Диссертация (1137741), страница 23

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 23)

yt ( j ) = Yt  . Pt 108(A1.25)From the first order condition of this problem we obtain the prices for both types offirms.For the firms, who set their price freely the price is defined as follows:pt (1) =WtP. − 1 [ (G ) (ht ( j )) −1 ] tP t(A1.26)For those, who set their price one period in advance the price is: Wtpt (2) = Et −1 Pt  ,P  −1  − 1 [ (Gt ) (ht ( j )) ] where M =(A1.27)is a markup over marginal cost. −1A1.2.

Log-linearisation of the model1.2.1 Linearisation of the demand sideLet us first consider the budget constraint of the borrower: 1 + it −1  Pt −1DCtb = − Dt −1 + t − Tt b .1 + rt 1 + rt −1  PtLog-linearizing it around the steady state, when Dt = D low = D and Yt = Y , we obtain:(Cbt)− C b = − ( Dt −1 − D ) + D(1 + it −1 Pt− log1 + logPt −11 + rt −1 )Pt +111 +Dt − D ) + Yt b − Y b − D− log(1 + r )  ,( log (1 + it ) + Et log1+ r1+ r Pt(A1.28)where Yt b = Wt htb .In the steady state the real interest rate is given by the discount factor of the patientconsumer, so that1 + rt = (1 + it )=1, inflation at the steady state is zero,1+ rPt= 1 , andPt −1Pt.

Moreover, we have 1 + it −1 = 1 + rt −1 at the steady state.Pt +1Ctb = Yt b +  Dt − Dt −1 +  D t −  D  ( it − Et t +1 − r ) ,109(A1.29)where Ctb = logCtbDYbP, Yt b = log t , r = log  −1 ,  t = log t , it = log (1 + it ) ,  D =andYPt −1YYYt b = Wt + htb .Combining first order conditions from the utility maximization problem of a saver, thefollowing Euler equation of a saver is obtained: P U cs ( Cts ) =  (1 + it ) EtU cs ( Cts+1 )  t  . Pt +1 (A1.30)Log-linearizing this equation and dividing it by U ccs Y we get:(Cst−CY) = E (Ctst +1−CY)+U cs( rt − r ) ,U ccs YCts = Et Cts+1 −  ( it − Et t +1 − r ) ,where Cts = log(A1.31)(A1.32)U csCts=−and.YU ccs YLog-linearizing the aggregate consumption, Ct =  s Cts + (1 −  s )Ctb , we get:(Ct − C ) =  s (Cts − C ) + (1 −  s )(Cts − C ) .(A1.33)Dividing it by Y , we obtain:Ct =  s Cts + (1 −  s )Ctb ,where Ct = log(A1.34)Ct sCsCb, Ct = log t , Ctb = log t .YYYTaking into account that log-linearized aggregate output is:Yt = Ct + Gt ,(A1.35)Yt =  s Cts + (1 −  s )Ctb + Gt .(A1.36)we get:1101.2.2 Derivation of the Phillips curveFrom the first order condition of firm maximization problem we obtain the prices forboth types of firms:pt (1) =WtP, − 1 [ (G ) (ht ( j )) −1 ] tpt (2) = Et −1 (where M = −1(A1.37)P tWtP) , − 1 [ (G ) (ht ( j )) −1 ] t(A1.38)P tis a markup over marginal cost.

Log-linearizing this equation we get:log p1t = Wt + log Pt + (1 − )ht −  gtP , where Wt = logWth, ht = log t .WYAn unexpected inflation can be expressed as follows: t − Et −1 t = log Pt − Et −1 log Pt =1− (log pt (1) − log Pt ) =1− (Wt + (1 −  )ht −  gtP ) .(A1.39)From the first order conditions we obtain the labor supply for two types of consumers:Wt =Wt = hs (hts ),(A1.40) hb (htb ).U cb (Ctb )(A1.41)U cs (Cts )Log-linearizing these conditions we obtain:Wt =  s hts (i) +  s −1 Cts ,(A1.42)Wt =  b htb (i) +  b−1 Ctb ,(A1.43)U cbhhs h b hhb h s U cshtbbbwhere  = s ,  = b ,  = s ,  = b , ht (i ) = log .YhU ccYU ccYhsIt is assumed that  =  =  and  =  =  .sbsbBy aggregating these conditions we obtain:Wt =  ht +  −1 Ct , t − Et −1 t =1− ( ht +  −1 Ct + (1 −  )ht −  gtP ) , where ht =111(A1.44)1Yt − Pg . t(A1.45)From market equilibrium condition Ct = Yt − Gt = Yt − gtU − gtP :t =1[(1 +  ) Yt − (1 +  ) gtP +  −1 (Yt − gtP − gtU )] + Et −1 t ,1− (A1.46)1[((1 +  ) +  −1 )Yt − (1 +  ) gtP −  −1Gt ] + Et −1 t ,1− (A1.47)t = t =  (1 +  )Yt −  gtP −  Gt + Et −1 t ,(A1.48)1  −1(1 +  ) ,  =where  =. 1− 1+ 1.2.3 Derivation of labor income of the borrowerLog-deviation of a borrower’s labor income is defined by the wage and hours worked:Yt b = Wt + htb(A1.49)Combining Euler equation of a saver together with the labor supply of a borrower weobtain:htb =  −1Wt −  −1 −1 Ctb .(A1.50)From the resource constraint borrower's consumption is:Ctb = b −1Yt − b −1 s Cts − b −1 Gt ,(A1.51)Labor demand from the production function is:ht =  −1Yt −  −1 g p .(A1.52)Substituting it into the aggregate labor supply and combining it with the resourceconstraint we obtain:Wt = ( −1 +  −1 )Yt −  −1 g p −  −1 Gt .(A1.53)Taking into account (A1.50), (A1.51) and (A1.53), log-deviation of borrower’s laborincome according to (A1.49) is:Yt b = Yt +  −1 ( −1b −1 s − 1)Gt − (1 +  ) −1 gtP ++ −1 b −1 s −1 ( Et Cts+1 −  (it − Et t +1 − r )),where  = (1 +  −1 )( −1 +  −1 ) −  −1 −1 b−1 .112(A1.54)A1.3 Analysis of the model1.3.1 Short run dynamics of the modelIn the short-run the budget constraint of a borrower transforms into: D high − DˆˆC = Y − D +  D S −  D  ( it −  L − r ) , where D =,YbSbSYSb = YS +  −1 ( −1b −1 − 1)GS −  −1b −1 s −1 (iS − r ) , andDt(A1.55)andDt −1are thedeviations from the steady state so that: D Dt −  D Dt −1 =  DDt − DD − D D  Dt − Dt −1−  D t −1=.DDYD(A1.56)In period t debt limit falls from Dhigh to Dlow = D , in t − 1 we have Dhigh and in periodt Dlow .

Therefore, this part of the budget constraint becomes: Dt − Dt −1  D high − D=.YYThe budget constraint of a saver transforms into the following:Cˆ Ss = Cˆ Ls −  (iS −  L − r ) .(A1.57)1.3.2 Derivation of the output in the short run s ( +  −1 ) + b D b ˆbˆYS = −(iS − r ) −D−TSb +1 − b1 − b1 − bb1 +  −1 −1 s − b −1(1 +  ) −1b P+ D S +GS −gS .1 − b1 − b1 − b(A1.58)After applying the definition of the natural interest rate: ( +  −1 ) + b D YˆS = − s(iS − rSn ) ,1 − b(A1.59)where the natural interest rate is:rSn = r −b s ( +  ) +  b D −1Dˆ + D b − s ( +  −1 ) +  b D  Sb1 +  −1 −1 s − b −1(1 +  ) −1b−T+G−g SP .SS−1−1−1 s ( +  ) + b D  s ( +  ) + b D  s ( +  ) + b D The Phillips curve has the following representation:113(A1.60) S = ( +  −1)YS −  gSP −  GS ,where  =(A1.61)1 −1(1 +  ) ,  = .1− 1− Combining the equations for the AS and the AD we obtain: s ( +  −1 ) + b D b ˆbˆYS = −(iS − r ) −D−TSb +1 − b1 − b1 − b+b1 +    s − b(1 +  ) b P D ( (1 +  )YS −  gSP −  Gt ) +GS −gS ,1 − b1 − b1 − b−1−1−1−1(A1.62) ( +  −1 ) + b D bYˆS = − s(iS − r ) −Dˆ −1 − b ( +  D (1 +  ))1 − b ( +  D (1 +  ))b1 +  −1 −1 s − b ( −1 +  D ) D b + (1 +  ) −1b P−TSb +GS −gS1 − b ( +  D (1 +  ))1 − b ( +  D (1 +  ))1 − b ( +  D (1 +  )).(A1.63)1.3.3 The case of a positive nominal interest rateSubstituting the definition of the positive nominal interest rate it = rt n +   t in theexpression for output and implementing the definition of the natural interest rate we get: ( +  −1 ) + b D YˆS = − s(iS − rSn ) ,1 − brSn = r −b s ( +  ) +  b D −1Dˆ +(A1.64) D b − s ( +  −1 ) +  b D  Sb1 +  −1 −1 s − b −1(1 +  ) −1b−T+G−g SP ,SS−1−1−1 s ( +  ) + b D  s ( +  ) + b D  s ( +  ) + b D  ( +  −1 ) + b D  n ( +  −1 ) + b D YˆS = − s(rS +   S − rSn ) = − s  S .1 − b1 − b(A1.65)(A1.66)Substituting the definition of the inflation in the short-run from the Phillips curve weobtain: ( +  −1 ) + b D YˆS = − s [ (1 +  )YS −  g SP −  GS ] ,1 − b114(A1.67) ( +  −1 ) + b D YˆS = − s [ (1 +  )YS − ( +  )GS ] ,1 − bmult G = ( +  )(  s ( +  −1 ) + b D  ).1 − b +   (1 +  )(  s ( +  −1 ) + b D  )(A1.68)(A1.69)An impact of the share of productive spending  has on the fiscal multiplier is:mult G  (  s ( +  −1 ) + b D  )=.1 − b +   (1 +  )(  s ( +  −1 ) + b D  )(A1.70)An impact of the productivity of government investment  is:mult G  (  s ( +  −1 ) + b D  )=.1 − b +   (1 +  )(  s ( +  −1 ) + b D  )(A1.71)1.3.4 The case of a zero lower boundIf deleveraging shock is sufficiently high and zero lower bound becomes binding weget:YˆS =  −bbDˆ −TSb + ,1 −  b (  +  D (1 +  ))1 −  b (  +  D (1 +  ))1 +  −1 −1 s − b ( −1 +  D ) D b +  D b + (1 +  ) −1b P+GS −gS ,1 − b ( +  D (1 +  ))1 − b ( +  D (1 +  ))(A1.72) s ( +  −1 ) + b D =r.1 − b ( +  D (1 +  ))After simplifying (A1.72) the following equation for the log-deviation of output can beobtained:YˆS =  −bbDˆ −TSb +1 −  b (  +  D (1 +  ))1 −  b (  +  D (1 +  ))1 +  −1 −1 s − b ( −1 +  D ) −b ( D (1 +  ) + (1 +  ) −1 )+GS .1 − b ( +  D (1 +  ))(A1.73)The multiplier is:mult G =1 +  −1 −1 s − b ( −1 +  D ) −b ( D (1 +  ) + (1 +  ) −1 ).1 − b ( +  D (1 +  ))115(A1.74)The impact the share of productive spending,  , has on the fiscal multiplier is:mult G −b ( D (1 +  ) + (1 +  ) −1 )=.1 − b (  +  D (1 +  ))(A1.75)the impact of productivity of government investment,  , is:mult G −b ( D (1 +  ) + (1 +  ) −1 )=.1 − b ( +  D (1 +  ))(A1.76)Appendix 22.1 Derivation of lifetime budget constraint of a Ricardian agentBy integrating the budget constraint of a Ricardian agent we obtain:a ( , t ) + a ( , t ) =  c R ( , )e( r (t )+  )(t − ) d ,RH(A2.1)twhere the following NPG condition has been imposed:lim a ( , )e( r (t )+  )( −t ) = 0. →(A2.2)Ricardian consumer maximizes expected lifetime utility subject to its lifetime budgetconstraint:L =  (1 −  ) ln c R ( , ) +  ln g  e(  +  )(t − ) d +  (t )  a ( , t ) + aHR ( , t ) −  c R ( , )e( r ( t )+  )(t − ) d ttTaking into account first order condition (1 −  ) / c R ( , )e(  +  )(t − ) =  (t )e( r (t )+  )(t − ) and(1 −  ) / c R ( , t ) =  (t ) for  = t we obtain:ttR(  +  )( t − )d =  c R ( , )e( r (t )+  )(t − ) d , c ( , t )ec R ( , t ) −e(  +  )(t − )  = a ( , t ) + aHR (t ) t+c R ( , t ) = (  +  ) a (, t ) + aHR (t )  .(A2.3)Aggregate consumption of Ricardian agents can be obtained by aggregating the individualconsumption:tC (t ) =   eR− ( −t )tc ( , t )d =   e ( −t ) (  +  ) a ( , t ) + aHR (t ) d =R−116(A2.4)t t= (  +  )    e  ( −t ) a ( , t )d +   e  ( −t ) aHR (t )d  = (  +  )  A(t ) + AHR (t )  .− −2.2.

Derivation of the aggregate Euler equation (the case of homogeneous agents)The equation (3.20) can be simplified as follows:C (t ) L(t )c (t , t ) −  C (t )=  r (t ) −   +.C (t )C (t )(A2.5)As new generations are born without any financial assets ( a (t , t ) = 0 ), thus from (3.8)c (t , t ) = (  +  )a H (t , t ) and taking into account (3.19) we get: L(t )c (t , t ) −  C (t ) L(t )a H (t , t ) −  ( A(t ) + A H (t ))= ( +  ).C (t )C (t )(A2.6)Aggregate human wealth is defined as follows:t −AH (t ) = a H ( , t ) d +t − −t at−H( , t ) 0t − d ,(A2.7)where:tt a H ( , t ) =  (1 − t L )W N ( , t )e R (t , )+  (t − ) d +  z e R (t , )+  (t − ) d ,t − (A2.8)R(t , ) =  r ( s)ds.t a ( , t ) =  (1 − t L )W N ( , t )e R (t , ) +  (t − ) d −0t − Ht +−tWeR ( t , ) +  ( t − )d +ze (A2.9)R ( t , ) +  ( t − )d .+tFor the simplicity of the analysis let us assume the constant interest rate r (so thatequations could be applicable to the steady state). a ( , t ) =  (1 − t L )W N ( , t )e( r +  )(t − ) d −0t − Ht()tz− W 1 − e − ( r +  )( + −t ) +e− ( r +  )( + −t ) .r+r (t ) + (A2.10)We know that age dependent wage can be written as follows:W N ( , t ) = E ( −  ) FN ( k (t ),1) = 0 e − ( − ) FN ( k (t ),1),117(A2.11) (1 − tL)W N ( , t )e( r +  )(t − ) d = e ( −t ) 0 (t ),(A2.12)twhere  0 (t ) is defined as follows:0 (t ) = 0  (1 − t L )FN (k (t ),1)e( r + +  )(t − ) d .(A2.13)tSubstituting this definition into the expressions of human wealth of workers andretirees, noted above, we get:t −−L( , t ) a H ( , t ) d =t − t − e −  te−  ( −t )z 0 (t ) +d =er +  (A2.14) z − = L(t ) 0 (t )e−( + ) −e .r+ +t L( , t ) at−t=  e −  tet−H( , t ) 0t − d =  ( −t )tz0 (t ) − W (1 − e− ( r +  )( + −t ) ) +e− ( r +  )( + −t )  d =er+r+(A2.15) t (t + z ) −   e− r − e− nL  = L(t ) 0 (t ) 1 − e− ( + ) − W (1 − e− ) + We  .Lr+r+ n − r  +()The aggregate human wealth is:− r 0 (t )− e− nL−  tW + z  eA (t ) = L(t ) +er +   nL − r  +H ,(A2.16)where it was taken into account that: tW (1 − e− ) = ze− − d (t )  tW + z =Fromtheexpressionforworking-agehouseholds,z − d (t ).1 − e−takingintoaccounttW = d + (tW + z )e − : t + z  −( r (t )+  ) −ta H (t , t ) = 0 (t ) +  W−e− W = er (t ) +  r (t ) +  () t + z  − r (t ) − nLd (t )= 0 (t ) + e−   W−e−. er (t ) +  r (t ) +  ()(A2.17)After substituting this expression in the equation for AH and eliminating  0 (t ) we get: L(t )a H (t , t ) = ( +  ) AH (t ) −  L(t ),118(A2.18)where  = e−   z − d (t )   e− r − e− nL−d (t )+ (r + +  )− Lr+ 1− e   r +    n − r.Taking into account the expression for a H (t , t ) and taking into account (3.19) we get:C (t ) L(t ) + ( +  ) A(t )= r (t ) −  +  + n L − (  +  ).C (t )C (t )(A2.19)2.3.

Характеристики

Список файлов диссертации