## Abstract

At the zero lower bound, the New Keynesian model predicts that output and inflation collapse to implausibly low levels and that government spending and forward guidance have implausibly large effects. To resolve these anomalies, we introduce wealth into the utility function; the justification is that wealth is a marker of social status, and people value status. Since people partly save to accrue social status, the Euler equation is modified. As a result, when the marginal utility of wealth is sufficiently large, the dynamical system representing the zero-lower-bound equilibrium transforms from a saddle to a source, which resolves all the anomalies.

## I. Introduction

A current issue in monetary economics is that the New Keynesian model makes several anomalous predictions when the zero lower bound (ZLB) on nominal interest rates is binding: an implausibly large collapse of output and inflation (Eggertsson & Woodford, 2004; Eggertsson, 2011; Werning, 2011), an implausibly large effect of forward guidance (Del Negro, Giannoni, & Patterson, 2015; Carlstrom, Fuerst, & Paustian, 2015; Cochrane, 2017), and an implausibly large effect of government spending (Christiano, Eichenbaum, & Rebello, 2011; Woodford, 2011; Cochrane, 2017).

Several papers have developed variants of the New Keynesian model that behave well at the ZLB (Gabaix, 2016; Diba & Loisel, 2019; Cochrane, 2018; Bilbiie, 2019; Acharya & Dogra, 2019), but these variants are more complex than the standard model. In some cases, the derivations are complicated by bounded rationality or heterogeneity. In other cases, the dynamical system representing the equilibrium—normally composed of a Euler equation and a Phillips curve—includes additional differential equations that describe bank-reserve dynamics, price-level dynamics, or the evolution of the wealth distribution. Moreover, a good chunk of the analysis is conducted by numerical simulations. Hence, it is sometimes difficult to grasp the nature of the anomalies and their resolutions.

It may therefore be valuable to strip the logic to the bone. We do so using a New Keynesian model in which relative wealth enters the utility function. The justification for the assumption is that relative wealth is a marker of social status, and people value high social status. We deviate from the standard model only minimally: the derivations are the same, the equilibrium is described by a dynamical system composed of a Euler equation and a Phillips curve, and the only difference is an extra term in the Euler equation. We also veer away from numerical simulations and establish our results with phase diagrams describing the dynamics of output and inflation given by the Euler-Phillips system. The model's simplicity and the phase diagrams allow us to gain new insights into the anomalies and their resolutions.^{1}

Using the phase diagrams, we begin by depicting the anomalies in the standard New Keynesian model. First, we find that output and inflation collapse to unboundedly low levels when the ZLB episode is arbitrarily long-lasting. Second, we find a duration of forward guidance above which any ZLB episode, irrespective of its duration, is transformed into a boom. The boom is unbounded when the ZLB episode is arbitrarily long-lasting. Third, we find an amount of government spending at which the government-spending multiplier becomes infinite when the ZLB episode is arbitrarily long-lasting. Furthermore, when government spending exceeds this amount, an arbitrarily long ZLB episode prompts an unbounded boom.

The phase diagrams also pinpoint the origin of the anomalies: they arise because the Euler-Phillips system is a saddle at the ZLB. In normal times, by contrast, the Euler-Phillips system is a source, so there are no anomalies. In economic terms, the anomalies arise because household consumption (given by the Euler equation) responds too strongly to the real interest rate. Indeed, since the only motive for saving is future consumption, households are very forward-looking, and their response to interest rates is strong.

Once wealth enters the utility function, however, the Euler equation is “discounted”—in the sense of McKay, Nakamura, and Steinsson (2017)—which alters the properties of the Euler-Phillips system. People now save partly because they enjoy holding wealth; this is a present consideration, which does not require them to look into the future. As people are less forward-looking, their consumption responds less to interest rates, which creates discounting.

With enough marginal utility of wealth, the discounting is strong enough to transform the Euler-Phillips system from a saddle to a source at the ZLB and thus eliminate all the anomalies. First, output and inflation never collapse at the ZLB: they are bounded below by the ZLB steady state. Second, when the ZLB episode is long enough, the economy necessarily experiences a slump, irrespective of the duration of forward guidance. Third, government-spending multipliers are always finite, irrespective of the duration of the ZLB episode.

Apart from its anomalies, the standard New Keynesian model has several other intriguing properties at the ZLB—some labeled “paradoxes” because they defy usual economic logic (Eggertsson, 2010; Werning, 2011; Eggertsson & Krugman, 2012). Our model shares these properties. First, the paradox of thrift holds: when households desire to save more than their neighbors, the economy contracts and they end up saving the same amount as the neighbors. The paradox of toil also holds: when households desire to work more, the economy contracts and they end up working less. The paradox of flexibility is present too: the economy contracts when prices become more flexible. Finally, the government-spending multiplier is above 1, so government spending stimulates private consumption.

## II. Justification for Wealth in the Utility Function

Before delving into the model, we justify our assumption of wealth in the utility function.

The standard model assumes that people save to smooth consumption over time, but it has long been recognized that people seem to enjoy accumulating wealth irrespective of future consumption. Describing the European upper class of the early twentieth century, Keynes (1919) noted that “The duty of saving became nine-tenths of virtue and the growth of the cake the object of true religion … Saving was for old age or for your children; but this was only in theory—the virtue of the cake was that it was never to be consumed, neither by you nor by your children after you.” Irving Fisher added, “A man may include in the benefits of his wealth … the social standing he thinks it gives him, or political power and influence, or the mere miserly sense of possession, or the satisfaction in the mere process of further accumulation” (Fisher, 1930, 17). Fisher's perspective is interesting since he developed the theory of saving based on consumption smoothing.

Neuroscientific evidence confirms that wealth itself provides utility, independent of the consumption it can buy. Camerer, Loewenstein, and Prelec (2005, 32) note that “brain-scans conducted while people win or lose money suggest that money activates similar reward areas as do other ‘primary reinforcers’ like food and drugs, which implies that money confers direct utility, rather than simply being valued only for what it can buy.”

Among all the reasons that people may value wealth, we focus on social status: we postulate that people enjoy wealth because it provides social status. We therefore introduce relative (not absolute) wealth into the utility function.^{2} The assumption is convenient: in equilibrium, everybody is the same, so relative wealth is 0. And the assumption seems plausible. Adam Smith, David Ricardo, John Rae, John Stuart Mill, Alfred Marshall, Thorstein Veblen, and Frank Knight all believed that people accumulate wealth to attain high social status (Steedman, 1981). More recently, a broad literature has documented that people seek to achieve high social status and that accumulating wealth is a prevalent pathway to do so (Weiss & Fershtman, 1998; Heffetz & Frank, 2011; Fiske, 2010; Anderson, Hildreth, & Howland, 2015; Cheng & Tracy, 2013; Ridgeway, 2014; Mattan, Kubota, & Cloutier, 2017).^{3}

## III. New Keynesian Model with Wealth in the Utility Function

We extend the New Keynesian model by assuming that households derive utility not only from consumption and leisure but also from relative wealth. To simplify derivations and be able to represent the equilibrium with phase diagrams, we use an alternative formulation of the New Keynesian model, inspired by Benhabib, Schmitt-Grohe, and Uribe (2001), and Werning (2011). Our formulation features continuous time instead of discrete time, self-employed households instead of firms and households, and Rotemberg (1982) pricing instead of Calvo (1983) pricing.

### A. Assumptions

The economy is composed of a measure 1 of self-employed households. Each household $j\u2208[0,1]$ produces $yj(t)$ units of a good $j$ at time $t$, sold to other households at a price $pj(t)$. The household's production function is $yj(t)=ahj(t)$, where $a>0$ represents the level of technology and $hj(t)$ is hours of work. Working causes a disutility $\kappa hj(t)$, where $\kappa >0$ is the marginal disutility of labor.

The goods produced by households are imperfect substitutes for one another, so each household exercises some monopoly power. Moreover, households face a quadratic cost when they change their price: changing a price at a rate $\pi j(t)=p\u02d9j(t)/pj(t)$ causes a disutility $\gamma \pi j(t)2/2$. The parameter $\gamma >0$ governs the cost to change prices and thus price rigidity.

### B. Euler Equation and Phillips Curve

The equilibrium is described by a system of two differential equations: a Euler equation and a Phillips curve. The Euler-Phillips system governs the dynamics of output $y(t)$ and inflation $\pi (t)$. Here we present the system; formal and heuristic derivations are in online appendices A and B; a discrete-time version is in online appendix C.

Because consumption-savings choices depend not only on interest rates but also on the marginal rate of substitution between wealth and consumption, future interest rates have less impact on today's consumption than in the standard model: the Euler equation is discounted. In fact, the discrete-time version of equation (4) features discounting exactly as in McKay, Nakamura, and Steinsson (2017) (see online appendix C).

The wealth-in-the-utility assumption adds one parameter to the equilibrium conditions: $u'(0)$. Accordingly, we compare two submodels:

**Definition 1.**

The NK model is the standard model; the WUNK model is the extension proposed in this paper. When prices are fixed ($\gamma \u2192\u221e$), condition (9) becomes $u'(0)>0$; when prices are perfectly flexible ($\gamma =0$), condition (9) becomes $u'(0)>\u221e$. Hence, at the fixed-price limit, the WUNK model only requires an infinitesimal marginal utility of wealth; at the flexible-price limit, the WUNK model is not well defined. In the WUNK model, we also impose $\delta >\sigma +\epsilon -1\delta \gamma $ in order to accommodate positive natural rates of interest.^{4}

### C. Natural Rate of Interest and Monetary Policy

The central bank aims to maintain the economy at the natural steady state, where inflation is at 0 and output is at its natural level.

In normal times, the natural rate of interest is positive, and the central bank is able to maintain the economy at the natural steady state using the simple policy rule $i(\pi (t))=rn+\varphi \pi (t)$. The corresponding real policy rate is $r(\pi (t))=rn+\varphi -1\pi (t)$. The parameter $\varphi \u22650$ governs the response of interest rates to inflation: monetary policy is active when $\varphi >1$ and passive when $\varphi <1$.

When the natural rate of interest is negative, however, the natural steady state cannot be achieved—because this would require the central bank to set a negative nominal policy rate, which would violate the ZLB. In that case, the central bank moves to the ZLB: $i(t)=0$, so $r(t)=-\pi (t)$.

What could cause the natural rate of interest to be negative? A first possibility is a banking crisis, which disrupts financial intermediation and raises the interest-rate spread (Woodford, 2011; Eggertsson, 2011). The natural rate of interest turns negative when the spread is large enough: $\sigma >\delta -u'(0)yn$. Another possibility in the WUNK model is a drop in consumer sentiment, which leads households to favor saving over consumption, and can be parameterized by an increase in the marginal utility of wealth. The natural rate of interest turns negative when the marginal utility is large enough: $u'(0)>(\delta -\sigma )/yn$.

### D. Properties of the Euler-Phillips System

^{5}The diagrams are displayed in figure 1.

We begin with the Phillips curve, which gives $\pi \u02d9$. First, we plot the locus $\pi \u02d9=0$, which we label “Phillips.” The locus is given by equation (3): it is linear, upward sloping, and goes through the point $[y=yn,\pi =0]$. Second, we plot the arrows giving the directions of the trajectories solving the Euler-Phillips system. The sign of $\pi \u02d9$ is given by equation (1): any point above the Phillips line (where $\pi \u02d9=0$) has $\pi \u02d9>0$, and any point below the line has $\pi \u02d9<0$. So inflation is rising above the Phillips line and falling below it.

We next turn to the Euler equation, which gives $y\u02d9$. Whereas the Phillips curve is the same in the NK and WUNK models, and in normal times and at the ZLB, the Euler equation is different in each case. We therefore proceed case by case.

^{6}

Second, the differential equation shows that any point above the Euler line has $y\u02d9<0$, and any point below it has $y\u02d9>0$. Hence, in all four quadrants of the phase diagram, the trajectories move away from the steady state. We conclude that the Euler-Phillips system is a source. At the ZLB, the Euler-Phillips system therefore behaves very differently in the NK and WUNK models.

With passive monetary policy in normal times, the phase diagrams of the Euler-Phillips system would be similar to the ZLB phase diagrams—except that the Euler and Phillips lines would intersect at $[y=yn,\pi =0]$. In particular, the Euler-Phillips system would be a saddle in the NK model and a source in the WUNK model.

For completeness, we also plot sample solutions to the Euler-Phillips system. The trajectories are obtained by linearizing the Euler-Phillips system at its steady state.^{7} When the system is a source, there are two unstable lines (trajectories that move away from the steady state in a straight line). At $t\u2192-\u221e$, all other trajectories are in the vicinity of the steady state and move away tangentially to one of the unstable lines. At $t\u2192+\u221e$, the trajectories move to infinity parallel to the other unstable line. When the system is a saddle, there is one unstable line and one stable line (trajectory that goes to the steady state in a straight line). All other trajectories come from infinity parallel to the stable line when $t\u2192-\u221e$ and move to infinity parallel to the unstable line when $t\u2192+\u221e$.

The next propositions summarize the results:

**Proposition 1.**

Consider the Euler-Phillips system in normal times. The system admits a unique steady state, where output is at its natural level, inflation is 0, and the ZLB is not binding. In the NK model, the system is a source when monetary policy is active but a saddle when monetary policy is passive. In the WUNK model, the system is a source whether monetary policy is active or passive.

**Proposition 2.**

Consider the Euler-Phillips system at the ZLB. In the NK model, the system admits a unique steady state, where output is above its natural level and inflation is positive; furthermore, the system is a saddle. In the WUNK model, the system admits a unique steady state, where output is below its natural level and inflation is negative; furthermore, the system is a source.

The propositions give the key difference between the NK and WUNK models: at the ZLB, the Euler-Phillips system remains a source in the WUNK model, whereas it becomes a saddle in the NK model. This difference will explain why the WUNK model does not suffer from the anomalies plaguing the NK model at the ZLB. The phase diagrams also illustrate the origin of condition (9). In the WUNK model, the Euler-Phillips system remains a source at the ZLB as long as the Euler line is steeper than the Phillips line (figure 1D). The Euler line's slope at the ZLB is the marginal utility of wealth, so that marginal utility is required to be above a certain level—which is given by condition (9).

The propositions have implications for equilibrium determinacy. When the Euler-Phillips system is a source, the equilibrium is determinate: the only equilibrium trajectory in the vicinity of the steady state is to jump to the steady state and stay there; if the economy jumped somewhere else, output or inflation would diverge following a trajectory similar to those plotted in figures 1A, 1B, and 1D. When the system is a saddle, the equilibrium is indeterminate: any trajectory jumping somewhere on the saddle path and converging to the steady state is an equilibrium (figure 1C). Hence, in the NK model, the equilibrium is determinate when monetary policy is active but indeterminate when monetary policy is passive and at the ZLB. In the WUNK model, the equilibrium is always determinate, even when monetary policy is passive and at the ZLB.

Accordingly, in the NK model, the Taylor principle holds: the central bank must adhere to an active monetary policy to avoid indeterminacy. From now on, we therefore assume that the central bank in the NK model follows an active policy whenever it can ($\varphi >1$ whenever $rn>0$). In the WUNK model, by contrast, indeterminacy is never a risk, so the central bank does not need to worry about how strongly its policy rate responds to inflation. The central bank could even follow an interest-rate peg without creating indeterminacy.

The results that pertain to the NK model in propositions ^{2} and ^{3} are well known (Woodford, 2001). The results that pertain to the WUNK model are close to those obtained by Gabaix (2016, proposition 3.1), although he does not use our phase-diagram representation. Gabaix finds that when bounded rationality is strong enough, the Euler-Phillips system is a source even at the ZLB. He also finds that when prices are more flexible, more bounded rationality is required to maintain the source property. The same is true here: when the marginal utility of wealth is high enough, such that condition (9) holds, the Euler-Phillips system is a source even at the ZLB; and when the price-adjustment cost $\gamma $ is lower, condition (9) imposes a higher threshold on the marginal utility of wealth. Our phase diagrams illustrate the logic behind these results. The Euler-Phillips system remains a source at the ZLB as long as the Euler line is steeper than the Phillips line (figure 1D). As the slope of the Euler line is determined by bounded rationality in the Gabaix model and by marginal utility of wealth in our model, these need to be large enough. When prices are more flexible, the Phillips line steepens, so the Euler line's required steepness increases: bounded rationality or marginal utility of wealth need to be larger.

## IV. Description and Resolution of the New Keynesian Anomalies

We now describe the anomalous predictions of the NK model at the ZLB: an implausibly large drop in output and inflation and implausibly strong effects of forward guidance and government spending. We then show that these anomalies are absent from the WUNK model.

### A. Drop in Output and Inflation

. | Timeline . | Natural Rate of Interest . | Monetary Policy . | Government Spending . |
---|---|---|---|---|

A. ZLB episode | ||||

ZLB | $t\u2208(0,T)$ | $rn<0$ | $i=0$ | – |

Normal times | $t>T$ | $rn>0$ | $i=rn+\varphi \pi $ | – |

B. ZLB episode with forward guidance | ||||

ZLB | $t\u2208(0,T)$ | $rn<0$ | $i=0$ | – |

Forward | $t\u2208(T,T+\Delta )$ | $rn>0$ | $i=0$ | – |

guidance | ||||

Normal times | $t>T+\Delta $ | $rn>0$ | $i=rn+\varphi \pi $ | – |

C. ZLB episode with government spending | ||||

ZLB | $t\u2208(0,T)$ | $rn<0$ | $i=0$ | $g>0$ |

Normal times | $t>T$ | $rn>0$ | $i=rn+\varphi \pi $ | $g=0$ |

. | Timeline . | Natural Rate of Interest . | Monetary Policy . | Government Spending . |
---|---|---|---|---|

A. ZLB episode | ||||

ZLB | $t\u2208(0,T)$ | $rn<0$ | $i=0$ | – |

Normal times | $t>T$ | $rn>0$ | $i=rn+\varphi \pi $ | – |

B. ZLB episode with forward guidance | ||||

ZLB | $t\u2208(0,T)$ | $rn<0$ | $i=0$ | – |

Forward | $t\u2208(T,T+\Delta )$ | $rn>0$ | $i=0$ | – |

guidance | ||||

Normal times | $t>T+\Delta $ | $rn>0$ | $i=rn+\varphi \pi $ | – |

C. ZLB episode with government spending | ||||

ZLB | $t\u2208(0,T)$ | $rn<0$ | $i=0$ | $g>0$ |

Normal times | $t>T$ | $rn>0$ | $i=rn+\varphi \pi $ | $g=0$ |

This table describes the three scenarios analyzed in section IV: the ZLB episode, in section IVA; the ZLB episode with forward guidance, in section IVB; and the ZLB episode with government spending, in section IVC. The parameter $T>0$ gives the duration of the ZLB episode; the parameter $\Delta >0$ gives the duration of forward guidance. We assume that monetary policy is active ($\varphi >1$) in normal times in the NK model; this assumption is required to ensure equilibrium determinacy (Taylor principle). In the WUNK model, monetary policy can be active or passive in normal times.

We start with the NK model. We analyze the ZLB episode by going backward in time. After time $T$, monetary policy maintains the economy at the natural steady state. Since equilibrium trajectories are continuous,^{2} the economy also is at the natural steady state at the end of the ZLB, when $t=T$.^{8}

We then move back to the ZLB episode, when $t<T$. At time 0, the economy jumps to the unique position leading to $[y=yn,\pi =0]$ at time $T$. Hence, inflation and output initially jump down to $\pi (0)<0$ and $y(0)<yn$, and then recover following the unique trajectory leading to $[y=yn,\pi =0]$. The ZLB therefore creates a slump, with below-natural output and deflation (figure 2A).

Critically, the economy is always on the same trajectory during the ZLB, irrespective of the ZLB duration $T$. A longer ZLB only forces output and inflation to start from a lower position on the trajectory at time 0. Thus, as the ZLB lasts longer, initial output and inflation collapse to unboundedly low levels (figure 2C).

Next, we examine the WUNK model. Output and inflation never collapse during the ZLB. Initially inflation and output jump down toward the ZLB steady state, denoted $[yz,\pi z]$, so $\pi z<\pi (0)<0$ and $yz<y(0)<yn$. They then recover following the trajectory going through $[y=yn,\pi =0]$. Consequently the ZLB episode creates a slump (figure 2B), which is deeper when the ZLB lasts longer (figure 2D). But unlike in the NK model, the slump is bounded below by the ZLB steady state: irrespective of the duration of the ZLB, output and inflation remain above $yz$ and $\pi z$, respectively, so they never collapse. Moreover, if the natural rate of interest is negative but close to 0, such that $\pi z$ is close to 0 and $yz$ to $yn$, output and inflation will barely deviate from the natural steady state during the ZLB—even if the ZLB lasts a very long time.

The following proposition records these results:^{9}

**Proposition 3.**

Consider a ZLB episode between times 0 and $T$. The economy enters a slump: $y(t)<yn$ and $\pi (t)<0$ for all $t\u2208(0,T)$. In the NK model, the slump becomes infinitely severe as the ZLB duration approaches infinity: $limT\u2192\u221ey(0)=limT\u2192\u221e\pi (0)=-\u221e$. In the WUNK model, in contrast, the slump is bounded below by the ZLB steady state $[yz,\pi z]$: $y(t)>yz$ and $\pi (t)>\pi z$ for all $t\u2208(0,T)$. In fact, the slump approaches the ZLB steady state as the ZLB duration approaches infinity: $limT\u2192\u221ey(0)=yz$ and $limT\u2192\u221e\pi (0)=\pi z$.

In the NK model, output and inflation collapse when the ZLB is long-lasting, which is well known (Eggertsson & Woodford, 2004, fig. 1; Eggertsson, 2011, fig. 1; Werning, 2011, proposition ^{2}). This collapse is difficult to reconcile with real-world observations. The ZLB episode that started in 1995 in Japan lasted for more than twenty years without sustained deflation. The ZLB episode that started in 2009 in the euro area lasted for more than ten years; it did not yield sustained deflation either. The same is true of the ZLB episode that occurred in the United States between 2008 and 2015.

In the WUNK model, in contrast, inflation and output never collapse. Instead, as the duration of the ZLB increases, the economy converges to the ZLB steady state. That ZLB steady state may not be far from the natural steady state: if the natural rate of interest is only slightly negative, inflation is only slightly below 0 and output only slightly below its natural level. Gabaix (2016, proposition 3.2) obtains a closely related result: in his model, output and inflation also converge to the ZLB steady state when the ZLB is arbitrarily long.

### B. Forward Guidance

We turn to the effects of forward guidance at the ZLB. We consider a three-stage scenario, as in Cochrane (2017). Between times 0 and $T$, there is a ZLB episode. To alleviate the situation, the central bank makes a forward-guidance promise at time 0: that it will maintain the policy rate at 0 for a duration $\Delta $ once the ZLB is over. After time $T$, the natural rate of interest is positive again. Between times $T$ and $T+\Delta $, the central bank fulfills its forward-guidance promise and keeps the policy rate at 0. After time $T+\Delta $, monetary policy returns to normal. This scenario is summarized in table 1B.

We begin with the NK model. We go backward in time. After time $T+\Delta $, monetary policy maintains the economy at the natural steady state. Between times $T$ and $T+\Delta $, the economy is in forward guidance (figure 3A). Following the logic of figure 2, we find that at time $T$, inflation is positive and output above its natural level. They subsequently decrease over time, following the unique trajectory leading to the natural steady state at time $T+\Delta $. Accordingly, the economy booms during forward guidance. Furthermore, as forward guidance lengthens, inflation and output at time $T$ become higher.

We look next at the ZLB episode, between times 0 and $T$. Since equilibrium trajectories are continuous, the economy is at the same point at the end of the ZLB and the beginning of forward guidance. The boom engineered during forward guidance therefore improves the situation at the ZLB. Instead of reaching the natural steady state at time $T$, the economy reaches a point with positive inflation and above-natural output, so at any time before $T$, inflation and output tend to be higher than without forward guidance (figure 3B).

Forward guidance can actually have tremendously strong effects in the NK model. For small durations of forward guidance, the position at time $T$ is below the ZLB unstable line. It is therefore connected to trajectories coming from the southwest quadrant of the phase diagram (figure 3B). As the ZLB lasts longer, initial output and inflation collapse. When the duration of forward guidance is such that the position at time $T$ is exactly on the unstable line, the position at time 0 is on the unstable line as well (figure 3C). As the ZLB lasts longer, the initial position inches closer to the ZLB steady state. For even longer forward guidance, the position at time $T$ is above the unstable line, so it is connected to trajectories coming from the northeast quadrant (figure 3D). Then, as the ZLB lasts longer, initial output and inflation become higher and higher. As a result, if the duration of forward guidance is long enough, a deep slump can be transformed into a roaring boom. Moreover, the forward-guidance duration threshold is independent of the ZLB duration.

In comparison, the power of forward guidance is subdued in the WUNK model. Between times $T$ and $T+\Delta $, forward guidance operates (figure 4A). Inflation is positive, and output is above its natural level at time $T$. They then decrease over time, following the trajectory leading to the natural steady state at time $T+\Delta $. The economy booms during forward guidance, but unlike in the NK model, output and inflation are bounded above by the forward-guidance steady state.

Before forward guidance comes the ZLB episode (figures 4B and 4C). Thanks to the boom engineered by forward guidance, the situation is improved at the ZLB: inflation and output tend to be higher than without forward guidance. Yet, unlike in the NK model, output during the ZLB episode is always below its level at time $T$, so forward guidance cannot generate unbounded booms (figure 4D). The ZLB cannot generate unbounded slumps either, since output and inflation are bounded below by the ZLB steady state (figure 4D). Actually, for any forward-guidance duration, as the ZLB lasts longer, the economy converges to the ZLB steady state at time 0. The implication is that forward guidance can never prevent a slump when the ZLB lasts long enough.

Based on these dynamics, we identify an anomaly in the NK model, which is resolved in the WUNK model (proof details in online appendix D):

**Proposition 4.**

Consider a ZLB episode during $(0,T)$ followed by forward guidance during $(T,T+\Delta )$:

- •
In the NK model, there exists a threshold $\Delta *$ such that a forward guidance longer than $\Delta *$ transforms a ZLB episode of any duration into a boom: let $\Delta >\Delta *$; for any $T$ and for all $t\u2208(0,T+\Delta )$, $y(t)>yn$ and $\pi (t)>0$. In addition, when forward guidance is longer than $\Delta *$, a long-enough forward guidance or ZLB episode generates an arbitrarily large boom: for any $T$, $lim\Delta \u2192\u221ey(0)=lim\Delta \u2192\u221e\pi (0)=+\u221e$; and for any $\Delta >\Delta *$, $limT\u2192\u221ey(0)=limT\u2192\u221e\pi (0)=+\u221e$.

- •
In the WUNK model, in contrast, there exists a threshold $T*$ such that a ZLB episode longer than $T*$ prompts a slump, irrespective of the duration of forward guidance: let $T>T*$; for any $\Delta $, $y(0)<yn$ and $\pi (0)<0$. Furthermore, the slump approaches the ZLB steady state as the ZLB duration approaches infinity: for any $\Delta $, $limT\u2192\u221ey(0)=yz$ and $limT\u2192\u221e\pi (0)=\pi z$. In addition, the economy is bounded above by the forward-guidance steady state $[yf,\pi f]$: for any $T$ and $\Delta $, and for all $t\u2208(0,T+\Delta )$, $y(t)<yf$ and $\pi (t)<\pi f$.

The anomaly identified in the proposition corresponds to the forward-guidance puzzle described by Carlstrom, Fuerst, and Paustian (2015, fig. 1) and Cochrane (2017, fig. 6).^{10} These papers also find that a long-enough forward guidance transforms a ZLB slump into a boom.

In the WUNK model, this anomalous pattern vanishes. In the New Keynesian models by Gabaix (2016), Diba and Loisel (2019), Acharya and Dogra (2019), and Bilbiie (2019), forward guidance also has more subdued effects than in the standard model. Besides, New Keynesian models have been developed with the sole goal of solving the forward-guidance puzzle. Among these, ours belongs to the group that uses discounted Euler equations.^{11} For example, Del Negro, Giannoni, and Patterson (2015) generate discounting from overlapping generations; McKay, Nakamura, and Steinsson (2016) from heterogeneous agents facing borrowing constraints and cyclical income risk; Angeletos and Lian (2018) from incomplete information; and Campbell et al. (2017) from government bonds in the utility function (which is closely related to our approach).

### C. Government Spending

Finally, we consider the effects of government spending at the ZLB. We first extend the model by assuming that the government purchases goods from all households, which are aggregated into public consumption $g(t)$. To ensure that government spending affects inflation and private consumption, we also assume that the disutility of labor is convex: household $j$ incurs disutility $\kappa 1+\eta hj(t)1+\eta /(1+\eta )$ from working, where $\eta >0$ is the inverse of the Frisch elasticity. The complete extended model, derivations, and results are presented in online appendix E.

In this model, the Euler equation is unchanged, but the Phillips curve is modified because the marginal disutility of labor is not constant and because households produce goods for the government. The modification of the Phillips curve alters the analysis in three ways.

First, the steady-state Phillips curve becomes nonlinear, which may introduce additional steady states. We handle this issue as in the literature: we linearize the Euler-Phillips system around the natural steady state without government spending and concentrate on the dynamics of the linearized system. These dynamics are described by phase diagrams similar to those in the basic model.

Third, public consumption enters the Phillips curve, so government spending operates through that curve. Indeed, since $\eta >0$ in equation (11), government spending shifts the steady-state Phillips curve upward. Intuitively, given private consumption, an increase in government spending raises production and thus marginal costs. Facing higher marginal costs, producers augment inflation.

We now study a ZLB episode during which the government increases spending in an effort to stimulate the economy, as in Cochrane (2017). Between times 0 and $T$, there is a ZLB episode. To alleviate the situation, the government provides an amount $g>0$ of public consumption. After time $T$, the natural rate of interest is positive again, government spending stops, and monetary policy returns to normal. This scenario is summarized in table 1C.

^{12}We construct the equilibrium path by going backward in time. At time $T$, monetary policy brings the economy to the natural steady state. At the ZLB, government spending helps, but through a different mechanism from forward guidance. Forward guidance improves the situation at the end of the ZLB, which pulls up the economy during the entire ZLB. Government spending leaves the end of the ZLB unchanged: the economy reaches the natural steady state. Instead, government spending shifts the Phillips line upward and, with it, the field of trajectories. As a result, the natural steady state is connected to trajectories with higher consumption and inflation, which improves the situation during the entire ZLB.

Just like forward guidance, government spending can have very strong effects in the NK model. When spending is low, the natural steady state is below the ZLB unstable line (figure 5B). It is therefore connected to trajectories coming from the southwest quadrant of the phase diagram—just as without government (figure 5A). Then, if the ZLB lasts longer, initial consumption and inflation fall lower. When spending is high enough that the unstable line crosses the natural steady state, the economy is also on the unstable line at time 0 (figure 5C). Finally, when spending is even higher, the natural steady state moves above the unstable line, so it is connected to trajectories coming from the northeast quadrant (figure 5D). As a result, initial output and inflation are higher than previously. And as the ZLB lasts longer, initial output and inflation become even higher, without bound.

Based on these dynamics, we isolate another anomaly in the NK model, which is resolved in the WUNK model (proof details in online appendix F):

**Proposition 5.**

- •
In the NK model, there exists a government spending $g*$ such that the government-spending multiplier becomes infinitely large when the ZLB duration approaches infinity: for any $s>0$, $limT\u2192\u221em(g*,s)=+\u221e$. In addition, when government spending is above $g*$, a long-enough ZLB episode generates an arbitrarily large boom: for any $g>g*$, $limT\u2192\u221ec(0;g)=+\u221e$.

- •In the WUNK model, in contrast, the multiplier has a finite limit when the ZLB duration approaches infinity: for any $g$ and $s$, when $T\u2192\u221e$, $m(g,s)$ converges toMoreover, the economy is bounded above for any ZLB duration: let $cg$ be private consumption in the ZLB steady state with government spending $g$; for any $T$ and for all $t\u2208(0,T)$, $c(t;g)<max(cg,cn)$.$1+\eta u'(0)\delta \gamma a\epsilon \kappa \xb7\epsilon \epsilon -1\eta /(1+\eta )-(1+\eta ).$(13)

The anomaly that a finite amount of government spending may generate an infinitely large boom as the ZLB becomes arbitrarily long-lasting is reminiscent of the findings by Christiano, Eichenbaum, and Rebelo (2011, fig. 2), Woodford (2011, fig. 2), and Cochrane (2017, fig. 5). They find that in the NK model, government spending is exceedingly powerful when the ZLB is long-lasting.

In the WUNK model, this anomaly vanishes. Diba and Loisel (2019) and Acharya and Dogra (2019) also obtain more realistic effects of government spending at the ZLB. In addition, Bredemeier, Juessen, and Schabert (2018) obtain moderate multipliers at the ZLB by introducing an endogenous liquidity premium in the New Keynesian model.

## V. Other New Keynesian Properties at the ZLB

Beside the anomalous properties described in section IV, the New Keynesian model has several other intriguing properties at the ZLB: the paradoxes of thrift, toil, and flexibility and a government-spending multiplier greater than 1. We now show that the WUNK model shares these properties.

### A. Paradox of Thrift

**Proposition 6.**

At the ZLB in the WUNK model, the paradox of thrift holds: an unexpected and permanent increase in the marginal utility of wealth reduces output and inflation but does not affect relative wealth.

The paradox of thrift was first discussed by Keynes, but it also appears in the New Keynesian model (Eggertsson, 2010, 16; Eggertsson & Krugman, 2012, 1486). When the marginal utility of wealth is higher, people want to increase their wealth holdings relative to their peers, so they favor saving over consumption. But in equilibrium, relative wealth is fixed at 0 because everybody is the same; the only way to increase saving relative to consumption is to reduce consumption. In normal times, the central bank would offset this drop in aggregate demand by reducing nominal interest rates. This is not an option at the ZLB, so output falls.

### B. Paradox of Toil

**Proposition 7.**

At the ZLB in the WUNK model, the paradox of toil holds: an unexpected and permanent reduction in the disutility of labor reduces output, inflation, and hours worked.

The paradox of toil was discovered by Eggertsson (2010, 15) and Eggertsson and Krugman (2012, 1487). It operates as follows. With lower disutility of labor, real marginal costs are lower, and the natural level of output is higher: producers would like to sell more. To increase sales, they reduce their prices by reducing inflation. At the ZLB, nominal interest rates are fixed, so the decrease in inflation raises real interest rates—which renders households more prone to save. In equilibrium, this lowers output and hours worked.^{13}

### C. Paradox of Flexibility

We then examine a decrease in the price-adjustment cost ($\gamma $). The steady-state Euler equation is not affected, but the steady-state Phillips curve is. Equation (3) shows that decreasing the price-adjustment cost leads to a counterclockwise rotation of the Phillips line around the natural steady state. This moves the economy downward along the Euler line, so output and inflation decrease (figure 7C). The following proposition records the results:

**Proposition 8.**

At the ZLB in the WUNK model, the paradox of flexibility holds: an unexpected and permanent decrease in price-adjustment cost reduces output and inflation.

The paradox of flexibility was discovered by Werning (2011, 13–14) and Eggertsson and Krugman (2012, 1487–1488). Intuitively, with a lower price-adjustment cost, producers are eager to adjust their prices to bring production closer to the natural level of output. Since production is below the natural level at the ZLB, producers are eager to reduce their prices to stimulate sales. This accentuates the existing deflation, which translates into higher real interest rates. As a result, households are more prone to save, which in equilibrium depresses output.

### D. Above-One Government-Spending Multiplier

We finally look at an increase in government spending ($g$), using the model with government spending introduced in section IVC. From equation (11) we see that increasing government spending shifts the Phillips line upward, which moves the economy upward along the Euler line: both private consumption and inflation increase (figure 7D). Since private consumption increases when public consumption does, the government-spending multiplier $dy/dg=1+dc/dg$ is greater than one. The ensuing proposition gives the results (proof details in online appendix F):

**Proposition 9.**

At the ZLB in the WUNK model, an unexpected and permanent increase in government spending raises private consumption and inflation. Hence the government-spending multiplier $dy/dg$ is above one; its value is given by equation (13).

Christiano, Eichenbaum, and Rebelo (2011), Eggertsson (2011), and Woodford (2011) also show that at the ZLB in the New Keynesian model, the government-spending multiplier is above 1. The intuition is the following. With higher government spending, real marginal costs are higher for a given level of sales to households. Producers pass the cost increase through into prices, which raises inflation. At the ZLB, the increase in inflation lowers real interest rates—as nominal interest rates are fixed—which deters households from saving. In equilibrium, this leads to higher private consumption and a multiplier above one.

## VI. Empirical Assessment of the WUNK Assumption

In the WUNK model, the marginal utility of wealth is assumed to be high enough that the steady-state Euler equation is steeper than the steady-state Phillips curve at the ZLB. We assess this assumption using US evidence.

### A. Natural Rate of Interest

A large number of macroeconometric studies have estimated the natural rate of interest, using different statistical models, methodologies, and data. Recent studies obtain comparable estimates of the natural rate for the United States: around $2%$ per annum on average between 1985 and 2015 (Williams, 2017, fig. 1). Accordingly, we use $rn=2%$ as our estimate.

### B. Output-Gap Coefficient in the New Keynesian Phillips Curve

Many studies have estimated New Keynesian Phillips curves. Mavroeidis, Plagborg-Moller, and Stock (2014, sec. 5) offer a synthesis for the United States. They generate estimates of the New Keynesian Phillips curve using an array of US data, methods, and specifications found in the literature. They find significant uncertainty around the estimates, but in many cases, the output-gap coefficient is positive and very small. Overall, their median estimate of the output-gap coefficient is $\lambda =0.004$ (table 5, row 1), which we use as our estimate.

### C. Time Discount Rate

Since the 1970s, many studies have estimated time discount rates using field and laboratory experiments and real-world behavior. Frederick, Loewenstein, and O'Donoghue (2002, table 1) survey 43 such studies. The estimates are quite dispersed, but the majority of them point to high discount rates, much higher than prevailing market interest rates. We compute the mean estimate in each of the studies covered by the survey and then compute the median value of these means. We obtain an annual discount rate of $\delta =35%$.

There is one immediate limitation with the studies discussed by Frederick, Loewenstein, and O'Donoghue: they use a single rate to exponentially discount future utility. But exponential discounting does not describe reality well because people seem to choose more impatiently for the present than for the future—they exhibit present-focused preferences (Ericson & Laibson, 2019). Recent studies have moved away from exponential discounting and allowed for present-focused preferences, including quasi-hyperbolic ($\beta $-$\delta $) discounting. Andersen et al. (2014, table 3) survey 16 such studies, concentrating on experimental studies with real incentives. We compute the mean estimate in each study and then the median value of these means; we obtain an annual discount rate of $\delta =43%$. Accordingly, even after accounting for present focus, time discounting remains high. We use $\delta =43%$ as our estimate.^{14}

### D. Assessment

We now combine our estimates of $rn$, $\lambda $, and $\delta $ to assess the WUNK assumption. Since $\lambda $ is estimated using the quarter as the unit of time, we reexpress $rn$ and $\delta $ as quarterly rates: $rn=2%/4=0.5%$ per quarter, and $\delta =43%/4=10.8%$ per quarter. We conclude that condition (14) comfortably holds: $\delta -rn=0.108-0.005=0.103$, which is much larger than $\lambda /\delta =0.004/0.108=0.037$. Hence, the WUNK assumption holds in US data.

The discount rate used here (43% per annum) is much higher than discount rates used in macroeconomic models (typically less than 5% per annum). This is because our discount rate is calibrated from microevidence, while the discount rate in macroeconomic models is calibrated to match observed real interest rates.

This discrepancy occasions two remarks. First, the wealth-in-the-utility assumption is advantageous because it accords with the fact that people exhibit double-digit discount rates and yet are willing to save at single-digit interest rates. In the standard model, by contrast, the discount rate equals the real interest rate in steady state, so the model cannot accommodate double-digit discount rates.

Second, the WUNK assumption would also be satisfied with annual discount rates below 43%. Indeed, condition (14) holds for discount rates as low as 27% because $\delta -rn=(0.27/4)-0.005=0.062$ is greater than $\lambda /\delta =0.004/(0.27/4)=0.059$. An annual discount rate of 27% is at the low end of available microestimates: in 11 of the 16 studies in Andersen et al. (2014, table 3), the bottom of the estimate range is above 27%, and in 13 of the 16 studies, the mean estimate is above 27%.

## VII. Conclusion

This paper proposes an extension of the New Keynesian model that is immune to the anomalies that plague the standard model at the ZLB. The extended model deviates only minimally from the standard model: relative wealth enters the utility function, which adds an extra term only in the Euler equation. Yet when the marginal utility of wealth is sufficiently high, the model behaves well at the ZLB: even when the ZLB is long-lasting, there is no collapse of inflation and output, and both forward guidance and government spending have limited, plausible effects. The extended model also retains other properties of the standard model at the ZLB: the paradoxes of thrift, toil, and flexibility and a government-spending multiplier greater than 1.

Our analysis would apply more generally to any New Keynesian model representable by a discounted Euler equation and a Phillips curve (Del Negro, Giannoni, & Patterson, 2015; Gabaix, 2016; McKay, Nakamura, & Steinsson, 2017; Campbell et al., 2017; Beaudry & Portier, 2018; Angeletos & Lian, 2018). Wealth in the utility function is a simple way to generate discounting; but any model with discounting would have similar phase diagrams and properties. Hence, for such models to behave well at the ZLB, there is only one requirement: that discounting is strong enough to make the steady-state Euler equation steeper than the steady-state Phillips curve at the ZLB; the source of discounting is unimportant. In the real world, several discounting mechanisms might operate at the same time and reinforce each other. A model blending these mechanisms would be even more likely to behave well at the ZLB.

## Notes

^{1}

Our approach relates to the work of Michaillat and Saez (2014), Ono and Yamada (2018), and Michau (2018). By assuming wealth in the utility function, they obtain non-New-Keynesian models that behave well at the ZLB. But their results are not portable to the New Keynesian framework because they require strong forms of wage or price rigidity (exogenous wages, fixed inflation, or downward nominal wage rigidity). Our approach also relates to the work of Fisher (2015) and Campbell et al. (2017), who build New Keynesian models with government bonds in the utility function. The bonds-in-the-utility assumption captures special features of government bonds relative to other assets, such as safety and liquidity (Krishnamurthy & Vissing-Jorgensen, 2012). While their assumption and ours are conceptually different, they affect equilibrium conditions in a similar way. These papers use their assumption to generate risk-premium shocks (Fisher) and to alleviate the forward-guidance puzzle (Campbell et al.).

^{2}

Cole, Mailath, and Postlewaite (1992, 1995) develop models in which relative wealth does not directly confer utility but has other attributes such that people behave as if wealth entered their utility function. In one such model, wealthier individuals have higher social rankings, which allows them to marry wealthier partners and enjoy higher utility.

^{3}

The wealth-in-the-utility assumption has been found useful in models of long-run growth (Kurz, 1968; Konrad, 1992; Zou, 1994; Corneo & Jeanne, 1997; Futagami & Shibata, 1998), risk attitudes (Robson, 1992; Clemens, 2004), asset pricing (Bakshi & Chen, 1996; Gong & Zou, 2002; Kamihigashi, 2008; Michau, Ono, & Schlegl, 2018), life-cycle consumption (Zou, 1995; Carroll, 2000; Francis, 2009; Straub, 2019), social stratification (Long & Shimomura, 2004), international macroeconomics (Fisher, 2005; Fisher & Hof, 2005), financial crises (Kumhof, Ranciere, & Winant, 2015), and optimal taxation (Saez & Stantcheva, 2018). Such usefulness lends further support to the assumption.

^{4}

Indeed, using equations (2) and (9), we see that in the WUNK model, This implies that the natural rate of interest, given by equation (5), is bounded above: For the WUNK model to accommodate positive natural rates of interest, the upper bound on the natural rate must be positive. This requires the time discount rate to be large enough.

^{5}

The properties are rederived using an algebraic approach in online appendix D.

^{6}

We also check that the intersection of the Euler and Phillips lines has positive output (online appendix D).

^{7}

Technically the trajectories only approximate the exact solutions, but the approximation is accurate in the neighborhood of the steady state.

^{8}

The trajectories are continuous in output and inflation because households have concave preferences over the two arguments. If consumption had an expected discrete jump, for example, households would be able to increase their utility by reducing the size of the discontinuity.

^{9}

The result that in the NK model output becomes infinitely negative when the ZLB becomes infinitely long should not be interpreted literally. It is obtained because we omitted the constraint that output must remain positive. The proper interpretation is that output falls much, much below its natural level—in fact, it converges to 0.

^{10}

In the literature, the forward-guidance puzzle takes several forms. The common element is that future monetary policy has an implausibly strong effect on current output and inflation.

^{11}

Other approaches to solve the forward-guidance puzzle include modifying the Phillips curve (Carlstrom, Fuerst, & Paustian, 2015), combining reflective expectations and temporary equilibrium (Garcia-Schmidt & Woodford, 2019), combining bounded rationality and incomplete markets (Farhi & Werning, 2019), and introducing an endogenous liquidity premium (Bredemeier, Kaufmann, & Schabert, 2018).

^{12}

There is a small difference with the phase diagrams of the basic model: private consumption $c$ is on the horizontal axis instead of output $y$. But $y=c$ in the basic model (government spending is 0), so the phase diagrams with private consumption on the horizontal axis would be the same as those with output.

^{13}

An increase in technology ($a$) would have the same effect as a reduction in the disutility of labor: it would lower output, inflation, and hours.

^{14}

There are two potential issues with the experiments discussed in Andersen et al. (2014). First, many are run with university students instead of subjects representative of the general population. There do not seem to be systematic differences in discounting between student and nonstudent subjects, however (Cohen et al., 2020, 334). Hence, using students is unlikely to bias the estimates reported by Andersen et al. Second, all the experiments elicit discount rates using financial flows, not consumption flows. As the goal is to elicit the discount rate on consumption, this could be problematic; the problems could be exacerbated if subjects derive utility from wealth. To assess this potential issue, suppose first (as in most of the literature) that monetary payments are consumed at the time of receipt and that the utility function is locally linear. Then the experiments deliver estimates of the relevant discount rate (Cohen et al., 2020, 323--325). If these conditions do not hold, the experimental findings are more difficult to interpret. For instance, if subjects optimally smooth their consumption over time by borrowing and saving, then the experiments only elicit the interest rate faced by subjects and reveal nothing about their discount rate (Cohen et al., 2020, 322--323). In that case, we should rely on experiments using time-dated consumption rewards instead of monetary rewards. Such experiments directly deliver estimates of the discount rate. Many such experiments have been conducted; a robust finding is that discount rates are higher for consumption rewards than for monetary rewards (Cohen et al., 2020, 312--315). Hence, the estimates presented in Andersen et al. are, if anything, lower bounds on actual discount rates.

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## Author notes

We thank Sushant Acharya, Adrien Auclert, Gadi Barlevi, Marco Bassetto, Jess Benhabib, Florin Bilbiie, Jeffrey Campbell, Edouard Challe, Varanya Chaubey, John Cochrane, Behzad Diba, Gauti Eggertsson, Erik Eyster, François Gourio, Pete Klenow, Olivier Loisel, Neil Mehrotra, Emi Nakamura, Sam Schulhofer-Wohl, David Sraer, Jon Steinsson, Harald Uhlig, and Ivan Werning for helpful discussions and comments. This work was supported by the Institute for Advanced Study and the Berkeley Center for Equitable Growth.

A supplemental appendix is available online at https://doi.org/10.1162/rest_a_00893.