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

2.6 The representative agent model

In the representative-agent Ramsey model we are, of course, unable to model pensions
and to differentiate between working hours and effective labor input. Everything else
is unchanged.

The representative household maximizes his infinite life-time utility

^X β^tu(ct, Mt/Pt, 1 - nt)
subject to

kt+1 + Mt+¹ = (1 - δ + rt(1 - τ[(πt∕π)(w_tn_t + rtkt)]))kt

Pt

+ -+ + (1 - τ[(∏t/π)(w_tn_t + r_tkt)])w_tn_t + trt + Ωt - ct.

Pt

His decision variables in period t = 0 are M1, k1, c0, and n0.

In this model, there are two predetermined state variables, the stock of capital kt and
beginning-of-period real money balances

mt := Mt/Pt-1 ^ Mt = —,∏t := ^pl-.                               (24)

Pt    ^πt       Pt-1

Using these definitions, we can write the first-order conditions as follows:

λt = βEtλt +1 (1 - δ + rt +1(1 - τ0[(∏t∕π)(Wtnt + rtkt)](∏t∕∏))) ,

(25b)

^λt+1 + ^{(1 - γ} ) ⁽ ct+1)^{γ (1 σ} )
^πt+1

(1-γ)(1-σ)-1
( mt +1 ∕^πt +1)

^πt+1

(25d)

n-t^ = ηo(1 - nt) ^η = λtwt (1 - τ0 [(∏t∕π)(w_tn_t + rtkt)](π_t∕π)).

(25e)

10

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