Interest Rates and Monetary Assets in an Economy with Risks, Study notes of Economics

Download Interest Rates and Monetary Assets in an Economy with Risks and more Study notes Economics in PDF only on Docsity! A Neoclassical Theory of Liquidity Traps Sebastian Di Tella∗ Stanford University August 2017 Abstract I propose a flexible-price model of liquidity traps. Money provides a safe store of value that prevents interest rates from falling during downturns and depresses in- vestment. This is an equilibrium outcome — prices are flexible, markets clear, and inflation is on target — but it’s not efficient. Investment is too high during booms and too low during liquidity traps. The optimal allocation can be implemented with a tax or subsidy on capital and the Friedman rule. 1 Introduction Liquidity traps occur when the role of money as a safe store of value prevents interest rates from falling during downturns and depresses investment. They can be very persistent, and are consistent with stable inflation and large increases in money supply. Liquidity traps are associated with some of the deepest and most persistent slumps in history. Japan has arguably been experiencing one for almost 20 years, and the US and Europe since the 2008 financial crisis. This paper puts forward a flexible-price model of liquidity traps and studies the optimal policy response. The baseline model is a simple AK growth model with log utility over consumption and money and incomplete idiosyncratic risk sharing. Markets are otherwise complete, prices are flexible, and the central bank follows an inflation-targeting policy. During downturns idiosyncratic risk goes up and makes risky capital less attractive. Without money, the real interest rate would fall and investment would remain at the first best level. ∗I’d like to thank Manuel Amador, Pablo Kurlat, Chris Tonetti, Chad Jones, Adrien Auclert, Arvind Krishnamurthy, Narayana Kocherlakota, Bob Hall, and John Taylor. email: sditella@stanford.edu. 1 But money provides a safe store of value that prevents interest rates from falling and depresses investment. Money improves idiosyncratic risk sharing and weakens agents’ pre- cautionary saving motive. This keeps the equilibrium real interest rate high and investment depressed relative to the economy without money. The value of money is the present value of expenditures on liquidity services. During normal times when the real interest rate is high this value is relatively small, but it becomes very large when interest rates fall. In particular, if risk is high enough the real interest rate can be very negative without money, but must be positive if there is money. The value of money grows and raises the equilibrium real interest rate until this condition is satisfied, depressing investment along the way. The result is a liquidity trap. The liquidity trap is an equilibrium outcome — prices are flexible and markets clear. The zero lower bound on nominal interest rates is not binding — money makes the natural rate positive and the central bank is able to hit its inflation target. Money is superneutral and Ricardian equivalence holds. And it’s not a transitory phenomenon — it lasts as long as the bad fundamentals. This is not a model of short-run fluctuations, but rather of persistent slumps associated with liquidity traps. The competitive equilibrium is inefficient. During booms there’s too much investment and too little risk sharing. During liquidity traps there is too little investment and too much risk sharing. But the optimal allocation doesn’t require monetary policy — it can be implemented with the Friedman rule and a tax or subsidy on capital. When investment is too low, subsidize it. When it’s too high, tax it. Money is superneutral, so the optimal monetary policy is the Friedman rule. To study the efficiency of the competitive equilibrium, I first microfound the incomplete idiosyncratic risk sharing with a fund diversion problem with hidden trade. The competitive equilibrium is the outcome of allowing agents to write privately optimal contracts in a competitive market. I then characterize the optimal allocation by a planner who faces the same environment, and ask how it can be implemented with a policy intervention. The inefficiency in this economy comes from hidden trade.1 Private contracts don’t internalize that when they make their consumption and investment decisions, they are affecting prices, such as the interest rate, and therefore the hidden-trade incentive-compatibility constraints of other contracts. Liquidity traps are caused by safe assets with a liquidity premium. Agents can trade 1See Farhi et al. (2009), Kehoe and Levine (1993), Di Tella (2016). 2 with money and no intermediaries. Di Tella (2017) shows that risk shocks that increase idiosyncratic risk can help explain the concentration of aggregate risk on the balance sheet of financial intermediaries that drives financial crises. This paper shows that these risk shocks may also be responsible for liquidity traps. There is also a large literature modeling money as a bubble in the context of OLG or incomplete risk sharing models (Samuelson (1958), Bewley et al. (1980), Aiyagari (1994), Diamond (1965), Tirole (1985), Asriyan et al. (2016), Santos and Woodford (1997)). More recently, Brunnermeier and Sannikov (2016b) study the optimal inflation rate in a similar environment with incomplete risk sharing.4 Money is a bubble, an asset that pays no divi- dends and yet has positive market value and yields the appropriate return, which imposes strong constraints on interest rates. In contrast, there are no bubbles in this paper, and the focus is on how risk shocks can produce liquidity traps. Money derives its value from the liquidity services it provides, as measured by its liquidity premium. This allows a more flexible account of inflation and interest rates and links the value of money to fundamentals, which is useful when trying to understand how the value of monetary assets reacts to shocks or to policy interventions. I study some of the differences and similarities of the two approaches in Section 5. Buera and Nicolini (2014) provide a flexible-price model of liquidity traps, based on borrowing constraints and lack of Ricardian equivalence. Aiyagari and McGrattan (1998) study the role of government debt in a model with uninsurable labor income and binding borrowing constraints. In contrast, here Ricardian equivalence holds and agents have the natural borrowing limit. Government debt without a liquidity premium has no effects on the economy; changing the amount of government debt with a liquidity premium only affects the liquidity premium on government debt and other assets, but not the real side of the economy. The contractual environment micro-founding the incomplete idiosyncratic risk sharing with a fund diversion problem with hidden trade is based on Di Tella and Sannikov (2016), who study a more general environment.5 Di Tella (2016) uses a similar contractual environ- ment to study optimal financial regulation, but does not allow hidden savings or investment. Instead, it focuses on the externality produced by hidden trade in capital assets by financial 4Brunnermeier and Sannikov (2016a) also features a similar environment where money is a bubble, but focuses on the role of intermediaries. 5Cole and Kocherlakota (2001) study an environment with hidden savings and risky exogenous income, and find that the optimal contract is risk-free debt. Here we also have risky investment. 5 intermediaries, That externality is absent in this paper because the price of capital is always one (capital and consumption goods can be transformed one-to-one). There is also a large literature micro-founding the role of money as a means of exchange in a search-theoretic framework (Kiyotaki and Wright (1993), Lagos and Wright (2005), Aiyagari and Wallace (1991), Shi (1997)). Here I use money in the utility function as a simple and transparent way to introduce money into the economy. While these shortcuts are never completely harmless, the purpose of this paper is not to make a contribution to the theory of money, but rather to understand the role it can play as a safe store of value during liquidity traps. Money has value because it is useful for transactions (that’s what money in the utility function is meant to capture), but its most relevant feature in this model is that this value is safe (it has no idiosyncratic risk). 2 Baseline model I use a simple AK growth model with money in the utility function and incomplete idiosyn- cratic risk sharing. The equilibrium is always a balanced growth path, and to keep things simple I will consider completely unexpected and permanent risk shocks that increase id- iosyncratic risk (comparative statics across balanced growth paths). In Section 4 I will introduce proper shocks in a dynamic model. 2.1 Setting The economy is populated by a continuum of agents with log preferences over consumption c and real money m ≡M/p U(c,m) = E [ ˆ ∞ 0 e−ρt ( (1− β) log ct + β logmt ) dt ] Money and consumption enter separately, so money will be superneutral. Money in the utility function is a simple and transparent way of introducing money in the economy.6 As we’ll see, what matters is that money has a liquidity premium. Agents can continuously trade capital and use it to produce consumption yt = akt, but it is exposed to idiosyncratic “quality of capital” shocks. The change in an agent’s capital 6In the Appendix I also solve the model with a cash-in-advance constraint. 6 over a small period of time is d∆k i,t = ki,tσdWi,t where ki,t is the agent’s capital (a choice variable) and Wi,t an idiosyncratic Brownian motion. Idiosyncratic risk σ is a constant here, but we will look at comparative statics of the equilibrium with respect to changes in σ. This is meant to capture a shock that makes capital less attractive and drives up its risk premium. Later we will introduce a stochastic process for σ and allow for aggregate shocks to σ. Idiosyncratic risk washes away in the aggregate, so the aggregate capital stock kt evolves dkt = (xt − δkt)dt (1) where xt is investment. The aggregate resource constraint is ct + xt = akt (2) where ct is aggregate consumption. Money is printed by the government and transferred lump-sum to agents. In order to eliminate any fiscal policy, there are no taxes, government expenditures, or government debt; later I will introduce safe government debt and taxes. For now money is only currency, but later I will add deposits and liquid government bonds. The total money stock Mt evolves dMt Mt = µMdt The central bank chooses µM endogenously to deliver a target inflation rate π. This means that in a balanced growth path µM = π + growth rate. Markets are incomplete in the sense that idiosyncratic risk cannot be shared. They are otherwise complete. Agents can continuously trade capital at equilibrium price qt = 1 (consumption goods can be transformed one-to-one into capital goods, and the other way around) and debt with real interest rate rt = it − π, where it is the nominal interest rate. There are no aggregate shocks for now; I will add them later and assume that markets are complete for aggregate shocks. Total wealth is wt = kt +mt + ht, which includes the capitalized real value of future money transfers ht = ˆ ∞ t e− ´ s t rudu dMs ps (3) 7 definition of h we obtain after some algebra and using the No-Ponzi conditions,8 mt + ht = mt + ˆ ∞ t e−r(s−t) dMs ps = ˆ ∞ t e−r(s−t)msids = mti r − (x̂− δ) (10) Because of log preferences, we get mti = ρβ(kt +mt + ht) which yields λ ≡ mt + ht kt +mt + ht = ρβ r − (x̂− δ) (11) Finally, use the Euler equation (5) and the definition of σc in (7) to obtain (8). How big is the value of liquidity λ? In normal times when r ≫ x̂ − δ, the value of monetary assets λ is small, close to the expenditure share on liquidity services β. To fix ideas, use a conservative estimate of β = 1.7%.9 But when the real interest rate r is small relative to the growth rate of economy x̂ − δ, the value of monetary assets can be very large (in the limit λ → 1). This happens when idiosyncratic risk σ is large – while capital is discounted with a large risk premium, money is discounted only with the risk-free rate, which must fall when idiosyncratic risk σ is large. Figure 1 shows the non-linear behavior of λ as a function of σ. This is an important insight — the value of liquidity may be small in normal times, but can become quite large during periods of low interest rates such as liquidity traps. It’s all about the discounting, so to speak. Proposition 1. For any β > 0, the value of monetary assets λ is increasing in idiosyncratic risk σ, ranges from β when σ = 0 to 1 as σ → ∞. Furthermore, idiosyncratic consumption risk σc = (1 − λ)σ is also increasing in σ, and ranges from 0 when σ = 0 to √ ρ when σ → ∞. For β = 0, λ = 0. 2.3 Non-monetary economy As a benchmark, consider a non-monetary economy where β = 0. In this case, m̂ = ĥ = 0 and therefore λ = 0. The BGP equations simplify to r = a− δ − σ2 and x̂ = a− ρ. Higher idiosyncratic risk σ, which makes investment less attractive, is fully absorbed 8Write mt + ´ ∞ t e−r(s−t) dMs ps = mt + ´ ∞ t e−r(s−t)dms + ´ ∞ t e−r(s−t)msπsds = limT→∞ e−r(T−t)mT + ´ ∞ t e−r(s−t)ms(rs + πs)ds, and use the No-Ponzi condition to eliminate the limit. 9As Section 2.4 shows, β is the expenditure on liquidity premium across all assets, including deposits and treasuries. Say checking and savings accounts make up 50% of gdp and have an average liquidity premium of 2%. Krishnamurthy and Vissing-Jorgensen (2012) report expenditure on liquidity provided by treasuries of 0.25% of gdp. Consumption is 70% of gdp. This yields β = 1.7%. 10 0.1 0.2 0.3 0.4 0.5 0.6 σ 0.15 0.10 0.05 0.05 0.10 r 0.1 0.2 0.3 0.4 0.5 0.6 σ 0.02 0.02 0.04 0.06 x 0.1 0.2 0.3 0.4 0.5 0.6 σ 0.05 0.10 0.15 0.20 0.25 σc 0.1 0.2 0.3 0.4 0.5 0.6 σ 1 2 3 4 5 m , m +h  Figure 2: Real interest rate r, investment x̂, idiosyncratic consumption risk σc, and mon- etary assets m̂ (solid) and m̂+ ĥ (dashed) as functions of idiosyncratic risk σ, in the non- monetary economy (dashed orange) and the monetary economy (solid blue). The lower bound on the real interest rate −π is dashed in black. Parameters: a = 1/10, ρ = 4%, π = 2%, δ = 1%, β = 1.7% by a lower real interest rate r (and therefore lower nominal interest rate i = r + π), but a constant investment rate x̂ and growth x̂ − δ. Figure 2 shows the equilibrium values of r and x̂ in a non-monetary economy for different σ (dashed line). Proposition 2. Without money (β = 0), after an increase in idiosyncratic risk σ the real interest rate r falls but investment x̂ is unaffected. We can understand the response of the non-monetary economy to higher risk σ in terms of the risk premium and the precautionary motive. Use the Euler equation (5) and asset pricing equation (6) to write r = a− δ − σcσ ︸︷︷︸ risk pr. (12) x̂ = a− ρ ︸ ︷︷ ︸ first best + σ2c ︸︷︷︸ prec. mot. − σcσ ︸︷︷︸ risk pr. = a− ρ (13) 11 Larger risk σ makes capital less attractive, so the risk premium α = σcσ goes up. Other things equal this depresses investment. But with higher risk the precautionary saving motive σ2c also becomes larger. Agents face more risk and therefore want to save more. Other things equal, this lowers the real interest rate and stimulates investment. Without money σc = (1 − λ)σ = σ, so the precautionary motive and the risk premium cancel each other out and we get the first best level of investment x̂ = a−ρ for any level of idiosyncratic risk σ (this doesn’t mean that this level of investment is optimal with σ > 0). This is a well known feature of preferences with intertemporal elasticity of one.10 For our purposes, it provides a clean and quantitatively relevant benchmark where higher id- iosyncratic risk that makes investment less attractive is completely absorbed by lower real interest rates which completely stabilize investment. But notice in Figure 2 that the real interest rate r could become very negative; in particular, we may need r ≤ x̂ − δ. This is not a problem without money because capital is risky, but it will be once we introduce money, which is safe, because its value would blow up if r ≤ x̂− δ. 2.4 Monetary economy If there is money, β > 0, an increase in idiosyncratic risk σ creates a liquidity trap. The real interest rate still goes down, but not as much as in the non-monetary economy. Instead, investment x̂ falls.11 Figure 2 also shows the competitive equilibrium of the monetary economy. The value of monetary assets λ plays a prominent role. Monetary assets serve as a safe store of value and improve risk sharing. Their value grows when risk σ is high, which keeps the real interest rate from falling and depresses investment. In particular, without money the real interest rate could be very negative, but with money it must remain above the growth rate of the economy. This is an equilibrium outcome: agents are optimizing, prices are flexible, all markets are clearing, and inflation is on target π. Real money balances m̂ become larger with high σ, because as the nominal interest rate i = r+ π falls the demand 10In the Appendix I solve the model with general Epstein-Zin preferences. Although the real interest rate r always falls with higher risk σ, without money investment x̂ may go up or down depending on whether intertemporal elasticity is lower or higher than one. But Proposition 14 shows that for natural parametrizations, the role of money is the same as in the baseline model with log preferences: it prevents interest rates from falling during downturns and depresses investment, producing a liquidity trap. 11Since output is fixed in the short-run, lower investment implies higher consumption. This is a well understood feature of this simple environment. This is not a model of high-frequency business cycles; it’s a model of persistent slumps produced by liquidity traps. 12 value of expenditures on liquidity services m × i. With log preferences a higher nominal interest rate i reduces real money holdings m proportionally, so that m× i doesn’t change. As a result, λ is not affected and neither is any real variable.15 In contrast to New Keynesian models with nominal rigidities, the zero lower bound on the nominal interest rate, i = r+π ≥ 0, doesn’t really play any essential role in the liquidity trap. While the presence of money creates this lower bound on interest rates, it also raises the equilibrium interest rate so that the zero lower bound is not binding. As Figure 2 shows, the zero lower bound is not binding except for very large levels of idiosyncratic risk σ.16 When the zero lower bound is binding, the central bank is simply unable to deliver the promised inflation target. But the focus and contribution of this paper is the wide parameter region where the zero lower bound is not binding, and yet we have a liquidity trap. What’s more, since money is superneutral, changing the inflation target will always “fix” the zero lower bound problem, but it will have no effects on the real side of the liquidity trap. In fact, under the optimal monetary policy, i ≈ 0, the zero lower bound is never a problem. The role of the liquidity premium Is the mechanism here really about money, or is it actually about safe assets? Here I’ll show that it’s about safe assets with a liquidity premium. Agents can trade risk-free debt, but it doesn’t produce a liquidity trap. As I’ll show below, we can add safe government debt (with or without a liquidity premium) and deposits (private debt with a liquidity premium). The bottom line is that they only produce a liquidity trap to the extent that they have a liquidity premium. To understand the role of the liquidity premium, notice that safe assets without a liquidity premium must be backed by payments with the same present value. Agents may hold the safe assets, but they are also directly or indirectly responsible for the payments backing them. The net value is zero, so they cannot function as a safe store of value. Assets with a liquidity premium instead have a value greater than the present value of payments backing them. The difference is the present value of the liquidity premium. This 15In the Appendix I solve the model with a) a CES demand structure for money and b) a cash-in-advance constraint. In both cases inflation targets have real effects because the expenditure share on liquidity services depends on the nominla interest rate. 16Of course, this depends on the inflation target π. If π is sufficiently negative the ZLB will be binding for all σ. 15 is what makes them a store of value that can improve idiosyncratic risk sharing and create a liquidity trap. Even with positive net value, if idiosyncratic risk sharing was perfect, agents would spend on liquidity services exactly their endowment of liquidity, so liquid assets would wash out. This is why with σ = 0 we can ignore money, as shown in Proposition 3. But with incomplete idiosyncratic risk sharing liquid assets can improve risk sharing. Agents with a bad shock can sell part of their liquid assets to agents with a good shock to reduce the volatility of their consumption (at the cost of accepting volatility in the money services they enjoy). We can see this formally if we integrate an individual agent i’s dynamic budget con- straint (4) to obtain17 E Q̃ [ ˆ ∞ 0 e− ´ t 0 rudu(cit +mitit)dt ] ≤ w0 = k0 + ˆ ∞ 0 e− ´ t 0 rudumtitdt (16) Here for simplicity I assume every agent owns an equal part of the aggregate endowment of capital and monetary assets. On the left hand side we have the present value of his expenditures on consumption goods and money services. On the right hand side we have the aggregate wealth in the economy, k0+m0+h0. The left hand side is evaluated with an equivalent martingale measure Q̃ that captures the market incompleteness; i.e. such that Wit + ´ t 0 (αu/σ)du is a martingale. A risky consumption plan costs less because it can be dynamically supported with risky investment in capital that yields an excess return α. The endowment of money on the rhs is safe, however. With perfect risk sharing, σ = 0, we have αt = 0, so market clearing ´ mitdi = mt means that money drops out of the budget constraint in equilibrium; i.e. EQ̃ [ ´∞ 0 e− ´ t 0 rudumititdt ] = ´∞ 0 e− ´ t 0 rudumtitdt. Money is worth more than the payments backing it because it has a liquidity premium (that’s why it appears on the rhs), but agents spend on holding money exactly that amount, so it cancels out of the budget constraint and has no effects on the equi- librium. But if idiosyncratic risk sharing is imperfect, the excess return then is positive, αt > 0. Then even if in equilibrium agents must hold all the money, ´ mitdi = mt, the present value of expenditures on money services under Q̃ is less than the value of the endowment of money services (which is not risky), EQ̃ [ ´∞ 0 e− ´ t 0 rudumititdt ] < ´∞ 0 e− ´ t 0 rudumtitdt. As a result, money does not drop out of the budget constraint, and they can use the extra 17The intertemporal budget constraint (16) is equivalent to the dynamic budget constraint (4) with incomplete risk sharing if shorting capital kt < 0 is allowed. This is not required in equilibrium of course. 16 value to reduce the risk in their consumption cit. To make this clear, agents could choose safe money holdings mit = mt if they wanted, in which case money would indeed drop out. This corresponds to never trading any money; just holding their endowment. But they are better off trading their money contingent on the realization of their idiosyncratic shocks. They get a risky consumption of money services mi, but reduce the risk in their consumption ci. Government debt, deposits, and Ricardian equivalence. Now let’s introduce safe government debt and bank-issued deposits. Both may have a liquidity premium.18 The bottom line is that government debt and deposits only produce a liquidity trap if they have a liquidity premium. Let bt we the real value of government debt, and dτ lump-sum taxes. The government’s budget constraint is dbt = bt(i b t − π)dt− dτt − dMt pt dmt = dMt pt − πmtdt where ibt is the nominal interest rate on government bonds; I allow for the possibility that ibt < it so that government debt also has a liquidity premium. The government has a no-Ponzi constraint limT→∞ e− ´ T 0 rsds(bT +mT ) = 0. Integrating both equations we obtain mt + bt = ˆ ∞ t e− ´ s t rudu ( msis + bs(is − ibs) ) ds+ ˆ ∞ t e− ´ s t rududτs (17) The government’s total debt is bt+mt, and it must cover it with the present value of future taxes plus what it will receive because its liabilities bt and mt provide liquidity services. When agents hold money, they are effectively paying the government mtit for its liquidity services (the forgone interest); when they hold government debt they are paying bt(it− ibt). In particular, if government debt is as liquid as money, ibt = 0, the only thing that matters is the sum (mt + bt)it. There are also banks that can issue deposits dt that pay interest idt < it. Banks are owned by households. The net worth of a bank is nt and follows the dynamic budget 18Krishnamurthy and Vissing-Jorgensen (2012) show that US Treasuries have a liquidity or convenience yield over equally safe private debt. 17 liquidity trap? Here I’ll show that while issuing equity improves risk sharing, it does not produce a liquidity trap. In the baseline model agents cannot issue any equity. Let’s say instead that they must retain a fraction φ ∈ (0, 1) of the equity, and can sell the rest to outside investors. Issuing outside equity improves idiosyncratic risk sharing, of course. Outside investors can fully diversify across all agents’ equity, creating a safe market index worth (1 − φ)kt. If agents could sell all the equity, φ = 0, we would obtain the first best with perfect risk sharing; with φ > 0 we have incomplete idiosyncratic risk sharing. Since agents can finance an extra unit of capital partly with outside equity, the effective risk of capital for an agent is φσ. In fact, we can obtain the competitive equilibrium by replacing σ by φσ in (5)-(9). The dynamic budget constraint is now21 dwt = (rtwt + ktαt − ct −mtit)dt+ ktφσdWt The risk premium is αt = σc(φσ), and the volatility of consumption is σc = kt/(kt +mt + ht)× (φσ) = (1− λ)(φσ). The value of liquidity is given by λ = ρβ ρ−((1−λ)φσ)2 . But while equity improves risk sharing, it does not produce a liquidity trap. In partic- ular, without money, β = 0, an increase in idiosyncratic risk σ is fully absorbed by lower real interest rates r = a − δ − (φσ)2, but investment remains at the first best x̂ = a − ρ. The reason is that issuing equity makes capital less risky and therefore more attractive. It improves risk sharing in a way that affects the marginal risk from an extra unit of capital and the average risk in agent’s portfolio equally. As a result, it dampens the risk premium σcφσ = (φσ)2 and the precautionary motives σ2c = (φσ)2 equally, canceling out. And the value of the safe market index is equal to the present value of the dividends, so it’s not a pos- itive net value. The aggregate wealth in the economy is still given by the right hand side of (16), but the total value of capital is split into inside and outside equity kt = φkt+(1−φ)kt. In particular, the value of the market index does not blow up to infinity as r approaches the growth rate x̂− δ, as the value of liquidity does.22 21Equity can be diversified so its return must be r. In equilibrium agents are holding wt = nt+mt+ht+et where nt = φkt is the inside equity in their firm that they retain, and et = (1−φ)kt is the diversified outside equity in other agents’ firms. Total equity nt + et = kt; since there are no adjustment costs, Tobin’s q is 1 here. Both inside and outside equity yield r, but the inside equity has idiosyncratic risk (outside equity also has id. risk but it gets diversified). Agents therefore also get a wage or bonus as CEO of their firm to compensate them for the undiversified idiosyncratic risk, ktαt. 22Total equity is always worth total capital, whose price takes into account its uninsurable idiosyncratic risk. As σ grows and r drops, insider wages or bonuses αkt increase to compensate for the idiosyncratic 20 0.1 0.2 0.3 0.4 0.5 0.6 σ -0.15 -0.10 -0.05 0.05 0.10 r 0.1 0.2 0.3 0.4 0.5 0.6 σ -0.02 0.02 0.04 0.06 x 0.1 0.2 0.3 0.4 0.5 0.6 σ 0.05 0.10 0.15 0.20 0.25 σc 0.1 0.2 0.3 0.4 0.5 0.6 σ 0.2 0.4 0.6 0.8 1.0 λ Figure 3: (Cashless limit) The real interest rate r, investment x̂, idiosyncratic consumption risk σc, and value of monetary assets λ as function of σ, for β = 5% (dotted green), β = 1.7% (solid blue – baseline case), β = 0.01% (dotted red), and β = 0 (dashed orange — non-monetary economy). Other parameters: a = 1/10, ρ = 4%, π = 2%, δ = 1%. Cashless limit. The liquidity trap does not hinge on a large expenditure share on liquid- ity services β — it survives even in the cashless limit β → 0. As explained in Section 2.2, the value of monetary assets λ is the present value of expenditures on liquidity discounted at the risk-free rate. When the real interest rate is high relative to the growth rate of the economy λ is small; close to the expenditure share on liquidity services β. But when the real interest rate is very close to the growth rate of the economy, λ can become very large regardless of how small β is. This can be seen very clearly in equation (11). It’s all about the denominator. So if we take the cashless limit, β → 0, the competitive equilibrium will not always converge to that of the non-monetary economy with β = 0. For σ such that in the non- monetary economy the real interest rate is above the growth rate, the monetary economy will indeed converge to the non-monetary one as β → 0. But for σ such that in the non- risk. 21 monetary economy the real interest rate is equal or below the growth rate of the economy, this cannot happen. As the real interest rate drops and approaches the growth rate of the economy, the value of monetary assets λ blows up to keep r above x̂ − δ, no matter how small β is. As a result, we get a liquidity trap even in the cashless limit β → 0, with high interests rates and depressed investment relative to the non-monetary economy. Figure 3 shows the convergence to the cashless limit. Proposition 5. If σ < √ ρ then as β → 0 the competitive equilibrium converges to that of a non-monetary economy with β = 0. But if σ ≥ √ ρ the liquidity trap survives even in the cashless limit β → 0. The real interest rate is high and investment low relative to the non-monetary economy with β = 0. It is important to make sure we are not violating any Ponzi conditions. Proposition 1 ensures that σ2c = ((1 − λ)σ)2 < ρ for all σ and any β > 0, so the Euler equation (5) guarantees that r > x̂ − δ . But what happens if β = 0? Then the only value of λ that satisfies the No-Ponzi condition is λ = 0. If σ ≥ √ ρ the limit of the monetary equilibrium as β → 0 would be an equilibrium of the non-monetary economy with β = 0 except for the No-Ponzi conditions. In other words, the monetary economy, which cannot have bubbles, converges to a bubbly equilibrium of the non-monetary economy. I will discuss the link with bubbles in detail in Section 5. The cashless limit also shows that the liquidity trap does not hinge on log preferences with a constant expenditure share on liquidity services β. In the Appendix I solve the model with CES preferences and demand elasticity of money η < 1, and with a cash-in-advance constraint. In both cases, the only modification is that the expenditure share on liquidity services β̃(i) becomes a function of the nominal interest rate, with β̃(i) → 0 as i → 0. I show that the liquidity trap survives in these settings. Even if we lower the inflation target to reduce the nominal interest rate i → 0, the liquidity trap survives, essentially for the same reason as in the cashless limit.23 3 Efficiency In this Section I study the efficiency properties of the monetary competitive equilibrium. Money provides a safe store of value that prevents the real interest rate from falling and 23In the CIA case we can actually set i = 0, but doing this requires r = x̂−δ if σ > √ ρ, so we get a bubble. This is not surprising, since as β → 0 the monetary economy approaches the non-monetary economy with a bubble. See the Appendix for details, and Section 5 for a discussion of the link with bubbles. 22 To understand this environment, write the local incentive compatibility constraints.27 σct = ρ(1− β)c−1 t ktσ "skin in the game" (22) µct = rt − ρ+ σ2ct Euler equation (23) αt = σctσ demand for capital (24) mt/ct = β/(1− β)i−1 t demand for money (25) The “skin in the game” constraint (22) says that the agent must be exposed to his own idiosyncratic risk to align incentives. The agent could always misreport a lower return and consume those funds, so incentive compatibility requires that the present value of his consumption goes down by ktσ after bad reported outcomes Yt. The skin in the game constraint is expressed in terms of the volatility of his consumption σct. If he steals a dollar, he won’t consume the dollar right away; he will consume it only at rate ρ(1− β).28 So his consumption must be exposed to his idiosyncratic shock as in (22). This is costly, of course. In the first best we would have perfect idiosyncratic risk sharing, σct = 0, but we need to expose the agent to risk to align incentives. The other IC constraints (23), (24), and (25) come from the agent’s ability to save at the risk-free rate, secretly invest in capital, and choose his money holdings, respectively. Ultimately they arise from agents’ ability to secretly trade amongst themselves. These constraints are binding. The principal would like to front-load the agent’s consumption to relax the idiosyncratic risk sharing problem, as can be seen in (22). By distorting the intertemporal consumption margin he can relax the risk sharing one. But the agent has access to hidden savings, so the principal must respect his Euler equation. Even then, if the agent couldn’t secretly invest in capital or choose his money holdings, the principal could use this to provide better incentives. In particular, he would like to promise less capital and risk in the future and after bad outcomes. This relaxes the agent’s precautionary motive and makes it cheaper for the principal to provide incentives. But he cannot do this because the agent can secretly invest in capital on his own. The same intuition goes for his money 27The competitive equilibrium and the planner’s allocation will be BGPs, but it is important to allow for time-varying allocations and prices. 28An equivalent derivation: the agent’s continuation utility if he doesn’t misbehave, Ut, follows a promise- keeping constraint dUt = ( ρUt − ( β log(ct) + (1− β) log(mt) )) dt+ σUtdWt. If he misreports he can imme- diately consume what he stole (he is indifferent at the margin) and obtain utility (1−β)c−1 t kt, so incentive compatibility requires σUt = (1−β)c−1 t ktσ. Because the agent can secretly save and invest, his continuation utility must be Ut = A+ 1 ρ log(ct), so we get σct = ρσUt. 25 holdings. The tradeoff between intertemporal consumption smoothing and idiosyncratic risk shar- ing captured in the skin in the game constraint is central to the liquidity trap. First, we’d like to see how this constraint manifests in the competitive equilibrium. Write σct = (1 − λ)σ = (kt/wt)σ; using ct = ρ(1 − β)wt, we obtain equation (22). Now, when idiosyncratic risk σ goes up and raises the value of monetary assets λ, it is moving the equi- librium along this IC constraint.29 In equilibrium this must be consistent with individual optimization, captured by the risk premium and the precautionary motive. As we’ll see, the planner will choose a different point on this IC constraint. All these conditions are only necessary, and are derived from considering local, single de- viations by the agent. Establishing global incentive compatibility is difficult in general, but in this environment it’s straightforward. Because the optimal contract coincides with the optimal portfolio problem where the agent essentially does what he wants, global incentive compatibility is ensured. 3.2 Planner’s problem The planner faces the same environment with moral hazard and hidden trade.30 An allo- cation is a plan for each agent (ci,mi, ki) and aggregate consumption c, investment x, and capital k satisfying the resource constraints (1), (2), ct = ´ 1 0 ci,tdi and kt = ´ 1 0 ki,tdi. An allocation is incentive compatible if there exist processes for real interest rate r, nominal interest rate i, and idiosyncratic risk premium α, such that (20) holds for each agent. An in- centive compatible allocation is optimal if there is no other incentive compatible allocation that weakly improves all agents’ utility and at least one strictly so. The local IC constraints are necessary for an incentive compatible allocation. But the thing to notice is that constraints (23), (24), and (25) involve prices that the planner doesn’t take as given. What these constraints really say is that all agents must be treated the same, or else they would engage in hidden trades amongst themselves. This is why the planner can improve over the competitive equilibrium. For example, the planner realizes that he 29It’s not the β that improves risk sharing; it is fixed as σ goes up, and the liquidity trap survives in the cashless limit with β → 0. It’s the distortions in the intertemporal consumption smoothing margin. 30It is natural to wonder if the planner could simply refuse to enforce debt contracts in order to eliminate hidden trade. Here I’m assuming the hidden trade is a feature of the environment that the planner cannot change; e.g. agents may have a private way of enforcing debt contracts. As we’ll see, the hidden trade constraints are already not binding for the planner, so he wouldn’t gain anything from doing this. And we wouldn’t learn a lot from pointing out that the planner could do better if he can change the environment. 26 can change the growth rate of all agents’ consumption at the same time, without creating any incentives to engage in hidden trades. So all agents get the same µc, σc, m/c, and k/c, and only differ in the scale of their contract, corresponding to how much initial utility they get. The only true constraint for the planner is the skin in the game constraint (22), which can be re-written using the resource constraints as σc = ρ(1− β) a− x̂t σ (26) The planner’s problem then boils down to choosing the aggregate consumption c, investment x, and real money balances m to maximize the utility of all agents. Using the aggregate resource constraints (1) and (2), and the incentive compatibility constraints, we can write the planner’s objective function E [ ˆ ∞ 0 e−ρt ((1− β) log(ci,t) + β log(mi,t)) dt ] = E [ ˆ ∞ 0 e−ρt ( log(k0) + log(a− x̂t) + β log ( βi−1 t 1− β ) + x̂t − δ − σ2ct 2 ρ ) dt ] (27) The planner’s problem then is to choose a process for x̂ and i to maximize (27) subject to (26). First, it is optimal to set i ≈ 0 (Friedman rule).31 This maximizes the utility from money, and costs nothing. Second, the FOC for x̂t is 1 ρ = 1 a− x̂t + ( ρ(1− β)σ a− x̂t )2 1 a− x̂t (28) =⇒ x̂ = a− ρ ︸ ︷︷ ︸ first best −σ2c (29) where recall that σc = ρ(1−β)σ a−x̂ . The lhs in (28) captures the benefit of having more capital forever. The rhs captures the cost of increasing investment. The first term is the utility loss from reducing consumption. The second term captures the loss from worse idiosyncratic risk sharing. A more backloaded consumption path makes fund diversion more attractive, and therefore tightens the IC constraint (22). If we didn’t have this second term (if we had 31Because of the log preferences we can’t set i = 0 because we would get infinite utility. But i = 0 is optimal in a limiting sense. 27 Both share the same expression for σc/σ = ρ(1− β) a− x̂ (32) which comes from the incentive compatibility constraint (22), and pins down the set of x̂ and σc/σ that are incentive compatible. But the planner and the competitive equilibrium disagree on which (x̂, σc/σ) pair to pick. For a fixed σ, we can interpret (30) and (31) as the desired investment x̂ for a given σc. Figure 5 captures the situation. For the planner, reducing investment below the first best is a way of improving idiosyncratic risk sharing, so the bigger σc is the more he will want to reduce investment. But in the competitive equilibrium it’s all about the difference between the precautionary motive σc × σc and the risk premium σc × σ, which is non-monotonic in σc. 34 At σc = 0 the precautionary motive and the risk premium are equal and cancel each other out, so investment wouldn’t be postponed in the competitive equilibrium, and the planner agrees with this. If we increase σc the precautionary motive falls behind the risk premium, so investment would fall in the competitive equilibrium, and it would do so by more than in the planner’s solution because σ > σc initially. However, after σc > 1 2σ, the difference between the precautionary motive and the risk premium starts to shrink. In fact, we know that if σc = σ the two would be exactly the same and we would be back to the first best investment. So investment in the competitive equilibrium as a response to σc is non-monotonic. For the planner, meanwhile, higher σc always increases his desire to reduce investment. The planner and the competitive equilibrium only agree on x̂ as a response to σc when σc = 0 or σc = 1 2σ. For σc/σ ∈ (0, 1/2) the planner wants more investment and less risk sharing than the competitive equilibrium; for σc/σ ∈ (1/2, 1 − β) the planner wants less investment and more risk sharing than the competitive equilibrium. Of course, σc is endogenous. Money improves risk sharing and dampens the precaution- ary motive relative to the risk premium, reducing investment below the first best. Whether investment in the competitive equilibrium is too high or too low depends on how much risk sharing money provides in equilibrium. If β ≥ 1/2 the value of monetary assets is very large, so there is too much risk sharing and too little investment in the competitive equilib- 34One may wonder how Figure 5 captures the situation if β = 0. In that case the IC curve shifts down so that it touches x̂ = a−ρ at σc/σ = 1 (in general it touches a−ρ at 1−β). This is of course the non-monetary equilibrium. There may also be another intersection between x̂CE and the IC curve for σc/σ < 1, but this violates the No-Ponzi conditions. It corresponds to a bubble equilibrium of the non-monetary economy. See Section 5 on bubbles and the discussion on the cashless limit in Section 2.4. 30 CE CE IC 1-β a-ρ 0.2 0.4 0.6 0.8 1.0 σc/σ -0.10 -0.05 0.05 x Figure 5: Thick black line is the locus of IC (x̂, σc). The blue line is the x̂ in the CE as a function of σc/σ corresponding to (31), the green line for the SP corresponding to (30). Solid is for σ = 0.2, dotted for σ = 0.5. Parameters: a = 1/10, ρ = 4%, π = 2%, δ = 1%, β = 1.7% rium for any σ. For the quantitatively relevant case with β < 1/2, the value of monetary assets is too small for low σ, so there is too little risk sharing and too much investment. But for high σ the value of monetary assets is too large, so there is too much risk sharing and too little investment. Proposition 8. If β ∈ (0, 1/2), there is a σ∗ = 2 √ ρ(1− 2β) > 0 such that for σ ∈ (0, σ∗) investment and consumption risk are too high in the competitive equilibrium, compared to the planner’s allocation; that is, x̂CE > x̂SP and σCEc > σSPc . For σ > σ∗ investment and consumption risk are too low in the competitive equilibrium; that is, x̂CE < x̂SP and σCEc < σSPC . If β ∈ [1/2, 1) investment and consumption risk are too low in the competitive equilibrium for any σ > 0. 31 3.4 Implementation of the optimal allocation We can implement the social planner’s optimal allocation as a competitive equilibrium with a tax on capital income τk, rebated lump-sum to agents (in addition to the Friedman rule, i ≈ 0).35 Total wealth now includes not only money, but also the lump-sum rebates wt = kt + mt + ht + ´∞ t e−rsτks ksds. The only equilibrium condition that changes is the asset pricing equation for capital (6) which becomes a− τk − δ − r ︸ ︷︷ ︸ α = σcσ (33) The competitive equilibrium still has an Euler equation r = ρ+(x̂−δ)−σ2c , and idiosyncratic risk sharing is given by36 σc = ρ(1− β) a− x̂ σ (34) Putting together the pricing equation for capital and the Euler equation, we obtain equi- librium investment x̂ = a− ρ− τk + σ2c − σcσ Taxing capital τk > 0 produces less investment. Recall that the planner’s solution has x̂ = a− ρ− σ2c , and σc = ρ(1−β) a−x̂ σ. So the wedge between the planner’s condition and the competitive equilibrium is given by τk−2σ2c +σcσ. If we set the tax on capital τk = 2σ2c − σcσ (35) we internalize the externality produced by hidden trade. This allows us to implement the planner’s allocation as a competitive equilibrium, provided that the required real interest rate is not too low. Of course, we also need to choose the inflation target to deliver the Friedman rule i = r + π ≈ 0. A Balanced Growth Path Equilibrium with tax τk and inflation π is an interest rate r, investment x̂, and real money m̂ such satisfying the Euler equation (5), the asset pricing equation (33), risk sharing equation (34), and money demand equation (9), as well as 35This is consistent with the hidden trade in the environment. The planner is taxing or subsidizing all capital, regardless of who holds it. 36This comes from the IC constraint (22). Alternatively, use ct = ρ(1 − β)wt and σc = σw = kt wt σ = ρ(1−β) ct ktσ = ρ(1−β) a−x̂ σ. 32 motion dψt = µψtdt+ σ̃RSψ dZ̃RSt (39) ψt is not exposed to TFP shocks because the economy is scale invariant to effective capital kt. Total wealth is wt = kt(1 + ψt), and we can recover λt = ψt 1+ψt . With this definition of ψ, the competitive equilibrium is a process for the real interest rate r and price of risk θTFP and θRS , investment x̂, and real money holdings m̂, all contingent on the history of shocks ZTFP and ZRS and satisfying the equilibrium conditions: rt = ρ+ (x̂t + g − δ) + µψt/(1 + ψt)− σ2ct − (σ̃TFPct )2 − (σ̃RSct )2 Euler (40) rt = a+ g − δ − σctσt − θTFPt σ̃TFP Asset Pricing (41) σc = σt/(1 + ψt) Id. risk (42) σ̃TFPc = θTFPt = σ̃TFP TFP risk (43) σ̃RSct = θRSt = σ̃RSψt /(1 + ψt) RS risk (44) m̂t = β/(1 − β)× (a− x̂t)/(rt + πt) Money (45) as well as limt→∞ E Q [ e− ´ s t ruduwt ] = 0 and it = rt + πt > 0. There is now a precautionary motive for idiosyncratic risk σ2ct, for aggregate TFP risk (σ̃TFPct )2, and for aggregate risk shocks (σ̃RSct )2. In addition, the growth rate of consumption in the Euler equation µct = x̂t+g−δ+µψt/(1+ψt) comes from writing ct = ρ(1−β)wt and wt = kt(1+ψt), and computing the drift of ct. Likewise, capital pays a risk premium for its idiosyncratic risk σctσt and for aggregate TFP risk θTFPt σ̃TFP = (σ̃TFP )2 (its return is not correlated with aggregate risk shocks, so there’s no risk premium for that). Idiosyncratic risk sharing is still given by σct = (1− λt)σ = σ 1+ψt — monetary assets provide a safe store of value that improves idiosyncratic risk sharing. But aggregate shocks cannot be shared. Since the economy is scale invariant to effective capital, TFP shocks don’t affect x̂t = xt/kt or ĉt = ct/kt, so σ̃TFPct = σ̃TFP . In contrast, risk shocks don’t affect the level of effective capital, but can affect the value of monetary assets and therefore x̂t and ĉt. So the price of the aggregate risk shock θRSt = σ̃RSct is endogenous and depends on how the shock affects ψt. Equation (44) comes from writing ct = ρ(1− β)wt and wt = kt(1 + ψt) and computing the exposure of consumption to aggregate risk shocks. If we know the behavior of ψt we can then obtain every other equilibrium object from 35 (40)-(45). In contrast to the baseline model, the value of monetary assets is not a constant — it will be characterized by an ODE. Reasoning as before, the value of monetary assets is equal to the present value of ex- penditures on liquidity services. Proposition 11. The equilibrium value of monetary assets satisfies mt + ht = E Q t [ ˆ ∞ t e− ´ s t rudumsisds ] (46) From this it is easy to obtain a BSDE for ψt m̂tit + µψt + ψt(x̂t + g − δ − rt) = θTFPt σ̃TFPψt + θRSt σ̃RSψt (47) and transversality condition limT→∞ E Q [ e− ´ T 0 rudukTψT ] = 0. It is worth noting that because of the log preferences, dynamics only matter through the value of monetary assets ψ. Without money we have ψt = 0, and the competitive equilibrium does not depend on the stochastic process for σt, as can be seen from inspecting equations (40)-(45) (only on the current σt). For any given value of σt the competitive equilibrium without money is the same as in the static economy with a constant σ = σt. With money, the only reason this is not the case is because ψt is forward-looking, as shown in equation (46). As we will see below, the planner’s optimal allocation for a given value of σt also coincides with the static case with a constant σ = σt. We can subtract equation (40) from (41) and re-arrange to obtain an expression for investment x̂t, analogous to (15): x̂ = a− ρ+ σ2ct − σctσ + ( σ̃RSψt 1 + ψt )2 − µψt 1 + ψt (48) Idiosyncratic and aggregate risk play very different roles. Money provides a safe store of value that improves idiosyncratic risk sharing. As before, it dampens the idiosyncratic precautionary motive σ2ct relative to the idiosyncratic risk premium σctσ and depresses investment. Aggregate risk, on the other hand, simply cannot be shared. As a result, aggregate TFP risk σ̃TFP reduces the equilibrium real interest rate r, but does not affect investment x̂, just as in the non-monetary economy. The risk premium σ̃TFPct σ̃TFPand precautionary motive (σ̃TFPct )2 produced by aggregate TFP risk exactly cancel each other 36 out even when there is money. Likewise, the exogenous growth rate of TFP g affects only the equilibrium real interest rate r, but has no effect on the value of liquidity ψ or investment x̂.38 The stochastic behavior of σt matters through the equilibrium behavior of the value of monetary assets ψt, which is forward-looking. Even conditional on the current value of ψt, if ψt+s is expected to be high in the future, this means the idiosyncratic precautionary motive will be weaker than the idiosyncratic risk premium and therefore consumption will be higher relative to capital; that is, ĉt+s will be higher. As a result, for a given interest rate rt, pinned down by the asset pricing equation (41), agents want less investment x̂t to achieve their desired intertemporal consumption smoothing. This is why µψt appears in (48). Likewise, aggregate risk shocks matter because they induce aggregate volatility in agents’ consumption. Since capital is not exposed to risk shocks, it is an attractive hedge. This is why σ̃RSψt appears in (48). Recursive Equilibrium. We look for a recursive equilibrium with σt as the state vari- able, so ψt = ψ(σt) can be characterized as the solution to an ODE derived from (47). Use Ito’s lemma to compute the drift and volatility of ψ, the Euler equation (40) to eliminate r and x̂ terms, and m̂tit = ρβwt/kt = ρβ(1 + ψt). Proposition 12. The equilibrium value of ψ(σ) solves the ODE ρβ(1 + ψ) + ψ′φ(σ̄ − σ) + 1 2ψ ′′σν2 1 + ψ = ψ(ρ− ( σ 1 + ψ )2) + ( ψ′√σν 1 + ψ )2 (49) With a solution ψ(σ) to (49) we can obtain all the other equilibrium objects using (40)- (45). If we also satisfy the transversality conditions, we have a competitive equilibrium. Figure 6 shows the competitive equilibrium in the dynamic model, with and without money. It has essentially the same properties as in the BGP economy with constant id- iosyncratic risk σ. Money improves idiosyncratic risk sharing but depresses investment. After a risk shock increases σt the value of monetary assets goes up, so the real interest rate falls less than it would without money and investment falls. The effects of the shock are as persistent as the shock itself, and inflation is always on target πt. The target itself doesn’t affect any real variable other than real money holdings mt through the interest 38This is a property of preferences with EIS of one. In general σ̃TFP and g could affect investment x̂ even without money, and can affect the value of liquidity ψ because r + θTFP σ̃TFP − g is not invariant to changes in g and σ̃TFP . 37 as in the stationary model, but it now has a dynamic dimension. As before, the planner and the competitive equilibrium share the same locus of incentive compatible combinations of investment x̂ and idiosyncratic risk sharing σc given by (56), but they choose different (x̂, σc) combinations. However, while the planner only cares about the current value of idiosyncratic risk σt , the competitive equilibrium picks a (x̂t, σct) that depends on the future behavior of idiosyncratic risk σt through the value of monetary assets today ψt. Figure 6 also shows the planner’s optimal allocation. An important difference with the static case is that it is possible for investment to be too low in the competitive equilibrium for very low σt, too high for intermediate σt, and again too low for high σt (and the same goes for σc through (56)). The reason for this is that while the planner’s allocation coincides with the static case with a constant σ = σt, the competitive equilibrium depends on the stochastic behavior of σt through the value of monetary assets ψt. If σt today is very low but is expected to go up to σ̄ very fast, the value of ψt will be closer to the value of monetary assets in the static case with σ = σ̄ rather than σ = σt. So the competitive equilibrium chooses a (x̂t, σct) on the locus on incentive compatible constraints given by (56) that is closer to what it would choose in the static case with σ = σ̄. This is captured in equation (48) through the role of µψt and σψt. 41 We can still implement the planner’s optimal allocation with a tax on capital τk that internalizes the externality produced by hidden trade. Introducing the tax only changes the asset pricing equation rt = a+ g − δ − τkt − σctσt − θTFPt σ̃TFP and therefore the equilibrium investment x̂t = a− ρ− τkt + σ2ct − σctσt − µĉt + (σ̃RSct )2 Comparing this expression to (57) we find the wedge τkt = 2σ2ct − σctσt − µĉt + (σ̃RSct )2 (58) Proposition 13. Let P be an optimal allocation with processes for investment x̂, aggregate consumption ĉ, and idiosyncratic consumption risk σc. Then rt = ρ+µct−σ2ct− (σ̃TFPct )2− (σ̃RSct )2, θTFPt = σ̃TFPc , θRSt = σ̃RSc , x̂ and m̂t = β 1−β a−x̂t rt+πt is a competitive equilibrium with 41In this solution, σψ → 0 as σ → 0, so it’s all really about the drift µψ which does not vanish as σ → 0. 40 τkt given by (58) and inflation target πt, provided that limt→∞ E Q [ e− ´ s t ruduktĉt ] = 0 and it = rt + πt > 0. The optimal inflation target satisfies it = rt + πt ≈ 0 (Friedman rule). 5 Discussion 5.1 Liquidity premium and bubbles. In the model the liquidity premium comes from the fact that liquid assets appear in the utility function, so agents are willing to hold them even if their yield is below the interest rate. This is the simplest and most transparent way of introducing assets with a liquidity premium, and it is meant to capture that money and other assets are useful for transactions. In the Appendix I also solve the model with a cash-in-advance constraint, and obtain the same results except that the expenditure share on liquidity depends on the nominal interest rate. This would also be the case with a more flexible, CES specification for money in the utility function. We could also use a more microfounded model of monetary exchange, as in Lagos and Wright (2005). As long as money has a liquidity premium, it will produce liquidity traps. The liquidity premium could also reflect the collateral value of some assets, such as government debt. If an asset serves to relax a financial friction, it will carry a convenience yield that will perform the same role as the liquidity premium. In fact, at least part of the spread on Treasuries is probably due to its collateral value.42 There is also a large literature modeling money as a bubble in the context of OLG or incomplete risk sharing models (Samuelson (1958), Bewley et al. (1980), Aiyagari (1994), Diamond (1965), Tirole (1985), Asriyan et al. (2016), Santos and Woodford (1997)). More recently, Brunnermeier and Sannikov (2016b) study the optimal inflation rate in a similar environment with incomplete risk sharing. The liquidity view in this paper has differences and similarities with the bubble view. A safe bubble can provide a safe store of value that improves idiosyncratic risk sharing and depresses investment, just like money does in this paper. But there are also important differences. To understand the link with bubbles, we can ignore the government bonds and deposits, bt = dt = 0, and assume money is not printed, dMt = 0, and therefore in a BGP inflation is simply minus the growth rate of the economy. Write equation (18) without the No-Ponzi 42See Krishnamurthy and Vissing-Jorgensen (2012). 41 condition wt = kt + ˆ ∞ t e− ´ s t rudumtitdt ︸ ︷︷ ︸ liquidity services + lim T→∞ e− ´ T 0 rsdsmT ︸ ︷︷ ︸ bubble In this paper, the No-Ponzi condition eliminates the last term limT→∞ e− ´ T 0 rsdsmT . Money is not a bubble – it derives its value from it’s liquidity premium, which arises because money provides liquidity services. In models with bubbles, instead, money doesn’t have a liquidity premium, so the nominal interest rate must be zero it = 0. Money is an asset that doesn’t pay any dividend, but still yields the arbitrage-free market return — rt if there is no aggregate risk. It has positive value only because the last term doesn’t vanish. In a BGP, this requires r = growth rate. But in both the liquidity and the bubble views, money provides a store of value that improves risk sharing, σc = kt wt σ < σ. Modeling money as an asset with a liquidity premium has several advantages. First, money does have a liquidity premium. The bubble view cannot explain why people hold money when they can hold safe nominal bonds that pay interest. If the bubble is really money, then the equilibrium nominal interest rate must be zero and the real interest rate must be equal to the growth rate of the economy. The liquidity view instead can provide a more flexible account of inflation and interest rates. Alternatively, the bubble may not really be money. It could be housing, the stock market, government debt, social security, or even tulips. This can potentially be very interesting, but it is difficult to determine if asset values really have a bubble component. In contrast, it’s relatively straightforward to establish that some assets have a liquidity premium. Linking the value of money to liquidity premiums also allows us to understand its behavior in response to shocks and policy interventions, since it is grounded in fundamentals. Finally, it’s worth pointing out that bubbles may have idiosyncratic risk. To the extent that agents cannot diversify this idiosyncratic risk, the bubble will not perform the same role as money, which is safe. For example, suppose there is a housing bubble, so that house prices are 10% above their fundamental value, but each agent must buy one house whose value has idiosyncratic risk. Then the bubble may not be able to produce a liquidity trap, since it is not a safe store of value. There is also a link between the liquidity view and the bubble view in the cashless limit. As explained in Section 2.4, the liquidity trap in this paper does not hinge on large expenditures on liquidity services (large β). This can be formalized by noting that the monetary economy does not converge to the non-monetary one as β → 0. The reason is 42 6 Conclusions Liquidity traps are associated with some of the deepest and most persistent slumps in modern history. This paper puts forward a flexible-price theory of liquidity traps. During downturns risk goes up and makes investment less attractive. Without money, the real interest rate would fall and investment would remain high. But the role of money as a safe store of value prevents interest rates from falling during downturns, and depresses investment. Money improves idiosyncratic risk sharing and drives a wedge between the risk premium on capital and agents’ precautionary saving motive, which keeps interest rates high and depresses investment. While the value of monetary assets is pretty small during normal times, during persistent slumps with low interest rates their value can become very large. The result is a liquidity trap. The liquidity trap is an equilibrium outcome — prices are flexible, markets clear, and inflation is on target — but it’s not efficient. In contrast to most of the literature on liquidity traps, the inefficiency does not arise from sticky prices and the zero lower bound on interest rates. It comes from the inability of private contracts to prevent agents from engaging in hidden trade. During booms investment is too high and money doesn’t provide enough risk sharing; during downturns investment is too depressed and money provides too much risk sharing. But the socially optimal allocation doesn’t require monetary policy, other than the Friedman rule. It’s much simpler. When investment is too high, tax it; when it’s too low, subsidize it. 7 Bibliography References Aiyagari, S. R. (1994). Uninsured idiosyncratic risk and aggregate saving. The Quarterly Journal of Economics, 659–684. Aiyagari, S. R. and E. R. McGrattan (1998). The optimum quantity of debt. Journal of Monetary Economics 42 (3), 447–469. Aiyagari, S. R. and N. Wallace (1991). Existence of steady states with positive consumption in the kiyotaki-wright model. The Review of Economic Studies 58 (5), 901–916. 45 Asriyan, V., L. Fornaro, A. Martin, and J. Ventura (2016, September). Monetary Policy for a Bubbly World. NBER Working Papers 22639, National Bureau of Economic Research, Inc. Bansal, R., D. Kiku, I. Shaliastovich, and A. Yaron (2014). Volatility, the macroeconomy, and asset prices. Journal of Finance. Bansal, R. and A. Yaron (2004). Risks for the long run: A potential resolution of asset pricing puzzles. The Journal of Finance 59 (4), pp. 1481–1509. Barillas, F., L. P. Hansen, and T. J. Sargent (2009). Doubts or variability? journal of economic theory 144 (6), 2388–2418. Bewley, T. et al. (1980). The optimum quantity of money. Models of monetary economies, 169–210. Bloom, N. (2009). The impact of uncertainty shocks. Econometrica 77 (3), pp. 623–685. Bloom, N., M. Floetotto, N. Jaimovich, I. Saporta-Eksten, and S. Terry (2012). Really uncertain business cycles. NBER working paper 18245. Brunnermeier, M. K. and Y. Sannikov (2016a). The i theory of money. Technical report, National Bureau of Economic Research. Brunnermeier, M. K. and Y. Sannikov (2016b, May). On the optimal inflation rate. Amer- ican Economic Review 106 (5), 484–89. Buera, F. and J. P. Nicolini (2014). Liquidity traps and monetary policy: Managing a credit crunch. Caballero, R. J. and A. Simsek (2017). A risk-centric model of demand recessions and macroprudential policy. Technical report, National Bureau of Economic Research. Campbell, J., S. Giglio, C. Polk, and R. Turley (2012, March). An intertemporal capm with stochastic volatility. working paper. Campbell, J., M. Lettau, B. G., and Y. Xu (2001). Have individual stocks become more volatile? an empirical exploration of idiosyncratic risk. The Journal of Finance 56 (1), pp. 1–43. 46 Christiano, L., R. Motto, and M. Rostagno (2014). Risk shocks. American Economic Review 104 (1). Cole, H. L. and N. R. Kocherlakota (2001). Efficient allocations with hidden income and hidden storage. The Review of Economic Studies 68 (3), 523–542. Di Tella, S. (2016). Optimal regulation of financial intermediaries. Working paper - Stanford GSB. Di Tella, S. (2017, August). Uncertainty shocks and balance sheet recessions. Forthcoming Journal of Political Economy . Di Tella, S. and Y. Sannikov (2016). Optimal asset management contracts with hidden savings. Working paper, Stanford GSB and Princeton University. Diamond, P. A. (1965). National debt in a neoclassical growth model. The American Economic Review 55 (5), 1126–1150. Eggertsson, G. B. et al. (2003). Zero bound on interest rates and optimal monetary policy. Brookings papers on economic activity 2003 (1), 139–233. Eggertsson, G. B. and P. Krugman (2012). Debt, deleveraging, and the liquidity trap: A fisher-minsky-koo approach. The Quarterly Journal of Economics 127 (3), 1469–1513. Eggertsson, G. B. and M. Woodford (2004). Policy options in a liquidity trap. The American Economic Review 94 (2), 76–79. Farhi, E., M. Golosov, and A. Tsyvinski (2009). A theory liquidity and regulation of financial intermediation. The Review of Economic Studies 76, 973–992. He, Z., B. T. Kelly, and A. Manela (2015). Intermediary asset pricing: New evidence from many asset classes. Available at SSRN 2662182 . He, Z. and A. Krishnamurthy (2013). Intermediary asset pricing. The American Economic Review 103 (2), 732–770. Karatzas, I. and S. Shreve (2012). Brownian motion and stochastic calculus, Volume 113. Springer Science & Business Media. 47 are left with a) λ → 0. From (14) and (15) we see that r and x converge to the values on the non-monetary economy with λ = β = 0. If instead σ ≥ √ ρ, we cannot have λ→ 0, because it implies that ρ− ((1−λ)σ)2 ≤ 0 at some point along the way (for λ small enough). Since λ > β always, this requires ρβ < 0, which is not true. So we have b) λ → 1 − √ ρ σ ≥ 0, and the inequality is strict if σ > √ ρ. From (14) and (15) we see that the real interest rate r is high and investment x̂ low relative to the economy without money (β = 0). Proposition 6 Proof. See Online Appendix Proposition 7 Proof. Combine (29) with σc = ρ(1−β)σ a−x̂ to obtain σc = ρ(1− β) ρ+ σ2c σ It follows that when σ = 0 we get σc = 0, and σc is increasing in σ. Rewrite it σc σ = ρ(1− β) ρ+ ( σc σ )2 σ2 It follows that σc/σ is decreasing in σ. The properties of x̂ follow from equation (29). Finally, write r = ρ+ x̂− δ − σ2c = a− δ − 2σ2c It follows that r falls with σ. Proposition 8 Proof. First take the β ∈ (0, 1/2) case. Rewrite (30) and (31) in terms of y = σc/σ x̂SP = a− ρ− y2σ2 Social Planner x̂CE = a− ρ− yσ2 × (1− y) Competitive Equilibrium 50 and the incentive compatible combinations x̂IC = a− ρ(1− β) y The competitive equilibrium lies at the intersection of x̂CE and x̂IC ; call the corresponding yCE ∈ [0, 1 − β]. The planner’s allocation lies at the intersection of x̂SP and x̂IC , call the corresponding ySP ∈ [0, 1− β]. We know that y = σc/σ can range from 0 to 1− β, in both the CE and SP (the upper bound comes from knowing that investment is below the first best in both the CE and the SP, and using x̂IC . x̂IC is increasing, strictly concave, and ranges from −∞ when y = 0 to the first best a− ρ when y = 1−β. It does not depend on σ, so it will be fixed when we do comparative statics. x̂CE and x̂SP do depend on σ. They both start at the first best a− ρ when y = 0. x̂SP is strictly decreasing and concave (it’s an inverted parabola) with vertex at (0, a − ρ). So it must cross x̂IC exactly once. x̂CE is a parabola with vertex at (12 , a − ρ − 1 4σ 2). Importantly, it intersects with x̂SP at this point. For σ > 0 they intersect at exactly two points, corresponding to y = 0 and y = 1/2, and this implies that x̂CE < x̂SP for all y ∈ (0, 1/2), and x̂CE > x̂SP for all y ∈ (1/2, 1 − β). Finally, x̂CE < a − ρ for all y ∈ (0, 1 − β). In particular, x̂CE(1− β) = a− ρ− σ2β(1− β) < a− ρ. Since x̂CE is strictly convex and x̂IC is strictly concave they intersect at tow points at most. Since x̂IC = ∞ for y = 0, x̂IC crosses x̂CE first from below, and the from above. But since x̂IC(1 − β) = 1 − ρ > x̂CE(1 − β), the second intersection has y > 1 − β, so it is not in the range on y. There is then only one valid intersection between x̂CE and x̂IC ; we called it yCE ∈ [0, 1 − β] and x̂IC < x̂CE for all y < yCE and x̂IC > x̂SP for all y > yCE in the range of y. Now the lower envelope of x̂CE and x̂SP , x̂L = min{x̂CE , x̂SP } coincides with x̂CE for y ∈ [0, 1/2] and with x̂SP for y ∈ [1/2, 1 − β]. This implies that if x̂IC first intersects with the lower envelope for y < 1/2, it must do so at yCE, and if it first intersects at y > 1/2, it must do so at ySP . In the first case, since x̂SP > x̂CE for y < 1/2, it is strictly decreasing, and goes from a − ρ for y = 0 to a − ρ − 1 4σ 2 = min x̂CE for y = 1/2; and x̂IC is strictly increasing and goes to 1− ρ; then it means that ySP < 1/2 as well and ySP > yCE. In the second case, obviously yCE > ySP > 1/2. If it first intersects at y = 1/2 then yCE = ySP . It follows immediately that if yCE < ySP and both below 1/2, then x̂CE < x̂SP and 51 σCEc < σSPc . On the other hand, if yCE > ySP and both above 1/2, then x̂CE > x̂SP and σCEc > σSPc . It only remains to see which will hold for a given σ. Since both x̂CE and x̂SP are decreasing for y ∈ (0, 1/2), and x̂IC is always increasing, it is enough to compare their values at y = 1/2. If x̂CE = x̂SP ≥ x̂IC at y = 1/2, then yCE and ySP are both in [1/2, 1 − β]. If instead x̂CE = x̂SP ≤ x̂IC at y = 1/2, then yCE and ySP are both in (0, 1/2). x̂CE = x̂SP ≥ x̂IC ⇐⇒ a− ρ− 1 4 σ2 ≥ a− 2ρ(1− β) ⇐⇒ σ2 ≤ σ∗ = 2 √ ρ(1− 2β) > 0 Finally, for the case β ∈ [1/2, 1), x̂CE < x̂SP for all y ∈ [0, 1 − β], regardless of σ, so x̂CE < x̂SP and σCEc < σSPc . Notice that in this case the formula for σ∗ < 0. Proposition 9 Proof. We already know that the planner’s allocation is a BGP with constant x̂ and σc. By setting the subsidy/tax τk according to (35) we ensure that r, x̂, and m̂ satisfy all the conditions for a BGP equilibrium. We can check that the value of total wealth w = kt +mt+ ht − ´∞ t e−rsτks ksds satisfies ct = ρ(1− β)wt, or equivalently σc = kt wt σ. Write wt kt = 1 + m̂i r − (x̂− δ) + τk r − (x̂− δ) wt kt = r − (x̂− δ) + ĉ β 1−β + τk r − (x̂− δ) = ρ− σ2c + ρβ σ σc + 2σ2c − σcσ ρ− σ2c wt kt = ρ+ ρβ σ σc + σ2c − σcσ ρ− σ2c = σ σc ( ρσc σ + ρβ + σ2c ( σc σ − 1) ρ− σ2c ) Use the planner’s FOC (28) and the skin in the game IC constraint (32) wt kt = σ σc ( ρσc σ + ρβ + (a− x̂− ρ)(σc σ − 1) 2ρ− a− x̂ ) = σ σc ( ρ(1 + β) + (a− x̂)ρ(1−β)−(a−x̂) a−x̂ 2ρ− a− x̂ ) 52 Proposition 12 Proof. From (47), we plug in θTFP = σ̃TFP , and θRS = σψ 1+ψ , as well as m̂i = ρβw/k = ρβ(1 + ψ) from (45) and x̂+ g − δ − r = −(ρ+ µψ 1 + ψ − ( σ 1 + ψ )2 − (σ̃TFP )2 − ( σψ 1 + ψ )2) Then use Ito’s lemma to obtain µψ = ψ′φ(σ̄ − σ) + 1 2 ψ′′σν2 σψ = ψ′√σν The ODE (49) has the µψ terms together, and the σ̃TFP terms cancel out. I also simplified the terms involving σψ into one term. Proposition 13 Proof. The equations for the competitive equilibrium are a modified version of (40)-(45), taking into account that total wealth now includes the present value of taxes/subsidies that are rebated lump-sum and the tax τk: rt = ρ+ (x̂t + g − δ) + µĉ,t − σ2ct − (σ̃TFPct )2 − (σ̃RSct )2 Euler (62) rt = a+ g − δ − τkt − σctσt − θTFPt σ̃TFP Asset Pricing (63) σct = kt wt σt = (1− β)ρktc −1 t σt Id. risk (64) σ̃TFPc = θTFPt TFP risk (65) σ̃RSct = θRSt RS risk (66) m̂t = β/(1 − β)× (a− x̂t)/(rt + πt) Money (67) Since the planner’s allocation satisfies (50)-(55) and the FOC (57), it satisfies also the equi- librium conditions (62)-(67). Equations (62), (65), (66), and (67) are immediate. Equation (63) follows from plugging the definition of x̂t from (57), τkt from (58), and θTFPt = σ̃TFPct into the Euler equation (51). (64) comes from the skin in the game constraint (50), using the fact that σUt = 1 ρ σct. Finally, limt→∞ E Q [ e− ´ s t ruduktĉt ] =⇒ limt→∞ E Q [ e− ´ s t ruduwt ] = 0 and it = rt + πt > 0 ensure that this is in fact an equilibrium. 55 8.2 CES utility for money and cash in advance In this Appendix I generalize the baseline model to allow a more flexible demand for money. In particular, we are interested in allowing the expenditure share on liquidity services to vary with the interest rate. First I generalize the log utility with a CES aggregator between money and consumption. This allows me to introduce an interest-elasticity of money demand η different from one (in particular, η < 1). This means that as i → 0 the expenditure share on liquidity vanishes. However, i must always be strictly positive. The second specification is a cash-in-advance model. Again, the expenditure share on liquidity vanishes as i→ 0, but in this case we can actually have i = 0. These cases are interesting on their own, since they allow a more flexible account of money demand. But they also help understand if the liquidity trap hinges on a constant expenditure share on liquidity, as in the baseline model. In light of the cashless limit result in Proposition 5, it should come as no surprise that the liquidity trap survives even when the expenditure share on liquidity vanishes. It survives even if i = 0 in the cash-in-advance model. CES utility. We can introduce a more flexible CES specification for money demand E [ ˆ ∞ 0 e−ρt log (( (1− β) 1 η c η−1 η t + β 1 ηm η−1 η ) η η−1) dt ] where η is the demand elasticity of money. With η = 1 we recover the baseline setting. The FOC for consumption and money demand are now slightly different: c = ρ 1− β 1− β + βi1−η w mi = ρ βi1−η 1− β + βi1−η w so the expenditure share on money services mi ρw = β̃(i) = βi1−η 1− β + βi1−η is a function of the nominal interest rate, and therefore the inflation target i = r + π. The 56 same reasoning as in the baseline yields m̂+ ĥ = m̂× (r + π) r − (x̂− δ) The real value of monetary assets is still the present value of liquidity services. But with η 6= 1, the target inflation rate π affects this value. If η < 1, a higher nominal interest rate i leads to higher expenditures on money services m̂i, and therefore higher real value of monetary assets (for fixed r and x̂); if η > 1 the opposite result holds. The system of equations takes the same form as before, except that the share of expen- ditures on money services β̃(r+ π) is a function of the nominal interest rate, and therefore the real interest rate and the inflation target. r = a− δ − (1− λ)σ2 (68) x̂ = a− ρ− ρ λ− β̃(r + π) 1− λ (69) with λ = ρβ̃(r + π) ρ− ((1 − λ)σ)2 (70) Now we need to solve the system of three equations simultaneously, because r appears in (70). We can verify that when σ ≈ 0, we obtain the RBC limit where we can ignore money. Indeed, in this case we get r = a − δ, x̂ = a − ρ, and λ = β̃(r + π). While the inflation target matters for monetary variables, the real interest rate and investment are the same as in the non-monetary economy. Higher inflation targets still lead to one-for-one increases in the nominal interest rate, leaving the real interest rate and investment unaffected. The reader may wonder if the liquidity trap survives with CES preferences. In the baseline model with log preferences, the expenditure share on liquidity services β is constant. This implies that as σ increases and the equilibrium real interest rate r drops, the value of monetary assets becomes very large (λ → 1). We are “dividing by zero”, so to speak. With CES preferences and demand elasticity of money η < 1, the share of expenditures on liquidity services β̃(i) falls as the equilibrium real interest rate r falls (inflation π is constant). But we can use the same reasoning as in the cashless limit of Proposition 5 to see that the liquidity trap must survive. In particular, if σ > √ ρ, the equilibrium 57 and x̂: r = a− δ −√ ρσ x̂ = a−√ ρσ The CIA constraint pins down only a lower bound on money, because with i = 0 agents are willing to hold money beyond their transaction needs, mt ≥ ct/v =⇒ m̂ ≥ (a− x̂)/v If they hold money beyond their transaction needs, money is acting like a bubble. So the conceptually cleanest decomposition is to set m̂ = (a− x̂)/v and then use (72) to pin down the value of the bubble bt b̂ = σ/ √ ρ− 1− (a− x̂)/v (73) Notice that b̂ could be negative. This is the case if v is very small, so money holding for transaction purposes m̂ must be large. This amount of money would create too much risk sharing, so we need a negative bubble.44 8.3 The role of intertemporal elasticity and risk aversion The baseline model has log preferences, which yield clean results and are quantitatively reasonable. In this Appendix I extend the baseline model to allow for EZ preferences to understand the role of intertemporal elasticity and risk aversion. Suppose agents have recursive EZ preferences with discount ρ, risk aversion γ, and intertemporal elasticity ψ. If ψ = 1/γ we have the standard CRRA preferences. If ψ = γ = 1 we have the baseline model with log preferences. The equilibrium equations are now modified as follows r = ρ+ (x̂− δ)/ψ − (1 + 1/ψ)(γ/2)σ2c Euler equation r = a− δ − γσcσ Asset Pricing σc = (1− λ)σ Risk Sharing 44We can interpret the negative bubble by taking the accounting value of money as m for the purpose of the CIA constraint, but the market value of money (the goods you could actually buy if you only had money) as m + b < m. But this is pushing the limits of the CIA constraint as a foundation for monetary trade. 60 m̂ = β 1− β a− x̂ r + π Money The expression for the value of liquidity, λ, must be solved simultaneously with r and x̂. λ = ρβ ρ+ (1/ψ − 1)(x̂ − δ)− (1 + 1/ψ)(γ/2)((1 − λ)σ)2 (74) We can check that if ψ = γ = 1 we recover the equation in the baseline model. First Best. If there is no idiosyncratic risk, σ = 0, we get closed form expressions for r and x̂ r = a− δ x̂ = (a− δ − ρ)ψ + δ Incomplete risk sharing and no money, β = 0. The non-monetary economy also allows for closed form expressions, because λ = β = 0 and σc = σ. r = a− δ ︸ ︷︷ ︸ first best −γσ2 x̂ = (a− δ − ρ)ψ + δ ︸ ︷︷ ︸ first best +ψ [ (1 + 1/ψ)(γ/2)σ2 ︸ ︷︷ ︸ precautionary − γσ2 ︸︷︷︸ risk pr. ] After a risk shock increases idiosyncratic risk σ, the real interest rate falls to accommodate the higher risk premium α = γσcσ = γσ2. But investment may go up or down, depending on the intertemporal elasticity ψ. If ψ > 1, investment falls when idiosyncratic risk σ goes up; if ψ < 1, investment raises. This can be understood in terms of the risk premium and precautionary motive. If ψ > 1, the precautionary motive is smaller than the risk premium, and the difference increases with σ ((1 + 1/ψ)/2 < 1). Intuitively, capital is less attractive because it is more risky, and since agents are very intertemporally elastic, they substitute towards consuming instead (accepting a big change in the growth rate of their consumption). But if ψ < 1, the precautionary motive dominates. Agents really want to smooth out their utility, and since they face more risk, they make it up by accumulating more capital. If ψ = 1, as in the baseline, the two effects cancel out and investment does not change when σ goes up. The important variable is the intertemporal elasticity. Risk aversion, γ, just makes the 61 idiosyncratic risk matter more. In fact, both enter jointly γσ2 in the equations. The role of intertemporal elasticity is well understood, and is the reason that the literature on time varying risk typically assumes high intertemporal elasticity, ψ > 1. Empirically, evidence about ψ is mixed, but ψ = 1 is considered a quantitatively reasonable benchmark. Incomplete risk sharing and money, β > 0. Now let’s see what happens when we add money. First, take the value of liquidity λ > 0 as given. Idiosyncratic risk sharing improves, σc = (1− λ)σ, so we get r = a− δ ︸ ︷︷ ︸ first best −γ(1− λ)σ2 x̂ = (a− δ − ρ)ψ + δ ︸ ︷︷ ︸ first best +ψ [ (1 + 1/ψ)(γ/2)σ2(1− λ)2 ︸ ︷︷ ︸ precautionary − γσ2(1− λ) ︸ ︷︷ ︸ risk pr. ] Money weakens the risk premium, so the real interest rate is higher than without money. Money also weakens the precautionary motive more than the risk premium, just as in the baseline model. But since investment can go up or down with risk, depending on ψ, it is useful to decompose the effect of higher risk into the effect without money, and what money adds relative to the non-monetary economy: r = a− δ − γσ2 ︸ ︷︷ ︸ non-monetary +λγσ2 ︸ ︷︷ ︸ ∆r (75) x̂ = (a− δ − ρ)ψ + δ + ψ [ (1 + 1/ψ)(γ/2)σ2 − γσ2 ] ︸ ︷︷ ︸ non-monetary + γσ2ψ ( (λ2 − 2λ)(1 + 1/ψ)/2 + λ ) ︸ ︷︷ ︸ ∆x̂ (76) The second terms are the effect of money on the real interest rate, ∆r, and investment, ∆x̂, relative to the economy without money. In general it is possible for investment in the monetary economy to be higher than in the non-monetary one. For very large ψ, ∆x̂ ≈ γσ2ψλ2/2 > 0. There are two forces at work. Remember that if ψ > 1, the risk premium dominates, so high risk σ can have a very large negative effect on investment x̂. Money improves risk sharing and weakens the risk premium α = γσ2(1 − λ), so it dampens the fall in investment from this channel. It also weakens the precautionary motive relative to the risk premium, which reduces investment just like in the baseline model. The two forces work in opposite directions. In the baseline 62 9 Online Appendix: Contractual setting In this Appendix I develop the contractual environment that yields the incomplete idiosyn- cratic risk sharing problem in the baseline model as the optimal contract. I also allow aggregate risk with complete risk sharing, which is the setting in the dynamic model in Section 4. The setting in the baseline model is a special case with no aggregate risk. The setting is essentially a special case of the environment in Di Tella and Sannikov (2016) with perfect misreporting (φ = 1 in the terms of that paper), generalized to allow for aggregate shocks. I discuss the similarities and differences below. 9.1 Setting The setting is as in the dynamic model in Section 4. The “capital quality” shock for an agent is ∆k i,t = σtki,tdWi,t + σ̃TFPdZTFPt (77) where ZTFP is an aggregate TFP shock. Aggregate TFP risk σ̃TFP is constant, but id- iosyncratic risk σt follows an autoregressive process dσt = µσ(σt)dt+ σ̃σ(σt)dZ RS t (78) where ZRS is the aggregate risk shock. ZTFP and ZRS are independent Brownian motions. There is a complete financial market with real interest rate r, nominal interest rate i, capital’s excess return α, and price of aggregate shocks θTFP and θRS, all adapted to the history of aggregate shocks ZTFP and ZRS . Let Q be the equivalent martingale measure associated with r, θTFP and θRS , and Q̃ the equivalent martingale measure associated with r, θTFP , θRS , and α.46 The agent receives consumption c and money holdings m from the principal, and man- ages capital k, all contingent on the history of aggregate shocks ZTFP and ZRS and the agent’s report of his idiosyncratic shock Y s. The idiosyncratic shock is not observable by the principal, so the agent can misreport at rate s, such that his reports are Y s t =Wt− ´ t 0 su σu du. Furthermore, the agent has access to hidden trade that allows him to choose his consump- tion c̃, money m̃, capital holdings k̃, and to trade aggregate risk σ̃TFPn and σ̃RSn .47 His 46That is, Q is defined by the SPD dξt/ξt = −rt − θTFPt dZTFPt − θRSt dZRSt and Q̃ by dξ̃t/ξ̃t = −rt − θTFPt dZTFPt − θRSt dZRSt − αt σt dWt. 47To keep things simple, allow k̃ < 0, but we can also restrict it to k̃ ≥ 0, as in Di Tella and Sannikov 65 hidden savings n start at n0 = 0 and satisfy the dynamic budget constraint dnt = (ntrt + ct − c̃t + (mt − m̃t)it + (k̃t − kt)αt + θTFPt σTFPnt + θRSt σ̃RSnt + ktst)dt (79) + (k̃t − kt)σtdWt + σ̃TFPnt dZTFPt + σ̃RSnt dZ RS with solvency constraint nt ≥ nt where nt is the natural debt limit nt = −max s∈S E Q̃ t [ ˆ ∞ t e ´ u t rτdτ (cu(Y s) +mu(Y s)iu + suku(Y s))du ] (80) where S = { s : E Q̃ [ ´∞ 0 e ´ u t rτdτ |cu(Y s) +mu(Y s)iu + suku(Y s)|du ] < ∞ } is the set of feasible stealing plans for a given contract. The natural debt limit nt is the maximum amount that the agent can pay back for sure at time t. The lender is not taking any risk as long as he enforces the natural debt limit. Lemma 1. Assume |n0| <∞. If nt ≥ nt always, then lim infT→∞ e− ´ t 0 rudunt ≥ 0 a.s. A contract C = (c,m, k) is admissible if EQ [ ´∞ 0 e− ´ t 0 rudu |ct +mtit + ktαt| dt ] <∞. It is always optimal to implement no misreporting or hidden trade.48 An admissible contract is incentive compatible if the agent chooses to report truthfully and not engage in hidden trade, (c,m, k, 0, 0, 0) ∈ argmax P U(c,m) st : (79) where P = (c̃, m̃, k̃, σ̃TFPn , σ̃RSn , s). An incentive compatible contract is optimal if it mini- mizes the cost of delivering utility to the agent J0(u0) = min (c,m,k)∈IC E Q [ ˆ ∞ 0 e− ´ t 0 rudu (ct +mtit − ktαt) dt ] st : U(c,m) ≥ u0 We pin down the agent’s initial utility u0 with a free-entry condition for principals. If the agent has initial wealth w0, he gives it to the principal in exchange for the full-commitment contract, and the principal breaks even, J0(u0) = w0. (2016). This doesn’t change the optimal contract. 48See Di Tella and Sannikov (2016). 66 9.2 Incentive compatibility and optimal contract Given contract C = (c,m, k), the agent’s problem is to choose a misreporting and hidden trade strategy P = (c̃, m̃, k̃, σ̃TFPn , σ̃RSn , s) to maximize his utility subject to his dynamic budget constraint. With the natural debt limit, the dynamic budget constraint is equivalent to the following intertemporal budget constraint E Q̃ [ ˆ ∞ 0 e− ´ t 0 rudu(c̃t + m̃tit)dt ] ≤ max s∈S E Q̃ [ ˆ ∞ 0 e− ´ t 0 rudu(ct(Y s) +mt(Y s)it + kt(Y s)st)dt ] (81) The rhs is the present value of the agent’s income from the principal, including what he “steals” from him, and is equal to (minus) the natural debt limit −n0. Of course, if the rhs is infinity the agent can achieve infinite utility. This corresponds to the case where the natural debt limit n0 = −∞ so the agent can get infinite utility under the dynamic constraint as well. Lemma 2. Assume |n0| < ∞. If (c̃, m̃, k̃, σ̃TFPn , σ̃RSn , s) and n satisfy the dynamic budget constraint (79) with nt ≥ nt always, then (c̃, m̃) satisfy the intertemporal budget constraint (81). If (c̃, m̃) satisfy the intertemporal budget constraint (81), then there are processes (k̃, σ̃TFPn , σ̃RSn , s) and n that satisfy the dynamic budget constraint (79) with nt ≥ nt always. We can split the agent’s problem into two parts. First, pick a misreporting strategy that maximizes the value of the rhs. Second, choose c̃ and m̃ to maximize utility subject to the intertemporal budget constraint (81). If s∗ = 0 is optimal, then ˆ t 0 e− ´ u 0 rτdτ (cu(Y s) +mu(Y s)iu + ku(Y s)su)du− e− ´ t 0 rudunt(Y s) must be a Q̃-martingale for s = 0 and a supermartingale for any other s. So we can write d ( e− ´ t 0 rudunt(Y s) ) = e− ´ t 0 rτdτ { (ct(Y s) +mt(Y s)it)dt+ σnt(Y s) (dY s t + αtdt) +σ̃TFPnt (Y s)(dZTFPt + θTFPt dt) + σ̃RSnt (Y s)(dZRSt + θRSt dt) } 67 Theorem 1. Let (c,m, k) be an optimal contract for initial utility u0, with cost J(u0). Then (c,m, k) solves the portfolio problem for w0 = J(u0). Conversely, let (c,m, k) solve the portfolio problem for some w0 > 0. If in addition limt→∞ E[e−rtwt] = 0, then (c,m, k) is an optimal contract for initial utility u0 with J(u0) = w0. Remark. The condition limt→∞ E[e−rtwt] = 0 must be satisfied in the competitive equilib- rium in the paper. 9.3 Comparison to Di Tella and Sannikov (2016) This setting is essentially the same as in Di Tella and Sannikov (2016), with hidden invest- ment and perfect misreporting (φ = 1 in the context of that paper). The main result here is Theorem 1, which is analogous to Lemma 28 in that paper. This is therefore a special case of the environment in that paper. But there are some differences. First, here I allow aggregate risk shocks that affect the investment environment. The setting in Di Tella and Sannikov (2016) is stationary. Second, in Di Tella and Sannikov (2016) the agent faces a no-debt solvency constraint nt ≥ 0 on his hidden savings n. Here I allow the agent to borrow up to the natural borrowing limit, using his income from the contract. As it turns out the optimal contract is the same. The no-debt borrowing constraint relaxes the IC constraints, but the principal does not use this freedom in the optimal contract. Intuitively, with nt ≥ 0 the principal could backload the agent’s consumption if he wanted. But what he really wants to do is to front load it. Finally, here I allow the agent to short capital in his hidden investment, k̃t < 0 and to overreport returns, st < 0. This is done for simplicity. In Di Tella and Sannikov (2016) hidden investment and misreporting must be non-negative, kt ≥ 0 and st ≥ 0, and the optimal contract is the same (for the special case with φ = 1). 9.4 Proofs Proof of Lemma 1 Proof. From the definition of the natural debt limit (80), if we take absolute value on both sides we get the following inequality |nt| ≤ St = E Q̃ t [ ˆ ∞ t e ´ u t rτdτ |cu(Y s∗) +mu(Y s∗)iu + s∗uku(Y s∗)|du ] <∞ 70 where s∗ is the misreporting process that achieves the maximum in (80). The process e− ´ t 0 ruduSt ≥ 0 is a Q̃-supermartingale with d ( e− ´ t 0 ruduSt ) = −e− ´ t 0 rudu|ct(Y s∗) +mt(Y s∗)it + s∗t kt(Y s∗)|dt+ Q̃-local mart. terms We also know that limT→∞ E Q̃ [ e− ´ T 0 ruduST ] = 0. To see this, write S0 = E Q̃ 0 [ ˆ T 0 e ´ u 0 rτdτ |cu(Y s∗) +mu(Y s∗)iu + s∗uku(Y s∗)|du ] + E Q̃ [ e− ´ T 0 ruduST ] and take the limit T → ∞, using the MCT on the first term. It follows that limT→∞ e− ´ T 0 ruduST exists and is zero almost surely (see Problem 3.16 in Karatzas and Shreve (2012)). Since |nt| ≤ St, the same is true for nt, and since nt ≥ nt, we obtain lim infT→∞ e− ´ t 0 rudunt ≥ 0 a.s. Proof of Lemma 2 Proof. In the first direction, use the dynamic budget constraint to compute E Q̃ [ e− ´ t 0 rudunt ] = E Q̃ [ ˆ t 0 e− ´ u 0 rτdτ (cu(Y s) +mu(Y s)iu + ku(Y s)su)du ] − E Q̃ [ ˆ t 0 e− ´ u 0 rτdτ (c̃u + m̃uiu)dt ] Subtract E Q̃ [ e− ´ t 0 rudunt ] < ∞ from both sides. Because n0 is the maximum value that the agent can get, we obtain an inequality: E Q̃ [ e− ´ t 0 rudu(nt − nt) ] ≤ max s E Q̃ [ ˆ ∞ 0 e− ´ u 0 rτdτ (cu(Y s) +mu(Y s)iu + ku(Y s)su)du ] − E Q̃ [ ˆ t 0 e− ´ u 0 rτdτ (c̃u + m̃uiu)du ] E Q̃ [ ˆ t 0 e− ´ u 0 rτdτ (c̃u + m̃uiu)du ] ≤ −n0 − E Q̃ [ e− ´ t 0 rudu(nt − nt) ] Take the limit t→ ∞ and use nt ≥ nt to obtain the intertemporal budget constraint (81). 71 In the other direction, define nt = nt + E Q̃ [ ˆ ∞ t e− ´ u t rτdτ (c̃u + m̃uiu)du ] ≥ nt Define Lt = E Q̃ t [ ´∞ t e− ´ u t rτdτ (c̃u + m̃uiu)du ] , so that ´ t 0 e − ´ u 0 rτdτ (c̃u+m̃uiu)du+e − ´ t 0 rτdτLt is a Q̃-martingale. Likewise, − ´ t 0 e − ´ u 0 rτdτ (cu(Y s∗)+mu(Y s∗)iu+ku(Y s∗)s∗u)dt+e − ´ t 0 rτdτnt is also Q̃-martingale, where s∗ is the misreporting process that achieves the maximum. So we can write dnt = ( ntrt + ct(Y s∗) +mt(Y s∗)it + kt(Y s∗)s∗t − (c̃t + m̃tit) ) dt + (σnt + σLt)(αtdt+ dWt) + (σ̃TFPnt + σ̃TFPLt )(θTFPt dt+ dZ̃TFPt ) + (σ̃RSnt + σ̃RSLt )(θ RS t dt+ dZ̃RSt ) Letting σnt + σLt = k̃t − kt, σ̃ TFP nt + σ̃TFPLt = σ̃TFPnt , and σ̃RSnt + σ̃RSLt = σ̃RSnt , we obtain the dynamic budget constraint (79). Proof of Lemma 3 Proof. Immediate from the argument in Section 9.2, noting that incentive compatibility requires |n0| <∞. Proof of Theorem 1 Proof. In the first direction, if (c,m, k) is an optimal contract, then it must satisfy the local IC constraints (82)-(86), which are the FOC for the consumption-portfolio problem. So c and m solve the optimal portfolio problem for some initial w0, with an associated wealth process w that satisfies the dynamic budget constraint (88) and wt ≥ 0. Now the IC constraint (87) pins down the corresponding k. We know that ct = (1 − β)ρwt in the portfolio problem, so (87) and (83) imply σc = αt σt = (1− β)ρ kt ct σt =⇒ kt wt = αt σ2t which is the expression for capital in the portfolio problem. Finally, we need to show that w0 = J0. Integrate the dynamic budget constraint (88) and take expectations under Q to 72