I.5 Structure of the thesis
1.2 The baseline macroeconomic model
1.2
The baseline macroeconomic model
The setup outlined in Section 2 implies the following structural relationships for the economy. The supply side is characterized by the forward-looking relationship
t =[yt+' (bt b
t)] +Ett+1: (1.18)
The aggregate supply equation links current ination, t, to the output gap, yt,
deviations in the tax ratebt from its steady-state value, and expected future ination,
Ett+1.24 The output gap yt is de ned as the di¤erence between actual and target
deviation (Yb
t ) of output from its steady-state value in response to a shock, with
e¢cient deviation being de ned as some function of that shock. Yb
t follows from the
approximation to welfare (see below and in the Appendix). b
t, which we shall refer
to as the e¢cient tax rate, is the deviation in the tax rate that would o¤set the cost- push (or cost-pull) pressure resulting from the increase in government spending. A cost-push (cost-pull) pressure follows a rise in government spending if the spending shock shifts the natural rate of output Ybn
t less (more) relative to what is desirable
from the viewpoint of welfare (the target deviationYb
t ).25 A policy ofb
t then brings
the natural rate in line with the target deviation and eliminates pressure on prices. Figure 1.1 provides a simple diagrammatic analysis to support the intuition behind
b
t. The government spending shock shifts the aggregate supply function to the right.
24 In general, we shall denote the t-period log-deviation of a variable xfrom its steady-state value xas
b
xt= logxxt . Prices are assumed to be stable in the steady state.
25 The natural rate of output is de ned as the level of output attained under exible prices. It is also the
level at which no price pressures would arise, if it prevailed in an economy with price rigidities. The deviations in the natural rate depend on real exogenous disturbancesit gathers the e¤ect of exogenous disturbances on real marginal costand, in our case, on tax policy. More details are provided in the Appendix to the chapter and are similar to Woodford (2003, Chapter 3).
1.2 The baseline macroeconomic model 52 P Y Y*(=Yn) Yn
}
cost-pull pressure tax rate G output price level P price level Y Yn Y*(=Yn) output tax rate}
cost-push pressure GFig. 1.1. Using tax policy to stabilize the price level
This shift reects at aggregate level a similar shift in the individuals labour supply schedule following the shock implied by (1.12). With no change in taxes, the level of output that would stabilize prices is denoted by Yn, which is the natural rate of
output in the absence of changes in taxation. In general, it is, however, welfare- e¢cient to stabilize output at a di¤erent level denoted Y
. If Y
is lower (higher) than Yn, as in the left (right) panel of Figure 1.1, cost-pull (cost-push) pressures will
arise. It is, however, possible to induce a backward (forward) shift in the aggregate supply schedule by raising (cutting) the rate of tax levied on wage income. A tax rise (cut) of b
exactly eliminates the pressures on prices and brings Yn in line with Y
. It remains to be discussed under which circumstances such policy action is feasible.
The derivation of this structural equation from the rms pro t-maximizing decision as well as the de nitions of parameter values are given in the Appendix.
1.2 The baseline macroeconomic model 53
A straightforward log-linearization of (1.11) gives us the intertemporal IS rela- tionship, which describes the demand side of the economy (see Appendix for details):
yt = Etyt+1
bit Ett+1 rbt
: (1.19)
bit = log1+1+iit; wherei is the steady state interest rate determined by the rate of time
preference. rb
t represents the deviation in the interest rate that is consistent with
b
Y
t . br
t depends on exogenous real variables only and hence cannot be a¤ected by
government policy.
The constraint (1.17), together with (1.11) and a transversality condition, im- plies the following scal sustainability condition expressed in terms of the gap vari- ables bbt 1 t 1yt+'t = (1 ) [fyyt+f(bt b t)] +Et h bbt t+1 1yt+1 i ; (1.20) in which 't is the scal stress term introduced in Benigno and Woodford (2003) as a composite measure of the consequences for scal solvency of the spending shock. More precisely, consider a situation in which the sticky-price economy had been in its steady state before the shock occurred and the policy response to the shock is given by the pair fbr
T;b
Tg
1
T=t, delivering a sequence of zero output gaps and no ination
for all t. Then 't is a ( rst-order-accurate) measure of the degree to which the scal sustainability requirement of balancing outstanding liabilities against current and future surpluses is not satis ed.
1.2 The baseline macroeconomic model 54
The Appendix explains the steps involved in deriving (1.20) from (1.17), using (1.11). Coe¢cientsfy andf are also de ned in the Appendix.
Finally, the policy maker conducts monetary and scal policy to minimize the quadratic loss function26
Lt =Et 1 X T=t T t 1 2qyy 2 T + 1 2q 2 T : (1.21)
Benigno and Woodford (2003, 2006b) explain the methodological background of de- riving a quadratic function such as (1.21) that is able provide a second-order accurate welfare-ranking of alternate policies when the steady state of the economy is inef- cient, whilst the structural equations, (1.18) and (1.20), together with the initial commitments (to be discussed below), are accurate only up to rst order. Again, the Appendix provides a detailed account on how this loss function follows from the household utility function. Under mild restrictions on the consumption share of na- tional income, and importantly, for realistic parameter values qy > 0 and q > 0;
which implies that the loss function is convex and hence, the second-order condi- tions would nd that the solution identi ed by the rst-order conditions constitutes a minimum.
26 Svensson (2002, 2003) refers to this objective as a general targeting rule, the variables inside as target