Welfare Implications of Asset Pricing Facts: Should

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Welfare Implications of Asset Pricing Facts: Should Central Banks Fill Gaps or Remove Volatility? Pierlauro Lopeza,1,∗ a

Banque de France

Abstract Twenty years of financial market data suggest a term structure of the welfare cost of economic uncertainty that is downward-sloping on average, especially during downturns. This evidence offers guidance in selecting a model to study the benefits of macroeconomic stabilization from a structural perspective. The addition of nonlinear external habit formation to a textbook monetary policy model can rationalize the evidence. The model is observationally equivalent in its quantity implications to a standard New Keynesian model with CRRA utility but the optimal policy prescription is overturned. In the model the central bank should focus on removing consumption volatility (a targeting of risk premia) rather than on filling the gap between consumption and its flexible-price counterpart (inflation targeting). JEL classification: E32; E44; E61; G12. Keywords: Welfare cost of business cycles, Macroeconomic priorities, Equity and bond yields, Optimal monetary policy, Financial stability.

1. Introduction What are the welfare implications of asset pricing facts? In a seminal contribution, Lucas (1987, 2003) proposed to consider the amount of growth that people would trade against the elimination of uncertainty around future consumption as a measure of the welfare cost of fluctuations, and as an indicator of the priority of stabilization policies. At least since Alvarez and Jermann (2004), it has been understood that a sufficiently complete financial market can reveal directly the marginal cost of fluctuations. I show that recent evidence about the term structures of equity and interest rates can reveal how fluctuations at different horizons contribute to the marginal cost of lifetime consumption fluctuations, and forms therefore a rich set of empirical features with implications for structural models for stabilization policy evaluation. ∗

Present address: Macro-Finance Division, 31 rue Croix des Petits Champs, 75001 Paris. Email address: [email protected] (Pierlauro Lopez) 1 I would like to thank Jarda Borovicka, John Cochrane, Patrick Kehoe, Ralph Koijen, Jordi Gal´ı, Arie Gozluklu, Urban Jermann, Antonio Mele, Fabio Trojani, and Harald Uhlig for very helpful advice and comments, as well as seminar participants at the 2016 meetings of the Society for Economic Dynamics, and at the 2014 and 2018 meetings of the European Finance Association for comments and discussions. The views presented here are those of the author and do not necessarily represent those of any institution of the Eurosystem.

First version: July 2015. This version: October 2018. Comments are most welcome

This information improves our understanding of the appropriate model to explain why uncertainty is costly, as calculations of welfare costs vary by orders of magnitude across alternative assumptions on preferences and cashflow processes.2 I develop a New Keynesian model with external habit formation that rationalizes the empirical measures of the welfare cost of uncertainty, and carry out an explicit welfare-theoretic analysis of the optimal monetary policy design. In the model, the representative consumer is sufficiently sensitive to cashflow fluctuations to value more a less volatile consumption path than less volatile inflation, thereby overturning a pervasive result in the New Keynesian literature. This analysis exemplifies the potentially dramatic implications of incorporating realistically large discount-rate variation into macroeconomics. 1.1. Decomposing the cost of uncertainty By focusing on fluctuations in the determinants of consumption, in particular wage and equity income, I show a tight link between the welfare cost of consumption fluctuations at different horizons and the risk premium commanded by claims to single market dividends payments, socalled dividend strips.3 The result relies on no-arbitrage relations and on a standard assumption on the consumption-labor tradeoff, does not require a parametric specification of consumer preferences, and allows for an observable measure of welfare costs at several horizons over the last two decades. The empirical analysis finds costs that are sizable, countercyclical, and horizon-dependent. The point estimates, reported in figure 1a, suggest a negatively sloped term structure of welfare costs, driven both by the negative slope of the term structure of equity and by the positive slope of the term structure of interest rates. At the margin people would trade an average of 0.5 percentage points of growth in next year’s consumption against the elimination of one-year ahead consumption risk. The volatility of the premium is similar in size; the cost increased to 2-3 percentage points during the last two recessions and compares to smaller benefits of long-run stability. Thus, I complement the analysis of Alvarez and Jermann (2004) through a different take on the two main challenges they face in measuring the marginal cost of uncertainty. First, I focus on fluctuations in the determinants of consumption and consider the endogeneity of the main source of income—labor income. In their calculations Alvarez and Jermann model an exogenous difference between consumption and dividends. However, the difference between consumption and dividends is endogenous from a consumer’s point of view; when an endogenous determinant of consumption such as labor income is marginally stabilized, the positive effect on utility is offset by the effect of the associated adjustment in labor hours. In this context, only claims to exogenous determinants of consumption such as equity income are necessary to measure the marginal cost of uncertainty.4 2

See Lucas (1987); Atkeson and Phelan (1994); Krusell and Smith (1999); Tallarini (2000); Otrok (2001); De Santis (2007); Gal´ı, Gertler and L´opez-Salido (2007); Barillas, Hansen and Sargent (2009); Barro (2009); Croce (2012); Ellison and Sargent (2012); Epstein, Farhi and Strzalecki (2014), among many others. 3 A recent and rapidly growing literature is focusing on risk pricing across maturities (see, for example, Lettau and Wachter, 2007, 2011; Hansen, Heaton and Li, 2008; Binsbergen, Brandt and Koijen, 2012; Binsbergen, Hueskes, Koijen and Vrugt, 2013; Backus, Chernov and Zin, 2014; Borovicka and Hansen, 2014; Belo, Collin-Dufresne and Goldstein, 2015; Dew-Becker and Giglio, 2016; Weber, 2018; Binsbergen and Koijen, 2017). 4 The cost of uncertainty can be measured using equity claims as long as claims to the residual determinant of consumption net of equity and wage income commands a relatively small risk premium. In any event, the measure can be interpreted as the cost of uncertainty in equity and labor income, which is a meaningful quantity on its own right.

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(a) Point estimates.

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Figure 1: Average term structures of equity (‘◦’ and dashed line), interest rates (‘×’ and dash-dotted line), and welfare costs (‘+’ and solid line); annualized expected real returns on the left axis for the term structures of equity and interest rates, and on the right axis for welfare costs. (Returns are deflated by the CPI.) The left subfigure plots point estimates during 1994:01-2013:12 (with block-bootstrap 95% confidence interval); E(re,m ) is the equity premium (average annualized excess return on a 6-month buy-and-hold strategy). The right subfigure plots model-implied average term structures in the dynamic equilibrium model presented in section 4. L{1,...,n} represents the amount of growth in the entire consumption process at a n-period horizon that people would trade against the elimination of consumption risk over the same horizon. The solid curve associates with a cost of lifetime uncertainty of 15 hundredths of a percent.

Second, the term structure of the cost of uncertainty answers roughly the question, ‘how much compensation do people command to bear n-year ahead cashflow uncertainty?’. This question compares to, ‘how much compensation do people command to bear uncertainty at business-cycle frequency in the entire cashflow process?’, which is the question studied by Alvarez and Jermann. Their answer depends on the parametric assumptions about the filter that separates the trend and the business-cycle frequencies of the cashflow process. The question I am interested in is nonparametric and complements their exercise by decomposing the marginal cost of uncertainty in the time domain. The empirical section uses bond data and extracts evidence about the term structure of equity from index option markets to infer the term structure of the marginal cost of uncertainty. Like Binsbergen, Brandt and Koijen (2012) and Golez (2014) this paper relies on option data and no-arbitrage relations to replicate synthetic single market dividend payments. I contribute to this literature by a strategy that avoids the need for an interest-rate proxy and that mitigates measurement error by excluding observations that violate the put-call parity relation. 1.2. Welfare implications of welfare cost measures Evidence of nontrivial welfare costs is not a sufficient condition to justify attributing priority to stabilizing fluctuations. The sufficiency of the condition can only be assessed in the context of a structural model. In fact, the literature opened by Lucas (1987) focuses on fluctuations in the level of cashflows, as opposed to fluctuations in its distance (gap) from some benchmark level, and is silent as to whether the current level of risk in the economy is Pareto optimal. By contrast, 3

structural macroeconomic models for policy evaluation typically prescribe stabilizing gaps between current and efficient cashflow levels (see Clarida, Gal´ı and Gertler, 1999, among many others). What welfare cost measures really say is that their empirical properties are key diagnostics of any macroeconomic model that seeks to assess the priority of different stabilization policies. In this respect, empirical measures of welfare costs are highly informative because it is well-known that explaining jointly a downward-sloping term structure of equity and an upward-sloping term structure of interest rates from a structural perspective is hard (e.g., Binsbergen and Koijen, 2017). Accordingly, I develop a structural model that captures the observed term structure of welfare costs and analyze its stabilization policy implications. The model relies on Campbell and Cochrane (1999) external habit formation and on nominal rigidities; in the model people fear instability especially in the short run because equity income is highly procyclical but partly mean reverting.5 Figure 1b reports the average term structure of welfare costs in the model. As in the data, the model-implied costs of short-run uncertainty are sizable and substantially larger than the cost of uncertainty over longer horizons, with a cost of lifetime uncertainty of around one-tenth of a percentage point of perpetual growth in consumption. Conveniently, the model builds on the textbook New Keynesian model (e.g., Gal´ı, 2008) used to study macroeconomic stabilization. The comparison can be made clear-cut. I can calibrate the linearized model so that quantities and inflation are the same as in the textbook model; a macroeconomist interested in the dynamics of quantities and inflation would not be able to discriminate between the two models. But the asset pricing implications differ dramatically. The two externalities—sticky prices and external habits—reconcile the two notions of stability in the literature. First, the central bank wants to close the consumption gap (or, equivalently, inflation). Second, it wants to remove consumption uncertainty (or risk premia variation). In the presence of supply shocks achieving both goals is unfeasible, so the optimal monetary policy trades them off. When comparing the two simple regimes under a parametrization consistent with measured welfare costs I find that removing uncertainty is a priority over filling the troughs. 2. The term structure of the welfare cost of fluctuations People live in a stochastic world, have finite resources, and decide how to allocate them across time. Financial markets are without arbitrage opportunities and sufficiently complete as to allow people to trade the full set of zero-coupon real bonds and equities. Identical risk-averse consumers j ∈ [0, 1] have time-t preferences Ut = Et U (C( j), N( j), X( j)), ∞ ∞ where C ≡ {Ct+n }∞ n=1 is consumption, N ≡ {Nt+n }n=1 is labor, and X ≡ {Xt+n }n=1 is any other factor 6 that influences utility. Without loss of generality let factor X(R j) depend on individual R 1 consumption 1 and labor only via aggregate consumption and labor, C = 0 C( j)d j and N = 0 N( j)d j. Since 5

To the best of my knowledge, and as discussed in Lopez, Lopez-Salido and Vazquez-Grande (2015), this is the only candidate structural framework in the extant literature that is able to explain jointly the observed term structure properties. This characteristic makes my results more compelling but is not crucial for the point I am making. Lopez et al. study a similar model but do not focus on welfare implications. 6 The definition of cost of uncertainty has a meaningful interpretation even if one relaxes the assumption of a representative consumer. The online appendix provides the detailed argument.

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there is a continuum of agents each of which has zero mass, this modeling strategy enables me to ask an individual how much consumption growth he would trade against stable consumption and labor streams without thereby having to affect all aggregate quantities, including factor X. The determinants of consumption, Ct = Dt + Wt Nt + et , include equity income Dt and labor income Wt Nt , with Wt the real wage rate, while et denotes any residual income. Let C t+n denote the consumption level that is hypothetically offered to the ith individual at time t + n, which I refer to as stable consumption. Stable consumption C t+n = Dt+n + Wt+n N t+n + et+n is defined as a stable stream of dividend income, Dt+n = Et (Dt+n ), of the path for labor N t+n that ensures a stable labor income Wt+n N t+n = Et (Wt+n Nt+n ), and of residual income. I parametrize stable consumption as C t+n (θ) = θC t+n + (1 − θ)Ct+n = θEt (Dt+n + Wt+n Nt+n ) + (1 − θ)(Dt+n + Wt+n Nt+n ) + et+n where the parameter θ ∈ [0, 1] represents the fraction of ex-post uncertainty in the main determinants of consumption that is removed. The associated labor level is N t+n (θ) = θN t+n + (1 − θ)Nt+n . Definition (Welfare cost of uncertainty). The map Lt : (θ, N) 7→ LN t (θ) defined by Et U



n  ∞  , {C } 1 + LN C (θ) , X t+n n∈N\N t+n n∈N t+n n=1 = t    ∞  = Et U C t+n (θ) n∈N , N t+n (θ) n∈N , {Ct+n }n∈N\N , {Nt+n }n∈N\N , Xt+n n=1 (1)

measures the cost of fluctuations, where the index set N ⊂ N ≡ {1, ..., ∞} indicates which coordinates are stabilized and allows for focusing on any window of interest. For example, the total cost LNt (1) measures how much extra growth the elimination of all uncertainty in equity and wage income is worth, and the marginal cost LtN ≡ ∂θ∂ LN t (0) represents how much extra growth a marginal stabilization is worth at the current level of uncertainty.7 I assume enough smoothness in preferences to guarantee that LN t is a differentiable map in θ ∈ [0, 1]. Additionally, suppose that the consumption-labor tradeoff is described by the condition Wt = −

∂Ut /∂Nt ∂Ut /∂Ct

(2)

i.e., the marginal rate of substitution between consumption and labor equals the relative price. This assumption is a standard optimality condition.8

7 The online appendix discusses the relationship between definition (1) and the definitions by Lucas (1987) and Alvarez and Jermann (2004). 8 There is ample evidence that labor wedges matter (Chari, Kehoe and McGrattan, 2007). This observation does not, however, rule out assumption (2), which is consistent with labor wedges generated by distortions between wages and the marginal product of labor on the firm side—including sticky prices—as well as by departures from CRRA utility on the consumption side. Assumption (2) is, however, inconsistent with some frictions such as sticky wages.

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Proposition 1. The marginal cost of uncertainty within any window of interest N, LtN , is LtN =

X Et (Mt,t+n )Et (Dt+n ) − Et (Mt,t+n Dt+n ) P n∈N n E t (Mt,t+nC t+n ) n∈N

(3)

where Mt,t+n = (∂Ut /∂Ct+n )/(∂Ut /∂Ct ) is the n-period stochastic discount factor. Under no(n) arbitrage, Dc,t = Et Mt,t+nCt+n is the price of a n-period consumption strip, D(n) d,t = E t Mt,t+n Dt+n is (n) the price of a n-period dividend strip, and Db,t = Et Mt,t+n is the price of a n-period zero-coupon real bond. Equation (3) expresses the marginal cost of uncertainty around all coordinates n ∈ N as a function of the price of a claim to the trend in dividend income and of the price and duration of claims to future dividends and consumption at all coordinates n ∈ N. Note that claims to labor income do not enter the expression because people set the marginal rate of substitution between consumption and labor equal to the wage rate, so the marginal effect of the adjustment in hours offsets exactly the benefits of a marginal stabilization in labor income. Definition (Term structure of the cost of uncertainty). The nth component of the term structure of the welfare cost of uncertainty is the risk premium for holding to maturity a portfolio long in a n-period dividend strip and short in a n-period zero-coupon bond, lt(n) =

 1 Et Re,(n) − 1 t→t+n n

e,(n) (n) where Rt→t+n = Dt+n D(n) b,t /Dd,t .

The motivation for calling the map lt : n 7→ lt(n) a term structure of the marginal cost of uncertainty is given by proposition 2. Given the prices of strips {D(n) } and the term structure components {l(n) } you can compute the marginal cost LN for any coordinate set N ⊂ N. Proposition 2. The marginal cost of uncertainty within any window of interest N, LtN , is the linear combination of the term components {lt(n) }, X X (n) N (n) LtN = αN ω l ≈ α ωN (4) t n,t t n lt n∈N

n∈N

P (n) P (n) N with scaling factor αN n∈N n Dd,t / n∈N n Dc,t , hence α = α = D/C in a stationary state, and t ≡ P P (n) (n) N where the weights ωN n∈N ωn,t = 1. n,t ≡ n Dd,t / n∈N n Dd,t are positive and such that Equation (4) shows how the first-order effect of distinguishing between consumption and dividends is that the cost of uncertainty around any coordinate set N scales the linear combination of the term structure components by a factor equal to the average dividend-consumption ratio. Intuitively, the scaling factor translates points of growth in dividend income into points of growth in consumption. This factor can be estimated at about 4.5% over the 1994-2013 period using data from the U.S. Flow of Funds Accounts on net dividends paid out by nonfarm nonfinancial corporates 6

(table F.103, line 3) and on personal consumption expenditures in nondurable goods and services (table F.6, lines 4 and 5). There is a powerful intuition behind these formulas. At the margin, people would trade (n) {n} Lt θ = α{n} t lt θ points of growth in the nth coordinate of consumption against the elimination of a fraction θ of the aggregate uncertainty around that coordinate, and that coordinate alone. Propositions 1 and 2 show how this tradeoff is precisely the one offered by the financial market. In fact, by holding to maturity a portfolio short in the n-period zero-coupon bond and long an equal amount in the n-period dividend strip, people can experience an average growth rate in the nth coordinate of dividend income of 1n (Et Re,(n) t→t+n − 1) by shouldering the uncertainty in n-period ahead dividend income. 3. Empirical measures of the cost of fluctuations Suppose that a full set of zero-coupon real and nominal bonds and a full set of put and call European options whose underlying is an aggregate equity index are traded on the market. In absence of arbitrage opportunities, put-call parity holds: Ct,t+n − Pt,t+n = Pt −

n X

D(t j) − X P(n) $b,t

(5)

j=1

where Ct,t+n and Pt,t+n are the nominal prices at time t of a call and a put European options on the market index with maturity n and nominal strike price X, P(n) = Et Mt,t+n Pt /Pt+n is the nominal P$b,t price of a n-period nominal zero-coupon bond, Pt = Pt Et ∞j=1 Mt,t+ j Dt+ j is the nominal value of the market portfolio, and D(n) t = Pt E t Mt,t+n Dt+n is the nominal price of the nth dividend strip, where P denotes the price level. Since the only unknowns in equation (5) are the prices of the dividend strips, {Dt(n) }, one can synthetically replicate them (Binsbergen et al., 2012). I follow Binsbergen et al. (2012) and Golez (2014) in synthesizing the evidence on dividend claims from put and call European options on the S&P 500 index. Standard index option classes, with twelve monthly maturities of up to one year, and Long-Term Equity Anticipation Securities (LEAPS), with ten maturities of up to three years, are exchange traded on the Chicago Board Options Exchange (CBOE) since 1990. The overall size of the index option market in the U.S. has grown rapidly over the years. During the first year of the sample Options Clearing Corp reports an average open interest of $60 billion for standard options and LEAPS with maturities of less than six months that gradually decreases across maturities to $200 million for options of two years or more. The corresponding figures in the last year of the sample are an open interest of $1,400 billion for maturities of less than six months and of $40 billion for maturities larger than two years. Like Golez, but unlike Binsbergen et al. (2012), the main analysis relies on end-of-day option data. I use a dataset provided by Market Data Express containing S&P 500 index option data for CBOE traded European-style options and running from January 1990 to December 2013. I obtain the daily S&P 500 price and one-day total return indices from Bloomberg and combine them to calculate daily index dividend payouts; I then aggregate the daily payouts to a monthly frequency without reinvestment. 7

There are three major difficulties when extracting options-implied prices through the put-call parity relation (see also Boguth et al., 2012). First, quotes may violate the law of one price for reasons that include measurement errors such as bid-ask bounce or other microstructural frictions. Second, the synthesized prices are sensitive to the choice of risk-free rate, which multiplies strikes in the put-call parity relation. Since strike prices are large numbers, any error in the interest rate will magnify in the synthetic prices. Third, end-of-day data quote the closing value of the index, whose components trade on the equity exchange, and the closing prices of derivatives that are exchange-traded on a market that continues to operate for 15 minutes after the equity exchange closes. An asynchronicity of up to 15 minutes may therefore drive a wedge between the reported quotes of the index value and the option prices and bias the synthetic prices. To address these difficulties, I combine options with different moneyness levels to extract both the risk-free rate and the strip price in a unique step.9 This approach produces the appropriate interest rate for synthetic replication, and it allows for spotting trade dates that violate the law of one price at some maturity. When I identify such violations, I drop the associated observations to mitigate microstructural noise. The appendix details the algorithm for synthetic replication. Finally, to measure real bond prices I rely on zero-coupon TIPS yields with maturities of up to ten years from G¨urkaynak et al. (2010). However, TIPS yields are either unavailable or unreliable during the 1990s. In fact, there is evidence of sizable liquidity premia in TIPS markets, especially at inception and during the recent crisis, that distort measures of real yields extracted from TIPS (D’Amico, Kim and Wei, 2018). In this context, D’Amico et al. find that a TIPS specific factor not captured by the first three principal components of nominal yields captures the liquidity premium commanded by TIPS. Accordingly, I regress TIPS of up to ten-year maturity on the 1999-2013 period on the first tree principal components of nominal yields, and interpret the projection as the real yield net of the TIPS liquidity premium.10 I then reconstruct real yields on the 1994-1999 period using the same projected coefficients.11 Both nominal and real government bonds are computed on the last trading day of the month. 9

The literature offers at least three alternatives to extract the term structure of equity. First, index options can be combined with some interest rate proxy as in the original intra-day approach of Binsbergen et al. (2012). The tick-level approach has the advantage of exploiting information from more data points and avoids asynchronicity issues, however, accuracy can be lost in the choice of an interest-rate proxy. Second, index options can be combined with the interest rate implied in index futures as in Golez (2014). However, CME S&P 500 futures have expiration dates only for eight months in a quarterly cycle over most of the available sample. Finally, the index dividend futures studied by Binsbergen et al. (2013) have the advantage to reveal strip prices without the need for synthetic replication, and they do so for longer maturities. However, for S&P 500 dividend futures exchange-trading started only in November 2015, and previously only proprietary datasets starting end 2002 are available covering over-the-counter trades. In this context, the exchange-traded nature of options mitigates concerns that the preferences embedded in their pricing do not reflect those of the average investor, which would complicate the macroeconomic interpretation of the derived cost of uncertainty. 10 This regression is consistent with the model of section 4 and with the no-arbitrage affine term structure models— e.g., Lettau and Wachter (2011)—that imply that up to three nominal yields capture all factors that drive real yields. 11 A less sophisticated proxy for real yields—nominal hold-to-maturity returns on Treasury yields deflated by holding-period inflation—results in similar results. In any event, none of the evidence suggests that real yields are significantly larger in absolute magnitude than nominal yields, so the real bond proxy problem is unlikely to affect by much the quantitative estimates of the term structure of welfare costs because at the observable end of the curve the contribution of bond yields is much smaller than the contribution of equity yields.

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Figure 2: Term structures of equity, real interest rates, and welfare costs over the last two decades; annualized real returns. The term structure of equity is synthesized from index options; the term structure of interest rates uses G¨urkaynak-Sack-Wright data about Treasuries and TIPS. Real interest rates over 1994-2013 are constructed as the projection of TIPS yields on the first three principal components of the term structure of nominal yields to filter out a liquidity premium and to construct missing data over 1994-1999. (See the appendix for details.) Regressors in panel 2d are the first two principal components of semestral equity yields and the first principal component of nominal yields; the semestral excess return on the index is regressed additionally on the market dividend yield. The shaded areas indicate business-cycle peaks and troughs as declared by the NBER (March-November 2001 and December 2007 to June 2009).

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3.1. Average costs of consumption fluctuations The evidence suggests a downward-sloping term structure of welfare costs, driven both by a negatively sloped term structure of equity and by a positively sloped term structure of interest rates. I follow Binsbergen et al. (2012) and focus on a semestral periodicity; the first strip pays off the next six months of dividends, the second strip the dividends paid out six to twelve months out, and so on. The measure of the hold-to-maturity return on the first semestral strip is the return on a six-month buy-and-hold strategy that pays off the next six months of dividends.12 Accordingly, I measure the hold-to-maturity return on the n-semester strip as the return for holding for n − 1 semesters a n-semester strip times the semestral return on the first semestral strip. To address the concerns raised by Boguth et al. (2012) that microstructural frictions might cause spuriously large arithmetic high-frequency returns on synthetic dividend claims, I report log returns on six-month buy-and-hold strategies and hold-to-maturity returns on strategies with maturities between 0.5 and 2 years, which Boguth et al. advocate as much less biased by microstructural effects related to highly leveraged positions. Figure 2a illustrates the size of average annualized monthly log returns on six-month strategies over different subsamples by plotting the cumulated real return on an investment strategy that goes long on January 31, 1996 by a dollar in a claim to the next n years of dividends, holds the investment for six months and then rolls over the position. Monthly average log returns are large and positive for short-duration equities and larger than the real return on the index. Figure 2b plots the analogous cumulated monthly returns on six-month bond strategies long a dollar on zero-coupon bonds with maturities between six months and ten years; average returns steadily increase in maturity across bonds, consistent with an upward-sloping average term structure of real interest rates. Table 1 reports the point estimates for the term structure of welfare costs. The first four term structure components at semestral frequency are of 11.9%, 10.7%, 8.5% and 6.3%, respectively. I then rely on proposition 2 to compute the costs of uncertainty around multi-period cashflows. Namely, I estimate average welfare costs of 0.53% associated with uncertainty one semester out, of 0.48% associated with up to one-year ahead uncertainty, of 0.43% with up to 18-month uncertainty, and of 0.37% with up to two-year ahead uncertainty, respectively. Additionally, table 1 shows the welfare cost of one-period ahead uncertainty for different periodicities, from semestral to biennial. These estimates complement the evidence about the term structure in a way that bypasses the somewhat arbitrary choice of the semestral periodicity of the strips. I find comparable results: the average cost of one-year ahead uncertainty over the two samples is of 0.37%, whereas the cost of uncertainty over the next two years is of 0.24%. A comparison of estimated costs of uncertainty with holding-period excess returns on the index reveals the economic significance of these estimates. To an approximation around ln R = 0, the 12

Golez (2014) raises concerns that equity prices of up to three-month maturity may be biased as a result of firms routinely preannouncing part of their dividend payouts, which would lower their riskiness. To mitigate such concerns, I roll over three times a two-month buy-and-hold strategy that goes long in the six-month strip rather than hold to maturity a six-month strip.

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equity premium relates to the term structures of equity and interest rates by Et (ln Re,m t+1 )



∞ X

ωn,t Et (ln Re,(n) d,t+1 )

n=1

lt(n)

(6)

1 1 (n) (n) (n) (n) ≈ Et (ln R(n) d,t→t+n − ln Rb,t→t+n ) = [E t (ln Rd,t+1 ) + ... + E t (ln Rd,t+1 )] − yb,t n n

(n) where yb,t = − ln(Et Mt,t+n )/n is the nth real yield, and weights ωn,t ≡ D(n) d,t /Pt add to one. Equation (6) implies that if the cost of 1-period ahead fluctuations is larger than the equity premium over the same period, then strip excess returns must be smaller than the equity premium at some longer duration. It follows that the term structure of welfare costs must slope downwards. Furthermore, an upward-sloping real term structure exacerbates this feature. Table 1 confirms this point. Figures 1a and 2c plot the point estimates for the term structures of equity, interest rates, and welfare costs. The figures also show block-bootstrapped one-sided critical values based on bootstrap-t percentiles corresponding to a five-percent size for the means of lt(n) and the six-month market return. (The block size of ten observations is slightly larger than the number of lags after which the correlogram of the underlying returns becomes negligible.) In both samples I can reject the hypothesis that the term structure components are trivial.

3.2. Time-variation in the cost of consumption fluctuations The components of the term structure of welfare costs are volatile and countercyclical. Present-value logic implies that equity yields contain information about expected returns. Since the same states would drive the risk premia that constitute the term structure of welfare costs, it follows that equity yields signal variation in the term structure of welfare costs. Motivated by theoretical models I consider a one- to three-factor specification for these risk premia.13 To capture the factors I extract the first two principal components of semestral equity yields, which capture 95% of their volatility, and the first principal component of bond yields, and use them to forecast the hold-to-maturity excess returns whose ex ante values constitute the welfare cost measures. Table 2 presents the predictive regressions and shows a standard deviation of expected returns about as large as the already sizable level. Since excess returns are forecastable, the cost of uncertainty varies over time. The cost of short-run cashflow uncertainty is substantial at some junctures of the business cycle. Figure 2d plots the estimated time series of the term structure of the welfare cost of uncertainty over time. The cost of uncertainty rises dramatically to 2-4% during the dot-com crash and the period immediately preceding the early 2000s recession as well as during the most recent recession. Moreover, the premium to hedge uncertainty six months out is considerably larger than the premium to hedge longer-run uncertainty. The estimated term structure remains downward-sloping during the downturns whereas it appears considerably flatter and even upward-sloping in normal times. 13

For example, the affine no-arbitrage model of Lettau and Wachter (2011) and the model of section 4 imply a term structure of welfare costs that can be revealed by the information set spanned by two equity yields and a bond yield.

11

0.5 years (n = 1) Mean

lt(n) 0.1189 (0.0714)

Mean

Lt{1,...,n} 0.0053 (0.0032)

Mean

e,m Et (rt+1 ) 0.0618 (0.0389)

1 year (n = 2)

1.5 years (n = 3)

0.1073 (0.0302)

0.0846 (0.0202)

0.0048 (0.0018)

0.0043 (0.0012)

2 years (n = 4)

0.5 years (n = 1)

1 year (n = 2)

1.5 years (n = 3)

2 years (n = 4)

0.0625 (0.0179)

lt(1) , 1 period = n semesters 0.1189 0.0831 0.0652 (0.0714) (0.0445) (0.0250)

0.0540 (0.0222)

0.0037 (0.0009)

Lt{1} , 1 period = n semesters 0.0053 0.0037 0.0029 (0.0032) (0.0020) (0.0013)

0.0024 (0.0010)

e,m Et (rt+1 ), 1 period = n semesters 0.0618 0.0565 0.0531 (0.0389) (0.0305) (0.0204)

0.0520 (0.0200)

Table 1: Options-implied average term structure of the welfare cost of uncertainty, 1994-2013. l(n) is the annualized cost of a marginal increase in uncertainty in n-semester ahead cashflows. L{1,...,n} is the annualized cost of a marginal increase in uncertainty in 1 to n-semester ahead cashflows. The right panel reports the cost of a marginal increase in one-period ahead uncertainty, L{1} , for different period lengths. The equity premium is the average buy-and-hold return on the S&P 500 index in excess over the corresponding risk-free rate, with dividends reinvested in real bonds; the ex-dividend return is the ex-dividend buy-and-hold return on the S&P 500 index. Boostrap standard errors use block sizes of ten observations to preserve time-series dependencies.

1 e,(2) 2 rt→t+2

e,(1) rt→t+1

const. pc1d,t

0.076 [0.032] 0.364 [0.104]

pc1b,t pc2d,t

0.076 [0.022] 0.342 [0.053] 0.143 [0.146] 0.778 [0.139]

0.085 [0.015] 0.139 [0.029]

0.085 [0.015] 0.139 [0.034] 0.010 [0.097] -0.049 [0.070]

1 e,(3) 3 rt→t+3

0.064 [0.015] 0.074 [0.022]

0.064 [0.013] 0.077 [0.024] -0.058 [0.078] -0.244 [0.049]

1 e,(4) 4 rt→t+4

0.046 [0.013] 0.067 [0.018]

0.046 [0.012] 0.069 [0.016] -0.123 [0.057] -0.129 [0.037]

d pm t R2

σ(Et r) E(r)

0.34 2.85

0.51 3.50

0.21 0.99

0.21 0.99

0.10 0.71

0.22 1.08

0.13 0.93

0.21 1.18

e,m rt+1

0.046 [0.036]

0.046 [0.034] -0.098 [0.052] 0.150 [0.158]

0.355 [0.160] 0.10 1.75

0.368 [0.221] 0.15 2.15

Table 2: Predictive regressions on hold-to-maturity semestral strip returns and on the semestral buy-and-hold market return. Annualized log returns in excess over the riskless return over the holding period. Regressors are the first two principal components of the semestral equity yields, pc1d and pc2d , the first principal component of up to ten-year bond yields, pc1b , and the market dividend yield, d pm . Monthly data, 1996m1-2013m12. Newey-West standard errors to correct for overlapping.

12

4. Welfare implications in a simple model This section investigates the optimal stabilization policy in a model that combines a textbook New Keynesian model economy (Gal´ı, 2008) with Campbell and Cochrane (1999) habits in consumption. Importantly, this model rationalizes the term structure evidence of section 3.14 The presence of two externalities (nominal rigidities and external habits) motivates a nontrivial policy tradeoff. This section computes the properties of the model with accurate numerical procedures—the online appendix describes the solution method—but I derive sharper insight from an affine approximation of the model or, equivalently, a linearization around the risky steady state (Lopez, Lopez-Salido and Vazquez-Grande, 2016). 4.1. Model I calibrate the model so that the linearized dynamics of inflation and economic activity under the Campbell-Cochrane pricing kernel coincide with those under a version of the model with CRRA utility. I abstract from capital formation, and focus on the optimal parametrization of a simple Taylor rule. This choice illustrates the point in the simplest possible setup. The two approximate models differ only in their asset pricing implications. The respective welfare implications turn out to be, however, dramatically different; under CRRA utility the optimal Taylor rule closes the consumption gap, while under habits it minimizes consumption volatility. The point of this theoretical analysis is how the policy prescriptions of the textbook New Keynesian model change once you incorporate a pricing kernel that produces realistic asset pricing facts—especially observed welfare cost measures. Like the standard New Keynesian literature, I therefore choose to focus on the unique bounded solution under an active Taylor rule despite the recent critiques by Cochrane (2011). (Throughout the exposition, lower-case letters denote natural logarithms.) 4.1.1. Firms Monopolistically competitive firms indexed by i ∈ [0, 1] maximize intertemporal profits E0

∞ X t=0

M0,t

Pt (i) Yt (i) − (1 − τ f )Wt Nt (i) − T tf Pt

!

subject to Calvo-type nominal price stickiness. The t-period stochastic discount factor M0,t is the households’—who own the firms. Firms operate the production technology: Yt (i) = At Nt (i)1−α 14

The online appendix shows the implications of some of the leading consumption-based asset pricing models for the term structure of the welfare cost of uncertainty, and confirms their difficulties in explaining the documented facts. I consider the habit formation model of Campbell and Cochrane (1999), the long-run risk model of Bansal and Yaron (2004), the long-run risk model under limited information of Croce, Lettau and Ludvigson (2015), the recursive preferences of Tallarini (2000) and Barillas et al. (2009), and the rare disasters model of Gabaix (2012).

13

where Yt is real output; Nt is the labor input, which they acquire at a unit cost equal to the real wage rate Wt ; and At is aggregate productivity. The government levies lump-sum taxes T tf on each firm to finance an employment subsidy, τ f = 1 − MC, with MC the real marginal cost, designed to offset any distortions caused by monopolistic competition in the steady state. Corporate profits Dt (i) = Yt (i)Pt (i)/Pt − (1 − τ f )Wt Nt (i) − T tf are paid out as dividends on market equity each period to households. R1 The ith good sells for the nominal price Pt (i), with Pt ≡ [ 0 Pt (i)1−ε di]1/(1−ε) the price index. Each firm i can reset prices at any given time only with probability 1 − η. Individual consumers bundle the continuum of goods via a CES aggregator with elasticity of substitution between goods, ε; their cost-minimizing plan gives rise to the demand curve for the ith good, Yt (i) = [Pt (i)/Pt ]−ε Yt , which constrains individual firms in Rtheir production choices. 1 Finally, price dispersion ∆t ≡ ln 0 [Pt (i)/Pt ]−ε/(1−α) di evolves according to the law of motion   ε " # (ε−1)πt − (1−α)(1−ε)   ε π +∆ 1 − ηe  ∆t = ln ηe 1−α t t−1 + (1 − η)  1−η

(7)

4.1.2. Households Identical consumers indexed by j ∈ [0, 1] trade in complete financial markets and choose consumption and labor to maximize intertemporal utility: U0 ( j) = E0

∞ X t=0

[Ct ( j) − Xtc ]1−γ [Ht ( j) − Xth ]1−γ +χ β 1−γ 1−γ

!

t

subject to the present-value budget constraint E0

∞ X t=0



X   B−1 ( j) M0,tCt ( j) = M0,t (1 − τh )Wt Nt ( j) + Dt + T th + E0 P0 t=0

with t-period real contingent claims prices M0,t . As in Benhabib et al. (1991) or Greenwood and Hercowitz (1991), households derive utility from consumption of two goods: Ct is real consumption purchased in the market, and Ht denotes the consumption produced at home, with production function Ht = At (1 − Nt ). Xtc and Xth represent external habit levels that are slow-moving averages of contemporaneous and past aggregate consumption. Hours worked in market production Nt are remunerated at the real wage rate Wt . Bt denotes holdings of one-period nominal debt issued by a fiscally passive government and with unit price exp(−it ) = Et Mt,t+1 Pt /Pt+1 . Dt is the equity income households receive from owning the aggregate firm. Parameter β is the subjective discount rate and parameter χ controls the steady-state effect of habits.15 There are at least two reasons to focus on home consumption rather than standard leisure. First, once accepted that people get used to an accustomed market consumption level, it is only natural to assume that people develop a habit also to home consumption. Second, home consumption frees Let χ = χ0 (S c /S h )1−γ , where χ0 is the counterpart to χ in the CRRA utility case, i.e., when Xtc = Xth = 0. I assume a calibration for χ0 to achieve a level of hours N = .5 in the flexible-price steady state. 15

14

up the elasticity of intertemporal substitution γ by making the economy consistent with balanced growth for any γ > 0. Consequently, I can recover the pricing kernel of Campbell and Cochrane (1999). The law of motion of habits is specified indirectly through the processes for surplus market consumption S c,t ≡ (Ct − Xtc )/Ct and surplus home consumption S h,t ≡ (Ht − Xth )/Ht . The habit levels thus specified are external to any individual consumer, as theR law of motion of the R surplus levels is driven by aggregate market and home consumption, Ct ≡ Ct ( j)d j and Ht ≡ Ht ( j)d j. To ensure well-behaved marginal utilities, consider the following dynamics for the logarithms of aggregate surplus levels: sc,t+1 = (1 − ρ s )sc + ρ s sc,t + Λc,t εct+1 sh,t+1 = (1 − ρ s )sh + ρ s sh,t + Λh,t εht+1

(8)

where εct ≡ (Et − Et−1 )ct and εht ≡ (Et − Et−1 )ht , with common persistence ρ s . I adopt a particular specification for the sensitivity functions Λc,t and Λh,t , and a specific calibration for the steady-state levels of the surplus variables: s ! p γ var(εc ) 1 Λc,t = max 0, 1 − 2(sc,t − sc ) − 1 , Sc = Sc 1 − ρ s − ξ1 /γ !−1 Sc 1 − Sh cov(εc , εh ) 1−S Λh,t = (1 + ξ2 ) Λc,t Sh = 1 + 1 − Sc Sh S var(εh ) for some free parameters ξ1 and ξ2 , whose key role I discuss below. Importantly, the two consumption habits specified indirectly by these surplus processes are approximately two predetermined, slow-moving averages of past market and home consumption, respectively, as required for a sensible notion of habit. Households grow used slowly to unanticipated movements in the two types of consumption. Finally, the government levies an income tax τh = 1 − S c /S h designed to offset any steady-state distortions caused by the habit externality, and rebates it in lump-sum fashion, T th . In this context, optimality implies the log stochastic real discount factor, m0,t = −t ln(β) − γ(ct − c0 ) − γ(st − s0 ) the equilibrium consumption-saving equation, it = − ln Et βe−γ∆ct+1 −γ∆st+1 −πt+1

(9)

and the labor supply equation, γ S c,t A1−γ χ t Ct Wt = h γ 1 − τ (1 − Nt ) S h,t

15

!γ (10)

4.1.3. Government Monetary policy is described by a simple Taylor rule for the nominal interest rate that reacts to inflation πt ≡ ln(Pt /Pt−1 ), and the output gap relative to trend, it = i∗ + φπ πt + φy [ct − atP − (1 − α)(n − ∆)]

(11)

for an interest rate level i∗ = E(rt ) + .5E[vart (πt+1 )] + E[covt (−mt+1 + πt+1 , −πt+1 )], with a real rate rt = − ln Et Mt+1 . Fiscal policy runs a balanced budget, T th = τh Wt Nt and T tf = τ f Wt Nt . 4.1.4. Market clearing R1 The market for goods and labor clear, Yt = Ct and Nt = 0 Nt (i)di. 4.1.5. Technology Log technology is composed of a permanent component atP and a transitory component aTt such that at = atP + aTt , with P at+1 = µ + atP + (1 − θ)et+1

aTt+1 = ρu aTt + θet+1 with average drift µ and persistence ρu , where θ indexes the extent to which a technology shock et ∼ Niid(0, σ2 ) has a permanent effect. For example, θ = 0 associates with random-walk technology, while θ = 1 associates with the typical trend-stationary specification (e.g., Gal´ı, 2008). Consistent with definition (1), θ indexes the amount of uncertainty in the consumption process that can be removed by monetary policy. 4.1.6. Competitive equilibrium Define the variables st ≡ sc,t and zt ≡ sh,t − sc,t . It turns out that there are particular values for the free parameters ξ1 and ξ2 such that the approximate equilibrium solution for the consumption gap and inflation are ct − at = (1 − α)n + ψc aTt ,

πt = ψπ aTt

to a first-order perturbation around the risky steady state. In words, average inflation is zero, and the equilibrium dependence of consumption and inflation on surplus consumption is zero. (To a first-order approximation price dispersion is a trivial process, ∆t = 0.) Therefore, by an appropriate choice of free parameters ξ1 and ξ2 the model is able to produce a separation between risk premia and quantity dynamics that preserves the implications for quantities and inflation of the basic New Keynesiam model with CRRA utility. The two parameters control the spillover of habits dynamics onto equilibrium quantities and inflation. Intertemporally, households balance strong intertemporal substitution motives with strong precautionary saving motives, thereby

16

disciplining, via ξ1 , the variation in the risk-free rate 1 rt = − ln(β) + γµ − γ(1 − ρ s − ξ1 /γ) + γψc aTt − ξ1 sˆt 2 and hence in households’ incentives to save and consume (9). Statically, households balance a strong aversion to fluctuations in market and home consumption across states, which is key to avoid excessive variation in the real wage rate (10). In fact, the choice of the sensitivity function for surplus home consumption implies approximately Zt S c,t /S c  S t ξ2 ≡ = Z S h,t /S h S in equilibrium, and hence ξ2 disciplines the variation in the marginal rate of substitution between consumption and labor. This property solves the puzzle in Lettau and Uhlig (2000) by preventing households from varying excessively labor hours to smooth out consumption. The particular parametrization required to achieve an approximate macro-finance separation is close to the point ξ1 = ξ2 = 0. Namely: (e−V0,3 − e−V0,2 )δη(1 − ρ s ) ξ2 = −V0,3 (e − δηρ s )(e−V0,2 − δη)

γ ξ1 = (1 + ψc θ)ψπ θσ2 , S

where V0,2 and V0,3 are second-order moments, and the approximate equilibrium coefficients are [γ(1 − ρu ) + φy ](1 − δρu ) + ϕ(φπ − ρu ) [γ(1 − ρu ) + φy ](1 − δρu ) + κ(φπ − ρu ) (κ − ϕ)(φπ − ρu ) ψπ = − [γ(1 − ρu ) + φy ](1 − δρu ) + κ(φπ − ρu ) ψc = −

Parameters κ and ϕ are transformations of other deep parameters and, in absence of risk, reduce to κ = (1 − δη)(1 − η)[γ(2 − α) + α]/η(1 − α + αε) and ϕ = 0. (The online appendix derives these formulas.) It follows that the equilibrium allocation is observationally equivalent to a model with CRRA households—an implication confirmed in figure 3a. A macroeconomist interested in the model’s implication for quantities and inflation would not be able to discriminate between this model and its CRRA version. 4.1.7. Quantitative implications Table 3 lists the calibration of the deep parameters of the model. The production side of the economy uses standard values in the New Keynesian literature from Gal´ı (2008). Parameters relating with the pricing kernel are calibrated as in Campbell and Cochrane (1999), with an EIS of 1/2, a rate of time preference to match the target mean risk-free rate, spillover parameters to produce macro-finance separation, and habit persistence to capture the persistence of observed price-dividend ratios. I estimate the three parameters that drive the dynamics of technology by a GMM strategy that 17

Parameter γ β ρs ξ1 ξ2

Preference block

New Keynesian block

Exogenous block

1−α ε 1/(1 − η) φπ φy µ ρu σ 1−θ

Value Inverse EIS Time preference Habit persistence Intertemporal spillover Static spillover

2 0.9937 0.9918 0.0001 −0.0117

Labor share in value added Elasticity of substitution in Dixit-Stiglitz aggregator Average price duration (in months) Policy response coefficient to inflation movements Policy response coefficient to output movements

2/3 6 9 1.5 0.5/12

Mean technology growth Persistence of the conditional mean of technology growth Conditional volatility of technology Fraction of permanent technology shocks

0.0012 0.743 0.0148 0.38

β matches an average real risk-free rate of 0.69% per year. ρ s matches a 12-month autocorrelation of price-dividend ratios of .906. µ matches an average annual per-capital real consumption growth of 1.39%. ρu , σ, and θ are estimated by GMM to match the 1- to 5-year ahead volatility of per-capita real consumption growth, real dividend growth, and inflation over 1994-2013. The calibration for ξ1 and ξ2 implies S c = 0.10 and S h = 0.35. Table 3: Deep parameters and their calibration (monthly frequency). Data for real consumption growth use BEA-NIPA data over the period 1994-2013 for per-capita personal consumption expenditure in nondurables and services, and are deflated by the CPI. Monthly simulated data are aggregated to an annual frequency and are matched to the corresponding data moments.

horizon (years), N up to cost of uncertainty, LN (p.a.)

0.5 0.0044

1 0.0032

1.5 0.0025

2 0.0021

5 0.0012

10 0.0010

Table 4: Model-implied marginal cost of fluctuations at all periodicities n ∈ N.

18

30 0.0015

minimizes the distance between model-implied moments and observed variances of growth rates over 1- to 5-year horizons for the three cashflows of interest—consumption, dividends, and inflation. Figure 3a reports the fit of the model in these implications for quantities and inflation, which captures relatively well the autocorrelation and volatility of cashflows observed over the 1994-2013 periods—the same period as I have evidence about welfare costs. The figure also reports the term structures of cashflow volatility in the model with CRRA utility. The implications for dividends of the model with and without habits are nearly identical, while the properties of consumption and inflation are indistinguishable, thereby confirming numerically the approximate equivalence discussed in section 4.1.6. Figures 1b and 3 and table 4 report the average term structures of the equity premium, of real interest rates and of the welfare cost of uncertainty implied by the model, as well as the interquartile range of the term structure of welfare costs. The model produces an average equity premium of around 6.2%, close to the observed value reported in figure 1a. The market premium is substantially lower than the premium commanded by short-term equities, consistent with observed strip returns. The representative household is particularly sensitive to fluctuations in market income, as countercyclical marginal costs exacerbate the procyclicality of corporate profits after a technology shock, but fears less uncertainty in the long-run because dividends and consumption are cointegrated. Crucially, this simple model is able to capture the main empirical properties of the term structure of welfare costs documented in section 3, including its countercyclical variation. 4.2. Optimal stabilization policy Aggregate (scaled) welfare Vt ≡ Ut e(γ−1)[(1−α)n+s] can be written as: Vt = (N =

1−α

S)

e(1−γ)(at + sˆt ) 1−γ

∞ X

! (Ct+ j S c,t+ j )1−γ (Ht+ j S h,t+ j )1−γ Et β +χ 1−γ 1−γ j=0    c 1−γ t −e   ∆ˆ t + ec1−α   1 − Ne   e(1−γ)(ect −ec−(1−α)∆) + (1 − α) ezˆt   + βE V t t+1   1 − N  

γ−1

j

(12)

where e ct = ct − at is the market consumption gap. The constrained-efficient allocation maximizes welfare subject to the definition of habits (8), to the law of motion of price dispersion (7), and to the pricing setting equations,  −γ ˆ ecˆ t ε(1 − τ f )χ0 N ecˆt +(1−γ)aTt −γ sˆt −γˆzt  1 − Ne∆t + 1−α    e = Et δηe + e 1−α (ε − 1)(1 − α)L 1−N " # 1−α+αε " # 1−α+αε (ε−1)πt (1−α)(1−ε) N (1−γ)(1−α) (1−γ)(ecˆ t +aTt )−γ sˆt 1 − ηe(ε−1)πt (1−α)(1−ε) `ˆt (ε−1)πt+1 +(1−γ)(1−θ)σεt+1 +`ˆt+1 1 − ηe e = Et δηe + e L 1−η 1 − ηe(ε−1)πt+1 `ˆt

ε ˆ 1−α πt+1 +(1−γ)(1−θ)σεt+1 +`t+1

where `t ≡ ln Γ1,t − (1 − γ)atP and δ ≡ βe(1−γ)µ .16 16

Note that a maximization in absence of the three pricing equations results in the Pareto optimal allocation—i.e., the competitive allocation under internal habits and flexible prices.

19

0.2 0.01

0.18 0.2 0.16 0.14

0.15

0.12 0.005 0.1

0.1 0.08

0.05 0.06 0.04

0

0

0.02 0 1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

8

9

10

n (in years)

(a) Average term structures of cashflow volatilities; (b) Average term structures of equity, interest rates, model (solid and dashed lines), model under CRRA and welfare costs. utility (dotted lines), and data (markers).

0.01

0.01

0.2

0.2

0.15

0.15

0.005

0.005

0.1

0.1

0.05

0.05

0

0

0

1

2

3

4

5

6

7

8

9

0

10

0

n (in years)

(c) First quartile of the term structure of welfare costs.

0

1

2

3

4

5

6

7

8

9

10

n (in years)

(d) Third quartile of the term structure of welfare costs.

Figure 3: Term structures of cashflows and asset prices. The nth cashflow term-structure component is defined as n−1 var[ln(Dt+n /Dt )] for a cashflow process D. Model-implied term structures of equity (dashed line), interest rates (dash-dotted line) and welfare costs (solid line); average and interquartile range. Annualized values on the left axis for the term structures of casfhlows, equity, and interest rates and on the right axis for welfare costs.

20

By examining the structural equations we can study the consequences of two simple regimes— inflation targeting and risk premia targeting. Implementation will be done by a Taylor rule with form (11). 4.2.1. Inflation targeting (or: close the consumption gap) Under an inflation targeting regime consumption equals the flexible-price level, and hence inflation is zero (πt = 0). We have: ct = at + (1 − α)n,

nt = n¯ ,

εct+1 = et+1 ∼ Niid(0, σ2 )

If follows that welfare (12) under this regime is: Vtit

P i e(1−γ)(at + sˆt ) h T it = 1 + (1 − α)e(1−γ)ˆzt e(1−γ)at + βEt Vt+1 1−γ

(13)

with highly volatile surplus consumption sˆitt+1 = ρ s sˆitt + Λ( sˆitt )σεt+1 . 4.2.2. Risk premia targeting (or: remove consumption volatility) A key property of the New Keynesian model is the stationarity of marginal costs, and hence of the consumption gap. It follows that a regime that removes as much consumption risk as possible has to preserve the property that consumption and technology are cointegrated. In any event, consumption has to inherit the permanent component of technology, which represents the maximum amount of consumption risk that monetary policy can remove. In the context of the model, removing consumption risk—i.e., ct = atP + (1 − α)(n − ∆)—is equivalent to stabilizing surplus consumption, and hence the maximum risk-return tradeoff ln Et Ret+1 c σt (mt+1 ) = γ(1 + Λ( sˆt ))σt (εt+1 ) ≥ e σt (rt+1 ) The policy is therefore equivalent to a regime that targets risk premia by stabilizing the maximal risk-return tradeoff. Under such a risk premia targeting regime consumption equals the permanent component of technology, and hence πt = −[κ/(1 − δρu )]aTt and εct = (1 − θ)et . If follows that welfare (12) under this regime is:   1−γ  aT t  ˆ   (1−γ)(atP + sˆt )  ∆ −  aT +ˆz 1 − Ne t 1−α    (γ−1)(1−α)∆ e e e t t   + βEt V rpt + (1 − α) (14) Vtrpt = t+1   1−γ  1 − N   rpt rpt with minimally volatile surplus consumption sˆrpt t+1 = ρ s sˆt + Λ( sˆt )(1 − θ)σεt+1 . The only remaining unknown is price dispersion ∆t , which depends on the dynamics of equilibrium inflation.

4.2.3. Approximate welfare criterion The appendix derives a quadratic approximation around the risky steady state of average (detrended) per-period welfare. Although the results use a global solution, approximate welfare is 21

useful to gain insight into its determinants. I consider average welfare losses L ≡ −E(Vt )/(CS )1−γ , L=

1 γ(2 − α) + α 1 ηε(1 − α + αε) (γ − 1)(2 − α) var(e ct ) + var(st ) var(πt ) + 2 2 (1 − η) (1 − α) 2 1−α 2 (1 − α)(γ − 1) − (γ − 1)cov(e ct , zt ) + var(zt ) + (γ − 1)(1 − α)cov(st , zt ) + t.i.p. 2

(15)

that hold up to a term of at least third order and a term independent of policy. There are two separate sources of strategic complementarities that generate departures from the efficient dynamics of the benchmark RBC model. First, in a sticky-price environment inflation volatility is approximately equivalent to cross-sectional dispersion in prices, which in turn associates with inefficient employment. Second, when habits are external people fail to internalize the fact that higher consumption also has a habit effect that means a higher marginal value of consumption (Ljungqvist and Uhlig, 2000). When facing good news about current and future states, people should not increase their consumption level as much as they would want to because they will become addicted to such a higher level. It follows that in an external-habit environment consumption fluctuates too much. Therefore, to maximize welfare criterion (15), a central bank would want to stabilize the consumption gap, ct − at —and hence stabilize inflation in the process—as well as consumption uncertainty, (Et − Et−1 )ct , to stabilize the price of risk, and hence risk premia. Since achieving both goals is unfeasible, the optimal monetary policy trades them off to minimize the welfare metric.17 4.2.4. Fill the gaps or remove volatility? Under the baseline calibration risk premia targeting dominates inflation targeting, as Lπ ≥ L x for any value of θ ∈ (0, 1). People hate so much consumption volatility that a policy that achieves the flexible-price equilibrium is suboptimal relative to a policy that removes as much consumption volatility as possible. Moreover, the inequality is strict for any θ ∈ (0, 1], with equality if and only if θ = 0, as random-walk technology negates any role to nominal rigidities; consumers cannot forecast future movements in the real rate, so they choose a random-walk consumption path, while firms cannot forecast future movements in marginal costs, so inflation is zero. Since some mean reversion is necessary to let nominal rigidities model a component in dividends that captures the downward-sloping term structure of welfare costs, it follows that the dominance of risk premia targeting over inflation targeting is deeply linked to the model’s ability to explain the empirical term structure properties. Table 5 exemplifies this result numerically under the baseline calibration. The welfare dominance of the risk premia targeting regime is overwhelming. Average welfare losses relative to the Pareto optimum can be one or two orders of magnitude larger under inflation targeting than under risk premia targeting. The welfare weight attached to the inflation objective dominates the welfare 17

A characterization of the optimal policy regime under discretion or commitment is beyond the scope of this paper. Similarly, I do not consider additional policy tools to remove the policy tradeoff—for example, Ljungqvist and Uhlig (2000) consider time-varying taxation to address the habit externality. The purpose of the present analysis is to point out how a model consistent with observed welfare cost measures can easily imply that people hate consumption fluctuations so much as to overturn the standard result that stable inflation is the macroeconomic priority.

22

θ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Baseline (1 − η)−1 = 6 (1 − η)−1 = 12 π = 2% p.a. γ = 1.5 γ=1 γ = 0.5

1 1 1 1 1 1 1

0.146 0.145 0.146 0.146 0.370 1.000 1.008

0.035 0.035 0.035 0.035 0.223 1.000 1.016

0.016 0.016 0.016 0.016 0.185 1.000 1.037

0.013 0.013 0.012 0.013 0.169 1.000 1.061

0.011 0.011 0.011 0.011 0.163 1.000 1.076

0.011 0.011 0.011 0.011 0.160 1.000 1.095

0.011 0.010 0.010 0.010 0.158 1.000 1.111

0.010 0.010 0.010 0.010 0.155 1.000 1.129

0.010 0.010 0.010 0.010 0.154 1.000 1.138

0.010 0.010 0.010 0.010 0.154 1.000 1.137

Table 5: Ratio of welfare losses L x /Lπ . Lπ : Welfare losses under a simple policy regime that targets inflation (or the consumption gap). L x : Welfare losses under a simple policy regime that targets risk premia (or consumption uncertainty). θ denotes the fraction of uncertainty in the consumption process that can be removed by monetary policy. Global solutions are projected onto the subspace spanned by a polynomial basis of order 4 collocated on a sparse grid of Chebyshev points.

weights attached to consumption and risk aversion stabilization, however, the volatility of surplus consumption is so large that removing its fluctuations becomes the priority. In particular, welfare under inflation targeting is much lower in bad surplus consumption states than under risk-premia targeting, and slightly higher in normal surplus consumption states. The extent of the difference between the two policies can be made most evident by noticing how the simple regimes can be implemented by an appropriate choice of the reaction parameters in Taylor rule (11). The central bank is able to implement the inflation targeting regime by choosing an extreme anti-inflationary response, φπ → ∞. The risk premia targeting regime can be implemented by choosing an extreme response to movements in detrended consumption, φy → ∞. In this sense the two policies are polar opposite. 4.3. Robustness This section carries out some exercises to explore the robustness of the model’s results. 4.3.1. Sensitivity to parameter values Table 5 reports the welfare exercise for different calibrations. First, I vary the degree of price rigidity. This parameter controls the weight attached to inflation volatility in welfare criterion (15). However, results barely move as I vary price stickiness—both upwards and downwards. Second, I consider trend inflation. Results change only slightly as the model moves away from a zero-inflation steady state and towards the mean inflation rate of 2% observed over the 1994-2013 period. Finally, I vary the elasticity of intertemporal substitution. Welfare criterion (15) suggests that a unitary elasticity of intertemporal substitution works as a threshold. Intuitively, higher volatility in log consumption has a detrimental effect because it increases the volatility of the consumption level, but it also has a beneficial effect by increasing the mean consumption level. At γ = 1 the two welfare effects offset each other. Table 5 confirms this intuition. The dominance of risk premia targeting remains strong for γ > 1 but vanishes at γ = 1, where the two regimes have equal outcome in terms of welfare, and reverts for γ < 1. Nevertheless, the model is no longer able to capture a realistic equity premium and sizable costs to short-run uncertainty for γ < 1. 23

1

1

15.4%

45.2%

4.0%

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0

0

-0.2

-0.2

-0.4

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-0.6

-0.6

-0.8

-0.8

5.5% -1 -1

84.2%

-0.8

33.9% -0.6

-0.4

-0.2

0

0.2

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0.6

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0.3% -1 -1

1

(a) Monthly rates.

-0.8

11.5% -0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

(b) Annual rates.

Figure 4: Block-bootstrap distribution of correlation coefficients between real consumption growth, real dividend growth, and inflation. Dots are coefficients for 100,000 bootstrap simulations. The circle is the sample estimate. Percentages represent the fraction of bootstrap simulations that fall in each quadrant. Blocks are of size 12 to preserve time-series dependencies. (Only 10,000 simulations are plotted.)

4.3.2. Is the nominal-real covariance plausible? A recent literature documents that the dynamics of inflation and consumption growth have changed precisely in the late 1990s—the period over which we have evidence about the term structure of welfare costs. For example, Boons et al. (2017) and Song (2017) document that predictive regressions of future real consumption growth on lagged inflation are characterized by a negative regression coefficient before the late 1990s and by a positive coefficient thereafter. In this context, the model presented in section 4 captures correctly the nominal-real covariance that the literature has documented as a distinguishing feature of the last two decades. The model’s emphasis on technology shocks as a key driving factor to rationalize the evidence about the term structure of welfare costs implies that high inflation signals good future real growth. This property is at odds with the pre-2000s post-war U.S. experience, but is in line with the evidence over the period of interest. Furthermore, the contemporaneous correlation coefficient over the 1994-2014 period between real consumption growth and inflation is −.09 using monthly growth and inflation rates and −.48 using annual rates. The bootstrapped p-value of the hypothesis that the correlation coefficient between consumption growth and inflation is positive, however, is higher than 20% for monthly rates—but it drops below 5% for annual rates.18 Nevertheless, we can find more statistical power when testing the joint hypothesis that the correlation coefficients between consumption growth and inflation and between dividend growth and inflation are both positive. I can reject that hypothesis at 5.5% (monthly rates) and 0.3% (annual rates) size levels based on the joint bootstrap distribution 18

Song (2017) also finds that the contemporaneous correlation between inflation and real consumption growth may have switched sign, turning positive, after 1998 but the result is not statistically significant. Furthermore, unmodeled reaction lags make evidence about this correlation as a discriminating factor not very powerful. The sign change in consumption growth predictive regressions by inflation for different forecasting horizons is instead robust to reaction lags.

24

of correlation coefficients—shown in figure 4. Inflation and consumption growth are positively correlated precisely in those subsamples in which dividend growth and inflation are negatively correlated.19 Therefore, the simple one-shock model presented in section 4 captures the key implications for the nominal-real covariance seen in the data during the period over which evidence about the term structure of welfare costs is available. 4.3.3. What if habits are internal? If habits are internal, the flexible-price equilibrium is the Pareto optimum. Nevertheless, if we take to the logical extreme the critique by Lettau and Uhlig (2000), a necessary diagnostic requirement for a model with habits—and, more generally, for any model that incorporates discountrate variation into a macro model—to be deemed admissible is that it can be calibrated to leave the implications for macroeconomic aggregates intact, or nearly so. In this context, the only way the internal-habit model can leave quantity and inflation dynamics unchanged relative to a CRRA specification is under γ = 1. But under a unitary elasticity of intertemporal substitution, the model reduces to a log utility CRRA specification in all its quantity, asset pricing, and welfare implications. (The appendix derives this result.) The trivial asset pricing implications of the macro-financially separate internal-habit model reject it forcefully. 5. Conclusion Lucas (1987, 2003) introduced the notion of cost of aggregate uncertainty as a thought experiment to provide an assessment of the tradeoff between growth and macroeconomic stability. Analogously, the term structure of the cost of uncertainty requires little structure to reveal the tradeoff between growth and macroeconomic stability at different time horizons. Recent derivative securities provide direct information about this term structure and, therefore, new insight into an old question (the tradeoff between growth and stability) and evidence to study a new question (the tradeoff between stability at different horizons). Asset markets suggest that potential gains from greater economic stability are not trivial, especially in the short run and during downturns such as in the early 2000s and during the recent financial crisis. The finding of sizable and volatile costs imposed by an increase in short-run uncertainty inscribes into a burgeoning literature that finds high and time-varying short-maturity risk premia as a pervasive phenomenon across different asset classes (Binsbergen and Koijen, 2017). The result that the marginal cost of uncertainty is a linear combination of risk premia makes one of the main tasks of macroeconomics—the assessment of the macroeconomic priorities (Lucas, 2003)—inextricably linked to finance. The negative slope and countercyclical variation of the estimated term structure of welfare costs cannot be easily captured by leading asset pricing theories and therefore represents a puzzling piece of evidence with seemingly crucial welfare consequences. For example, I showed that two models that are observationally equivalent in their quantity implications but differ in their asset pricing implications prescribe polar opposite policies. 19

Note that the joint hypothesis that consumption growth and inflation are positively correlated, dividend growth and inflation are positively correlated, and consumption growth and dividend growth are negatively correlated must be even less likely—with bootstrap p-values of 2.3% (monthly rates) and .15% (annual rates).

25

Appendix A. Theoretical results A.1. Proof of proposition 1 Differentiating (1) with respect to θ and evaluating at θ = 0, it follows that LtN =

X Et (U1,t+n )Et (Dt+n + Wt+n Nt+n ) − Et (U1,t+n [Dt+n + Wt+n Nt+n ]) P n∈N n E t (U 1,t+nC t+n ) n∈N X Et (−U2,t+n /Wt+n )Et (Wt+n Nt+n ) − Et (−U2,t+n Nt+n ) P − n∈N n E t (U 1,t+nC t+n ) n∈N

where U1,t ≡ ∂Ut /∂Ct and U2,t ≡ ∂Ut /∂Nt . Then, under assumption (2), the terms cointaining labor cancel each other, and hence LtN = αN t with αN t ≡

P n∈N

n Et (Mt,t+n Dt+n )/

X Et (Mt,t+n )Et (Dt+n ) − Et (Mt,t+n Dt+n ) P n∈N n E t (Mt,t+n Dt+n ) n∈N P n∈N

n Et (Mt,t+nCt+n ).

A.2. Proof of proposition 2 I can rewrite equation (3) as LtN =

X P n∈N

 n Et (Mt,t+n Dt+n ) 1  Et (Mt,t+n )Et (Dt+n ) × −1 n Et (Mt,t+n Dt+n ) n∈N n E t (Mt,t+nC t+n )

The absence of arbitrage opportunities ensures positive weights {ωN n,t }. The proposition then follows directly from the expression of the term structure components, lt(n) , and the definition of the hold-to-maturity return on an arbitrary payoff, X, maturing in n periods, (n) (n) R(n) x,t→t+n = Xt+n /D x,t , with no-arbitrage price D x,t = E t (Mt,t+n Xt+n ) at period t. Market equity is characterized by X = D and real bonds by X = 1. A.3. Derivation of the welfare criterion In the model with external habits and nominal rigidities, aggregate welfare is:   1−γ ∞ X  Ct Ht 1−γ  t  U0 = E0 β  +χ 1 − γ 1 − γ t=0 where Ct ≡ Ct − Xtc and Ht ≡ Ht − Xth . Consider the stationary transformation of utility, U0 A1−γ 0 with per-period detrended utility

= E0

∞ X

δt exp[(1 − γ)(at − a0 − tµ)]Vt

t=0

1 Ct Vt ≡ 1 − γ At 26

!1−γ

χ Ht + 1 − γ At

!1−γ

A second-order expansion around the risky steady state of detrended, per-period utility yields: ! ! 1 − γ 1 − γ 1−γ 2 1−γ 2 f ft ) + ft ) + t.i.p. Vt = Ce ln(Cet ) + ln(Cet ) + χH ln(H ln(H 2 2 ft ≡ Ht /At , up to a term independent with the detrended surplus consumption levels Cet ≡ Ct /At and H 1−γ f /Ce1−γ = 1 − α, given the efficient of policy. I can rewrite the ratio of partial derivatives χH employment subsidy and fiscal and monetary policies such that risky and deterministic steady states coincide. Using the second-order expansion at N = .5, ft ) + 1 ln(H ft )2 = sˆh,t − nˆ t + 1 sˆ2 − 1 nˆ 2 − nˆ t sˆh,t ln(H 2 2 h,t 2 t approximate, detrended, per-period average welfare can be rewritten as:   ! f1−γ    Vt H 1 − γ 1 − γ 2 2 ft ) + ft )  + t.i.p. = E ln(Cet ) + ln(Cet ) + χ ln(H ln(H E  1−γ 1−γ e e 2 2 C C ! 1−γ 2 2 (e ct + sˆc,t + 2e =E e ct + sˆc,t + ct sˆc,t ) 2 ! 1+γ 2 1−γ 2 nˆ + sˆ − (1 − γ)ˆnt sˆh,t + t.i.p. + (1 − α)E sˆh,t − nˆ t − 2 t 2 h,t (γ − 1)(2 − α) 1 γ(2 − α) + α var(e ct ) − var(st ) = −(1 − α)E(∆t ) − 2 1−α 2 (1 − α)(γ − 1) + (γ − 1)cov(e ct , zt ) − var(zt ) − (γ − 1)(1 − α)cov(st , zt ) + t.i.p. 2 R1 where e ct ≡ ct − at and ∆t ≡ ln 0 [Pt (i)/Pt ]−ε/(1−α) di, with the aggregate production relation (1 − α)ˆnt = e ct + (1 − α)∆t .20 By a standard argument, let S (t) ⊂ [0, 1] represent the set of firms Rnot reoptimizing their 1 posted price in period t, recall the definition of aggregare price level Pt ≡ ( 0 Pt (i)1−ε di)1/(1−ε) , and recognize that all resetting firms choose an identical price P∗t . It follows that 1=

Z S (t)

∆t

e =

Z S (t)

20

Nt =

R1 0

Nt (i)di =

R1 0

Pt (i) Pt Pt (i) Pt

!1−ε

P∗ di + (1 − η) t Pt

ε !− 1−α

!1−ε =

P∗ di + (1 − η) t Pt

[Yt (i)/At ]1/(1−α) di = (Yt /At )1/(1−α)

ε !− 1−α

R1 0

27

ηΠε−1 t

P∗ + (1 − η) t Pt

ε 1−α

∆t−1

= ηΠt e

!1−ε

P∗ + (1 − η) t Pt

ε !− 1−α

[Pt (i)/Pt ]−ε/(1−α) di, with the clearing condition Yt = Ct .

and hence a second-order expansion around a steady state with π = 0 implies ∆t = η∆t−1 +

1 ηε(1 − α + αε) 2 π 2 (1 − η)(1 − α)2 t

Therefore, I can rewrite average, per-period, detrended welfare as: E(Vt ) 1 ηε(1 − α + αε) 1 γ(2 − α) + α (γ − 1)(2 − α) =− var(πt ) − var(e ct ) − var(st ) 2 2 (1 − η) (1 − α) 2 1−α 2 Ce1−γ (1 − α)(γ − 1) + (γ − 1)cov(e ct , zt ) − var(zt ) − (γ − 1)(1 − α)cov(st , zt ) + t.i.p. 2 up to a term of at least third order and a term independent of policy. A.4. Internal habits The Pareto optimum (flexible prices, internal habits) implies the intertemporal and intratemporal rates of substitution !−1 1−γ 1−γ Ct+1 S c,t+1 + Λ(st )(Et+1 − Et )Mct+1 Ct+1 int. Mt+1 = β 1−γ Ct Ct1−γ S c,t + Λ(st−1 )(Et − Et−1 )Mct 1−γ 1−γ 1−γ h ∂Utint. /∂Nt Ct χAt (1 − Nt ) S h,t + ϕh Λ(st−1 )(Et − Et−1 )Mt = − int. 1−γ ∂Ut /∂Ct 1 − Nt Ct1−γ S c,t + Λ(st−1 )(Et − Et−1 )Mct

with the shadow values of market and home surplus consumption 1−γ Mct = Ct1−γ S c,t + βρ s Et Mct+1 1−γ + βρ s Et Mht+1 Mht = χHt1−γ S h,t

On the one hand, a positive market (home) consumption shock means a lower marginal value of market (home) consumption; on the other hand, a positive market (home) consumption shock increases the habit level and thereby increases the marginal value of market (home) consumption. It is straightforward to verify how a unitary elasticity of intertemporal substitution, γ = 1, produces constant shadow values of surplus market and home consumption. Under this parametrization we have !−1 ∂Utint. /∂Nt Ct+1 χCt int. Mt+1 = β , − int. = Ct ∂Ut /∂Ct 1 − Nt so all intertemporal and intratemporal effects of the habit are absent. The condition γ = 1 is actually necessary to grant a macro-finance separation (even approximately) when habits are internal, for any value of the spillover parameter ξ2 . The macro-financially separate Pareto optimum displays the same low and stable risk premia as under a log-utility specification. The associated trivial asset pricing implications are the reason I favor the external-habit specification. 28

B. Empirical results B.1. Data selection and synthetic replication I drop weekly, quarterly, pm-settled and mini options, whose nonstandard actual expiration dates are not tagged. Index mini options with three-year maturities are traded since the 1990s but standard classes appear only in the 2000s; for this reason I follow Binsbergen et al. (2012) and focus on options of up to two-year maturity. I eliminate all observations with missing values or zero prices and keep only paired call and put options. I use mid quotes between the bid and the ask prices on the last quote of the day and closing values for the S&P 500 index. On any date t, consider all available put-call pairs that differ only in strike price. For the ith strike price, Xi , i = 1, ..., I, define the auxiliary variable A(n) it ≡ Pt − Ct,t+n + Pt,t+n (n) = P(n) d,t + Xi P$b,t

Pn ( j) where the last equality holds by put-call parity, with P(n) j=1 Dt the no-arbitrage price of the d,t = next n periods of dividends. Therefore, if there are no arbitrage opportunities and the law of one (n) price holds then the map A(n) t : Xi 7→ Ait is strictly monotonic and linear. In practice, the relation does not always hold without error across all strike prices available; as long as more than two strikes are available for a given maturity, one can use the no-arbitrage relation to extract P(n) d,t and (n) P$b,t as the least absolute deviations (LAD) estimators that minimize expression I X A(n) − P(n) − X P(n) i $b,t it d,t i=1

for a given trade date t and maturity n. The cross-sectional error term accounts for potential measurement error (e.g., because of bid-ask bounce, asynchronicities, or other microstructural frictions). Over most of the sample the strikes and the auxiliary variables are in a nearly perfect linear relation except for a few points that violate the law of one price. The LAD estimator is particularly appropriate to attach little weight to those observations as long as their number is small relative to the sample size of the cross-sectional regression. Accordingly, I drop all trade dates and maturities that associate with a linear relation between Xi and A(n) it that fails to fit at least a tenth of the cross-sectional size (with a minimum of five points) up to an error that is less than 1% of the extracted dividend claim price.21 The procedure results in a finite number of matches, which I combine to calculate the prices of options implied dividend claims and nominal bonds by using the put-call parity relation. As shown in figure B.5d, the number of cross-sectional observations available to extract the options implied prices of bonds and dividend claims increases over time (from medians of around 25 observations 21

In many instances, non-monotonicities in the auxiliary variable are concentrated in deep in- and out-of-the-money options. Whenever I spot non-monotonicities for low and high moneyness levels I restrict the sample to strikes with moneyness levels between 0.7 and 1.1 before running the cross-sectional LAD regression.

29

per trading day up to more than 100) as the market grows in size and declines with the options maturity. Of the resulting extracted prices I finally discard all trading days that associate with prices (n) Pd,t that are nonincreasing in maturity, as they would represent arbitrage opportunities. Overall, my selection method based on law-of-one-price violations excludes almost a fifth of the available put-call pairs. Finally, to obtain monthly implied dividend yields with constant maturities, I follow Binsbergen et al. (2012) and Golez (2014) and interpolate between the available maturities. As advocated by Golez to reduce the distance between intra-day and end-of-day options implied prices and hence the potential effect of asynchronicities and other microstructural frictions, I then construct monthly prices using ten days of data at the end of each month.22 B.2. Errors in synthetic replication (n) Figure B.6 plots the auxiliary map A(n) : X 7→ P(n) t d,t + X P$b,t for selected trading days and maturities. As shown by the lower part of figure B.6, the typical map towards the end of the sample is virtually perfectly linear and monotonic as one moves along the strike prices, so cross-sectional errors are immaterial. In the middle of the sample, the relationship still holds with almost no error despite a lower number of strike prices available relative to the last years of the sample. However, note how the cross-sectional errors are clearly visible during the first years of the sample, in which the strikes available are relatively few. (The figure also reports the index price to better gauge the moneyness of the put and call options that associate with each cross-sectional data point.) Figure B.5c box-plots the size of the law-of-one-price violations present in the sample which, for the most part, concentrate around errors of less than 1%; larger violations associate with the first years of the sample—probably because of a relatively low liquidity—and to years of greater volatility such as 2001 or the last years of the sample. Data previous to 1994 are more problematic by this metric (see also Golez, 2014) and I therefore exclude them altogether from the sample. Figure B.5a plots the prices of the synthetic dividend claims. Figure B.5b shows how synthesized dividend strip prices are leading indicators of subsequent dividends, in line with their interpretation as risk-neutral expectations of dividends.

22

I find large correlations with the intra-day options implied prices extracted by Binsbergen et al. (2012) over the 1996-2009 period; the correlation of the 6-, 12-, 18- and 24-month equity prices with Binsbergen-Brandt-Koijen data are of .91, .95, .95 and .94, respectively, with a mean-zero difference in levels. End-of-month data using a one-day window have slightly higher volatilities and correlations between .80 and .95. The median or the mean over a three-day window centered on the end-of-month trading day increases correlations to .87-.95; the marginal increase in correlations for window widths of more than ten days is nearly imperceptible. Since my approach extracts very similar prices, I bring additional robustness to the synthetic prices extracted by Binsbergen et al.. The nearly white-noise deviation between their estimate and mine over the comparable sample are likely a mixture of asynchronicities and different proxies for the interest rate. (I find options implied interest rates with nearly perfect correlations with the corresponding LIBOR and Treasury rates but with different levels that lie about halfway between the two proxies.)

30

100 90 80

18 0.5 years 1 year 1.5 years 2 years

16

6-month dividend 1-year biannual strip scaled S&P500

14 70 12 60 10 50 8

40

6

30 20

4

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0

0 1996

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(a) Price of the next n years of dividends.

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(b) Dividend strips and realized dividends.

200 0.1

150

0.05

0 100 -0.05 50 -0.1

0

-0.15

(c) Law-of-one-price violations.

(d) Number of strikes per trade date.

Figure B.5: Implied dividend claim prices, index, dividends, and sample statistics. Law-of-one-price violations are expressed in percent of the associated options implied dividend claim price. Only observations not filtered out by the data selection criteria are included.

31

date = 5/25/1994; maturity = 0.068493 years

date = 5/25/1994; maturity = 1.5644 years 520

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date = 12/3/2003; maturity = 0.04918 years

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date = 11/7/2013; maturity = 1.6164 years

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(n) (n) Figure B.6: Auxiliary map A(n) t : X 7→ Pd,t + X P$b,t on given trading days and for given maturities. The dotted lines indicate the value of the index at each respective trading day.

32

References Alvarez, F., Jermann, U.J., 2004. Using asset prices to measure the cost of business cycles. Journal of Political Economy 112, 1223–56. Atkeson, A., Phelan, C., 1994. Reconsidering the costs of business cycles with incomplete markets. NBER Macroeconomics Annual 9, 187–218. Backus, D., Chernov, M., Zin, S.E., 2014. Sources of entropy in representative agent models. Journal of Finance 69, 51–99. Bansal, R., Yaron, A., 2004. Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59, 1481–1509. Barillas, F., Hansen, L.P., Sargent, T.J., 2009. Doubts or variability? Journal of Economic Theory 144, 2388–418. Barro, R., 2009. Rare disasters, asset prices, and welfare costs. American Economic Review 99, 243–64. Belo, F., Collin-Dufresne, P., Goldstein, R.S., 2015. Dividend dynamics and the term structure of dividend strips. Journal of Finance 70, 1115–60. Benhabib, J., Rogerson, R., Wright, R., 1991. Homework in macroeconomics: Household production and aggregate fluctuations. Journal of Political Economy 99, 1166–1187. Binsbergen, J.H.v., Brandt, M., Koijen, R.S.J., 2012. On the timing and pricing of dividends. American Economic Review 102, 1596–1618. Binsbergen, J.H.v., Hueskes, W.H., Koijen, R.S.J., Vrugt, E.B., 2013. Equity yields. Journal of Financial Economics 110, 503–19. Binsbergen, J.H.v., Koijen, R.S.J., 2017. The term structure of returns: Facts and theory. Journal of Financial Economics 124, 1–21. Boguth, O., Carlson, M., Fisher, A., Simutin, M., 2012. Leverage and the limits of arbitrage pricing: Implications for dividend strips and the term structure of equity risk premia. Manuscript. Boons, M., Duarte, F., de Roon, F., Szymanowska, M., 2017. Time-varying inflation risk and the cross section of stock return. Manuscript. Borovicka, J., Hansen, L.P., 2014. Examining macroeconomic models through the lens of asset pricing. Journal of Econometrics 183, 67–90. Campbell, J.Y., Cochrane, J.H., 1999. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107, 205–51. Chari, V.V., Kehoe, P.J., McGrattan, E.R., 2007. Business cycle accounting. Econometrica 75, 781–836. Clarida, R., Gal´ı, J., Gertler, M., 1999. The science of monetary policy: A New Keynesian perspective. Journal of Economic Literature 37, 1661–1707. Cochrane, J.H., 2011. Determinacy and identification with Taylor rules. Journal of Political Economy 119, 565–615. Croce, M.M., 2012. Welfare costs in the long run. Manuscript. Croce, M.M., Lettau, M., Ludvigson, S.C., 2015. Investor information, long-run risk, and the term structure of equity. Review of Financial Studies 28, 706–42. D’Amico, S., Kim, D.H., Wei, M., 2018. Tips from TIPS: The informational content of Treasury Inflation-Protected Security prices. Journal of Financial and Quantitative Analysis 53, 395–436. De Santis, M., 2007. Individual consumption risk and the welfare cost of business cycles. American Economic Review 97, 1488–1506. Dew-Becker, I., Giglio, S., 2016. Asset pricing in the frequency domain: Theory and empirics. Review of Financial Studies 29, 2029–68. Ellison, M., Sargent, T.J., 2012. Welfare cost of business cycles with idiosyncratic consumption risk and a preference for robustness. American Economic Journal: Macroeconomics 7, 40–57. Epstein, L.G., Farhi, E., Strzalecki, T., 2014. How much would you pay to resolve long-run risk? American Economic Review 104, 2680–97. Gabaix, X., 2012. Variable rare disasters: An exactly solved framework for ten puzzles in macro-finance. Quarterly Journal of Economics 127, 645–700. Gal´ı, J., 2008. Monetary Policy, Inflation, and the Business Cycle. An Introduction to the New Keynesian Framework. Princeton University Press, Princeton.

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Gal´ı, J., Gertler, M., L´opez-Salido, J.D., 2007. Markups, gaps, and the welfare cost of business fluctuations. Review of Economics and Statistics 89, 44–59. Golez, B., 2014. Expected returns and dividend growth rates implied in derivative markets. Review of Financial Studies 27, 790–822. Greenwood, J., Hercowitz, Z., 1991. The allocation of capital and time over the business cycle. Journal of Political Economy 99, 1188–1214. G¨urkaynak, R.S., Sack, B., Wright, J.H., 2010. The TIPS yield curve and inflation compensation. American Economic Journal: Macroeconomics 2, 70–92. Hansen, L.P., Heaton, J.C., Li, N., 2008. Consumption strikes back? Measuring long-run risk. Journal of Political Economy 116, 260–302. Hansen, L.P., Scheinkman, J.A., 2009. Long term risk: An operator approach. Econometrica 77, 177–234. Judd, K.L., Maliar, L., Maliar, S., Valero, R., 2014. Smolyak method for solving dynamic economic models: Lagrange interpolation, anisotropic grid and adaptive domain. Journal of Economic Dynamics and Control 44, 92–123. Krusell, P., Smith, A.A., 1999. On the welfare effects of eliminating business cycles. Review of Economic Dynamics 2, 245–72. Lettau, M., Uhlig, H., 2000. Can habit formation be reconciled with business cycle facts? Review of Economic Dynamics 3, 79–99. Lettau, M., Wachter, J.A., 2007. Why is long-horizon equity less risky? A duration-based explanation of the value premium. Journal of Finance 62, 55–92. Lettau, M., Wachter, J.A., 2011. The term structures of equity and interest rates. Journal of Financial Economics 101, 90–113. Ljungqvist, L., Uhlig, H., 2000. Tax policy and aggregate demand management under catching up with the Joneses. American Economic Review 90, 356–66. Lopez, P., Lopez-Salido, D., Vazquez-Grande, F., 2015. Nominal rigidities and the term structures of equity and bond returns. FEDS Board of Governors of the Federal Reserve System, 2015-064. Lopez, P., Lopez-Salido, D., Vazquez-Grande, F., 2016. Risk-adjusted linearizations of dynamic equilibrium models. Manuscript. Lucas, R.E.J., 1987. Models of Business Cycles. Oxford University Press, Oxford. Lucas, R.E.J., 2003. Macroeconomic priorities. American Economic Review 93, 1–14. Otrok, C., 2001. On measuring the welfare cost of business cycles. Journal of Monetary Economics 47, 61–92. Song, D., 2017. Bond market exposures to macroeconomic and monetary policy risks. Review of Financial Studies 30, 2761–2817. Tallarini, T.D.J., 2000. Risk-sensitive real business cycles. Journal of Monetary Economics 45, 507–32. Weber, M., 2018. Cash flow duration and the term structure of equity returns. Journal of Financial Economics 128, 486–503.

34

ONLINE APPENDIX I. Relationship with the definitions of Lucas (1987) and Alvarez and Jermann (2004) Definition (1) is slightly different from to the one studied by Alvarez and Jermann (2004). First, they measure the cost of fluctuations by the uniform compensation Ωt in  ∞  Et U { 1 + Ωt (θ) Ct+n }∞ n=1 , {Xt+n }n=1 = ∞  = Et U {θEtCt+n + (1 − θ)Ct+n }∞ n=1 , {Xt+n }n=1 and focus on Lucas’s (1987) total cost, Ωt (1), and on the marginal cost, ∂θ∂ Ωt (0). The definition I consider measures instead the cost of fluctuations by a compounded compensation, so it can be interpreted as the tradeoff between growth and macroeconomic stability (the working paper version of Alvarez and Jermann, 2004, makes this point). Second, I allow for considering the stabilization of only some coordinates of consumption—the set N in definition (1)—rather than of the whole stochastic process. This flexibility grants a direct focus on the relevant periodicity of economic uncertainty. Finally, I address the proxy problem by stabilizing the determinants of consumption rather than consumption itself. II. Extension to heterogeneous consumers The term structure of marginal costs of uncertainty remains well-defined even in a heterogeneousagent, incomplete-market setting. Consider agents with heterogeneous preferences, U( j), and an idiosyncratic consumption component, εt ( j), driving their consumption stream, Ct ( j) = Dt + Wt Nt + et + εt ( j). Agents can only trade the entire term structures of dividend strips and zero-coupon bonds and may therefore face uninsurable idiosyncratic risk. In a heterogenenous-agent, incomplete-market context the cost of fluctuations can be defined as   n    Et U j 1 + LNj,t (θ) Ct+n + εt+n ( j) = Et U j C t+n (θ) + εt+n ( j) , N t+n (θ) so Lt measures the cost of uncertainty around the systematic part of the jth agent’s consumption (Alvarez and Jermann, 2004). By the absence of arbitrage opportunities, the projection of the marginal rates of substitution on the payoff space is the same across people, so their valuations of available assets are equal. Since all agents have access to the entire term structures of strips and zero-coupon bonds, they end up equalizing their valuation of welfare costs. Therefore, the marginal cost of uncertainty around all coordinates n ∈ N is constant across agents j ∈ (0, 1), LNj,t

=

X n∈N

j n Et (Mt,t+n Dt+n )

! j 1 Et (Mt,t+n )Et (Dt+n ) − 1 = LtN × P j j n Et (Mt,t+n Dt+n ) n∈N n E t (Mt,t+nC t+n )

35

III. Term structures in some consumption-based asset pricing models Table III.6 and figure VI.7 show the implications of some of the leading consumption-based asset pricing models for the three term structures and for the welfare cost of uncertainty and the equity premium. I consider the habit formation model of Campbell and Cochrane (1999), the long-run risk model of Bansal and Yaron (2004), the long-run risk model under limited information of Croce et al. (2015), the recursive preferences of Tallarini (2000) and Barillas et al. (2009), the rare disasters model of Gabaix (2012), and the affine no-arbitrage model of Lettau and Wachter (2011). In studying the term structures in the different asset pricing models, I consider the original calibrations, which the authors choose to match some asset pricing facts. The appendix works out the details of each model. I refer to the original writings for a list of the stylized asset pricing facts each model replicates. III.1. Structural approach The habit formation model of Campbell and Cochrane (1999) predicts a flat term structure of interest rates and an upward-sloping term structure of equity. The term structure of interest rates is driven by a particular calibration of the time-varying risk aversion that produces a constant risk-free rate. The term structure of equity is instead driven by the positive correlation between the pricing factor—shocks to consumption growth—and dividend growth, and by the perfectly negative correlation between the pricing factor and the shocks to the price of risk, which decreases as consumption grows away from the external habit. Since dividend strips load negatively on shocks to the price of risk, and the more so the longer the maturity, people command a greater risk premium to bear long-run dividend strip risk. Under the baseline calibration, the model of Campbell and Cochrane predicts a marginal cost of all fluctuations of 0.20% and an equity premium of 6.8%. The long-run risk model of Bansal and Yaron (2004) generates an upward-sloping term structure of equity and a downward-sloping term structure of interest rates. Bansal and Yaron introduce rich dynamics in consumption growth, which is driven both by shocks to expected consumption growth and to consumption volatility. Epstein-Zin-Weil utility then makes all shocks to the consumption opportunity set show up as pricing factors. In the calibration of Bansal and Yaron, long-run dividend strips load more heavily on the shocks to the consumption opportunity set and therefore are more risky, as long as the elasticity of intertemporal substitution is larger than one. In the model the risk-free rate is driven by shocks to the predictable component of consumption, which is positively priced; since long-run zero-coupon bonds load less on this state than the risk-free rate, they provide long-run insurance. This property explains the downward-sloping term structure of interest rates. The quantitative implications of the long-run risk model is a marginal cost of all fluctuations of 0.34% and an equity premium of 4.9%. Croce et al. (2015) consider the long-run risk model of Bansal and Yaron (2004) and change the information structure. Under limited information, not all shocks to the cashflow opportunity set are observable; the shocks that are priced are a linear combination of both short-run and long-run cashflow shocks. Then, since long-run shocks have a relatively small volatility, long-run dividend strips load less on the shocks that are priced under limited information than short-run dividend strips. This strategy allows for generating a downward-sloping term structure of equity; however, the curvature is not enough quantitatively, at least under the baseline calibration, it still predicts a 36

downward-sloping term structure of interest rates, and it works in a world in which risk premia are not time-varying. The model predicts a marginal cost of all fluctuations of 0.39%, against a predicted equity premium of 6.6%. The ambiguity averse multiplier preferences in Barillas et al. (2009) and the recursive preferences of Tallarini (2000) yield two flat term structures which imply the equality between the equity premium and the cost of uncertainty around any coordinate set N ∈ N, up to a scale factor. The unitary elasticity of intertemporal substitution that characterizes the recursive preferences of Tallarini (2000) and the robust control literature implies constant dividend yields, as discount-rate effects exactly offset cashflow effects in pricing equity claims; the random walk in consumption in turn implies constant interest rates and hence a flat bond term structure. The multiplier preferences of Barillas et al. (2009) and the observationally equivalent model of Tallarini (2000) predict a marginal cost of all fluctuations of 0.09% and an equity premium of about 2.0%. Finally, the rare disasters model of Gabaix (2012) produces two flat term structures of holdingperiod returns but a slightly downward-sloping term structure of hold-to-maturity equity returns. The intuition behind the flat term structure of holding period returns in the rare disasters model is that different dividend strips have the same exposure to the disaster event, whose probability is independent of the cashflow shocks that are priced. However, the mean-reversion in the state that drives equity prices makes long-duration equities load slightly less on it if held to maturity than short-duration equities. The model implies a marginal cost of all fluctuations of 0.31% and an equity premium of 7.0%. III.2. Descriptive approach I turn to the exponential-Gaussian no-arbitrage model of Lettau and Wachter (2011), which is designed to capture a downward-sloping term structure of equity and an upward-sloping term structure of interest rates. Without micro-founding it, Lettau and Wachter directly specify a stochastic discount factor, whose existence is guaranteed by the no-arbitrage theorems. They assume a single conditional pricing factor perfectly related to short-run cashflow shocks and a single state driving the price of risk. To match the downward-sloping term structure of equity, they assume that the predictable component of cashflows is negatively related to the priced shocks. Long-run dividend strips thus contain a component that provides long-run insurance. They then assume a zero correlation between cashflow and discount-rate shocks to avoid that the negative loading of long-run dividend strips on the state that drives the price of risk offsets the long-run insurance effect. Finally, since only short-run cashflow shocks are priced, Lettau and Wachter manage to capture an upward-sloping term structure of interest rates by assuming that shocks to the state driving the risk-free rate are negatively correlated with the priced shocks. Since long-run zero-coupon bonds are less exposed to this state than short-run bonds are, the assumption generates a positive bond risk premium as the maturity increases. The model predicts a marginal cost of total uncertainty of 0.12% and an equity premium of 7.2%. Table III.7 reports the cost of short- and long-run fluctuations over different coordinate sets. An increase in consumption uncertainty by a fraction θ over a ten-year period has a marginal cost of more than 0.5θ percentage points of growth per year during the decade. These numbers are in 37

Campbell and Cochrane (1999) Bansal and Yaron (2004) Croce et al. (2015) Barillas et al. (2009) Gabaix (2012) Lettau and Wachter (2011)

LN

ln E(Re,m )

0.20 0.34 0.39 0.09 0.31 0.12

6.81 4.90 6.56 1.92 7.89 7.18

Table III.6: Mean marginal cost of lifetime uncertainty and equity premium (percent per year). LN

LN\N

0.746 0.671 0.550 0.388 0.125

0.124 0.117 0.105 0.086 0

N up to 1 year up to 5 years up to 10 years up to 20 years N

Table III.7: Marginal cost of fluctuations at all periodicities n ∈ N. Lettau and Wachter (2011) model-based estimates

line with the options implied estimates in table 1 and compare to smaller yet nontrivial marginal benefits of long-run stability, which tend to zero as the stabilization becomes asymptotic. The volatility of the term structure as captured by the model of Lettau and Wachter supports the evidence in table 2 and figure 2d. The standard deviation of the cost of fluctuations at short periodicities is large and decays over long horizons. In their model, the term structure of welfare costs follows a one-factor structure driven entirely by movements in the time-varying market price of risk, as shown in the online appendix. IV. Global solution of the model I use a Smolyak sparse-grid collocation method with an adaptive grid to project the global solution of the model onto the subspace spanned by a basis of Chebyshev polynomials of up to degree eight. I proceed in two steps. First, I solve for equilibrium quantities. Second, I project equilibrium term structures using no-arbitrage relations by iterating the Fredholm pricing equation to convergence. I adopt the fast Smolyak method proposed by Judd, Maliar, Maliar and Valero (2014) by constructing a Smolyak grid using disjoint rather than nested sets of grid points, and by relying on Lagrange interpolation rather than on the closed-form map between the function evaluated at the grid points and its interpolated values. While the analytical expressions are free from numerical errors, they still imply computationally expensive evaluations over nested sets of grid points and basis functions. I delimit the parallelotope within which I search the state space for the projected solution by simulating 100,000 periods and considering minimum and maximum values of the elements of the √ state vector, St = [aTt ; ζt ; ∆t−1 ; zt ], where ζt ≡ 1 − 2(st − s). (See below for a discussion of why ζ is a better choice of state than sˆ.) I initialize the guessed solution with the first-order risky steady 38

state perturbation for consumption and inflation. This approximate solution is then used to explore the ergodic set and to map the Smolyak hypercube into the state space; the grid points are placed in a way that covers exactly all quantiles of simulated data. Finally, Judd et al. (2014) modify the conventional Smolyak method to allow for asymmetric accuracy levels {µi }3i=1 across the three dimensions of the state space. This anisotropic version of the Smolyak method is particularly useful in my setting, as most nonlinearities in the decision variables owe to their dependence on surplus consumption; I can therefore increase the precision in that dimension alone while maintaining the overall dimensionality of the problem limited. IV.1. Competitive equilibrium The necessary conditions for a competitive equilibrium are:23 it = − ln Et β exp {−γ∆ct+1 − γ∆st+1 − πt+1 } γ(1 − ρ s − ξ1 /γ) it = − ln(β) + π + γµ − + φπ (πt − π) + φy [ct − atP − (1 − α)(n − ∆)] 2 wt = ln(χ) − ln(1 − τh ) + γct − γht + at + γsc,t − γsh,t = ln(χ0 ) + γct − γht + at − γˆzt  ht = at + ln 1 − exp {nt } yt = at + (1 − α)nt − (1 − α)∆t ct = yt !− ε Z 1 Pt (i) 1−α exp{∆t } ≡ di Pt 0  ε    ε ∗ = η exp πt + ∆t−1 + (1 − η) exp − (p − pt ) 1−α 1−α t  1 = η exp {(ε − 1)πt } + (1 − η) exp (1 − ε)(p∗t − pt ) ( ) Γ1,t 1 − α + αε ∗ (pt − pt ) = exp Γ2,t 1−α   ε Γ1,t = Et βη exp πt+1 Γ1,t+1 + exp {(1 − γ)ct − γ sˆt + mct + ln[ε/(ε − 1)] − ∆t } 1−α Γ2,t = Et βη exp {(ε − 1)πt+1 } Γ2,t+1 + exp {(1 − γ)ct − γ sˆt } ! ∂Yt f wt = mct − ln(1 − τ ) + ln ∂Nt sˆt+1 = ρ s sˆt + Λ( sˆt )(ct+1 − Et ct+1 )  √   S −1 1 − 2 sˆt − 1, sˆt ≤ 12 (1 − S 2 ) Λ( sˆt ) =   0 elsewhere ∆at+1 = µ − (1 − ρu )aTt + σεt+1 aTt+1 = ρu aTt + θσεt+1 23

Nt =

R1 0

Nt (i)di =

R1 0

[Yt (i)/At ]1/(1−α) di = (Yt /At )1/(1−α)

R1 0

39

[Pt (i)/Pt ]−ε/(1−α) di, with the clearing condition Yt = Ct .

with εt ∼ Niid(0, 1), χ ≡ χ0 (S c /S h )1−γ , 1 − τh = S c /S h , 1 − τ f = MC, where we defined mc ˆ t ≡ wt − pt − ln [∂Yt /∂Nt ], and hence mc ˆ t (i) = wt − pt − ln [∂Yt (i)/∂Nt (i)] = mc ˆ t + [ct − nt ] − α α αε [ct (i) − nt (i)] = mc ˆ t − 1−α ct − ∆t + 1−α ct (i) = mc ˆ t − 1−α [pt (i) − pt ] − ∆t . Rearrange and simplify until we are left with 1 exogenous aTt and 3 endogenous states [ sˆt , ∆t−1 , zˆt ]: aTt+1 = ρu aTt + θσεt+1 sˆt+1 = ρ s sˆt + Λ( sˆt )(e ct+1 − Ete ct+1 + σεt+1 ) " #− ε ε 1 − ηe(ε−1)πt (1−α)(1−ε) ∆t πt +∆t−1 1−α e = ηe + (1 − η) 1−η   1 1 e c +∆ zˆt+1 = ρ s zˆt + Λh,t ln(1 − e 1−α t+1 t+1 ) − Et ln(1 − e 1−α ect+1 +∆t+1 ) + σεt+1 − Λc,t (e ct+1 − Ete ct+1 + σεt+1 )

and with 3 endogenous jump variables [e ct , πt , `t ]: i∗ + π + φπ (πt − π) + φy (e ct − e c) = − ln Et βe−γ[µ+∆ect+1 −(1−ρu )at +σεt+1 ]−γ∆ sˆt+1 −πt+1 h i−γ 1 ε 1 χ0 T e`t = Et δηe 1−α πt+1 +(1−γ)[−(1−ρu )at +σεt+1 ]+`t+1 + e 1−α ect −γ sˆt −γˆzt 1 − e 1−α ect +∆t 1−α # 1−α+αε " " # 1−α+αε (ε−1)πt (1−α)(1−ε) 1 − ηe(ε−1)πt (1−α)(1−ε) T (1−γ)e ct −γ sˆt 1 − ηe e`t = Et δηe(ε−1)πt+1 +(1−γ)[−(1−ρu )at +σεt+1 ]+`t+1 + e 1−η 1 − ηe(ε−1)πt+1 T

(IV.1) (IV.2) (IV.3)

where `t ≡ ln Γ1,t − (1 − γ)at , and where ct ≡ ct − at . √ we eliminated the state at by defining e The approximation of the term 1 − 2 sˆt , which determines the price of risk, with powers of √ sˆt can only be done imperfectly, so we use ζt ≡ 1 − 2 sˆt as a state instead of sˆt ; powers of ξt can easily span powers of sˆt .24 Finally, using the definition of the price index, equilibrium dividends reduce to: # Z 1 Z 1" Pt (i) f f Wt Dt ≡ Dt (i)di = Yt (i) − (1 − τ ) Nt (i) − T t di Pt Pt 0 0 d t ]Ct = [1 − (1 − α) MC   −γ  γ(1−α)+α 1 = Ct 1 − χ0 e 1−α ect +∆t −γˆzt 1 − e 1−α ect +∆t Under inflation targeting equation πt = 0 replaces equation (IV.1) (under inflation targeting the nonlinear solution has a closed form). Under risk premia targeting I replace equation (IV.1) with equation e ct − e c = −aTt (IV.4) IV.2. Algorithm to solve for quantities 1. Setup: (a) Simulate the first-order solution of the model and select the smallest parallelotope in the state space that contains all realizations of the state variable St = [aTt ; ζt ; ∆t−1 ; zˆt ]. 24

I work with log variables as the solution turns out to be more stable. Furthermore, this specification has the additional advantage of nesting the affine approximation.

40

Choose the level of precision in each dimension µ = [µ1 ; µ2 ; µ3 ; µ4 ], and adapt a Smolyak hypercube to the parallelepiped. (b) Select the Smolyak grid: to construct the 4-dimensional, n-point Smolyak grid {S˚ i }ni=1 we use extrema of Chebyshev polynomials as unidimensional grid points. Select the Smolyak basis functions: we use the Chebyshev polynomial family as unidimensional basis functions. ˚ = [Ψi (S˚ j )]n , and (c) Evaluate the Smolyak basis functions at the collocation points, Ψ(S) i, j=1 ˚ −1 . precompute the inverse operator Ψ(S) ˚ 2. Guess the values of e c, π and ` at each collocation point S˚ i , i = 1, ..., n, denoted by fˆx (S) c, π, `. (We initialize our algorithm with the risk-adjusted for each decision variable x = e log-linearized solution.) By Lagrange interpolation, this gives an interpolant fˆx (S) = Ψ(S)bˆ x , ˚ −1 fˆx (S). ˚ 25 with bˆ x = Ψ(S) (Note that under risk premia targeting, fˆec is known.) 3. Compute the value of the optimal time t functions at each collocation point S˚ i , i = 1, ..., n, as follows: (a) Fix an i. This gives the state today, [aTt , ζt , ∆t−1 , zˆt ], and the values of e ct , πt and `t . (b) Derive ∆t from the law of motion for price dispersion using πt . Using St , weights and nodes from the degree-q Gauss-Hermite cubature rule,26 the innovations ε j,t+1 , j = 1, ..., q2 , and function fˆec guessed in stage 2, construct aTj,t+1 = ρu aTt + θσε j,t+1 and solve the fixed-point problem: e c j,t+1 zˆ j,t+1

  r ρ  ζ     T  s t 2 ˆ ˆ  = fec a j,t+1 , 1 − 2 min (1 − ζt ) + max 0, − 1 e c j,t+1 − Ete ct+1 + σε j,t+1 , .5 , ∆t , zˆ j,t+1  2 S     1 1 ζt S 1 − Sh max 0, − 1 ln(1 − e 1−α ec j,t+1 +∆ j,t+1 ) − Eˆ t ln(1 − e 1−α ect+1 +∆t+1 ) + σεt+1 = ρ s zˆt + 1 − S Sh S   ζ t c j,t+1 − Eˆ te ct+1 + σεt+1 ) − max 0, − 1 (e S

ct+1 ) = w0 f (e ct+1 ) according with the Gauss-Hermite formula, for given fˆec , where Eˆ t f (e q2 q2 with w ∈ R and e ct+1 = [e c j,t+1 ] j=1 . We proceed by fixed-point iterations (with dampening where necessary to improve the convergence properties) and stop at iteration N whenever q2 −6 ke c(N) c(N−1) t+1 − e t+1 k∞ < 10 . We subsequently derive the associated value of ζt+1 = [ζ j,t+1 ] j=1 . (c) Using S j,t+1 = [aTj,t+1 , ζ j,t+1 , ∆ j,t , zˆ j,t+1 ], weights and nodes from the degree-q GaussHermite cubature rule, the innovations ε j,t+1 , j = 1, ..., q2 , and the functions [ fˆec ; fˆπ ; fˆ` ] ˚ −1 remains fixed across iterations on the unknown function fˆ. Given the n Note how the inverse operator Ψ(S) ˆ values of an updated function f at each collocation point, we compute the new coefficient b and interpolate by simply evaluating the Chebyshev polynomial at the arbitrary point S. 26 Results are virtually equivalent for q ≥ 6; the two-dimensional Gauss-Hermite quadrature remains therefore computationally inexpensive and reliable, so we do not resort to monomial rules. 25

41

guessed in stage 2, compute expectations: Et βe−γ(µ+ect+1 −(1−ρu )at +σεt+1 )−γ sˆt+1 −πt+1 T

Et δηe

ε T 1−α πt+1 +(1−γ)(−(1−ρu )at +σεt+1 )+`t+1

Et δηe(ε−1)πt+1 +(1−γ)(−(1−ρu )at +σεt+1 )+`t+1 T

"

#− 1−α+αε 1 − ηe(ε−1)πt+1 (1−α)(1−ε) 1−η

(IV.5) (IV.6) (IV.7)

(Under risk premia targeting, I drop equation (IV.5).) cnew by substituting the Taylor (d) Using expectation (IV.5) and the guess for πt , calculate e t rule for it in equation (IV.1). (Under risk premia targeting, I drop this substep.) Using expectation (IV.6), the guess for πt and e ct , derive `tnew from equation (IV.2). ct and the value of `t , derive πnew Using expectation (IV.7), the value of e from equat tion (IV.3). 4. If the differences between the guessed values for e ct , πt and `t and their derived values in 3d are all close to zero at each collocation point, then stop. The stopping criterion I adopt is kxt − xtnew k2 < 10−6 , with xt = [e ct ; πt ; `t ]. Otherwise, update the guesses and go to stage 2. I tried several iteration schemes to perform this loop but the problem is sufficiently complex not to display the properties of a contraction mapping. We therefore rely on a derivative-free polytope method that searches directly the policy space for an optimal functional f x = [ fec ; fπ ; f` ]. IV.3. Algorithm to solve for asset prices The solution for quantities allows for projecting the pricing operator Pd,t : Fd,t+1 7→ Pd,t (Fd,t+1 ) = Et [Mt+1 (Dt+1 /Dt )Fd,t+1 ], while iterations on the recursion ! Dt+1 (n−1) (n) (0) Fd,t = Et Mt+1 F , Fd,t =1 Dt d,t+1 converge to the Perron-Frobenius eigenfunction, Fd (St ), of the pricing operator that solves the corresponding Fredholm equation, Fd,t = Pd,t (Fd,t+1 ). As emphasized by Hansen and Scheinkman (2009), this equation has a rich structure, as the pricing operator is element-wise positive and therefore allows for the application of the infinite-dimensional extension of Perron-Frobenius theory, which describes conditions for the existence and uniqueness of a positive eigenfunction. V. Risk-adjusted linearization of the model Throughout, I define ut ≡ −aTt /(1 − ρu ) and φ ≡ (1 − ρu )θ.

42

We have the expectational equations: 0 = ln Et exp[ f (yt , zt , yt+1 , zt+1 )]   φ  ln(β) − γµ + i∗ − π + φπ πˆ t + φye cˆ t − 1−ρy u ut − γ(∆e cˆ t+1 + ut + σεt+1 ) − γ∆ sˆt+1 − πˆ t+1   ε  ε  πˆ t+1 + (1 − γ)(ut + σεt+1 ) + ∆`ˆt+1 − w1,t ln(βe(1−γ)µ ηΠ 1−α ) + 1−α    ln(βe(1−γ)µ ηΠε−1 ) + (ε − 1)ˆπ + (1 − γ)(u + σε ) + ∆`ˆ + ∆wˆ t+1 t t+1 t+1 3,t+1 − w2,t     h i h i −γ   f (yt , zt , yt+1 , zt+1 ) =  ε(1−τ f ) χ0 Ne−∆ 1 ˆ 1 ˆ ˆ ˆ ct − γ sˆt − γˆzt − `t 1 − N exp( 1−αe ct + ∆t ) w1,t − ln 1 − ε−1 1−α L exp 1−αe     h i (Ne−∆ )(1−γ)(1−α)   ˆ ˆ e w − ln 1 − exp (1 − γ) c − γ s ˆ − ` − w ˆ 2,t t t t 3,t   W3 L 1−α+αε ε−1 w3,t − (1−α)(ε−1) {ln(1 − ηΠ exp[(ε − 1)ˆπt ]) − ln(1 − η)}

with state dynamics:   ρu ut     u  t+1    ρ s sˆt  sˆ   " # ε   (1−α)(ε−1)  t+1  =  ε−1 (ε−1)ˆ π  t ε ε  ∆t  ln ηΠ 1−α e 1−α πˆ t +∆t−1 + (1 − η) 1−ηΠ1−ηe     zˆt+1 ρ s zˆt     −φ 0     0 " #     Λ( sˆt ) e ct+1 Λ( sˆt ) 0   σεt+1  (Et+1 − Et ) e +  +   0 0   0 ht+1   (ϕh − 1) Λ( sˆt ) −Λ( sˆt ) ϕh Λ( sˆt ) S 1−S h . where ϕh ≡ 1−S Sh We use the algorithm in Lopez et al. (2016) to obtain a first-order approximation around the risky steady state.

Step 1. Write expectational equations in terms of a certainty-equivalent and an entropy terms:   φy e c π ut − γ(Et ∆e cˆ t+1 + ut ) + γ(1 − ρ s ) sˆt − Et πˆ t+1 + Vt e−xt (εt+1 +σεt+1 )−εt+1 1 − ρu   ε π ε ε ` 0 = ln(βe(1−γ)µ ηΠ 1−α ) + Et πˆ t+1 + (1 − γ)ut + Et ∆`ˆt+1 − w1,t + Vt e 1−α εt+1 +(1−γ)σεt+1 +εt+1 1−α  w3  π ` (1−γ)µ ε−1 0 = ln(βe ηΠ ) + (ε − 1)Et πˆ t+1 + (1 − γ)ut + Et ∆`ˆt+1 + Et ∆wˆ 3,t+1 − w2,t + Vt e(ε−1)εt+1 +(1−γ)σεt+1 +εt+1 +εt+1 0 = ln(β) − γµ + i∗ − π + φπ πˆ t + φye cˆ t −

x with εt+1 ≡ xt+1 − Et xt+1 .

Step 2. Conjecture the approximate solution e ct = e c + ψcu ut + ψc∆ ∆ˆ t−1 , πt = π + ψπu ut + ψπ∆ ∆ˆ t−1 ˆ and `t = ` + ψ`u ut + ψ`s sˆt + ψ`∆ ∆t−1 , and characterize the entropy terms, using ct+1 − Et ct+1 = (1 − ψcu φ)σεt+1 . Note that, in equilibrium, zˆt = ξ2 sˆt by the calibration of S h , which is such that # φψcu  S 1 − Sh  1− = − (1 + φψcu ) σet+1 = ξ2 (1 + φψcu )σet+1 = ξ2 εct+1 1 − S Sh 1−α "

ϕh εht+1



εct+1

(V.1)

where ξ2 is a free parameter. Therefore, state variables are effectively only 3. Also, in the end I pick a calibration for ξ1 and ξ2 such that the dependence of consumption and inflation on surplus consumption is zero, which justifies equation (V.1). 43

Step 3. Linearize: ∆t ≈ η∆ˆ t−1 = 0 w1,t

w2,t

! " # ε(1−τ f ) χ0 N −γ ε(1 − τ f ) χ0 N 1+γˆ ε−1 1−α L (1 − N) −γ ˆ e ≈ ln 1 − (1 − N) − c − γ(1 + ξ ) s ˆ − ` t 2 t t f) χ 0 N −γ 1 − α ε−1 1−αL 1 − ε(1−τ ε−1 1−α L (1 − N) ! N (1−γ)(1−α) h i N (1−γ)(1−α) W3 L ˆ t − γ sˆt − `ˆt − wˆ 3,t e − ≈ ln 1 − (1 − γ) c (1−γ)(1−α) W3 L 1 − N W3 L

w3,t ≈ −

1 − α + αε η πˆ t 1−α 1−η

Also, approximate equilibrium dividends reduce to: " # 1 − α γ(1 − α) + α + γ 1 − α e mc ˆ t =e ct − cˆ t − γξ2 sˆt det ≡ dt − at ≈ e ct − α α 1−α Step 4. Match coefficients. We pose the parametrization:27 i∗ = − ln β + γµ −

2 1 γ (1 − ψcu φ) − ψπu φ σ2 2 S

Then, using: 2   1 γ p e c π Vt e−xt (εt+1 +σεt+1 )−εt+1 = 1 − 2 sˆt (1 − ψcu φ) − ψπu φ σ2 2 S 2  ε π  1 ε ` Vt e 1−α εt+1 +(1−γ)σεt+1 +εt+1 = − ψπu φ + 1 − γ − ψ`u φ + ψ`s Λ( sˆt )(1 − ψcu φ) σ2 2 1−α " #2   w3 ε 1 − α + αε 1 (ε−1)επt+1 +(1−γ)σεt+1 +ε`t+1 +εt+1 − ψπu φ + 1 − γ − ψ`u φ + ψ`s Λ( sˆt )(1 − ψcu φ) + ψπu φ σ2 = Vt e 2 1−α (1 − α)(1 − η) The form of i∗ is assumed to be structural, so we can work with a constant inflation rate π = π∗ at the risky steady state even after policy changes. 27

44

we have: π=0   e c π ∂Vt e−xt (εt+1 +σεt+1 )−εt+1 φ y sˆt 0 = φπ πˆ t + φye cˆ t − ut − γ(Et ∆e cˆ t+1 + ut ) − Et πˆ t+1 + γ(1 − ρ s ) sˆt + sˆt =0 1 − ρu ∂ sˆt | {z } V s,1

2 1 ε 0 = ln(βe(1−γ)µ ηeV0,2 ) − w1 , V0,2 ≡ − ψπu φ + 1 − γ − ψ`u φ + ψ`s Λ(1 − ψcu φ) σ2 2 1−α  ε π  ` ∂Vt e 1−α εt+1 +(1−γ)σεt+1 +εt+1 ε sˆt Et πˆ t+1 + (1 − γ)ut + Et ∆`ˆt+1 − wˆ 1,t + 0= sˆt =0 1−α ∂ sˆt | {z } V s,2

! #2 ε 1 1 − α + αε − ψπu φ + 1 − γ − ψ`u φ + ψ`s Λ(1 − ψcu φ) σ2 0 = ln(βe ηe ) − w2 , V0,3 ≡ − 2 1 − α (1 − α)(1 − η)  w3  π ` ∂Vt e(ε−1)εt+1 +(1−γ)σεt+1 +εt+1 +εt+1 0 = (ε − 1)Et πˆ t+1 + (1 − γ)ut + Et ∆`ˆt+1 + Et ∆wˆ 3,t+1 − wˆ 2,t + sˆt sˆt =0 ∂ sˆt | {z } "

(1−γ)µ

V0,3

V s,3

 1 − βe  ηe w1,t = ln βe(1−γ)µ ηeV0,2 − (1−γ)µ βe ηeV0,2 (1−γ)µ   1 − βe ηeV0,3 w2,t = ln βe(1−γ)µ ηeV0,3 − βe(1−γ)µ ηeV0,3 1 − α + αε η w3,t = − πˆ t 1−α 1−η (1−γ)µ

V0,2

1+γˆ e ct − γ(1 + ξ2 ) sˆt − `ˆt 1−α h i (1 − γ)e cˆ t − γ sˆt − `ˆt − wˆ 3,t "

#

and hence ! χ0 ε(1 − τ f ) ` = ln + n − γ ln (1 − en ) ε − 1 (1 − α)(1 − βe(1−γ)µ ηeV0,2 ) h i 1 ` + ln(1 − βe(1−γ)µ ηeV0,3 ) n= (1 − γ)(1 − α) V.1. Steady-state monetary and fiscal policies to correct the risky steady state Along with the parametrization for the monetary policy rule 2 1 γ (1 − ψcu φ) − ψπu φ σ2 i = − ln β + γµ − 2 S ∗

we have that price and wage setting imply the risky steady state expression for marginal costs χ0 N γ(1−α)+α e(1−γ)(1−α)∆ (1 − N)−γ 1−α ε − 1 1 − δηeV0,2 = ε 1 − δηeV0,3

MC = (1 − τ f )

45

Under τ f = 1 − MC (constraint on steady-state fiscal policy): 1 − τ f = MC =

ε − 1 1 − δηeV0,3 ε 1 − δηeV0,2

Fiscal policy (through τh and τ f ) and monetary policy (through i∗ ) correct jointly the risky χ0 N γ(1−α)+α (1 − N)−γ = 1, and hence N = N¯ = .5, as guessed initially—risky and steady state, as 1−α deterministic steady states coincide. V.2. Macro-finance separation Matching coefficients, I find the equations: γ(1 − ρu ) + φy = [(γ(1 − ρu ) + φy )ψcu + (φπ − ρu )ψπu ](1 − ρu ) (1 − δη)(1 − η) 1 − α e−V0,3 (γ − 1)(1 − α) + e−V0,2 (1 + γ) ψπu = ψcu − (e−V0,2 − e−V0,3 )ψ`u 1−α η(e−V0,3 − δρu ) 1 − α + αε ε δηρu e−V0,2 − δη 1 + γ δη + ψ + ψcu ψ`u = (1 − γ) −V πu e 0,2 − δηρu 1 − α e−V0,2 − δηρu e−V0,2 − δηρu 1 − α 0 = γ(1 − ρ s ) + V s,1

!

e−V0,3 − δη [γ + ψ`s ] e−V0,2 − δη e−V0,2 − δη ψ`s = −γ(1 + ξ2 ) −V e 0,2 − δηρ s

[γ(1 + ξ2 ) + ψ`s ] =

Therefore, when γ ξ1 = − (1 − ψcu φ)ψπu φσ2 S (e−V0,2 − δηρ s )(e−V0,3 − δη) ξ2 = −V0,3 −1 (e − δηρ s )(e−V0,2 − δη) with 2 1 ε − ψπu φ + 1 − γ − ψ`u φ + ψ`s Λ(1 − ψcu φ) σ2 2 1−α " ! #2 1 ε 1 − α + αε − − ψπu φ + 1 − γ − ψ`u φ + ψ`s Λ(1 − ψcu φ) σ2 ≡ 2 1 − α (1 − α)(1 − η)

V0,2 ≡ V0,3

we verify the conjecture, as consumption and inflation do not depend on surplus consumption.

46

Let κ≡

ϕ≡

(1−δη)(1−η) 1−α η 1−α+αε

1+



e−V0,3 (γ−1)(1−α)+e−V0,2 (1+γ) 1−α



(e−V0,2 −e−V0,3 )(e−V0,2 −δη) 1+γ 1−α e−V0,2 −δηρu

δηερu (1−δη)(1−η) 1−α e−V0,2 −e−V0,3 η 1−α+αε (e−V0,3 −δρu )(e−V0,2 −δηρu ) 1−α

(1−δη)(1−η) 1−α (e−V0,2 −e−V0,3 )δη(1−γ) η 1−α+αε e−V0,2 −δηρu

1+

(1−δη)(1−η) 1−α δηερu e−V0,2 −e−V0,3 η 1−α+αε (e−V0,3 −δρu )(e−V0,2 −δηρu ) 1−α

 1 − δρu − δρu

e−V0,3

(1 − ρu )(1 − δρu ) e−V0,3 − δρu

Then, we have ψπu =

κ ϕ ψcu − 1 − δρu (1 − ρu )(1 − δρu )

and hence the approximate solution: [γ(1 − ρu ) + φy ](1 − δρu ) + ϕ(φπ − ρu ) 1 [γ(1 − ρu ) + φy ](1 − δρu ) + κ(φπ − ρu ) 1 − ρu (κ − ϕ)[γ(1 − ρu ) + φy ] 1 = [γ(1 − ρu ) + φy ](1 − δρu ) + κ(φπ − ρu ) 1 − ρu

ψcu = ψπu

Note that in absence of risk, σ = 0, all second-order moments are zero, and we end up with the 1−α γ(1−α)+α+γ deterministic steady-state coefficients—which have the same form but for κ = (1−δη)(1−η) η 1−α+αε 1−α and ϕ = 0. V.3. No habits A linear approximation around the risky steady state of the CRRA model yields identical expressions but for different risk corrections:  1 γ(1 + ψcu φ) + ψπu φ 2 σ2 2 2 1 ε ≡ − ψπu φ + 1 − γ − ψ`u φ σ2 2 1−α " ! #2 1 ε 1 − α + αε ≡ − − ψπu φ + 1 − γ − ψ`u φ σ2 2 1 − α (1 − α)(1 − η)

crra V0,1 ≡ crra V0,2 crra V0,3

In terms of quantities and inflation, the model with habits and the particular calibration for ξ1 and ξ2 and the model without habits are observationally indistinguishable to a first-order approximation around the risky steady state. Risk corrections will be slightly different but an estimation of the model without habits that matches the same second-order moments for consumption and inflation would result in identical equilibrium processes. (E.g., κ and ϕ can be kept the same by a slightly different choice of ε, η, and δ.)

47

VI. Internal habits The Pareto optimum (flexible prices, internal habits) can be characterized as the solution to a social planner problem. However, we appeal to the welfare theorems and decentralize the economy to build intuition and gain insight into the consumption and labor margins. Internal-habit consumers maximize the intertemporal objective max U0 = E0

∞ X t=0

(Ct − Xtc )1−γ − 1 (Ht − Xth )1−γ − 1 β +χ 1−γ 1−γ

!

t

subject to the budget constraint and the structural habit equations, Ht = At (1 − Nt ), and Ct − Xtc = Ct S t , Ht −

Xth

sˆt+1 = ρ s sˆt + Λ( sˆt )(Et+1 − Et ) ln(Ct+1 )

= Ht Zt ,

zˆt+1 = ρ s zˆt + (1 + ξ2 )Λ[ˆzt /(1 + ξ2 )](Et+1 − Et ) ln[Gt+1 (Ht+1 )]

We let yct ≡ Et ct+1 and yht ≡ Et ht+1 , and we set up the stochastic Lagrangian ( ∞ 1−γ X h i + χ(Ht S h,t )1−γ −1 t (C t S c,t ) W N + B − (1 + r ) B + D − C E0 β + Lint. t t t−1 t t t t t 1−γ t=0 h i h i + Mct ρ s sc,t−1 + Λ(st−1 )(ct − yct−1 ) − sc,t + Ntc yct − ct+1 h i) h i h h h h + Mt ρ s sh,t−1 + ϕh Λ(st−1 )(ht − yt−1 ) − sh,t + Nt yt − ht+1 where the Lagrange multiplier Lt is the marginal value of consumption. Optimality requires that the joint evolution of the processes satisfies ∂Utint. Λ(st−1 ) 1−γ = Ct−γ S c,t + (Et − Et−1 )Mct ∂Ct Ct ∂U int. ϕh Λ(st−1 ) 1−γ + (Et − Et−1 )Mht = − t = χAt Ht−γ S h,t ∂Nt 1 − Nt 1−γ Mct = Ct1−γ S c,t + βρ s Et Mct+1

Lint. = t Wt Lint. t

1−γ Mht = χHt1−γ S h,t + βρ s Et Mht+1

(VI.1) (VI.2) (VI.3) (VI.4)

where Mct and Mht are Lagrange multipliers associated with the market and home consumption habit equations that affect the marginal utility of wealth with a time-varying loading, and thereby implies the intertemporal and intratemporal rates of substitution int. Mt+1

Ct+1 =β Ct

!−1

1−γ 1−γ Ct+1 S c,t+1 + Λ(st )(Et+1 − Et )Mct+1 1−γ Ct1−γ S c,t + Λ(st−1 )(Et − Et−1 )Mct

1−γ 1−γ 1−γ h ∂Utint. /∂Nt Ct χAt (1 − Nt ) S h,t + ϕh Λ(st−1 )(Et − Et−1 )Mt − int. = 1−γ ∂Ut /∂Ct 1 − Nt Ct1−γ S c,t + Λ(st−1 )(Et − Et−1 )Mct

48

Flexible-price firms maximize period profits, Yt − Wt Nt , subject to the production technology, Yt = At Nt1−α , which results in the optimality condition Wt = (1 − α)Yt /Nt . After imposing market clearing, Yt = Ct , we can characterize the Pareto optimum by the equality between the intratemporal rate of substitution and the marginal product of labor, ∂Utint. /∂Nt Ct − int. = (1 − α) Nt ∂Ut /∂Ct It is straightforward to verify how a unitary elasticity of intertemporal substitution, γ = 1, produces constant shadow values of surplus market and home consumption. Under this parametrization we have !−1 Ct+1 int. Mt+1 = β Ct int. ∂U /∂Nt χCt − tint. = ∂Ut /∂Ct 1 − Nt so all intertemporal and intratemporal effects of the habit are absent. The condition γ = 1 is actually necessary to grant a macro-finance separation (even approximately) when habits are internal, for any value of the spillover parameter ξ2 .

49

(n) 1 n E(r d,t→ t+n ) (n) 1 E(r b,t→ t+n ) n (n)

0.2

E(l

0.2

)

0.15

0.15

0.1

0.1

0.05

0.05

0

0

−0.05

0

5

10

(n) 1 n E(r d,t→ t+n ) (n) 1 E(r b,t→ t+n ) n (n)

15

20

25

−0.05

30

E(l

0

5

10

n (in years)

0.15

0.1

0.1

0.05

0.05

0

0

0

5

10

15

30

(n) 1 n E(r d,t→ t+n ) (n) 1 E(r b,t→ t+n ) n (n)

0.2

)

0.15

−0.05

25

(b) Long-run risk (Bansal and Yaron, 2004).

(n) 1 n E(r d,t→ t+n ) (n) 1 E(r b,t→ t+n ) n (n)

E(l

20

n (in years)

(a) Habit formation (Campbell and Cochrane, 1999).

0.2

15

)

20

25

−0.05

30

E(l

0

5

10

n (in years)

15

20

)

25

30

n (in years)

(c) Ambiguity aversion (Barillas, Hansen and Sargent, (d) Long-run risk (limited information) (Croce, Lettau 2009), Epstein-Zin-Weil log utility (Tallarini, 2000). and Ludvigson, 2015). (n) 1 n E(r d,t→ t+n ) (n) 1 n E(r b,t→ t+n ) (n)

0.2

E(l

0.2

)

0.15

0.15

0.1

0.1

0.05

0.05

0

0

−0.05

0

5

10

(n) 1 n E(r d,t→ t+n ) (n) 1 n E(r b,t→ t+n ) (n)

15

20

25

−0.05

30

n (in years)

E(l

0

5

10

15

20

)

25

30

n (in years)

(e) Rare disasters (Gabaix, 2012).

(f) No-arbitrage (Lettau and Wachter, 2011).

Figure VI.7: The term structures of equity, interest rates and welfare costs of uncertainty in some consumption-based asset pricing models.

50