Aggregation and the PPP Puzzle in a Sticky Price Model

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					 Aggregation and the PPP Puzzle in a Sticky
               Price Model


              Carlos Carvalho                                   Fernanda Nechio
     Federal Reserve Bank of New York                          Princeton University


                                         May 2009

The views expressed in this presentation are those of the authors and do not necessarily re‡ect the
position of the Federal Reserve Bank of New York or the Federal Reserve System.
Real exchange rates and PPP

  Real exchange rate (RER): ratio of prices of basket of goods denominated in
  common currency

   – Q = EP
          P


  Purchasing power parity (PPP) hypothesis: “Q is constant”

   – Q = 1: “absolute PPP”

   – Q = constant 6= 1: “relative PPP”
                                                      1




                                    0.4
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FX (left axis)
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RER (left axis)
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                                                                            US Dollars / "French Franc-Euro"




                           Jan-00

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Price Ratio (right axis)




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                                    0.4
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The puzzle: data + model failure

   “PPP Puzzle” - Rogo¤ (1996)

    – High persistence and volatility of real exchange rates (deviations from PPP)

    – Hard to reconcile with economic models

    – In particular, when deviations arise from nominal shocks in the presence of
      price rigidity


   Quantitative framing of the puzzle:

    – Most estimates of half-life of deviations from PPP fall between 3 - 5 years

    – Standard sticky-price models with monetary shocks: half-life around 1 year


   “Too much” price rigidity needed to match persistence of the data
Our paper

  Build on ample evidence of heterogeneity in frequency of price adjustment (Bils
  and Klenow 2004 and others)

   – Average degree of price rigidity not the relevant statistic for aggregate
     dynamics (Carvalho 2006)


  Potential source of heterogeneity in dynamics of sectoral real exchange rates


  Multi-sector, two-country, sticky-price model:

   – Heterogeneity in frequency of price changes

   – Price discrimination and local currency pricing


  ) Heterogeneous sectoral real exchange rates
                       Summary of the model - K sectors

            Home                                                                        Foreign


                        Final                                 Final
                        Goods                                 Goods*
             C                             Y                                                  C*
                                                                                  Y*

Consumers                                                                                           Consumers*

                             Y H,k,j       Y*
                                            H,k,j           Y F,k,j            Y*
                                                                                F,k,j


            L           Intermediate                          Intermediate                    L*
                        Goods                                 Goods*




            Sector 1   ...      Sector k ... Sector K   Sector 1      ...   Sector k ... Sector K
                                 Y H,k,j                                     Y F,k,j
                                  Y*                                         Y*
                                                                              F,k,j
                                   H,k,j
                   Summary of the model - 1 sector

            Home                                       Foreign


                   Final              Final
                   Goods              Goods*
            C               Y                              C*
                                                  Y*

Consumers                                                        Consumers*

                    Y H,j   Y*
                             H,j     Y F,j     Y*
                                                F,j


            L      Intermediate       Intermediate         L*
                   Goods              Goods*
Summary of core quantitative results

   Quantitative model

    – Discipline our argument: distribution of price stickiness chosen to match
      micro data

        Average frequency of price changes implies average spells of 4.4 months

    – Half-life of deviations from PPP in heterogeneous economy: 3.8 years

    – Half-life of deviations from PPP in homogeneous economy: 1.2 years
Core results: intuition I

   Sectors with more stickiness are disproportionally important for aggregate dy-
   namics


   Extreme case: 1 ‡exible- and 1 sticky-price sector, no pricing complementarities

    – Aggregate RER dynamics driven only by sticky-price sector


   Mathematically: measures of persistence (and volatility) are convex in the
   frequency of price changes
Core results: intuition II

   E¤ects even stronger in the presence of pricing complementarities!


   Not-so-extreme version of “responders vs non-responders” of Haltiwanger and
   Waldman (1991)


   Bottom line in terms of persistence:

    – heterogeneous economy     homogeneous economy with more price sticki-
                     s
      ness than what’ implied by average frequency of price changes
The model

  Two countries, Home and Foreign, with identical, in…nitely lived consumers


  Commodities are labor, a consumption good and a continuum of intermediate
  goods


  Consumer supplies labor, invests in complete set of state-contingent assets, and
  consumes non-traded …nal good


  Competitive …nal good producers bundle intermediate goods; ‡exible prices


  Monopolistically competitive intermediate goods producers:

   – Price discriminate, setting prices in local currency

   – Sticky prices; adjustment frequency varies across sectors
The model - …nal good I

             0                                                                                          1
                 K
                 X          Z 1                                     K
                                                                    X           Z 1
  max PtYt   @         fk         PH;k;j;tYH;k;j;tdj +                     fk         PF;k;j;tYF;k;j;tdj A
                 k=1         0                                   k=1             0
  s:t:

                                  XK              1       1!    1
                  Yt =                     f Y
                                        k=1 k k;t
                                             1                              1   !
                                    1                              1                   1
                 Yk;t =           ! YH;k;t + (1                ! ) YF;k;t

                                            Z 1                 !
                                        1             1                1
             YH;k;t =             fk              YH;k;j;tdj
                                             0
                                            Z 1                !
                                        1             1                1
             YF;k;t =             fk              YF;k;j;tdj
                                             0
The model - …nal good II

   Demands
                                      !                 !                 !
                           PH;k;j;t          PH;k;t              Pk;t
           YH;k;j;t = !                                                            Yt
                           PH;k;t                Pk;t                Pt
                                             !                   !                 !
                                  PF;k;j;t              PF;k;t            Pk;t
           YF;k;j;t = (1    !)                                                          Yt
                                  PF;k;t                Pk;t                  Pt


   Price Indices:
                                                             1
                                 XK               1      1
                       Pt =              f P
                                      k=1 k k;t
                                                                          1
                                     1              1
                     Pk;t =      !PH;k;t + (1 ! ) PF;k;t 1
                                 Z 1            ! 1
                                                 1
                                       1
                    PH;k;t =         PH;k;j;tdj
                                  0
                                 Z 1            ! 1
                                                 1
                                       1
                    PF;k;t =         PF;k;j;tdj
                                  0
The model - intermediate …rms I

   Intermediate …rms:

    – divided into sectors that di¤er in the frequency of price changes

    – Calvo (1983) pricing:   k   = frequency of price changes in sector k

    – indexed by country, sector (k 2 K ), and by j 2 [0; 1]

    – sectoral weights fk

    – produce di¤erentiated varieties using labor
The model - intermediate …rms II

   Choose prices XH;k;j;t; XH;k;j;t, set in local currencies:
                1                        "                                           #
                X                            XH;k;j;tYH;k;j;s + EsXH;k;j;tYH;k;j;s
                                   s t
       max Et         t;s (1    k)
                s=t
                                                            WsNk;j;s

       s:t: YH;k;j;t + YH;k;j;t = Nk;j;t
            and demands


   Analogous for Foreign
The model - aggregate and sectoral RERs

  Aggregate:
                              EtPt
                        Qt
                               Pt



  Sectoral:
                              EtPk;t
                       Qk;t
                               Pk;t
Closing the model

  Speci…cations for monetary policy that ensure existence and uniqueness of RE
  equilibrium


  Equilibrium: optimality and market clearing conditions


  Solution: loglinearization around zero in‡ation steady-state
Counterfactual homogeneous economy

  Counterfactual homogeneous (one-sector) economy:

   – Everything the same, except:
                                                      PK
   – One sector with frequency of price changes   =    k=1 fk k
Parametrization

   36-sector model:

    – k = 1=k, k = 1; :::; 36

    – Micro evidence from Nakamura-Steinsson (2008):

        CPI data - no sales/substitutions

        fk = sum of expenditure weights for underlying categories

        36th sector: f36 = 4:2%

           = 0:226 ) Prices changes, on average, every 4:4 months
Quantitative results

             Persistence measures:    P (q )      P q 1 sec

                    CIR                79:9          20:2
                    SAC                0:98          0:95
                    LAR                0:94          0:86
                     HL                  45           14
                       1                :98          :96

                                                             1=2
              Volatility measure:    V (q )1=2   V q 1 sec

                                       0:10          0:04
Using our structural model - extras

   We provide structural interpretation to the (reduced-form) empirical literature
   on heterogeneity, aggregation and PPP

    – “PPP Strikes Back/Aggregation Bias” debate between Imbs et al. (2005),
      and Chen and Engel (2005) / Crucini and Shintani (2008)


   Study persistence in the cross-section of sectoral exchange rates ; touch base
   with Kehoe and Midrigan (2008) (this is work in progress)
Decomposition of e¤ect of heterogeneity


                                           XK                   XK
 P (qt) P qt sec  1             P (q t )        f P (q k ) +
                                             k=1 k
                                                                    f P (q k )
                                                                 k=1 k
                                                                                      P qt sec
                                                                                         1
 |          {z         }    |                {z           } |                  {z                }
total heterogeneity e¤ect             aggregation e¤ect              counterfactuality e¤ect
Applying decomposition

                        Data              Panel, agg.    Panel, sect.    Panel, sect.
                   Estim. method         Fixed E¤ects       OLS             MG
                                             P (q )      1 P P (q )      P q 1 sec
"              #                                         K         k
    Eurostat       Persist. meas.:
     data               CIR                 64:39          59:48           33:19
                        SAC                  0:98           0:97            0:97
                       LAR                   0:97           0:94            0:95
                         HL                   46           43:16             26


                          Economy          Heterog.        Heterog.      One-sector
                            Data          Single, agg.   Panel, sect     Single, agg.
                       Estim. method         OLS             OLS            OLS
"                  #                                     1  P
                                             P (q )      K    P (q k )    P q 1 sec
    Simulated
      data             Persist. meas.:
                            CIR              73:4           60:8             36:9
                            SAC              0:98           0:97             0:97
                           LAR               0:96           0:89             0:87
                             HL              43:1           31:6             21:8
Conclusion - I

   Multi-sector, two-country, sticky-price model can produce RER that is as per-
   sistent as in the data


   One-sector version of same economy with average frequency of price changes
   fails to do so


   We use our model to explain the apparently con‡icting …ndings in reduced-form
   empirical literature

    – Di¤erent papers measured di¤erent objects

    – As Chen and Engel (2005) and Crucini and Shintani (2008), we …nd aggre-
      gation e¤ect to be small (in the parametrized model and in the data)

    – As Imbs et al. (2005), we …nd total heterogeneity e¤ect to be large (in the
      parametrized model and in the data). This is due to the counterfactuality
      e¤ect
Conclusion - II

   What matters for understanding the gap between data and standard one-sector
   models is total heterogeneity e¤ect


   Heterogeneity in price stickiness goes a long way towards solving the PPP
   puzzle


   Such type of heterogeneity known to matter for policy:

    – In “closed economies”: Aoki (2001), Benigno (2004), Eusepi, Hobjn and
      Tambalotti (2008), Berriel and Sinigaglia (2008)

    – In open economies: ?

				
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posted:11/10/2009
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