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

6.4 The log-linear Ramsey model

The log-linear version of (25a) is given by

ʌ

[γ(1 — σ) — 1]^t = — (1 — γ)(1 — σ)mt + At + (1 — Y)(1 - σ)πt.

Log-linearizing (25e) delivers:

where τ⁰ (τ⁰⁰) is the marginal tax rate (the second derivative of the tax function)
computed at the steady state solution of gy = wn + rk. The cost-minimizing conditions
(16) and (15) provide two additional equations:

an t + Wj t = αk t — x t + zt,

^{( α —} 1) n t + r t = ^{( α —} 1)fc t ^{— χ} t + z t.

The log-linear version of the aggregate production function is given by:

( α — 1) n t + y t = αk t + z t-                                                        (43e)

The definition of gross investment i_t = y_t — c_t implies
ʌ

^[( y/i ) ^{— 1] ^} t ^{— (} y/i )^y t + it = 0 -                                                         (^43f)

Finally, the profit equation Ω_t = y_t (1 — g_t)) provides the following log-linear equation:

ʌ

^—yt + ^ω t = ^{(1 —} e )^g t-                                                             (⁴3g)

The five equations that determine the dynamics of the log-linear model are derived
from the economy’s resource constraint kt+1 = (1 — δ)kt + yt — ct, the Euler equations
for capital and money balances, (25b) and (25c), the definition of beginning-of-period

32

<<   31   32   33   34   35   36   37   >>

More intriguing information