Appendix to The Macroeconomics of Trend Inflation Journal of Economic Literature, September 2014

Transcription

1 Appendix o The Macroeconomics of Trend Inflaion Journal of Economic Lieraure, Sepember 204 Guido Ascari Universiy of Oxford and Universiy of Pavia Argia M. Sbordone Federal Reserve Bank of New York Sepember 24, 204 Absrac This Appendix conains a deailed derivaion of he Generalized New Keynesian GNK model wih a direc reference o he equaions in he published aricle indicaed as PA. We presen he model, derive he seady sae and he log-linear equaions, and provide background for he dynamic analysis in he PA. The views expressed here are solely hose of he auhors and do no necessarily reflec he posiion of he Federal Reserve Bank of New York or any oher par of he Federal Reserve Sysem.

2 Conens The GNK Model. Households Technology Firms pricing Recursive formulaion of he opimal price-seing equaion Aggregaion and price dispersion Moneary policy The complee non-linear sysem Deerminisic seady sae 6 2. The cos of price dispersion in seady sae The log-linear GNK model 8 3. The GNKPC in erms of marginal coss The GNKPC in erms of oupu The GNKPC in erms of oupu gap The log-linear model in he zero-inflaion seady sae Macroeconomic dynamics 5 5 The opimal rend inflaion 6

3 The GNK Model. Households The period uiliy funcion of he represenaive agen is assumed o be separable in consumpion C and labor N: U C, N C σ ϕ σ d ς N ne ϕ, where ς is a labour supply shock; he period by period budge consrain is given by: P C i B W N D B, 2 where i is he nominal ineres rae, B is one-period bond holdings, W is he nominal wage rae, N is he labor inpu, and D is disribued dividends. The represenaive consumer maximizes he expeced discouned ineremporal uiliy, subjec o he budge consrains; his yields he following firs-order condiions, where β is he discoun facor: Euler equaion : C σ βe P P i C σ, 3 Labor supply equaion: W P U N U C d ne ς N ϕ /C σ d n e ς N ϕ Cσ. 4.2 Technology Final good producers aggregae inermediae goods Y i, wih echnology: Y 0 Y i, di, 5 where indicaes he elasiciy of subsiuion among inermediae goods. Their opimal Pi, demand for inermediae inpus is equal o Y i, P Y. The producion funcion of inermediae goods producers is: Y i, A N i,, 6 where A is an exogenous process for he level of echnology, which we assume saionary. The labor demand of firm i is: N d i, Yi, Toal cos and marginal cos, in real erm, are respecively: A T C r i, W P N i, W P. 7 Yi, A, 8 In he PA he producion echnology is linear in labor α 0. Here we consider he more general case of decreasing reurns o labor 0 < α <.

4 and MCi, r A α Yi,. 9 P Noe ha marginal coss of firm i depend upon he quaniy produced by he firm, given decreasing reurns o scale. Since firms charging differen prices would produce differen levels of oupu, hey have differen marginal coss. Subsiuing he expression for Y i, in 9, hese are:.3 Firms pricing MCi, r A α W P W Pi, P α Y α. 0 The pricing model is based on Calvo 983. In each period here is a fixed probabiliy θ ha a firm can re-opimize is nominal price, which we denoe by Pi,. Wih probabiliy θ, insead, he firm can index is price o he previous period inflaion rae: P i, π ϱ P i,, where inflaion is π P P and ϱ 0, indicaes he degree of indexaion e.g., Chrisiano e al., The price seing problem is: max P i, P E D,j θ j j0 subjec o he demand consrain: Y i,j i, Π ϱ,j P j Y i,j W j P j P i, Π ϱ,j Y j, P j Yi,j A j, where D,j β j λ j λ 0 is he sochasic discoun facor, wih λ j denoing he j marginal uiliy of consumpion. Π,j indicaes cumulaive inflaion beween periods and j: { for j 0 Π,j P P 2 P P P j P j for j The firm s firs order condiion is: E θ j D,j j0 from which we ge: P α i, α Pi, Pi, Π ϱ j,j P j Y j W j P j Yj A j E j0 θj W D j,j P j E j0 θj D,j Π ϱ j,j P j Yj A j Π ϱ j,j P j Π ϱ j,j P j 0, Y j 2 2 This is a more general version han he baseline model analyzed in he PA, for which here is no indexaion. As we indicae in wha follows, he PA resuls are obained by seing ϱ 0. 2

5 Dividing boh sides by P and observing ha α, we ge: P i, P α E j0 θj W D j Yj,j P j A j E j0 θj D,j Π ϱ j,j Π,j Π ϱ j,j Π,j Noe ha 3 reduces o equaion 4 in he PA when seing α ϱ 0. The aggregae price level P di evolves according o: P θπ ϱ 0 P i, P i, θ P Y j. 3, 4 which is equaion 6 in he PA when ϱ 0. Dividing by P and leing π P /P and relaive price p i, P i, /P, we can wrie 4 as: θπ ϱ π θ p i,, 5 or: p i, ϱ θπ θ π, 6 which corresponds o equaion 30 in he PA for ϱ Recursive formulaion of he opimal price-seing equaion To derive he generalized new Keynesian Phillips curve GNKPC we use a recursive formulaion of he opimal price-seing equaion. Firs, we wrie 3 as: p i, α ψ, 7 φ where, denoing he real wage as w W /P, using he definiion of he discoun facor D,j β j λ j λ 0, and he fac ha λ j Cj σ σ Yj in equilibrium, he auxiliary variables ψ and φ are defined, respecively, as: ψ E θβ j Y j0 σ j A j w j Π ϱ j,j Π,j 8 φ E j0 θβ j Y σ j Π ϱ j,j. 9 Π,j We hen rewrie he infinie sums in 8 and 9 recursively, as ψ w A Y σ ϱ θβπ 3 E π ψ, 20

6 and φ Y σ Defining an economy-wide real marginal coss as: θβπ ϱ E π φ. 2 MC w A α α Y, 22 equaions 7, 20 and 2 correspond o in he PA, by seing α ϱ 0..4 Aggregaion and price dispersion Using 7 and he expression for inermediae inpus Y i,, aggregae labor demand is N d 0 N d i,di Yi, 0 A di 0 Pi, Y di P A } {{ } s Y s. A 23 Schmi-Grohé and Uribe 2007 show ha he variable s is bounded below by one, so ha in his model s represens he resource coss due o relaive price dispersion arising from posiive long-run inflaion: he higher s, he more labor is needed o produce a given level of oupu. If rend inflaion is zero π, s does no affec he real variables up o he firs order. In he PA we hus define s as he relaive price dispersion measure. As shown by Schmi-Grohé and Uribe 2007, under he assumpions of he Calvo model s can be rewrien as: P i, s θ θ 2 θ Collecing erms yields: P i, s θ P P θ P θ θ P i, π ϱ P i, 2 π ϱ πϱ 2 P θπ ϱ i, P π P θ θ P i, 2 π ϱ 2 P. The expression in curly brackes is exacly he definiion of s. Thus, i follows ha he equaion for s can be wrien recursively as: s θ p i, θ π s, 24 π ϱ which corresponds o equaion 34 in he PA, when seing α ϱ 0. 4

7 .5 Moneary policy The cenral bank follows a sandard Taylor rule, wih weigh φ π on deviaions of inflaion from arge π we assume ha arge inflaion and rend inflaion are he same and weigh φ y on oupu deviaions from seady sae oupu Ȳ : i ī π π φy φπ Y e v, 25 Ȳ where v is a moneary policy shock, and φ π, φ Y are non-negaive parameers..6 The complee non-linear sysem To summarize, he complee non-linear model, imposing equilibrium condiion C Y, is described by equaions 3, 4, 6, 7, 20, 2, 23, 24, and 25, which we reproduce here for convenience: Y σ ψ w A p i, i βe π Y σ p i, α Y w d n e ς N ϕ Y σ ϱ θπ θ φ Y σ σ π ψ φ ϱ θβπ E π θβπ ϱ E π φ N s Y A s θ p i, θ π ϱ π i ī π π φy φπ Y e v Ȳ ψ s The model has nine endogenous variables: Y, i, π, w, N, p i,, φ, ψ, s and hree exogenous shocks: A, ς, v. 3 3 A DYNARE code o simulae he model is available from he auhors upon reques. 5

8 2 Deerminisic seady sae We compue he non-sochasic seady sae of his sysem, characerized by a posiive rend inflaion π, as: π β i, 26 and w d n N ϕ Y σ, 27 p θ π ϱ i, 28 θ p α ψ i φ, 29 ψ wa Y σ θβ π ϱ Y σ, 30 φ, θβ π ϱ 3 Y N s, A 32 s θ θ π ϱ MC w Y p i, 33 α A. 34 Noe ha subsiuing 28 in 33 gives he relaion beween price dispersion and inflaion in seady sae: θ θ π ϱ s, θ π ϱ θ Seing α ϱ 0 his is equaion 35 in he PA. Plugging he values of w from 27 and N from 32 ino 34 we obain: MC d nn ϕ Y σ Y α A From his equaion i follows ha: Y ϕα σ d n s ϕ ϕ Y A Y σ A ϕ MC d n s ϕ Y α A ϕ A d n s ϕ, µ d n s ϕ Y ϕα σ A ϕ. 35 where we indicaed by µ he seady sae mark-up µ /MC. The expression for seady sae oupu is hen: Y A ϕ d n s ϕ µ ϕσα σ 6 Y π. 36

9 Seing α 0, his is equaion 36 in he PA. In his model superneuraliy doesn hold. We noe his by indicaing seady sae oupu in 36 as Y π ; his noaion will help disinguishing his seady sae level of oupu from he oupu level in a seady sae wih zero inflaion. We can also obain a relaion beween opimal price and marginal cos in seady sae. Using 29 and 27 we ge: wa Y σ p α θβ π ϱ i θβ π ϱ MC, Y σ θβ π ϱ θβ π ϱ where he second equaliy follows from observing ha, from 34: wa Y σ Y σ MC. Therefore: p θβ π ϱ α i MC. θβ π ϱ When α 0, he above equaion gives he raio MC, as in equaion 37 of he PA. Assuming also ϱ 0, we obain he simple decomposiion of he mark-up discussed in he PA: µ P p MC i θ π βθ π Pi MC θ βθ π. 37 }{{}}{{} price adjusmen gap marginal markup As menioned in foonoe 40 of he PA, i is easy o show ha wih posiive rend inflaion π > he marginal markup is increasing in β : βθ π βθ π θ π βθ π θ π βθ π β βθ π 2 θ π π βθ π 2 > 0. Hence, he higher he discouning he lower β, he lower is he marginal mark-up and, oher hings equal, he lower is he average mark-up and he higher he seady sae oupu. 2. The cos of price dispersion in seady sae From 32: Y A s N ÃN. where we defined à A. A change in s causes a variaion of Ã. By definiion, denoing s by à he percenage change in Ã, we have ha a change in s from s 0 o s yields: à Ā Ā s s 0 Ā s 0 s0 s p i 7, 38 s

10 by using he fac ha s s s 0 s 0, hence s s 0 s. Expression 38 allows o map percenage changes in he seady sae value s, which are due o changes in rend inflaion, ino an equivalen percenage change in aggregae produciviy. If s 0, so ha s s : Ã s. If, in addiion, α 0 : Ã s, showing ha, as discussed in foonoe 37 of he PA, decreasing reurns o labor 0 < α < make price dispersion more sensiive o changes in rend inflaion, and herefore deliver a larger equivalen change in produciviy for any change in s. 3 The log-linear GNK model We now ake a log-linear approximaion around he deerminisic seady sae defined above for a generic rend inflaion rae π. Defining by ˆx he log deviaion of any variable x from is seady sae, from 3, 4, 23, and 25, we obain, respecively: Ŷ E Ŷ σ î E ˆπ, 39 ŵ ϕ ˆN σŷ ς, 40 ˆN ŝ Ŷ Â, 4 î φ πˆπ φ y Ŷ v. 42 To obain an expression for ŝ we sar by log-linearizing 24 o obain: ŝ θ p i s ϱ θ π π s ˆπ s ˆp i, θ π ϱ π ϱ ϱ θ π π s s s ŝ. s ˆπ Nex, subsiuing he expression for seady sae price dispersion s from 33, we obain: ŝ θ π ϱ ˆp i, θ π ϱ ϱ ˆπ ˆπ ŝ. 43 Nex, we log-linearize expression 7 o ge: α ˆp i, ˆψ ˆφ, 44 where expressions for ˆψ and ˆφ are obained by log-linearizing equaions 20 and 2, respecively. 8

11 From 20 we ge: ˆψ wa Ȳ ψ σ σ ϱθβ ϱ π ŵ wa Ȳ ˆπ σ wa Ȳ Ŷ ψ θβ ϱ π ψ σ Â E ˆπ θβ π ϱ E ˆψ. Subsiuing in he value for ψ from 30, his expression becomes: ˆψ θβ π ϱ θβ π ϱ ŵ E ˆψ Â E ˆπ σ Ŷ ϱ ˆπ. 45 Analogously, from 2 we ge: ˆφ σ Ȳ σ Ŷ ϱ θβ π φ ϱ ˆπ θβ π ϱ E ˆπ θβ π ϱ E ˆφ which, upon subsiuing in he value of seady sae φ from 3, becomes: ˆφ θβ π ϱ σ θβ π Ŷ ϱ ϱ ˆπ E ˆφ E ˆπ. Finally, he log-linearizaion of equaion 6 gives: 0 θ ϱ π ϱ ˆπ θ p i ˆp i,, θ π ϱ ˆπ which, subsiuing p i from 28, becomes: θ π ϱ ˆπ ϱˆπ θ π ϱ ˆp i, or ˆp i, 46 θ π ϱ θ π ϱ ˆπ ϱˆπ. 47 To sum up, he log-linearized GNK model consiss of equaions 39, 40, 4, 43, 47, 44, 45, 46, and 42. By combining equaions 40, 47, 44, 46, 4, and 43, one can subsiue ou he variables ˆN, ˆp i,, ŵ and ˆφ, and wrie he sysem in more compac form as a 9

12 5-equaion sysem in he variables Ŷ, ˆπ, ˆψ, ŝ and î, driven by he hree exogenous shocks: Â, ς, and v. In he nex wo subsecions we show ha he supply side of he model can be alernaively wrien as a GNKPC in erms of marginal coss, in erms of oupu, or in erms of oupu gap. 3. The GNKPC in erms of marginal coss To ease noaion, we define he quasi-difference of inflaion as ˆπ ϱˆπ. Equaions 47, 45 and 46 can hen be wrien, respecively, as: ˆp i, θ π ϱ θ π ϱ 48 ˆφ ˆψ θβ π ϱ θβ π ϱ ŵ E ˆψ θβ π ϱ σ Ŷ Â σ Ŷ E θβ π ϱ E ˆφ E Subsiuing 48 ino 44 and rearranging, we obain: ˆφ ˆψ α θ π ϱ θ π ϱ. Using his expression o subsiue ˆφ and ˆφ in 50, and rearranging, we ge: ˆψ α θ π ϱ θ π ϱ θβ π ϱ σ Ŷ 5 θβ π ϱ E ˆψ α θ π ϱ θ π ϱ E E. Now noe ha, from 49, ˆψ is funcion of he real wage a componen of he marginal cos; subsiuing 5 in 49 and rearranging, we obain: α θ π ϱ θβ π ϱ σ Ŷ θβ π ϱ θβ π ϱ θβ π ϱ θ π ϱ ŵ Â θβ π ϱ θβ π ϱ σ Ŷ θβ π ϱ E ˆψ 52 E α θ π ϱ E. 0

13 From 22, he log-linear expression for he marginal cos is: mc ŵ α α Â Ŷ, 53 so ha he expression in he second square brackes of 52 is: ŵ Â σ Ŷ mc σ Ŷ. Subsiuing his expression in 52 and, recalling ha ˆπ ϱˆπ,we obain an expression of he dynamics of inflaion in erms of marginal coss. In he case of he PA, α ϱ 0, his expression is: θ π θβ π ˆπ θ π mc 54 β π θ π E ˆπ β π θ π σ Ŷ E ˆψ, which is equaion 4 in he PA. Furhermore, using expression 53 for he marginal coss, 5 can be wrien as: ˆψ Ŷ θβ π ϱ mc σ θβ π ϱ E ˆψ E, 55 which, when α ϱ 0, is equaion 42 in he PA: ˆψ θβ π mc σ Ŷ θβ π E ˆψ E ˆπ. 56 Finally, combining 43 and 48, we obain an expression for price dispersion as funcion of inflaion and pas price dispersion: ŝ θ π ϱ θ π ϱ θ π ϱ θ π ϱ θ π ϱ θ π ϱ θ π ϱ θ π ϱ θ π ϱ ŝ. ŝ For α ϱ 0, and observing ha in ha case π, his expression simplifies o: θ π ŝ π π θ π θ π ŝ, 57 which is expression 45 in he PA.

14 3.2 The GNKPC in erms of oupu To express he GNKPC as an inflaion-oupu relaionship, we firs subsiue he expressions for labor inpu 4, ino ha for real wage, 40, o obain: ϕ ŵ ϕŝ σ Ŷ ϕ ς. 58 Â We hen go back o eq. 52, and, using 58, observe ha he erm in square brackes is ŵ Â σ Ŷ ϕŝ ϕ Ŷ Â ς so ha eq. 52 can be rewrien as: α θ π ϱ θ π ϱ ϕŝ ϕ θβ π ϱ θβ π ϱ θβ π ϱ θβ π ϱ σ Ŷ Ŷ Â ς θβ π ϱ E ˆψ θβ π ϱ θβ π ϱ α E θ π ϱ E. When α ϱ 0, π, and his expression simplifies o: θ π θ π π θβ π σ Ŷ θβ π ϕŝ ϕ Ŷ Â ς θβ π π E ˆψ θβ π π E π θβ π θ π E π. Adding and subracing θβ π σ Ŷ o he righ hand side yields: θ π θ π π θβ π π σ Ŷ θβ π ϕŝ ϕ σ Ŷ ϕ Â ς θβ π π E ˆψ θβ π π θ π E π, which can be wrien as: π λ π Ŷ b π E π κ π ϕŝ ϕ Â ς b 2 π σ Ŷ E ˆψ. 59 This is equaion 43 in he PA, where he coeffi ciens are defined as follows: λ π κ π ϕ σ, κ π θ π θβ π, b θ π π β π θ π, and b 2 π β θ π π. 2

15 3.3 The GNKPC in erms of oupu gap In order o define an oupu gap we need o define he flexible price equilibrium in his model. Firs we define he seady sae oupu under flexible prices as Ȳ ; from 36, since under flexible prices s and θ > 0, we derive: Ȳ d n µ A ϕ ϕασ, 60 where now µ. In he seady sae under flexible prices oupu does no depend on inflaion, and i is also equal o oupu in he zero inflaion seady sae of a sicky price model. Taking shocks ino accoun, he flexible price equilibrium is: Y n Ȳ ϕ A ϕασ e ς A α ϕασ, where we assumed ς 0. Taking logs: log Y n log Ȳ ϕ ϕ α σâ ϕ α σ ς. 6 We define he oupu gap in a zero-inflaion seady sae as Ỹ log Y log Y n. We can hen decompose he log-deviaion of oupu from seady sae in our model: Ŷ log Y log Y π, where Y π is he seady sae level of oupu as derived in expression 36 above, as follows: Ŷ log Y log Y π log Y log Y n log Y n Ỹ ϕ ϕ α σâ log Ȳ log Ȳ log Y π ϕ α σ ς Ỹ, 62 where we have indicaed he long-run oupu gap by Ỹ log Y π log Ȳ.4 This equaion corresponds o equaion 63 in he PA. We now derive he GNKPC in erms of oupu gap for he case of he PA, where α ϱ 0. Saring from equaion 59, we subsiue in i he expression for Ŷ: ˆπ β π θ π E ˆπ θ π θβ π ϕŝ ϕ Â ϕ σ Ỹ ϕ θ π β θ π π σ Ỹ ϕ σ ϕâ ϕ σ ς Ỹ σ ϕâ E ˆψ ϕ σ ς Ỹ ς and, rearranging, we ge: ˆπ κ π ϕŝ λ π Ỹ Ỹ b π E ˆπ b 2 π Ỹ Ỹ ϕ σ ϕâ ϕ σ ς σ E ˆψ, 63 4 The expression for Ỹ can be easily derived from expressions 36 and 60. 3

16 where λ π, κ π, b π, and b 2 π were defined previously. This is equaion 64 in he PA. Similarly, upon subsiuion of Ŷ, he expression for ˆψ becomes: ˆψ θβ π which is 65 in he PA. ϕŝ ϕ θβ π E ˆψ E ˆπ, Ỹ Ỹ σ σ ϕâ σ ϕ σ ς Noe ha he slope of he GNKPC depends on κ π θ π θβ π, which is θ π decreasing wih rend inflaion since θβ π < and θ π <. 3.4 The log-linear model in he zero-inflaion seady sae When prices are sable i.e. a zero inflaion: π, i is sraighforward o show ha he sandard resuls are obained. The log-linear model in his case assuming also no indexaion: ϱ 0, simplifies o: ˆψ θβ ŵ Ŷ E Ŷ σ î E ˆπ θβ ˆψ ŵ ϕ N σŷ ς ˆp i, θ θ ˆπ α ˆp i, ˆψ ˆφ ˆπ Â σ Ŷ ˆφ θβ σ Ŷ θβ ˆφ ˆπ ˆN ŝ Ŷ Â ŝ θ ˆp i, θ ˆπ ŝ î φ πˆπ φ y Ŷ. Price dispersion becomes as in eq. 46 in he PA: ŝ θ ˆπ θ ˆπ θŝ θŝ so he dynamics of ŝ does no affec firs order dynamics. 64 4

17 The NKPC in erms of marginal coss is: θ θβ ˆπ mc βe ˆπ λ mc βe ˆπ. α θ Using he expression for mc derived by combining 53 and 58, he NKPC in erms of oupu becomes: ˆπ λ σ ϕ α Ŷ λ ϕ λς βe ˆπ. Â Finally, wih he usual subsiuions, he NKPC in erms of oupu gap becomes: ˆπ κỹ βˆπ, where κ λ σ ϕα, and Ỹ ϕ σϕαâ Ŷ σϕα ς. 4 Macroeconomic dynamics For his par we used in he PA a simplified version of he baseline model where, in addiion o he assumpions of consan reurn o scale and no indexaion α ϱ 0, we also assumed σ and ϕ 0. Wih hese assumpions he log-linear equaions 59, 45 and 39 simplify o: π λ π Ŷ Â ς b π E ˆπ b 2 π E ˆψ 65 ˆψ θβ π Ŷ Â ς θβ π E ˆψ ˆπ 66 Ŷ E Ŷ î E ˆπ. 67 These are equaions 53 o 55 in he PA. The policy rule 42 is unchanged, and he noaion for he coeffi ciens is as indicaed previously; wih he addiional assumpions above i is also he case ha λ π κ π. Noe ha because of he assumpion of linear uiliy in labor ϕ 0, price dispersion, ŝ, does no affec he dynamics of he above sysem. Technology, labor supply, and moneary shocks are specified as AR processes, respecively as: Â ρ A Â u A u A i.i.d.n0, ς ρ ζ ς u ζ u ζ i.i.d.n0, ν ρ v ν u ν u ν i.i.d.n0,, which are equaions 50 o 52 in he PA. The sysem can be easily solved wih he mehod of undeermined coeffi cien. The soluion is given by he wo equaions: ˆπ λ π λ π φπ φ y ς Â v, 68 φ y 5

18 which is equaion 57 in he PA, and: Ŷ φ y φ π λ π λ π Â ς v φ π 69 which is equaion 58 in he PA. To look a he effecs of rend inflaion on he impac of he shocks on oupu and inflaion, denoing by π i and y i, for i A, ς, v, he impac of shock i on variables π and Y, respecively, we compue he effec on he echnology shocks impac as: π A π λ π λ π φπ φy π λ π 2 λ π φπ π φ y < 0 and y A π λ π φy φπ λ π π φy φ π φ y φ π λ π 2 < 0. π λ π I follows ha an increase in π reduces he impac effec of a echnology labor supply shock on inflaion and oupu because i flaens he Phillips curve by reducing λ π. Noe ha he effec of ς is jus he opposie of he one of A. For he moneary shocks we compue: π v π λ π φ yφ πλ π π φ y λ π φ y λ π φ π 2 < 0, π so ha an increase in rend inflaion decreases he absolue value of he negaive response λ π of inflaion. Moreover, since y v φ yφ πλ π, and π < 0, i follows ha an increase in rend inflaion increases he absolue value of he negaive response of oupu. 5 The opimal rend inflaion In his secion we derive he resuls of Yun 2005 in our GNK model, ha we analyze in secion 3.5 of he PA. Again assuming, as in he PA, consan reurn o scale and no indexaion α ϱ 0, subsiuing he expression for p i, in 5 ino 24 yields: s θ θπ θπ θ s. 70 Following Yun 2005, he Ramsey problem can be defined as maximizing he uiliy funcion of he represenaive consumer under he resource consrain, A N C s and he evoluion of price dispersion s described in 70. The firs-order condiion wih 6

19 respec o inflaion is hus i jus involves 70: Hence: 0 θ 0 θπ 2 θπ θ θπ θ θ π 2 θ θπ s θπ s θπ π s. θ θπ π s, 7 θ which is equaion 2.5 in Yun 2005, p. 94. Subsiuing his equaion ino 70 we ge: s π s θ θ s. 72 Noing ha 7 can be wrien as: and dividing 72 by 73 we ge: or: π θ θ s, 73 s π s θ θ s θ θ s π s, π π s s. 74 This is equaion 2.6 in Yun p. 95, and 68 in he PA. Plugging his equaion ino 73 gives equaion 2.7 in Yun and 69 in he PA: Now recall equaion 24, which under ϱ 0 is: s s θ θ s. 75 s θ p i, θπ s, and use equaion 74 o wrie: θπ s p θ i,, and equaion 7 o ge: Using again 74, i hen follows: s π s p i,. p i, s, 76 which is equaion 2.2 in Yun 2005, p. 96, and equaion 70 in he PA. 7