AAn Economical Business-Cycle Model
Pascal Michaillat and Emmanuel SaezDecember 2019
In recent decades, in developed economies, slack on the product and labor marketshas fluctuated a lot over the business cycle, while inflation has been very stable. Atthe same time, these economies have been prone to enter long-lasting liquidity trapswith stable positive inflation and high unemployment. Motivated by these observa-tions, this paper develops a simple policy-oriented business-cycle model in which (1)fluctuations in aggregate demand and supply lead to fluctuations in slack but not ininflation; and (2) the aggregate demand structure is consistent with permanent liquid-ity traps. The model extends the money-in-the-utility-function model by introducingmatching frictions and including real wealth into the utility function. Matching fric-tions allow us to represent slack and to consider a general equilibrium with constantinflation. Wealth in the utility function enriches the aggregate demand structure tobe consistent with permanent liquidity traps. We use the model to study the effectsof various aggregate demand and supply shocks, and to analyze several stabilizationpolicies—such as conventional monetary policy, helicopter drop of money, tax onwealth, and government spending.
Michaillat: Brown University. Saez: University of California–Berkeley. First draft: November 2013. We thank RegisBarnichon, Giancarlo Corsetti, Wouter den Haan, Emmanuel Farhi, Jordi Gali, Yuriy Gorodnichenko, EtienneLehmann, Karel Mertens, Yoshiyasu Ono, Kevin Sheedy, Harald Uhlig, Carl Walsh, Johannes Wieland, and MichaelWoodford for helpful discussions and comments. This work was supported by the Economic and Social ResearchCouncil [grant number ES/K008641/1], the Institute for New Economic Thinking, the Berkeley Center for EquitableGrowth, and the British Academy. a r X i v : . [ ec on . T H ] D ec . Introduction In the United States since the 1980s, slack on the product and labor markets has fluctuated a lotover the business cycle, while inflation has been very stable. Figure 1 displays two measures ofslack on the product market (the rate of idle capacity and the rate of idle labor), one measuresof slack on the product market (the rate of unemployment), and the core inflation rate. Themeasures of slack are very countercyclical, whereas core inflation is very stable around 2%. TheGreat Recession is a good example of this pattern: from the beginning of 2008 to the middle of2009, the rate of idle labor increased from 19% to 33%, the rate of idle capacity from 24% to 40%,the rate of unemployment increased from 5% to 10%, while the core inflation rate only fell from2.1% to 1.2%. Moreover, economies with low and stable inflation seem prone to enter long-lasting liquiditytraps after negative shocks. The ZLB episode that started in 1995 in Japan has been lasting formore than twenty years. The ZLB episode that started in 2009 in the euro area has been lastingfor more than 10 years. And the ZLB episode that occurred in the United States after the GreatRecession lasted 8 years, from 2008 to 2015.Motivated by these two observations we develop a model of the business cycle in whichfluctuations in demand and supply lead to fluctuations in slack but not in inflation, and in whichliquidity traps may be long-lasting or even permanent. Our model offers a perspective on businesscycles which differs from that of the standard New Keynesian model. In that model fluctuationsin demand and supply lead to fluctuations in inflation but not in slack, and long-lasting liquiditytraps generate an array of anomalies. We then use our model to analyze several monetary andfiscal policies. We contrast the effects of these policies in and out of liquidity traps.Our model has a simple structure since it only adds two elements to the money-in-the-utilitymodel of Sidrauski (1967). The first element is matching frictions on the market where self-employed households sell labor services to other households. In modeling matching frictions wefollow Michaillat and Saez (2015, 2019b). Michaillat and Saez (2015) also provide a broad range of A possible explanation for the stability of inflation is that monetary policy maintains inflation constant. But thisseems implausible. First, the mandate of monetary policy is to stabilize both slack and inflation, so it is unlikely thatmonetary policy solely focuses on stabilizing inflation. Second, there is good empirical evidence that in the shortrun monetary policy does not have much influence on inflation: most empirical studies find that monetary policybarely contributes to short-run price movements. For instance, Christiano, Eichenbaum, and Evans (1999) find thatinflation responds only modestly and with long delays to monetary policy: in vector autoregressions, the response ofinflation to monetary policy is not statistically significant, and it has a lag of two years. The models in Michaillat and Saez (2015, 2019b) do not feature interest-bearing assets, so they cannot be usedto think about monetary policy or liquidity traps. Michaillat and Saez (2019b, sec. 2) highlight the similarities anddifferences between the matching framework used here and the canonical Diamond-Mortensen-Pissarides model.Most of the differences arise from the need to have a framework adapted to address run-of-the-mill business-cycleand policy questions. This social norm could determined bycommunication from the central bank in the long run; but in the short run, which is the horizonof the model, nothing affects the inflation rate. This approach to modeling prices in a matchingmodel is inspired by Hall (2005). The advantage of the approach is that accepting fixed inflationis bilaterally efficient for sellers and buyers: in any trade, buyer and seller prefer transacting atthe price given by fixed inflation than either not transacting (since there is a surplus from anytransaction) or transacting with somebody else later on (since searching for a new trade partneris costly). Accordingly, the assumption of fixed inflation rests on solid theoretical grounds in ourmatching model, unlike assumptions of fixed prices in non-matching models (Barro 1977). In thisgeneral equilibria with constant inflation, market tightness adjusts to equalize aggregate supplyand demand.The second element is the presence of real wealth in the utility function. This assumptionallows us to obtain a well-behaved model in liquidity traps, as showed in Michaillat and Saez(2019c). In fact, a liquidity trap with positive inflation—and thus negative real interest rate—and high unemployment is a possible steady-state equilibrium. Permanent liquidity traps existbecause the consumption Euler equation is modified with wealth in the utility. The motivationfor the wealth-in-the-utility assumption is that people seem to care about real wealth not onlyas future consumption but for its own sake. People may value wealth because it is commonlyused to rank people in societies and thus high wealth provides high social status (Weiss andFershtman 1998; Heffetz and Frank 2011; Fiske 2010; Anderson, Hildreth, and Howland 2015;Cheng and Tracy 2013; Ridgeway 2014; Mattan, Kubota, and Cloutier 2017). People may alsodesire to accumulate wealth as an end in itself. Recent neuroscientific evidence suggests thatthese considerations matter (Camerer, Loewenstein, and Prelec 2005). The wealth-in-the-utilityassumption has also been found useful across a broad range of fields, which provides additionalalbeit indirect support for it. For example, the assumption has been used in models of long-run growth (Kurz 1968; Konrad 1992; Zou 1994; Corneo and Jeanne 1997; Futagami and Shibata Inflation has become extremely hard to forecast after 1984. Stock and Watson (2008) find that it has becomeexceedingly difficult to improve systematically upon simple univariate forecasting models. In particular, it is difficultto describe the sluggish dynamics of inflation with standard accelerationist Phillips curves (for example, Rudd andWhelan 2007; Gordon 2011; Ball and Mazumder 2011). The behavior of inflation over the past two decades led Hall(2011) to argue that inflation is exogenous for all practical purposes. The model in Michaillat and Saez (2019c) is New Keynesian so it does not feature any slack or unemployment.
994 1999 2004 2009 2014 0%10%20%30%40% 0%2.5% 5%7.5% 10%
Core inflation (right scale)Idle capacity (left scale)Idle labor (left scale) Unemployment(right scale)
Figure 1.
Slack and inflation in the United States, 1994–2014
The rate of idle capacity is one minus the rate of capacity utilization in the manufacturing sector measured bythe Census Bureau from the Survey of Plant Capacity. The rate of idle labor is one minus the operating rate in themanufacturing sector measured by the Institute for Supply Management. The rate of unemployment is the civilianunemployment rate measured by the Bureau of Labor Statistics from the Current Population Survey. The rate of coreinflation is the percent change from year ago of the personal consumption expenditures index (excluding food andenergy) constructed by the Bureau of Economic Analysis as part of the National Income and Product Accounts. Therates of idle capacity and idle labor are quarterly series. The rates of unemployment and core inflation are quarterlyaverages of seasonally adjusted monthly series. . Model
The model extends the money-in-the-utility-function model of Sidrauski (1967) by adding matchingfrictions on the market for labor services and wealth in the utility function. The economy consistsof a measure 1 of identical households who hold money and bonds, produce labor services, andpurchase labor services from other households for their own consumption. Households can issue or buy riskless nominal bonds. Bonds are traded on a perfectly competitivemarket. At time t , households hold B ( t ) bonds, and the rate of return on bonds is the nominalinterest rate i ( t ) .A quantity M ( t ) of money circulates at time t . Money is issued by the government throughopen market operations: the government buys bonds issued by households with money. At anytime t , the quantity of bonds issued equals the quantity of money put in circulation: − (cid:219) B ( t ) = (cid:219) M ( t ) .Initially, − B ( ) = M ( ) . Therefore, at any time t ,(1) − B ( t ) = M ( t ) . The representative household is net borrower: B ( t ) ≤
0. At time t , the revenue from seignorage is S ( t ) = − B ( t ) · i ( t ) = i ( t ) · M ( t ) . The government rebates this revenue lump sum to households. Without public spending or taxes,the government’s budget is therefore balanced at any time.Finally, money is the unit of account. At time t , the price level is p ( t ) , the rate of inflation is π ( t ) = (cid:219) p ( t )/ p ( t ) , and the quantity of real money in circulation is m ( t ) = M ( t )/ p ( t ) . The market for labor services is modeled as in Michaillat and Saez (2015, 2019b).
Informal description.
As the market for labor services is not standard, we begin by describing itinformally, borrowing from Michaillat and Saez (2019b, sec. 2). To simplify the analysis, we abstract from firms and assume that all production directly takes place withinhouseholds. Michaillat and Saez (2015) show how the model can be extended to include a labor market and a productmarket, distinct but formally symmetric. In such extension, firms hire workers on the labor market and sell theirproduction on the product market. ormal description.
Households sell labor services on a market with matching frictions. House-holds would like to sell k units of services at any point in time. The capacity k of each household isexogenous. Households also consume labor services, but they cannot consume their own services,so they trade with other households. To buy labor services, households post v ( t ) help-wantedadvertisements at time t .A matching function h with constant returns to scale gives the number of trades at time t : y ( t ) = h ( k , v ( t )) . The matching function is twice differentiable, strictly increasing in both arguments, and withdiminishing marginal returns in both arguments. It also satisfies 0 ≤ h ( k , v ( t )) ≤ min { k , v ( t )} . In each trade, one unit of labor service is bought at price p ( t ) > x is defined by x ( t ) = v ( t )/ k . With constant returns to scale in matching,the market tightness determines the probabilities to trade for sellers and buyers. At time t , onelabor service is sold with probability f ( x ( t )) = y ( t ) k = h ( , x ( t )) , and one help-wanted advertisement leads to a trade with probability q ( x ( t )) = y ( t ) v ( t ) = h (cid:18) x ( t ) , (cid:19) . We denote by 1 − η and − η the elasticities of f and q : 1 − η ≡ x · f (cid:48) ( x )/ f ( x ) > η ≡− x · q (cid:48) ( x )/ q ( x ) >
0. We abstract from randomness at the household’s level: at time t , a householdsells f ( x ( t )) · k labor services and purchases q ( x ( t )) · v ( t ) labor services with certainty.Households are unable to sell all their labor services since f ( x ( t )) ≤
1. Households are idle afraction 1 − f ( x ( t )) of the time. The rate of idleness can be interpreted as the unemployment ratein this economy of self-employed workers. Since h is strictly increasing in its two arguments, f isstrictly increasing and q is strictly decreasing in x . This means that when the market tightnessis lower, it is harder for households to sell their labor services but easier for them to buy laborservices from others.Posting help-wanted advertisements is costly. The flow cost of an advertisement is ρ ≥ ρ · v ( t ) recruiting services are spent at time t . Theserecruiting services represent the resources devoted to matching with an appropriate worker.Recruiting services are purchased like any other labor services. As output of labor services is Such a matching function is h ( k , v ) = ( k − ζ + v − ζ ) − / ζ with ζ > c ( t ) , and recruiting, we have y ( t ) = c ( t ) + ρ · v ( t ) . Only laborservices for consumption enter households’ utility function; labor services for recruiting do not.Thus it is consumption and not output that matters for welfare. The number of help-wanted advertisements is related to consumption by q ( x ( t )) · v ( t ) = y ( t ) = c ( t ) + ρ · v ( t ) Therefore, the desired level of consumption determines the number of advertisements: v ( t ) = c ( t )/[ q ( x ( t )) − ρ ] . Hence, consuming one unit of services requires to purchase 1 + ρ · v ( t )/ c ( t ) = + τ ( x ( t )) units of services where τ ( x ( t )) = ρq ( x ( t )) − ρ . The function τ is positive and strictly increasing for all x ∈ [ , x m ) where x m > ρ = q ( x m ) . Furthermore, lim x → x m τ ( x ) = + ∞ . The elasticity of τ is η · ( + τ ( x )) .We characterize the efficient market tightness x ∗ , which maximizes consumption given thematching frictions. In equilibrium,(2) c ( t ) = y ( t ) + τ ( x ( t )) = f ( x ( t )) + τ ( x ( t )) · k . Since 1 /( + τ ( x )) = − ρ / q ( x ) and q ( x ) = f ( x )/ x , we obtain(3) c ( t ) = [ f ( x ( t )) − ρ · x ( t )] · k . This equation says that ρ · x ( t ) · k = ρ · v ( t ) units of services are dissipated in matching frictions.As established by Michaillat and Saez (2015), the tightness that maximizes consumption, x ∗ = argmax [ f ( x ) − ρ · x ] · k , is uniquely defined by f (cid:48) ( x ∗ ) = ρ . An equivalent definition is τ ( x ∗ ) = ( − η )/ η . This definition will be useful when we study the Phillips curve arising from costly priceadjustment in section 5. The efficient tightness is the tightness underlying the condition of Hosios(1990) for efficiency in a matching model.The market can be in three regimes. The market is slack if a marginal increase in tightnessincreases consumption, tight if a marginal increase in tightness decreases consumption, andefficient if a marginal increase in tightness has no effect on consumption. Equivalently, themarket is slack if x ( t ) < x ∗ , efficient if x ( t ) = x ∗ , and tight if x ( t ) > x ∗ . If tightness is efficient on This definition of consumption is different from that in national accounts, where y ( t ) would be called consump-tion, but defining consumption as output net of recruiting costs is common in the matching literature (for example,Gertler and Trigari 2009). ecruitingLabor services M a r k e t t i gh t ne ss x Unsold capacityConsumption Output Capacity x* Efficient tightness x m Figure 2.
Capacity, output, unsold capacity, and consumptionaverage, then business cycles are a succession of slack and tight episodes.Figure 2 summarizes the relation between market tightness and different quantities. Capacity, k , is a vertical line, independent of tightness. Output, y = f ( x ) · k , is increasing in tightnessas it is easier to sell services when tightness is high. Consumption, c = f ( x ) · k /[ + τ ( x )] = [ f ( x ) − ρ · x ] · k , first increases and then decreases in tightness. At the efficient tightness, theconsumption curve is vertical. The difference between output and consumption are recruitingservices, ρ · v = ρ · k · x . The difference between capacity and output is idle capacity, ( − f ( x )) · k . Households spend part of their income on labor services and save part of it as money and bonds.The law of motion of the representative household’s assets is (cid:219) B ( t ) + (cid:219) M ( t ) = p ( t ) · f ( x ( t )) · k − p ( t ) · [ + τ ( x ( t ))] · c ( t ) + i ( t ) · B ( t ) + S ( t ) . Here, M ( t ) are money balances, B ( t ) are bond holdings, p ( t ) is the price of services, [ + τ ( x ( t ))] · c ( t ) is the quantity of services purchased, f ( x ( t )) · k is the quantity of services sold, and S ( t ) islump-sum transfer of seignorage revenue from the government. Let A ( t ) = M ( t ) + B ( t ) denotenominal financial wealth at time t . The law of motion can be rewritten as (cid:219) A ( t ) = p ( t ) · f ( x ( t )) · k − p ( t ) · [ + τ ( x ( t ))] · c ( t ) − i ( t ) · M ( t ) + i ( t ) · A ( t ) + S ( t ) . a ( t ) = A ( t )/ p ( t ) denote real financial wealth at time t and s ( t ) = S ( t )/ p ( t ) real transfer ofseignorage. Since (cid:219) a ( t )/ a ( t ) = (cid:219) A ( t )/ A ( t ) − π ( t ) , we have (cid:219) a ( t ) = (cid:0) (cid:219) A ( t ) − π ( t ) · A ( t ) (cid:1) / p ( t ) , and thelaw of motion can be rewritten as(4) (cid:219) a ( t ) = f ( x ( t )) · k − [ + τ ( x ( t ))] · c ( t ) − i ( t ) · m ( t ) + r ( t ) · a ( t ) + s ( t ) , where r ( t ) ≡ i ( t ) − π ( t ) is the real interest rate at time t . This flow budget constraint is standardbut for two differences arising from the presence of matching frictions on the labor market. First,income k is discounted by a factor f ( x ( t )) ≤ f ( x ( t )) of k is actually sold.Second, consumption c ( t ) has a price wedge 1 + τ ( x ( t )) ≥ + τ ( x ( t )) units of services.Households experience utility from consuming labor services and holding real money bal-ances and real wealth. Their instantaneous utility function is u ( c ( t ) , m ( t ) , a ( t )) , where u is strictlyincreasing in its three arguments, strictly concave, and twice differentiable. The assumptionsthat real money balances and real wealth enter the utility function are critical to obtain a nonde-generate IS-LM system, and obtain permanent liquidity traps. The utility function of a householdat time 0 is the discounted sum of instantaneous utilities(5) ∫ + ∞ e − δ · t · u ( c ( t ) , m ( t ) , a ( t )) dt , where δ > [ x ( t )] + ∞ t = denotes the continuous-timepath of variable x ( t ) . Definition 1.
The representative household’s problem is to choose paths for consumption, realmoney balances, and real wealth [ c ( t ) , m ( t ) , a ( t )] + ∞ t = to maximize (5) subject to (4) , taking as giveninitial real wealth a ( ) = and the paths for market tightness, nominal interest rate, inflation,and seignorage [ x ( t ) , i ( t ) , π ( t ) , s ( t )] + ∞ t = . Concretely, the model can be seen as the Sidrauski model with two additions. First, real wealth a ( t ) enters the utility function. Second, matching frictions lower labor income by a factor f ( x ( t )) and increase the effective price of consumption by a factor 1 + τ ( x ( t )) . Because x ( t ) is taken asgiven by households, the model can be solved exactly as the original Sidrauski model. To solvethe household’s problem, we set up the current-value Hamiltonian: H ( t , c ( t ) , m ( t ) , a ( t )) = u ( c ( t ) , m ( t ) , a ( t )) + λ ( t ) · { f ( x ( t )) · k − [ + τ ( x ( t ))] · c ( t ) − i ( t ) · m ( t ) + r ( t ) · a ( t ) + s ( t )} c ( t ) and m ( t ) , state variable a ( t ) , and current-value costate variable λ ( t ) .Throughout we use subscripts to denote partial derivatives. The necessary conditions for aninterior solution to this maximization problem are H c ( t , c ( t ) , m ( t ) , a ( t )) = H m ( t , c ( t ) , m ( t ) , a ( t )) = H a ( t , c ( t ) , m ( t ) , a ( t )) = δ · λ ( t ) − (cid:219) λ ( t ) , and the transversality condition lim t → + ∞ e − δ · t · λ ( t ) · a ( t ) =
0. Given that u is concave in ( c , m , a ) and that H is the sum of u and a linear function of ( c , m , a ) , H is concave in ( c , m , a ) and theseconditions are also sufficient.These three conditions imply that u c ( c ( t ) , m ( t ) , a ( t )) = λ ( t ) · [ + τ ( x ( t ))] (6) u m ( c ( t ) , m ( t ) , a ( t )) = λ ( t ) · i ( t ) (7) u a ( c ( t ) , m ( t ) , a ( t )) = [ δ − r ( t )] · λ ( t ) − (cid:219) λ ( t ) . (8)Equations (6) and (7) imply that the marginal utilities from consumption and real money balancessatisfy(9) u m ( c ( t ) , m ( t ) , a ( t )) = i ( t ) + τ ( x ( t )) · u c ( c ( t ) , m ( t ) , a ( t )) . In steady state, this equation yields the LM curve. It represents a demand for money. The demandfor real money is declining with i ( t ) because i ( t ) is the implicit price of holding money payingzero nominal interest instead of bonds paying a nominal interest rate i ( t ) .Equations (6) and (8) imply that the marginal utilities from consumption and real wealthsatisfy(10) [ + τ ( x ( t ))] · u a ( c ( t ) , m ( t ) , a ( t )) u c ( c ( t ) , m ( t ) , a ( t )) + [ r ( t ) − δ ] = − (cid:219) λ ( t ) λ ( t ) , where (cid:219) λ ( t )/ λ ( t ) can be expressed as a function of c ( t ) , m ( t ) , a ( t ) , x ( t ) , and their derivativesusing (6). This is the consumption Euler equation. In steady state, this equation yields the IScurve. It represents a demand for saving in part from intertemporal consumption-smoothingconsiderations and in part from the utility provided by wealth. If there are no matching costs ( ρ = τ ( x ) =
0) and if the utility only depends on consumption .4. Equilibrium with constant inflation We now define and characterize the equilibrium with constant inflation.
Definition 2.
An equilibrium with constant inflation π consists of paths for market tightness,consumption, real money balances, money supply, real wealth, nominal interest rate, and pricelevel, [ x ( t ) , c ( t ) , m ( t ) , M ( t ) , a ( t ) , i ( t ) , p ( t )] + ∞ t = , such that the following conditions hold: (i) [ c ( t ) , m ( t ) , a ( t )] + ∞ t = solve the representative household’s problem; (ii) monetary policy determines [ M ( t )] + ∞ t = ; (iii) themoney market clears; (iv) the bond market clears; (v) actual tightness on the market for servicesequals the tightness taken as given by households for their optimization problem; and (vi) [ p ( t )] + ∞ t = is a continuous function of time t satisfying the differential equation (cid:219) p ( t ) = π · p ( t ) with initialcondition p ( ) = . Conditions (i)–(v) are standard equilibrium conditions in a Walrasian economy. They im-pose that households optimize taking as given prices and trading probabilities, and that tradingprobabilities are realized. These would be the equilibrium conditions in the Sidrauski model.Conditions (i)–(v) generate six independent equations. Since the equilibrium consists of sevenvariables, the equilibrium definition is incomplete with these conditions only. This incomplete-ness arises from the presence of matching frictions on the market for labor services, whichadds one aggregate variable—the market tightness. We therefore need a pricing mechanism tocomplete the equilibrium definition. It is common in the matching literature to use bargainingas a pricing mechanism. Here, we choose instead a pricing mechanism that generates constantinflation. Namely, we impose that the price process is exogenous and grows at constant inflationrate π . (The initial condition p ( ) = π =
0, the price is constant over time. This criterion seemsappropriate to describe the short run in the United States because inflation has been very sluggishthere since the 1990s.
Proposition 1.
An equilibrium with constant inflation π consists of paths of market tightness,consumption, real money balances, money supply, real wealth, nominal interest rate, and pricelevel, [ x ( t ) , c ( t ) , m ( t ) , M ( t ) , a ( t ) , i ( t ) , p ( t )] + ∞ t = , that satisfy the following seven conditions: (i) equa-tion (9) holds; (ii) equation (10) holds; (iii) [ M ( t )] + ∞ t = is determined by monetary policy; (iv) m ( t ) = M ( t )/ p ( t ) ; (v) a ( t ) = ; (vi) equation (3) holds; and (vii) (cid:219) p ( t ) = π · p ( t ) with p ( ) = . ( u a = u m = [ r ( t ) − δ ] · ϵ = (cid:219) c ( t )/ c ( t ) , where ϵ ≡ − u (cid:48) ( c )/[ c · u (cid:48)(cid:48) ( c )] is the intertemporal elasticity of substitution. In a Walrasian market, agents behave under the assumption that they can buy or sell any quantity at the postedprice; that is, they take as given a trading probability of 1. The equilibrium requirement that supply equals demandensure that agents can actually buy or sell the quantity they desire in equilibrium, ensuring that the trading probabilityof 1 is realized in equilibrium. a ( t ) =
0, and the condition that actual tightnessequals posted tightness yields (3).
3. Properties of the equilibrium
In this section we analyze the equilibrium with constant inflation. We represent the steady stateof the equilibrium with an IS curve and a LM curve depicted in a (consumption, interest rate)plane, and an AD curve and an AS curve depicted in a (consumption, market tightness) plane.This graphical representation is useful to analyze the comparative static effects of shocks andpolicies in section 4. We also study the transitional dynamics of the equilibrium.To obtain closed-form expressions for the curves, we assume that the utility function isseparable in consumption, real money, and real wealth: u ( c , m , a ) = ϵϵ − · (cid:16) c ϵ − ϵ − (cid:17) + ϕ ( m ) + ω ( a ) . (11)The curvature of utility over consumption is measured by ϵ ≥
1. The function ϕ is strictly concaveand strictly increasing on [ , m ∗ ] and constant on [ m ∗ , + ∞) . The quantity m ∗ ∈ ( , ∞) is a blisspoint in real money balances; the bliss point is required to obtain liquidity traps. The function ω is strictly concave and strictly increasing on (−∞ , + ∞) . As wealth is zero in aggregate, thekey parameter is the marginal utility of wealth at zero, ω (cid:48) ( ) . We assume that ω (cid:48) ( ) ∈ ( , + ∞) ; apositive marginal utility of wealth is also required to obtain liquidity traps. The functions ϕ and ω are depicted in figure 3. We define the IS, LM, AD, and AS curves that we use to represent the steady state.
Definition 3.
The LM curve c LM is a function of nominal interest rate, market tightness, andreal money balances defined by c LM ( i , x , m ) = (cid:20) i [ + τ ( x )] · ϕ (cid:48) ( m ) (cid:21) ϵ for all i ∈ [ , + ∞) , all x ∈ [ , x m ] , and all m ∈ [ , m ∗ ) . When real money balances are above themoney bliss point ( m ≥ m ∗ ), the LM curve determines a unique nominal interest rate: i LM ( x , m ) = for all x ∈ [ , x m ] and all m ∈ [ m ∗ , + ∞) . In this situation the economy is in a liquidity trap. eal money m U t ili t y m* Money bliss point ( m ) A. Utility over real money
Real wealth a Zero aggregate wealth U t ili t y ! ( a ) B. Utility over real wealth
Figure 3.
Utility functions over money and wealthThe LM curve is the amount of consumption that solves equation (9). The LM curve is definedseparately for m below and above the bliss point because when m is above the bliss point, ϕ (cid:48) ( m ) = i = Definition 4.
The IS curve c IS is a function of nominal interest rate, market tightness, andinflation defined by c IS ( i , x , π ) = (cid:20) δ + π − i [ + τ ( x )] · ω (cid:48) ( ) (cid:21) ϵ for all i ∈ [ , δ + π ] , all x ∈ [ , x m ] , and all π ∈ [− δ , + ∞) . If marginal utility of wealth is zero( ω (cid:48) ( ) = ), the IS curve determines a unique interest rate: i IS ( x , π ) = π + δ for all x ∈ [ , x m ] and all π ∈ [− δ , + ∞) . The IS curve is the amount of consumption that solves equation (10) when (cid:219) λ ( t ) =
0. Althoughthe IS curve is expressed with inflation and nominal interest rate, it only depends on the realinterest rate, r = i − π . The IS curve is defined separately when the marginal utility of wealthis positive or zero because when the marginal utility of wealth is zero, (10) is degenerate andimposes r = δ .The properties of the IS and LM curves are illustrated in figure 4. First, panel A showsthat the LM curve is upward sloping in a ( c , i ) plane. This property follows the standard logic.Money does not pay interests; therefore, demand for real money is decreasing with i as a higher i increases the opportunity cost of holding money. Demand for real money is increasing in c as ahigher c reduces the marginal utility of consumption, which makes real money more attractiverelative to consumption. Given that real money balances are constant, an increase in i requires The linear IS and LM curves in figure 4 correspond to the case with log utility over consumption ( ϵ =
14n increase in c to maintain equilibrium. Through the same logic, an increase in real moneybalances shifts the LM curve out, as illustrated in panel C.Second, panel A shows that the IS curve is downward sloping in a ( c , i ) plane. The intuition isthe following. For a given inflation, a higher i leads to a higher r and a higher marginal value ofsavings through bonds via the wealth effect r · ω (cid:48) ( ) , which makes holding wealth more attractive.Since wealth is zero in equilibrium, c must decline for households to be indifferent betweensaving and consumption. This logic also implies that an increase in inflation, which reduces r for a given i , shifts the IS curve out, as showed in panel D. By the same logic, a decrease in themarginal utility of wealth shifts the IS curve out, as showed in panel E. An increase in the discountrate has the same effect, as showed in panel F.Third, the IS and LM curves shift outward when market tightness decreases, as illustrated inpanel B. The logic is that a lower tightness reduces the effective price of labor services, ( + τ ( x ))· p ,which makes consumption of labor services more desirable relative to holding bonds or money.However, the nominal interest rate defined by the intersection of the IS and LM curves doesnot depend on tightness: the IS and LM curves shift by commensurate amounts such that theequilibrium interest rate remains the same.Fourth, the LM curve prevents the nominal interest rate from falling below zero because themarginal utility of money ϕ (cid:48) ( m ) is nonnegative. If the nominal interest rate were negative, moneywould strictly dominate bonds. When real money is at or above the bliss point m ∗ , the LM curvebecomes horizontal at i =
0, as illustrated in panel A of figure 5. Real money balances do notaffect the LM curve any more. This situation of liquidity trap has important implications to whichwe will come back.Fifth, without utility of wealth, the IS curve becomes horizontal at i = δ + π as depictedin panel B of figure 5. The intuition is well known: steady-state consumption is constant sohouseholds hold bonds only if the return on bonds, r = i − π , equals the subjective discount rate, δ . With utility of wealth, r < δ as households also experience utility from wealth holding.The equilibrium interest rate is given by the intersection of the IS and LM curves. The equality c IS ( i , x , π ) = c LM ( i , x , m ) implies that the equilibrium nominal interest rate is i = ϕ (cid:48) ( m ) ϕ (cid:48) ( m ) + ω (cid:48) ( ) · ( δ + π ) . (12)At that interest rate households are indifferent between money and bonds. The equilibrium realinterest rate is r = ϕ (cid:48) ( m ) ϕ (cid:48) ( m ) + ω (cid:48) ( ) · δ − ω (cid:48) ( ) ϕ (cid:48) ( m ) + ω (cid:48) ( ) · π . c0 0 ⇡ + i c AD ( x, ⇡, m ) c IS ( i, x, ⇡ ) c LM ( i, x, m ) A. Equilibrium i a =i b ci c a c b B. Decrease in market tightness cii a c a c b i b C. Increase in real money balances cii a c a c b i b D. Increase in inflation cii a c a c b i b E. Decrease in marginal utility of wealth cii a c a c b i b F. Increase in discount rate
Figure 4.
IS and LM curves in ( c , i ) plane16 IS ( i, x, ⇡ ) ci c AD ( x, ⇡, m ) i LM ( x, m ) A. Zero marginal utility of money ⇡ + c AD ( x, ⇡, m ) i IS ( x, ⇡ ) c LM ( i, x, m ) ci B. Zero marginal utility of wealth
Figure 5.
IS and LM curves in special casesNext, we construct the AD curve by plugging (12) into the LM curve:
Definition 5.
The AD curve c AD is a function of market tightness, inflation, and real moneybalances defined by (13) c AD ( x , π , m ) = (cid:20) δ + π [ + τ ( x )] · ( ϕ (cid:48) ( m ) + ω (cid:48) ( )) (cid:21) ϵ for all x ∈ [ , x m ] , all π ∈ [− δ , + ∞) , and all m ∈ [ , ∞) . The AD curve represents the consumption level obtained at the intersection of the IS andLM curves. The AD curve is downward sloping in a ( c , x ) plane, as illustrated in panel A offigure 8. The logic for this property is displayed in panel B of figure 4, where c a = c AD ( x a , π , m ) , c b = c AD ( x b , π , m ) with x a > x b , and clearly c a < c b . In fact, all the properties of the AD curvefollow from the mechanisms illustrated in figure 4 and discussed above. For instance, the ADcurve shifts out after an increase in the discount rate, an increase in the inflation rate, or adecrease in the marginal utility of wealth, as these changes shift the IS curve out. The AD curvealso shifts out after an increase in real money balances, as this change shifts the LM curve out.Last, we define the AS curve: Definition 6.
The AS curve is a function of market tightness defined by c AS ( x ) = [ f ( x ) − ρ · x ] · k for all x ∈ [ , x m ] . The AS curve is the consumption level arising from the matching process on the labor market,17 ˙ ˙ = F ( ) Figure 6.
Phase diagram of the equilibrium with constant inflationplotted in figure 2. The AS curve is showed in panel A of figure 8. An increase in capacity shiftsthe AS curve out.
The steady state of the equilibrium with constant inflation is as follows:
Proposition 2.
The steady state of the equilibrium with constant inflation π consists of mar-ket tightness, consumption, real money balances, level of money supply, growth rate of moneysupply, and nominal interest rate, ( x , c , m , M ( ) , (cid:219) M / M , i ) , such that c LM ( i , x , m ) = c IS ( i , x , π ) , c AD ( x , π , m ) = c AS ( x ) , c = c AS ( x ) , M ( ) is set by monetary policy, (cid:219) M / M = π ; and m = M ( )/ p ( ) = M ( ) . The steady state consists of 6 variables determined by 6 conditions. In steady state, the pricegrows at a constant, exogenous inflation rate π . The money supply, M ( t ) , must also grows at rate π but monetary policy does not control π . Hence, changing the growth rate of M ( t ) is not withinthe scope of the analysis under constant inflation. Since the price level is unaffected by monetarypolicy, monetary policy controls real money balances by controlling the level of money supply.When m is large enough ( m > m ∗ ), the steady state is in a liquidity trap, with a nominalinterest rate at 0. This steady state is unique. Hence, the model easily accommodates permanentliquidity traps. This is a desirable feature of the model since low-inflation economies seem proneto enter long liquidity traps after a large negative shock: the ZLB episode that started in 1995 in18apan lasted for more than twenty years without sustained deflation; the ZLB episode that startedin 2009 in the euro area lasted for more than 10 years without sustained deflation either; the sameis true of the ZLB episode that occurred in the United States between 2008 and 2015. Here we describe the transitional dynamics toward the steady state. The dynamical system describ-ing the equilibrium is characterized in proposition 1. We focus here on one single endogenousvariable: the costate variable λ ( t ) . All the variables can be recovered from λ ( t ) .In equilibrium, wealth is zero so the law of motion for the costate variable from equation (8)is ω (cid:48) ( ) = ( δ + π − i ( t )) · λ ( t ) − (cid:219) λ ( t ) . Both money supply and price grow at a constant rate π so real money balances are constant: m ( t ) = M ( )/ p ( ) = m . Hence, equation (7) implies that i ( t ) · λ ( t ) = ϕ (cid:48) ( m ) , and the law of motion of the costate variable in equilibrium is (cid:219) λ ( t ) = ( δ + π ) · λ ( t ) − ω (cid:48) ( ) − ϕ (cid:48) ( m ) ≡ F ( λ ( t )) . The steady-state value of the costate variable satisfies F ( λ ) = λ = ( ω (cid:48) ( ) + ϕ (cid:48) ( m ))/( δ + π ) .The nature of the dynamical system is given by the sign of F (cid:48) ( λ ) . Since F (cid:48) ( λ ) = δ + π >
0, thesystem is a source. We represent the phase diagram for the system in figure 6.As there is no state variable, the system jumps from one steady state to the other in responseto an unexpected shock—the transitional dynamics are immediate. This is illustrated in panel Aof figure 7 where the equilibrium jumps from λ a to λ b at time t when an unexpected shock occur.The values λ a and λ b are the steady-state values of λ for the parameters values before and aftertime t . Accordingly, comparative-statics analysis is sufficient to completely describe the behaviorof the system after unexpected shocks.The transitional dynamics are a bit different in response to an expected shock. This is illus-trated in panel B of figure 7. An announcement is made at time t that a shock changing thesteady-state value of λ from λ a to λ b will occur at time t . A key property of the system is thatabsent new information, λ is a continuous variable of time so λ can only jump at time t but notat time t . Assume that λ a > λ b . Then λ jumps down at time t . The amplitude of the jump issuch that at time t , λ = λ b . Between t and t , λ falls because (cid:219) λ = F ( λ ) <
0. We conclude that attime t , λ jumps down part of the way toward its steady-state value, and that it keeps on fallingslowly toward its new steady-state value until the expected shock occurs. The implication is thateven with expected shocks, comparative statics give the correct sign of the adjustments occurringwhen the announcement of the shock is made and in the long run.19 b a ˙ A. Unexpected shock b a ˙ B. Expected shock
Figure 7.
Response of the equilibrium with constant inflation to unexpected and expected shocks
4. Shocks and policies
In this section we use comparative statics to describe how the economy responds to aggregatedemand and supply shocks and to various monetary and fiscal policies. As discussed in theprevious section, comparative statics completely describe the response of the economy to anunexpected permanent shock because the equilibrium jumps from one steady state to another inresponse to such a shock. The comparative statics are summarized in table 1 and illustrated infigure 8.
We first analyze aggregate demand shocks. We parameterize an increase in aggregate demandby an increase in the subjective discount rate or a decrease in the marginal utility of wealth. Apositive aggregate demand shock shifts the IS curve out, as depicted in panels E and F of figure 4,and it therefore raises interest rates. Note that interest rates are independent of tightness, asillustrated in panel B of figure 4, so the general-equilibrium response of interest rates to theaggregate demand shock is the same as the partial-equilibrium response depicted in panels Eand F of figure 4.Since the IS curve shifts out, the AD curve also shifts out, as depicted in panel B of figure 8.Hence, the increase in aggregate demand leads to increases in market tightness and output. Sincetightness is higher, the unemployment rate falls. Consumption increases if the labor market isslack and decreases if the labor market is tight. If the labor market is efficient, the aggregatedemand shock has no first-order effect on consumption.20 m c, y, ky kxx c y k c AS ( x ) c AD ( x, ⇡, m ) A. Steady-state equilibrium x c, y, ky kx b c b y b kc a y a x a c AD ( x, ⇡, m ) c AS ( x ) B. Increase in aggregate demand x c, y, ky kx b c b y b c a y a x a k b k a c AS ( x ) c AD ( x, ⇡, m ) C. Increase in capacity x c, y, ky kx a c a y a c b y b x b k c AS ( x ) c AD ( x, ⇡, m ) D. Increase in mismatch
Figure 8.
Steady-state equilibrium and aggregate demand and supply shocks in a ( c , x ) plane Next we analyze aggregate supply shocks. We consider two types of shocks: a shock to the pro-duction capacity and a mismatch shock.An increase in capacity is illustrated in panel C of figure 8. This increase shifts out the ASand output curves, while the AD curve is unchanged. Hence, consumption increases, tightnessdecreases, and the unemployment rate increases. We can show that output increases. Interestrates do not change.Following Michaillat and Saez (2015), we parameterize an increase in mismatch as a decreasein matching efficacy along with a commensurate decrease in matching costs: h ( k , v ) becomes σ · h ( k , v ) and ρ becomes σ · ρ with σ <
1. The efficient tightness and the function τ are notaffected by mismatch. Panel D of figure 8 illustrates an increase in mismatch. The AD curvedoes not change, but the AS and output curves shift inward. As a result, consumption decreases,tightness increases, and output decreases. We can show that the unemployment rate increases.21 able 1. Comparative statics: aggregate shocks and policies with constant inflation
Effect on:Tightness Consumption Output Unemployment rate Interest ratesIncrease in: x c y − f ( x ) i , r Aggregate demand + + / 0 / − + − + Capacity − + + + + − − + + + / 0 / − + − − - in liquidity trap 0 0 0 0 0Helicopter money + + / 0 / − + − ?Wealth tax + + / 0 / − + − + Government spending + + / 0 / − + − + / /− ” indicates that consumption increases when thelabor market is slack, does not change when the labor market is efficient, and decreases when the labor market istight. In the column on interest rates, “?” indicates that the response of the interest rate can be positive or negativedepending on the utility functions ω and ϕ . Given that inflation is constant, both nominal and real interest ratemove in the same way. In the row on government purchase, consumption means total consumption—personal plusgovernment consumption. Private consumption always falls when government consumption increases. Interest rates do not change.The comparative statics are the same in a liquidity trap and away from it because the ADand AS curves retain the same properties in a trap. This property distinguishes our model fromNew Keynesian models, in which aggregate supply shocks have paradoxical effects in liquiditytraps. In these models, a negative aggregate supply shock is contractionary in normal times butexpansionary in a liquidity trap (Eggertsson 2010; Eggertsson and Krugman 2012), even withwealth in the utility function (Michaillat and Saez 2019c). Whether these paradoxical effects appearin the data is debated: using a variety of empirical tests, Wieland (2019) rejects the predictionthat negative aggregate supply shocks are expansionary in a liquidity trap. The predictions of ourmodel are consistent with Wieland’s findings.
The only lever that monetary policy chooses is the level of money supply, M ( ) . A change in M ( ) leads to a change in real money balances. Monetary policy cannot change the growth rate of M ( t ) ,which must satisfy the steady-state requirement that (cid:219) M ( t )/ M ( t ) = π . We study the comparativestatic effects of an increase in real money balances.Away from a liquidity trap, an increase in real money balances shifts out the LM curve,22s showed in panel C of figure 4, and hence shifts out the AD curve, as showed in panel B offigure 8. Higher money supply therefore leads to lower interest rates, higher tightness, lowerunemployment rate, and higher output. The effect on consumption depends on the state of thelabor market.As long as the nominal interest rate is positive, monetary policy can control the AD curve andthus fully accommodate shocks. Suppose that the economy starts with tightness at its efficientlevel, which maximizes consumption, and that the government wants to use monetary policy tokeep tightness at this level. A negative aggregate demand shock lowers tightness and requiresan increase in real money balances, and conversely, a positive aggregate demand shock raisestightness and requires a decrease in real money balances. Here monetary policy absorbs aggregatedemand shocks, thus preventing inefficient economic fluctuations. A positive aggregate supplyshock, either an increase in capacity or a decrease in mismatch, lowers tightness and requiresan increase in real money balances, and conversely, a negative aggregate supply shock raisestightness and requires a decrease in real money balances. Here monetary policy exacerbates theeffect of aggregate supply shocks on output to achieve efficient economic fluctuations.In a liquidity trap, monetary policy cannot accommodate shocks anymore because real moneybalances do not influence the LM curve and thus cannot control the AD curve. This situationis illustrated in panel A of figure 5. Monetary policy becomes ineffective. Of course, monetarypolicy could still be effective if it could change inflation. We know that higher inflation stimulatesthe IS curve and thus the AD curve, even in a liquidity trap, as showed in panel D of figure 4. Butwe assume here that monetary policy has no effect on inflation, consistent with the empiricalevidence presented by Christiano, Eichenbaum, and Evans (1999). Conventional monetary policy is not effective in a liquidity trap. We now present other policiesthat remain effective in this situation.We start by analyzing a helicopter drop of money, first discussed by Friedman (1969). Moneycomes from two sources: a quantity M b ( t ) = − B ( t ) of money is issued by buying bonds throughopen market operations, and a quantity M h ( t ) of money is printed and given directly to householdsthrough a helicopter drop. Total money supply is M ( t ) = M b ( t ) + M h ( t ) . Real money balancesare m b ( t ) = M b ( t )/ p ( t ) and m h ( t ) = M h ( t )/ p ( t ) and m ( t ) = M ( t )/ p ( t ) . Real wealth is no longerzero because helicopter money contributes to real wealth: real wealth is a ( t ) = ( B ( t ) + M b ( t ) + M h ( t ))/ p ( t ) = m h ( t ) .With helicopter money, our analysis carries over by adjusting the marginal utility of wealth23 i c a c b i a i b A. Helicopter drop of money cii a c a c b i b B. Tax on wealth
Figure 9.
Unconventional policies in the IS-LM diagramfrom ω (cid:48) ( ) to ω (cid:48) ( m h ) . The IS curve now depends on helicopter money: c IS ( i , x , π , m h ) = (cid:20) δ + π − i [ + τ ( x )] · ω (cid:48) ( m h ) (cid:21) ϵ . Since the function ω is concave, an increase in helicopter money shifts the IS curve outward in a ( c , i ) plane, as showed in panel A of figure 9. It also shifts the LM curve outward at it raises realmoney balances. The AD curve depends on both total and helicopter money: c AD ( x , π , m , m h ) = (cid:20) δ + π [ + τ ( x )] · (cid:0) ϕ (cid:48) ( m ) + ω (cid:48) ( m h ) (cid:1) (cid:21) ϵ . A helicopter drop of money shifts the AD curve out in a ( c , x ) plane, as in panel B of figure 8.Although open-market money cannot stimulate the AD curve in a liquidity trap, helicoptermoney stimulates the AD curve even in a liquidity trap. Helicopter money is effective in a liquiditytrap because it stimulates both the LM and the IS curve, and the IS channel is immune to theliquidity trap. In contrast, open-market money only stimulates the LM curve, and the LM channeldoes not operate in a liquidity trap.One drawback of a helicopter drop of money is that it is harder to reverse than open marketoperations. Effectively, reversing a helicopter drop of money requires to take away money fromhouseholds with no compensation—taxing money held by households and destroying it. The efficacy of a helicopter drop of money requires concave utility of wealth. With linear utility of wealth, ω (cid:48) ( m h ) is constant and helicopter money does not shift the IS curve. In that case, a helicopter drop of money is ineffective. .5. Tax on wealth Another way to stimulate aggregate demand in a liquidity trap is to tax wealth at rate τ a ( t ) . Thewealth tax applies to the entire wealth, bond holdings plus money balances. The tax raises norevenue as the aggregate wealth is zero. But the tax changes the law of motion of the consumer’swealth and the consumption Euler equation. The law of motion becomes (cid:219) a ( t ) = f ( x ( t )) · k − [ + τ ( x ( t ))] · c ( t ) − i ( t ) · m ( t ) + ( r ( t ) − τ a ( t )) · a ( t ) + s ( t ) . Therefore, the Euler equation becomes [ + τ ( x ( t ))] · u a ( c ( t ) , m ( t ) , a ( t )) u c ( c ( t ) , m ( t ) , a ( t )) + ( r ( t ) − τ a ( t ) − δ ) = − (cid:219) λ ( t ) λ ( t ) , and the IS curve admits a new expression: c IS ( i , x , π , τ a ) = (cid:20) δ + τ a + π − i [ + τ ( x )] · ω (cid:48) ( ) (cid:21) ϵ . An increase in the wealth tax shifts the IS curve outward in a ( c , i ) plane, as showed in panel B offigure 9. The LM curve remains the same. The AD curve is now a function of the wealth tax: c AD ( x , π , m , τ a ) = (cid:20) δ + τ a + π [ + τ ( x )] · ( ϕ (cid:48) ( m ) + ω (cid:48) ( )) (cid:21) ϵ . An increase in the wealth tax shifts the AD curve outward in a ( c , x ) plane, as in panel B of figure 8.Since the wealth tax acts on the IS curve and not the LM curve, the wealth tax is effective in aliquidity trap. The intuition for the effectiveness of the tax is simple: taxing wealth makes holdingwealth more costly and hence less desirable, hereby stimulating current consumption. The last policy that we consider is the purchase of д ( t ) units of services by the government.We begin by assuming that government purchases are financed by a lump-sum tax τ ( t ) . Thegovernment’s budget constraint imposes that p ( t ) · д ( t ) = τ ( t ) . We assume that governmentspending enters separately into households’ utility function such that д ( t ) does not affect theconsumption and saving choices of the households. Accordingly, the IS and LM curves remain25he same, and д only enters in the AD curve:(14) c AD ( x , π , m , д ) = (cid:20) δ + π [ + τ ( x )] · ( ϕ (cid:48) ( m ) + ω (cid:48) ( )) (cid:21) ϵ + д + τ ( x ) . We abuse notation and keep the labels c AD and c AS for the AD and AS curves, even though c ispersonal consumption whereas the AD and AS curves measure total consumption—personal plusgovernment consumption. An increase in government spending shifts the AD curve outward, asshowed in panel B of figure 8.Government spending remains effective in a liquidity trap because they do not rely on the LMcurve. However, government spending is not especially effective in a liquidity trap. What mattersfor the effectiveness of government spending are the slopes of the AD and AS curves. In that, ourmodel sharply differs from the New Keynesian model, which predicts that government multipliersare much larger in a liquidity trap (Christiano, Eichenbaum, and Rebelo 2011; Woodford 2011).Following the logic described in Michaillat (2014), Michaillat and Saez (2019b), and Ghassibeand Zanetti (2019), however, the government multiplier is higher when the economy is slack thanwhen it is tight. What matters for the size of the multiplier in our model is the amount of slack inthe economy and not the liquidity trap. Our findings are consistent with the empirical findingthat multipliers seem higher when unemployment is higher or output is lower (Auerbach andGorodnichenko 2012; Candelon and Lieb 2013; Fazzari, Morley, and Panovska 2015). For instance,estimating regime-switching SVARs on US data, Auerbach and Gorodnichenko (2012, table 1)find that while the output multiplier is 0.6 in expansions and 1 on average, it is as high as 2.5 inrecessions.An increase in government spending leads to higher output but not always to higher totalconsumption. Following the usual logic, total consumption increases when the market is slack,decreases when the market is tight, and does not change when the market is efficient. Governmentconsumption always crowds out personal consumption. Crowding out arises because an increasein government spending shifts the AD curve outward and raises market tightness; therefore, itis more expensive for households to purchase goods: the effective price ( + τ ( x )) · p increases.Households reduce consumption because of the increase in effective price. Crowding out is partialwhen the market is slack, one-for-one when the market is efficient, and more than one-for-onewhen the market is tight.Finally, there is a simple interaction between fiscal and monetary policy. As long as monetarypolicy is able to maintain the market at efficiency, fiscal policy should follow public-financeconsiderations: the economy is always efficient so there is no reason to use government spendingfor stabilization purposes. If monetary policy cannot maintain the economy at efficiency, fiscal26olicy can play a role to stabilize the economy, in addition to public-finance considerations. Thiswould happen for instance when the economy is in a liquidity trap and monetary policy cannotstimulate aggregate demand. Michaillat and Saez (2019b) formalize this discussion and providean formula for optimal stimulus spending when unemployment is inefficient.
5. Model with Phillips curve
In the previous sections inflation was constant. Although the approximation that inflation isconstant seems useful and realistic to describe the short run, this approximation may be unsat-isfactory to describe the medium run. In the medium run, a Phillips curve likely describes thejoint dynamics of inflation and slack. In this section, we propose a version of the model that ismore appropriate to describe the medium run. This version combines directed search as in Moen(1997) with costly price adjustment as in Rotemberg (1982). In equilibrium, inflation dynamicsare described by a Phillips curve.To simplify the exposition, we specialize the utility (11) by setting ϵ = ϕ ( m ) = ln ( m ) :(15) u ( c , m , a ) = ln ( c ) + ln ( m ) + ω ( a ) . By using log utility over money, we set the money bliss point to infinity and ensure that theeconomy never enters a liquidity trap. Studying the properties of the equilibrium with Phillipscurve in a liquidity trap would be challenging: as in New Keynesian models, the analysis ofliquidity traps with a Phillips curve raises difficult issues. It is possible that these issues couldbe tackled by assuming wealth in the utility function, following the logic in Michaillat and Saez(2019c); we leave this analysis for future work.Finally, we assume that the money supply remains constant over time: M ( t ) = M for all t .Unlike in a New Keynesian where monetary policy follows an interest-rate rule, monetary policyis completely passive here. We begin by solving the representative seller’s problem when buyers direct their search towardsthe most attractive markets but adjusting prices is costly to sellers. Buyers choose the marketwhere they buy labor services based on the price, p , and tightness, x , in that market. What mattersfor buyers is the effective price they pay, p · ( + τ ( x )) . When a seller sets a price, she takes intoaccount the effect of her price on the tightness she faces, which in turn determines how muchlabor services she sells. The solution of the seller’s problem yields a Phillips curve.27s in Moen (1997), we assume that sellers post their price p ( t ) and that buyers arbitrage acrosssellers until they are indifferent across sellers. This means that search for labor services is notrandom but directed. For a given price p ( t ) , the tightness that a seller faces is given by(16) [ + τ ( x ( t ))] · p ( t ) = e ( t ) where e ( t ) is the effective price in the economy. The effective price is taken as given by buyersand sellers. This condition simply says that buyers are indifferent between all sellers. Sellers canchoose high prices and get few buyers or low prices and get many buyers. If a seller chooses aprice p ( t ) , her probability to sell therefore is F ( p ( t )) ≡ f ( x ( t )) = f (cid:18) τ − (cid:18) e ( t ) p ( t ) − (cid:19) (cid:19) . A useful result is that the derivative of F is F (cid:48) ( p ) = −( − η ) · f ( x )/( η · τ ( x ) · p ) . Absent anyprice-adjustment cost, sellers choose p ( t ) to maximize p ( t ) · f ( x ( t )) subject to (16); that is, theychoose x ( t ) to maximize f ( x ( t ))/( + τ ( x ( t ))) = f ( x ( t )) − ρ · x ( t ) ; thus, they set f (cid:48) ( x ( t )) = ρ and x ( t ) = x ∗ is efficient. This is the central efficiency result of Moen (1997).We add price-adjustment costs to the directed search setting. We follow the price-adjustmentspecification of Rotemberg (1982). Sellers incur a cost ( (cid:219) p ( t )/ p ( t )) · κ ( t )/ κ ( t ) = κ · p ( t ) · y ( t ) . This cost is quadratic in the growth rate of prices, (cid:219) p ( t )/ p ( t ) ,and scaled by the size of the economy p ( t ) · y ( t ) and a cost parameter κ . If κ =
0, prices adjust atno cost.The representative seller takes e ( t ) , κ ( t ) , and i ( t ) as given and chooses a price level p ( t ) , aprice growth rate π ( t ) , and a tightness x ( t ) to maximize the discounted sum of nominal profits(17) ∫ + ∞ e − I ( t ) · (cid:20) p ( t ) · f ( x ( t )) · k − κ ( t ) · π ( t ) (cid:21) dt , subject to (16) and to the law of motion for the price level(18) (cid:219) p ( t ) = π ( t ) · p ( t ) . The seller’s discount rate is I ( t ) = ∫ t i ( s ) ds . To solve the seller’s problem, we express x ( t ) as afunction of p ( t ) using f ( x ( t )) = F ( p ( t )) and set up the current-value Hamiltonian H ( t , π ( t ) , p ( t )) = p ( t ) · F ( p ( t )) · k − κ ( t ) · π ( t ) + µ ( t ) · π ( t ) · p ( t ) π ( t ) , state variable p ( t ) , and current-value costate variable µ ( t ) . The neces-sary conditions for an interior solution to this maximization problem are H π ( t , π ( t ) , p ( t )) = H p ( t , π ( t ) , p ( t )) = i ( t ) · µ ( t ) − (cid:219) µ ( t ) , together with the appropriate transversality condition.The first condition implies that(19) κ ( t ) p ( t ) · π ( t ) = µ ( t ) . Recall that r ( t ) = i ( t ) − π ( t ) denotes the real interest rate. The second condition implies that(20) (cid:219) µ ( t ) = r ( t ) · µ ( t ) + (cid:20) − ηη · τ ( x ( t )) − (cid:21) · f ( x ( t )) · k . In a symmetric equilibrium, κ ( t )/ p ( t ) = κ · y ( t ) = κ · f ( x ( t )) · k so the first optimalitycondition simplifies to(21) κ · f ( x ( t )) · k · π ( t ) = µ ( t ) . As the elasticity of f ( x ) is 1 − η , log-differentiating (21) with respect to time yields (cid:219) µ ( t ) µ ( t ) = ( − η ) · (cid:219) x ( t ) x ( t ) + (cid:219) π ( t ) π ( t ) Combining this equation with (20) yields(22) (cid:219) π ( t ) = (cid:20) r ( t ) − ( − η ) · (cid:219) x ( t ) x ( t ) (cid:21) · π ( t ) + κ · (cid:20) − ηη · τ ( x ( t )) − (cid:21) . This differential equation describes sellers’ optimal pricing; it underlies the Phillips curve.
Here we derive the dynamical system describing the equilibrium. The system is composed ofthree equations: a consumption Euler equation, a Phillips curve, and a law of motion for themarginal utility of money.The consumption Euler equation describes the solution to the household’s problem. It is givenby (10), but it is convenient to rewrite it as a differential equation in x . Using (15), (10) becomes ω (cid:48) ( ) · f ( x ( t )) · k + r ( t ) − δ = − (cid:219) λ ( t ) λ ( t ) . f ( x ( t )) · k = / λ ( t ) . Log-differentiating this equation with respect to time, we obtain − (cid:219) λ ( t ) λ ( t ) = ( − η ) · (cid:219) x ( t ) x ( t ) which yields the Euler equation(23) ( − η ) · (cid:219) x ( t ) x ( t ) = r ( t ) − ( δ − ω (cid:48) ( ) · f ( x ( t )) · k ) . We now turn to the Phillips curve. To ease notation, we denote the tightness gap as G ( x ( t )) = − − ηη · τ ( x ( t )) . The function G increases in x , is positive if x > x ∗ , negative if x < x ∗ , and zero if x = x ∗ . Itmeasures how far the market is from efficiency. Combining (22) with the Euler equation (23) toeliminate (cid:219) x ( t ) yields the Phillips curve(24) (cid:219) π ( t ) = [ δ − ω (cid:48) ( ) · f ( x ( t )) · k ] · π ( t ) − κ · G ( x ( t )) The two differences with the usual Phillips curve in New Keynesian models is that the tightnessgap, G ( x ( t )) , replaces the usual output gap and the effective discount rate δ − ω (cid:48) ( ) · f ( x ) · k replaces the usual discount rate δ . Using the Phillips curve, we can express inflation as thediscounted sum of future tightness gaps: π ( t ) = κ · f ( x ( t )) ∫ ∞ t G ( x ( s )) · f ( x ( s )) · e R ( t )− R ( s ) ds , with R ( t ) = ∫ t r ( s ) ds . This expression is obtained by integrating the differential equation (20)and using (21).A last equation is required to describe the dynamics of the real interest rates, r ( t ) . Thisequation is based on the dynamics of real money balances. Let ψ ( t ) = ϕ (cid:48) ( M / p ( t )) = p ( t )/ M denote the marginal utility of real money balances. Since M is fixed and p ( t ) is a state variable, ψ ( t ) is a state variable. As ψ ( t ) = p ( t )/ M , the law of motion of ψ ( t ) is(25) (cid:219) ψ ( t ) = π ( t ) · ψ ( t ) . Using (7), we find that i ( t ) = ψ ( t ) · f ( x ( t )) · k and r ( t ) = i ( t ) − π ( t ) = f ( x ( t )) · k · ψ ( t ) − π ( t ) .30ence we can rewrite the Euler equation as(26) (cid:219) x ( t ) = x ( t ) − η · [ f ( x ( t )) · k · ( ψ ( t ) + ω (cid:48) ( )) − δ − π ( t )] . The dynamical system of (25), (24), and (26) describes the behavior over time of the vector [ ψ ( t ) , π ( t ) , x ( t )] representing the equilibrium. It is a nonlinear system of differential equations. Inthis system, x ( t ) and π ( t ) are jump variables and ψ ( t ) is a state variable. Proposition 3 determinesthe properties of the dynamical system: Proposition 3.
The vector [ ψ ( t ) , π ( t ) , x ( t )] describing the equilibrium satisfies the dynamicalsystem { (25) , (24) , (26) } . The dynamic system admits a unique steady state. This steady state hasno inflation, efficient tightness, and an interest rate below the subjective discount rate: π = , x = x ∗ , and i = ψ · y ∗ = δ − ω (cid:48) ( ) · y ∗ , where y ∗ = f ( x ∗ ) · k is the efficient output level.Around the steady state, the dynamic system is a saddle, and the stable manifold is a line. Sincethe system has one state variable ( ψ ) and two jump variables ( x and π ), this property implies thatthe equilibrium is determinate. At the steady state, the stable manifold is tangent to the vector z = [ ψ / γ , , ( y ∗ · ψ − γ ) · x ∗ · κ ] , where γ = ( δ / ) · (cid:104) − (cid:112) + /( κ · δ · ( − η )) (cid:105) < . Theresponses of the equilibrium to small shocks are determined by z and summarized in table 2. Proof . In steady state, (cid:219) ψ = (cid:219) π = (cid:219) x =
0. Since ψ ( t ) >
0, (25) implies that π =
0. There is noinflation in steady state, which is not surprising because there is no money growth. Since π = G ( x ) = x = x ∗ . The market tightness is efficient in steady state. This meansthat prices always adjust in the long run to bring the economy to efficiency. The mechanismis that the price level determines real money balances and thus aggregate demand—this is theLM channel discussed in section 4. This channel operates as long as the economy is not in aliquidity trap. Last, (26) with π = ( ψ + ω (cid:48) ( )) · y ∗ = δ with y ∗ = f ( x ∗ ) · k . Thus ψ = δ / y ∗ − ω (cid:48) ( ) and i = ψ · y = δ − ω (cid:48) ( ) · y ∗ .To study the stability properties of the system around its steady state, we need to determinethe eigenvalues of the Jacobian matrix (cid:74) of the system evaluated at the steady state. Simplecomputations exploiting the fact that in steady state π = x = x ∗ , G ( x ∗ ) = G (cid:48) ( x ∗ ) = / x ∗ , and ( ψ + ω (cid:48) ( )) · f ( x ∗ ) · k − δ =
0, imply that (cid:74) = ψ ψ · y ∗ − κ · x ∗ y ∗ · x ∗ − η − x ∗ − η δ . ⇡ xx ⇤ Projection of stable manifold A. Projection in the ( x , π ) plane ⇡ Projection of stable manifold B. Projection in the ( ψ , π ) plane Figure 10.
Response of the equilibrium with Phillips curve to an unexpected shockThe characteristic polynomial of (cid:74) is P ( X ) = ( X − ψ · y ∗ ) · (cid:20) − X + δ · X + κ · ( − η ) (cid:21) , so (cid:74) admits three real eigenvalues: γ = ψ · y ∗ > γ = δ · (cid:20) + (cid:114) + κ · δ · ( − η ) (cid:21) > γ = δ · (cid:20) − (cid:114) + κ · δ · ( − η ) (cid:21) < . Therefore, the system is a saddle path around the steady state, and the stable manifold is a line.Since the system has one state variable ( ψ ) and two jump variables ( x and π ), this property impliesthat the system does not suffer from dynamic indeterminacy. Suppose that the economy is at itssteady state. In response to an unexpected and permanent shock at t =
0, both x and π jump to theintersection of the new stable line and the plane { ψ = ψ } , where ψ denotes the old steady-statevalue of ψ . The economy remains on the plane { ψ = ψ } , orthogonal to the ψ axis, right after theshock because the state variable ψ cannot jump. This intersection is unique so the response ofthe system to the shock is determinate.Finally, we compute the eigenvector z associated with the negative eigenvalue, γ . The stableline is tangent to z at the new steady state. Hence, this vector allows us to describe qualitativelythe response of the equilibrium to aggregate demand and supply shocks, and monetary policy.32he eigenvector is defined by (cid:74) z = γ z . Simple calculation shows that this eigenvector is z = (cid:20) ψγ , , ( y ∗ · ψ − γ ) · x ∗ · κ (cid:21) . Using this vector, we obtain the responses to unexpected and permanent shocks described intable 2. We now justify these responses.We begin with two preliminary observations. First, irrespective of the shock, π and x alwayskeep the same values in steady state, at π = x = x ∗ . The steady states are therefore allaligned along a line { π = , x = x ∗ } , parallel to the ψ axis. Second, as γ <
0, the coordinatesof the eigenvector z along dimensions π and x are both positive—the coordinates are 1 > ( y ∗ · ψ − γ ) · x ∗ · κ >
0. This property is depicted in panel A of figure 10. Hence, x and π always move together in response to shocks—either they both increase, or they both decrease.In response to unexpected shocks, our Phillips curve therefore predicts a positive correlationbetween market tightness and inflation, or a negative correlation between unemployment andinflation, in the spirit of the traditional Phillips curve.Next, consider a positive aggregate demand shock. For concreteness, assume that the marginalutility for wealth, ω (cid:48) ( ) , decreases. In steady state, the marginal utility for money satisfies ψ = δ / y ∗ − ω (cid:48) ( ) so a lower ω (cid:48) ( ) implies a higher ψ . Since the coordinate of the eigenvector z alongdimension ψ is ψ / γ < π and x necessarily jump up on impact. This jump is depicted inpanel B of figure 10. The responses of y and i follow because y = f ( x ) · k and i = ψ · y . Themechanism is that after the shock, prices cannot adjust immediately so the economy becomestight; unemployment is lower and output is higher than efficient. As sellers face a tight market,they increase prices—inflation is positive. As prices rise, real money balances decrease and ψ increases. This adjustment continues until the new steady state is reached. A monetary policyshock defined as a change in money supply has the same effects on all variables except i .Last, a positive aggregate supply shock has exactly the same effect on tightness and inflationas a negative aggregate demand shock. For concreteness, assume that capacity, k , increases.In steady state, the marginal utility for money satisfies ψ = δ /( f ( x ∗ ) · k ) − ω (cid:48) ( ) so a higher k implies a lower ψ , exactly like a negative aggregate demand shock. Hence, π and x necessarilyjump down on impact after a positive aggregate supply shock. The response of output is morecomplicated because y = f ( x ) · k and x jumps down whereas k jumps up. However, we canexploit the eigenvector z to prove that the jump of x is always larger than that of k . Hence, y jumps down on impact and increases during the dynamic adjustment toward its new highersteady-state value. The response of i follows because i = ψ · y . (cid:4) With directed search and costly price adjustment, prices converge slowly toward efficiency.33 able 2.
Dynamic response of the equilibrium with Phillips curve to shocks
Short-run / long-run effect on:Tightness Inflation Price Output Interest rateIncrease in: x π p y i
Aggregate demand + / 0 + / 0 0 / + + / 0 + / + Money supply + / 0 + / 0 0 / + + / 0 − / 0Aggregate supply − / 0 − / 0 0 / − − / + − / − The symbol “ X / Y ” indicates that the response of a variable to a shock is X on impact and Y in steady state. Transitionfrom impact to new steady state is monotonic for x , π , and p . An increase in aggregate demand is an increase inthe subjective discount rate or a decrease in the marginal utility of wealth. An increase in aggregate supply is anincrease in capacity or a decrease in mismatch. The mechanism is simple. If prices are too high, new markets are created with lower prices buthigher tightness. If prices are too low, new markets are created with higher prices but lowertightness. Sellers and buyers have incentives to move to these new markets because they are moreefficient so there is a larger surplus to share. Sellers are compensated for the lower price witha higher probability to sell. Buyers are compensated for the higher matching wedge by a lowerprice. Lazear (2010) finds evidence of such behavior for price and tightness in the US housingmarket.This analysis could be used to formalize the conflict between price adjustment and inflationadjustment discussed by Tobin (1993). In response to a shock, the price adjustment requires aninflation change that further destabilizes the economy, possibly making the recession worse orexacerbating the overheating. This suggests that, even if price adjustments are fairly fast, thetemporary inflation changes could amplify short-run fluctuations in tightness and output. Forinstance, after a negative aggregate demand shock inflation jumps down to allow prices to fall. Thisfall in prices increase real money balances and stimulates aggregate demand, which eventuallybrings the economy at efficiency. But a decrease in inflation has a temporary negative effect onaggregate demand and tightness. This negative effect is illustrated in panel D of figure 4, wherewe show that a decrease in inflation depresses the IS curve and thus the AD curve. The economicmechanism is that lower inflation implies higher real interest rates, which lead households towant to accumulate more wealth and hence consume less.With costly price adjustment, assuming away liquidity traps, monetary policy can accommo-date all shocks. A monetary expansion, defined as an increase in money supply, can absorb a neg-ative aggregate demand shock so that output, inflation, and tightness remain at their steady-statelevel at the time of the shock. Conversely, monetary tightening can absorb a positive aggregatedemand shock. A monetary expansion can accommodate a positive aggregate supply shock so34hat the economy jumps immediately to its new steady state with zero inflation, efficient tightness,and higher output. Conversely, monetary tightening can accommodate a negative supply shock.In all cases, monetary policy should be based on tightness rather than output as efficient outputvaries with some shocks such as supply shocks while efficient tightness does not. On the one hand, if the price-adjustment cost is infinite then inflation equals zero. This can beseen in the first-order condition (19), where π ( t ) = κ ( t ) → + ∞ . This corresponds to themodel studied in sections 2–4. On the other hand, at the limit without price-adjustment cost,sellers always select a price to maintain the tightness at its efficient level as in Moen (1997). Thiscan be seen by combining the first-order condition (19), where µ ( t ) = κ ( t ) =
0, with thefirst-order condition (20), where τ ( x ( t )) = ( − η )/ η if µ ( t ) =
0; that is, x ( t ) = x ∗ when κ ( t ) = m ∗ < + ∞ , a large negative aggregate demandshock could bring the economy into a liquidity trap, whereby real money balances are abovethe bliss point but tightness is still below its efficient level. In that case, the directed searchmechanism implies that sellers want to lower their price to increase the tightness they face, eventhough this does not increase tightness in general equilibrium. The economy may fall into aninstantaneous deflationary spiral with no steady-state equilibrium. Hence, liquidity traps areworse with flexible prices than with constant inflation, in line with the results in Eggertsson andKrugman (2012). Alternatively, instead of choosing the level of M , the central bank could set the nominal interest rate i to followan interest-rate rule of the form i = α · π with α >
1. A negative shock increasing slack leads to negative inflationthat prompts the central bank to lower i . Under this monetary policy, the economy immediately adjusts to shocks toremain at x = x ∗ and π =
0. New Keynesian models have the same property. Increasing inflation could push the economy out the trap. However, after the shock has happened and theeconomy is in a liquidity trap, conventional monetary policy cannot influence inflation anymore even with flexibleprices. Helicopter drops or the wealth tax could still successfully pull the economy out of the liquidity trap.
951 1970 1985 2000 2019 0% 3% 6% 9%12% U ne m p l o y m en t r a t e EfficientActual
Figure 11.
Unemployment gap in the United States (source: Michaillat and Saez 2019a)
6. Conclusion
In this paper we develop a model of business cycles and use it to study a broad range of stabilizationpolicies. The main attributes of the model are that (1) cyclical fluctuations in demand and supplylead to fluctuations in slack but not in inflation; and (2) liquidity traps may be permanent and mayfeature positive inflation and high unemployment. Hence, in the model, market forces are notable to move inflation around to maintain the economy at efficiency; neither are market forcesable to bring the economy out of liquidity traps.Since our model is quite different from the standard New Keynesian model, it offers newinsights for the conduct of monetary policy and fiscal policy. An advantage of the matching theoryof unemployment is that it lends itself well to welfare analysis. With current unemployment andvacancy rates, and estimates of three sufficient statistics (slope of the Beveridge curve, recruitingcost, and nonpecuniary value of unemployment), it is possible to construct a real-time measure ofthe unemployment gap—something that nobody has been able to do with the output gap. Figure 11depicts the unemployment gap constructed by Michaillat and Saez (2019a) from the Beveridgecurve in the United States. The graph shows that the US unemployment gap is almost alwayspositive and highly countercyclical—indicating that the labor market tends to be inefficientlyslack, especially in slumps.The unemployment-gap series in figure 11 indicates that there is much scope for monetaryand fiscal policy to stabilize the economy over the business cycle. Equipped with our economical36usiness-cycle model, we could compute the optimal response of monetary and fiscal policy suchdeviations from efficiency. These optimal responses are fairly simple to characterize.First, if that is at all possible, monetary policy should completely fill the unemploymentgap. So by observing the current unemployment gap as well as the impact of monetary policyon unemployment, we could obtain a simple prescription for optimal monetary policy. A largeliterature is measuring the effect of monetary policy on unemployment (for example, Bernankeand Blinder 1992; Romer and Romer 2004; Coibion 2012). Combining this evidence with themeasure of the unemployment gap in figure 11, we could determine the optimal monetary policyat any point int time, and assess the conduct of monetary policy by the Federal Reserve.Of course, if the unemployment gap is too large, monetary policy may not be sufficient tocompletely fill it. If the nominal interest rate needs to fall significantly, it will eventually runagainst the zero lower bound; at this point, alternative stabilization policies are required. Onepolicy that is commonly used in that case is to increase public expenditure through a stimuluspackage. This is exactly what happened during the Great Recession in the United States (Wilson2012). Combining the formula for optimal stimulus spending developed by Michaillat and Saez(2019b), available estimates for the government-spending multipliers (for example, Auerbachand Gorodnichenko 2012; Ghassibe and Zanetti 2019), and the measure of the unemploymentgap, we could compute the optimal stimulus package for the United States in various situationsand under various calibrations. We could highlight general principles for the design of optimalstimulus spending, discussing in particular the role of taxation.Beside government spending, we could study other possible stabilization policies. We coulddescribe for instance how forward guidance operates in our model, following the analysis ofMichaillat and Saez (2019c) in the New Keynesian model. An advantage of having a modifiedEuler equation through the introduction of wealth in the utility is that the model is not subjectto any type of forward-guidance puzzle. Therefore, the analysis of forward guidance would berelevant and applicable to the real world. Other policies that could be effective in a liquidity trapinclude a helicopter drop of money or a tax on wealth.
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