The object of our analysis is a monetary dynamic general equilibrium model econ-omy which is made up of a government sector and a private sector. As in Lucas and Stokey (1983) there is no capital. The government sector consists of a monetary authority and a fiscal authority who take their decisions independently. The fiscal authority collects consumption taxes2 τtc in order to finance an exogenously given stream of public expenditures gt. For the time being, we let gt = g be determin-istic and constant over time. The policy instrument controlled by the monetary authority is the supply of money Mt+1g (the superscript g is used to distinguish an aggregate variable from an individual variable where necessary), whereby seignorage revenues from money creation Mt+1g − Mtg are used to purchase g from the private households. Hence, the two government authorities interact via a consolidated gov-ernment budget constraint, but decision power remains decentralized among the two independent institutions. Finally, we assume that the fiscal authority, besides its tax policy, issues nominal one-period bonds Bt+1g , whereby the quantity of bonds traded is determined by the following flow budget constraint for the government sector which has to be satisfied for all t ≥ 0:
Mt+1g + Bt+1g + Ptτtcct ≥ Mtg + Bgt(1 + Rt) + Ptg (1.1) Here, Pt is the price level prevailing at time t, while Rt is the nominal interest rate paid on the bonds issued at date t − 1. The initial stock of money M0g and the initial debt liabilities Bg0(1 + R0) are given. However, we will impose the additional consis-tency condition that, in the rational expectations equilibrium, there is no surprise inflation in the initial (t = 0) period; thus, by linking the nominal interest rate R0to the equilibrium rate of inflation in the first period, we prevent the authorities from taking advantage of the inelasticity of the amount of oustanding nominal balances M0 and B0 in the initial period.
On the private side, the economy is inhabited by a continuum of identical infinitely-lived households whose preferences over sequences of consumption ct and
2An alternative specification would be to use labor taxation, τtn. However, against the back-ground of the general equivalence results for different forms of taxation, the choice of the tax instrument assigned to the fiscal authority is irrelevant so that our specification is without loss of generality.
labor nt can be represented by the following additively-separable expression:
∞
X
t=0
βt{u(ct) − v(nt)} , (1.2)
where the discount factor β satisfies 0 < β < 1. In what follows, we will assume u(ct) = log(ct) and v(nt) = αnt.3 Each consumer faces the following sequence of budget constraints:
Mt+1+ Bt+1 ≤ Mt− Pt(1 + τtc)ct+ Bt(1 + Rt) + Wtnt, (1.3) where Wtis the nominal wage and Bt+1 and Mt+1are nominal government debt and nominal money balances taken over from period t to period t + 1. We assume that each consumer faces a no-Ponzi condition that prevents him from running explosive consumption/debt schemes:
limT →∞βTBT +1
PT ≥ 0 (1.4)
As a shortcut for introducing a well-defined money demand we additionally as-sume that the gross-of-tax consumption expenditure in period t must be financed using currency carried over from period t − 1, which implies the following cash-in-advance (CIA) constraint:
Mt≥ Pt(1 + τtc)ct (1.5)
The timing structure underlying this CIA constraint follows Svensson (1985). Specif-ically, we assume that newly injected money transfers are not available for purchasing private consumption until the next period. Consequently, the purchases of goods have to be undertaken before nominal balances can be reshuffled optimally, and the CIA constraint may not allow to realize the desired consumption. Moreover, the information about the money injection leads to an immediate price reaction. Hence, the effects of inflation are twofold: First, expected inflation leads to a distortion via its effect on the nominal interest rate. Second, surprise inflation is distortionary since the households are constrained in their consumption decisions by the value of
3The assumption of linear disutility of labor is made to sharpen the discussion, but implies also that the government sector cannot affect the real interest rate. Conversely, the assumption of log utility from consumption allows to focus on the role of nominal debt as a source of time inconsistency rather than on the effects due to private holdings of nominal money balances. That is, we abstract from seignorage on base money and focus on the implications of changing the real value of nominal debt. Importantly, this focus is consistent with the situation in most developed economies where government debt is arguably more important than money holdings as a source of dynamically inconsistent incentives. See also Nicolini (1998) for an instructive exposition of the nature of the time inconsistency of monetary policy and Martin (2006) for results with a more general specification of preferences.
the money balances taken over from the previous period.4 Importantly, therefore, monetary policy has wealth effects and is not neutral. For a benevolent monetary authority this means that it trades off the reduction in the households’ current utility and the increase in future utility which results from the reduction in future distortions when it considers whether or not to carry out a surprise inflation.
The productive side of the model economy is extremely simple since there is no capital. In each period, labor nt can be transformed into private consumption ct or public consumption g at a constant rate, which we assume to be one. Then, the equilibrium real wage is wt ≡ WPt
t = 1 for all t ≥ 0, and aggregate feasibility is reflected by the following linear resource constraint:
ct+ g ≤ nt (1.6)
We are now ready to define a competitive equilibrium for given government policy choices {τtc, Mt+1g , Bt+1g , g}∞t=0.
Definition 1.1 A competitive equilibrium for this economy is composed of the gov-ernment sector’s policies {τtc, Mt+1g , Bt+1g , g}∞t=0, an allocation {ct, nt, Bt+1, Mt+1}∞t=0, and prices {Rt+1, Pt}∞t=0 such that:
1. given B0g(1 + R0) and M0g, the policies and the prices satisfy the sequence of budget constraints of the government sector described in expression (1.1);
2. when households take B0(1 + R0), M0 and prices as given, the allocation solves the household problem of maximizing (1.2) subject to the private budget con-straint (1.3), the CIA concon-straint (1.5) and the no-Ponzi condition (1.4);
3. markets clear, i.e.: Btg = Bt, Mtg = Mt, and g and the allocation satisfy the economy’s resource constraint (1.6) for all t ≥ 0.
On the basis of our assumptions on household preferences it is straightforward to show that in the competitive equilibrium allocation of this economy the household budget constraint (1.3) and the aggregate resource constraint (1.6) are both satisfied at equality. Moreover, the first order conditions of the Lagrangean representing the household’s constrained optimization problem are both necessary and sufficient conditions to characterize the solution to the household problem. Finally, when Rt+1 > 0, the CIA constraint (1.5) is binding, and the competitive equilibrium allocation for given government policies can be determined from the government budget constraint (1.1), the aggregate resource constraint (1.6) and the following conditions that must hold for all t ≥ 0:
Mt = Pt(1 + τtc)ct (1.7)
4This second effect is not present in a CIA economy with the alternative timing structure underlying the model of Lucas and Stokey (1983) where monetary surprises are lump-sum.
u0(ct)
Since our focus is on an environment where there is no explicit commitment technol-ogy, we now seek to find a time consistent policy rule which is sequentially optimal from the two authorities’ perspectives. The sequential decisions about policy are decentralized. In each period, the two authorities move simultaneously and take their respective counterpart’s policy choice as well as their successors’ future policy mappings as given. We define a policy rule to be the combination of a fiscal and a monetary policy rule; each of these latter rules is determined independently by the subsequent incarnations of the respective authority. We limit the analysis to Markov-stationary policy rules, where a policy rule is a mapping that returns pol-icy choices as a time-invariant function of the current (payoff-relevant) state of the economy. We denote the policy function by ϕ(zg) = (ϕf(zg), ϕm(zg)), where ϕf(zg) and ϕm(zg) are the fiscal and monetary parts of the rule which give the respective policy instruments τc and M0g as functions of the aggregate state zg ≡ BgM(1+R)g , the government sector’s real debt burden inherited from the past.5 In order to identify the equilibrium policy rule, we therefore need to find the optimal time-invariant strategies in the strategic game between the two authorities.
The policy problem in the present economy can be described as an infinite-horizon dynamic game of almost perfect information whose build-ing block is a two-player6 simultaneous-moves stage game G(ztg; ϕ) =
f, m; Af(ztg), Am(ztg), ˆV (ztg; π, ϕ), ˆW (zgt; π, ϕ)
, where zgt is the payoff-relevant state variable and π and ϕ denote arbitrary policy rules in place in the current period (π) and from the next period onwards (ϕ). We are now going to define the components of the stage game.
As already hinted above, the game is not of a repeated variety due to the pres-ence of the endogenous state variable ztg ≡ Bgt(1+RMg t)
t which can be manipulated over time and is informative about (i) the composition of the nominal claims with which
5For most of what follows, we will switch to recursive notation where primes denote variables pertaining to the next time period.
6To be precise, we deal with a three-player game where the two policy authorities play in a Stackelberg relation towards the third player, the continuum of private agents. Although an individual household’s choices have no strategic weight, their presence - via their formation of expectations - is key to the construction of an equilibrium.