Business Cycle Dynamics of a New Keynesian Overlapping Generations Model with Progressive Income Taxation

The working agent receives income from effective labor e(s, j)n_t^s,j and capital k_t^s,j as
well as government transfers tr_t and profits Ω_t which are spent on consumption cs^, and
next-period capital k_t^s₊⁺₁^1,j and money M_t^s₊⁺₁^1,j . He pays taxes on his nominal income
_Ptyts,j_:

Pty_t^s,j = Ptrtk_t^s,j + Ptwte(s, j)n_t^s,j .

The government adjust the tax income schedule at the beginning of each period for the
average rate of inflation in the economy which is equal to the non-stochastic steady
state rate π. Therefore, nominal income is divided by the price level, P_t-1π, and the
tax schedule τ (.) is a time-invariant function of (deflated) income with τ⁰ > 0. Notice
that when we have unanticipated inflation, π_t = pp-i > π, the real tax burden increases
as the agent’s real income moves into a higher tax bracket, the so-called “bracket creep“
effect.

The nominal budget constraint of the retired worker is given by

P ( s∙⁺⁺1 ^{3             sj} ʌ    ( ^{s +}1 ³    M^s3 ʌ I P s,³

Pt k_t+1 - (1 - δ)k_t   + M_t+1  - M_t   + Ptc_t

= Ptrtk^sS³ + Pent + Pttrt + PtΩt - PtTt

s = T + 1, . . . , T + T^R, j = 1, . . . , ne,
with the capital stock and money balances at the end of the life at age s = T + T ^R being
equal to zero, k_t^{T +T R +1,j} = M_t^{T +T R +1,j} ≡ 0 for all productivity types j ∈ {1, . . . , ne},
because the household does not leave bequests. Furthermore, since retirement at age
T + 1 is mandatory, n_t^{T +1,j} = n_t^{T +2,j} = . . . = n_t^{T+TR ,j} ≡ 0. Pent are nominal pensions
and are distributed lump-sum. They are not taxed. Again, the government adjusts
pensions each period for expected inflation according to Pen_t = pen P_t-1π, where pen
is constant through time. If inflation is higher then expected, πt > π , the real value of
pensions declines.

The real budget constraint of the s-year old household with productivity type j is given
by

(^{1 +} rt ^{- δ})ks³ + mt, + wte(s^,j)ns, + trt + ^ωt ^- τt yS⅛^πp} - cs^,

s= 1,...,T,

(1 + r, - δ ) kSⁱ + ^m∏j + ^pn^π + trt + Ω t - τt ( ^yjt ´ - cS^,,

s=T+1,...,T+T^R,

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