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The Zero Bound on Interest Rates and Optimal Monetary Policy Gauti Eggertsson and Michael Woodford 2003 Keywords: monetary policy “To preview our results, we find that the zero bound does represent an important constraint on what monetary stabilization policy can achieve, at least when certain kinds of real disturbances are encountered in an environment of low inflation. We argue that the possibility of expansion of the monetary base through central-bank purchases of a variety of types of assets does little if anything to expand the set of feasible equilibrium paths for inflation and real activity that are consistent with equilibrium under some (fully credible) policy commitment.” “The key to dealing with this sort of situation in the least damaging way is to create the right kind of expectations regarding the way in which monetary policy will be used subsequently, at a time when the central bank again has room to maneuver. We use our intertemporal equilibrium model to characterize the kind of expectations regarding future policy that it would be desirable to create, and discuss a form of price-level targeting rule that — if credibly committed to by the central bank — should bring about the constrained-optimal equilibrium.” Liquidity trap definition: “when interest rates have fallen to a level below which they cannot be driven by further monetary expansion” I. - - Is “Quantitative Easing” a Separate Policy Instrument? QE: “expansion of the monetary base” In this section, build a model that actually has money “Our key result is an irrelevance proposition for open market operations in a variety of types of assets that might be acquired by the central bank, under the assumption that the open market operations do not change the expected future conduct of monetary or fiscal policy” “The point of our result is to show that the key to effective central-bank action to combat a deflationary slump is the management of expectations,” not QE though Model: MIU, flexible wages, Calvo-staggered prices, monopolistic competition in the goods market A. Households Utility function: money-in-the-utility - Where Ct is a Dixit-Stiglitz aggregate of consumption over a continuum of goods with elasticity of substitution theta - Dixit-Stiglitz price aggregator (1) - Mt is monetary base Ht(j) is quantity of labor supplied of type j (each industry employs industry-specific labor with its own wage) - u is concave, increasing in consumption, and increasing in real balances up to a satiation point m̅ which is a function of consumption: m̅ = m̅(Ct; ξt) - “The existence of a satiation level is necessary in order for it to be possible for the zero interest-rate bound ever to be reached; we regard Japan’s experience over the past several years as having settled the theoretical debate over whether such a level of real balances exists.” [unless JGBs have a liquidity premium] - Don’t assume additive separability between consumption and real balances; thus money can affect aggregate demand by affecting the marginal utility of consumption - v is convex and increasing in labor - ξt is the vector of exogenous disturbances Budget constraint - Assume complete financial markets, no limits on borrowing - Budget constraint: - Qt,T is the “stochastic discount factor by which the financial markets value random nominal income at date T in monetary units at date t” δt is the opportunity cost of holding money, which is it/(1+it) where it is the nominal interest rate on a riskless one period obligation purchased in period t Wt is the nominal value of the household’s financial assets, including money, at the start of period t Πt(i) is the profits (revenue minus wage costs) in period t of the supplier of good i wt(j) is the nominal wage earned by laborer type j in period t Tth is net nominal tax liabilities of each household in period t Optimization - Euler equation: (2) - FOC on money demand, assuming no corner solution of zero money demand: - If both consumption and money are normal goods, then from these can solve for unique level of real balances L* L* = L(Yt,it; ξt) - And define L(Y, 0; ξt) = m̅(Y; ξt), the minimum level of real balances for which um = 0, so that the function L is continuous at i = 0 - Equilibrium condition can be written as: (3) (4) - Along with complementary slackness, i.e.: 1. it ≥ 0 and Mt/Pt = L(Yt,it; ξt) OR 2. it = 0 and Mt/Pt ≥ L(Yt,it; ξt) - Also required is a no-Ponzi condition and a transversality condition - (5, 6) Where Dt is the (nominal) sum of total government liabilities, both money and debt B. Firms An infinite number of monopolistically competitive industries - Production function in each industry is yt(i) = Atf(ht(i)) - A is exogenous productivity common to all industries - ht(i) is labor specific to industry i - Supplier of good i sets price for that good which it will supply to meet demand each period, hiring labor as necessary - Can be shown that under profit maximization, taking labor supply and quantity demanded as given from household optimization, profits each period are - ptj is the common price charged by other firms in industry j ξtilde is exogenous shocks If prices were fully flexible, pt(i) would be chosen each period to maximize this equation. Instead: Calvo pricing - Let 0<α<1 be the fraction of industries with prices that remain unchanged each period - In any industry that revises its prices in period t, “the new price pt* will be the same” - Price is implicitly defined by FOC from maximizing discounted sum of expected profits: (7) - Stochastic discount factor is [from UMP]: - Price aggregator law of motion becomes with Calvo pricing: (8) (9) C. Monetary authority Monetary policy - Assume interest-rate targeting rule similar to Taylor rule - - - (10) Assume ϕ is nonnegative (i.e. doesn’t violate ZLB constraint) This rule implies the central bank sets the monetary base to hit the prescribed interest rate [thought interest rate targeting made money supply indeterminate?????????????????] However, if the rule prescribes a zero interest rate, then this only gives a lower bound on the money supply If it is zero, suppose the monetary base rule is (11) Where ψ satisfies: o ψ(Pt/Pt-1, Yt) ≥ 1 o ψ(Pt/Pt-1, Yt) = 1 if ϕ(Pt/Pt-1) = it > 0 QE can be understood as the choice of a function φ that is greater than one sometimes What assets does the central bank buy when it varies the monetary base? k types of securities At end of period t, vector of nominal values of central bank assets is Mtωtm where ωtm is a vector of “central bank portfolio shares” These shares are determined by policy rule (12) Where the vector-valued function ωm has the property that its components sum to one Since it depends on the same arguments as ϕ, assets bought could change at ZLB Payoffs to securities in each state of the world are given by (state-contingent) vectors at and bt and matrix Ft - A vector of asset holdings zt-1 gives at end of period t-1 to owner a quantity at’zt-1 of money, a quantity bt’zt-1 of the consumption good, and a vector Ftzt-1 of securities that may be traded in period t This lets us deal with many types of assets: “For example, security j in period t-1 is a one-period riskless nominal bond if bt(j) and Ft(. , j) are zero in all states, while at(j) > 0 is the same in all states. Security j is instead a one-period real (or indexed) bond if at(j) and Ft(. , j) are zero, while bt(j) > 0 is the same in all states. It is a two-period riskless nominal pure discount bond if instead at(j) and bt(j) are zero, Ft(i; j) = 0 for all i ≠ k, Ft(k, j) > 0 is the same in all states, and security k in period t is a one-period riskless nominal bond.” Thus gross nominal return Rt(j) on the jth asset between period t-1 and t is - Where qt is the vector of nominal asset prices in period t trading No arbitrage condition ensures that equilibrium prices must satisfy: - (14) Nominal transfer to the Treasury each period, i.e. seigniorage, becomes: where Rt is the vector of returns from (13) - - (13) (15) D. Fiscal authority Fiscal authority: followd rule for evolution of total government liabilities Dt - (16) Dt is both the monetary base, the rule for which is specified above, and other types of securities, which are a subset of the k securities purchased by the central bank Let ωjtf indicate the share of government debt at the end of period t that is of type j Let Bt = Dt – Mt be the total nominal value of non-monetary liabilities Let Tth be the nominal value of the primary budget surplus (taxes minus transfers, since no government consumption) The flow government budget constraint then is: - Government determines the composition of its debt according to function: - Again where components of vector-valued ωtf sum to one - (17) E. Equilibrium and the irrelevance proposition Equilibrium: a collection of stochastic processes {pt*, Pt, Yt, it, qt, Mt, ωtm, Dt, ωtf}, “with each endogenous variable specified as a function of the history of exogenous disturbances to that date”, that satisfy the aggregate demand block of the model (2)-(6), aggregate supply conditions (7) and (9), conditions (10)-(12) specifying monetary policy, and conditions (16)-(17) specifying fiscal policy each period Irrelevance proposition: “The set of paths for the variables{pt*, Pt, Yt, it, qt, Mt, ωtm, Dt, ωtf} that are consistent with the existence of a rational-expectations equilibrium are independent of the specification of the functions ψ in equation (1.11), ωm in equation (1.12), and ωf in equation (1.17)” - Recall ψ is the QE function, ωm is the composition of assets purchased, and ωf is the composition of fiscal authority liabilities - The set of restrictions on the set of processes can be written in a form that does not involve the irrelevant variables Mt, ωm, or ωf and so does include the functions ψ, ωm, or ωf - Proof: - Note first, for all m ≥ m̅(C;.), it is true that u(C, m;.) = u(C, m̅;.) because it’s a satiation point - Differentiating, we see that uc(C, m;.) does not depend on the exact value of m if it exceeds the satiation point - Thus at the ZLB we can replace uc(Yt, Mt/Pt;.) that appears several times in our equilibrium relations by the following λ function, using the fact that Mt/Pt = L at all levels of real balances at which uc depends on the level of real balances - i.e. rewrite uc as a function that doesn’t include Mt/Pt We can similarly replace um(Yt, Mt/Pt;.)(Mt/Pt) that appears in (5) by - Since Mt/Pt = Lt when real balances are lower than satiation, and um = 0 when real balances are higher Finally we can express nominal profits as the following, after having substituted λ(.,.;.) for the marginal utility of real income in the wage demand function that is used in deriving the profit function - - Thus we can rewrite the system of equations that are the equilibrium relations as follows, none of which include ψ, ωm, ωf - Along with relations (9), (10), and (16) as before Thus generalized QE (LSAP, Operation Twist) does not matter if policy rules do not change - “If the commitments of policymakers regarding the rule by which interest rates will be set, on the one hand, and the rule by which total private sector claims on the government will be allowed to grow, on the other, are fully credible, then it is only the choice of those commitments that matters.” - Also no portfolio balance effect, despite distinguishing among different assets o “Our conclusion differs from that of the literature on portfolio balance effects for a different reason. The classic theoretical analysis of portfolio balance effects assumes a representative investor with mean-variance preferences. This has the implication that if the supply of assets that pay off disproportionately in certain states of the world is increased (so that the extent to which the representative investor ’ s portfolio pays off in those states must also increase), the relative marginal valuation of income in those particular states is reduced, resulting in a lower relative price for the assets that pay off in those states. But in our generalequilibrium asset pricing model, there is no such effect. The marginal utility to the representative household of additional income in a given state of the world depends on the household’s consumption in that state, not on the aggregate payoff of its asset portfolio in that state. And changes in the composition of the securities in the hands of the public do not change the state-contingent consumption of the representative household — this depends on equilibrium output” - The irrelevance proposition is in the spirit of Wallace (1981)’s irrelevance proposition for open market operations. Wallace’s analysis is usually useless, because money is only a store of value in his model, so an equilibrium where short-term Treasuries have a higher return than money is not possible, something that is very often observed. But at the ZLB, the marginal liquidity services provided by money is zero, so Wallace’s analysis is correct What about the assumption of complete markets and no debt constraints? - “The absence of borrowing limits is also innocuous, at least in the case of a representative-household model, because in equilibrium the representative household must hold the entire net supply of financial claims on the government. As long as the fiscal rule (equation 16) implies positive government liabilities at each date, any borrowing limits that might be assumed can never bind in equilibrium. Borrowing limits can matter more in the case of a model with heterogeneous households. But in this case the effects of open-market operations should depend not merely on which sorts of assets are purchased and which sorts of liabilities are issued to finance those purchases, but also on how the central bank’s trading profits are eventually rebated to the private sector (that is, with what delay and how distributed across the heterogeneous households), as a result of the specification of fiscal policy. The effects will not be mechanical consequences of the change in composition of assets in the hands of the public, but instead will result from the fiscal transfers to which the transaction gives rise; it is unclear how quantitatively significant such effects should be.” The important assumption is that OMOs have no consequence for future interest rate policy or on the evolution of total government debt liabilities Dt at present or in the future - “In our view it is more important to note that our irrelevance proposition depends on an assumption that interest rate policy is specified in a way that implies that these openmarket operations have no consequences for interest rate policy, either immediately (which is trivial, because it would not be possible for them to lower current interest rates, which is the only effect that would be desired), or at any subsequent date. We have also specified fiscal policy in a way that implies that the contemplated open-market operations have no effect on the path of total government liabilities {Dt} either, whether immediately or at any later date. Although we think these definitions make sense, as a way of isolating the pure effects of open-market purchases of assets by the central bank from either interest rate policy on the one hand or fiscal policy on the other, those who recommend monetary expansion by the central bank may intend for this to have consequences of one or both of these other sorts.” For example, a helicopter drop is not just a change in the function φ in the policy rule - First, the expansion is supposed to be permanent - In this case, the function ϕ that defines interest rate policy is also being changed in a way that will be relevant at a future date when money supply no longer exceeds satiation - Second, a helicopter drop implies a change in fiscal policy as well - The operation increases the nominal value of total government liabilities, and this is often presumed to be permanent as well - - - “First of all, it is typically supposed that the expansion of the money supply will be permanent. If this is the case, then the function φ that defines interest rate policy is also being changed, in a way that will become relevant at some future date, when the money supply no longer exceeds the satiation level. Second, the assumption that the money supply is increased through a helicopter drop rather than an open-market operation implies a change in fiscal policy as well. Such an operation would increase the value of nominal government liabilities, and it is generally at least tacitly assumed that this is a permanent increase as well.” “This explains the apparent difference between our result and that obtained by Auerbach and Obstfeld (2003) in a similar model. These authors assume explicitly that an increase in the money supply at a time when the zero bound binds carries with it the implication of a permanently larger money supply, and that there exists a future date at which the zero bound ceases to bind, so that the larger money supply will imply a different interest rate policy at that later date. Clouse and others (2003) also stress that maintenance of the larger money supply until a date at which the zero bound would not otherwise bind represents one straightforward channel through which open-market operations while the zero bound is binding could have a stimulative effect, although they discuss other possible channels as well.” Thus, helicopter drops are not subject to the irrelevance proposition Thus future policy is important - “And this question can be addressed without explicit consideration of the role of openmarket operations by the central bank of any kind. Hence we shall simplify our model — abstracting from monetary frictions and the structure of government liabilities altogether — and instead consider how it is desirable for interest-rate policy to be conducted, and what kind of commitments about this policy it is desirable to make in advance” II. How Severe a Constraint is the Zero Bound? Central bank is still powerful at ZLB since it’s the expected path of short term rates, not the current level alone, which is important; but ZLB is a genuine constraint Log-linearize model above around zero inflation SS since this is optimal (not Friedman’s deflation, since abstracting from transaction frictions) - Under zero inflation, “it is easily seen” real rate of interest (and also nominal rate of interest) is r̅ = β-1 – 1 > 0 [since interest rate will equal rate of time preference rho] - Because log linearization only works for small disturbances, and 0 might be too far away, we have to abstract and say that there is some interest paid on money im > 0 that cannot be reduced, and is the lower bound As Woodford has shown in Interest and Prices, the log linear approximated equilibrium relations can be summarized by two equations each period: - A forward looking IS relation - And a forward looking AS relation (NK Phillips Curve) - Where πt = log(Pt/Pt-1), i.e. inflation xt is the “welfare-relevant” output gap it is now the continuously compounded nominal interest rate, corresponding to log(1+it) using the notation of the previous section ut is exogenous “cost push disturbances” that shift the NKPC rnt is exogenous variation in the Wicksellian natural rate that shifts the IS relation (24) is log linearized equation (2) (25) is log linearized equation (7)-(9) eliminating log(p*t/Pt) We can omit the log linear equation for money demand since we don’t care about it “The other equilibrium requirements of the earlier discussion can be ignored in the case that we are interested only in possible equilibria that remain forever near the zeroinflation steady state, because they are automatically satisfied in that case” For the inflation target π* to be hit, the real interest rate must equal the Wicksellian rate However, the ZLB prevents this from holding if 𝑟𝑡𝑛 < −𝜋 ∗ - E.g. if target inflation is zero, if natural rate is negative, it won’t be able to be hit - Higher inflation target thus allows central bank to be less likely to have this problem - But inflation still distortionary Suppose for illustration the natural rate of interest is unexpectedly -2% in the first period then reverts back to steady state r̅ =4% > 0 with constant probability 10% every period (expected liquidity trap length of 10 periods) - With 2% inflation target, natural rate can be hit by nominal rate This is why some call for a higher inflation target. But if inflation target doesn’t have to be fixed, we can have even more optimal policy: III. The Optimal Policy Commitment What is the optimal policy to satisfy (24) and (25)? - Assume, for now that policy is fully credible - Purely forward-looking policy (i.e. neglecting past conditions) would do no good The government loss function can be derived from a second-order Taylor expansion of the utility function of the representative household (see again Interest and Prices) - Again, this is abstracting from transaction frictions; if they exist, the loss function includes an additional term proportional to (it – im)2, a la Friedman rule However, because Minimize loss function under the constraint (from ZLB and IS equation): - Giving the Lagrangian - The FOCs are - This cannot be solved algebraically, due to nonlinearities of (31); must use numerical methods [fuck] Optimal policy - Is history dependent – note Lagrange multipliers are lagged (t-1) - Depends on keeping interest rate lower than otherwise in the future - Even promising inflation 10 years down line, i.e. not that soon, has an effect - However, higher inflation in the future will have distortionary effects, and there is a tradeoff; so even under optimal policy with commitment, there will be some negative effect today - Further, although there will be a higher price level in the future, the price level will ultimately be stabilized Suppose for illustration again natural rate negative in period zero then reverts to SS with fixed probability. Under optimal policy: (again where lines indicate different paths of natural rate) - Optimal policy has you create a boom and inflation once the natural rate returns to SS Price level increases, but eventually stabilizes Contrast interest rate paths in comparison with a zero-inflation target: Full comparison where natural rate returns to SS, stochastically, at period 15: - Policy rate is kept at zero five periods longer - As a consequence, deflation and output gap largely avoided IV. Implementing optimal policy Implementation of optimal policy - Can be committed to via a policy rule that is a function of what has occurred before that date - Because describing policy as the (state-contingent) interest rate path leaves variable indeterminate, not an appropriate policy rule - Instead, describe rule as function of deviations of inflation, output gap from desired paths - Only needs to observe price level and output gap; not equilibrium interest rate Each period there is a predetermined price level target pt*; the central bank chooses the interest rate it to achieve, if possible - And otherwise sets it = 0. 𝑝̃𝑡 is the output gap adjusted price level index defined as (see Interest and Prices) - If 𝜆 = 𝜅, then the natural rate of output follows a deterministic trend and this is NGDP targeting; unlikely though The target for the next period is then determined by - Where Δ𝑡 is the shortfall in period t - This achieves the optimal commitment solution “If the price-level target is not reached, because of the zero bound, the central bank increases its target for the next period. This, in turn, increases inflation expectations further in the trap, which is exactly what is needed to reduce the real interest rate.” The best way to make a rule credible is to conduct policy over time that demonstrates the commitment - “The best way of making a rule credible is for the central bank to conduct policy over time in a way that demonstrates its commitment. Ideally, the central bank’s commitment to the price-level targeting framework would be demonstrated before the zero bound came to bind (at which time the central bank would have frequent opportunities to show that the target did determine its behavior).” - In normal times, when natural rate of interest is nonnegative, Δ𝑡 = 0 A simpler version of this rule approximates this and does not have any special proviso for the liquidity trap: - “More easily communicated to the public”; fully optimal if ZLB never binding Constant price level target: the target for the gap adjusted price level is fixed at all times Comparison with optimal policy: (welfare loss relative to strict inflation target of zero) If this was instead inflation targeting instead of level targeting (first differencing: 𝜆 𝜋𝑡 + 𝜅 (𝑥𝑡 − 𝑥{𝑡−1} ) = 0), when the ZLB binds this would be a disaster. Central bank would deflate after leaving liquidity trap - i.e., the level targeting (history dependence) aspect is key V. Other issues What if ZLB binds forever? - Consistent with all equilibrium conditions - …EXCEPT transversality condition (6) (which recall is equivalent to the condition that households hit their intertemporal budget constraint) - To answer the question of whether this is possible requires a more complete specification of the fiscal policy rule - With a Ricardian policy rule a la Benhabib, Schmitt-Grohe, and Uribe (2001), you can get multiple equilibria including a perpetual liquidity trap - However with a fiscal policy rule that does not allow for rapid contraction in government liabilities, e.g. a balanced budget rule, however, such a case is not possible - Also e.g. a rule not to contract monetary base in combination with a commitment for a nonnegative asymptotic present value of public debt Other aspects of expectations management - What policies might contribute to desirable expectations - - Demonstrate resolve by following rule before crisis hits Manipulating money supply “can be helpful, even though irrelevant to interest rate control as a way of communicating to the private sector the central bank’s belief about where the price level ought properly to be (and hence the quantity of base money that the economy ought to need)” i.e. QE can be an important communication device Thus Svensson’s “fool-proof method” is just a communication and commitment device The time inconsistency problem and output gap-adjusted price level targeting - If the central bank is credibly able to commit to OGA-PLT, then ZLB not too serious - However, if the central bank is not able to credibly commit to future actions, then central bank each period will end up targeting strict inflation of zero - Result is a prolonged contraction - “Deflationary bias,” as opposed to inflationary bias of Kydland and Prescott Preventing time inconsistency - If central bank cares about distortions caused by taxes, then tax cuts financed by money printing can be an effective commitment mechanism - Purchases of real assets (e.g. real estate) can be thought of as another way of increasing nominal government liabilities, which would affect inflation incentives similar to deficit - Purchases of foreign exchange: seigniorage off of foreigners