Debt and incomplete financial markets: a case for nominal GDP targeting.

Sheedy, Kevin D.

ABSTRACT For many households borrowing is possible only by accepting a financial contract that specifies a fixed repayment stream. However, the future income that will repay this debt is uncertain, so risk can be inefficiently distributed. This paper shows that when debt contracts are written in terms of money, a monetary policy of nominal GDP targeting improves the functioning of financial markets. By insulating households' nominal incomes from aggregate real shocks, this policy effectively achieves risk sharing by stabilizing the ratio of debt to income. The paper also shows that when there is price stickiness, the objective of improving risk sharing should still receive considerable weight in the conduct of monetary policy relative to stabilizing inflation.

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At the heart of any argument for a monetary policy strategy lies a view of what are the most important frictions or market failures that monetary policy should seek to mitigate. The canonical justification for inflation targeting as optimal monetary policy rests on the argument that pricing frictions in goods markets are of particular concern (see, for example, Woodford 2003). With infrequent price adjustment owing to menu costs or other nominal rigidities, high or volatile inflation leads to relative price distortions that impair the efficient operation of markets and that directly consume time and resources in the process of setting prices. Inflation targeting is the appropriate policy response to such frictions, because it is able to move the economy closer to, or even replicate, what the equilibrium would be if prices were flexible. In other words, inflation targeting is able to undo or partially circumvent the frictions created by nominal price stickiness. (1)

This paper argues that nominal price stickiness might not be the most serious friction that monetary policy has to contend with. While the use of money as a unit of account in setting infrequently adjusted goods prices is well documented, money's role as a unit of account in writing financial contracts is equally pervasive. Moreover, just as price stickiness means that nominal prices fail to be fully state contingent, financial contracts are typically not contingent on all possible future events, an example being debt contracts that specify a fixed stream of nominal repayments.

The problem that noncontingent debt contracts raise for risk-averse households is that when they are borrowing for long periods, there will be considerable uncertainty about the future income from which the fixed debt repayments must be made. The issue is not only one of idiosyncratic uncertainty--households also do not know the future course the economy will take, which will affect their labor income. Will there be a productivity slowdown, a deep and long-lasting recession, or even a "lost decade" of poor economic performance to come? Or will unforeseen technological developments or terms-of-trade movements boost future incomes and good economic management successfully steer the economy on a path of steady growth? Borrowers do not know what aggregate shocks are to come, but they must commit to a stream of contractual repayments prior to this information being revealed.

The inability to use more complex financial contracts that make repayments contingent on any future events means that financial markets are incomplete, in the sense that households do not have access to insurance against future risks that could affect their ability to repay debt. Financial contracts might not be fully contingent for a variety of reasons, but one explanation could be that transaction costs make it prohibitively expensive to write and enforce complicated and lengthy contracts. Since many agents, such as households, would find it difficult to issue liabilities with state-contingent repayments resembling equity or derivatives, they must instead rely on simple debt contracts if they are to borrow. Thus, in a way that is similar to the way menu costs can make prices sticky, transaction costs can render financial markets incomplete.

This paper studies the implications for optimal monetary policy of such financial-market incompleteness in the form of nominal noncontingent debt contracts. The argument can be understood in terms of which monetary policy strategy is able to undo or mitigate the adverse consequences of financial-market incompleteness, just as inflation targeting can be understood as a means of circumventing the problem of nominal price stickiness. For both noncontingent nominal financial contracts and nominal price stickiness, it is money's role as a unit of account that is crucial, and in both cases, optimal monetary policy is essentially the choice of a particular nominal anchor that makes money best perform its unit-of-account function. But in spite of this formal similarity, the optimal nominal anchor turns out to be very different when the friction is noncontingent debt contracts rather than sticky prices.

The simplicity of noncontingent debt contracts can be seen as coming at the price of bundling together two fundamentally different transfers: a transfer of consumption from the future to the present for borrowers, but also a transfer of aggregate risk to borrowers. The future consumption of borrowers is paid for from the difference between their uncertain future incomes and their stream of fixed debt repayments. The more debt they have, the more their future income is effectively leveraged, leading to greater consumption risk. The flip side of borrowers' leverage is that savers are able to hold a risk-free asset, reducing their consumption risk.

To see the sense in which this bundling together of borrowing and a transfer of risk is inefficient, consider what would happen in complete financial markets. Individuals would buy or sell state-contingent bonds (Arrow-Debreu securities) that make payoffs conditional on particular events (or, equivalently, write loan contracts with different repayments depending on what happens in the future). Risk-averse borrowers would want to sell relatively few bonds paying off in future states of the world where GDP and thus incomes are low, and sell relatively more that would pay off in good states of the world. As a result, contingent bonds paying off in bad states would be relatively expensive and those paying off in good states relatively cheap. These price differences would entice savers to shift away from noncontingent bonds and take on more risk in their portfolios. Given that the economy has no risk-free technology for transferring goods over time, and since aggregate risk cannot be diversified away, the efficient outcome is for risk-averse households to share aggregate risk. Complete financial markets allow this risk sharing to be unbundled from decisions about how much to borrow or save.

The efficient financial contract between risk-averse borrowers and savers in an economy subject to aggregate income risk (abstracting from idiosyncratic risk) turns out to closely resemble an "equity share" in GDP. In other words, borrowers' repayments should move in line with GDP, falling during recessions and rising during booms. This means the ratio of debt liabilities to GDP should be more stable than it would be in a world where nominal debt liabilities are fixed in value while GDP fluctuates.

With noncontingent debt contracts, monetary policy has a role to play in promoting risk sharing, because these contracts are typically denominated in terms of money. Hence, the real degree of state contingency in financial contracts depends on the conduct of monetary policy. If the incompleteness of financial markets were the only source of inefficiency in the economy, then the optimal monetary policy would aim to make nominally noncontingent debt contracts mimic--through variation in the real value of the monetary unit of account--the efficient financial contract that would be chosen with complete financial markets.

Since the efficient financial contract between borrowers and savers resembles an equity share in GDP, it follows that a goal of monetary policy should be to stabilize the ratio of debt liabilities to GDP. With noncontingent nominal debt liabilities, this can be achieved by having a noncontingent level of nominal income, in other words, a monetary policy that targets nominal GDP. Nominal income thus replaces nominal goods prices as the optimal nominal anchor. While the central bank cannot eliminate uncertainty about future real GDP, it can in principle make the level of future nominal GDP (and hence the nominal income of an average household) perfectly predictable. Removing uncertainty about future nominal income thus alleviates the problem of the stream of nominal debt repayments being fixed.

A policy of nominal GDP targeting is generally in conflict with inflation targeting, because any fluctuations in real GDP would lead to fluctuations in inflation of the same size and in the opposite direction. Recessions would feature higher inflation and booms would feature lower inflation or even deflation. These inflation fluctuations could be helpful because they would induce variation in the real value of the monetary unit of account, making it and the noncontingent debt contracts expressed in terms of it behave more like equity. This would promote efficient risk sharing. A policy of strict inflation targeting would fix the real value of the monetary unit of account, converting nominally noncontingent debt into real noncontingent debt, which would imply an uneven and generally inefficient distribution of risk.

The inflation fluctuations that would occur with nominal GDP targeting would entail relative-price distortions if goods prices were sticky, so the benefit of risk sharing would probably not be achieved without some cost. Whether nominal GDP targeting is preferable to inflation targeting is ultimately a quantitative question, whether the inefficiency due to the suboptimal risk sharing of noncontingent debt contracts is more important than the inefficiency caused by relative-price distortions. Using a calibrated model with both nominal debt contracts and sticky prices, optimal monetary policy is found to place a weight of approximately 90 percent on promoting risk sharing and 10 percent on stabilizing inflation.

This paper is related to a number of areas of the literature on monetary policy and financial markets. First, there is the empirical work of Bach and Stephenson (1974), Cukierman, Lennan, and Papadia (1985), and more recently, Doepke and Schneider (2006), who all document the effects of inflation in redistributing wealth between debtors and creditors. The novelty here is in studying the implications for optimal monetary policy in an environment where inflation fluctuations with such distributional effects may actually be desirable precisely because of the incompleteness of financial markets.

The basic idea of this paper has many precedents in the history of monetary economics, arguably extending back at least to Bailey (1837) (a survey of the literature is given in Selgin 1995). The contribution here is in the modeling and the quantitative analysis of optimal monetary policy with nominal debt contracts, not in the fundamental ideas. In the modern literature on this question, Selgin (1997) describes the ex-ante efficiency advantages of falling prices in good times and rising prices in bad times when financial contracts are noncontingent, although there is no formal modeling of the argument. (2)

Pescatori (2007) studies optimal monetary policy in an economy with rich and poor households, in the sense of there being an exogenously specified distribution of assets among otherwise identical households. In that environment, both inflation and interest rate fluctuations have redistributional effects on rich and poor households, and the central bank optimally chooses the mix between them. A related paper is that of Lee (2010), who develops a model where heterogeneous households choose less than complete consumption insurance because of the presence of convex transaction costs in accessing financial markets. Inflation fluctuations expose households to idiosyncratic labor-income risk, because households work in specific sectors of the economy, and sectoral relative prices are distorted by inflation when prices are sticky. This leads optimal monetary policy to put more weight on stabilizing inflation. In contrast to those two papers, the argument here is that inflation fluctuations can actually play a positive role in completing otherwise incomplete financial markets.

Most recently, and closest in approach to this paper, Koenig (2013) has advanced the risk-sharing argument for nominal GDP targeting in the context of a two-period model and has also studied the robustness of the results to the possibility of default when there are bankruptcy costs. In comparison, the model presented here is more suited to quantitative monetary policy analysis, because it brings together incomplete financial markets and all the features of the workhorse New Keynesian model in a tractable framework.

While the papers above focus on nominal debt contracts in the context of household borrowing, the idea that inflation fluctuations may have a positive role to play when financial markets are incomplete is now long established in the literature on government debt (and has also been recently applied by Allen, Carletti, and Gale 12011] in the context of the real value of the liquidity available to the banking system). Bohn (1988) developed the theory that noncontingent nominal government debt can be desirable because when combined with a suitable monetary policy, inflation can change the real value of the debt in response to fiscal shocks that would otherwise require fluctuations in distortionary tax rates. (3)

Quantitative analysis of optimal monetary policy of this kind was developed by Chari, Christiano, and Kehoe (1991) and expanded further by Chari and Kehoe (1999). One finding was that inflation needs to be extremely volatile to complete financial markets. As a result, both Schmitt-Grohe and Uribe (2004) and Siu (2004) argued that once some nominal price rigidity is considered so that inflation fluctuations have a cost, the optimal policy becomes very close to strict inflation targeting. This paper shares the focus of that literature, using inflation fluctuations to complete markets, but comes to a different conclusion regarding the magnitude of the required inflation fluctuations and whether the costs of those fluctuations outweigh the benefits. First, in this paper the benefits of completing markets are linked to the degree of risk aversion and the degree of heterogeneity among households, which are generally unrelated to the benefits of avoiding fluctuations in distortionary tax rates and which prove to be large in the calibrated model. Second, the earlier results assumed government debt with a very short maturity. With longer-maturity debt (in this paper, household debt), the costs of the inflation fluctuations needed to complete financial markets are much reduced. (4)

There is also a literature on household debt that emphasizes alternative frictions, such as credit constraints and interest-rate spreads (Iacoviello 2005; Curdia and Woodford 2009; Guerrieri and Lorenzoni 2011; Eggertsson and Krugman 2012). Finally, the paper is related to the literature on nominal GDP targeting (Meade 1978; Bean 1983; Hall and Mankiw 1994, and more recently, Sumner 2012) but proposes a different argument in favor of that policy.

The plan of the paper is as follows. Section I sets out a basic model and identifies which monetary policies can support risk sharing when financial markets are incomplete. Section II introduces a dynamic stochastic general equilibrium (DSGE) model that includes both incomplete financial markets and sticky prices, and hence a trade-off between mitigating the incompleteness of financial markets and avoiding relative-price distortions. Optimal monetary policy subject to this trade-off is studied in section III. Section IV shows how the full model can be calibrated and presents a quantitative analysis of optimal monetary policy. Finally, section V draws some conclusions.

I. A Model of a Pure Credit Economy

The analysis begins with a simplified model that studies household borrowing and saving in a finite-horizon endowment economy with incomplete financial markets. A full DSGE model with incomplete markets together with labor supply, production, and sticky prices is presented in section II.

I.A. Assumptions

The economy contains two groups of households, "borrowers" (b) and "savers" (s), each making up 50 percent of a measure-one population. Household types are indexed by [tau] [member of] {b, s}. There are three time periods t [member of] {0, 1, 2}. All households have preferences represented by the utility function:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sub.[tau],t] is per-person consumption by households of type [tau] at time t. All households have the same subjective discount factor [delta] and the same coefficient of relative risk aversion [alpha]. Real GDP [Y.sub.t] is an exogenous endowment. The level of GDP in period 0 is nonstochastic, but subsequent real GDP growth rates [g.sub.t] = ([Y.sub.t] - [T.sub.t-1])/[Y.sub.t-1] are uncertain. Household types are distinguished by the shares they receive of this endowment at different dates. The ratio of borrowers' per-person incomes to per-person real GDP is denoted by the parameter [[psi].sub.t] and hence household incomes at time t are:

(2) [Y.sub.b,t] = [[psi].sub.t][Y.sub.t] and [Y.sub.s,t] = (2 - [[psi].sub.t])[Y.sub.t].

The income shares [[psi].sub.t] are known with certainty in period 0. Given that both household types have the same time preferences, the households labeled as borrowers will indeed choose to borrow from the savers in equilibrium when the sequence {[[psi].sub.0], [[psi].sub.1],, [[psi].sub.2]} is increasing. In other words, borrowers are those households with initially low incomes relative to savers, while savers anticipate having a relatively low income in the future. In what follows, the analysis is simplified by assuming the particular monotonie sequence of income shares below:

(3) [[psi].sub.0] = 1 [[delta].sup.2]E [[(1 +[g.sub.1]).sup.1-[alpha]] [(1+[g.sub.2]).sup.1-[alpha]], [[psi].sub.1] and [[psi].sub.2] = 2

It is assumed that the subjective discount factor [delta] is sufficiently low relative to expected real GDP growth so that [[psi].sub.0] [greater than or equal to] 0.

Financial markets are incomplete in the sense that only nominal bonds can be issued or held by households. It is assumed that borrowers cannot issue liabilities with state-contingent nominal payoffs, and all non-contingent bonds are denominated in terms of money. It is assumed that a single type of bond is traded at each date. Each bond issued in period 0 is a promise to repay one unit of money in period 1, and [gamma] units of money in period 2. The parameter y determines the duration, or average maturity, of the bonds ([gamma] = 0 is one-period debt, while larger values of [gamma] represent longer-term debt contracts). Each bond issued in period 1 is simply a promise to repay one unit of money in period 2. Note that in period 1, an outstanding bond from period 0 is equivalent to [gamma] newly issued bonds (old bonds are therefore counted in terms of new-bond equivalents from period 1 onwards). Households can take positive or negative positions in bonds (save or borrow) with no limit on borrowing except being able to repay in all states of the world. There is no default, and so all bonds are risk-free in nominal terms.

Households begin with no initial assets or debts and must leave no debts at the end of period 2. The net bond position per person of type-[tau] households at the end of period t is denoted by [B.sub.[tau],l] the nominal bond price is [Q.sub.t] at date t, and the price of goods in terms of money is [P.sub.t] The flow budget identities are:

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with the term (1 + [gamma][Q.sub.1]) being the sum of the period-1 coupon payment from period-0 bonds and the market value of the period-2 repayment, the latter equivalent in value to [gamma] newly issued bonds. Money in this economy is simply a unit of account used in writing financial contracts. Monetary policy is assumed to determine the inflation rate [[pi].sub.t] = ([P.sub.t] - [P.sub.t-1])/[P.sub.t-1] at each date.

I.B. Equilibrium

Maximizing the utility function (equation 1) subject to the budget identities (equation 4) implies Euler equations that must hold for both household types [tau] [member of] {b, s} at dates t [member of] {0, 1}:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In equilibrium, the goods and bond markets must clear at all dates:

(6) [C.sub.b,t]/2 + [C.sub.s,t]/2 = [Y.sub.t];

(7) [B.sub.b,t]/2 + [B.sub.s,t]/2 = 0

In what follows, let [c.sub.[tau],l] [equivalent to] [C.sub.[tau],t]/[Y.sub.t], denote the ratios of consumption to GDP, and define the variables [d.sub.t], [l.sub.t], [r.sub.t] and [[rho].sub.t] as follows:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As will be confirmed in equilibrium, [B.sub.b,t] [less than or equal to] 0 and [B.sub.s,t] [greater than or equal to] 0, so [d.sub.t] can be interpreted as the gross debt-to-GDP ratio (the beginning-of-period value of debt liabilities per person relative to GDP), and [l.sub.t] as the end-of-period value of all bonds issued per person relative to GDP, referred to as the loans-to-GDP ratio. The variable [r.sub.t] is the ex-post real return on holding bonds between periods t - 1 and t. Note that this is not the same as the interest rate on those bonds, which refers to the ex-ante expected real return [[rho].sub.t]. Finally, it is convenient to express the equations in terms of the yield-to-maturity, denoted by [j.sub.t] rather than the bond price [Q.sub.t]. Given the coupon payments on the bonds issued in periods 0 and 1, the price-yield relationships are:

(9) [Q.sub.o] = 1/1 + [j.sub.0] + [gamma]/[(1 + [j.sub.0]).sup.2] and [Q.sub.1] = 1/1 + [j.sub.1].

With the definitions (equation 8), the equations of the model are:

(10) [d.sub.t] = (1 + [r.sub.1]/ 1 + [g.sub.t])[l.sub.t-1];

(11) [c.sub.b,t] = [[psi].sub.t] - 2([d.sub.t] - [l.sub.t]), [c.sub.s,t] = (2-[[psi].sub.t]) + 2([d.sub.t]-[l.sub.t])

with [d.sub.0] and [l.sub.2] = 0 and

(12) [c.sup.-[alpha].sub.[tau],t] = [delta][E.sub.t] [(1 + [r.sub.t+1])[(1+[g.sub.t+1]).sup.-[alpha]] [C.sup.- [alpha].sub.[tau],t+1]],

where equation 10 follows directly from equation 8, equation 11 is derived from the budget identities (equation 4) and the market-clearing condition (equation 7), and equation 12 is derived from the Euler equations (5). Finally, equation 8 and the definition of the yield-to-maturity in equation 9 imply:

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is assumed that the parameter restriction (equation 3) always holds in what follows.

In the case where there is no uncertainty about the path of real GDP ([g.sub.1] = [bar.[g.sub.1]] and [g.sub.2] = [bar.[g.sub.2]], where g, and g2 are nonstochastic), and where there are no unexpected changes in inflation ([[pi].sub.1], = [bar.[[pi].sub.1]] and [[pi].sub.2] = [bar.[[pi].sub.2]). the system of equations 10-12 and equation 13 has the following solution:

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(15) [bar.[d.sub.1]] = [delta]/2 [(1 + [bar.[g.sub.2]]).sup.1-[alpha]], and [bar.[d.sub.2]] = 1/2; and

(16) [bar.[l.sub.0]] = [[delta].sup.2]/2 [(1 + [bar.[g.sub.1]).sup.-[alpha]][(1 + [bar.[g.sub.2]).sup.-[alpha]] and [bar.[l.sub.1]] [delta]/2 [(1 + [bar.[g.sub.2]).sup.-[alpha]]

The equilibrium interest rates (equal here to the ex-post real rates of return [bar.[r.sub.1]] and [bar.[r.sub.2]) are identical to what would prevail if there were a representative household. The choice of income shares in equation 3 means that in the absence of shocks, borrowers and savers would have the same levels of consumption. Given the levels of income and consumption, the implied final debt-to-GDP ratio is 50 percent, and given equation 3, the debt-to-GDP and loans-to-GDP ratios at earlier dates are discounted values of the final debt-to-GDP ratio (adjusted for any real GDP growth). This steady state is independent of monetary policy. The values of the inflation rates [bar.[[pi].sub.1]], and [bar.[[pi].sub.2]] together with the real interest rates [bar.[r.sub.1]], and [bar.[r.sub.2]] determine the nominal bond yields [bar.[j.sub.0]] and [bar.[j.sub.1]].

I.C. The Complete Financial Markets Benchmark

Consider a hypothetical economy that has complete financial markets but which is otherwise identical to the economy described above. Households now have access to a complete set of state-contingent bonds (traded sequentially, period-by-period), denominated in real terms without loss of generality. Let [F.sup.*.sub.[tau]t+1] denote the contingent bonds per person held between periods t and t + 1 by households of type [tau] (the asterisk signifies complete financial markets). The prices of these securities in real terms relative to the conditional probabilities of the states at time t are denoted by [K.sub.t+1], so [E.sub.t][K.sub.t+1] [F.sup.*.sub.[tau]t+1] is the date-r cost of the date-t + 1 payoff [F.sup.*.sub.[tau]t+1].

In this version of the model, the flow budget identities (equation 4) are replaced by

(17) [C.sup.*.sub.[tau],l] + [E.sub.t] [[K.sub.t+1][F.sup.*.sub.[tau],t+1]] = [Y.sub.[tau],t] + [F.sup.*.sub.[tau],t]

together with initial and terminal conditions [F.sup.*.sub.[tau],0] = 0 and [F.sup.*.sub.[tau],3] = 0. The Euler equations for maximizing utility (equation 1) subject to equation 17 are:

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which hold in all states of the world. The market-clearing condition [F.sup.*.sub.b,t]/2 + [F.sup.*.sub.[tau]2]/2 = 0 replaces equation 7.

To relate the economy with complete markets to its incomplete-markets equivalent, consider the following definitions of variables [d.sup.*.sub.t] [l.sup.*.sub.t] and [r.sup.*.sub.t] which will be seen to be the equivalents of the debt-to-GDP ratio [d.sub.t] the loans-to-GDP ratio [l.sub.t] and the ex-post real return [r.sub.t] in the incomplete-markets economy (as given in equation 8):

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Debt in an economy with complete financial markets refers to the total gross value of the contingent bonds repayable in the realized state of the world. Loans refers to the value of the whole portfolio of contingent bonds issued by borrowers, and the (gross) ex-post real return is the state-contingent value of the bonds repayable relative to the value of all the bonds previously issued.

The definitions in equation 19 directly imply that equation 10 must hold in terms of [c.sup.*.sub.[tau],l] [d.sup.*.sub.l] [l.sup.*.sub.l], and [r.sup.*.sub.[tau],l]. The budget identities (equation 17) and the contingent bond-market clearing conditions imply that equation 11 holds in terms of the variables defined in equation 19. Since [E.sub.t][(1 + [r.sup.*.sub.t+1])[K.sup.t+1]] = 1 follows from the definition in equation 19, the first-order conditions (equation 18) imply that equation 12 must hold in terms of [c.sup.*.sub.[tau],l] and [r.sup.*.sub.t]. Hence, the block of equations 10-12 applies to both the incomplete- and completemarkets economies.

However, the first-order condition (equation 18) with complete markets has stronger implications than equation 12. It also requires

(20) [c.sup.*.sub.b,t+1]/[c.sup.*.sub.b,t] = [c.sup.*.sub.s,t+1]/[c.sup.*.sub.s,t]

to hold in all states of the world. This equation states that consumption growth rates must always be equalized between borrowers and savers; in other words, households use complete financial markets to share risk. This is not generally an implication of the equilibrium conditions in equations 10-12 and equation 13 with incomplete financial markets. Furthermore, since a complete-markets economy has no restriction on the types of assets that households can buy and sell, equation 13 that determines the ex-post real return on a portfolio of nominal bonds is now irrelevant to determining the equilibrium of the complete-markets economy. The relevant ex-post real return is now determined implicitly by the portfolio of contingent securities that ensures that the risk-sharing condition (equation 20) holds.

The complete-markets equilibrium can be obtained analytically by solving the system of equations 10-12 and equation 20 (again under the parameter restriction in equation 3). The equilibrium consumption-GDP ratios are [c.sup.*.sub.[tau],t] = 1, so there is full risk sharing between borrowers and savers; this means that all households' consumption levels perfectly co-move in response to shocks (the consumption levels are equal owing to the parameter restriction in equation 3). Complete financial markets therefore allocate consumption efficiently across states of the world, as well as over time. The equilibrium values of other variables have similar expressions to those found in the nonstochastic case (equations 14-16), except that now, uncertain outcomes are replaced by conditional expectations:

(21) [d.sup.*.sub.1] = [delta]/2 [E.sub.1] [[(1 + [g.sub.2]).sup.1-[alpha]], and [d.sup.*.sub.2] = 1/2;

(22) [l.sup.*.sub.0] = [[delta].sup.2]/2 E[[(1 + [g.sub.1]).sup.1-[alpha]] [(1 + [g.sub.2]).sup.1-[alpha]]], and [l.sup.*.sub.t] = [delta]/2 [E.sub.1] [[(1 + [g.sub.2]).sup.1-[alpha]]].

The final debt-to-GDP ratio [d.sup.*.sub.2] is nonstochastic, and earlier debt-to-GDP ratios depend only on conditional expectations of future real GDP growth rates. Since the realization of shocks in period 1 can change these conditional expectations, [d.sup.*.sub.1] and [l.sup.*.sub.1] fare stochastic in general. The expressions for [l.sup.*.sub.0], [l.sup.*.sub.1], and [d.sup.*.sub.1] can be interpreted as the present discounted values in periods 0 or 1 of a payoff proportional to the final debt-to-GDP ratio [d.sup.*.sub.2] evaluated using prices of contingent securities (which are equal to households' common stochastic discount factor according to equation 18). Intuitively, the complete-markets portfolio is an equity share in future real GDP. This supports risk sharing by allowing the repayments of borrowers to move exactly in line with their incomes.

Finally, observe that the complete-markets equilibrium has a particularly simple form in two special cases. If the utility function is logarithmic ([alpha] = 1) or real GDP follows a random walk ([g.sub.t] is i.i.d.), then [d.sup.*.sub.t] and [l.sup.*.sub.t] are all nonstochastic:

(23) [d.sup.*.sub.1] = [beta]/2, [d.sup.*.sub.2] = 1/2, [l.sup.*.sub.0] = [[beta].sup.2]/2, and [l.sup.*.sub.1] = [beta]/2,

where [beta] = [delta]E[[(1 + [g.sub.t]).sup.1-[alpha]]].

The complete-markets equilibrium is entirely independent of monetary policy in all cases.

I.D. Replicating Complete Financial Markets

The block of equations 10-12 is common to the economy's equilibrium conditions irrespective of whether financial markets are complete or not. The only difference is that the incomplete-markets economy includes equation 13 as an equilibrium condition instead of equation 20 in the complete-markets economy. Since equation 13 includes the inflation rates [[pi].sub.1] and [[pi].sub.2], the ability of monetary policy to engineer a suitable state-contingent path for inflation means that the ex-post real returns in equation 13 can be chosen to generate the same consumption allocation as implied by the risk-sharing condition (equation 20). This is equivalent to ensuring the actual debt-to-GDP ratio d, mimics its hypothetical equilibrium value [d.sup.*.sub.1] in the economy with complete financial markets.

The monetary policy that replicates complete financial markets in this way turns out to be a nominal GDP target. Nominal GDP is denoted by [N.sub.t] = [P.sub.t][Y.sub.t], and its growth rate is denoted by [n.sub.t] = ([N.sub.t] - [N.sub.t-1])/[N.sub.t-1]. Since it is assumed that monetary policy can determine a state-contingent path for inflation [[pi].sub.t], and since real GDP growth [g.sub.t] is exogenous, monetary policy can equally well be specified as a sequence of nominal GDP growth rates [n.sub.t]. Equation 13 for the ex-post real returns on nominal bonds can be written in terms of nominal GDP growth as follows:

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consider first the replication argument in either of the special cases of log utility ([alpha] = 1) or real GDP following a random walk ([g.sub.t] is an i.i.d. stochastic process). In these special cases, the complete-markets debt-to-GDP ratios (equation 23) that monetary policy is aiming to replicate are nonstochastic. Suppose monetary policy sets a nonstochastic path for nominal GDP, that is, [n.sub.1] = [[bar.n].sub.1] and [n.sub.2] = [[bar.n].sub.2] for some constants [[bar.n].sub.1] and [[bar.n].sub.2]. If the replication is successful, [c.sub.[tau],t] = 1, and so both households' Euler equations (12) are satisfied when 1 = [delta][E.sub.t] [(1 + [r.sub.t+1])[(1 + [g.sub.t+1]).sup.-[alpha]]], With [n.sub.2] = [[bar.n].sub.2] and equation 24, this requires, for t = 1:

1 + [j.sub.1]/1 + [[bar.n].sub.2] = 1/[delta][E.sub.1][[(1 + [g.sub.2]).sup.1-[alpha]]] = 1/[beta],

using the definition of [beta] from equation 23. It then follows from equation 24 that (1 + [r.sub.2])/(1 + [g.sub.2]) = 1/[beta]. From the complete-markets solution (equation 23) together with equation 10, (1 + [r.sup.*.sub.2])/(1 + [g.sub.2]) = 1/[beta], so this monetary policy ensures that the ex-post real return [r.sub.2] on nominal bonds is identical to that on the complete-markets portfolio [r.sup.*.sub.2] for all realizations of shocks. Similarly, with [n.sub.1] = [[bar.n].sub.1] and the solution [gamma]/(1 + [j.sub.1]) = [beta][gamma]/(1 + [[bar.n].sub.2]) from above, the Euler equation at t = 0 requires:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so (1 + [r.sub.1])/(1 + [g.sub.1]) = 1/[beta], which coincides with (1 + [r.sup.*.sub.1])/(1 + [g.sub.1]) = 1/[beta]. Therefore, [r.sub.1] = [r.sup.*.sub.1] and [r.sub.2] = [r.sup.*.sub.2] under this monetary policy, which establishes that [d.sub.t] = [d.sup.*.sub.t] and [c.sub.[tau],t] = [c.sup.*.sub.[tau],t]. A nominal GDP target with any nonstochastic rates of nominal GDP growth succeeds in replicating the complete-markets equilibrium and supporting risk sharing among borrowers and savers. The intuition is that if the numerator of the debt-to-GDP ratio (expressed in monetary units) is fixed because nominal debt liabilities are not state-contingent, the ratio can be stabilized by targeting the denominator (expressed in monetary units), that is, ensuring that nominal incomes are predictable.

There is one other special case in which a monetary policy that makes nominal GDP growth perfectly predictable manages to replicate complete financial markets, even when the complete-markets debt-to-GDP ratio [d.sup.*.sub.1] is stochastic. This is the case where borrowers do not need to issue any new debt and do not need to refinance any existing debt after the initial time period. In the model, this corresponds to the limiting case of pure long-term bonds, where [gamma] [right arrow] [infinity] ([gamma] is the ratio of the period-2 and period-1 coupon payments on a bond issued in period 0). If monetary policy ensures that (1+[n.sup.1]) (1+[n.sup.2]) = [(1+[bar.n]).sup.2] for some nonstochastic [bar.n], then the equilibrium of the economy will coincide with the hypothetical complete-markets equilibrium. Unlike the earlier special cases, here it is not necessary that both the period 1 and 2 nominal GDP growth rates be nonstochastic; it is only necessary that the cumulated growth rate over both time periods be perfectly predictable. Intuitively, with long-term debt, monetary policy needs only to ensure that nominal incomes are predictable when debt is actually repaid.

When [gamma] is finite, some existing debt must be repaid by borrowers in period 1, requiring them to issue some new bonds if they are to continue to borrow until period 2. This exposes them to risk coming from uncertainty about the interest rate that will prevail in period 1. A monetary policy that aims to replicate complete financial markets must then address refinancing risk as well as income risk. In general, this requires a target for nominal GDP growth that changes when shocks occur, although as will be seen, there will still be a long-run target for nominal GDP that is invariant to shocks. If the maturity parameter [gamma] is positive and finite, then the monetary policies that replicate complete financial markets are characterized by the following nominal GDP growth rates in periods 1 and 2:

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[bar.n].sub.1] and [[bar.n].sub.2] are any nonstochastic growth rates (these would be the actual nominal GDP growth rates in the absence of shocks). Note that any such monetary policy has the implication that

(26) (1 + [n.sub.1])(1 + [n.sub.2]) = (1 + [[bar.n].sub.1])(1 + [[bar.n].sub.2]),

so the long-run target for nominal GDP must be nonstochastic. In the special cases of [alpha] = 1 or [g.sub.t] being i.i.d., which were analyzed earlier, the requirements on nominal GDP growth rates in equation 25 reduce simply to [n.sub.1] = [[bar.n].sub.1] and [n.sub.2] = [[bar.n].sub.2]. Intuitively, the problem of refinancing risk is absent in these special cases, albeit for a different reason in each case. When real GDP growth rates are independent over time, there is no news that changes expected future real GDP growth, and thus no reason for the equilibrium real interest rate to vary over time. On the other hand, with log utility, the real interest rate changes by the same amount and in the same direction as any revision to expectations of future growth. Higher real interest rates are then exactly offset by expectations of an improvement in future incomes relative to current incomes. This leaves monetary policy needing only to provide insurance against fluctuations in current incomes, and for this any predictable nominal GDP growth rate suffices.

As discussed above, in the special case of pure long-term debt ([gamma] [right arrow] [infinity]), equation 26 is needed, but equation 25 need not hold. In the special case of pure short-term debt ([gamma] = 0), it can be shown that only the condition on [n.sub.1] in equation 25 is needed, together with the restriction that [n.sub.2] take on any value that is perfectly predictable in period 1.

I.E. Discussion

The importance of the arguments for nominal GDP targeting in this paper obviously depends on the plausibility of the incomplete-markets assumption in the context of household borrowing and saving. It seems reasonable to suppose that households will not find it easy to borrow by issuing Arrow-Debreu state-contingent bonds, but might there be other ways of reaching the same goal? Issuing state-contingent bonds is equivalent to households' agreeing to loan contracts with financial intermediaries that specify a complete menu of state-contingent repayments. But such contracts would be much more time consuming to write, harder to understand, and more complicated to enforce than conventional noncontingent loan contracts, as well as making monitoring and assessment of default risk a more elaborate exercise. Moreover, unlike firms, households cannot issue securities such as equity that feature state-contingent payments but do not require a complete description of the schedule of payments in advance.

Another possibility is that even if households are restricted to non-contingent borrowing, they can hedge their exposure to future income risk by purchasing an asset with returns that are negatively correlated with GDP. But there are several pitfalls to this. First, it may be unclear which asset has a reliably negative correlation with GDP (even if "GDP securities" of the type proposed by Shiller [1993] were available, borrowers would need a short position in these). Second, the required gross positions for hedging may be very large. Third, a household already intending to borrow will need to borrow even more to buy the asset for hedging purposes, and the amount of borrowing may be limited by an initial down payment constraint and subsequent margin calls. In practice, a typical borrower does not have a significant portfolio of assets except for a house, and housing returns probably lack the negative correlation with GDP required for hedging the relevant risks.

In spite of these difficulties, it might be argued that the case for the incomplete-markets assumption is overstated because the possibilities of renegotiation, default, and bankruptcy introduce some contingency into apparently noncontingent debt contracts. However, default and bankruptcy allow for only a crude form of contingency in extreme circumstances, and these options are not without their costs. Renegotiation is also not costless, and evidence from consumer mortgages in both the recent U.S. housing bust and the Great Depression suggests that the extent of renegotiation may be inefficiently low (White 2009; Piskorski, Seru, and Vig 2010; Ghent 2011). Furthermore, even ex-post efficient renegotiation of a contract with no contingencies written in ex-ante need not actually provide for efficient sharing of risk from an ex-ante perspective. In a more general model where the incompleteness of financial markets is endogenized, the inflation fluctuations induced by nominal GDP targeting may also play a role in minimizing the costs of contract renegotiation or default when the economy is hit by an aggregate shock.

It is also possible to assess the completeness of markets indirectly through tests of the efficient risk-sharing condition, which is equivalent to a perfect correlation between the consumption growth rates of different households. These tests are the subject of a large literature (Cochrane 1991; Nelson 1994; Attanasio and Davis 1996; Hayashi, Altonji, and Kotlikoff 1996), which has generally rejected the hypothesis of full risk sharing.

Finally, even if financial markets are incomplete, the assumption that contracts are written in terms of specifically nominal noncontingent payments is important for the analysis. The evidence presented in Doepke and Schneider (2006) indicates that household balance sheets contain significant quantities of nominal liabilities and assets (for assets, it is important to account for indirect exposure via households' ownership of firms and financial intermediaries). Furthermore, as pointed out by Shiller (1997), indexation of private debt contracts is extremely rare. This suggests that the model's assumptions are not unrealistic.

The workings of nominal GDP targeting can also be understood from its implications for inflation and the real value of nominal liabilities. Indeed, nominal GDP targeting can be equivalently described as a policy of inducing a perfect negative correlation between the price level and real GDP and ensuring these variables have the same volatility. When real GDP falls, inflation increases, which reduces the real value of fixed nominal liabilities in proportion to the fall in real income; when real GDP rises, the opposite takes place.

It is perhaps surprising that optimal monetary policy in a non-representative-agent model should feature inflation fluctuations, given the long tradition of regarding inflation-induced unpredictability in the real values of contractual payments as one of the most consequential costs of inflation. As discussed by Clarida, Gali, and Gertler (1999), there is a widely held view that the difficulties this induces in long-term financial planning ought to be regarded as the most significant cost of inflation, more significant than the relative price distortions, menu costs, and deviations from the Friedman rule that have been stressed in representative-agent models. The view that unanticipated inflation leads to inefficient or inequitable redistributions between debtors and creditors clearly presupposes a world of incomplete markets, otherwise inflation would not have these effects. How then to reconcile this argument with the result that the incompleteness of financial markets suggests nominal GDP targeting is desirable because it supports efficient risk sharing? (Again, were markets complete, monetary policy would be irrelevant to risk sharing because all opportunities would already be exploited.)

While nominal GDP targeting does imply unpredictable inflation fluctuations, the resulting real transfers between debtors and creditors are not an arbitrary redistribution--they are perfectly correlated with the relevant fundamental shocks: unpredictable movements in aggregate real incomes. Since future consumption uncertainty is affected by income risk as well as risk from fluctuations in the real value of nominal contracts, long-term financial planning is not necessarily compromised by inflation fluctuations that have known correlations with the economy's fundamentals. An efficient distribution of risk requires just such fluctuations, because the provision of insurance is impossible without the possibility of ex-post transfers that cannot be predicted ex-ante. Unpredictable movements in inflation orthogonal to the economy's fundamentals (such as would occur in the presence of monetary-policy shocks) are inefficient from a risk-sharing perspective, but there is no contradiction with nominal GDP targeting because such movements would only occur if policy failed to stabilize nominal GDP.

It might be objected that if debtors and creditors really wanted such contingent transfers then they would write them into the contracts they agree to, and it would be wrong for the central bank to try to second-guess their intentions. But the absence of such contingencies from observed contracts may simply reflect market incompleteness rather than what would be rationally chosen in a frictionless world. Reconciling the noncontingent nature of financial contracts with complete markets is not impossible, but it would require both substantial differences in risk tolerance across households and a high correlation of risk tolerance with whether a household is a saver or a borrower. With assumptions on preferences that make borrowers risk-neutral or savers extremely risk-averse, it would not be efficient to share risk, even if no frictions prevented households from writing contracts to implement it.

There are a number of problems with this alternative interpretation of the observed prevalence of noncontingent contracts. First, there is no compelling evidence to suggest that borrowers really are risk-neutral or that savers are extremely risk-averse relative to borrowers. Second, while there is evidence suggesting considerable heterogeneity in individuals' risk tolerance (Barsky and others 1997; Cohen and Einav 2007), most of this heterogeneity is not explained by observable characteristics such as age and net worth (even though many characteristics such as these have some correlation with risk tolerance). The dispersion in risk tolerance among individuals with similar observed characteristics also suggests there should be a wide range of types of financial contract with different degrees of contingency if markets were complete. Risk-neutral borrowers would agree to noncontingent contracts with risk-averse savers, but contingent contracts would be offered to risk-averse borrowers.

Another problem with this interpretation based on complete markets but different risk preferences relates to the behavior of the price level over time. While nominal GDP has never been an explicit target of monetary policy, the implication of nominal GDP targeting--a countercyclical price level--has been largely true in the United States during the postwar period (Cooley and Ohanian, 1991), albeit with a correlation coefficient much smaller than one in absolute value, and a lower volatility relative to real GDP. Whether by accident or design, U.S. monetary policy has had some of the features of nominal GDP targeting, resulting in real values of fixed nominal payments positively co-moving with real GDP (but by less) on average. In a world of complete markets with extreme differences in risk tolerance between savers and borrowers, efficient contracts would undo the real contingency of payments brought about by the countercy-clicality of the price level, for example through indexation clauses. But as discussed in Shiller (1997), private nominal debt contracts have survived in this environment without any noticeable shift toward indexation. Furthermore, both the volatility of inflation and the correlation of the price level with real GDP have changed significantly over time; for example, the high volatility of the 1970s may be contrasted with the "Great Moderation," and the countercyclicality of the postwar price level may be contrasted with its procyclicality during the interwar period. The basic form of noncontingent nominal contracts has remained constant in spite of this change.

Finally, while the policy recommendation of this paper goes against the long tradition of citing the avoidance of redistribution between debtors and creditors as an argument for price stability, there is a similarly ancient tradition in monetary economics (which can be traced back at least to Bailey 1837) of arguing that money prices should co-move inversely with productivity to promote "fairness" between debtors and creditors. The idea is that if money prices fall when productivity rises, those savers who receive fixed nominal incomes are able to share in the gains, while the rise in prices at a time of falling productivity helps to ameliorate the burden of repayment for borrowers. This is equivalent to stabilizing the money value of incomes, in other words, nominal GDP targeting. The intellectual history of this idea (the "productivity norm") is thoroughly surveyed in Selgin (1995). Like the older literature, this paper places distributional questions at the heart of monetary policy analysis, but it studies policy through the lens of mitigating inefficiencies in incomplete financial markets rather than through looser notions of fairness.

II. Incomplete Financial Markets in a Monetary DSGE Model

This section develops a model that allows optimal monetary policy with incomplete financial markets to be studied in an infinite-horizon production economy that includes price stickiness as an additional friction.

II.A. Households

In this model, there are equal numbers of two types of households, referred to as borrowers and savers ([tau] [member of] {b, s}). A representative household of type x has preferences given by the following utility function

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sub.[tau],t] is per-household consumption of a composite good and [H.sub.[tau],t] is hours of labor supplied. The two types are now distinguished by their subjective discount factors, with [[delta].sub.[tau],t] being the discount factor of type-[tau] households between time t and t+1. Both types have a constant coefficient of relative risk aversion given by [alpha], and a constant elasticity of intertemporal substitution given by [[alpha].sup.-1]. The household-specific Frisch elasticity of labor supply is [[eta].sub.[tau]]. Each household of type [tau] receives real income [Y.sub.[tau],t] at time t, to be specified below. The discount factor [[delta].sub.[tau],t], of type-[tau] households is assumed to be the following:

(28) [[delta].sub.[tau],t] = [[delta].sub.[tau]] ([C.sub.[tau],t]/[Y.sub.[tau],t]), where [[delta].sub.t](c) = [[DELTA].sub.[tau]][c.sup.-(1-[lambda])[alpha]],

and where the parameters [[DELTA].sub.b], [[DELTA].sub.s], and [lambda], are such that 0 < [[DELTA].sub.b] < [[DELTA].sub.s] < [infinity] and 0 < [lambda] < 1. It is assumed individual households of type [tau] take [[delta].sub.[tau],t] as given, that is, they do not internalize the effect of their own consumption on the discount factor.

There are two differences compared to a representative-household model with a standard time-separable utility function. First, there is heterogeneity in discount factors because borrowers are more impatient than savers ([[DELTA].sub.b] < [[DELTA].sub.s]), all else equal. This is the key assumption that will give rise to borrowing and saving in equilibrium by the households that have been referred to as borrowers and savers. Second, discount factors display the marginal increasing impatience ([lambda] < 1) property of Uzawa (1968), in that the discount factor is lower when consumption is higher (relative to income), all else equal. This assumption is invoked for technical reasons because it ensures that the wealth distribution will be stationary around a well-defined nonstochastic steady state. That households take discount factors as given is assumed for simplicity and is analogous to models of "external" habits (see, for example, Abel 1990).

The composite good [C.sub.[tau],t] is a CES aggregate (with elasticity of substitution [epsilon]) of a measure-one continuum of differentiated goods indexed by [iota] [member of] [0, 1], which is the same for both types of households. Households allocate spending [C.sub.[tau],t] ([iota]) between goods to minimize the nominal expenditure [P.sub.t] [C.sub.[tau],t] required to obtain [C.sub.[tau],t] units of the consumption aggregator:

(29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [P.sub.t]([iota]) is the nominal price of good [iota]. Households of type [tau] face a real wage [w.sub.[tau],t] for their labor. All households own equal (nontradable) shareholdings in a measure-one continuum of firms, with firm [iota] paying real dividend [J.sub.t]([iota]). All households are assumed to face a common lump-sum tax [T.sub.t] in real terms. Real disposable income for households of type [tau] is thus

(30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

II.B. Incomplete Financial Markets

The only liability that can be issued by households is a noncontingent nominal bond. The nominal bond has the following structure. One newly issued bond at time t makes a stream of coupon payments in subsequent time periods, paying 1 unit of money (a normalization) at time t + 1, then [gamma] units at t + 2, [[gamma].sup.2] units at t + 3, and so on (0 [less than or equal to] [gamma] < [infinity]). The geometric structure of the coupon payments means that a bond issued at time t - l is after its time-r coupon payment equivalent to a quantity [[gamma].sup.l] of new date-t bonds. It therefore suffices to track the overall quantity of bonds in terms of new-bond equivalents, rather than the quantities of each vintage separately. (5) The flow budget identity at time t of households of type [tau] is:

(31) [C.sub.[tau],t] = [Q.sub.t] [B.sub.[tau],t]/[P.sub.t] = [Y.sub.[tau],t] + (1 + [gamma][Q.sub.t]) [B.sub.[tau],t-1]/[P.sub.t]

where [B.sub.[tau],t] denotes the outstanding quantity of bonds (in terms of new-bond equivalents) held (or issued, if negative) by type-[tau] households at the end of period t. The term 1 + [gamma][Q.sub.t] refers to the coupon payment plus the resale value of bonds acquired or issued in the past.

II.C. Firms

Firm [iota] [member of] [0, 1] is the monopoly producer of differentiated good [iota]. Goods are produced using an aggregator of labor inputs. Production of good [iota] is denoted by [Y.sub.t]([iota]), firm [iota]'s labor usage by [H.sub.t]([iota]), and [w.sub.t] denotes the wage cost per unit of [H.sub.t]([iota]). The firm pays out all real profits at time t as dividends [J.sub.t]([iota]):

(32) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first equation following the definition of profits is the production function, with [A.sub.t] denoting the common exogenous productivity level and where the parameter [xi] determines the extent of diminishing returns to labor ([xi] [greater than or equal to] 0). The final equation in (32) is the demand function that arises from the household expenditure minimization problem (equation 33).

The labor input [H.sub.t] ([iota]) is an aggregator of labor supplied by the two types of households. Firms receive a proportional wage-bill subsidy at rate [[epsilon].sup.-1]. Firms choose labor inputs [H.sub.[tau],t] ([iota]) to minimize the post-subsidy cost [w.sub.t] [H.sub.t] ([iota]) of obtaining a unit of the aggregate labor input [H.sub.t] ([iota]):

(33) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The labor aggregator has a Cobb-Douglas functional form, implying a unit elasticity of substitution between different labor types.

II.D. Sticky Prices

Price adjustment is assumed to be staggered according to the Calvo (1983) pricing model. In each time period, there is a probability [sigma] that firm [iota] must continue to use its previous nominal price [P.sub.t-1]([iota]). If at time t a firm does receive an opportunity to change price, it sets a reset price denoted by [[??].sub.t]. The reset price is set to maximize the current and expected future stream of profits. Future profits conditional on continuing to charge [[??].sub.t] are multiplied by the probability [[sigma].sup.l] that the reset price will actually remain in use l periods ahead, and then are discounted using the real interest rate [[rho].sub.t].

(34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

II.E. Money and Monetary Policy

The economy is "cashless" in that money is not required for transactions, but money is used as a unit of account in writing financial contracts and in pricing goods. Monetary policy is assumed to be able to determine a path for the price level [P.sub.t].

II.F. Fiscal Policy

The only role of fiscal policy is to provide the wage-bill subsidy to firms by collecting equal amounts of a lump-sum tax from all households. It is assumed that the fiscal budget is in balance, so taxes [T.sub.t] are set at the level required to fund the current subsidy: (6)

(35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

II.G. Market Clearing

Market clearing in goods, labor, and bond markets requires:

(36) [1/2] [C.sub.b,t]([iota]) + [1/2] [C.sub.s,t] ([iota]) = [Y.sub.t]([iota]), for all [iota] [member of] [0,1];

(37) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(38) [1/2] [B.sub.b,t] + [1/2] [B.sub.s,t] = 0.

II.H. Equilibrium

The derivation of the equilibrium conditions is presented in the online appendix. (7) The analysis of incomplete financial markets follows the method used for the simple model in section I, while other aspects of the model are standard features of New Keynesian models with sticky prices. The consumption-to-GDP ratios are [c.sub.[tau],t] [equivalent to] [C.sub.[tau],t]/[Y.sub.t], and the debt-to-GDP ratio [d.sub.t], loans-to-GDP ratio [l.sub.t], ex-post real return [r.sub.t], and real interest rate [[rho].sub.t] are defined as in equation 8. The yield-to-maturity [j.sub.t] on the nominal bonds is defined by:

(39) [Q.sub.t] = [[infinity].summation over (l-1)] [[gamma].sup.l-1]/[(1 + [j.sub.t]).sup.l], implying [j.sub.t] = 1/[Q.sub.t] - 1 + [gamma].

As in the simple model of section I, the analysis will also make use of an otherwise identical model where financial markets are complete, where the equivalents of [d.sup.*.sub.t], [l.sup.*.sub.t], and [r.sup.*.sub.t] are as defined in equation 19.

In a steady state where exogenous productivity [A.sub.t] is growing at a constant rate, there is a steady-state rate of real GDP growth [bar.g]. The steady-state consumption-GDP ratios are given by

(40) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 0 < [theta] < 1. The term [theta] depends on the relative patience [[DELTA].sub.b]/[[DELTA].sub.s] of the two household types and the utility-function parameters [alpha] and [lambda]. The steady-state discount factors and real interest rate are:

(41) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and it is assumed that [bar.g] is low enough to ensure that 0 < [beta] < 1. In the steady state, the discount factors of the two types are aligned at [delta], which is effectively an average of the patience parameters [[DELTA].sub.b] and [[DELTA].sub.s]. The steady-state debt ratios can be written in terms of [beta] and [theta] as follows:

(42) [bar.d] = [theta]/2(1 - [beta]), and [bar.l] = [beta][theta]/2(1 - [beta]).

The model can be parameterized directly with [beta] and [theta] rather than the two patience parameters [[DELTA].sub.b] and [[DELTA].sub.s] (leaving [alpha], [lambda], and [bar.g] to be chosen separately). The term [beta] plays the usual role of the discount factor in a representative-household economy given its relationship with the real interest rate (with an adjustment for steady-state real GDP growth). The term [theta] quantifies the extent of heterogeneity between borrower and saver households, which is related to the amount of borrowing and saving that occurs in equilibrium, and hence to the debt-to-GDP ratio in equation 42. Given equation 40, [theta] can be interpreted as the "debt service ratio" because it is the net fraction of income transferred by borrowers to savers. As will be seen, [theta] is a sufficient statistic for the extent of heterogeneity in the economy, with [theta] [right arrow] 0 being the limiting case of a representative-household economy ([[DELTA].sub.b] [right arrow] [[DELTA].sub.s]).

Given that prices are sticky, attention is restricted to a zero-inflation steady state. Rather than specify the bond coupon parameter [gamma] directly, it is convenient to set the steady-state fraction of debt that is not refinanced each period, denoted by [mu]. This fraction is [mu] = [gamma]/(1 + [bar.g]). Finally, a parameter restriction on the Frisch elasticities of borrowers and savers is imposed, which implies that the wealth distribution has no effect on aggregate labor supply (up to a first-order approximation):

(43) [[eta].sub.b] = (1 - [theta])[eta]/(1 + [theta][eta]), and [[eta].sub.s] = (1 + [theta])[eta]/(1 - [theta][eta]),

where [theta] is the steady-state debt service ratio defined in equation 40 and [eta] is the effective aggregate Frisch elasticity of labor supply (it is assumed that [eta] < 1/[theta]).

III. Optimal Monetary Policy

This section studies the features of optimal monetary policy in the DSGE model with incomplete financial markets of section II. An exact analytical solution is not available in general, so this section resorts to finding the log-linear approximation of the optimal policy (the first-order perturbation around the nonstochastic steady state), which can be found analytically. The method is the familiar linear-quadratic approach whereby the appropriate welfare function is approximated up to second-order accuracy and is minimized subject to first-order accurate approximations of the equilibrium conditions, which act as the constraints on monetary policy. Derivations of all results are found in the appendix.

The notational convention below is that variables in a sans serif font (for example, d) denote log deviations of the equivalent variables in roman letters (for example, d) from their steady-state values (log deviations of interest rates, inflation rates, and growth rates are log deviations of the corresponding gross rates; for variables that have no steady state, the sans serif letter simply denotes the logarithm of that variable).

III.A. Constraints

Monetary policy analysis can be performed by studying just four endogenous variables: the "debt gap" [[??].sub.t], the inflation rate [[pi].sub.t] [equivalent to] [P.sub.t] - [P.sub.t-1], the nominal bond yield [j.sub.t], and the output gap [[??].sub.t]. The debt gap [[??].sub.t] [equivalent to] [d.sub.t] - [d.sup.*.sub.t] is the deviation of the actual debt-to-GDP ratio d, from the "natural debt-to-GDP ratio" [d.sup.*.sub.t] that is, the debt-to-GDP ratio that would prevail with complete financial markets (* signifies complete financial markets). The natural debt-to-GDP ratio [d.sup.*.sub.t] is a multiple of the discounted sum of expectations of future real GDP growth [g.sub.t] = [Y.sub.t] - [Y.sub.t-1]: (8)

(44) [d.sup.*.sub.t] = (1 - [alpha]) [[infinity].summation over (l=1)] [[beta].sup.l][E.sub.t][g.sub.t+l].

With complete financial markets, the consumption of borrowers and savers would co-move perfectly with each other and with real GDP (their consumption ratios would be [c.sup.*.sub.b,t] = 1 - [theta] and [c.sup.*.sub.s,t] = 1 + [theta]). Since the debt-to-GDP ratio completely describes the distribution of financial wealth when there is a representative borrower and a representative save[r.sub.t] the debt gap [[??].sub.t] is a sufficient statistic for the deviation of the consumption allocation from the risk sharing provided by complete financial markets:

(45) [c.sub.b,t] = - [[theta](1 - [beta][lambda])/(1 -[theta])(1 - [beta])] [[??].sub.t], and [c.sub.s,t] = [[theta](1 - [beta][lambda])/(1 -[theta])(1 - [beta])] [[??].sub.t].

A positive debt gap corresponds with the consumption of savers growing faster than that of borrowers, and a negative debt gap corresponds with the consumption of savers growing more slowly than that of borrowers.

The output gap [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the deviation of the actual level of output [Y.sub.t] from the natural level of output [[??].sub.t], that is, the level of output that would prevail with fully flexible prices (^ signifies flexible prices). (9) The growth rate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the natural level of output is:

(46) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a multiple of the exogenous growth rate of total factor productivity [A.sub.t] as in the textbook New Keynesian model.

There are three constraints on monetary policy imposed by the equilibrium conditions involving the debt gap [[??].sub.t], inflation [[pi].sub.t], the nominal bond yield [j.sub.t] and the output gap [[??].sub.t]:

(47) [E.sub.t][[??].sub.t+1] = [lambda][[??].sub.t];

(48) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(49) [kappa]([[pi].sub.t], - [beta][E.sub.t][[pi].sub.t,+1]) = v[[??].sub.t], where [kappa] [equivalent to] [sigma] (1 + [epsilon][xi])/(1 - [sigma]) (1 - [sigma][beta]),

and v [equivalent to] [alpha] + [xi] + 1 + [xi]/[eta].

These equations include one exogenous variable [[??].sup.*.sub.t], which depends only on the exogenous growth rate g, of the natural level of output from equation 46:

(50) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Comparison with equation 44 shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[??].sup.*.sub.t] is the natural debt-to-GDP ratio when output is always at its natural level, hence the current natural debt-to-GDP ratio differs from its past value only because [[??].sup.*] differs from real GDP growth [[??].sub.t] Since debt repayments net of new borrowing must be zero to support risk sharing between borrowers and savers, this means that [[??].sup.*.sub.t] can be interpreted as the real return on the complete-markets portfolio in the case where output is always equal to its natural level (the real return that ensures the actual debt-to-GDP ratio is always equal to the natural debt-to-GDP ratio when the output gap is zero).

The first constraint (equation 47) restricts the predictable component of the future debt gap [E.sub.t][[??].sub.t+1] to be a multiple [lambda] of the current debt gap. This is an implication of consumption smoothing. The debt-to-GDP ratio is a sufficient statistic for the wealth distribution in the economy, with a rise increasing the wealth of savers and decreasing the wealth of borrowers. Households react to changes in financial wealth by smoothing out the response of consumption over time, which means any changes in financial wealth are persistent. This persistence is tempered by the marginal increasing impatience of the Uzawa discount factors (equation 28) when [lambda] < 1, ensuring stationarity of [[??].sub.t].

The second constraint (equation 48) determines the unpredictable component of the debt gap [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The first two terms on the left-hand side are equal to the ex-post real return [r.sub.t] on nominal bonds between date t - 1 and t:

(51) [r.sub.t] = [j.sub.t-1] - [beta][mu][j.sup.t] / 1 - [beta][mu] - [[pi].sub.t].

In the case of short-term debt ([mu] = 0), this collapses to the usual ex-post Fisher equation [r.sub.t] = [j.sub.t-1] - [[pi].sub.t]. With longer term debt ([mu] > 0), a rise in the current nominal yield j, reduces the real value of existing nominal assets or liabilities. Equation 48 shows that the unpredictable component of the debt gap [[??].sub.t] - [E.sub.t-1] [[??].sub.t] depends positively on the difference between the actual ex-post real return on nominal bonds [r.sub.t] and the complete-markets-portfolio real return [[??].sup.*.sub.t] (assuming output is always at its natural level). Intuitively, if the actual real return is too high compared to the complete-markets portfolio, the debt gap rises; the opposite happens if the real return is too low. The remaining terms in the middle of the left-hand side of equation 48 are present because the debt gap [[??].sub.t] is defined as the deviation of [d.sub.t] from [d.sup.*.sub.t], that is, in terms of the natural debt-to-GDP ratio associated with the actual sequence of real GDP growth rates (which equation 45 shows is what is relevant for risk sharing), rather than the hypothetical sequence of real GDP growth rates [[??].sub.t] that would occur with flexible prices. These additional terms in equation 48 reflect the deviation of [d.sup.*.sub.t] from [[??].sup.*.sub.t] owing to price stickiness.

The third constraint (equation 49) is the standard New Keynesian Phillips curve [[pi].sub.t] = [beta][E.sub.t][[pi].sub.t+1] + (v/[kappa])[[??].sub.t] relating current inflation to the output gap and expected future inflation. The coefficients in the Phillips curve are identical to those found in the textbook New Keynesian model (see Woodford 2003). (10) The elasticity of real marginal cost with respect to the output gap [[??].sub.t] is v, and k captures the extent of nominal and real rigidities.

The textbook New Keynesian model comprises the Phillips curve equation 49 and an "IS curve," where the IS curve is essentially a consumption Euler equation. Although the model here does not feature a representative household, it turns out that the usual IS curve is an implication of the two equations 47 and 48 together: (11)

(52) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The term [[rho].sub.t] = [E.sub.t][r.sub.t+1] is the (ex-ante) real interest rate (the expected value of the real return [r.sub.t] on nominal bonds from equation 51). This reduces to [[rho].sub.t] = [j.sub.t] - [E.sub.t][[pi].sub.t+1] in the special case of short-term debt ([mu] = 0), which is what is assumed in the textbook New Keynesian model. The term [[??].sub.t], is the natural real interest rate (the ex-ante interest rate that would prevail if prices were flexible, not to be confused with the ex-post real return [[??].sup.*.sub.t] that would prevail with complete financial markets), which also has an identical form to the textbook model. Note that the nominal yield j, is a weighted average of expectations of current and future sums of real interest rates and inflation rates:

(53) [j.sub.t] = (1 - [beta][mu])[[infinity].summation over (l=0)] [([beta][mu]).sup.l][E.sub.t][[[rho].sub.t+l] + [[pi].sub.t+l+1]],

where the weights on future expectations depend on [beta] and the maturity parameter [mu] (the equation above uses the no-Ponzi condition in equation 48, which is necessary to rule out bubbles when [mu] > 0 because bonds have no terminal date). (12)

An implication of equation 52 is that [[rho].sub.t] = [alpha][E.sub.t][g.sub.t+1]. Using this, the expression for the natural debt-to-GDP ratio in equation 44 can be given a more intuitive form:

(54) [d.sup.*.sub.t] = (1 - [beta]) [[infinity].summation over (l=0)][[beta].sup.l][E.sub.t] [[Y.sub.t-l] - [l-1.summation over (J=0)] [[rho].sub.t+j]] - [Y.sub.t],

which states that the natural debt-to-GDP ratio moves one-for-one with the ratio of the present discounted value of all current and future real GDP to current real GDP. Intuitively, financial wealth must co-move with the present value of all nonfinancial income to support risk sharing between borrowers and savers; in other words, the complete-markets portfolio is an equity share in real GDP.

The three constraints in equations 47-19 leave one degree of freedom for monetary policy to affect the four endogenous variables. For simplicity, monetary policy can be thought of as selecting a state-contingent path for the inflation rate [[pi].sub.t] or any other nominal variable such as nominal GDP growth. Given a path for the inflation rate [[pi].sub.t], the Phillips curve (equation 49) determines the output gap [[??].sub.t]. The two equations (47 and 48) can then be solved for the debt gap [[??].sub.t] in terms of the inflation path and an exogenous shock [p.sub.t] that depends on the natural real GDP growth rate [[??].sub.t]:

(55) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The next step in characterizing optimal monetary policy is to specify the objective function.

III.B. Objective Function

The objective function for monetary policy is taken to be the social welfare function with Pareto weights that support the complete-markets consumption allocation (meaning that the weights of borrowers and savers are proportional to [(1 - [theta]).sup.[alpha]] and [(1 + [theta]).sup.[alpha]] respectively). The social planner takes the discount factors (equation 28) as given, so there is no tension with the household decision problem where these discount factors were also taken as given. The welfare function is scaled by a function of initial output with flexible prices so that its units are percentage equivalents of initial output.

A second-order approximation of the welfare function (starting from date [t.sub.0]) around the nonstochastic steady state is given by the negative of the loss function,

(56) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where terms that are independent of monetary policy are ignored. The loss function includes the squared debt-to-GDP gap [[??].sub.t], which is a sufficient statistic for the welfare loss due to deviations from risk sharing. The coefficient is increasing in risk aversion [alpha] and the degree of heterogeneity between borrower and saver households as measured by [theta]. Intuitively, greater risk aversion increases the importance of the risk sharing that is obtained from complete financial markets, while greater heterogeneity between households leads to larger financial positions of borrowers and savers, so a given percentage change in financial wealth has a larger impact on consumption.

The second term in the loss function (equation 56) is the squared inflation rate, which is a sufficient statistic for the welfare loss due to relative-price distortions. This is a well-known property of Calvo pricing. The coefficient is increasing the price elasticity of demand [epsilon], the probability of price stickiness [sigma], and the output elasticity of marginal cost [xi], which affects the degree of real rigidity in the economy. The third term in the loss function is the squared output gap. This represents the losses from aggregate output and employment deviating from their efficient levels, and its coefficient is simply the elasticity v of real marginal cost with respect to the output gap. Both loss-function coefficients are identical to those found in the textbook New Keynesian model.

Optimal monetary policy minimizes the loss function (equation 56) subject to the constraints in equations 47-49. It is assumed that the central bank is able to commit to a policy and that this commitment has been made at some date [t.sub.0] far in the past. (13) The model developed here has essentially added one variable (the debt gap) and one equation to the textbook three-equation and three-variable New Keynesian model; the extra variable also appears in the utility-based loss function. This provides a parsimonious and tractable framework for studying how the incompleteness of financial markets affects optimal monetary policy.

III.C. Special Case I: Complete Markets or a Representative Household

Suppose that the assumption that households can only buy or sell nominal bonds is changed to allow them access to complete financial markets. All other features of the model in section II are unchanged. In this version of the model, the value of contingent debt liabilities would always equal the natural debt-to-GDP ratio, so the debt gap [[??].sub.t] would always be zero. Hence [[??].sub.t] = 0 replaces equation 48, and equation 47 automatically holds. It follows that the task of optimal monetary policy is to minimize only the inflation and output gap terms in the loss function (equation 56). Given the Phillips curve (equation 49) with its "divine coincidence" property that stabilizing inflation is equivalent to stabilizing the out-gut gap, optimal monetary policy is strict inflation targeting [[pi].sub.t] = 0. Since [[??].sub.t] = 0 and [[??].sub.t] = 0 in this special case, the policy achieves a first-best allocation of resources.

Alternatively, maintain the assumption of incomplete financial markets, but consider the special case of a representative household where the difference between households labeled 'borrowers' and those labeled 'savers' is eliminated. This difference is the degree of impatience represented by the parameters [[DELTA].sub.b] and [[DELTA].sub.s] in equation 28. If [[DELTA].sub.b] = [[DELTA].sub.s], then equation 40 shows that this corresponds to a case where [theta] = 0. The coefficient of the debt gap in the loss-function equation 56 is zero, so optimal monetary policy again needs only to minimize the inflation and output gap terms subject to the Phillips curve (equation 49). Strict inflation targeting is again optimal and results in a first-best allocation of resources, corresponding to the finding in the textbook New Keynesian model.

III.D. Special Case II: Fully Flexible Prices

Suppose goods prices are fully flexible, which means that [sigma] = 0 and hence [kappa] = 0 using equation 49. The coefficient of inflation in the loss function (equation 56) is zero in this case, because inflation does not lead to relative-price distortions. Furthermore, the Phillips curve (equation 49) implies the output gap is zero ([[??].sub.t] = 0) irrespective of the behavior of inflation. Consequently, the only concern of optimal monetary policy is to minimize the first term involving the debt gap in the loss function (equation 56). Since inflation affects the ex-post real return on nominal bonds (and thus the debt gap) through equation 48, it is feasible to obtain [[??].sub.t] = 0, which achieves a first-best allocation of resources.

Since inflation has a zero coefficient in the loss function (equation 56), there are generally many possible inflation paths equally consistent with [[??].sub.t] = 0 and thus the same zero value of the loss function. An inflation path must satisfy [[summation].sup.[infinity].sub.l=0] [([beta][mu]).sup.l]([E.sub.t][[pi].sub.t+l] - [E.sub.t-1][[pi].sub.t+l]) - [p.sub.t] = 0 which follows from equation 55 with [[??].sub.t] = 0 and [kappa] = 0. With a zero output gap ([g.sub.t] = [[??].sub.t]), the exogenous Shock [p.sub.t] from equation 55 is:

(57) [p.sub.t] = [[infinity].summation over (l=0)][[beta].sub.l] (l - [alpha](1 - [[mu].sup.l]))([E.sub.t][g.sub.t+t] - [E.sub.t-1][g.sub.t+1]).

The first part of this expression is news about the sequence of current and expected future real GDP growth rates [[summation].sup.[infinity].sub.l=0][[beta].sub.t][E.sub.t][g.sub.t+l] (discounting using the steady-state discount factor [beta]). This captures any change in the anticipated stream of nonfinancial income that can be used by borrowers to service their debts.

Using the formula for the real interest rate [[rho].sub.t] = [alpha][E.sub.t][g.sub.t+1], the second part of equation 57 is seen to be news about the sequence of current and expected future real interest rates [[summation].sup.[infinity].sub.l=0] [[beta].sup.l+1](1 - [[mu].sup.l+1])[E.sub.t][[rho].sub.t+l]. An increase in this term is associated with a rise in the current or anticipated future cost of continuing to borrow. The interest rate l periods ahead is multiplied not only by the discount factor [[beta].sup.l+1] but also by 1 - [[mu].sup.l+1]. The reason is that changes in (ex-ante) real interest rates affect borrowers only to the extent that existing debt is refinanced or new debt is issued, and the (steady-state) fraction of existing debt that remains un-refinanced l periods from now is [[mu].sup.l+1]. Overall, the terms in equation 57 capture the shock to the stream of nonfinancial income, adjusted for any mitigating changes (when [alpha] < 1) or aggravating changes (when [alpha] > 1) in real interest rates triggered by this news.

Let [N.sub.t] = [P.sub.t] + [Y.sub.t] denote the level of (log) nominal GDP. Adopting the nominal GDP target below ensures that the debt gap is zero and therefore replicates complete financial markets:

(58) [N.sup.t] = - (1 - [mu])/(1 - [beta][mu])[d.sup.*.sub.t].

Monetary policy should target a constant level of nominal GDP if the natural debt-to-GDP ratio [d.sup.*.sub.t] is constant; and, if the natural debt-to-GDP ratio changes, the target for the level of nominal GDP should be adjusted in the opposite direction. In the case of short-term debt ([mu] = 0), this target reduces simply to [N.sub.t] = -[d.sup.*.sub.t], so the nominal GDP target should move one-for-one in the opposite direction to the natural debt-to-GDP ratio. Intuitively, if the nominal value of debt liabilities is completely predetermined (as it is with one-period debt), the debt-to-GDP ratio should be adjusted to the natural debt-to-GDP ratio by having the denominator of the debt-to-GDP ratio (nominal GDP) move inversely with the natural debt-to-GDP ratio. With longer-maturity debt ([mu] > 0), the nominal GDP target does not need to adjust as much to changes in the natural debt-to-GDP ratio.

Equation 58 shows that a constant nominal GDP target is optimal when either the natural debt-to-GDP ratio [d.sup.*.sub.t] is constant, or when debt has a sufficiently long maturity so that no refinancing is necessary ([mu] = 1). To understand this, first consider the special case where real GDP follows a random walk, so that real GDP growth g, is an i.i.d. stochastic process, with equation 44 implying that [d.sup.*.sub.t] = 0. Equation 57 shows that the exogenous shock [p.sub.t] is simply equal to current real GDP growth [g.sub.t] in this case. Following a constant nominal GDP target implies [[pi].sub.t] = -[g.sub.t], which means the inflation rate is also serially uncorrelated. The term [[summation].sup.[infinity].sub.l=0][([beta][mu]).sup.l][E.sub.t][[pi].sub.t+l] reduces to [[pi].sub.t], which confirms (using equation 55) that the nominal GDP target [N.sub.t] = 0 is optimal.

Intuitively, if shocks to real GDP are permanent, the current percentage change in real GDP is equal to the percentage change in the present discounted value of all current and future real GDP. (There is no change in ex-ante real interest rates because consumption smoothing calls for an equal-sized adjustment of consumption, as confirmed by the equation [[rho].sub.t] = [alpha][E.sub.t][g.sub.t+1].) A constant nominal GDP target moves the price level unexpectedly in the opposite direction to real GDP, which changes the real value of all nominal debt liabilities (of whatever maturity) by the same percentage as the stream of real nonfinancial income, which is what is required for risk sharing.

With different dynamics of real GDP, current real GDP growth is not the same as the percentage change in the present discounted value of the stream of nonfinancial income. Now consider the case where utility is logarithmic in consumption ([alpha] = 1), where equation 57 implies [p.sub.t] = [[summation].sup.[infinity].sub.l=0][([beta][mu]).sup.l]([E.sub.t-1][[g.sub.t+l] - [E.sub.t-l][[g.sub.t+l]). Since a constant nominal GDP target means [[pi].sub.t] = - [g.sub.t], the innovation to [[summation].sup.[infinity].sub.l=0][([beta][mu]).sup.l][E.sub.t][[pi].sub.t+l] is equal to - [p.sub.t], so the policy succeeds in replicating complete financial markets.

To understand the intuition, suppose all debt is refinanced each period, where only unexpected inflation leads to a change in the real value of existing debt (though the argument applies to any maturity of debt). The nominal GDP target ensures that any surprise change in current real GDP will be matched by an equal percentage change in the real value of debt. However, changes in current real GDP are not necessarily an accurate reflection of changes in the value of the stream of current and future nonfinancial income, which is what ultimately matters for the ability of borrowers to service their debts. For example, an expected recovery of real GDP following an initial negative shock means that the loss of current income overstates the fall in the present discounted value of current and future real GDP. However, such predictable variation in real GDP growth rates also leads to changes in ex-ante real interest rates, with an expected rise in real GDP increasing interest rates. In the example of the expected recovery, the position of borrowers is worsened by the higher interest rates because they must refinance their debts.

With logarithmic utility, the percentage-point change in the real interest rate is equal to the percentage-point change in expected real GDP growth, so with an expected recovery the higher real interest rate exactly cancels out the benefit of the higher income anticipated in the future. A constant nominal GDP target is optimal because the required adjustment of the real value of debts for risk sharing is, therefore, simply equal to the unexpected change in current real GDP. This argument applies more generally for any maturity parameter [mu] when utility is logarithmic. The longer the maturity of debt, the less impact changes in real interest rates will have on borrowers, because there is less refinancing, but the greater the effect of predictable inflation on the real value of existing debt. These two forces exactly cancel each other out with logarithmic utility, and a constant nominal GDP target remains optimal.

In the special case of very long maturity debt that requires no refinancing (p = 1), observe that the shock [p.sub.t] in equation 57 reduces to [p.sub.t] = [[summation].sup.[infinity].sub.l=0][([beta][mu]).sup.l]([E.sub.t][g.sub.t+l] - [E.sub.t-1][g.sub.t+l]). Unsurprisingly, the terms related to changes in ex-ante real interest rates are absent, because there is no refinancing risk for borrowers. With [mu] = 1, [[summation].sup.[infinity].sub.l=0][([beta][mu]).sup.l][E.sub.t][[pi].sub.t+l] = [[summation].sup.[infinity].sub.l=0][([beta][mu]).sup.l][E.sub.t][[pi].sub.t+l] so a constant nominal GDP target that implies [[pi].sub.t] = -[g.sub.t] succeeds in achieving [[??].sub.t] = 0.

Intuitively, given debt's long maturity, both predictable and unpredictable inflation have exactly the same effect on its real value. Hence, in the absence of refinancing risk, whatever the dynamics of real GDP are, inflation movements in the opposite direction that mirror real GDP growth ensure that the real value of debt moves one-for-one with the present discounted value of the stream of nonfinancial income. More generally, for 0 < [mu] < 1, the longer the maturity of debt, the less important is refinancing risk, and the smaller the difference between the effects of unpredictable and predictable inflation on the real value of debt. When real GDP displays predictable dynamics, it is these two factors that explain why the nominal GDP target (equation 58) needs to move with the natural debt-to-GDP ratio, and thus why the adjustment of the target is smaller when [mu] is larger.

As discussed, when [mu] > 0, the optimal monetary policy is not unique because many different inflation paths can lead to the same real value of debt. Rather than a nominal GDP target (equation 58) that adjusts to changes in the natural debt-to-GDP ratio, it is often possible to find a time-invariant weighted nominal GDP target [N.sub.[omega],t] [equivalent to] [P.sub.t] + [omega][Y.sub.t] = 0 that achieves the same outcome for some relative weight w on real GDP. If real GDP growth is a stationary and invertible stochastic process that can be expressed as [g.sub.t] = [[summation].sup.[infinity].sub.l=0] [v.sub.l][[epsilon].sub.t-l] a white-noise shock [[member].sub.t] and a sequence of coefficients {[v.sub.l]} then the following monetary policy will also replicate complete financial markets:

(59) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that [[omega].sup.*] is equal to one (an unweighted nominal GDP target) in any of the special cases discussed above (namely, [v.sub.l] = 0 for all l [greater than or equal to] 1, or [alpha] = 1, or [mu] = 1). It might be tempting to interpret equation 59 as a form of flexible inflation targeting, but it is important to note that [Y.sub.t] is the level of real GDP, not the output gap. It is crucial that the measure of real GDP in equation 59 is not corrected for any changes in potential output.

III.E. Special Case III: Sticky Prices and Inelastic Labor Supply

With a zero Frisch elasticity of labor supply ([eta] = 0), equation 49 shows that v [right arrow] [infinity] (v is the elasticity of real marginal cost with respect to the output gap). The Phillips curve (equation 49) then implies that the output gap must be zero ([[??].sub.t] = 0) even though prices are sticky. While the coefficient of the output gap in the loss function (equation 56) becomes large, the product of this with the squared output gap tends to zero. Thus, optimal monetary policy is concerned only with the first two terms in the loss function (equation 56), the debt gap [[??].sub.t] and the inflation rate [[pi].sub.t]. However, there is a trade-off when stabilizing these variables because inflation that affects the real value of debt now leads to relative price distortions because prices are sticky. The optimal monetary policy subject to this trade-off is:

(60) [[??].sub.t] = [lambda][[??].sub.t-1] -(1 - [chi])[p.sub.t], and [[pi].sub.t], = [mu][[pi].sub.t-1] - [chi](1 - [beta][[mu].sup.2])[p.sub.t],

where

(61) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and where [p.sub.t] is as given in equation 57. With sticky prices, replicating complete financial markets through variation in inflation is costly, so the central bank tolerates some deviation from full risk sharing. To the extent that [chi] in equation 61 is less than one, a shock [p.sub.t] leads to fluctuations in the debt gap [[??].sub.t]. These fluctuations are persistent (the debt gap is an AR(1) process) because of the constraint in equation 47: the serial correlation of the debt gap must be [lambda]. Once a nonzero debt gap arises at time t, because of consumption smoothing there is no predictable future policy action that can undo its future consequences.

Equation 60 shows that if [chi] were equal to 1, the debt gap would be completely stabilized, and if [chi] were 0, inflation would be completely stabilized. Since 0 < [chi] < 1, optimal monetary policy can be interpreted as a mixture of strict inflation targeting and a policy that replicates complete financial markets. As the responses of [[??].sub.t] and [[pi].sub.t], to the shock [p.sub.t] are linearly related to [chi], the terms [chi] and 1 - [chi] can be interpreted respectively as the weights on supporting risk sharing and avoiding relative-price distortions. Comparing equations 56 and 61, it can be seen that [chi] is positively related to the ratio of the coefficients of [[??].sup.2.sub.t] and [[pi].sup.2.sub.t] in the loss function divided by (1 - [beta][[lambda].sup.2]) and (1 - [beta][[mu].sup.2]).

Greater risk aversion ([alpha]) or more heterogeneity ([theta]) and hence more borrowing increase the coefficient of [[??].sup.2.sub.t] and thus [chi]; a larger price elasticity of demand ([epsilon]), stickier prices ([sigma]), or more real rigidities ([xi]) increase the coefficient of [[pi].sup.2.sub.t] and thus reduce [chi]. The optimal trade-off is also affected by the constraints in equations 47 and 48, which explains the presence of the terms (1 - [beta][[lambda].sup.2]) and (1 - [beta][[mu].sup.2]) in the formula for [chi]. A greater value of [lambda] increases the persistence of the debt gap, which makes fluctuations in d, significantly more costly than suggested by the loss function coefficient alone. The parameter [mu] affects the link between bond yields and the debt gap. It is evident that an increase in [mu] leads to a higher value of [chi], the intuition for which is related to the optimal behavior of inflation. Finally, note that while the optimal policy responses depend on the stochastic process for real GDP growth (which determines the shock [p.sub.t] according to equation 57), the optimal weight [chi] does not.

Equation 60 shows that optimal monetary policy features inflation persistence (the optimal behavior of inflation is an AR(1) process). The optimal serial correlation is given by the debt maturity parameter [mu], which is the steady-state fraction of un-refinanced debt in each time period. The result is that inflation should return to its average value at the same rate at which debt is refinanced. At the extremes, one-period debt ([mu] = 0) corresponds to serially uncorrelated inflation, while perpetuities ([gamma] = 1, for which [mu] [approximately equal to] 1) correspond to near random-walk persistence of inflation.

To understand this, note that with one-period debt, the current bond yield [j.sub.t] disappears from the constraint (equation 48); thus, the only way that policy can affect [[??].sub.t] is through an unexpected change in current inflation. With longer-maturity debt, the range of policy options increases. Changes in current bond yields [j.sub.t] are also relevant in addition to current inflation, and the bond yield is affected by expectations of future inflation (equation 53). This can be seen from equation 55 where the sum [[summation].sup.[infinity].sub.l=0] [([beta][mu]).sup.l] ([E.sub.t][[pi].sub.t+l] - [E.sub.t-1][[pi].sub.t+l]) affects the debt gap, with [[mu].sup.l] being the fraction of existing debt that will not have been refinanced after l time periods. It is possible to use inflation that is spread out over time to influence the debt gap when [mu] > 0, not only inflation surprises.

Furthermore, this 'inflation smoothing' is optimal because the welfare costs of inflation are convex (inflation appears in the loss function [equation 56] as [[pi].sup.2.sub.t]), so the costs of a given cumulated amount of inflation are smaller when spread out over a number of quarters or years than when all the inflation occurs in just one quarter. This is analogous to the "tax smoothing" argument of Barro (1979). Interestingly, the argument shows that high degrees of inflation persistence need not be interpreted as a failure of policy. In contrast with the tax smoothing analysis, it is generally not optimal for inflation to display random-walk or near-random-walk persistence unless debt contracts are close to perpetuities. As the maturity parameter [mu] is reduced and thus [beta][mu] falls significantly below one, expectations of inflation far in the future have a smaller effect on bond yields than inflation in the near future. The further in the future inflation is expected to occur, the less effective it is at influencing real returns and thus the debt-to-GDP ratio.

Even if optimal monetary policy places a substantial weight [chi] on risk sharing compared to relative-price distortions, in what sense does monetary policy still resemble a nominal GDP target? It turns out that optimal monetary policy retains the essential feature of nominal GDP targeting in generating a negative co-movement between prices and output. However, because of the desire to smooth inflation, the central bank should not generally aim to stabilize nominal GDP (or a weighted measure of nominal GDP) exactly on a quarter-by-quarter basis. Instead, optimal policy can be formulated as a long-run target for weighted nominal GDP together with the inflation smoothing rule [E.sub.t][[pi].sub.t+1] = [mu][[pi].sub.t], implied by equation 60. When real GDP is nonstationary, optimal monetary policy features cointegration between the price level and output. If real GDP growth is the stationary and invertible stochastic process [g.sub.t] = [[summation].sup.[infinity].sub.l=0] [v.sub.l][[member of].sub.t-l], then

(62) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [[omega].sup.*] as given in equation 59. This cointegrating relationship can be interpreted as a long-run target for weighted nominal GDP, because there is some linear combination of the price level and real output that is invariant to shocks in the long run.

III.F. The General Case

In the general case where labor supply is elastic ([eta] > 0), monetary policy can also affect the output gap [[??].sub.t] through the Phillips curve (equation 49), and the ex-ante real interest rate [[rho].sub.t] through the IS curve in equation 52 (implied by equations 47 and 48). This is important not only because of a concern for stabilizing the output gap, given that it appears in the loss function (equation 56), but also because the output gap affects the debt gap [[??].sub.t] through the constraint (equation 48). The IS curve (equation 52) implies that the output gap and the ex-ante real interest rate [[rho].sub.t] are linked by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[??].sub.t] is the natural real interest rate. The coefficient [alpha] of the output gap in this equation is the source of the coefficient [alpha] of [[??].sub.t] in equation 48.

The solution to the optimal monetary policy problem in the general case is:

(63) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(64) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(65) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As before, the debt gap [[??].sub.t] is described by the AR(1) process in equation 63 with autoregressive root [lambda]. The response to the shock [p.sub.t] (from equation 55) is scaled by 1 - [chi], where [chi] in equation 64 is a number between zero and one that indicates the extent to which monetary policy replicates full risk sharing. Inflation [[pi].sub.t] is now the ARMA(2,1) process in equation 65 with autoregressive roots [mu] (as before) and x (satisfying 0 < x < 1), and a positive moving-average root. The response of inflation to the shock [p.sub.t] is scaled by [chi], where a value of [chi] of one indicates the inflation response that would replicate complete financial markets.

To understand the new aspects of the optimal monetary policy problem with elastic labor supply, note that policy can now influence three variables that have implications for the debt gap and thus the extent to which monetary policy supports risk sharing: inflation, the ex-ante real interest rate, and real GDR Inflation affects the ex-post real return on nominal bonds and thus the value of existing debt, as described earlier. The ex-ante real interest rate affects the ongoing costs of servicing debt relative to the stream of current and future labor income (formally, the ex-ante real interest rate influences the debt gap by changing the level of the debt-to-GDP ratio consistent with risk sharing). Real GDP (and hence the output gap) affects the denominator of the debt-to-GDP ratio. In the solution for the debt gap [[??].sub.t] in equation 55, these three channels correspond to the three terms appearing after the shock [p.sub.t].

An increase in the output gap [[??].sub.t] has the effect of directly boosting real GDP growth at time t and thus reducing the debt-to-GDP ratio, but the impact on the debt gap [[??].sub.t] is more subtle. Since monetary policy has only a temporary influence on real GDP, extra growth now reduces overall growth in the future by exactly the same amount. Given the link between growth expectations and the debt-to-GDP ratio consistent with risk sharing (the natural debt-to-GDP ratio [d.sup.*.sub.t] in equation 54), the effect of the output gap on the debt gap actually depends on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], not only on [[??].sub.t]. With the New Keynesian Phillips curve (equation 49), it is seen that this term is equal to (1 - [beta]) ([kappa]/v) [[pi].sub.t], the reciprocal of the long-run Phillips curve slope multiplied by current inflation, which explains the final term in equation 55. Since it is reasonable to assume the discount factor [beta] is close to one, this term is negligible for all practical purposes; it is not exactly zero because future growth is discounted relative to current growth, so by bringing growth forward, there is still a small positive effect. Monetary policy therefore cannot have a sustainable impact on the burden of debt simply through its temporary effect on the level of real GDP.

However, it does not follow that expansionary or contractionary monetary policy has no effect on the debt burden beyond its implications for ex-post real returns through inflation. There remains the option of changing ex-ante real interest rates. Intuitively, expansionary monetary policy that reduces the ex-ante real interest rate is effectively a transfer from savers to borrowers, what might be labeled "financial repression," even though the means by which monetary policy affects real interest rates is no different from the textbook New Keynesian model.

Changing ex-ante real interest rates thus provides monetary policy with an alternative to influencing the debt gap through the effect of inflation on ex-post real returns. In contrast to the latter, which is effective only while debt contracts are not refinanced, the former is effective only when refinancing does take place. For debt refinanced l periods after a shock at time t, the impact of monetary policy on the date-i debt burden is determined by the discounted sum of real interest gaps [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from t + l onwards, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Given the New Keynesian Phillips curve (equation 49) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implied by the IS curve (equation 52), these terms reduce to ([alpha][kappa]/v)[[beta].sup.l+1][E.sub.t][[[pi].sub.t+l+1]- [[pi].sub.t+l]]. This formula reveals that the gradient of the inflation path over time is an indicator of financial repression through ex-ante real interest rates, in contrast to the level of inflation that matters for the ex-post real return. With a (steady-state) fraction (1 - [mu])[[mu].sup.l] of debt issued prior to date t being refinanced at t + l, the overall effect of changes in ex-ante real interest rates on the debt burden is given by the unexpected change in [alpha][kappa]/v)(1-[mu])[beta] [[summation].sup.[infinity].sub.l=0] [([beta][mu]).sup.l][E.sub.t][[[pi].sub.t+l+1]- [[pi].sub.t+l]], which explains the penultimate term in equation 55.

This analysis shows that the more the inflation trajectory is smoothed, the smaller is the effect of monetary policy on ex-ante real interest rates, and the longer the maturity of debt, the less impact ex-ante real interest rate changes have on the debt gap. As has been seen earlier, when the maturity of debt increases, a policy of smoothing out changes in inflation is increasingly effective at influencing the ex-post real return on nominal debt at a low welfare cost in terms of relative price distortions (the Phillips curve in equation 49 implies that inflation smoothing also helps reduce output gap fluctuations). Significant financial repression will not be optimal with long-maturity debt, because it would require an inflation trajectory with a nonzero slope further in the future. Given the Phillips curve, the required inflation path would entail output gap fluctuations over a longer horizon, increasing the losses from following such a policy. But for short-maturity debt, where only immediate inflation surprises can affect ex-post real returns, which are much more costly than smoothed-out inflation, financial repression provides an additional tool for stabilizing the debt gap, with the losses from following this policy being small when the short-run Phillips curve is relatively flat.