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

IV. Quantitative Analysis of Optimal Monetary Policy

This section presents a quantitative analysis of the nature of the optimal monetary policy characterized in section III.

IV.A. Calibration

Let T denote the length in years of one discrete time period in the model. The numerical results presented here assume a quarterly frequency (T= 1/4). The parameters of the model are [beta], [theta], [alpha], [lambda], [eta], [mu], [epsilon], [xi], and [sigma]. As far as possible, these parameters are set to match features of U.S. data. (14) The baseline calibration targets and the implied parameter values are given in table 1 and justified below.

Consider first the parameters [beta] and [theta] (equations 40 and 41 show that the choice of these parameters is equivalent to specifying the patience parameters [[DELTA].sub.b] and [[DELTA].sub.s]). These parameters are calibrated to match evidence on the average price and quantity of household debt. The "price" of debt is the average annual continuously compounded real interest rate r paid by households for loans. As seen in equation 41, the steady-state growth-adjusted real interest rate is related to [beta]. Let g denote the average annual real growth rate of the economy. Given the length of the discrete time period in the model, 1 + [bar.[rho]] = [e.sup.rT] and 1 + [bar.g] = [e.sup.gT]. Hence, using equation 41, [beta] can be set thus:

(66) [beta] = [e.sup.-(r-g)T].

From 1972 through 2011 there were average annual nominal interest rates of 8.8 percent on 30-year mortgages, 10 percent on 4-year auto loans, and 13.7 percent on 2-year personal loans, while the average annual change in the personal consumption expenditure (PCE) price index over the same period was 3.8 percent. The average credit-card interest rate between 1995 and 2011 was 14 percent. For comparison, 30-year Treasury bonds had an average yield of 7.7 percent over the periods 1977-2001 and 2006-11. The implied real interest rates are 4.2 percent on Treasury bonds, 5 percent on mortgages, 6.2 percent on auto loans, 9.9 percent on personal loans, and 12 percent on credit cards. The baseline real interest rate is set to the 5 percent rate on mortgages, since these constitute the bulk of household debt. The sensitivity analysis considers values of r from 4 percent up to 7 percent. Over the period 1972-2011, used to calibrate the interest rate, the average annual growth rate of real GDP per capita was 1.7 percent. Together with the baseline real interest rate of 5 percent, this implies that [beta] [approximately equal to] 0.992 using equation 66. Since many models used for monetary policy analysis are typically calibrated assuming zero real trend growth, for comparison the sensitivity analysis also considers values of g between 0 percent and 2 percent.

The relevant quantity variable for debt is the ratio of gross household debt to annual household income, denoted by D. This corresponds to what is defined as the loans-to-GDP ratio [bar.l] in the model (the empirical debt ratio being based on the amount borrowed rather than the subsequent value of loans at maturity) after adjusting for the length of one time period (T years), hence D = [bar.l]T. Using the expression for [bar.l] in equation 42 and given a value of [beta], the equation can be solved for the implied value of the debt service ratio [theta]:

(67) [theta] = 2(1 - [beta])D/[beta]T.

Note that in the model, all GDP is consumed, so for consistency between the data and the model's prediction for the debt-to-GDP ratio, either the numerator of the ratio should be total gross debt (not only household debt), or the denominator should be disposable personal income or private consumption. Since the model is designed to represent household borrowing, and because the implications of corporate and government debt may be different, the latter approach is taken.

In the United States, as in a number of other countries, the ratio of household debt to income has grown significantly in recent decades. To focus on the implications of the levels of debt recently experienced, the model is calibrated to match average debt ratios during the five years from 2006 to 2010. The sensitivity analysis considers a wide range of possible debt ratios from 0 percent up to 200 percent. Over the period 2006-10, the average ratio of gross household debt to disposable personal income was approximately 124 percent, while the ratio of debt to private consumption was approximately 135 percent. Taking the average of these numbers, the target chosen is a model-consistent debt-to-income ratio of 130 percent, which implies (using equation 67) a debt service ratio of [theta] [approximately equal to] 8.6 percent.

For the coefficient of relative risk aversion [alpha], the survey evidence presented by Barsky and others (1997) suggests considerable risk aversion, but most likely not in the high double-digit range for the majority of individuals. Overall, the weight of evidence from this and other studies suggests a coefficient of relative risk aversion above one, but not significantly higher than 10. A conservative baseline value of 5 is adopted, and the sensitivity analysis considers values from zero up to 10.

One approach to calibrating the discount factor elasticity parameter [lambda] (from equation 28) is to select a value on the basis of its implications for the marginal propensity to consume from financial wealth. Let m denote the increase in per-household (annual) consumption of savers from a marginal increase in their financial wealth. It can be shown that m, [lambda], and [beta] are related as follows:

(68) [lambda] = 1 - mT/[beta].

Parker (1999) presents evidence to suggest that the marginal propensity to consume from wealth lies between 4 and 5 percent (for a survey of the literature on wealth and consumption, see Poterba 2000). However, it is argued by Juster and others (2006) that the marginal propensity to consume varies between different forms of wealth. They find that the marginal propensity to consume is lowest for housing wealth and larger for financial wealth. Given the focus on financial wealth in this paper, the baseline calibration assumes m [approximately equal to] 6 percent, which using equation 68 implies [lambda] [approximately equal to] 0.993. The sensitivity analysis considers marginal propensities to consume from 4 to 8 percent.

The range of available evidence on the Frisch elasticity of labor supply [eta] is discussed by Hall (2009), who concludes that a value of approximately 2/3 is reasonable. However, both real business cycle and New Keynesian models have typically assumed Frisch elasticities significantly larger than this, often as high as 4 (see King and Rebelo 1999; Rotemberg and Woodford 1997). The baseline calibration adopted here uses a Frisch elasticity of 2, and the sensitivity analysis considers a range of values for q from completely inelastic labor supply up to 4. With the assumption (equation 43) on the differences between the Frisch elasticities of borrowers and savers that ensures the wealth distribution has no impact on the aggregate supply of labor, the baseline calibration amounts to setting [[eta].sub.b] [approximately equal to] 1.6 and [[eta].sub.s] [approximately equal to] 2.6.

The debt maturity parameter [mu] (which given [mu] = [gamma]/(1 + [bar.n]) stands in for the parameter [gamma] specifying the sequence of coupon payments) is set to match the average maturity of household debt contracts. In the model, the average maturity of household debt is related to the duration of the bond that is traded in incomplete financial markets. Formally, duration [T.sub.m] refers to the average of the maturities (in years) of each payment made by the bond weighted by its contribution to the present value of the bond. Given the geometric sequence of nominal coupon payments parameterized by [gamma], the bond duration (in steady state) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let j denote the average annualized nominal interest rate on household debt, with 1 + [bar.j] = [e.sup.jT]. In the optimal policy analysis, the steady-state rate of inflation is zero ([bar.[pi]] = 0), hence nominal GDP growth is [bar.n] = [bar.g], and so [mu] = [gamma]/ (1 + [bar.g]). It follows that [gamma] and [mu] can be determined by

(69) [gamma] = [e.sup.jT] (1 - T/[T.sub.m]), and [mu] = [e.sup.-gt] [gamma].

Doepke and Schneider (2006) present evidence on the average duration of household nominal debt liabilities. Their analysis takes account of refinancing and prepayment of loans. For the most recent year in their data (2004), the duration lies between 5 and 6 years, and the duration has not been less than 4 years over the entire period covered by the study (1952-2004). This suggests a baseline duration of [T.sub.m] [approximately equal to] 5 years, which using equation 69 implies [mu] [approximately equal to] 0.967. The sensitivity analysis considers the effects of having durations as short as one quarter (one-period debt) and as long as 10 years.

There are two main strategies for calibrating the price elasticity of demand [epsilon]. The direct approach draws on studies estimating consumer responses to price differences within narrow consumption categories. A price elasticity of approximately 3 is typical of estimates at the retail level (see, for example, Nevo 2001), while estimates of consumer substitution across broad consumption categories suggest much lower price elasticities, typically lower than one (Blundell, Pashardes, and Weber 1993). Indirect approaches estimate the price elasticity based on the implied markup 1/([epsilon] - 1), or as part of the estimation of a DSGE model. Rotemberg and Woodford (1997) estimate an elasticity of approximately 7.9 and point out this is consistent with markups in the range of 10 to 20 percent. Since it is the price elasticity of demand that directly matters for the welfare consequences of inflation rather than its implications for markups as such, the direct approach is preferred here and the baseline value of [epsilon] is set to 3. A range of values is considered in the sensitivity analysis, from the theoretical minimum elasticity of 1 up to 10.

The production function is given in equation 32. If e denotes the elasticity of aggregate output with respect to hours, then the elasticity [xi], of real marginal cost with respect to output can be obtained from e using [xi] = (1 - e)/e. A conventional value of e [approximately equal to] 2/3 is adopted for the baseline calibration (this would be the labor share in a model with perfect competition), which implies [xi] [approximately equal to] 0.5. An important implication of [xi], is the strength of real rigidities, which are absent in the special case of a linear production function ([xi] - 0). The sensitivity analysis considers values of [xi] between 0 and 1.

In the model, [sigma] is the probability of not changing price in a given time period. The probability distribution of survival times for newly set prices is (1 - [sigma])[[sigma].sup.l], and hence the expected duration of a price spell [T.sub.p] (in years) is [T.sub.p] = T[[summation].sup.infinity].sub.l=1] l(1 - [sigma])[[sigma].sup.l] = T/(1 - [sigma]). With data on [T.sub.p], the parameter [sigma] can be inferred from:

(70) [sigma] = 1 - T/[T.sub.p].

Following Nakamura and Steinsson (2008), the baseline duration of a price spell is taken to be 8 months ([T.sub.p] [approximately equal to] 8/12), implying [sigma] [approximately equal to] 0.625. The sensitivity analysis considers average durations from 3 to 15 months.

IV.B. Results

Consider an economy hit by an unexpected permanent fall in potential output. Flow should monetary policy react? In the basic New Keynesian model with sticky prices but either complete financial markets or a representative household, the optimal monetary policy response to a total-factor-productivity shock is to keep inflation on target and allow actual output to fall in line with the loss of potential output. Using the baseline calibration from table 1 and the solution (equations 63-65) to the optimal monetary policy problem, figure 1 shows the impulse responses of the debt-to-GDP gap [[??].sub.t], inflation [[pi].sub.t], the output gap [[??].sub.t], and the bond yield [j.sub.t] under the optimal monetary policy and under a policy of strict inflation targeting for the 30 years following a 10 percent fall in potential output.

With strict inflation targeting, the debt-to-GDP gap rises in line with the fall in output (10 percent) because the denominator of the debt-to-GDP ratio falls while the numerator is unchanged. The effects of this shock on the wealth distribution and hence on consumption are long lasting. The serial correlation of the debt gap is equal to [lambda] [approximately equal to] 0.993, implying an average duration of approximately 36 years. Intuitively, with a marginal propensity to consume from financial wealth of 6 percent per year, consumption smoothing leads to persistence in the wealth distribution for far longer than typical business-cycle frequencies. Strict inflation targeting does ensure that the output gap is completely stabilized (the "divine coincidence"), and with no change in real interest rates or inflation, bond yields are completely unaffected.

The optimal monetary policy response is in complete contrast to strict inflation targeting. Optimal policy allows inflation to rise, which stabilizes nominal GDP over time in spite of the fall in real GDP. This helps to stabilize the debt-to-GDP ratio, moving the economy closer to the outcome with complete financial markets where borrowers would be insured against the shock and the value of debt liabilities would automatically move in line with income. The rise in the debt-to-GDP gap is very small (around 1 percent) compared to strict inflation targeting (10 percent). The rise in inflation is very persistent, lasting around two decades. The higher inflation called for is significant, but not dramatic: for the first two years, it is around 2-3 percentage points higher (at an annualized rate), for the next decade around 1-2 percent higher, and for the decade after that, around 0-1 percent higher. The serial correlation of inflation is due almost entirely to the autoregressive root [mu] [approximately equal to] 0.967 (the other autoregressive root is x [approximately equal to] 0.29, and the moving-average root is 0.41, which are much smaller and not far from canceling out as a common factor). The average duration of inflation is approximately 7 years, which is longer than typical business-cycle frequencies. Inflation that is spread out over time is still effective in reducing the debt-to-GDP ratio because debt liabilities have a long average maturity. It is also significantly less costly in terms of relative-price distortions to have inflation spread out over a longer time than the typical durations of stickiness of individual prices: this is the inflation-smoothing argument that drives the optimality of the autoregressive root [mu].

The rise in inflation does affect the output gap for the first one or two years, but this is short-lived because the duration of the real effects of monetary policy through the traditional price-stickiness channel is brief compared to the relevant time scale of decades for the other variables. The effect is also quantitatively small because inflation is highly persistent, the rise in expected inflation closely following the rise in actual inflation, so the Phillips curve implies little impact on the output gap. Finally, nominal bond yields show a persistent increase. It might seem surprising that yields do not fall as monetary policy is loosened, but the bonds in question are long-term bonds, and the effect on inflation expectations is dominant (there is a small fall in real interest rates because the rise in bond yields is less than what is implied by the higher expected inflation, but there is no significant "financial repression" effect).

The term [chi] from equation 64 provides a precise measure of the relative importance of risk sharing versus inflation stabilization under the optimal monetary policy (the response of the debt gap is a multiple 1 - [chi] of what it would be under strict inflation targeting, while the response of inflation is a multiple [chi] of what it would need to be to support full risk sharing). The baseline calibration leads to a policy weight [chi] on debt gap stabilization of approximately 89 percent and a policy weight 1 - [chi] on inflation stabilization of 11 percent.

The baseline calibration thus implies that addressing the problem of incomplete financial markets is quantitatively the main focus of optimal monetary policy rather than other objectives such as inflation stabilization. What explains this, and how sensitive is this conclusion to the particular calibration targets? Consider the exercise of varying each calibration target individually over the ranges discussed in section IV. A, holding all other targets constant. For each new target, the implied parameters are recalculated and the new policy weight [chi] is obtained. Figure 2 plots the values of [chi] (the optimal policy weight on risk sharing) obtained for each target.

As can be seen in figure 2, over the range of reasonable average real GDP growth rates and real interest rates there is almost no effect on the optimal policy weight. The results are most sensitive to the calibration targets for the average debt-to-GDP ratio and the coefficient of relative risk aversion. The average debt-to-GDP ratio proxies for the parameter [theta], which is related to the difference in patience between borrowers and savers. It is not surprising that an economy with less debt in relation to income has less of a concern with the incompleteness of financial markets, because in such a case the impact of shocks is felt more evenly by borrowers and savers. In the limiting case of a representative-household economy, the average debt-to-GDP ratio tends to zero, and the degree of completeness of financial markets becomes irrelevant. Risk sharing receives more than half the weight in the optimal policy as long as the calibration target for the average debt-to-GDP ratio is not below 50 percent.

It is also not surprising that the results are sensitive to the coefficient of relative risk aversion. Since the only use for complete financial markets in the model is risk sharing, if households were risk-neutral then there would be no loss from these markets being absent, as long as saving and borrowing remained possible in incomplete financial markets. The baseline coefficient of relative risk aversion is higher than the typical value of 2 found in many macroeconomic models (although that number is usually relevant for intertemporal substitution in those models, not for attitudes to risk), but it is low compared to the values often assumed in finance models that seek to match risk premia. The optimal policy weight on risk sharing exceeds 0.5 if the coefficient of relative risk aversion exceeds 1.3, so lower degrees of risk aversion do not necessarily overturn the conclusions of this paper.

The next most important calibration target is the price elasticity of demand. A higher price elasticity increases the welfare costs of inflation. Welfare ultimately depends on quantities, not prices, but the price elasticity determines how much quantities are distorted by dispersion of relative prices. To reduce the optimal policy weight on the debt gap below one-half it is necessary to assume price elasticities in excess of 10. Such values would be outside the range typical in IO and microeconomic studies of demand, with 10 itself being at the high end of the range of values used in most macroeconomic models. The typical value of 6 often found in New Keynesian models only reduces [chi] to approximately 71 percent.

The results are largely insensitive to the marginal propensity to consume from financial wealth, which is used to determine the parameter [lambda] in the specification of the endogenous discount factors. The Frisch elasticity of labor supply has a fairly small but not insignificant effect on the results, with the optimal policy weight on risk sharing increasing with the Frisch elasticity. A higher elasticity increases the welfare costs of shocks to wealth distribution by distorting the labor supply decisions of different households, as well as making it easier for monetary policy to influence the real value of debt by changing the ex-ante real interest rate in addition to inflation. An elastic labor supply does mean that inflation fluctuations lead to output gap fluctuations, which increases the importance of targeting inflation, but the first two effects turn out to be more important quantitatively.

The results are somewhat more sensitive to the average duration of a price spell and the elasticity of real marginal cost with respect to output. The first of these determines the importance of nominal price rigidities. Greater nominal rigidity leads to more dispersion of relative prices from a given amount of inflation, and thus reduces the optimal policy weight on the debt gap. A higher output elasticity of marginal cost implies that the production function has greater curvature, so a given dispersion of output levels across otherwise identical firms represents a more inefficient allocation of resources. However, considering the range of reasonable values for the duration of price stickiness does not reduce [chi] below 65 percent, and the range of marginal cost elasticities does not lead to any [chi] value below 80 percent.

The effects of the calibration target for the average duration of household debt are more subtle. It might be expected that the longer the maturity of household debt, the higher the optimal policy weight on risk sharing. This is because longer-term debt allows inflation to be spread out further over time, reducing the welfare costs of the inflation, yet still having an effect on the real value of debt. However, the sensitivity analysis shows that the optimal policy weight is a non-monotonic function of debt maturity: either very short-term or long-term debt maturities lead to high values of [chi], while debt of around 1.5 years maturity has the lowest value of [chi] (approximately 75 percent).

This apparent puzzle is resolved by recalling that there are two ways monetary policy can affect risk sharing: inflation to change the ex-post real return on nominal debt, and changes in the ex-ante real interest rate ("financial repression"). As has been discussed, the first method is effective at a lower cost for long debt maturities. When labor supply is inelastic, the second method is not available, and the value of [chi] is then indeed a strictly increasing function of debt maturity (with the value of [chi] falling to 15 percent for the shortest-maturity debt). When the ex-ante real interest rate method is available, it is most effective compared to the first method (taking account of the costs in terms of inflation and output gap fluctuations) when debt maturities are short.

Finally, it is possible to calculate the magnitude of the losses from following a policy of strict inflation targeting rather than the optimal policy described above. With strict inflation targeting, equation 55 shows that the innovation to the debt gap is given by the negative of the shock [p.sub.t], with the effect on the debt gap in subsequent periods being -[lambda] [p.sub.t], -[[DELTA].sup.2] [p.sub.t], and so on. The welfare loss (as an equivalent percentage of GDP) from strict inflation targeting according to the loss function (equation 56) is therefore equal to [p.sup.2.sub.t] multiplied by the coefficient of [[??].sup.2.sub.t] in equation 56 divided by (1 - [beta][[lambda].sup.2]).

Using the baseline calibration, a 1-percent shock to the debt gap results in a total loss equivalent to 0.023 percent of one year's GDP, a 5-percent shock results in a 0.58-percent GDP loss, and a 10-percent shock results in a 2.3-percent loss. These losses are not inconsiderable for large shocks, but are negligible for small shocks. With a higher relative risk aversion of 10, the losses from the 1-percent, 5-percent, and 10-percent shocks would be 0.078 percent, 2.0 percent, and 7.8 percent of GDP, respectively. The expected loss per year is obtained by averaging these over the probability distribution of [p.sub.t], shocks occurring during a year, which can be derived from the stochastic process for

real GDP using equation 55. Even though losses from large shocks are significant, fortunately the U.S. economy only rarely experiences shocks of the order of magnitude seen during the financial crisis. Using the 2.1-percent standard deviation of annual real GDP growth over the period 1972-2011 suggests that the average annual loss from strict inflation targeting would lie in the range 0.1-0.3 percent of GDP.

If the average welfare loss from the lack of risk sharing under strict inflation targeting is so small, how is it possible that concerns over risk sharing receive such a high weight relative to inflation stabilization in the optimal monetary policy? The small expected loss might suggest that there should be little willingness to pay to obtain insurance. However, note that the optimal policy only deviates significantly from inflation targeting when large shocks occur (figure 1 is drawn for a 10-percent shock to potential output). The inflation fluctuations called for in a typical year are around five times smaller than those shown in figure 1 and would likely involve (annualized) inflation being not much more than 0.4 percent from its average, for which the welfare losses are vanishingly small.

This means it is possible to put a high weight on replicating complete financial markets even when the expected gains from risk sharing are small because, unlike an insurance premium, a non-negligible cost of inflation fluctuations is incurred only when large shocks occur, which is also when the gains from risk sharing are large. Combined with inflation smoothing to keep down the welfare losses from relative-price distortions when nominal debt has a long average maturity, this means the benefits of greater risk sharing from a long-term nominal GDP target can outweigh the costs even without assuming double-digit coefficients of relative risk aversion.

V. Conclusion

This paper has shown how a monetary policy of nominal GDP targeting facilitates efficient risk sharing in incomplete financial markets where contracts are denominated in terms of money. In an environment where risk derives from uncertainty about future real GDP, strict inflation targeting would lead to a very uneven distribution of risk, with leveraged borrowers' consumption highly exposed to any unexpected change in their incomes when monetary policy prevented any adjustment of the real value of their liabilities. Strict inflation targeting does provide savers with a risk-free real return, but fundamentally, the economy lacks any technology that delivers risk-free real returns, so the safety of savers' portfolios is simply the flip side of borrowers' leverage and high levels of risk. Absent any changes in the physical investment technology available to the economy, aggregate risk cannot be annihilated, only redistributed.

That leaves the question of whether the distribution of risk is efficient. The combination of incomplete markets and strict inflation targeting implies a particularly inefficient distribution of risk when households are risk averse. If complete financial markets were available, borrowers would issue state-contingent debt where the contractual repayment was lower in a recession and higher in a boom. These securities would resemble equity shares in GDP, and they would have the effect of reducing the leverage of borrowers and hence distributing risk more evenly. In the absence of such financial markets, in particular because of the inability of households to sell such securities, a monetary policy of nominal GDP targeting could effectively replicate complete financial markets even when only noncontingent nominal debt was available. Nominal GDP targeting operates by stabilizing the debt-to-GDP ratio. With financial contracts specifying liabilities fixed in terms of money, a policy that stabilizes the monetary value of real incomes ensures that borrowers are not forced to bear too much aggregate risk, converting nominal debt into real equity.

While the model is far too simple to apply to the recent financial crises and deep recessions experienced by a number of economies, one policy implication does resonate with the predicament of several economies faced with high levels of debt combined with stagnant or falling GDPs. Nominal GDP targeting is equivalent to a countercyclical price level, so the model suggests that higher inflation can be optimal in recessions. In other words, while each component of the word "stagflation"--"stagnation" and "inflation"--is bad in itself, if stagnation cannot immediately be remedied, some inflation might be a good idea to compensate for the inefficiency of incomplete financial markets. And even if policymakers were reluctant to abandon inflation targeting, the model does suggest that they would have the strongest incentives to avoid deflation during recessions (a procyclical price level). Deflation would raise the real value of debt, which combined with falling real incomes would be the very opposite of the risk sharing stressed in this paper, and even worse than an unchanging inflation rate.

It is important to stress that the policy implications of the model in recessions are matched by equal and opposite prescriptions during an expansion. Thus, it is not just that optimal monetary policy tolerates higher inflation in a recession--it also requires lower inflation or even deflation during a period of high growth. Pursuing higher inflation in recessions without following a symmetrical policy during an expansion is both inefficient and jeopardizes an environment of low inflation on average. Therefore, the model also argues that more should be done by central banks to "take away the punch bowl" during a boom, even were inflation to be stable.

ACKNOWLEDGMENTS I am grateful to Carlos Carvalho, Wouter den Haan, Monique Ebell, Cosmin Ilut, John Knowles, Greg Mankiw, Albert Marcet, Matthias Paustian, George Selgin, the editors, and my discussants for helpful suggestions and comments. The paper has also benefited from the comments of Brookings Panel participants and seminar participants at Banque de France, University of Cambridge, CERGE-EI, Ecole Polytechnique, University of Lausanne, University of Maryland, National Bank of Serbia, National University of Singapore, Federal Reserve Bank of New York, University of Oxford, PUC-Rio, Sao Paulo School of Economics, University of Southampton, University of St. Andrews, University of Warwick, the Anglo-French-Italian Macroeconomics Workshop, Birmingham Econometrics and Macroeconomics Conference, Centre for Economic Performance Annual Conference, Econometric Society North American Summer Meeting, EEA Annual Congress, ESSET, ESSIM, Joint French Macro workshop, LACEA, LBS-CEPR conference "Developments in Macroeconomics and Finance," London Macroeconomics Workshop, Midwest Macro Meeting, and NBER Summer Institute in Monetary Economics. I have no relevant material or financial interests to declare regarding the content of this paper.

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COMMENT BY

JAMES BULLARD (1)

Modern rationales for monetary stabilization policy rely mainly on the sticky price friction. Sticky prices are thought to prevent the market solution from being fully optimal and therefore suggest a role for monetary policy intervention. Generally speaking, leading renditions of this idea lead to the monetary policy advice that prices should be stabilized along a price level path. In this fascinating paper, Kevin Sheedy studies an alternative rationale for monetary stabilization policy. In the alternative, the friction is nonstate contingent nominal contracting (NSCNC), and it is this defect of credit markets that keeps the market solution from being fully optimal. The monetary policy advice associated with this rationale is somewhat different from that associated with sticky prices. Rather than keeping prices stable along a price level path, the advice calls for deliberate movements in the price level in order to offset shocks to the growth rate of national income--countercyclical price level movements.

Sheedy has laid out considerable intuition for the alternative rationale. I would go so far as to say that he has set the standard for future analyses in this area. The paper includes valuable commentary on an extensive related literature, and it includes a calibrated model with both sticky price and NSCNC frictions included. In the calibrated case, the more important of the two frictions is associated with nonstate contingent nominal contracting.

Is it surprising that the NSCNC friction can be more important from a policymaking perspective than the sticky price friction? Perhaps not. According to Atif Mian and Amir Sufi (2011), the ratio of household debt to GDP in the United States was about 1.15 before debt rose in the 2000s, when it ballooned to 1.65 or more. In today's dollars, the latter ratio would mean about \$19.5 to \$28 trillion in debt, comprising mostly mortgage debt. Improper functioning in these markets might be quite costly for the economy, so it is certainly plausible that the nonstate contingent nominal contracting friction could be quite important.

My discussion is organized around three questions. First, given that some may view the model here as somewhat special, would these results hold in a model with many more heterogeneous participants interacting in a large private credit market? Based on a general equilibrium life cycle model with many period lives, the tentative answer seems to be yes, so that Sheedy's results may have more general applicability than it might first appear. The second question is: What are some of the key issues on which future research in this area may wish to focus? And finally, what does this paper have to say in framing the ongoing global monetary policy debate on the wisdom of nominal GDP targeting?

IS THE MODEL SPECIAL? The Sheedy model has two types of households: relatively patient and relatively impatient. Since there are just two types of agents interacting in a credit market, there is only one set of marginal conditions that requires "repair." The policymaker has just one tool, the price level, which in certain circumstances neatly fixes the marginal conditions. A natural question is whether these results would carry over to a more realistic environment with more heterogeneity in the private credit market. My tentative answer is that the results do carry over to a somewhat different class of models with a greater degree of heterogeneity, and therefore that the Sheedy results have greater applicability than may be initially apparent.

One way to investigate this is to consider a stripped-down, endowment general equilibrium life-cycle economy. (2) In order to stress that business cycle questions can be addressed with such a model, I will use a "quarterly" specification, with households living 241 periods. One interpretation would be that households enter economic life around age 20, die around age 80, and are most productive in the middle period, around age 50. (3) To this standard framework we can add the key assumption made by Sheedy, namely that loans are dispersed and repaid in the unit of account--that is, in nominal terms--and are not contingent on income realizations. This is the NSCNC friction. Agents in the economy I am describing are endowed with an identical productivity profile over their lifetime. This productivity profile begins at zero, rises to a peak at the middle period of life, and then declines to zero, exhibiting perfect symmetry. Agents can sell the productivity units they have in the labor market at a competitive wage.

Such a model, which is very standard in textbooks, produces very uneven income over the life cycle. People near the beginning or end of the life cycle earn little or no labor income, while those in the middle of life earn a lot. If the productivity profile were exactly triangular, then 50 percent of the households would earn 75 percent of the income. All of these cohorts will wish to use the credit market to smooth consumption relative to income.

A second key assumption in the Sheedy paper is that there is an aggregate shock, and that this shock is the only source of uncertainty. (4) Let us think of this as a Markov process for the aggregate gross rate of real wage growth, which can take on values of high, medium, or low with equal probability and where the medium value is the average of the three possibilities. Real national income is then the real wage multiplied by the sum of the productivity endowments. Therefore, the growth rate of real wages is also the growth rate of real output. The policymaker completely controls the price level, which is just a unit of account in this model. (5) An important within-period timing protocol is embedded: (i) nature chooses the growth rate, (ii) the policymaker chooses a price level, and (iii) households make decisions to consume and save. This timing protocol is what allows the policymaker to potentially offset incoming shocks.

The model I have described is simple, but it is interesting in light of what Sheedy teaches us about the effects of the NSCNC friction. The version I have described has 241 households, all credit market users, each with a different level of asset holding depending on their position in the life cycle. To calculate the full stochastic equilibrium, one has to keep track of the distribution of asset holdings over time, a fact that has made models in this class less intensively studied than their representative agent cousins. Yet Sheedy's key insights apply to this model as well, even though there are now many more agents and the policymaker still only has one tool, the price level.

Consider a nonstochastic balanced-growth path of the model I have described, in which the economy simply grows at the average rate forever. Assume also that the policymaker "gets out of the way" and simply sets the price level to unity every period. On this balanced growth path, consumption is exactly equalized across the cohorts living at a point in time. The real interest rate is exactly equal to the real output growth rate. (6) The private credit market completely solves the point-in-time income inequality problem. Sheedy provides some excellent intuition for results like this, which fuel the findings later: The exact consumption equality across cohorts living at a given date means all households have an "equity share" in the economy. That is, despite their very uneven incomes at a point in time, they all consume an equal fraction of national income available at that date. Equity share contracts are known to be optimal when preferences are homothetic, as they are in the economy I have described. In the stochastic case, the main idea is to replicate this equity share outcome.

COUNTERCYCLICAL PRICE LEVEL MOVEMENTS. In the stochastic case, the NSCNC friction means that markets are incomplete. Households are not allowed to contract based on actual realized returns. There is no default or renegotiation--loans must be repaid. However, because of the timing protocol, the policymaker can potentially provide state-contingent movements in the price level after observing the shock each period, and therefore restore the complete credit markets outcome.

The nature of this policy involves countercyclical price-level movements. A period associated with a high real growth shock is also a period with a lower-than-normal price level, and conversely, low growth is associated with a higher-than-normal price level. This policy restores complete markets because, in the economy I have described, each cohort living at date t would consume the same amount, and this amount would be higher or lower according to whether the growth rate was particularly high or low at that date. In this sense, Sheedy's results may have important applications in a wide class of life-cycle economies, probably the most important class of heterogeneous agent economies in macroeconomics.

The countercyclical price-level policy seems very different from one focused on not allowing the price level to deviate far from a price-level path. We might think of the price-level targeting policy here as maintaining P (t) equal to unity at all times. (7) As Sheedy stresses in the paper, such a policy would be inappropriate given the NSCNC friction.

DIRECTIONS FOR FUTURE RESEARCH. The Sheedy model has little to say about average inflation rates. This is important, since nominal GDP targeting is sometimes casually discussed in a way that suggests a rationalization for higher average inflation. The Sheedy model calls for higher-than-average inflation at certain points in time, notably in bad times, but also calls for lower-than-average inflation in good times, leaving the average rate of inflation unchanged in the long run.

It is sometimes asserted in discussions of nominal GDP targeting that one can simply target nominal GDP and not worry about the decomposition between real output and the price level. I do not see much support for this view in the logic behind the Sheedy analysis. (8) The typical statement might be that the policymaker could target a nominal GDP aggregate gross growth rate, perhaps without knowing the exact value of the average real growth rate of the economy. This indeed succeeds in obtaining the counter-cyclical price-level movements necessary under the NSCNC friction to complete the credit market. But it does not succeed in maintaining the average rate of inflation at a desirable level for the economy. In particular, such an approach would suggest that the balanced growth path with a gross real growth rate of 1.02 and a gross inflation rate of 1.02 was equally as desirable as the balanced growth path with 1.00 and 1.04, respectively. Yet this would only be true if there were no welfare costs of inflation in the model.

The literature on the welfare cost of inflation is well established and argues for lower average inflation as opposed to higher values, all else equal. The Sheedy model does not provide any reason to choose the higher average inflation value. It only provides a reason to generate higher-than-average inflation in response to certain shocks and lower-than-average inflation in response to other shocks. I conclude from this that proper implementation of the Sheedy nominal GDP targeting strategy would require knowledge of the average real growth rate for the economy. One would have to know when real growth was "below normal" or "above normal" in order to know when to generate the required price-level movement to maintain complete credit markets. If the policymaker did not know the average growth rate of the economy and targeted only a nominal GDP growth rate, the policymaker could end up with an average inflation rate considerably different from the desired level. This could undo all the good done by the complete credit markets policy.

I think further research on the trade-off between the benefits of targeting a pure nominal quantity and the costs of inadvertently generating higher-than-desirable inflation could be a fruitful area of future research. I caution potential researchers, however: The literature on the welfare costs of inflation tends to find that the welfare losses from higher average inflation are much larger than the welfare gains reported in Sheedy's paper from improved monetary stabilization policy.

Many have argued that the NSCNC friction is not as compelling as it may first appear. This is because we do observe default in actual economies, and because of this there is a certain state-contingency in actual contracting that is assumed away in models like Sheedy's. Research along the lines of Sheedy's that could make better contact with the issue of default could provide helpful insight. More subtly, the mere threat of default can radically shape equilibrium outcomes, even in models where no default occurs in equilibrium. For an example of how endogenous debt constraints change one's view of potential equilibria in a life cycle setting like the one described above, see Costas Azariadis and Luisa Lambertini (2003). More research in this area would be desirable as well, especially if it could shed more and sharper light on the likely importance or unimportance of what seems to be non-state-contingent contracting in actual economies.

Finally, Sheedy's model has no money demand, treating the role of money only as a unit of account. What Sheedy is advocating is a policy that focuses on completing the credit market and ignores households that are holding money balances as a large fraction of their wealth. The people who are in this latter situation may be hurt economically by a monetary policy sharply focused on credit markets. In the United States, some estimates suggest 10 to 15 percent of the population is unbanked, and another 10 to 15 percent may be nearly unbanked. These households tend to be poor and to use cash intensively, and they may be shut out of credit markets. Research on models that include this group may provide a better balance in assessing the best role for monetary policy.

This paper has considerable potential to sharpen the ongoing debate on nominal GDP targeting, an idea that has not often had an explicit modem macroeconomic model behind it. Those interested in studying nominal GDP targeting can proceed from the Sheedy model and study the many additional issues that could arise if policymakers adopted the idea of countercyclical price-level movements as optimal monetary policy. Others can investigate the extent to which NSCNC may or may not be as important a friction as it appears to be, perhaps because of the way credit default is conceptualized and modeled. Both types of research would likely improve our understanding of the NSCNC friction and monetary policy's role in alleviating it.

REFERENCES FOR THE BULLARD COMMENT

Azariadis, Costas, and Luisa Lambertini. 2003. "Endogenous Debt Constraints in Life Cycle Economies." Review of Economic Studies 70, no. 3: 1-27.

Koenig, Evan. 2013. "Like a Good Neighbor: Monetary Policy, Financial Stability, and the Distribution of Risk." International Journal of Central Banking 9: 57-82.

Mian, Atif, and Amir Sufi. 2011. "House Prices, Home Equity-Based Borrowing, and the U.S. Household Leverage Crisis." American Economic Review 101, no. 5: 2132-56.

Sheedy, Kevin. 2013. "Debt and Incomplete Markets: A Case for Nominal GDP Targeting." Discussion Paper no. 1209, Centre for Economic Performance, May.

(1.) Any views expressed are my own and do not necessarily reflect the views of others on the Federal Open Market Committee. I appreciate the valuable comments I have received on these remarks from the editors.

(2.) See Sheedy (2013) for a three-period overlapping-generations version of this paper.

(3.) I have in mind a model with a long list of simplifying assumptions: identical within-cohort agents, no population growth; inelastic labor supply; time-separable log preferences; no discounting; no capital; no default; flexible prices; no borrowing constraints; and no government other than the central bank.

(4.) In models like the one I am describing, it is also popular to include idiosyncratic uncertainty, but that is not necessary for the argument presented in the Sheedy paper.

(5.) For a two-period example along this line, see Koenig (2013).

(6.) This is due to the symmetric endowment pattern combined with other simplifying assumptions.

(7.) Assuming a net inflation target of zero.

(8.) The total real output in the economy at a date t would be the real wage multiplied by the sum of endowments, and the latter would cancel in this expression.

COMMENT BY

IVAN WERNING (1)

This paper by Kevin Sheedy argues that risk sharing should be an important goal in the conduct of monetary policy. It makes two distinct contributions in this direction. First, it presents a tractable model in which inflation affects risk sharing, applying this to derive implications for monetary policy. Nominal GDP targeting is shown to achieve optimal risk sharing in incomplete market settings with flexible prices. Second, it pits the new risk sharing goal for monetary policy against the traditional stabilization role. For a calibrated New Keynesian economy featuring sticky prices, the paper finds that significant weight should be placed on the risk sharing goal, affecting the reaction to technology shocks.

These ideas are important, and the effort to push standard representative-agent macroeconomic models to incorporate heterogeneous agents is laudable and, here, accomplished very elegantly. The paper helps create a bridge between the monetary policy literature, typically divorced of risk-sharing considerations, with a literature focusing on risk sharing that is typically divorced of monetary and nominal considerations.

Sheedy definitely succeeds at making one think about risk sharing and monetary policy in a more systematic way. As a discussant, I found little to disagree with within the confines of the paper's setting. However, I do believe that a few important elements are missing and that they need to be incorporated to assess the appropriateness of risk sharing as a goal for monetary policy.

I will begin by restating the main idea of risk sharing with flexible prices in a simple static model. I then incorporate heterogeneous risk exposures and idiosyncratic uncertainty, two features that I believe are crucial to any discussion of risk sharing. These may weaken the case for nominal GDP targeting in particular, although not necessarily for inflation-induced risk sharing in general. Finally, I briefly touch on elements that may affect the trade-off between risk sharing and stabilization. I conclude that, rather than being competing goals, risk sharing and stabilization may be complementary ones.

RISK SHARING AND NOMINAL GDP TARGETING. Let me reduce the argument for inflation-induced risk sharing and nominal GDP targeting to its bare essentials, within a static risk-sharing model.

Two agents, B (borrowers) and S (savers), have a common utility function u(c). Income is distributed proportionally, with [y.sub.B] = [[psi].sub.B]Y and [y.sub.s] = [[psi].sub.S]Y, assuming that [[psi].sub.B] > [[psi].sub.S].

Let us assume, momentarily, that a conditional transfer T(Y) is available. The planning problem is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[lambda].sub.B] and [[lambda].sub.S] are Pareto weights. From now on I specialize to equal weights [[lambda].sub.B] = [[lambda].sub.S] = 1, since nothing of interest is lost by doing so.

The expectation above is taken over aggregate income K However, the maximization can be performed for each realization of aggregate income Y,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The optimum equalizes consumption by setting

(1) T(Y) = [[psi].sub.B] - [[psi].sub.S] / 2 Y.

One can implement this optimal state-contingent transfer using nominal debt, D, and a state-contingent price level, P(s), satisfying

T(Y) = D / P(Y),

or substituting using our solution (equation 1),

Y x P(Y) = 2 / [[psi].sub.B] - [[psi].sub.S] D,

a constant value for nominal spending. Optimal policy can be characterized as targeting nominal GDP.

HETEROGENEOUS EXPOSURE TO AGGREGATE RISK. Following the paper, I assumed above that individual income moves in proportion with aggregate income--the elasticity of individual income with respect to aggregate income is unity. Consider instead

[y.sub.B] = [[phi].sub.B] (Y),

[y.sub.S] = Y - [[phi].sub.B] (Y),

for some function [[phi].sub.B] (*). The elasticity of the borrower's income to aggregate income may now depart from one. By the same arguments I obtain

T(Y) = [[phi].sub.B] (Y) - [1/2] Y.

This shows that, in general, T(Y) is no longer proportional to aggregate income Y. As long as T(Y) does not change signs, I can implement the transfers by T(s) = D/P(s) for some P(s) > 0. Let us assume this is the case. By implication, it is no longer the case that nominal spending Y x P(Y) is constant. Instead,

([[phi].sub.B] (Y) - 1/2 Y) P(Y) = D.

For example, if the income of borrowers is more responsive to aggregate income, so that the elasticity of [[phi].sub.B] (Y) is greater than one, then the price level P(Y) should also have an elasticity greater than one in absolute value. That is, the price level should be more responsive than nominal GDP targeting.

IDIOSYNCRATIC UNCERTAINTY. The paper abstracts from idiosyncratic income risk. This is unfortunate, because it is well appreciated that the uncertainty households face trumps aggregate uncertainty.

To incorporate idiosyncratic uncertainty, let us assume

[y.sub.Bi] = [[epsilon].sub.Bi] and [y.sub.Si] = [[epsilon].sub.Si]Y,

where [[epsilon].sub.Bi] and [[epsilon].sub.Si], are idiosyncratic realizations for individual i within each respective group. The case without idiosyncratic uncertainty is now a special case where Var [[epsilon]] = 0. One important example of [epsilon] may be a specification that captures unemployment risk, with [epsilon] = 0 when the agent is unemployed.

The planning problem is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order condition is

(2) [E.sub.[epsilon]] [u'([[epsilon].sub.B] Y - T(Y))|s] = [E.sub.[epsilon]] [u'([[epsilon].sub.S]Y + T(Y))|Y],

which equalizes average marginal utility after each realization of aggregate income Y. As before, as long as T(Y) does not change signs we can implement the transfers by T(s) = D/P(s) for some P(s) > 0. Let us assume this is the case.

I want to investigate whether

(3) T(s) = [tau]Y (s)

for some [tau]. For this purpose, it is helpful to assume a homogeneous utility function u(c) = [c.sup.1-[sigma]]/(1 - [sigma]). Substituting the guess (equation 3) into equation 2, one finds that validating the guess requires

[E.sub.[epsilon]] [([[epsilon].sub.B] - [tau]).sup.-[sigma]] | Y] = [E.sub.[epsilon]] [[([[epsilon].sub.S] + [tau]).sup.-[sigma]] | Y]

to hold for some fixed [tau] for all realizations of Y This is not generally possible, except in the special case where [[epsilon].sub.B] and [[epsilon].sub.S] are independent of Y. There is a large empirical literature documenting the fact that idiosyncratic uncertainty varies over the business cycle. When idiosyncratic shocks are not independent of aggregate ones, the optimal policy for P(Y) will not target nominal GDP.

IS INFLATION THE RIGHT TOOL? Idiosyncratic uncertainty also highlights how imperfect inflation--or any tool that depends only on aggregates and does not condition on idiosyncratic shocks--is for risk-sharing purposes. Other policies, such as progressive taxes or unemployment insurance benefits, do provide insurance against idiosyncratic uncertainty.

Equally relevant, the paper assumes that borrowers take on debt that is no-contingent and free of risk. However, as the recent crisis reminds us, both secured and unsecured consumer credit is not risk-free, and borrowers default on both forms of debt, providing a form of state contingency that is tailored to idiosyncratic conditions.

Overall, for these reasons, inflation is a relatively coarse tool for dealing with the uncertainty that households face. It may be argued, however, that once other available instruments are exhausted, there remains a residual role to be played by inflation. Knowing just how significant that role should be is crucial if one is going to have monetary policy deviate from its traditional role.

(1.) My views were enriched by exchanges with Adrien Auclert and Matt Rognlie.

GENERAL DISCUSSION Robert Shimer asked an empirical question: how tightly is the consumption decline during a recession linked to an individual's debt load? He thought the individuals who experienced the largest declines in consumption might have no debt load on account of their nonparticipation in credit markets. Shimer thought that smoothing out inflation would not be very effective at helping this group of people. It seemed to him that in principle this was something one could address using existing data.

He also agreed with discussant Ivan Werning that using inflation alone in the model is not going to work. As he understood the model, even the optimal rate of inflation was still not going to have better than a second-order benefit. If it turned out that the modeling was far off because of incomplete risk sharing, one could imagine achieving a first-order benefit from inflation. But which way it goes is going to depend on how consumption declines during recessions or is related to debt holdings, and Shimer acknowledged that he did not know how that works.

Johannes Wieland asked the author to clarify the trade-off between inflation targeting and nominal GDP targeting. He wondered if it would be better to target inflation volatility instead of GDP.

Gerald Cohen found the concept of a natural rate of debt to GDP to be rather frightening. He said he had not been able to find a theoretical justification for any particular ratio of debt to GDP. Ever since 2008, people have been talking about the economy needing to be deleveraged, but Cohen said that whenever he asked others what the optimal ratio of debt to GDP would be, they could only wave their hands or, in his view, invent a number. People once talked about a ratio of 130 percent, but looking in retrospect today most people think that number is too high, and now a common figure is 90 percent. Cohen's feeling was that if one is going to target an optimal ratio one ought to have a good theory behind it.

Justin Wolfers wondered why financial incompleteness existed at all. He speculated that when people obtained a mortgage they might often wonder why it was that the real value of the payoffs would be allowed to decline over the course of the mortgage. After all, one could simply write a debt contract in real terms, and if half the mortgage industry began to do that the rest would certainly follow suit.

Kevin Sheedy responded to the comments made during this brief discussion. First, he pointed out that he used a 10 percent fall in potential output in his paper--certainly a huge shock but a reasonable size relative to people's expectations of the trends prior to the financial crisis. And when he did, he found that such a shock led to only a 2-percentage-point increase in inflation over a decade. Since a shock of that magnitude is rare, Sheedy suggested that there would be little impact on inflation volatility. In his view, then, during a "great moderation" period there would not be any tension between targeting nominal GDP and targeting inflation. So the policy of nominal GDP targeting would be a good one when the economy needed it, when it was hit with really big real shocks, and when the need was not there one would not have to tolerate a lot of inflation volatility. This is key to explaining why the weight on risk sharing is so high.

Additionally, Sheedy noted, with long-term debt contracts inflation is smoothed out over time, so there is less relative price distortion and less aggregate volatility. So although the benefits of the policy may be small when one considers other factors, such as idiosyncratic risk, the costs of achieving the policy would also be relatively small. Lastly, he agreed with some discussants that financial market frictions might be entirely removed at some point in the future, but he did not believe that would occur soon. Given the prevalence of nominal debt contracts, he believed the case for nominal GDP targeting was still strong.

KEVIN D. SHEEDY

London School of Economics

(1.) In addition to the theoretical case, the more practical merits of implementing inflation targeting are discussed by Bemanke and others (1999).

(2.) Persson and Svensson (1989) is an early example of a model--in the context of an international portfolio allocation problem--where it is important how monetary policy affects the risk characteristics of nominal debt.

(3.) There is also a literature that emphasizes the impact of monetary policy on the financial positions of firms or entrepreneurs in an economy with incomplete financial markets. De Fiore, Teles, and Tristani (2011) study a flexible-price economy where there is a costly state verification problem for entrepreneurs who issue short-term nominal bonds. Andres, Arce, and Thomas (2010) consider entrepreneurs facing a binding collateral constraint who issue short-term nominal bonds with an endogenously determined interest rate spread. Vlieghe (2010) also has entrepreneurs facing a collateral constraint, and even though they issue real bonds, monetary policy still has real effects on the wealth distribution because prices are sticky, so incomes are endogenous.

(4.) This point is made by Lustig, Sleet, and Yeltekin (2008) in the context of government debt.

(5.) Woodford (2001) uses this modeling device to study long-term government debt. See Garriga, Kydland, and Sustek (2013) for a richer model of mortgage contracts.

(6.) The wage-bill subsidy is a standard assumption which ensures the economy's steady state is not distorted (Woodford 2003). A balanced-budget rule is assumed to avoid any interactions between fiscal policy and financial markets.

(7.) Online appendixes for this volume may be found at the Brookings Papers website, www.brookings.edu/bpea, under "Past Editions."

(8.) Note that the natural debt-to-GDP ratio is not independent of monetary policy when monetary policy is able to affect real GDP growth.

(9.) The assumption (equation 43) on the Frisch elasticities of borrowers and savers ensures that the level of output with flexible prices is independent of the wealth distribution, and thus the completeness of financial markets, up to a first-order approximation. The general case is taken up in an earlier working paper (Sheedy 2014).

(10.) If the assumption in equation 43 is relaxed then the debt gap [[??].sub.t] will appear in the Phillips curve. The consequences of this are taken up in an earlier working paper (Sheedy 2014), but they are not found to be quantitatively important.

(11.) This is because the model has the feature that the marginal propensities to consume from financial wealth are the same for borrowers and savers up to a first-order approximation.

(12.) With both short-term and long-term bonds satisfying the expectations theory of interest rates [j.sub.t] = (1 - [beta][mu])[[SIGMA].sup.[infinity].sub.l=0] [([beta][mu]).sup.l][E.sub.t][i.sub.t+l] where [i.sub.t] is the short-term interest rate, then the usual ex-ante Fisher equation [i.sub.t] = [[rho].sub.t] + [SIGMA] + [E.sub.t],[[pi].sub.t+1] would hold.

(13.) Time consistency issues and the discretionary policy equilibrium are studied in an earlier working paper (Sheedy 2014).

(14.) All the data referred to below were obtained from Federal Reserve Economic Data (http://research.stlouisfed.org/fred2).
```
Table 1. Baseline Calibration: Targets and Parameter Values

Calibration targets (a)

Real GDP growth                   g            1.7%

Real interest rate                r            5%

Debt-to-GDP ratio                 D            130%

Coefficient of relative risk
aversion

Marginal propensity to consume    m            6%

Frisch elasticity of labor
supply

Average duration of debt          [T.sub.m]    5

Price elasticity of demand

Marginal cost elasticity w.r.t.
output

Average duration of price         [T.sub.p]    8/12
stickiness

Implied parameter values (b)

Discount factor                   [beta]       0.992

Debt service ratio                [theta]      8.6%

[alpha]      5

Discount factor elasticity        [lambda]     0.993

[eta]        2

Debt maturity parameter           [mu]         0.967

[epsilon]    3

[xi]         0.5

Calvo pricing parameter           [sigma]      0.625

Sources: See discussion in section IV.A.

(a.) The calibration targets are specified in annual time units;
the parameter values assume a quarterly model (T = 1/4).

(b.) The parameters are derived from the calibration targets
using equations 66-70.
```