Supporting Materials for: Two Monetary Models with Alternating Markets

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1 Supportng Materals for: Two Monetary Models wth Alternatng Markets Gabrele Camera Chapman Unversty Unversty of Basel YL Chen Federal Reserve Bank of St. Lous 1 Optmal choces n the CIA model On date t, gven hstory S t, the constrant of the frm s F h F t S t )) = c F 1tS t ) + c F 2tS t ) 1) where c F 1t and c F 2t denote cash and credt goods, p jt s the nomnal spot prce of good j = 1, 2, and w t s the nomnal spot wage on t. Nomnal profts net dollar nflows) are dstrbuted as dvdends n the afternoon, and on the mornng of t are p 1t S t )c F 1tS t ) + p 2t S t )c F 2tS t ) w t S t )h F t S t ). 2) Snce the frm sells for cash and for credt, payments accrue as follows: n the mornng, t receves cash payments for cash-goods sales, and n the afternoon t receves payments for the mornng s credt sales. Let q t S t ) denote the date 0 prce of a clam to one dollar delvered n the afternoon of t, contngent on S t = state-contngent nomnal bond). The frm s date 0 proft-maxmzaton problem s: gven state-contngent prces q t S t ), choose sequences of output and labor c F 1tS t ), c F 2tS t ), h F t S t )) to solve Maxmze: q t S t ) { p 1t S t )c F 1tS t ) + p 2t S t )c F 2tS t ) w t S t )h F t S t ) } ds t subject to: c F 1tS t ) + c F 2tS t ) = F h F t S t )). 3) 1

2 Substtutng for c F 1tS t ) from the constrant, the FOCs for all t, S t are h F t S t ) : p 1t S t )F h F t S t )) w t S t ) = 0 c F 2tS t ) : p 1t S t ) p 2t S t ) = 0. Consequently, for all t, S t we have p 1t S t ) = p 2t S t ) = p t S t ) and p t S t )F h F t S t )) = w t S t ). 4) An agent who contracts on date 0 maxmzes the expected utlty β t Uc 1t S t ), c 2t S t ), h t S t ))f t S t )ds t where we assume U s a real-valued functon, twce contnuously dfferentable n each argument, strctly ncreasng n c j, decreasng n h, and concave. Maxmzaton s subject to two constrants. One s the cash n advance constrant p 1t S t )c 1t S t ) M t S t 1 ) for all t and S t, where M t S t 1 ) are money balances held at the start of t, brought n from the afternoon of t 1, when the shock s t was not yet realzed. Gven ths uncertanty, money may be held to conduct transactons and for precautonary reasons. The other constrant s the date 0 nomnal ntertemporal budget constrant: {qt S t ) [ p 1t S t )c 1t S t ) + p 2t S t )c 2t S t ) w t S t )h t S t ) M t S t 1 ) +M t+1 S t ) Θ t ]} ds t Π + M. The date 0 sources of funds are M ntal money holdngs =ntal labltes of the central bank) and the frm s nomnal value Π. The left hand sde s the date 0 present value of net expendture. It s calculated by consderng the prce of money delvered 2

3 n the afternoon of t, q t S t ). There are two elements: 1. Mornng net expendture: w t S t )h t S t ) wages earned, pad n the afternoon; M t S t 1 ) p 1t S t )c 1t S t ) unspent balances avalable n the afternoon; p 2t S t )c 2t S t ) purchases of credt goods settled n the afternoon. These funds are avalable n the afternoon of t, where the date-0 value of one dollar s q t S t ). 2. Afternoon net expendtures: the agent receves Θ t transfers and exts the perod holdng M t+1 S t ) money balances, so net expendture s M t+1 S t ) Θ t, wth date 0 value q t S t ). Gven that values can be hstory-dependent, we ntegrate over S t. Agents choose sequences of state-contngent consumpton, labor and money holdngs c 1t S t ), c 2t S t ), h t S t ), and M t+1 S t ) to maxmze the Lagrangan: L := β t Uc 1t S t ), c 2t S t ), h t S t ))f t S t )ds t + λπ + M) λ {qt S t )[p 1t S t )c 1t S t ) + p 2t S t )c 2t S t ) w t S t )h t S t ) M t S t 1 ) + M t+1 S t ) Θ t ]} ds t + µt S t )[M t S t 1 ) p 1t S t )c 1t S t )]ds t, 5) where µ t S t ) s the Kühn-Tucker multpler on the cash constrant on t, gven S t. Omttng the arguments from U and f where understood, n an nteror optmum the FOCs for all t and S t are: c 1t S t ) : β t U 1 f t S t ) λp 1t S t )q t S t ) µ t S t )p 1t S t ) = 0 p 1t S t )c 1t S t ) M t S t 1 ) c 2t S t ) : β t U 2 f t S t ) λp 2t S t )q t S t ) = 0 6) h t S t ) : β t U 3 f t S t ) + λw t S t )q t S t ) = 0 M t+1 S t ) : λq t S t ) + λ q t+1 S t+1 )ds t+1 + µ t+1 S t+1 )ds t+1 = 0. 3

4 Gven p 2t S t ) = p 1t S t ) = ps t ) and 4) we get U 3 U 2 = F h t S t ); S t ) for all t, S t U 1 = λq ts t ) + µ t S t ) U 2 λq t S t ) for all t, S t. 7) 2 The prce dstorton n the LW model u Under barganng, 1c 1 ) s the margnal beneft from spendng a dollar. Ths rato z c 1 ; θ) becomes u 1c 1 ), wth p 1 = η c 1 ), when θ = 1. To see ths, note that f θ = 1, then p 1 /p 2 p 2 z = η. If θ < 1 we have z > η. Indeed, η ; hence, θ + 1 θ)η <. From the defnton of zc 1 ; θ) we have z = θ + 1 θ)η η + A where A > 0. The Fgure plots ψc 1, θ) to llustrate how Nash barganng dstorts prces, relatve to compettve prcng, dependng on the buyer s barganng power θ, and the rate of nflaton. As θ approaches one, the prce dstorton vanshes for any rate of nflaton, and the Nash barganng prce dstorton vanshes. 3 Proof of Lemma 1 Consder an equlbrum wth hstory-ndependent prces p 1t S t ) = p 1t and w 1t S t ) = w 1t, as n LW, 2005). 6 To prove the frst part of the Lemma let s t = 1 and µ t S t ) = 0. From the frst and thrd expressons n 12) we have β t c 1t S t )) = λp 1t q t = λw 1t q t = β t η h 1t S t )), for all t, S t, From market clearng h F 1tS t ) = δh 1t S t ) = δc 1t S t ) = c F 1tS t ). 7 Hence, u 1 c 1tS t )) η c 1t S t )) = 1 for all t, S t. That s c 1t S t ) = c 1 for all t and all agents such that s t = 1. To prove the second part of the Lemma let s t = 1 and µ t S t ) > 0. Update by one 4

5 Fredman Rule Zero Inflaton 10% Inflaton θ Fgure 1: Illustratng the barganng prce dstorton usng ψc 1, θ) Notes: The three curves correspond to ψc 1 ; θ) assumng as n the calbraton n LW, 2005) that η = 1, u 1 c 1 ) = c 1+b) 1 a b 1 a 1 a, a = 0.3, b = 0, δ = 0.5 and r = 1.04γ 1, wth γ = β =Fredman rule), γ = 1 =zero nflaton) and γ = 1.1 = 10% nflaton). perod the frst expresson n the FOCs 12) to get β t+1 c 1,t+1 S t+1 ))fs t+1 )f t S t ) = λq t+1 fs t+1 )f t S t ) + µ t+1 S t+1 ), f s t+1 = 1 where we substtuted f t+1 S t+1 ) = fs t+1 )f t S t ). Now substtute c 1,t+1 S t+1 ) = M t+1 S t+1 ) snce µ t+1 S t+1 ) > 0. The expresson above has the status of an equalty only f s t+1 = 1. In that case, we can ntegrate both sdes wth respect to s t+1, 5

6 condtonal on s t+1 = 1. For the left-hand-sde we get β t+1 1 {s t+1 =1} u 1c 1,t+1 S t+1 ))fs t+1 )f t S t )d st+1 = βt+1 = βt+1 = βt+1 ) 1 {s t+1 =1} fs t+1)f t S t )d st+1 ) f t S t ) 1 {s t+1 =1} fs t+1)d st+1 Mt+1 S t ) Mt+1 S t ) Mt+1 S t ) ) f t S t )δ 8) For the rght-hand-sde we get 1 {s t+1 =1} [λq t+1fs t+1 )f t S t ) + µ t+1 S t+1 )]ds t+1 = λq t+1 f t S t ) + µ t+1 S t+1 )ds t+1 Φ = λq t f t S t ) Φ, 9) where the last step follows from the last lne n 12) and Φ := 1 {s t+1 =0} [λq t+1fs t+1 )f t S t ) + µ t+1 S t+1 )]ds t+1 = 1 {s t+1 =0} [λq t+1fs t+1 )f t S t )]ds t+1, snce µ t+1 S t+1 ) = 0 when s t+1 = 0 = λq t+1 f t S t ) 1 {s t+1 =0} fs t+1)ds t+1 = λq t+1 f t S t )1 δ), snce 1 {s t+1 =0} fs t+1)ds t+1 = 1 δ = β t+1 u 2c 2,t+1 ) f t S t )1 δ), from 12). p 2,t+1 Equatng the expectatons of both sdes from 8) and 9) we have β t+1 Mt+1 S t ) ) δ = λq t Substtutng Φ n the equaton above we get β t+1 Mt+1 S t ) ) Φ f t S t ) δ = λq t βt+1 u 2c 2,t+1 ) 1 δ), p 2,t+1 6

7 or equvalently, snce u 2c 2,t+1 ) = 1 for all t + 1 and S t+1, we have β t+1 [ Mt+1 S t ) ) δ + 1 δ ] = λq t. p 2,t+1 Ths mples that f s t+1 = 1, then c 1,t+1 S t+1 ) = M t+1s t ) = M t+1 = c 1,t+1 for all t and S t and for all agents, because q t s ndependent of S t. The dstrbuton of money s degenerate because there are no wealth effects due to the lnear dsutlty from producng credt goods. Agents equally reach the same cash holdngs by adjustng ther labor supply h 2. By market clearng, h F 2t = h 2td = c 2t where h 2t satsfes the agents budget constrant. Now substtute λq t = βt u 2c 2t ) p 2t = βt p 2t from 12) and wrte the equaton above as 13). Fnally, from the frm s problem, we have η h 1t ) = w 1t w 2t = p 1t p 2t. 4 Comparng notatons n LW and our model In LW, UX) s the utlty receved from consumng X CM goods u 2 c 2 ) n our notaton). labor s lnear. The technology to produce CM goods s lnear and the dsutlty from In the DM, a porton ασ δ n our notaton) of agents desres to consume but cannot produce) and an dentcal porton can produce but does not consume; uq) s the utlty receved from consumng q DM goods u 1 c 1 ) n our notaton); c s the dsutlty from labor n the DM η n our notaton); the nomnal prce s d q per unt of consumpton p 1 n our notaton); the real prce s φd q, where φ s 1 n our notaton. Wth bndng cash constrants d = M and φm where M s p 2 q the agent s money holdngs. We also have φm zq) where 0 < θ 1 s the buyer s barganng power. The nomnal nterest rate s r n our notaton). 7

8 5 Detals about the quanttatve exercse Preferences specfcaton: Preferences over goods are defned by u 1 c 1 ) = c 1 + b) 1 a b 1 a 1 a and u 2 c 2 ) = B log c 2, for some a > 0, b 0, 1) and B > 0. Consumpton c 2 satsfes 6), labor dsutlty satsfes η = 1, so c 1 satsfes γ β 1 = δ[τu 1c 1 ) 1]. 10) The welfare cost of nflaton: Defne ex-ante welfare W γ := u 2 c 2 ) c 2 + δ[u 1 c 1 γ)) c 1 γ)]. Consderng the compensatng varaton, welfare at zero nflaton s denoted W 1 := u 2 c 2 ) c 2 + δ[u 1 c 1 1)) c 1 ]. The welfare cost of γ 1 nflaton s the value 1 where satsfes W 1 W γ = 0. The markup: In LW the markup vares wth the barganng power and t generally vares wth c 1 but not always; consder ηh) = hx, x 1 and θ = 1). In the x calbraton labor dsutlty s lnear so the markup concdes wth the relatve prce p 1, whch s zc 1; θ). p 2 c 1 The share of DM output : The share of DM output n LW s easly constructed, gven that n the calbrated model everyone s matched n the DM α = 1 n LW). 8

9 DM output s δc 1 and CM output s c 2 B, n the calbrated model. Hence, total output s Y = δc 1 + B and the DM output share s δc 1 t ncreases as nflaton falls Y because real money balances ncrease); ths also gves us the share of cash goods to total goods n the CIA model. Ths share s used to calculate average markups. In the calbraton, when θ = 0.5 we have τ = ψc 1 ; θ) =.719,.846,.928 for, respectvely, γ =.1, 0, 1 β ; the correspondng average sales tax rates are:.025,.037,.034. β Instead, when θ = 0.343, we have τ = ψc 1 ; θ) =.511,.672,.802; the correspondng average sales tax rates are:.014,.019,.013. As nflaton decreases the markup n cash trades, 1, falls; yet, the average markup ncreases because the share of cash goods to τ total output rses. 9