Liquidity trap and secular stagnation

Transcription

1 Liquidity trap and secular stagnation Philippe Bacchetta University of Lausanne Swiss Finance Institute CEPR Kenza Benhima University of Lausanne CEPR Yannick Kalantzis Banque de France February 2015 PRELIMINARY AND INCOMPLETE Abstract In this paper we analyze the link between the ZLB and slow growth in a model with heterogeneous agents and explicit money demand. While the model is neoclassical with small shocks, a large deleveraging shock in the spirit of Eggertsson and Krugman (2012) has permanent eects even with exible prices. It aects supply rather than demand and implies a long-term decrease in potential output and an increase in cash holding. The basic reason is that in a liquidity trap, saving is allocated to cash rather than physical capital. With short-term price stickiness, monetary policy in the form of an expansion in money supply is eective in reducing unemployment in the short-run, but not in aecting the long term output level. An increase in debt may help exiting the ZLB, but it may lower the capital stock because of higher interest rates. 1

2 1 Introduction Since 2008, many advanced economies have hit the zero lower bound (ZLB). In addition to low growth and high unemployment, a striking feature of recent years in these economies has been the low level of investment. This has led to downward revisions of estimates of potential output in coming years. As time went by, observers have begun to speculate whether these episodes might prove long-lasting. In a famous speech, Larry Summers suggested that the world economy might be suering of secular stagnation. A popular explanation for this situation is the deleveraging shock that followed the nancial crisis of Deleveraging has been shown to decrease the interest rate, making it likely to hit the ZLB and fall into a liquidity trap (Eggertsson and Krugman 2012). With nominal rigidities, this can lead to low levels of output and employment in the short run, as has been studied by the New-Keynesian literature (Eggertson and Woodford 2003, Werning 2012). However, why hitting the ZLB should lead to secular stagnation, a permanent state of lower output, is still not well understood. This paper explores this link between liquidity traps and secular stagnation. We build a model of heterogeneous agents with nancial frictions. In this framework, we consider a deleveraging shock, modeled as in Eggertsson and Krugman (2012) by a tightening of borrowing constraints. This shock decreases the supply of saving instruments and triggers a decrease in the interest rate as an equilibrium adjustment. Contrary to most of the recent literature, we explicitly introduce money in the model. At the ZLB, outside money becomes a valuable saving instrument, which can accommodate the need for saving instruments and substitutes to private bonds in the portfolio of savers. This translate into lower physical investment and a decrease in potential output in the new steady state, even in the absence of nominal rigidities. With a constant money supply, the adjustment mechanism consists in a lower price level, which increases real money holdings to satisfy the demand of saving instruments. This adjustment is made possible by a Pigou-Patinkin eect, which is absent from models of cashless economies or models of representative agents. The recent period of low growth and high unemployment has also been accompanied by a strikingly low level of investment. This has led to downward revisions of estimates of potential output in coming years. As time went by, observers have begun to speculate whether this 2

3 Currency and checkable deposits (Households, Nonprofit organisations and Nonfinancial business) (Left) Fed fund rate (right) Figure 1: Fed funds rate and cash holdings in the US Source: Federal Reserve Board of Saint-Louis and Flow of Funds :Q1 2006:Q1 2007:Q1 2008:Q1 2009:Q1 2010:Q1 2011:Q1 2012:Q1 2013:Q1 2014:Q Growth of capital stock Growth of hours Figure 2: Capital and labor in the US Source: Fernald (2014). 3

4 situation might prove long-lasting. In a famous speech, Larry Summers suggested that the world economy might be suering of secular stagnation. At the same time, many advanced economies have hit the zero lower bound (ZLB) in recent year. There is an active debate on the factors explaining this period of slow growth. 1 A popular explanation is the deleveraging shock that followed the nancial crisis of Deleveraging has been shown to decrease the interest rate, making it likely to hit the ZLB and fall into a liquidity trap (Eggertsson and Krugman, 2012). With nominal rigidities, this can lead to low levels of output and employment in the short run, due to lower demand. 2 However, the long-run impact on output growth is not well understood and is absent in most existing models. The objective of this paper is to explore the potential link between liquidity traps and secular stagnation. We build a model of heterogeneous agents with nancial frictions and initially assume exible prices. In this framework, we consider a deleveraging shock, modeled as in Eggertsson and Krugman (2012) by a tightening of borrowing constraints. This shock decreases the supply of saving instruments and triggers a decrease in the interest rate as an equilibrium adjustment. Contrary to most of the recent literature, we explicitly introduce money in the model. At the ZLB, outside money becomes a valuable saving instrument, which can substitutes to private bonds in the portfolio of savers. This translate into lower physical investment and a decrease in potential output in the new steady state, even in the absence of nominal rigidities. With a constant money supply, the adjustment mechanism consists in a lower price level, which increases real money holdings to satisfy the demand of saving instruments. This adjustment is made possible by a Pigou-Patinkin eect, which is absent from models of cashless economies or models of representative agents. Our approach is consistent with the facts illustrated in Figures 3 and 2. Figure 1 shows the dramatic increase in cash holdings by private agents once the ZLB is hit, as observed in the US in Figure 2 shows that the ZLB is associated with a fall in investment. In the data, while labour starts to fall in , capital growth only decreases in In our model, these two facts are linked: in the liquidity trap, the cash buildup diverts resources from physical investment. We introduce money in a model with credit-constrained entrepreneurs and workers in the 1 See Teulings and Baldwin (2015) for an interesting collection of essays on this issue. 2 Eggertson and Woodford (2003), Werning 2012 or Benigno et al. (2014) show this in New-Keynesian models. 4

5 Figure 3: Fed funds rate and cash holdings in the US Currency and checkable deposits (Households, Nonprofit organisations and Nonfinancial business) (Left) Fed fund rate (right) Reserve Board of Saint-Louis and Flow of Funds. Source: Federal :Q1 2006:Q1 2007:Q1 2008:Q1 2009:Q1 2010:Q1 2011:Q1 2012:Q1 2013:Q1 2014:Q Growth of capital stock Growth of hours (2014). Source: Fernald Figure 4: Capital and labor in the US 5

6 spirit of Bacchetta and Benhima (2015) and Woodford (1990). Entrepreneurs alternate between investing phases and producing phases. Because they face credit constraints, they have a tendency to save in their producing phase in order to be able to invest in their investing phase. Agents can save either in nominal bonds or in money. But as long as the nominal interest rate is positive, money is dominated as an asset and is held only for transaction purposes. At the ZLB, bonds and money become substitutes and money can be held for saving purposes as well. As in Eggertsson and Krugman (2012), a deleveraging shock generates a decrease in the natural interest rate that makes the nominal interest rate hit the ZLB. However, in our model, the fall in interest rate lasts as long as the deleveraging shock. If this shock is permanent, then the ZLB persists in the steady state. Indeed, there is a lack of nancial assets in the economy that prevents entrepreneurs from moving away from their credit constraints through saving. Eggertsson and Mehrota (2014) has a similar implication in an OLG model where the horizon of agents is shorter. We show that the consequences of a deleveraging shock are very dierent outside the ZLB and at the ZLB. Outside the ZLB, a deleveraging shock has no eect on long-run capital accumulation and output. However, large deleveraging shocks that bring the economy at the ZLB has a negative eect on long-run capital accumulation and output. Unlike the previous literature, the ZLB generates a secular stagnation even in the absence of nominal rigities. This is because of a liquidity trap eect. At the ZLB, in the presence of a constant long-run growth rate of money (and hence constant long-run ination), the real interest rate becomes inelastic. At the same time, money becomes a potential saving vehicle. As a result, while in the absence of ZLB the real interest rate would adjust to discourage savings, leaving the equilibrium longrun capital stock unchanged, in the ZLB savings are channelled to money, which crowds out investment. Other papers do model the ZLB with investment but they do not have the liquidity trap eect because: (1) they do not explicitly allow agents to save in money; (2) they have representative agents so the Ricardian equivalence holds; (3) changes in money holdings and transfers cancel out implicitly. Our model has heterogeneous agents and transfers do not automatically oset changes in money demand at the individual level, so changes in money demand have real eects. 6

7 Our mechanism can be contrasted with other mechanisms of recessions at the ZLB. Our mechanism is based on supply-side eects, while other papers focus on demand-side eects. To generate demand-side eects, nominal rigidities are crucial. In Eggertsson and Mehrota (2014), long-run nominal rigities are necessary to generate a secular stagnation. With rigid prices, negative demand shocks (such as deleveraging shocks) have strong recessionary eects, especially at the ZLB where these eects are not dampened by an adjustment in the interest rate. Regarding policy implications, in a context where demand eects are strong, scal policy can help ght the recession (e.g., Christiano et al., 2011) When secular stagnation is driven by the ZLB and nominal rigidities, the short-run policy implications are also valid in the long run. This is not necessarily true in the more standard case where demand eects and nominal rigidities are absent in the long run. We explore the eects of scal and monetary policy in our context. Because we introduce money explicitly, our framework also allows us to study the eects of quantitative easing and of government bonds. For example, an increase in government debt may help exiting the ZLB by increasing the natural interest rate. But higher interest rates may lower the capital stock. When we introduce short-term price stickiness in the model, we nd that quantiative easing is eective in reducing unemployment in the short-run, contrarily to what is found in other models. On the other hand, monetary policy is not eective in aecting the long term output level. [Literature review TBA] The rest of the paper is organized as follows. Section 2 presents the basic model with innitely-lived entrepreneurs and workers and Section 3 analyzes its dynamics. Section 4 describes the steady state outside of a liquidity trap and shows that it is unaected by the deleveraging shock. Section 5 analyzes the steady state in a liquidity trap to show that steady state output and capital decrease with the size the deleveraging shock. Section 6 examines the steady-state impact of various policies. Section 7 studies the short-run eect of a deleveraging shock and extends the model to sticky prices. Finally, Section 8 concludes. 7

8 2 A model with money Consider a monetary economy with entrepreneurs and workers. As our analysis focuses on the steady state, we rst assume that prices are exible. The model is a variant of Bacchetta- Benhima (2014). There are three types of assets: a nominal bond, money and capital. While the bond yields a nominal interest rate i and can be positive or negative, money bears no interest rate and is non-negative. There are two groups of entrepreneurs with log utility, investing and producing every other period. In their investment phase, entrepreneurs typically borrow so we denote them by L (for loans). In their producing phase, entrepreneurs typically save so we denote them by D (for deposits). We denote workers by w. All agents have log-utility and maximize for i = w, D, L. U i = β s log(c i t+s) s=0 Entrepreneurs In period t, L entrepreneurs start with wealth Ω L t = i t D t + M D t + T L t where P is the price level, D and M D are respectively the nominal bonds and the nominal money holdings inherited from their preceding producing phase, and T L are nominal transfers from the government. Denote the real interest rate by r t+1 = i t+1 /Π t+1, Where Π t+1 = +1 / is ination. L entrepreneurs consume c L t, borrow L t+1, save a monetary amount M L t+1, and engage in a productive process by investing K t+1, so Ω L t = c L t L t+1 + M L t+1 + K t+1 (1) Ω D t In period t, D entrepreneurs hire l t workers at cost w t to produce, so they start with wealth = K α t l 1 α t + (1 δ)k t w t l t i t L t + M L t + T D t, where K, L and M L are respectively the capital stock, nominal debt and the nominal money holdings inherited from their preceding investment phase, and T D are nominal transfers from the government. They consume c D t, save D t+1 in interest-bearing deposits and M D t+1 in money, so that Ω D t = c D t + D t+1 + M D t+1 (2) 8

9 Entrepreneurs face borrowing constraints that will be eective for L entrepreneurs so that i t+1 L t+1 +1 φk t+1 (3) Workers The budget constraint of workers is: w t l t + M w t + T t w i t L w t = c w t + M w t+1 Lw t+1. (4) They are also subject to a cash-in-advance (CIA)constraint: c w t M w t + T w t (5) And they face a borrowing constraint: i t+1 L w t+1 +1 L w (6) L w represents the net supply of bonds by workers. For the moment, assume labor supply is equal to 1. The wage is equal to marginal productivity so that K α t l 1 α t αf(k t ) in equilibrium, where f(k) = K α. Workers earn (1 α)f(k t ). w t l t simplies to Asset scarcity, binding constraints and the demand for money In equilibrium, with strong borrowing constraints, the economy is asset-scarce and we will have βr < 1 in the long run, so the borrowing constraints will typically be binding for workers and L entrepreneurs at the vicinity of the steady state, which we assume throughout. When i > 1, entrepreneurs hold the minimum amount of money, which is zero, as money has a strictly lower return than nominal bonds, so M D = M L = 0. Similarly, for workers, the CIA constraint will also be binding. Otherwise, if i = 1, both workers and L entrepreneurs would still hold the minimum amount of money, as they would save the minimum for βr < 1, so M L = 0 and the CIA constraint is binding. 3 However, for D entrepreneurs, who typically save, bonds and money become perfect substitutes and they start holding money for saving purposes. Note that entrepreneurs, contrary to workers, are not subject to a CIA constraint, 3 Note that the CIA constraint might still be lax in the initial period or when there is an unexpected shock. 9

10 so their demand for cash will be zero outside the zero lower bound. This assumption is without loss of generality and is made for the purpose of clarity. We will consider later an extension where entrepreneurs have a demand for cash for transaction purposes. The government budget constraint and money supply Denote M t the monetary mass at the beginning of period t. We have M t = M D t + M L t + M w t where M D t is money held by saving entrepreneurs, M L t is money held by borrowing entrepreneurs and M w t is money held by workers. For simplicity, we assume that money is created through transfers to workers, so we set T D t = T L t = 0. The government has a constant real supply of bonds L g. ( M t+1 = M t + Tt w + 1 Π ) t+1 P i Lg t. (7) t+1 Later, we can study the eects of quantitative easing and of the public supply of bonds L g and look at alternative monetary policy (helicopter drops to entrepreneurs, open market operations, etc.). We can also study scal policy by introducing government spending. Finally, suppose the monetary authority keeps the growth of money M t+1 /M t = θ constant. This assumption enables us to pin down steady-state ination easily, as it will be equal to θ. We will thus be able to study the long-run eects of monetary policy. We make the following parametric assumption: Assumption 1 θ > β. Assumption 1 implies that the gross nominal interest rate i = rθ would be strictly greater than 1 in a steady state where the borrowing constraint would not bind so that r = β 1. With a binding borrowing constraint, r < β 1 and it will possible to have steady states with i = rθ = 1. Assumption 1 is naturally satised as long as θ 1, that is with a non-negative steady-state ination. The bonds market The market for nominal bonds clears so that L t + L w t + L g t = D t. We dene L = L w + L g as the supply of bonds by the rest of the economy, which includes workers and the government. We rst assume L g to be constant, so changes in L come exclusively from changes in L w. We will therefore be able to contrast the eects of a deleveraging shock 10

11 on entrepreneurs (a drop in φ) to the eects of a deleveraging shock on workers (a drop in L driven by L w ). Later, we can study the eects of the public supply of bonds (a change in L driven by L g ). Note that in equilibrium, L is also the net position of entrepreneurs. If L > 0, then entrepreneurs are net creditors. If L < 0, then entrepreneurs are net borrowers. We can consider cases where L it is positive or negative in order to study situations where entrepreneurs are net creditors or net borrowers. 3 Dynamics The dynamics is governed by the Euler equation of savers (D entrepreneurs), the aggregate budget constraint of entrepreneurs, and the money market equilibrium. 3.1 Euler equation of savers Savers (D entrepreneurs) are typically unconstrained so their Euler equation (with log-utility) is satised: c L t+1 c D t = β i t+1 Π t+1 With log-utility, we also have c L t+1 = (1 β)ω L t+1 and c D t = (1 β)ω D t where Ω L t+1 = i t+1 D t M D t+1 +1 Ω D t = αf(k t ) + (1 δ)k t i t L t Here we have used the entrepreneurs' income denition where we have replaced wl by (1 α)f(k) and set Mt L = 0, as L entrepreneurs never hold any money. Since the L entrepreneurs' and the workers' credit constraints (3) and (6) are binding and in equilibrium L+L w +L g = D, then the Euler equation yields φk t+1 + L + M D t+1/+1 αf(k t ) + (1 δ φ)k t = βr t+1 (8) 11

12 where we replaced i/π with r. Note that, everything else equal, an unexpected deleveraging shock on entrepreneurs (a drop in φ) has a negative eect on the real interest rate through two channels. First, it reduces the supply of bonds (eect on the numerator). In subsequent periods, as D entrepreneurs start the period with less debt, it increases their income and hence their demand for bonds. These two eects aect negatively the equilibrium interest rate. A deleveraging shock on workers (a drop in L) has similar eects but only through a reduction in the bond supply. 3.2 Aggregate budget constraint With log-utility, entrepreneurs save a fraction β of their wealth Ω i, i = D, L. The budget constraints of L and D entrepreneurs (1) and (2) become respectively βω L t = K t+1 L t+1 / and βω D t = M D t+1/ + D t+1 /. Summing these two constraints and using L + L w + L g = D, we nd ( M D K t+1 + Π t+1 t+1 + L ) [ = β αf(k t ) + (1 δ)k t + M t D +1 i t+1 ] + L This equation represents the aggregate budget constraint of entrepreneurs. It depends on their net position vis-à-vis the rest of the economy L, as D L = L w + L g. It is independent of φ, because the net position of entrepreneurs as a whole ultimately depends on the net supply of bonds by the rest of the economy L. (9) 3.3 Money market equilibrium Equilibrium in the market for money implies Mt D + Mt L + Mt w = M t. As argued earlier, the CIA and borrowing constraints are always binding for workers. If i > 1, neither type of entrepreneur holds any money, so M L = M D = 0. We therefore refer to equilibria where i > 1 as cashless equilibria. If i = 1, then D entrepreneurs start saving in money, so M D 0. We thus refer to equilibria where i = 1 as liquidity-trap equilibria. Only workers and D entrepreneurs can have a positive demand for money so the money market equilibrium therefore implies that the money supply is equal to the money demand by workers plus the money demand by D entrepreneurs. In the absence of unexpected shocks, the budget constraints of workers and the government (4) and (7), along with the binding CIA 12

13 constraint for workers (5) yield: M t+1 = (1 α)f(k t ) ( 1 Π ) t+1 L w + M t+1 D. (10) i t+1 In the initial period, or when the economy is hit by an unexpected shock, we have: M t+1 = (1 α)f(k t ) ( 1 Π ) t+1 L w + M ( t+1 D M w + max t + Tt w i t+1 M ) t+1 w + Tt+1 w, 0 βi t+1 (11) where the last term is the excess money holding demanded by workers in order to smooth consumption. 4 Cashless steady states When i t+1 > 1, no money is held by entrepreneur so Mt+1 D = 0. In that case, the classical dichotomy applies. The dynamics of real variables follows a two-dimensional system in (K t+1, r t+1 ): φk t+1 + r t+1 = L β[αf(k t ) + (1 δ φ)k t ] [ ( K t+1 = β αf(k t ) + (1 δ)k t + L 1 1 )] βr t+1 (12) (13) where (12) and (13) refer respectively to the savers' Euler equation (8)and the aggregate budget constraint of entrepreneurs (9). Then the price level comes from the money market equilibrium: = M t+1 (1 α)f(k t ) ( 1 1 r t+1 ) Lw (14) Given the real rate r t+1, the nominal rate follows from i t+1 = r t+1 +1 /. Notice that equations (13) and (12) depend on L, the net supply of money from the rest of the economy, regardless of whether this net supply comes from L w, the net supply from workers, or from L g, the net supply from the government. We consider two cases: rst, the case where entrepreneurs are in eective autarky, that is L = 0, which happens if workers cannot borrow at all ( L w = 0) and there are no government bonds ( L g = 0); second, the case where entrepreneurs 13

14 BC Euler, low φ Euler, high φ 12 K r 1/theta 1/beta Figure 5: Cashless steady state for dierent φs ( L = 0) are net creditors ( L > 0) or net borrowers vis-à-vis the rest of the economy ( L < 0). 4.1 Entrepreneurs are in autarky When L = 0, the dynamics of capital accumulation (13) becomes independent of the real interest rate r t+1 and of the intensity of the borrowing constraint φ. Proposition 1 (Cashless steady state - No net supply of bonds) If φ < 1, then there 2β exists a locally constrained steady state. If, additionally, φ > 1, then entrepreneurs hold no θ+β money in this steady state. It is dened by a value K that is independent from φ, M D = 0, and i = θr > 1 which depends positively on φ and θ. Proof 1 In a steady state, the money market equilibrium implies that Π = θ. As a result, i = r θ. If there exists a steady state with i > 1, then it satises (13) and (12), which yields ( ) 1/(1 α) K αβ = 1 β(1 δ) and r = φ 1 βφ. i > 1 then implies φ > 1. We nally check that θ+β βr < 1 as φ < 1/2β, which insures that the steady state if locally constrained and follows (13), (12) and (14). Here, a permanent negative credit shock on rms, through a decrease in φ, has no eect on investment, but it has a negative eect on the real interest rate r. Credit shocks therefore do not aect investment in the long run and do not generate secular stagnation. To understand, consider Figure 5, which represents the equilibrium steady state of the economy. The Euler 14

15 curve gives the long-run stock of capital K implied by the Euler equation of D entrepreneurs (Equation (12)) for a given real interest rate r. It is akin to a long-run saving equation. When the interest rate is higher, the saving demand is higher, implying a higher stock of capital through the saving-investment equilibrium. When φ is lower (like after a deleveraging shock), the demand for saving by D entrepreneurs is higher, which can only be accommodated through a higher supply of bonds by L entrepreneurs. As the pledgeable share of capital φ decreases, this is achieved through a higher collateral K. The BC curve represents the longrun capital stock implied by the budget constraint of entrepreneurs (Equation (13)). It is akin to an investment curve. It is independent of r, for two reasons. First, this is due to log-utility: agents save a fraction of their income each period, as the income eect osets the substitution eect. Second, as L = 0, entrepreneurs are neither net creditors nor net borrowers, so the interest rate does not generate any income or expense. As the long-run investment capacities of rms are inelastic to r, a decrease in φ does not aect the long-run equilibrium capital stock. D entrepreneurs are more willing to save for a given interest rate as the Euler curve shifts upward, but the interest rate adjusts downward, leaving the equilibrium capital unchanged. Notice that a decrease in r implies a proportional decrease in i in the absence of a change in the ination rate θ. Therefore, a low steady-state ination makes the economy at the risk of hitting the ZLB following a contraction of credit. In fact, i hits the zero-lower bound if r 1/θ. In that case, the long-run equilibrium does not follow Figure 5. This case is treated in the next section. Similarly, a high enough φ brings the equilibrium interest rate at 1/β. beyond this, the credit constraint is not biding anymore. To summarize, in cashless steady states where entrepreneurs have no net position vis-à-vis the rest of the economy, credit shocks that aect φ do not have long-run eects on capital accumulation. This is true as long as these credit shocks are small and do not bring the natural real interest rate below 1/θ, which would make the ZLB binding. Note that log-utility matters here it implies that the BC curve does not depends on the interest rate. It provides however a useful benchmark that will make it easier to study the ZLB case. 4 4 In fact, the case of log-utility is a realistic one when it comes to modeling the saving behavior of agents, as a unitary elasticity of intertemporal substitution is well within the estimated ranges. 15

16 4.2 Entrepreneurs are net creditors or net debtors Figure 6 represents the case with a net supply or demand for bonds from the rest of the economy ( L 0). We consider cases where the economy is asset-scarce so that βr < 1 in the long run. Therefore, for the L < 0, case, we choose a high φ and for the L > 0, case, we choose a low φ. The two cases have therefore comparable supplies of bonds, but in the former the supply comes from entrepreneurs and in the latter it comes from the rest of the economy. Consider rst the case where bonds are supplied mostly by entrepreneurs and hence entrepreneurs are net creditors ( L < 0 and φ high). Notice that the BC curve is now downward sloping. As entrepreneurs are net debtors vis-à-vis the rest of the economy, a higher interest rate generates higher costs, which reduces their investment capacity. the Euler curve is still upward sloping. A decrease in L (an increase in the net demand for bonds from the rest of the economy) shifts the Euler curve upwards (just as an decrease in φ): a lower supply of bonds by the rest of the economy is compensated by a higher supply of bonds by L entrepreneurs through a higher collateral. As BC is downward sloping, this increases the equilibrium capital stock and decreases the equilibrium interest rate. A deleveraging shock on workers (a fall in L) has therefore the same eect on the interest rate as a deleveraging shock on entrepreneurs (a fall in φ). When entrepreneurs are net debtors, this fall in the interest rate has a positive eect on entrepreneurs' resources and it stimulates investment. This also implies that workers' deleveraging can bring the economy to the ZLB. Figure 6 also represents the case where bonds are supplied mostly by the rest of the economy ( L > 0 and φ low). As entrepreneurs are now net creditors, BC is upward sloping: a higher interest rate generates more resources for entrepreneurs and increases their investment capacities. Importantly, now Euler is downward sloping, and a decrease in L (a less negative L) shifts Euler downward. Indeed, when the interest rate increases, or when L decreases, the demand for saving increases. However, when φ is low, an increase in the collateral cannot accommodate an increase in saving. Equilibrium is therefore achieved through a lower income and hence a lower capital stock. This is a variant of the paradox of thrift whereby an increase in saving reduces investment in equilibrium. As a result, a decrease in L has a negative eect on the equilibrium capital stock and on the interest rate. A deleveraging shock aecting workers (a fall in L) has therefore the same eect on the interest rate as in the case with L < 0. However,it depresses 16

17 BC, low Lbar BC, high Lbar Euler, low Lbar Euler, high Lbar K 6.5 K r 1/theta 1/beta r 1/theta 1/beta Entrepreneurs are net debtors ( L < 0) - High φ Entrepreneurs are net creditors ( L > 0) - Low φ Figure 6: Cashless steady state for dierent Ls investment. Again, workers' deleveraging can bring the economy to the ZLB. The next section studies in detail the steady state at the ZLB. 5 Liquidity-trap steady states When i t+1 = 1, then money becomes attractive to entrepreneurs and M D t+1 0. In that case (8), (9), and (10) become: φk t+1 + m D t+1 + L αf(k t ) + (1 δ φ)k t = β Π t+1 (15a) K t+1 + Π t+1 (m D t+1 + L) = β [ αf(k t ) + (1 δ)k t + m D t + L ] (15b) = M t+1 (1 α)f(k t ) (1 Π t+1 ) L w + Π t+1 m D t+1 (15c) where m D t = M D t /. As before, we consider rst the case without a net supply or demand for bonds from the rest of the economy ( L = 0). Proposition 2 (Liquidity-trap steady state - No net supply of bonds) If φ < 1 θ+β, then there exists a steady state where i = 1 entrepreneurs hold money. It is dened by ˆK which depends positively on φ with ˆK < K, ˆm D > 0 which depends negatively on φ, and ˆr = 1/θ. ˆK 17

18 is increasing in θ if φ > 1 2θ. Proof 2 In a steady state, the money market equilibrium implies that Π = θ. As a result, i = r θ. If φ < 1, then the steady state without money does not exist, as the implied θ+β nominal interest rate i would be below one. If there exists a steady state with i = 1, then it satises (15), which yields ˆK 1 α = αβ 2. We can show that this is lower than θ β 2 (1 δ) φ(θ 2 β 2 K ) as φ < 1, and that it is increasing in φ. Similarly, we can show that it is increasing in θ if θ+β φ > 1/2θ. Besides, ˆm D = β 1 [1 φ(θ + β)] ˆK, which is strictly positive as φ < 1 θ+β. Contrary to a deleveraging shock in cashless steady states, a deleveraging shock that is large enough to make the economy fall in a liquidity trap has permanent, negative eects on capital and output. Indeed, in the liquidity trap, as the real interest rate cannot adjust (it is xed at 1/θ), a credit contraction (a fall in φ) has a negative eect on investment. Here, savings are inelastic to the interest rate, but the allocation of savings between bonds and money is not. In fact, D entrepreneurs are indierent between bonds and money. A fall in credit reduces the supply of bonds as it prevents L entrepreneurs from borrowing. Whereas outside the liquidity trap the interest rate could adjust to stimulate the supply of bonds, thus reestablishing equilibrium, here the excess supply of saving is used to increase rms' real money holdings. The long-run investment then becomes sensitive to permanent credit shocks. A permanent credit shock is then consistent with secular stagnation, as in Eggertsson and Mehrota (2014). However, in Eggertsson and Mehrota (2014), this was driven by a fall in consumption demand, which can lead to long-run stagnation only in the presence of nominal rigidities. rigidity. Here, the eects come from investment and hold in the absence of any nominal Note that the eect of ination on steady-state investment is ambiguous. It is positive if φ is not too low. In that case, rms have a relatively high level of debt and a debt-deation mechanism is at play, as debt is denominated in nominal terms. If φ is low, then this eect is smaller. In that case, ination has a negative real-balance eect on rms, which reduces their investment capacities. Figure 7 represent the long-run equilibrium for K and m for dierent values of φ and L. Consider rst the left panel. With L = 0, the BC curve is independent of φ. Importantly, it 18

19 m D BC Euler, low φ Euler, high φ m D BC, low Lbar BC, high Lbar Euler, low Lbar Euler, high Lbar K Eect of a change in φ ( L = 0) K Eect of a change in L Figure 7: Long-run equilibrium at the ZLB implies a negative relationship between K and m. This is due to decreasing marginal returns. When the stock of capital increases, the corresponding increase in income does not allow to sustain the capital increase unless rms hold less money. On the opposite, holding more money takes resources out of capital. A decrease in φ shifts the Euler curve upwards. Indeed, a deleveraging shock increases entrepreneurs' savings. For a given capital stock, as the interest rate is constant, the bond supply cannot accommodate this increase in savings and entrepreneurs can only invest in cash. As a result, a deleveraging shock increases the equilibrium money holdings by rms, which crowds out capital. Consider now the right panel. As money and bonds are perfectly substitutable at the ZLB, an increase in L is oset by a decrease in m. This appears in the gure through an identical shift in the BC and Euler curves, which leaves the equilibrium capital stock unchanged. Note that both a decrease in L or φ, which correspond to deleveraging shocks, increase equilibrium money. With high L or φ, m becomes negative. However, m cannot be negative. Equilibria with negative money therefore do not exist. They actually correspond to cashless equilibria, described in the previous section. The following proposition summarizes the conditions under which the steady state is a rst-best steady state, a cashless steady state or a liquidity trap. Proposition 3 (Steady state) Let Λ 1 = ˆKβ 1 [1 φ(θ + β)] and Λ 2 = K β 1 [1 2φβ]. 19

20 For L Λ 1, the steady state is in a liquidity trap with K = ˆK and m D = Λ 1 L. For Λ 1 < L < Λ 2, the steady state is cashless with K = ψ K ( L) and r = ψ r ( L), where ψ r ( L) is strictly increasing, and ψ K ( L) is U-shaped with a minimum reached at some Λ (0, Λ 2 ). For L = Λ 2, the steady state is at the rst best with K = K and r = β 1. After a deleveraging shock on workers, L decreases and the economy can fall in a liquidity trap, like for a deleveraging shock on entrepreneurs. However, L is also composed of government bonds L g. An increase in government bonds can therefore bring the economy out of the liquidity trap. This is discussed in the next section on policy implications. 6 Policy implications In this section, we study the policy implications of the model. 6.1 Fiscal policy Public supply of bonds In section 4, we have seen that a decrease in L has a negative eect on the equilibrium real interest rate and hence can bring the economy to a liquidity trap. An increase in the supply of government bonds, by increasing L, can have an opposite eect and bring the economy out of the liquidity trap. However, marginal changes in L have no eect as long as the economy is in the liquidity trap, at least in the long run, as shown in section 5. Indeed, money and bonds are substitutes, so changes in the supply of bonds is matched by a decrease in the demand for real money balances. To accommodate for the change in the demand for real money, prices increase. Following a deleveraging shock that brings the economy in a liquidity trap, is an increase in government spending that is large enough to take the economy out of the liquidity trap then a good instrument to stimulate output? Not necessarily. In the case of a deleveraging shock on workers, an osetting increase in government bonds can restore the previous level of output, as L w and L g have identical eects. Things are dierent if the deleveraging shock aects entrepreneurs. 20

21 K φ=0.4 φ=0.5 φ=0.6 φ=0.7 φ=0.8 Lbar Figure 8: Long-run capital stock 21

22 This is illustrated in Figure 8. It represents the long-run capital stock as a function of L, for dierent values of φ. The gure shows that a decrease in φ brings the economy to the ZLB, as seen above. When the economy is in the ZLB, a marginal increase in L has no eect on the capital stock as bonds and money are perfect substitutes. However, a strong increase in L brings the economy out of the ZLB. Whether this has a positive of negative eect on capital depends on the sign of L. If L > 0, then getting out of the ZLB has a positive eect on investment. The eect is negative if L < 0. Indeed, as the economy gets out of the ZLB, the real interest rate increases. When entrepreneurs are net creditors ( L > 0), this has a positive eect on rms's investment capacities. When entrepreneurs are net debtors, it has a negative eect on investment capacities. Government spending Government spendings do not have an eect per se. Their eect depends on the way they are nanced. Government spendings have no eect on entrepreneurs if they are nanced through taxes on workers. However, if government spending are nanced through government bonds, then our previous analysis applies. If, alternatively, government spendings are nanced partially through taxes on entrepreneurs, then they would have negative eects on their investment capacities and hence on capital and output. 6.2 Monetary policy Ination The liquidity-trap steady state arises whenever the cashless steady-state interest rate is below 1/θ, where θ is steady-state ination. A high steady state ination can therefore prevent the appearance of liquidity traps. This result is similar to Krugman (1998). However, as suggested by Proposition 2, small increases in θ, i.e. changes that leave the economy in the liquidity trap, have an ambiguous eect on capital. If rms are highly leveraged, then the eect is positive because it deates rms' debt and improve their nancing capacities. Quantitative easing [TBA] 22

23 7 The short-run eect of a deleveraging shock and the Pigou-Patinkin eect In this section, we discuss the short-run eect of a large deleveraging shock that brings the economy to the ZLB and the potential policy implications. Consider the following thought experiment. Suppose we are outside the ZLB at time t 1. Then, at time t, φ unexpectedly decreases to φ, something we might refer to as deleveraging. For simplicity, suppose L g = L w = L = 0. For a small shock, φ is still high enough so that the economy stays in a non-zlb equilibrium. From (13), this does not aect capital accumulation. From (12), adjustment comes from a lower real rate. As prices are not aected by the shock (see equation (14)), the nominal interest rate drops. Intuitively, the shock decreases the supply of saving instruments by investing entrepreneurs. However, as explained before, demand for saving instruments is unaected: saving of D entrepreneurs are still fully channeled to investment by L entrepreneurs. The equilibrium reaction is then a higher price of saving instruments, i.e. a lower interest rate. The lower interest rate increases the collateral of L entrepreneurs which allow them to create the missing saving instruments. For a large enough shock, that is when φ is suciently low, the nominal rate i t+1 reaches the lower bound i t+1 = 1 and cannot adjust to create the missing saving instruments. However, outside money now becomes a valuable saving instrument that can substitute to bonds in the portfolio of savers: D entrepreneurs start demanding money. As outside money is initially held by workers, saving by D entrepreneurs is now channeled to worker consumption instead of investment by L entrepreneurs. The equilibrium adjustment now comes from a drop in the price level, which increases the real supply of outside money in the economy. A deleveraging shock is then deationary. This works through a Pigou-Patinkin eect: the lower price level increases workers' real money holdings and makes them consume more, selling some of their money to D entrepreneurs who want to hoard cash. When the shock hits, workers might also want to save part of the valuation gains on their money holding. They can do so either by keeping excess money holding or holding nominal assets. Without loss of generality, we can 23

24 assume that they keep excess money holding, making the price level decrease even more to clear the money market. In the short run, the economy then falls into a ZLB equilibrium with lower investment and higher consumption, for a given predetermined level of output. In the long run, lower accumulation of capital leads to a lower steady state output as described in Proposition 2. This dynamics is illustrated in Figure 9 in the case L = 0 and θ = 1. At t = 0, the economy is assumed to be in a steady state corresponding to φ = 1, that is where the ZLB θ+β is marginally binding. At t = 1, φ unexpectedly decreases to a permanently lower level. As described above, the tighter borrowing constraint leads to less borrowing by L entrepreneurs (a) and positive money holding by D entrepreneurs (b), which crowds out capital (c). Adjustment comes from a permanent decrease in the price level (d). Expected ination (the real interest rate) slightly increase (decrease), but not enough to channel saving to investment and restore the initial level of investment. This is dierent from models with a representative agent such as Krugman (1998) or models of moneyless economies such as Eggertsson and Krugman (2012): in these works, there is no Pigou eect and the price level decreases enough to generate ination expectations that overcome the ZLB. By contrast, with heterogeneous agents and an explicitly modeled money demand, the price decrease has distributional eects that change the allocation of resources between investment and consumption, with long-term eect on future output. Sticky prices How would this adjustment mechanism play out with sticky prices? Suppose goods produced by entrepreneurs are sold to a retail sector with monopolistic competition and some form of nominal rigidity in the short run (e.g. Calvo pricing). Suppose in addition that households now have an elastic labor supply. Then, the price level cannot fully adjust at time t when the shock hits. The adjustment comes instead from employment and the level of output. With a lower production, the demand for saving instruments, including money, by D entrepreneur decreases, as can be seen from Equations (15), as well as the demand for money by workers. This restores equilibrium on the market for money. As time goes by and prices gradually adjust, employment and output increases back to their exible-price level and the economy converges to the steady state described by Proposition 2. Therefore, with sticky prices, a deleveraging shock moves the economy to the ZLB and creates a negative output gap 24

25 (a) L t+1 / (b) m D t+1 (c) K t (d) (e) Π t (f) r t+1 Figure 9: Transitory dynamics after a permanent tightening of the borrowing constraint in the short run, as in the existing literature. Contrary to the existing literature, the economy stays at the ZLB with a lower capital stock and level of output, even after prices have adjusted and the output gap has closed. Policy implications for the short run What does this imply for short-run policy? Consider rst scal policy. As the nominal interest rate is inelastic, scal policy has strong positive demand eects in the presence of nominal rigidities. This is in line with previous literature. However, in the long-run, these positive demand eects disappear and only the supply-side eects described above remain. Consider now monetary policy. When the shock hits, a monetary expansion taking the form of transfers to workers can move the economy close to the exible-price allocation. From Equations (15), it is clear that monetary policy can increase money supply to target any desired level of employment at a given price level. This result stands in sharp contrast to existing work, for instance Krugman (1998), where money creation taking the form of transfers has no eect at the ZLB with pre-set prices (see footnote 11 of this work). It comes from the non-ricardian 25

26 structure of the model, which gives rise to the Pigou-Patinkin eect described above. However, if a policy of monetary transfers can be very eective in closing the output gap in the short-run of this model, it has no eect in the long run and therefore cannot prevent secular stagnation. As explained before, preventing secular stagnation requires increasing the steady state ination target θ or the supply of bonds L to the entrepreneurial sector. 8 Conclusions [TBA] 26

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