Диссертация (1137741), страница 17
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However, according to Walras' Lawonly two market equilibrium conditions are needed to ensure equilibrium on these markets:y (t ) = cT (t ) + i (t ) + g (t ),(3.41)a (t ) = k (t ) + a G (t ).(3.42)77The system of dynamic equations is non-linear and cannot be solved analytically, so weare solving the system of equations T3.1.1-T3.1.3 numerically, taking into account T3.1.11T3.1.12.80 We check whether the resulting set of possible equilibria satisfies the stabilitycondition of the equilibrium.
81To distinguish the socially optimal policy mix of instruments (income tax and socialcontributions) we define the social welfare function as the present value of the utility of allcurrently living and future generations weighted by their share of the population.
Socialwelfare is a sum of the welfare of young generations and retirees, where is a time of birthand is a current moment. −SW (t ) = L( , )[(1 − ) ln c NR ( , ) + ln g ]e( + )(t − ) d d +t − −+(1 − ) L( , )[(1 − ) ln c R ( , ) + ln g ]e( + )( t − ) d d +t − + (3.43)L( , )[(1 − ) ln cNR( , ) + ln g ]e( + )( t − )d d +t − +(1 − ) t L( , )[(1 − ) ln c R( , ) + ln g ]e( + )(t − ) d d .−Taking into account equation (3.8) and (3.10), individual consumption of the young andold generations is expressed as functions of their individual human wealth, which depends onthe effective capital.
82 Simplifying equation (3.43) the following welfare function can beoyobtained, where a H and a H is the human wealth of the young and old generations of bothtypes of consumers:SW (t ) = e− (ln(( + )aHo ) + ( + 1) / ) + ln g −− (1 − e− )(ln(( + )aHy ) − ln g ) − −1 (1 − )(1 − e − e− )) ,=(3.44)enLt, = r − .nL − − All calculations were conducted using Matlab. Setting k (t ) = 0 , and aG (t ) = 0 , we use a root-finding method(the bisection method) to define the level of capital per capita to bring the growth of per capita consumption tozero, c R (t ) = 0 . All possible combinations of feasible model parameters and policy instruments are considered80to determine the steady-state level of k * .81The stability condition ensures that the determinant of the Jacobian matrix of the log-linearized dynamicsystem of equations T3.1.1, T3.1.2 and T3.1.4 is negative.
In this case the model is locally saddle-point stable.The constraint on the public debt ensures that public debt does not exceed 1.5 times the output.82For more details see Appendix 2.78It is known that in the OLG model with Ricardian agents and a dynamically efficientequilibrium, a PAYG pension system worsens social welfare. 83 If the level of pensions ischosen optimally from the maximization of social welfare, it is optimal to set zero pensionsand social contributions. Taking this result into account in the maximization problem the levelof pensions and the retirement age were fixed in order to analyse the optimal choice ofincome taxes and social contributions.It is worth mentioning that the equilibrium with diminishing labor productivity can bedynamically inefficient if the speed of the decrease in productivity, , is large enough.
In thiscase labor income is high during the youth and falls rapidly with age, so agents save a lotduring youth which makes capital stock too large, leading to an overaccumulation of capital.The necessary condition for dynamic efficiency is , which corresponds to a positiveinterest rate. The calibration used in this chapter satisfies the condition of dynamic efficiency.3.4. CalibrationThe optimal choice of the retirement age, social contributions and pensions illustratesthat the pension system is redundant. This result coincides with the results of neoclassicalmodels, with forward-looking agents having an incentive to save on their own in order tosmooth their consumption.
That is why we investigate the optimal mix of income tax andsocial contributions with a fixed level of pensions and a fixed retirement age. Table 3.2contains the parameters used for the baseline calibration.The share of pensions, , is fixed at 30% of the median lifetime wage, while theoptimal size of the mandatory social contributions, , as a share of the median wage, andincome tax are chosen optimally from the maximization of social welfare. 84 The share ofgovernment expenditure, , is fixed at 25% of GDP, a common value for the OECDcountries.
The productivity declines with age at the speed of , which equals 1.25%,meaning that the worker is half as productive at retirement age than he was at the beginning oflife. For the results presented below, the death rate equals 1.25%, giving a life expectancyof 80 years. The birth rate varies from 1% to 3%. This range allows us to analyse bothnegative and positive population growth. In considered steady states the endogenous interestrate is higher than the population growth (the so-called Aaron condition is satisfied).8384See Blanchard and Fischer (1989).The results are robust to the change in the share of the median wage paid to pensions.79Table 3.2. Baseline calibrationVariableRate of time preferenceBirth rateProbability of deathOutput elasticity of capitalSpeed of decline in the labor efficiencyCapital depreciation rateRetirement ageShare of non-Ricardian agentsSymbolValue0.0150.01250.01250.330.01250.0360[0; 0.9]SourceHeijdra, Ligthart (2006)85Heijdra, Ligthart (2006) 86Heijdra, Ligthart (2006)Standard87Nickel et al.
(2008) 88Heijdra et al. (2017)89Bettendorf, Heijdra (2006) 90A range of possible valuesThere is no consensus in the literature concerning the number of non-Ricardian agentsin the economy. The estimates vary from 50% in Campbell and Mankiw (1989) for the US to0.24-0.37 for the Euro area estimated by Coenen and Straub (2005). 91 At the same time Marto(2014) estimated the share of non-Ricardian agents as equal to 0.58 for the Portugueseeconomy.
As for Russia, Tiffin and Hauner (2008) assume that 40% of Russian householdsare liquidity-constrained in the GIMF model calibrated for Russia. 92In the next two sections we consider the optimal fiscal instruments in an economywhere all agents are Ricardian (Section 3.5) and in an economy with two types of agents,Ricardian and non-Ricardian (Section 3.6).3.5. Optimal fiscal policy instruments in a homogeneous economyThis section first shows how optimal social contributions and income tax rate depend onthe retirement age and population growth.
We start by searching for the optimal proportion ofthe median wage that goes to social contributions, and move to the determination of thechoice for both social contributions and income tax rate (Subsection 3.5.1). In Subsection3.5.2 the optimal choice of these two instruments is considered under a balanced andunbalanced pension system. Subsection 3.5.3 provides a robustness check, covering the85The value belongs to the interval considered by Heijdra and Ligthart (2006): from 0.01 to 0.04.In Heijdra and Ligthart (2006) the birth rate varies from 0.01 to 0.04, while the death rate varies from 0 to0.03. In the baseline calibration the birth rate equals 0.015 and the death rate is 0.0187In Heijdra and Ligthart (2006) elasticity with respect to capital equals 0.35.88It is consistent with the value used in Nickel et al.
(2008) where this parameter equals 0.014, death rate at 0.01.89This value is consistent with the literature on overlapping generations in a continuous time. In Heijdra et al.(2017) capital depreciation rate equals to 0.036. Heijdra and Ligthart (2006) use depreciation rate which is equalto 0.06 with the higher rate of time preference being equal to 0.04.90Although Bettendorf and Heijdra (2006) do not use calibration, in the numeric example the restriction on theretirement age postulates that it should exceed 48.5 or 56.8 depending on the birth and death rates considered.91Here the estimates for posterior means are provided. These estimates are rather low, taking into account thefact that Coenen and Straub (2005) used the value of 0.5 as a mean of the prior distribution.92Global Integrated Monetary and Fiscal model (GIMF), presented in Kumhof, Laxton (2007, 2009, 2013),Kumhof et al.
(2009, 2010).8680optimal choice of fiscal instruments under different life expectancy and labor productivity.The results of this section are presented in Mamedli (2017).3.5.1. Policy instruments under unbalanced pension system dependingon the retirement ageIn order to investigate how the optimal social contributions depend on the retirementage we first consider the case with a fixed income tax ( t L = 40% ).
Quantitative resultssuggest that a higher retirement age leads to a lower rate of social contributions (Figure3.1). The value of social contributions tW is also affected by the change in the median wage.Median wage increases with the higher retirement age due to the higher per capita capital anddecreases with the higher retirement age due to the decreasing productivity of labor.
The lattereffect is stronger, so median wage decreases with the higher retirement age. As a result, lowerpensions correspond to a higher retirement age. The optimal level of decreases with higherpopulation growth as it leads to a higher share of young generation members who pay socialcontributions.Ψπ = 5535%π = 60π = 6530%25%20%15%10%5%0%-0.5%0.0%0.5%1.0%1.5%2.0%nLFigure 3.1. Optimal social contributions rate under different retirement ageSource: estimates based on the model.Next, we consider the optimal choice of two instruments: income tax, tL , and the rate ofsocial contributions, tW .
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