Long-Term Debt and Optimal Policy in the Fiscal Theory of the Price Level

The fiscal theory states that “nominal debt / price level = present value of real primary surpluses”; with long-term debt, “the nominal value of the debt on the left-hand side… is not fixed; it depends on nominal bond prices which in turn depend on expected future price levels” (Section 1, pp. 69-70). Cochrane’s motivating example: if bad news lowers the present value of future surpluses, and only short-term debt is outstanding, the price level must rise today to reequilibrate the valuation equation; but if long-term bonds are outstanding, their price can fall instead, leaving today’s price level unaffected – “lower bond prices today correspond to expectations of higher price levels in the future, so long-term debt means that bad news about future surpluses can result in future rather than current inflation” (p. 70).

If the government neither issues new debt nor repurchases outstanding debt before maturity (e.g., it lets a perpetuity’s coupons simply come due), “prices are determined by bonds that fall due at each date divided by that date’s surplus,” and “shocks to future deficits have no influence at all on the current price level” – the entire shock is instead absorbed by long-term bond prices, i.e., by future inflation (Section 3.2, p. 78). Cochrane generalizes this with the “k-period debt” case, in which the government issues only k-period discount bonds each period: the resulting price level solves a k-period difference equation in which “only every kth term matters” – for 5-year debt, only surpluses expected in years 5, 10, 15, etc. affect today’s price level, with k=1 recovering the standard “all future deficits matter” result and k to infinity recovering the “only today’s surplus matters” result (Section 3.3, p. 79).

Yes, but only if long-term debt is outstanding: with only short-term debt, bonds are claims to the same fixed real resources, so new debt sales just lower bond prices one-for-one with no effect on revenue or the current price level (“a unit-elastic demand curve”); with long-term debt outstanding, new issuance dilutes existing long bonds’ claim on that fixed future resource stream, so the government can raise real revenue and lower today’s price level by selling more long-term debt, at the cost of higher inflation later (Section 1, pp. 70-71). Cochrane works a concrete example: selling an extra 10-year bond at time 0 and letting it mature lowers the price level at time 0, leaves it roughly unaffected at intermediate dates if debt is genuinely long-term, and raises it by a full one unit at year 10 when the bond must be redeemed “from the same set of resources” (Section 5, Figure 3, pp. 85-86) – and he notes the identical price path could equally be produced by rolling over a one-period bond for ten years before repaying it, since “all that matters to the price path is when the debt is expected to be repaid.”

Cochrane finds “short maturity structures are preferred when the present value of the surplus varies by less than the surplus itself; while long maturity structures are preferred when surpluses build up following a shock so that the present value varies by more than the surplus itself” (Section 6, pp. 90-92). For a transitory (small, mean-reverting) AR(1) or AR(2) surplus process, short-term debt is optimal because it makes the price level track the already-smooth present value of surpluses; for a hump-shaped, persistently building surplus process, in which the present value moves by more than the surplus itself, the model calls for the longest feasible maturity structure – Cochrane’s calculated optimum stays “never much above” roughly one year of average maturity in his baseline calibration, even in this favorable case (Section 6.2, Figure 5, pp. 91-92).

Active, state-contingent adjustment of long-term debt sales gives the government a second, independent tool for smoothing inflation: it can respond to a negative surplus shock by selling additional long-term debt, “trading a lower price level today for a higher price level in the future,” which postpones and spreads out the inflationary consequences of the shock rather than absorbing them all at once (Section 6.3, pp. 92-93). Cochrane notes this creates a second, distinct motivation for holding long-term debt even in cases where short-term debt would be the optimal fixed maturity structure: “this option gives another motivation for long-term debt, since state-contingent debt sales can postpone a shock to the price level if long-term debt is outstanding.”

Simple fiscal-theory comparative statics predict the real value of government debt should move together with the surplus (both driven by the same present-value relationship), but in U.S. data “high surpluses pay down the debt” – the opposite correlation, which Canzoneri, Cumby, and Diba (1998) used to argue against the fiscal theory under a simple AR(1) surplus process (Section 7, pp. 97-98). Cochrane shows this negative correlation is not evidence against the theory but a natural consequence of how a government facing a recession-driven deficit chooses to respond: it can (1) let inflation erode existing debt, (2) sell more long-term debt with no change in future surpluses (postponing but not reducing the fiscal burden), or (3) sell more debt while promising to raise future surpluses to pay it off. Only the third option – the one governments actually appear to follow, and the one that produces the least inflation volatility of the three – generates the observed pattern in which “a negative surplus shock today is followed by an increased surplus in the future,” and hence real debt rises with deficits and falls with surpluses, “not an accounting identity” but “the government’s choice to do so rather than to finance deficits by inflating away the value of outstanding debt” (Section 7, pp. 98-99).

Cochrane reports the fit is good in direction but “too successful”: his optimal policies produce less inflation volatility than is actually observed in the data, and the specific approximate-solution method used could not pin down much about the optimal state-contingent variation in maturity structure (Section 8, pp. 106-107). He proposes two ways to read this shortfall, both with precedent in the optimal-monetary-policy literature: either the true model needs additional frictions (he specifically points to price stickiness, which would revive an inflation-output tradeoff and give the smoothing objective an explicit welfare-theoretic foundation, following Woodford’s 1998 extension), or some of the actually observed inflation volatility reflects policy mistakes rather than the outcome of a genuinely optimal smoothing policy.

Per the abstract, shifting the objective from price-level variance to inflation variance produces a much less volatile inflation series, but at the cost of introducing a unit root into the price level – i.e., price-level shocks are no longer mean-reverting (Abstract, p. 69). Cochrane connects this tradeoff to observed monetary history: minimizing price-level variance is “a plausible characterization of monetary policy objectives in the prewar, gold-standard regime” (Section 6.1, p. 90), in which the price level was expected to return to a fixed long-run anchor, whereas the much lower inflation volatility (paired with a non-mean-reverting price level) that the alternative objective produces is “consistent with the stabilization of U.S. inflation after the gold standard was abandoned,” when policy shifted toward stabilizing the rate of inflation rather than the level of prices.

Cochrane restates, briefly, the stock-market analogy that anchors his broader defense of the fiscal theory: the basic valuation equation “is, like [a] stock example… an equilibrium valuation equation, not a constraint,” since nothing forces Microsoft to raise future earnings just because its stock price rises, and nothing forces the government to raise future taxes in response to an “off-equilibrium” deflation (Section 1, “A Few Comments on the Fiscal Theory,” pp. 73-75). He extends this to nominal debt and surpluses as separate policy instruments even in a fully cashless economy – the government can contemplate doubling outstanding debt, know that prices will double in response, and choose to do so anyway, “just as Microsoft knows that its share price will halve if it does a split” – explicitly distinguishing this deliberate policy choice (a rare, currency-reform-like event) from the far more typical case, central to this paper’s whole analysis, in which additional debt sales come paired with an implicit or explicit promise of higher future surpluses.