Journal of Monetary Economics 59 (2012) 417–421

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Discussion

Comment on ‘‘The market price of fiscal uncertainty’’ by Croce, Nguyen and Schmid Anastasios G. Karantounias Federal Reserve Bank of Atlanta, 1000 Peachtree St NE, Atlanta, Georgia 30309, United States

1. Summary The authors (CNS henceforth) consider an endogenous growth economy with exogenous government expenditures, a linear tax on labor income and a household that is afraid that the probability model of productivity and government expenditure shocks is misspecified. In particular, CNS use a stochastic version of an expanding input variety model a la Romer (1990) and explore the implications of two fiscal regimes: a balanced-budget regime and a fiscal regime that allows the accumulation of debt and involves primary deficits or surpluses when labor is below or above its steady-state value respectively. CNS find that with full confidence in the model, the fiscal regime that allows for counter-cyclical deficits dominates in terms of welfare the balanced-budget regime. With doubts about the probability model though, the opposite happens. The explanation of this intriguing result is based on the formation of agents’ expectations. Agents with concerns about misspecification use continuation utility in order to adjust pessimistically their expectations about the future. Countercyclical deficit policies, despite their welfare-enhancing potential in the short-run, posit long-run uncertainty in the sense of creating more persistent and volatile dynamics in continuation utility, towards which agents are averse. Agents adjust downwards their assessment of the present value of profits of a new input variety, innovate less and as a result, a lower growth rate in the economy is obtained. In this comment, I will highlight some relevant ingredients of the CNS economy and make some points towards optimal fiscal policy.

2. The mechanisms This section illustrates schematically some of the mechanisms of CNS.

2.1. Preferences, pricing kernel and profits Let the household rank consumption and leisure plans by using the multiplier preferences of Hansen and Sargent (2001), vt ¼ ð1bÞ½c ln ct þð1cÞln lt  þ b min ½Et mt þ 1 vt þ 1 þ yEt mt þ 1 ln mt þ 1  mt þ 1 Z 0

ð1Þ

subject to Et mt þ 1 ¼ 1. These preferences describe a decision maker that expresses doubts about a reference probability model (associated with the expectation operator E) and considers alternative probability models, captured by the conditional likelihood ratio mt þ 1 . The decision maker is averse to model ambiguity and forms worst-case scenarios. y is a E-mail address: [email protected] 0304-3932/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmoneco.2012.05.004

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positive penalty parameter that captures doubts about the reference model. When y ¼ 1, multiplier preferences reduce to expected utility. Performing the minimization in (1) delivers the worst-case beliefs of the household: mt þ 1 ¼

expðsvt þ 1 Þ , Et expðsvt þ 1 Þ

ð2Þ

where s  1=y o 0. The exponential tilting formula (2) shows that the cautious household is assigning high probability to low utility events in its effort to take decisions that are robust to model misspecification. Plugging the worst-case distortion into the utility recursion delivers the familiar risk-sensitive recursion vt ¼ ð1bÞ½c ln ct þ ð1cÞln lt  þ

b

s

ln Et expðsvt þ 1 Þ:

The stochastic discount factor under the reference model takes the form  1 c Lt þ 1 ¼ b t þ 1 mt þ 1 ct

ð3Þ

ð4Þ

or, taking logs, ln Lt þ 1 ¼ ln bxt þ 1 þ s½vt þ 1 s1 ln Et expðsvt þ 1 Þ, where xt þ 1 stands for consumption growth, xt þ 1  ln ct þ 1 ln ct . Doubts about the model introduce the (risk-adjusted) innovation in continuation utility in the logarithm of the stochastic discount factor. Profits. In the endogenous growth economy of CNS, the expansion of intermediate input varieties is determined by Vt, the present value of profits of a new variety that survives each period with probability ð1dÞ. Vt obeys the recursion V t ¼ Pt þð1dÞEt Lt þ 1 V t þ 1 1 X ¼ Et ð1dÞi Lt,t þ i Pt þ i , i¼0

where Lt,t þ i is the corresponding stochastic discount factor for multi-period payoffs. Using (4) we get  1 1 X c ðbð1dÞÞi t þ i Pt þ i , V t ¼ E~ t ct i¼0

ð5Þ

where E~ t refers to conditional expectation with respect to the worst-case model of the household, making precise the notion that agents compute the present value of profits under their pessimistic expectations. In particular, the conditional expectation of a random variable X t þ 1 under the worst-case model is given by E~ t X t þ 1  Et mt þ 1 X t þ 1 . 2.2. A log-linear setup The relevant object for the calculation of the worst-case beliefs of the household (and therefore of the pricing kernel and profits) is continuation utility vt þ 1 . In order to show its explicit dependence on the state of the economy, I am going to employ a log-linear Gaussian setup as in Hansen et al. (2008). Assume that the evolution of the state of the economy Xt is described by X t ¼ AX t1 þ C Et ,

ð6Þ

where A a stable matrix and Et  Nð0,IÞ. For example, Xt includes productivity and government expenditure shocks, the mass of intermediate input varieties and whatever is necessary to capture government policy in the model. Assume that consumption growth, leisure and period profits depend linearly on Xt, xt þ 1 ¼ mc þ U 0c X t þ 1 , ln lt ¼ ml þ U 0l X t , ln Pt ¼ mp þ U 0p X t : In order to obtain stationarity, define a scaled version of continuation utility, zt  vt c ln ct , that follows the recursion zt ¼ ð1bÞð1cÞln lt þ

b

s

ln Et expðsðzt þ 1 þ cxt þ 1 ÞÞ:

The particular log-utility structure in combination with the log-linear Gaussian dynamics allows for a closed form solution for zt ¼ dz þ D0 X t , where dz ¼ ð1bÞ1 ½ð1bÞð1cÞml þ bcmc þ 12bsðD þ cU c Þ0 CC 0 ðD þ cU c Þ, D ¼ ðIbA0 Þ1 ½ð1bÞð1cÞU l þ bcA0 U c :

A.G. Karantounias / Journal of Monetary Economics 59 (2012) 417–421

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The worst-case conditional distortion can be expressed in terms of the stationary variables zt and consumption growth xt and ultimately in terms of the innovation Et þ 1 , mt þ 1 ¼

expðsðzt þ 1 þ cxt þ 1 ÞÞ expðsðD þ cU c Þ0 C Et þ 1 Þ ¼ : Et expðsðzt þ 1 þ cxt þ 1 ÞÞ Et expðsðD þ cU c Þ0 C Et þ 1 Þ

The product of mt þ 1 times the reference density Nð0,IÞ delivers the worst-case conditional density of the shocks Et þ 1 , which in this setup turns out to be Gaussian with mean m~ t  E~ t Et þ 1 equal to

m~ t ¼ m~ ¼ sC 0 ðD þ cU c Þ

ð7Þ

and variance equal to I. Note that only the conditional mean (and not the covariance matrix) is distorted. Furthermore, the worst-case conditional mean of Et þ 1 does not depend on t.1 Similarly, the stochastic discount factor becomes 1 ln Lt þ 1 ¼ lnbxt þ 1 þ s½ðD þ cU c Þ0 C Et þ 1  sðD þ cU c Þ0 CC 0 ðD þ cU c Þ 2 and the conditional mean of consumption growth and profits E~ t xt þ 1 ¼ mc þ U 0c AX t þU 0c C m~ E~ t ln Pt þ 1 ¼ mp þ U 0p AX t þ U 0p C m~ : By calculating each term in (5) we can get an explicit formula for the present value of profits. For example, expected discounted profits next period are   ct þ 1 1 1 ln E~ t Pt þ 1 ¼ mp mc þðU p U c Þ0 AX t þ ðU p U c Þ0 CC 0 ðU p U c Þ þ ðU p U c Þ0 C m~ : 2 ct We are ready now to make some remarks on continuation utility and its effect on the pessimistic expectations of the household. The expression for the matrix D shows that the continuation utility channel is amplified when there is persistence in the state of the economy (captured by the matrix A), which is the essence of the long-run risk literature in asset pricing. In order to make sense of the formulas that describe the worst-case model, assume for the sake of the exposition that there is no leisure (c ¼ 1) and that the state Xt is one-dimensional with persistence parameter A ¼ r o 1 and standard deviation C ¼ sE . Furthermore, assume that consumption growth depends positively on Xt (U c 40), as would be the case for a productivity shock. Then the worst-case mean of Et þ 1 would be negative,

m~ ¼ 

1

1

y 1br

U c sE o0:

Therefore, the household’s worst-case scenario involves a productivity process with a lower conditional mean. Note that the higher the persistence parameter r, the larger the distortion that the cautious household is assigning. Furthermore, if we thought of Xt as the process of government expenditures (and therefore U c o 0), then m~ 40, i.e. the cautious household would assign a higher conditional mean on government expenditures. In both cases the conditional mean of consumption growth would be lower. Similar points can be made about expected profits. In the actual economy of CNS the policy functions for consumption growth, leisure and profits as well as the dynamics of the state Xt are endogenous objects. Still, the exogenous log-linear setup outlined above may be useful to illustrate some of the points of CNS. For example, the main argument of CNS for the dominance of a balanced-budget regime with doubts about the model is based on the matrix A and the induced present-value formula for consumption growth and leisure as captured in D. Their second fiscal regime that allows running deficits/surpluses and debt accumulation alters in effect the dimensionality of the state Xt (by adding the debt-to-GDP ratio as a state variable), the matrix A (making the state more persistent) and therefore the size of D. In a setup where agents doubt the model, the change in the persistence properties of the state – resulting from a tax rate that does not balance the budget every period – leads to more pronounced worstcase models. The worst-case conditional mean in turn affects the present value of profits and therefore the growth prospects of the economy, which forms the basic explanation for the dominance of the balanced-budget regime. In contrast, with full confidence in the model, the continuation utility channel is muted, leading to the inferiority of balancedbudget policies. 3. Towards optimal taxation What are the relevant distortions in the endogenous growth economy of CNS and how should a Ramsey planner design tax policies optimally, taking into account the short- and long-run considerations that arise naturally in an environment with doubts about the model? 1

This is a special feature of the log-linearity in the period utility function.

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3.1. Three main features I will first list three main features of the CNS economy that lie behind the deviation of the market equilibrium from the first-best and will play an important role at the second-best problem, i.e. in an environment where lump-sum taxes are not available. Monopoly power. The typical distortion in the expanding input variety model without government expenditures and taxes comes from the monopoly power of the intermediate input producers. Their pricing above marginal cost leads to a demand for intermediate inputs by the final good sector that is smaller than the socially efficient level. This static distortion leads to a lower growth rate than in the first-best, becoming therefore the source of a dynamic distortion. By eliminating the monopoly distortion the socially efficient growth rate can be obtained. This can be achieved by taxing lump-sum the household and using the proceeds in order to subsidize the purchase of intermediate inputs by the final good producers, so that the marginal product of an intermediate input equals marginal cost. See Acemoglu (2008) for further analysis of these types of policies. Taxation of labor income. The endogenous growth economy of CNS features taxation of labor income in order to finance government expenditures. Government expenditures are a fraction kt of the growing output in the economy, g t ¼ kt yt . Note that when government expenditures are a fraction of output, then labor income taxation is not necessarily distortionary. This can be easily seen in an exogenous growth economy without capital. In that setup, the first-best solution would require the equality of the marginal rate of substitution between consumption and leisure to a fraction of the marginal product of labor, MRSc,l ¼ ð1kt ÞF L . Essentially, the effective or social marginal rate of transformation is altered under this government expenditure structure.2 Thus, the first-best could be achieved by imposing a tax on labor income tt ¼ kt . In the endogenous growth economy of CNS, this feature will show up as well. This does not mean necessarily that the first-best can be achieved with taxes on labor income because it may very well be necessary to use a combination of linear labor taxes, subsidies on the purchase of intermediate inputs and lump-sum taxes in order to achieve the first-best, which makes the second-best problem non-trivial. Externalities. An important element that determines the endogenous growth rate is the marginal rate of transformation of final goods to new varieties 1=Wt . CNS assume that Wt depends negatively on the investment intensity in R&D. The freeentry condition in the R&D sector equates the marginal rate of transformation to the present discounted value of profits. Market participants do not internalize the effect of their investment intensity on Wt and ultimately on the growth rate in the economy, a feature that leads to a discrepancy between the equilibrium and the first-best allocation. This externality will be relevant for the second-best problem.

3.2. Robustness Doubts about the probability model affect equilibrium prices and the endogenous growth rate of the economy by affecting the present value of profits of a new input variety. From an optimal policy perspective, what has to be kept in mind is that policy instruments like the labor tax will affect continuation utilities and therefore the agent’s pessimistic expectations about shocks in the economy. For example, a high tax rate reduces the utility of the agent and therefore induces him to assign a higher probability on this contingency. How would a Ramsey planner manage these pessimistic expectations? Karantounias (forthcoming) shows that in an exogenous growth economy with model uncertainty and without capital, the policymaker wants to manage the pessimistic beliefs of the household in order to affect properly the equilibrium price of government debt. In particular, he tries to reduce the return on government debt and increase the return on government assets. Furthermore, due to the fact that the pessimistic household is forward-looking – since it takes into account the entire future evolution of consumption and leisure in forming worst-case scenarios – future tax rates are affecting current beliefs and therefore asset prices. As a result, the optimal tax rate has to take into account the entire history of debt or asset positions and becomes persistent, regardless of the stochastic properties of exogenous shocks. Thus, the forward-looking nature of the worst-case beliefs implies a particular intertemporal composition of tax rates, in other words a particular ‘‘long-run risk’’ structure. In the endogenous growth economy of CNS, the manipulation of debt returns through the household’s pessimistic expectations will also emerge. However, the three features listed previously will alter non-trivially the policy problem. Furthermore, there is a distinct channel that is associated with the present value of profits of a new input variety. The planner will have to determine the intertemporal composition of taxes with the objective to both reduce returns on government debt and to optimally affect the creation of new varieties by indirectly affecting the present value of profits. For example, a high tax rate at a particular state of the world, by inducing a higher worst-case probability, will make the agent discount less profits at this state. These issues deserve further investigation and an optimal taxation analysis would be a welcome contribution.

2

See Teles (2011).

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Acknowledgments I would like to thank Marvin Goodfriend, Christopher Sleet and Stanley Zin for giving me the opportunity to discuss this paper. All errors are my own. The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. References Acemoglu, D., 2008. Introduction to Modern Economic Growth. Princeton University Press. Hansen, L.P., Heaton, J.C., Li, N., 2008. Consumption strikes back? Measuring long-run risk. Journal of Political Economy 116 (2), 260–302. Hansen, L.P., Sargent, T.J., 2001. Robust control and model uncertainty. American Economic Review 91 (2), 60–66. Karantounias, A.G. Managing pessimistic expectations and fiscal policy. Theoretical Economics, forthcoming. Romer, P.M., 1990. Endogenous technological change. Journal of Political Economy 98 (5), 71–102. Teles, P., 2011. Exogenous public spending in the optimal taxation problem. Mimeo, Banco de Portugal and Universidade Catolica Portuguesa.