The tax smoothing models of Barro [6], Aiyagari et.al. [2], and Lucas and Stokey [19] explain how issuing risk-free or state-contingent government debt allows a benevolent government to finance an exogenous process for government expenditures by levying flat rate taxes across time and states in the least distorting ways. By assuming that all taxes are lump sum and that government purchases of durable and nondurable public expenditures are chosen each period by majorities, we focus on a different reason for government debt: to realign the interests of majorities of voters in ways that will induce them to choose more efficient allocations of public and private goods. By studying the effects of our two-parameter class of constitutions that allow majorities to issue debt to finance only durable public goods, we thus offer a reason for government debt that is complementary to the one studied by Barro and Lucas and Stokey. Our model provides a framework for evaluating proposals, for example, to substitute a golden rule for the maximum percentage of GDP deficit rule stated in the Maastricht Treaty.
37Chari and Miller [13] rely on these models in their informal discussion to advocate the adoption of the golden rule at the federal level.
38We view extensions of Rogoff and Sibert [31] and Rogoff [30] as particularly good candidates because they imply a tendency to choose projects with immediate benefits over projects with delayed benefits.
39Among these, the most natural extension of models of partisan politics such as Alesina and Tabellini [3],
Persson and Svensson [23], or Tabellini and Alesina [38] would generate overspending in both types of goods, unless government capital is perceived as a less partisan good than nondurable consumption, as assumed by Peletier, Dur and Swank [22] and by Azzimonti-Renzo [4].
Appendix
A Proofs
A.1 Proof of Proposition 1 (completion)
Our proof works by construction. We proved earlier that all competitive equilibria share the same values of pt, Kt, wt, and rt, t ≥ 0, and that these are all constant and independent of {Gt, Γt}∞t=0. This implies that, for any history ht, there is a unique way to set p(ht), K(ht), w(ht), and r(ht). We set investment to the only values that are consistent with (5) and (6):
i(ht) =K(ht)− (1 − δk)K(ht−1) and γ(ht) = Γt− (1 − δΓ)Γt−1. For any non-null history, we set B recursively:
B(ht) = (1− α)B(ht−1)
1 +n +xγ(ht)
p(ht) . (28)
We setτ(ht) according to (21), and aggregate consumption such that
∞ s=0
λscs(ht) =F
K(ht−1) 1 +n , 1
− i(ht)− Gt− γ(ht). (29)
Subject to (29), individual consumption is not uniquely pinned down.
To construct individual consumption, we partition the set of histories into sequences{hv}∞v=t, t ≥ −1, with the property that, within elements of the same partition, hv = (hv−1, G(hv−1), Γ(hv−1)), while for each initial element of a sequence eithert = −1 or ht= (ht−1, G(ht−1), Γ(ht−1)). Within each partition, a history represents a path that is induced by the functions G(.) and Γ(.) from the history ht. Along each of these paths, we set the functions cs(.), ks(.), and bs(.) arbitrarily, subject to the following restrictions:
(i) Given the taxes and prices specified above, the resulting sequences must satisfy the house-hold budget constraints (13) for an economy that starts in period t with initial conditions specified by (ks(ht−1), bs(ht−1))N−1s=0 ;
(ii) if N = ∞, the implied asset allocation must be essentially bounded from below;
(iii) the implied real allocation must be bounded;
(iv) individual consumption schedules are such that (29) holds in the aggregate.
Examples of functions that satisfy these requirements are the following:40 ks(hv) =K(hv)/(1 − λN), bs(hv) =B(hv)/(1 − λN), and
By iterating equation (28) forward and using the facts that p(hv) and Γ(hv) are constant over time, we can show thatB(hv) is bounded, which implies that the consumption plans of the households are bounded.41
It is now straightforward to see that the mapping that we have just constructed generates a competitive equilibrium starting from any history hj, given the levels of public spending and capital implied by (24). Starting from any initial capital level, we previously showed that the values assigned to p(ht), K(ht), w(ht) and r(ht), t ≥ j, are the unique choices that ensure that the household problem has a (bounded) solution, that factor prices equal marginal products, and that (17) holds. With these choices, households are indifferent among all possible consumption profiles. Given the previously determined variables and (21), equation (28) describes the unique
40IfN = ∞, λN should be set to 0, and (33) and (34) do not apply.
41IfN = +∞, the proposed asset allocation also satisfies the no-Ponzi games restriction and the transversality condition (19).
sequence of government debt that satisfies the government budget constraint and (19), and (29) describes the unique sequence of aggregate consumption that is consistent with the resource constraint (4). The conditions on the consumption of individual households ensure that the individual budget constraints are met. Whenever the future levels of G and Γ depend only on the future shocks and not on any history, the mapping constructed above implies that the current choices of G and Γ do not affect private capital or factor prices in any period, nor do they affect subsequent values of public spending and capital. These are precisely the conditions under which households will vote forG and Γ according to (24), which implies that the mapping satisfies condition (ii) of a political-economic equilibrium. Finally, condition (iii) is met because it was imposed in the construction of the mapping.
To check that all Markov equilibria lead to the same ex ante welfare for all generations, notice that prices, taxes, and the provision of the public goods are the same in all Markov equilibria and in all histories. As a consequence, the individual household’s optimization problem is identical in all of the equilibria, and the multiplicity of the Markov equilibria comes only from the fact that households are indifferent among many equivalent optimal consumption plans. QED.
A.2 Proposition 2
Proposition 2 Generically in the death process, there exists a value x∗ such that, if x = x∗, the allocation induced by any Markov equilibrium is Pareto optimal. A sufficient condition for x∗ to be unique is that Qs is decreasing in x for all ages.
We already observed that in a competitive equilibrium private capital and investment are equal to their (unique) value implied by any Pareto-optimal allocation. We also proved that the balanced-budget restriction onGt ensures that (9) is satisfied. We need only to prove that there exists a value x∗ such that the solution to (24) satisfies (10). First, note that Qs is linear in x.
Matching the first-order condition of (24) with respect to Γ to (10) requires
median(Q0s+Q1sx) = 1 − β(1 − δΓ) (35)
where
For a generic death process, we have median(Q1s)= 0. In this case, the left-hand side of equation (35) is a continuous function of x, and it diverges to infinity as x diverges, but with opposite signs as x → −∞ or x → +∞. This implies that a solution to equation (35) exists. If Q1s < 0 for all ages, the median cost strictly decreases withx and the solution is unique. QED.