Диссертация (1137741), страница 24

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Derivation of aggregate non-interest incomeAggregate non-interest income is defined as follows:tWI (t ) = L(, t )WI (, t )d,(A2.20)−where WI ( , t ) is defined as follows:(1 − tL )W N ( , ) − tWWI ( , ) = N(1 − tL )W ( , ) + zfor  −    ,for  −    .(A2.21)WI ( , t ) is split into two parts, non-interest income of the retirees and non-interestincome of the young:t −WI (t ) =L( , t ) (1 − t L )WI ( , t ) + z  d +t L(, t ) (1 − tL)WI ( , t ) − tW  d.(A2.22)t −−After applying the expression for W N ( , t ) = E (t −  ) FN (k N (t ,1)) = 0 e − (t − ) FN (k N (t ,1)) ,which comes from taking into account the definition of the efficiency index E( − ) and thefact that the wage of a worker born at  is equal to the marginal product of labor, adjusted forhis or her productivity we obtain:t −WI (t ) = L( , t ) (1 − t− L( , t ) (1 − tt−)0 e− (t − ) FN (k N (t ,1)) + z  d +(A2.23)t+LL)0 e− ( t − )FN (k N (t ,1)) − tW  d.After rearranging we get:tt −WI (t ) = (1 − t L ) FN (k N (t ,1))   L( , t )0 e− (t − ) d +  L( , t )0 e− (t − ) d  +t − −119(A2.24)t −+z+zL( , t )d − tWt−t −t −ttL( , t )d = (1 − t L ) FN (k N (t ,1))  L( , t )0e− (t − ) d +− L( , t )d − t  L( , t )d.W−t−Noting that n ( , t ) = E (t −  ) = 0 e − (t − ) and applying the notion of N (t ) from (3.27)we get:WI (t ) = (1 − t L ) FN (k N (t ,1))t −t0L(t ) + z  L( , t )d − tW  L( , t )d. +−t −(A2.25)Applying (3.17) we get:t −t0 −  tWI (t ) = (1 − tL ) FN (k N (t ,1))L(t ) + z  ed − tW  e − t d = +−t −0= (1 − tL ) FN (k (t ,1))L(t ) − (1 − e− )tW en t + ze− en t . +L(A2.26)LTaking into account the fact that the pension system is run on a non-balanced manner,(3.39), the equation of the aggregate income can be simplified as follows:WI (t ) = (1 − t L )0F (k (t ,1)) L(t ) + D(t ). + N N(A2.27)2.4.

Social welfare  −SW = t − L( , )[lnc ( , )]e(  +  )(t − ) d d + t L( , )[lnc ( , )]e (  +  )( t − )d d .(A2.28)−Taking into account the Euler equation:c ( , ) = c ( , t )e( r −  )( −t ) ,(A2.29)c ( , ) = c ( , t )e( r −  )( −t ) ,(A2.30)(  +  )[a ( , ) + a H ( , )] = (  +  )[a ( , t ) + a H ( , t )]e− ( r (t ) −  )( −t ) .(A2.31)Applying a ( , ) = 0 and simplifying we obtain the consumption of generation  :c ( , ) = (  +  )(a ( , ) + a H ( , )) = (  +  )a H ( , )e− ( r −  )( −t ) .(A2.32)z k   ( −t ) a ( , t ) =,  (1 −  )(1 − t L )   e +r + +  n r+(A2.33)Ho1120twtw + z − ( r +  )( + −t ) k   ( −t ) a ( , t ) =+e.  (1 −  )(1 − t L )   e −r + +  n r+ r+1Hy(A2.34)Substituting the expressions of human wealth from above we get the following socialwelfare function as a function of the steady state level of capital per capita:SW (t ) =enLt + 1 −(ln((  +  )aoH ) + (r −  ))e − ln((  +  )a yH )(1 − e− ) −nL −  − (A2.35)− (r −  )(1 − e − e− ) −1 ) 2.5.

Estimation resultsTable A2.5.1. Steady states with the optimal choice of income tax rate (  = 25% )The share of non-Ricardian consumers 10%20%30%40%50%60%70%80%90%tL38.0%38.0%38.0%38.0%38.0%38.0%38.0%38.0%38.0%k11.52.0%11.22.1%10.82.2%10.42.4%9.92.5%9.42.7%8.92.9%8.33.2%7.63.6%rTable A2.5.2 Optimal choice of income tax rate and social contributions (  = 25% )The share of non-Ricardian consumers 10%20%30%40%50%60%70%80%90%tL43%42%42%41%41%39%38%38%37%5%6%6%6%6%48%48%48%50%121VariablesY(y)KN (kN )K (k )LN (n )WNWKrNA ( a, a )output (per efficiency unit of labor)capital stock (per efficiency unit of labor)aggregate (per capita) capital stockaggregate population sizeaggregate (per capita) labor supply in efficiency units of labornominal wagenominal rental rate on capitalprofitreal interest rateaggregate (individual, per capita) real financial wealthAG ( a G , aG )AHi ( aHi , aHi )aggregate (individual, per capita) government debtZ ( z, z )nominal (real) transfer to retireesTW ( tW , tW )nominal (real) tax on the youngC i ( c i , ci )aggregate (individual, per capita) consumption of Ricardian, non-Ricardianagentstotal aggregate (per capita) consumptionC T ( cT )Iaggregate (individual, per capita) real human wealthgross investmentG (g)aggregate (per capita) government consumptionD(d )deficit of the pension fund (per capita)ration of consumption (deficit of the pension fund, government debt) percapita to output per capitac y ( d y , aGy )122.

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