The government budget constraint is stated in the following equation, where the left-hand side is tax revenue and debt in period _t_ which finances the expenditures on the right-hand side which is the reimbursement of last periods debt with interest and the public pension benefit for retirees and health care expenditures — all denoted in units of effective labor, where �� is public debt as a share of GDP per worker. Technically there is one equation and three free (policy) variables, τ, Ψ, and ε, where τ is the aggregate contribution rate.

Capital market clears when �� ����� ���, i.e. savings yields interests either in form of capital in firms or by government repaying debt with the same real interest rate. This immediately implies that a higher debt ratio result in lower capital accumulation and therefore the government pays a higher real interest rate.

# _<u>Calibration</u>_

The model is calibrated for Brazil with the following data: Time periods of the model are assumed to be 20 years; as evident from Figure 5a, the population growth rate equals 56 percent (growth rate over the period 1970-1990; UN Population Division); in order to calibrate for life expectancy, the length of the model’s first period is normalized at unity while the length of the second period equals 61 percent (estimated as the share of total life spent in retirement: average retirement age equals 57.83 (Queiroz and Figoli, 2010), average entry-age into the labor force is 13.3 (Leme & Malaga, 2001), life expectancy equals 69.3 (average life expectancy at birth over the period 1998-2088, UN Population Division)); the share of capital in output equals 18 percent (average gross capital formation over the period 1990-2010; IMF WEO); For the base year 2010 in the simulations a debt/GDP ratio of 41 percent (Banco Central do Brazil); the contribution rates to the health system is residually estimated based on 1.46 percent for health benefits (based on health spending/GDP over the period 1984-2009; Tafner, 2010); the contribution rate to the social PAYG pension system is calibrated to 31 percent (11 percent from workers and 20 percent from employers; Queiroz and Figoli, 2010).

The average replacement rate, weighted by the share of the population receiving pension benefits at various rates, for pensions is estimated to be 45 percent. The replacement rate to public servants is assumed to be 95 percent since older public servants participated in a plan of virtually 100 percent replacement rate, while younger generations receive slightly less; the replacement rate for nonpublic servants is estimated residually based on pension payment data for the two groups; the weighted average is constructed based on the share of the population is each group; the effective weighted average is found to equal 45 percent; sources: DATAPREV, SUB, Plano Tabular da DIIE, and Ministerio da Fazenda. Productivity growth is assumed to follow output growth and equals 168 percent (growth over the period 1988-2008: World Bank GDP); discount rate equals 0.9.

# _<u>Simulations</u>_

The model simulations start in 1990 so the past 20 years of demographic transition is taken into account in projecting the economic implications from the year 2010 to 2050. The model is simulated in Matlab using Dynare software (Jorgensen, 2006). The results should be interpreted as the 18