Диссертация (1137741), страница 16
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Demography and aggregate household sectorThe framework allows us to consider non-zero population growth, by distinguishing theprobability of death 0 , and the probability of birth, 0 .75 The population size L(t ) ,which consists of Ricardian agents ( 1 − ) and non-Ricardian agents ( ), and growsproportionally with the net growth rate n L :L(t )= − = nL .L(t )(3.13)Taking into account a normalization L(0) = 1 , the population size is:L(t ) = e nLt .(3.14)The size of the generation born at t is proportional to the current size of the population:L( , ) = L( ).(3.15)The size of each generation falls exponentially with the probability of death :L( , t ) = e ( −t ) L( , ), t .(3.16)The current size of the generation born at time can be obtained by substituting (3.14)and (3.15) into (3.16):L( , t ) = e e− t .75This framework was developed by Buiter (1988).72(3.17)The aggregate variables are defined as the integral of the variable values, specific foreach living generation, weighted by the size of that generation.
Aggregate consumption can bedefined as follows:tC (t ) = L(, t )c (, t )d = CK(t ) + (1 − )C R (t ),(3.18)−where L( , t ) and c ( , t ) are given by (3.8), (3.10) and (3.17), respectively.By aggregating (3.8) by all living generations of Ricardian consumers we obtain (3.19),which postulates that the aggregate consumption of Ricardian agents is proportional to theirwealth, where A(t ) is aggregate financial wealth and AH (t ) is aggregate human wealth:76C R (t ) = ( + ) A(t ) + AHR (t ) .(3.19)The growth rate of aggregate consumption is obtained from (3.18), taking into account(3.7) and (3.17):C R (t ) L(t )c R (t , t ) − C R (t )= r (t ) − +,C R (t )C R (t )(3.20)where r (t ) − is the growth of individual consumption of Ricardian agents, while the secondterm represents the so-called generational turnover (Bettendorf and Heijdra, 2006), whichdepends on the demographic parameters.
Aggregate consumption of Ricardian agentsincreases with the arrival of new agents and decreases with the death of the older generation.The growth rate of the aggregate consumption of Ricardian agents can be simplified to: 77 (t ) =C R (t ) L(t ) + ( + ) A(t )= r (t ) − + + n L − ( + ),RC (t )C R (t )(3.21) e− z R − d (t ) e− r (t ) − e− nL −d (t )+ ( r (t ) + + ) .− Lr (t ) + 1 − e r (t ) + n − r (t ) (3.22)The aggregate consumption growth of Ricardian agents exceeds the growth of theirindividual consumption ( r (t ) − ) if net population growth is positive ( nL 0 ), and laborproductivity decreases over time ( 0 ).
At the same time, aggregate consumption growth ofRicardian agents can be lower than the individual consumption growth if newborns consumeless or if the redistribution of wealth from the young to the old through the pension systemtakes place. In contrast to Bettendorf and Heijdra (2006), depends on the deficit of the7677For more details see Appendix 2.For more details see Appendix 2.73pension fund ( d (t ) – per capita deficit of the pension fund).78 Deficit of the pension fund isdefined by the difference between the sum of pensions paid by the pension fund and the sumof social contributions paid to the pension fund.
It is defined below, in equation (3.39).Aggregate financial wealth owned by Ricardian consumers is defined as follows:tA(t ) = L(, t )a (, t )d.(3.23)−The definition of aggregate savings can be found by differentiating equation (3.23) forthe aggregate financial wealth with respect to time and taking into account that the newborngeneration does not have any financial wealth, a (t , t ) = 0 :A(t ) = − A(t ) +t L(, t )a (, t )d.(3.24)−By substituting (3.2) to (3.24) we get:79A(t ) = r (t ) A(t ) + WI (t ) − C R (t ),(3.25)0(1 − t L ) FN (k N (t ),1)) L(t ) + D(t ), +(3.26)WI (t ) =where FN (k N (t ),1) is the marginal product of labor and D(t ) is the deficit of the pensionsystem.The aggregate labor supply at time t measured in efficiency units is proportional to thepopulation size in the corresponding period.
It is obtained from (3.5), (3.6), (3.14) and (3.17):tN (t ) =L( , t )n ( , t )d =−0L(t ). +(3.27)3.2.3. FirmsAs opposed to Bettendorf and Heijdra (2006), we consider a closed economy with anendogenous interest rate, which is important in the estimation of pensions. The output isproduced according to the Cobb-Douglas technology:Y = F ( K , N ) = K N 1− ,where K and N represent capital and effective units of labor.Producers maximize profit, choosing the optimal level of capital and labor:7879Bettendorf and Heijdra (2006) assume thatFor more details see Appendix 2. is equal to zero.74(3.28)t(t ) = Y (t ) − W N ( , t ) L( , t )d − W K (t ) K (t ),(3.29)−where W K (t ) is a capital rent and W N ( , t ) is the wage at time t of the worker of generation.The first order conditions are:W K (t ) = FK (k N (t ),1),(3.30)W N ( , t )W (t ) = FN (k N (t ),1),E ( − )(3.31)Nwhere FK = F / K N and FN = F / N .
W N (t ) is the wage per unit of effective labor andk N (t ) = K (t ) / N (t ) is the capital effective-labor ratio. The produced output is allocated to thetotal private consumption C , investment I and government expenditure G :Y (t ) = C (t ) + I (t ) + G(t ).(3.32)The optimal investment decision is based on the maximization of the net present valueof cash flows from the investor's capital stock subject to the capital accumulation identity:V (t ) = W K ( ) K ( ) − I ( ) e− R (t , ) d ,(3.33)tK (t ) = I (t ) − K (t ),(3.34)where R(t , ) = r ( s)ds is the discount factor.tKThe first order condition (3.35), specifies that the rental rate W equals the return oncapital r (t ) taking into account the capital depreciation rate :W K (t ) = r (t ) + .(3.35)3.2.4.
Public sectorGovernment budget identity defines the accumulation path of public debt AG (t ) , whichdepends on the current government expenditure G(t ) , labor tax revenues and additionalexpenditure, which we assume come from covering the total deficit of pension fund D(t ) . Itcan be written as follows:.AG (t ) = r (t ) AG (t ) + G (t ) − t LW N (t ) N (t ) + D(t ).Taking into account the transversality condition:75(3.36)lim AG ( )e− R (t , ) = 0,(3.37)A (t ) = tLW N ( ) N ( ) − G( ) − D( ) e− R (t , ) d .(3.38) →where the public debt is:GtThe key difference with the paper of Heijdra and Bettendorf (2006) is the assumptionthat the PAYG pension system can be run on an unbalanced-budget basis, with a deficit D(t ) :tW (1 − e − ) L(t ) = ze − L(t ) − D (t ) .(3.39)The left-hand side of (3.39) represents the total social contributions paid by the younggeneration, while on the right-hand side are total pensions paid to the old and the deficit (orsurplus) of the pension fund if the sum of social contributions and pensions do not match.
Foreasier comparison, we assume that the government determines the value of socialcontributions by setting the value of , which is the share of the median wage that goes tosocial contributions, paid until retirement. This makes the value of social contributionsdepend on the retirement age:tW = 0 FN (k (t ),1) e1 1+ e− − ln 2 .(3.40)The value of pensions can be expressed the same way, as a share of the median wagethat goes to pensions, :z = 0 FN (k (t ),1) e1 1+ e− − ln 2 .(3.41)3.3. Model summary and social welfareTable 3.1 below summarizes the key equations of the model in per capita terms, whereNRTRKNGthe endogenous variables are k , y , g , c , c , c , a , a , r , W , W , , n , tW , z . Theparameters are , , n L , , , , and .
The policy instruments are the retirement age , the share of the median wage that goes to social contributions and pensions, and ,public expenditure as a share of output, , and income tax tL .76Table 3.1. Summary of the ModelDescriptionDynamic equations:Analytical representationCapitalk (t ) = ny(t ) − cT (t ) − g (t ) − (nL + )k (t )(Т.3.1.1)Private consumption ofRicardian agentsPublic debtAssetsc R (t ) = (r (t ) − + )c R (t ) − ( + )( (t ) + ( + ) a(t ))(Т.3.1.2)a G (t ) = ( r (t ) − nL ) a G (t ) + g (t ) − nt LW N (t ) + d (t )(Т.3.1.3)a (t ) = ( r (t ) − nL ) a (t ) − c (t ) − n(1 − )(1 − t L )W (t ) + d (t )(Т.3.1.4) e− z − d (t ) e− r (t ) − e− nL ,−d (t )+ ( r (t ) + + ) − Lr (t ) + 1 − e r (t ) + n − r (t ) (Т.3.1.5)RNStatic equations:Pension fund (t ) =where tW (1 − e−) = ze−− d (t ) −1Rental rate k (t ) W K (t ) = k N (t ) −1 = n (Т.3.1.6)Interest rater (t ) = W K (t ) − (Т.3.1.7)WageW (t ) = (1 − ) y (t )(Т.3.1.8)NOutput k (t ) y (t ) = k N (t ) = n Supplied efficiency unitsn=Wealtha(t ) = k (t ) + a G (t )(Т.3.1.11)Total consumptioncT (t ) = c R (t ) + c NR (t )(Т.3.1.12)(Т.3.1.9)0 +(Т.3.1.10)Equation T3.1.1 corresponds to the accumulation of per capita capital, and it is obtainedby combining (3.34) and (3.35).
Equation T3.1.2 represents an optimal path of consumptionper capita, obtained from (3.21) in per capita terms. Equation T3.1.3 is the government budgetconstraint expressed in per capita terms, derived from the government budget constraint(3.36). The last dynamic equation, Equation T3.1.4, represents the accumulation of per capitaassets and is obtained from (3.25), taking into account (3.26) and (3.39).Definition 1. Given the set of policy variables , , , , tL that satisfy the governmentbudget constraint, the setWtK, rt , yt , ctT , ctR , ctNR , kt , a Rt , atG , dtdefines equilibrium, if itsatisfies the optimal conditions of households and firms, (3.7) and (3.30)-(3.31), andequilibrium conditions for goods and capital markets.
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