ + d log C), Ψ (L) ) + λ Λ L + ( θ )λ µ Cov Ω ()( Ψ () + Ψ ( ) (d log Λ + d log C), Ψ (L) ) λ Ψ L. 86

12 To complete the proo, note that d log C Ψ ( ) = d log C. (95) In other words, ths s a vector o all ones multpled by the scalar d log C, and hence t drops out o the covarance operators, snce the covarance o a vector o ones wth any other vector under any probablty dstrbuton s always equal to zero. Hence, d Λ L =( θ 0 ) Cov b( Ψ () + Ψ ( ) d log Λ, Ψ (L) ) + λ Λ L + ( θ )λ µ Cov Ω ()( Ψ () + Ψ ( ) d log Λ, Ψ (L) ) λ Ψ L. Ths can be combned wth Theorem 2.3 to complete the proo. Proo o Example 3.6. We start wth the matrces lsted n (25) and (26), so that ( θ 0 )Cov b ( Ψ (L), Ψ (L) ) ( θ 0 )Cov b ( Ψ (K), Ψ (L) ) Γ = ( θ 0 )Cov b ( Ψ (K), Ψ (L) ) ( θ 0 )Cov b ( Ψ (K), Ψ. (96) (K) δ () = ( θ 0 )Cov b ( Ψ (), Ψ (L) ) ( θ 0 )Cov b ( Ψ (), Ψ (K). (97) From the structure o the problem, we can explctly wrte the value o Γ as ollows: Γ = ( θ 0 ) b 3 (b µ + b 2 µ ) 2 b 3(b µ + b 2 µ ) 2 b 3 ( b 3 )µ 3 b 3 ( b 3 )µ 3. (98) Frst we loo at the cases or =, 2 (the case where a actor o producton s shared). We note that the case s symmetrc or =, 2 by the structure o the networ and the problem. For these values, we have that δ () = ( θ 0 ) b (µ (b µ + b 2 µ 2 )) b b 3 µ 3. (99) 87

13 Generally (or all cases), n order to solve the system n equaton (24), wrte (Λ Γ) = Invert ths to get (Λ Γ) = det Γ The determnant s (b µ + b 2 µ )( 2 0)b 3 ) ( θ 0 )b 3 (µ b + b 2 µ ) 2 ( θ 0 )b 3 ( b 3 )µ 3 b 3 µ ( 3 0)( b 3 )) b 3 µ ( 3 0)( b 3 )) ( θ 0 )b 3 (µ b + b 2 µ ) 2 ( θ 0 )b 3 ( b 3 )µ 3 (b µ + b 2 µ )( 2 0)b 3 ) det Γ =(b µ + b 2 µ 2 )( ( θ 0)b 3 )b 3 µ 3 ( ( θ 0)( b 3 )) ( θ 0 ) 2 b 2 3 ( b 3)µ 3 (µ b + b 2 µ 2 )), = Λ L Λ K (( ( θ 0 )b 3 )( ( θ 0 )( b 3 )) ( θ 0 ) 2 b 3 ( b 3 )). (00). (0) = Λ L Λ K ( ( θ 0 )b 3 ( θ 0 )( b 3 ) + ( θ 0 ) 2 b 3 ( b 3 ) ( θ 0 ) 2 b 3 ( b 3 )) = Λ L Λ K ( ( θ 0 )(b 3 + b 3 )) = Λ L Λ K θ 0. Plug ths bac nto (0) and smply, (Λ Γ) Λ K ( ( θ 0 )( b 3 )) ( θ 0 )b 3 Λ L = Λ L Λ K θ 0 Λ K ( θ 0 )( b 3 ) Λ L ( ( θ 0 )b 3 ). (02) Returnng to the specc case where =, (Λ Γ) δ () = ( ( θ 0 )( b 3 )) (θ 0 )b 3 θ 0 Λ L Λ K θ 0 ( θ 0)( b 3 ) ( θ 0 )b 3 θ 0 Λ L Λ K θ 0 b (θ 0 )(µ Λ L ) b (θ 0 )Λ K. (03) Multplyng the values n (03), and usng the dentty n (24), d log Λ =(Λ Γ) δ (), = b (θ 0 ) θ 0 ( µ Λ L )θ 0 + b 3 ( θ 0 ) µ Λ L + b 3 (θ 0 ) b 3 (θ 0 ) (θ 0 )( b 3 )( µ Λ L ) ( ( θ 0 )b 3 ). 88

14 Combne ths wth (3) gves ( θ 0 [ µ d log C = b ( b 3 )b ] ) θ 0 + b 3 ( θ 0 ) µ θ 0 Λ L Λ L [ θ 0 b 3 b (θ 0 )( b 3 ) ( µ θ 0 whch urther smples to Λ L ) ( ( θ 0 )b 3 ), d log C ] = b + b (θ 0 ) [ ( b 3 ) µ. (04) d log A Λ L Ths gves the desred result or rm. Note, as mentoned beore, a symmetrc result holds or rm 2. In the case o rm 3, we have that δ (3) = (θ 0 ) b 3 (b µ + b 2 µ 2 ) b 3 µ 3 ( b 3) ].. (05) The results rom (02) gve the blueprnt or solvng or the value o d log C d log A 3 ths, we can conclude that d log C = b 3 d log A 3 b + b 2 b 3 whch urther smples to T ( ( θ 0 )( b 3 )) θ 0 Λ L (θ 0 )b 3 Λ K θ 0 ( θ 0)( b 3 ) θ 0 Λ L ( θ 0 )b 3 Λ K θ 0 b 3 (b µ + b 2 µ 2 ) b 3 µ 3 ( b 3) as well. From, (06) = b 3 b 3 b 3 T [ b 3 ( θ 0 ) θ 0 ( θ0 )( b 3 ) ] + (θ 0 ) 2 θ 0 b 3 ( b 3 ) (θ 0 ) 2 θ 0 b 3 ( b 3 ) + [ ( θ 0 )( b 3 ) ] ( b 3 )(θ 0 ) θ 0. (07) Ths smples to gve the requred result d log C d log A 3 = b 3. (08) 89

15 Proo o Proposton 5.. The ndustry level prce s then gven by p = ( M M ( c(w, p) µ m = c(w, p) µ ( (m E A ) ε Φ(z, m ) d m d z A z ) ε. z ) ε ) ε, By Shephard s lemma, we now that, d log p = ( = ) + F d log w F α F + Ω d log p, (09) where α F and Ω are rm-level expendtures on actor F and ndustry as a share o costs (excludng entry costs). Rewrte ths n matrx orm to get d log p = (I Ω) ( α d log w e ) = Ψ( α d log w e ), (0) where e s the th standard bass vector. Let the household s aggregate consumpton good to be the numerare, so that the household s deal prce ndex P c s always equal to one. Then we now that Combne ths wth the prevous expresson to get Note that, b Ψ = λ and b Ψ α = Λ. Hence, d log P c = b d log p = 0. () b Ψe + b Ψ α d log w = 0. (2) λ + Λ d log w = 0. (3) Now, note that Λ = w L F P c C. (4) 90

16 From ths, we now that d log Λ = d log w + d log L d log C. (5) Substtute ths nto the prevous expresson to get λ + Λ d log Λ Λ d log L + d log C = 0, (6) where we use the act that Λ =. Proo o Proposton 5.2. Let the cost uncton o entrant n ndustry producng y unts be gven by c(w, p)h(y /A), (7) where A s ndustry level shoc and subscrpts have been suppressed. Free entry then mples that µc(w, p)h (y/a) y c(w, p)h(y/a) = c(w, p), (8) A where s the entry cost n unts o the nput good. We can smply ths to µh (y/a) y h(y/a) =. (9) A Ths equaton pns down the ecent scale o operaton y = Ay (, µ). The cost o producng q unts o ndustry output s then gven by nc(w, p) (y (µ, )) + n c(w, p), (20) such that ny = q. Substtute the constrant nto ths to get C(w, p) q A = Hence the ndustry s cost uncton s lnear q/a as needed. Fnally, q y A c(w, p) (y (µ, )) + q c(w, p). (2) Ay C(w, p)q/a = q c(w, p) (y (µ, )) + q c(w, p) = nl w y A w Ay + n l w. (22) 9

17 Thereore, the ndustry level cost uncton obeys Shephard s Lemma and we can replcate the rest o the proo rom Proposton 5. Proo o Proposton 5.3. ( ) d log p = ( = ) d log A + s (z) α (z) d F(z) d log w z ( ) ( ) + s (z) ω (z) d F(z) d log p + s (z) β (z) d log r (z) d F(z) z ε s (z) (z ) d log z. where s (z) s rm s share o sales n ndustry, and β (z) s the cost share o rm on ts cttous xed actor. We can solve ths system to get z d log p = (I Ω) ( e d log A + α d log w + ξ κ ), (23) where ξ = ( s z (z) β (z) d log r (z) d F(z) ) and κ = ε s (z) (z ) d log z, and the elements o Ω and α are dened approprately. Let output be the numerare so that b d log p = 0. (24) Hence, λ + Λ d log w + λ ξ λ κ = 0. (25) For each actor, we now that d log w = d log Λ + d log C. (26) Substtute ths n to get λ + Λ d log Λ + Λ d log C + λ z s (z) β (z) d log Λ r (z) d F(z) + Fnally, observe that Λ + λ z s (z) β (z) =. The proo or marup shocs s very smlar. z λ s (z) β (z) d log C λ κ = 0. Proposton B.. Consder an economy wth a cash-n-advance constrant, and nomnal rgdtes. 92

18 Then, where and d log C Λ d log L = λ d log A λ e s d log µ + d H( Λ, Λ), d log µ = (e s Ψe s ) e s Ψ(d log A α d log w), (27) d log w = d log Λ + d log M d log L. In the specal case where some racton δ n ndustry are lexble. Then, λ (s) d log µ = ( b (I Ω) b (I δ Ω) δ ) d log A ( b (I δω) δα) d log w, (28) = ( λ b (I δ Ω) δ ) d log A ( b (I δω) δα) d log w. Proo o Proposton B.. Order the producers so that the rst s producers are the ones wth stcy prces. For a vector x, denote x (s) = e sx. From the cash n advance constrant, we now that d log C = d log M d log P c, = d log M b d log p, = d log M b Ψ ( α d log w d log A ) + b Ψe s d log µ, = d log M Λ d log w + λ d log A λe s d log µ, = d log M Λ ( d log Λ + d log M d log L ) + λ d log A λe s d log µ, = d log M Λ ( d log Λ d log L ) d log M + λ d log A λe s d log µ, whch mples that d log C Λ d log L = λ d log A λ (s) d log µ Λ d log Λ. (29) To get the marups necessary to eep p (s) stcy, we mpose d log p (s) = d log µ + e s Ω log p d log A (s) = 0. (30) Ths mples d log µ = e s Ω d log p α (s) d log w + d log A (s). (3) 93

19 On the other hand, we have d log p = Ψ ( α d log w d log A ) + Ψe s d log µ. (32) Substtutng ths bac nto the prevous expresson gves d log µ = e s Ω Ψ α d log w e s Ω Ψe s d log µ + e s Ω Ψ d log A e s α d log w + e s d log A. (33) Solve ths to get d log µ = (e s(i + Ω Ψ)e s ) e s(i + Ω Ψ) ( d log A d log w ), = (e s Ψe s ) e s Ψ ( d log A d log w ). Proo o Proposton 4.. The labor-lesure condton and the cash-n-advance condton mply that Hence, or Thereore, L /(ν) = ( ) ( w w =. (34) P c C M) d log L = d log w d log M = d log Λ d log L + d log M d log M, (35) ν d log L = To nsh, apply Propostons 3. and 3.2. ν d log Λ. (36) + ν d log w = d log Λ + d log M (37) ν + Proo o Proposton 4.2. Wth log utlty n consumpton and nnte Frsch elastcty o labor supply we have that d log w d log P c = d log C, (38) or n other words, substtutng n the cash n advance constrant d log w = d log M. (39) 94

20 Furthermore, the cash n advance constrant mples that d log C = d log M d log P c, = d log M Λ d log w + λ d log A λ e s d log µ, = d log M d log M + λ d log A λ e s d log µ, = λ d log A λ e s d log µ, substtutng equaton (27) rom Proposton B. completes the proo = λ d log A λ (s) (e s Ψe s ) e s Ψ(d log A α d log w), = λ d log A λ (s) (e s Ψe s ) e s Ψ(d log A α d log M). In the specal case where some racton δ n ndustry are lexble. Then, λ (s) d log µ = ( b (I Ω) b (I δ Ω) δ ) d log A ( b (I δω) δα) d log w, (40) = ( λ b (I δ Ω) δ ) d log A ( b (I δω) δα) d log w. At the ndustry level, equaton (40) shows that the changes n marups can be nterpreted as some racton o the rms n each ndustry change ther marup n response to shocs. Proo o Proposton 5.4. On the other hand, On the other hand, d log A ( ) e ( = s = s µ ) = s µ 0. (4) d τ e d 2 2 log A ( = (σ )Var d τ 2 s (µ ) s µ )2 = (σ 2)Var s (µ ) s µ. (42) Consder an ndustry where: all rms use the same upstream nput bundle wth cost C; rms transorm ths nput nto a rm-specc varety o output usng constant return to returns to scale technology; each rm has productvty a and charges a marup µ ; the varetes are combned nto a composte good by a compettve downstream ndustry accordng to a CES producton uncton wth elastcty σ on rm. 95

21 We denote the quantty o composte good produced as Frm charges a prce Q = [ b σ The resultng demand or rm s varety s where the prce ndex s gven by Total prots are gven by p σ σ ] σ σ. (43) p = µ a C. (44) q = ( p P ) σ b Q, (45) P = [ ] b p σ σ. (46) Π = (p C)( p P ) σ b Q. (47) We solve out the prce ndex and prots explctly and get P = b ( µ a ) σ σ C, (48) σ ( µ Π = ) a a µ a [ ( ] ) σ σ µ b a b CQ. (49) For completeness we can also solve or the sales o each rm as a racton o the sales o the ndustry λ = p q PQ = b ( µ a ) σ b ( µ a ) σ. (50) We want to understand how to aggregate ths ndustry nto homogenous ndustry 96

22 wth productvty A and marup µ. These varables must satsy Π = P = µ C, (5) A ( µ A A) CQ. (52) Ths mples that A and µ are the solutons o the ollowng system o equatons µ A = b ( µ a ) σ σ, (53) σ ( µ A A) ( µ = ) a a µ a [ ( ] ) σ σ µ b a b. (54) The soluton s A = [ ( ] ) σ σ µ ( b a µ ) µ a ( ) σ µ σ b a σ µ a b, (55) µ = [ ( µ ) σ] σ b a [ ( ] ) σ σ µ ( b a µ ) µ a ( ) σ µ σ b a σ µ a b. (56) We can also rewrte ths n a useul way as A = [ ( ] ) σ σ µ b a µ ( µ a ) σ b b ( µ a ) σ = [ ( ] ) σ σ µ b a, (57) µ λ 97

23 µ = µ ( µ a ) σ b b ( µ a ) σ =. (58) µ λ Dene the ecency o each rm to be e = µ. Then, consder a steady state where s µ = s e =. Consder a transormaton e = τ + ( τ)e. Ths transormaton eeps µ constant. On the other hand, On the other hand, d log A ( ) e ( = s = s µ ) = s µ 0. (59) d τ e d 2 2 log A ( = (σ )Var d τ 2 s (µ ) s µ )2 = (σ 2)Var s (µ ) s µ. (60) C Relabellng As mentoned prevously, our results can be appled to any nested CES economy, wth any arbtrary pattern o nested substtutabltes and complementartes among ntermedate nputs and actors. For concreteness, we descrbe the relabelng or one specc example. Let the household s consumpton aggregator be C C = b ( c c ) σ σ σ σ. (6) Suppose each producer produces usng a CES aggregator o value-added VA and ntermedate nputs X: y y = A α ( VA VA ) θ θ + ( α ) ( X X θ ) θ θ θ. 98

24 Value-added conssts o derent actors VA VA = F l ν = where l s actor o type used by. Intermedate nput conssts o nputs rom other producers X X = N = l η η η η ε ( ) ε ε x ε ω x. Denote the matrx o ν usng V and the matrx o ω as Ω. We rewrte ths economy n the standard orm we requre, and put hats on the new expendture shares. The new nput-output matrx ˆΩ has dmenson (+3N+F) (+3N+F). The rst ndustry s the household s consumpton aggregator, the next N ndustres are the orgnal ndustres, the next N ndustres produce the value-added o the orgnal ndustres, the next N ndustres produce the ntermedate nputs o the orgnal ndustres, and the nal F ndustres correspond to the actors. Under the relabelng, we have ˆb = (, 0,..., 0),, ˆΩ = b dag(a) dag( a) V Ω , (62) wth and ˆθ = (σ, θ, ν, ε, ), (63) ˆµ = (, µ,,..., ). (64) 99

25 D Aggregaton o cost-based Domar weghts In ths Appendx we show that recoverng cost-based Domar weghts rom aggregated data s, n prncple, not possble. We also show how the Basu-Fernald decomposton can detect changes n msallocaton even n acyclc economes (where the equlbrum s ecent, and there s no possblty o reallocaton o resources). Example 2. also shows the alure o the aggregaton property mpled by Hulten s theorem. The easest way to see ths s to consder aggregatng the nput-output table or the economy n Example 2.. For smplcty, suppose that marups are the same everywhere so that µ = µ or all. Snce there s no possblty o reallocaton n ths economy, and snce marups are unorm, ths s our best chance o dervng an aggregaton result, but even n ths smplest example, such a result does not exst. Suppose that we aggregate the whole economy S = {,..., N}. Then, n aggregate, the economy conssts o a sngle ndustry that uses labor and nputs rom tsel to produce. In ths case, the nput-output matrx s a scalar, and equal to the ntermedate nput share o the economy Ω SS = µ N µ N, (65) and the aggregate marup or the economy s gven by µ. Thereore, λ S constructed usng aggregate data s λ S = (I µω) = µ N µ. (66) µ However, we now rom the example that d log C d log A = S λ = N λ S = µ N µ, (67) µ except n the lmtng case wthout dstortons µ. Thereore, even n ths smplest case, wth homogenous marups and no reallocaton, aggregated nput-output data cannot be used to compute the mpact o an aggregated shoc. To compare our decomposton wth that o Basu and Fernald (2002), consder the acyclc economy n Fgure 3 wth two actors L and L 2. Now, we now that d log C d log A 2 = λ 2 = ( α ). (68) 00

26 The Basu-Fernald decomposton or ths economy gves d log C d log A 2 = λ 2 + R M + µ v R M = α µ + µ v R M, (69) where the rst term (n ths case, the sales-share o 2) s the pure technology eect, and R M s the reallocaton o ntermedate nputs, even though n ths economy, there s no capacty or reallocatng resources and the equlbrum s ecent. L L 2 HH 2 Fgure 3: Acyclc economy where the sold arrows represent the low o goods. The low o prots and wages rom rms to households has been suppressed n the dagram. The two actors n ths economy are L and L 2. E Extra Examples Example E.. Next, consder the mnmal example wth two elastctes o substtuton, whch demonstrates the prncple that changes n msallocaton are drven by how each node swtches ts demand across ts supply chan n response to a shoc. To ths end, we apply Proposton 3. to the economy depcted n Fgure 4. d log C d log A 3 = λ 3 Λ L ( (θ0 ) [ b (ω 3 µ (ω 3µ 3 + ω 4 µ +(θ )µ λ [ ω3 µ 3 ω 3 ( ω3 µ 3 + ω 4 µ 4 4 )) ω 3b Λ L ] The term multplyng (θ 0 ) captures how the household wll sht ther demand across and 2 n response to the productvty shoc, and the relatve degrees o msallocaton n and 2 s supply chans. The term multplyng (θ ) taes nto account how wll sht ts demand across 3 and 4 and the relatve amount o msallocaton o labor between 3 and 0 )]).

27 L HH Fgure 4: An economy wth two elastctes o substtuton. 4. Not surprsngly, nstead we shoc ndustry, then only the household s elastcty o substtuton matters, snce ndustry wll not sht ts demand across ts nputs n response to the shoc to ndustry 2: d log C d log A =b Λ L (θ 0 ) [ b µ (ω 3µ 3 + ω 4 µ 4 ) b Λ L ]. Ths llustrates the general prncple n Proposton 3. that an elastcty o substtuton θ matters only s somewhere downstream rom. 02