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)nts,j and capital kts,j as
well as government transfers
trt and profits Ωt which are spent on consumption cs, and
next-period capital
kts++11,j and money Mts++11,j . He pays taxes on his nominal income
Ptyts,j:

Ptyts,j = Ptrtkts,j + Ptwte(s, j)nts,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, Pt-1π, and the
tax schedule
τ (.) is a time-invariant function of (deflated) income with τ0 > 0. Notice
that when we have unanticipated inflation,
πt = 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    Ms3 ʌ I P s,3

Pt kt+1 - (1 - δ)kt   + Mt+1  - Mt   + Ptct

PtySj

Pt-1


(4)


= PtrtksS3 + Pent + Pttrt + PtΩt - PtTt

s = T + 1, . . . , T + TR, 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,
ktT +T R +1,j = MtT +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, ntT +1,j = ntT +2,j = . . . = ntT+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
Pent = pen Pt-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 yS⅛πp} - cs,

ks+1,j+  s+1,j

kt+1 +mt+1 =


s= 1,...,T,

(1 + r, - δ ) kSi + mj + pnπ + trt + Ω t - τt ( yjt ´ - cS,,

s=T+1,...,T+TR,



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