An Open Economy Model for the Monetary Policy Analysis (A Concise Title)

1 The Model

1.1 Households

Households maximize their utility subject to the budget constraint and the law of motion for the capital accumulation. The optimization problem has the form

  m (l s )1     jt jt   dt log(c jt  hc jt1 )  log t    pt 1       c i W   p p m jt b jt ext b jt  (1 ) t c  t i     q a d   c p jt p jt p p p  j,t1,t j,t1 j,t1,t     t t t t t  t L  s 1 1  E 0    t   jt  (1 w )w jt l jt  (rt u jt (1 k )  t  k  t [u jt ])k jt 1  t0  W   1 m 1 b 1 ex b 1     jt1  T  F  R jt1  RW t jt1  a   L t t t1 L t1 L L jt     t pt  t pt  t pt  t       i     L  L jt     Q jt  t k jt  (1  )k jt1  t 1 S t  i jt    i       jt1   

L where the index j denotes j  th household,  is the discount parameter,  t denotes L the population growth with the population shock t , dt is the process for intertemporal d preference shock t , c jt is per capita consumption, h is the habit formation parameter,  is

m jt the parameter controlling money demand, denotes the per capita real money balances, pt    is the process for labor supply shock t , is the labor supply parameter,  is s the inverse of the Frisch labor supply elasticity, l jt is the per capita hours worked,  jt is c the Lagrangian multiplier for the budget constraint,  C is the tax rate on consumption, pt is the price level of the final consumption good (or level of consumers' prices), pt is the price i level of the domestic intermediate good, pt is the price level of the investment final good, i jt W is per capita investment, b jt is the level of outstanding debt, ext b jt is an amount of foreign government bonds in the domestic currency, ext is nominal exchange rate, the term q a d  j,t1,t j,t1 j,t1,t denotes the insurance against the idiosyncratic wage-adjustment risk where a jt1 is the amount of Arrow securities with real price q j,t1,t in event  j,t1,t ,  W is the tax rate on wage income, w jt is the overall real wage index, rt is the real rental price of capital, u jt is the intensity of use of capital,  k is the tax rate on capital income, t is  the investment-specific technology with the shock t ,  is the depreciation rate of capital, 1 t [u jt ] is the physical cost of use of capital in resource terms, k jt is the per capita capital,

Tt is the lump-sum transfer, Ft are the profits of the firms in the economy, Rt is the domestic W nominal interest rate, Rt is the foreign nominal interest rate, Q jt is the Lagrangian multiplier

 L i jt  for the capital constraint, and S t  is the investment adjustment cost function.  i jt1 

The first order conditions (FOC) for consumption, investment, capital and its utilization, domestic and foreign bonds, and real money balances can be derived directly from the optimization problem.1 The FOC for the labor supply and wages is a two-step problem. The first is the derivation of the relation between these two variables (a labor bundler's optimization). The second is the wage optimization where the relevant part of the Lagrangian is

s 1   (l )   w  E    L  d   jt   ts1 (1 )w l s , t  w    t t jt  w jt jt   0  1 s1  ts      w w  s.t. l s   ts1 jt  l d , jt   t  s1  ts wt  where  w is the Calvo wage parameter,  t is the inflation of the domestic intermediate good,

 w is the wage indexation parameter,  is the elasticity of substitution among different types d of labor, and lt is the per capita labor demand.

1.2 Intermediate Goods Producing Firms

The domestic intermediate firms solve the two-step problem. First, given input prices, firms minimize their costs subject to the production function

min w l d  r k ,s.t. y  A k  (l d )1  z . d t it t it1 it t it1 it t lit ,kit 1 where yit is the per capita domestic intermediate production, At is the neutral technology A level with the associated shock t ,  is the labor share in production,  is the parameter associated with the fixed cost of production, and zt is the per capita long run growth.

Second, the domestic intermediate firms maximize profits under the monopolistic competition and the Calvo setup. They choose the optimal price subject to the demand function

   L  t   p pit  max Et  p      ts1  mct yit , p    it  0 t  s1 pt 

1 The FOC for the real money balances and Arrow securities are not necessary; the former because of the inflation targeting regime with interest rate rule and separable utility in its arguments, the latter because of the assumption of complete markets. See (Burriel et al., 2010) and (Erceg et al., 2000).    p    p it  s.t. yit   ts1  yt ,  s1 pt  where  p is the Calvo parameter,  p is the domestic intermediate indexation parameter, mct denotes the real marginal cost, and  is the elasticity of substitution among different types of the domestic intermediate goods.

Since the import intermediate firms use a single foreign good as the only input there is ~W M ext pt ~W a straightforward specification for the real marginal cost mct  M , where pt is pt ~W the price of the foreign homogenous good in the foreign currency, ext pt is the foreign price M in the domestic currency, and pt is the price of imported goods in the domestic currency. Under the monopolistic competition and the Calvo pricing, the price-setting problem has the form

  M    p   L  t  M M it M  M max Et  M     ts1  M  mct yit , pM    it  0 t  s1 pt    M M   p  s.t. y M    M M it  y M , it  ts1  M  t  s1 pt  M where pit is the price of imported goods in the domestic currency,  M is the Calvo M M parameter,  t is the imported goods inflation,  M is the indexation parameter, yt is the intermediate imported good, and  M is the elasticity of substitution among different types of imported goods.

1.3 Final Goods Producing Firms

Consumption, investment and export firms ( s  c,i, x ) purchase both intermediate composite inputs and produce final goods.2 They maximize profits subject to production functions

  L  t s d M M max Et    pt st  pt st  pt st , sd s M  t t  0 t

 s 1  s 1 1  s 1  1  d M s  s   s   s s.t. st  ns  s st   1 ns  s st 1 t   ,   d M s where st and st denote domestic and imported consumption, investment or export, pt 3 denotes corresponding prices ,  s denotes the elasticities of substitution, ns is the home bias s in the aggregation, and t denotes adjustment costs.

2 The government sector uses domestic inputs only. 3 Note that consumption and investment prices are set in domestic currency whereas export prices are set in s c i ~ x foreign currency. Thus, pt  pt , pt , pt . To capture the imperfect exchange rate pass-through, the model contains monopolistic competition in the intermediate import and final goods export sectors and the local currency pricing (LCP) assumption. Under LCP, importers’ prices are sticky in domestic currency whereas exporters’ prices are sticky in foreign currency. Thus, there is a second stage optimization problem of exporting firms associated with their market power and the Calvo price-setting   ~ x    ~  x p  L  t  x it x  x max Et  x     ts1  x  mct yit , ~p x   ~  it  0 t  s1 pt    ~ x x  ~  x p  s.t. y x    x it  y x , it  ts1  ~ x  t  s1 pt  ~ x where pit is the price of the exported goods in the foreign currency,  x is the Calvo ~ x parameter,  t is the export price inflation in the foreign currency,  x is the indexation p mc x  t ~ x parameter, t ~ x denotes the real marginal costs in the export sector, ext pt is ext pt x the price of the exported goods in the domestic currency, yt is the demand for exports, and

 x is the elasticity of substitution among different types of exported goods.

1.4 Policy Authorities

c The central bank targets the four period-ahead year-on-year headline inflation 4t4 according to a Taylor-type rule

1 R  d  y    L yt        t zt  R  R  R  4c   y d   t  t1  t4  t1 exp  m ,    TAR      t  R R 4   d    t4   L y           TAR where R is the steady state of the nominal interest rate, 4 is the inflation target,  R ,  

 d and  y are Taylor rule parameters,  L is the steady-state population growth rate, y is the m steady-state growth rate of output, and t is the monetary policy shock.

The fiscal policy has a simple form g gt   g ct  t where gt is the per capita level of real government consumption,  g is a parameter linking g government spending to the real consumption and t is the shock.

1.5 The Rest of the World

The rest of the world is represented by the EU and is modelled exogenously. The domestic importers purchase the packed foreign composite good, bundled from the production of a continuum of foreign firms. Similarly, production of domestic exporters is aggregated by the bundler allowing working with one price of exports. The world demand for domestic exports is given by4 ~ x W  ext pt    p x  v x  t  yW , t t ~W t  ext pt     pt  x where  t is the export prices dispersion,  W is the elasticity of substitution among different W types of world trade goods, and yt is the world demand.

The equations for the foreign (EU) interest rates, demand5, and inflation are modelled via processes W W W RW log R W log R (1 W )log R ,  t    R  t1    R   t W W W yW log y W log y (1 W )log y ,  t    y  t1    y   t

~ W ~ W ~ W W log    W log   (1  W )log    ,  t    t1     t ~ W where  t is the foreign inflation of the world homogenous good.

The model is closed with a debt-elastic premium W prem W ex b , t  b t t prem where premt is an AR process with shock t .

4 The volume of exports is a function of the relative price of exports to world prices, both in domestic currency. 5 Note that the process for foreign demand is on levels to be consistent with other measurement equations which are on levels too. 2 Model Moments

Table 1 Data and Model Moments (q-o-q, %) Variable Mean Std. D. Data Model Data Model CPI inflation 0.8071 0.5377 0.7677 2.2419 Nominal interest rate 1.3315 0.6904 1.1053 2.0510 Nominal exchange rate -0.5006 -0.5159 2.7831 4.3221 Nominal wages 1.7886 1.6429 0.8005 2.8838 Real consumption 0.6504 1.0993 0.8197 2.4851 Foreign demand 2.0723 2.0213 3.3148 5.3099 Foreign nominal interest rate 0.8055 0.9853 0.3094 0.3292 Foreign inflation 0.4312 0.4963 1.0226 1.0215

Table 2 Cross Correlations Data/Model CPI NIR NER RC CPI inflation 1.00 0.82 0.10 -0.21 Nominal interest rate 0.75 1.00 0.21 -0.22 Nominal exchange rate 0.31 0.30 1.00 -0.12 Real consumption 0.06 0.04 -0.06 1.00 3 ADF Test

Table 3 ADF t-statistic Parameter Technologies No Technologies

nc -5.009370 -1.496169

nx -5.548387 -1.527146

ni -4.617531 -0.077851

 g -3.810984 -2.034182  -4.617531 -2.173670

W -4.123811 -1.934761 h -2.969596 -1.951361  -4.714276 -2.926210  -6.112995 -3.574608

 M -5.407886 -3.689742

 x -6.201202 -1.873562

 c -4.590722 -1.875038

 i -4.436917 -1.157988  -4.712341 -2.937060

M -5.394803 -3.691788

 x -6.287101 -2.114388

w -4.709789 -2.905656

 M -5.399268 -3.691025

 x -6.423782 -2.062927

 w -4.716944 -2.938541

Note: Critical values for 1% (-3.566), 5% (-2.937), 10% (-2.615), 90% (-0.4393), 95% (-0.0499), 99% (0.6942).