Optimal Negative Interest Rates in the Liquidity Trap

Optimal Negative Interest Rates in the Liquidity Trap Davide Porcellacchia 8 February 2017 Abstract The canonical New Keynesian model features a zero lower bound on the interest rate. In the simple setting of this paper, where agents are subject to idiosyncratic liquidity risk and therefore a banking system emerges to insure such risk as in Diamond and Dybvig (1983), the zero lower bound on nominal interest rates vanishes and thus negative interest rates on bank reserves become a viable policy option. However, negative interest rates come with a new tradeoff for the central bank: between keeping the banking system fully functional by setting a high enough interest rate on reserves and achieving full employment, which requires a strictly negative nominal interest rate when large adverse demand shocks hit the economy. The paper s main finding is that it is always optimal for the central bank to set a strictly negative interest rate on reserves in response to negative demand shocks that push the economy in the liquidity trap. 1 Introduction Banks play an important role in the economy as providers of liquidity-risk insurance. However, this role is abstracted from in the canonical models that are used to analyse optimal monetary policy in the liquidity trap (e.g. Eggertsson and Woodford (2003)). This paper shows that when maturity transformation is explicitly modelled, even in the presence of money as an alternative store of value with zero net nominal return, negative interest rates on reserves are not only possible but optimal in the liquidity trap. As in Krugman (1998), I model the liquidity trap as a systemic increase in agents discount factor in a neoclassical growth model with nominal rigidities. Liquidity risk and the insurance thereof is in the spirit of Diamond and Dybvig (1983). Agents are unable to insure each other directly from idiosyncratic liquidity shocks. Thus, banks can make a profit by offering deposit contracts that serve this purpose. The policy tradeoff that this paper analyses is between stimulating demand in the liquidity trap by means of negative interest rates and ensuring that healthy banks can provide valuable liquidity-risk insurance. PhD candidate in the department of Economics of the London School of Economics. 1

1.1 Key assumptions Idiosyncratic liquidity shocks: agents discount factor β i is stochastic. As in Diamond and Dybvig (1983) agents can be early or late consumers. Notice that in this model I assumes that the agents type is publicly observable once it is realised. No commitment: Agents cannot make intertemporal contracts because they cannot commit to honouring them in the following period. Hence, there is a role for banks to pool savings in the form of deposits and then pay high returns to those depositors with higher marginal utility of consumption. Price rigidity: Following Krugman (1998), I assume a price rigidity. For simplicity, I assume the extreme form P 0 = P 1 = 1. 2 Agents 2.1 Consumers There is a unit measure of consumers j [0, 1] who supply labour inelastically and choose a path for consumption (C j0, C j1 ) by investing in capital K j1, money M j1 0, and deposits D j1 in order to maximise The individual s flow budget constraints are u(c j0 ) + E 0 [ βj u(c j1 ) ] (1) K j1 + M j1 + D j1 + C j0 = (1 + r 0 δ) K 0 + W 0 N 0 T 0 (2) C j1 = W 1 N j1 + (1 + r 1 δ) K j1 + (1 + d j ) D j1 + M j1 T 1 (3) The discount factor β j = β ɛ j is a stochastic variable with { ɛ H with probability θ ɛ j = ɛ L with probability 1 θ. (4) ɛ H > ɛ L. The liquidity shock ɛ j is realised for each agent at time one and it is idiosyncratic with E(ɛ j ) = 1. 2.1.1 Key optimality conditions Consumers make their saving decision based on the financial asset offering the best risk-adjusted return, so that u (C 0 ) = max{v K, V D, V M } (5) 2

where (V K, V D, V M ), the time-0 marginal value of respectively capital, deposits, and money, are V K = β (1 + r 1 δ) ɛ H u (C H1 ) X (6) Define V D = β [(1 + i D H) θ ɛ H u (C H1 ) + (1 + i D L ) (1 θ) ɛ L u (C L1 ) ] (7) V M = β ɛ H u (C H1 ) X (8) X = θ + (1 θ) ɛl u (C L1 ) ɛ H u (C H1 ) the index of liquidity-risk insurance. X is increasing with the extent of liquidity insurance, with X = 1 indicating perfect insurance. 2.2 Monti-Klein Banks Banks that collect deposits, invest, and pay back to deposit-holders on the basis of their type emerge naturally in this framework. I assume no barriers to entry, and therefore a competitive banking market. Banks emerge because agents are unable to insure each other against liquidity shocks because of their inability to commit. Pooling savings in a bank in the form of deposits is a commitment device that allows a transfer in the future period from depositors hit by a low liquidity shock to depositors hit by a high liquidity shock. Banks are of the Monti-Klein type. They choose their asset portfolio among capital K B1, money M B1 0, and reserves R 1 and finance these asset holdings with deposits (I do not consider bank equity), as according to the budget constraint (9) K B1 + M B1 + R = D (10) There is a reserve requirement on banks proportional to their deposits R ρ D 1, ρ (0, 1] (11) and, since agents are unable to commit to paying into the bank in period one if they turn out to be L-type, then the deposit contract must have 1 + d j 0, j {H, L} (12) Subject to the above constraints, banks maximise profits Π B1, defined as [ ] Π B1 = (1 + r 1 δ) K B1 + M B1 + (1 + i) R 1 θ (1 + d H ) + (1 θ) (1 + d L ) D (13) If profits are less than zero, banks shut down. 3

2.2.1 Key optimality and equilibrium conditions Given this paper s focus on negative interest rates on reserves (that penalise financial intermediation), I consider a banking system of minimum size to achieve the equilibrium level of liquidity insurance. The main implication of this is that equation 12 with j = L is binding in equilibrium, so that 1 + d L = 0 (14) This means that the deposit contract provides the maximum level of liquidity insurance per unit of deposit, consistent with the lack of a commitment device for consumers. In fact, banks pay all the returns of their investments to agents that become H-types and pay nothing to L-types. Notice that this is a contract that agents faced with idiosyncratic liquidity risk will be happy to enter ex ante. 2.3 Firms Firms are standard. In each period, they produce output Y with labour N and capital K according to constant-returns-to-scale production function Subject to equation 15, firms statically maximise their profits Y = f(k, N) (15) Π F = Y W N r K (16) I assume that firms operate in a perfectly competitive market. 2.3.1 Key optimality and equilibrium conditions As is standard, in equilibrium firms equate marginal product and marginal cost of the factors of production so that f 1 (K, N) = r (17) f 2 (K, N) = W (18) As is usual in the literature, in this model the liquidity trap is a rationing equilibrium. In such equilibrium, competition continues to make firms pay factors of production at their marginal product. However, firms will not hire and rent all workers and capital supplied, because they find it impossible to sell their production beyond a certain level. 2.4 Government The government issues money M and reserves R and carries out monetary policy by setting the nominal interest rate on reserves i. 4

Notice that, regardless of the interest rate, banks hold reserves because of the reserve requirement. Nonetheless, monetary policy affects the amount of wealth that agents are willing to hold in intermediated finance and therefore the general attractiveness of saving vs. consuming. I assume that in each period the government balances its budget by means of lumpsum taxes T imposed equally on all consumers, regardless of their type. The government s budget constraints therefore are 0 = M 1 + M B1 + R 1 + T 0 (19) M 1 + M B1 + (1 + i) R 1 = T 1 (20) 3 Autarchy Before discussing the equilibrium concept more carefully in the next section, here I study the equilibrium allocation that would prevail if banks were not allowed to operate. Proposition 1: Without banks agents cannot diversify their liquidity risk, so that X aut = θ + (1 θ) ɛl ɛ H < 1 (21) Notice that in this model banks do not operate if i is set too low. 4 Equilibrium with banking: market-clearing vs. rationing Equilibrium is defined as the vector of all endogenous quantities and prices such that Consumers maximise utility taking prices as given subject to budget constraints, Banks maximise profits in a perfectly competitive market, Firms maximise profits in a perfectly competitive market. In this section, I determine the equilibrium value of endogenous variables that are of interest for the optimal monetary policy problem: ( C 0, C H1, C L1, K 1 K 1 +K B1, K 0, Ñ0, X ). The total level of capital available for production in period 1 is denoted K 1. K0 and Ñ0 are the quantities of capital and labour employed in production in period zero, which in a rationing equilibrium may be smaller than capital and labour supplied. Proposition 2: There exists a threshold ˆβ > 1 such that if β > ˆβ, then the equilibrium features rationing (i.e. at least one of K 0 < K 0 and Ñ0 < 1 is true). I call this liquidity 5

trap. In the case of this model, β is simply a parameter. So, this is a simple comparative statics exercise. However, a high level of β > ˆβ > 1 represents a systemic shock that increases consumers propensity to save. 4.1 Market-clearing equilibrium In the standard market-clearing equilibrium, money is an inferior asset in terms of returns and hence consumers do not hold any. Consumers hold capital and therefore save according to u (C 0 ) = β [1 + f 1 ( K 1, 1) δ ] ɛ H u (C H1 ) X (22) where X is the index of liquidity-risk insurance defined as X = θ + (1 θ) ɛl u (C L1 ) ɛ H u (C H1 ) In equilibrium, the extent of liquidity-risk insurance is determined by X = max {1 ρ[ f 1 ( K 1, 1) δ i ] } 1 + f 1 ( K 1, 1) δ, Xaut with one indicating full liquidity insurance and the degree of liquidity insurance decreasing as the value of X decreases. It is clear that setting the interest rate on reserves i equal to the net rate of return on capital f 1 ( K 1, 1) δ gives full liquidity insurance in equilibrium while lower levels of i implement partial liquidity insurance as agents invest their wealth directly rather than investing in intermediated finance that is effectively taxed. The resource constraints that hold in equilibrium are (23) (24) K 1 + C 0 = (1 δ) K 0 + f(k 0, 1) (25) θ C H1 + (1 θ) C L1 = (1 δ) K 1 + f( K 1, 1) (26) Notice that in the market-clearing equilibrium, which takes place when β < ˆβ, there is full employment of the factors of production, so that K 0 = K 0 (27) where K 0 is the exogenous initial quantity of capital in the economy, and Ñ 0 = 1 (28) 6

4.2 Liquidity-trap equilibrium As according to proposition 2, if β > ˆβ, the economy enters the liquidity trap which is a rationing equilibrium. Consumers find it optimal to hold money as a financial investment. Thus, they save according to u (C 0 ) = β ɛ H u (C H1 ) X (29) where X is the index of liquidity-risk insurance defined as X = θ + (1 θ) ɛl u (C L1 ) ɛ H u (C H1 ) (30) In equilibrium, the extent of liquidity-risk insurance is determined by 1 { X = max 1 + ρ i, X aut} (31) with one indicating full liquidity insurance and the degree of liquidity insurance decreasing as the value of X decreases. The equation shows that if the government sets a nominal interest rate on reserves of zero, then there is full liquidity-risk insurance in the economy. Lower levels of insurance are associated with lower levels of the interest rate i. Also, if i < 1 Xaut ρ, then the economy reverts to autarchy as consumers shun deposits altogether. In equilibrium, it must be that for anyone in the economy to hold capital. The resource constraints that hold in equilibrium are f 1 ( K 1, 1) = δ (32) K 1 + C 0 = (1 δ) K 0 + f( K 0, Ñ0) (33) θ C H1 + (1 θ) C L1 = (1 δ) K 1 + f( K 1, 1) (34) Notice that, under the scenario where β is higher than the threshold ˆβ, there is rationing in equilibrium. This means that at least one of the following inequalities holds strictly: In equilibrium, ( K 0, Ñ0) are not determined individually. K 0 K 0 (35) Ñ 0 1 (36) 1 Figure 1 in the appendix provides a graphical analysis of the determination of X in equilibrium. 7

5 Optimal monetary policy In this section, I study the optimal monetary policy problem. The government s policy instrument is the nominal interest rate on bank reserves and the government aims to maximise agents aggregate welfare. 5.1 Market-clearing equilibrium Proposition 3: If β ˆβ, then optimal monetary policy sets i = f 1 ( K 1, 1) δ. This implements the first-best efficient allocation. In the market-clearing equilibrium, the government should pay the market rate of return on capital on bank reserves. 5.2 Liquidity trap 5.2.1 Optimisation problem The government chooses a value for its policy instrument i to maximise aggregate welfare defined as the sum of all consumers utility functions (for simplicity, I adopt a CES specification for utility) C 1 γ 0 1 + β 1 γ [θ ɛ H C1 γ H1 Optimisation is subject to the Euler equation where X is an index of liquidity-risk insurance ] 1 γ + (1 θ) ɛ L C1 γ L1 1 γ (37) C γ 0 = β ɛ H C γ H1 X (38) X = θ + (1 θ) ɛl ɛ H ( CH1 C L1 ) γ (39) Furthermore, it is subject to the following equation that holds because people hold deposits too (unless they are better off in autarchy) and subject to the resource constraint X = max { 1 + ρ i, X aut} (40) θ C H1 + (1 θ) C L1 = (1 δ) K 1 + f( K 1, 1) (41) Notice that, by equation 40, the government can directly determine the degree of liquidity-risk insurance by choosing to pay interest on reserves at the market rate (i.e. i = 0) or to pay a negative interest rate on reserves. 8

5.2.2 Result Proposition 4: If β > ˆβ and therefore the economy is in the liquidity trap, it is optimal for the government to set a strictly negative nominal interest rate on bank reserves. The resulting allocation is second best, with partial liquidity-risk insurance.. The policy tradeoff in the model s liquidity trap is between having more consumption (i.e. less rationing) at time zero and ensuring that banks provide valuable liquidityrisk insurance. In fact, by taxing the banking system with a negative interest rate on reserves, the government discourages consumers from saving excessively. However, harming financial intermediaries is undesirable too, as banks provide valuable liquidityrisk insurance. In the relevant (C 0, X) space, the government can pick a point that belongs to the feasibility curve, defined as the combination of the Euler equation (equation 38), the definition of X (equation 39), and the resource constraint (equation 41).The slope of the feasibility curve is dx u (C 0 ) ɛ L u (C L1 ) [θ C H1 + (1 θ) C L1 ] = dc 0 β θ ɛ H u (C H1 ) [ɛ H u (C H1 ) C L1 ɛ L u (C L1 ) C H1 ] (42) which is weakly negative if X θ+(1 θ) ( ɛ L ɛh ) 1 γ+1 > X aut and strictly positive otherwise. The government s preferences can be represented by an indifference map in the (C 0, X) space. The indifference curves are obtained by combining aggregate welfare (equation 37) with the definition of X (equation 39), and the resource constraint (equation 41). It can be shown that the slope of the indifference curves is given by dx γ u (C 0 ) ɛ L u (C L1 ) [θ C H1 + (1 θ) C L1 ] = dc 0 β θ ɛ H u (C H1 ) C H1 C L1 [ɛ H u (C H1 ) ɛ L u (C L1 )] < 0 (43) At the optimum, the government will pick the combination (C 0, X ) where the feasibility curve and an indifference curve are tangent. It can be shown that this point has X < 1, which implies a negative interest rate on reserves. 2 6 Conclusion I study the liquidity trap in a neoclassical growth model with nominal rigidities, as in Krugman (1998), and with idiosyncratic liquidity shocks, in the spirit of Diamond and Dybvig (1983). The model is simple and tractable. Nominal rigidities create a role for 2 Figure 2 in the appendix provides a graphical analysis of the optimal monetary policy problem. 9

monetary policy and the liquidity shocks give a role to maturity transformation performed by a banking sector. The banking sector is subject to a reserve requirement and policymakers set the interest rate on reserves. I find that, even in the absence of costs to holding wealth in money, in this simple economy there is no zero lower bound on the policy interest rate. Moreover, when the economy is hit by a large adverse demand shock, it is unambiguously optimal for the central bank to set a strictly negative nominal rate on reserves. When the economy is not in a liquidity-trap state, it is optimal to set the policy interest rate equal to the (weakly) positive net rate of return on capital prevailing in the economy. 10

7 Appendix V K = V M u C 0 V D (i < 0) X aut X i < 0 1 X Figure 1: Equilibrium determination of X X 1 X Feasibility Indifference θ + 1 θ ε L ε H 1 γ+1 X aut = θ + 1 θ ε L ε H C 0 C 0 Figure 2: The optimal monetary policy problem 11

References [1] Diamond, Douglas W & Dybvig, Philip H, 1983. Bank Runs, Deposit Insurance, and Liquidity. Journal of Political Economy [2] Eggertsson, Gauti, & Michael Woodford, 2003. The Zero Bound on Interest Rates and Optimal Monetary Policy. Brookings Papers on Economic Activity [3] Krugman, Paul, 1998. It s Baaack: Japan s Slump and the Return of the Liquidity Trap. Brookings Papers on Economic Activity. 12