Optimal Monetary Policy at the Zero Lower Bound on Nominal Interest Rates in a Cost Channel Economy

The nominal interest rates were at zero level in the recent past in many countries across the globe. It has been widely debated recently what a central bank should do to stimulate the economy when the nominal interest rate is at the zero lower bound (ZLB). The optimal monetary policy literature suggests that monetary policy inertia, i.e. committing to continue zero interest regime even after the ZLB is not binding, is a way to get the economy out of recession. In this paper, I examine whether this result holds when monetary policy has not only the conventional demand-side e ect but also a supply-side e ect on the economy. To accomplish this objective, I incorporate the cost channel of monetary policy into an otherwise standard new Keynesian model and evaluate the optimal monetary policy at the ZLB. The study revealed some important insights in the conduct of the optimal monetary policy in a cost channel economy at the ZLB. First, the discretionary policy requires central banks to keep interest rates at the zero lower bound for longer in a cost channel economy compared ∗Email: lasitha.pathberiya@uqconnect.edu.au. †I would like to thank Dr. Yuki Teranishi for sharing the Matlab code used in the analysis of their paper Jung et al. (2005). This code was helpful to develop the Matlab code for the present paper.


Introduction
The zero lower bound on nominal interest rates (ZLB) is no longer just a theoretical interest.
Nominal interest rates were at zero lower bound in the recent past in many countries across the globe, including the USA and Japan. 1 It has been widely debated recently what a central bank should do to stimulate the economy when the aggregate demand is weak, even when the nominal interest rate is at zero level.
2 Optimal monetary policy literature suggests monetary policy inertia, i.e. committing to continue zero interest regime even after the ZLB is not binding, is a way to get the economy out of recession. In this paper, I examine whether this result holds when monetary policy has not only the conventional demand-side eect but also a supply-side eect on the economy.
In the optimal monetary policy literature, there are two main policies which attempt to stabilise the economy in terms of ination and output following a shock to the economy. They are known as discretionary policy and commitment policy. Under discretion, the central bank takes the current state of the economy and private sector expectations as given. Under this policy, the central bank optimises in each period; therefore, any promises given by the bank are not credible. On the other hand, under commitment policy, the central bank chooses a path for current and future ination as well as output and commit to that. Therefore, under commitment, if the central bank is credible, it can adjust private sector expectations [see Walsh (2010, p357-364)]. Nobel laureates Finn E. Kydland and Edward C. Prescott, in their seminal paper Rules Rather than Discretion: The Inconsistency of Optimal Plans [Kydland and Prescott (1977)], showed how an announcement of commitment to a low ination regime by monetary authorities might create lower private sector inationary expectations. They 1 Four central banks in Europe, including the European Central Bank, Swedish Riksbank and recently the central bank of Japan have pushed short-term nominal interest rates below the zero lower bound. This phenomenon is unprecedented and conned only to those central banks. In the present study, I consider short-term nominal interest rates are to be constrained by the zero lower bound.
2 During the past decade, the central banks around the world, including the Federal Reserve Bank of the USA had to resort to unconventional monetary policies due to ZLB constraint. Two main such unconventional policies that have been considered are forward guidance and balance sheet policies. However, in this paper, my focus is only on conventional monetary policy with the interest rate instrument.
argued that if this monetary policy is then changed and interest rates are reduced to give a short-term lift to employment, credibility of policy makers will be lost and conditions may worsen.
The commitment of the central bank to future actions and informing the public of them in the form of forward guidance 3 at the ZLB is supported by a large body of literature, starting with Krugman (1998). Although John Maynard Keynes was the rst to raise the question of zero lower bound on nominal interest rates in the context of the Great Depression, it was of only theoretical interest until Japan faced the ZLB constraint in reality in the 1990s.
Krugman, in his seminal work in 1998, recommended that central banks commit to credible promises to the public to have higher ination in the future. Since then, scholars such as Jung, Teranishi and Watanabe (2005, henceforth JTW), Eggertsson and Woodford (2003), Adam and Billi (2006) and Nakov (2008), among others using more complex dynamic forward looking models have conrmed the ndings of Krugman.
However, the optimal monetary policy literature at the ZLB thus far has abstracted an important characteristic of the economy, i.e. the cost channel of monetary policy. Most recently Chattopadhyay and Ghosh (2016) conduct a study on optimal monetary policy at the ZLB in a cost channel economy with varying degree of interest rate pass-through. Their methodology is similar to the methodology use in this paper and they report results similar to mine.

4
The cost channel is said to be present in an economy if the changes in nominal interest rates aect the supply-side of the economy. It has been found by many recent studies 5 that the cost channel is an important channel of monetary policy in the USA and other developed countries. Ravenna and Walsh (2006), utilising a new Keynesian forward looking model, 3 Forward guidance is issuing explicit statements by central banks about the outlook for future policy, in addition to their announcements about the immediate policy actions that they are undertaking. The literature divides forward guidance into two types. First is Odyssean forward guidance, where the monetary authority publicly commits to a future action. The other is Delphic forward guidance, where the monetary authority merely forecasts macroeconomic performance and likely monetary policy actions. 4 Both papers have been written in the same time period, but independently. 5 For example, see Barth and Ramey (2001), Christiano et al. (2005), Kim and Lastrapes (2007), Henzel et al. (2009), Ravenna and Walsh (2006) and Chowdhury et al. (2006). theoretically showed that the cost channel aects optimal monetary policy in important ways.
They showed that both the output gap and ination are allowed to uctuate in response to productivity and demand shocks under optimal monetary policy in a cost channel economy, among other ndings.
Presence of the cost channel could aect the policy outcomes of new Keynesian studies which examine zero lower bound policies. In general, the cost channel makes changes directly to the current ination as well as the current output gap due to changes in nominal interest rates, other things being equal. In addition to that, when the cost channel is active in the model, it accelerates future inationary expectations if the monetary authority commits to a tight monetary policy. This again raises current ination. Consequently, the presence of the cost channel may aect the optimal monetary policy at the ZLB.
The main objective of this paper is to examine the central bank policy options at the ZLB in a cost channel economy. Specically, this study inquires when the central bank should exit the zero nominal interest rate regime. In this regard, I consider both discretionary and commitment polices, although the study mainly focuses on the commitment policies. I consider a variation of the standard new Keynesian model to accomplish the above objective.
To carry out simulations, I calibrate the model to the economy of the USA.
The main ndings are as follows: a) the discretionary policy requires central banks to keep interest rates at the zero lower bound for longer in a cost channel economy compared to no-cost channel economies. b) Under commitment policy, the simulation exercise shows that the central bank is able to terminate the zero interest rate regime earlier in cost channel economies than otherwise. c) The cost channel generates substantially high welfare losses, under both discretionary and commitment policies.
The rest of the study is structured as follows. In section 2, I review the relevant literature on the optimal monetary policy at the ZLB and the optimal monetary policy with the cost channel. Section 3 describes the model, steady states and optimal dynamic paths. Model simulations and results are given in section 4. Section 5 concludes the study. Although there are few policy options within the traditional interest rate rule to get out of the ZLB constraint, the most accepted solution is the commitment to a policy. Krugman (1998) puts it as follows: monetary policy will in fact be eective if the central bank can credibly promise to be irresponsible, to seek a higher future price level. He argues that under these conditions the liquidity trap boils down to a credibility problem. Private agents expect any monetary expansion carried out by the central bank at the ZLB would be reverted immediately once the economy has recovered. Such expectations may not stimulate the economy in the recession. As a solution, Krugman suggests that the central bank should commit to a policy of high future ination over an extended horizon. Following Krugman's work, many proved in more complex dynamic models that the commitment to a policy plan which is facilitated by forward guidance is one way of getting out of the slump. Eggertsson and Woodford (2003) studied optimal commitment policy with ZLB in an inter-temporal model in which the natural rate of interest is allowed to take two dier-ent values. The natural rate of interest was assumed to become negative unexpectedly in the beginning and then move to a positive level with certain probability in every period. They explored how the existence of the zero lower bound aects the optimal conduct of monetary policy with regard to both ination and output. Eggertsson and Woodford recommended a form of price-level targeting rule that should bring about the constrained optimal equilibrium if the central bank is credible. JTW considered a similar set up to Eggertsson and Woodford (2003) with perfect foresight, however, they considered an exogenous AR(1) process to the natural rate of interest. Both Eggertsson and Woodford (2003) and JTW found that at the ZLB, under commitment, the central bank should continue zero nominal interest rates even after the natural rate of interest returns back to the positive level. Doing so, the central bank can stimulate the economy by generating higher inationary expectations. Extending this work, recently, Hasui et al. (2016) consider the optimal commitment policy in an economy with ination persistence. They argue that ination persistence changes the central bank's objective from achieving target ination rate to ination smoothing. Therefore, agents expect an accommodative monetary policy, in turn, increasing inationary expectations. This produces an acceleration in ination and allows the central bank to terminate the ZLB policy earlier compared to an economy without ination persistence. Nakov (2008) solved numerically a stochastic general equilibrium model with the ZLB. He extended the work of Eggertsson and Woodford (2003) and JTW with an explicit occasionally binding ZLB. Previous studies analysed the economy given that the economy is at the ZLB following a large demand shock. Nakov found that uncertainty plays an important role in the dynamics of variables such as ination, output gap and interest rates in the presence of ZLB. For example, he found that under discretionary policy ination falls short of its target for any value of natural interest rate. That is, average value of ination rate is below the target, implying deationary bias.
Although many scholars suggest continuing zero level of interests for a longer time at the ZLB to increase inationary expectations in a recession, some raise doubts. Levin et al.  (2015) found that forward guidance is highly sensitive to the complete markets assumption in standard new Keynesian models. They showed that if the agents face uninsurable income risk and borrowing constraints, such agents adjust their responses to changes in future interest rates. This is due to precautionary savings. Accordingly, forward guidance has less power to stimulate the economy.
The above line of research assumes that the central bank is fully credible, such that private agents believe the commitments. Bodenstein et al. (2012) relaxed the assumption of the fully credible central bank. In a new Keynesian set up, he found that at the ZLB, the central bank faces a severe time-inconsistency problem. Initially, a promise to keep the nominal interest rate low for an extended period raises inationary expectations. Further, it lowers current and future real interest rates, and thus stimulates current output. However, once the economy has emerged from the slump, it is not optimal to keep interest low any longer. Accordingly, he found that if a central bank's announced promises are not credible, then the economy goes through a deeper recession than otherwise.
All the studies specied above are based on the central bank optimising the social welfare.
There is another line of research which studies the performance of simple monetary policy rules at the ZLB. Here, the monetary authority commits to a particular type of rule such as the Taylor rule [Taylor (1993)]. Studies such as Fuhrer andMadigan (1997), Eggerston andWoodford (2003), Wolman (2005), Coenen et al. (2004) and Nakov (2008) examine this problem. These studies, in general, show that if the target ination rate is closer to zero, simple policy rules such as Taylor rule, can generate signicant welfare losses. However, Eggerston and Woodford (2003) and Wolman (2005) showed that the policy rules formulated in terms of a price level target can considerably reduce these welfare losses. In contrast, recently, Hasui et al. (2016)showed that the performance of price-level target in an economy with ination persistence is substantially low. Ravenna and Walsh (2006) were the rst to show that the presence of the cost channel alters the optimal monetary policy problem in important ways. They showed that the interest rate changes carried out to stabilise the output gap lead to ination uctuations when a cost channel is present. As a consequence, the output gap and ination uctuate in response to productivity and demand disturbances, even when the central bank is setting policy optimally. They assumed that a cost channel is present in the economy when rms' marginal cost depends directly on the nominal interest rate. Following Ravenna and Walsh (2006), others analysed the optimal monetary policy with the cost channel from dierent perspectives and found that the cost channel is important when analysing the optimal monetary policy. Chattopadhyay and Ghosh (2016), written independent of this paper, consider optimal monetary policy in a cost channel economy. They report similar results to this paper under both optimal discretionary and optimal commitment policies using a new Keynesian model at the ZLB. In addition to the two optimal policies, they consider a policy called 'T-only' policy. Under T-only policy, the central bank chooses and announces the optimal exit time of zero interest rates regime and promises to exercise the discretionary policy following the exit. Chattopadhyay and Ghosh show that this policy closely replicates the commitment policy both under presence and absence of the cost channel.

Optimal Monetary Policy with the Cost Channel
Fiore and Tristani (2013) studied optimal monetary policy in a model of the credit channel with the cost channel of monetary policy. Using a second-order approximation of the welfare function, they showed that welfare is directly aected not just by the volatility of ination and the output gap, as in the standard case where there is no nancial frictions, but also by the volatility of the nominal interest rate and credit spreads. Credit spreads aect optimal monetary policy through the cost channel. Higher credit spreads make borrowing costly for rms by increasing marginal cost of production. Overall, the authors have concluded that the monetary authorities ought to pay attention to nancial market friction.
Tillmann (2009) studied the optimal monetary policy with an uncertain cost channel.
He concluded that, the larger the degree of uncertainty about the cost channel, the smaller the interest rate response to ination. He incorporated uncertainty of the cost channel into the model since the eectiveness of the cost channel varies signicantly over time and across countries. Therefore, the monetary authority may not be certain about the eectiveness of the true role of the cost channel at a given time. The framework of his study is new Keynesian, which has a policy maker who plays a zero-sum game against an evil agent who sets the parameters such that the welfare loss is maximised. In the model, an uncertain policy maker should overestimate the quantitative importance of the cost channel when setting interest rates. In this sense, the policy maker is less aggressive than under certainty.
Studies show that optimal monetary policy in the presence of the cost channel leads to an increased indeterminacy region. Surico (2008) studied the conditions that guarantee equilibrium determinacy in a standard sticky price new Keynesian model augmented with a cost channel. Surico showed that a central bank that assigns a positive weight to the output gap in the reaction function makes the economy more prone to multiple equilibria compared to the standard case. His results are robust to forward-looking, current, and backwardlooking policy rules. Surico suggested that, when the cost channel is empirically important, trying to limit cyclical swings in real activity may result in undesired volatility of ination and output.
The next section presents the model, derives steady states and analyses the optimal dynamic path following a negative shock to the economy.

The Model
I consider a new Keynesian forward looking inter-temporal model to study the cost channel economy at the ZLB. This model is most suitable for the present analysis as it incorporates private sector expectations explicitly into the model. The model is based on JTW and Ravenna and Walsh (2006). I extend these authors' models to incorporate both the cost channel and the ZLB. The basic model is standard; however, a brief exposition is presented here to self-contain the analysis. The exposition is based on Ravenna and Walsh (2006); however, their model has been simplied by ignoring the government and taste shocks.
Following them, I assume the cost of labour must be nanced at the beginning of the period.
However, their assumption that the full labour cost has to be nanced externally at the beginning of the year has been relaxed.
The model economy consists of three main sectors, namely, households, production and monetary authority. Financial intermediaries are also part of the economy, where rms borrow money to nance their wage bill. These players interact with each other in assets, goods and labour markets.

Households
There is a large number of identical innitely-lived households in the economy. The preferences of a representative household are dened over a composite good C t and time devoted to employment N t . Households maximise the expected present discounted value of utility:

Households
Optimal Monetary Policy at the ZLB in a Cost Channel Economy where β ∈ (0, 1) is a subjective rate of discount, σ > 0 is the coecient of relative risk aversion and η > 0 is elasticity of labour supply. The composite consumption good consists of dierentiated goods produced by monopolistically competitive nal goods producers. There is a continuum of such producers of measure 1. C t is dened thusly: , where c jt is the consumption of the good produced by rm j and θ(> 1) is the elasticity of substitution between varieties. The price elasticity of demand for the individual goods is determined by θ. As θ increases, the dierent goods becomes closer substitutes. According to this specication, consumer demand and the aggregate price index are given by c jt = respectively. The price of the nal good of rm j at time t is P jt .
Households receive their labour income at the beginning of the period at the nominal wage rate of W t . They enter the period t with cash holdings of M t and make deposits D t at the nancial intermediary. Accordingly, household's consumption expenditures are restricted by the following cash-in-advance constraint: and budget constraint: where Π t is the prot income received from owning nancial intermediaries and R t is the gross nominal interest rate. It is also assumed that households are subject to a solvency constraint that prevents them from engaging in Ponzi-type schemes.
By maximising household utility subject to budget constraint, the following rst order The next section describes the production sector of the economy.

Production Sector
Firms in this model use no capital in the production process. They have to pay wages at the beginning of the period, before realising sales proceeds. The production technology is given by y jt = A t N jt , where y jt is total demand for good j in period t, N jt is employment by rm j in period t and A t is an exogenous aggregate productivity factor. The staggered price setting of Calvo (1983) is used assuming each rm resets its price in any given period only with probability 1 − ω. Firms set their prices independent of other rms and of the time elapsed since the last adjustment. By considering the optimal price chosen by each rm, it is well-known, as shown by Gali (2002) and others, that this standard production sector specication leads to the following ination adjustment equation, mostly known as the new Keynesian Phillips curve (NKPC): where π t is the rate of ination between time t−1 and t,ψ t is the percentage deviation of real marginal cost around its steady state 6 (which is same for all the rms) and κ = (1−ω)(1−ωβ) ω .
The cost channel model deviates from the standard new Keynesian model in the specication of the marginal cost. The marginal cost is dierent in the cost channel model than in 6 Throughout this paper, a hat sign (ˆ) denotes the percentage deviation of the concerned variable around its steady state.
Optimal Monetary Policy at the ZLB in a Cost Channel Economy the standard model due to the borrowing of the wage bill. What follows is the derivation of the corresponding real marginal cost with regard to the cost channel model.
Assume a rm takes out a loan worth JW t N t from nancial intermediaries to cover part of its nominal wages of W t N t . Accordingly, J (J ∈ [0, 1]) denotes the portion of the wage bill covered by rms using external loans at time t. If J = 1, rms borrow the full wage bill externally. If J = 0, that means the rm does not take out loans externally to cover the wage bill.
Accordingly, the real marginal cost is given thusly: The log linearised real marginal cost (see Appendix A for derivation) is: where x t is the output gap given by (Ŷ t −Ŷ f t ). The percentage deviation of output around its steady state isŶ t , andŶ f t is the percentage deviation of exible price output around its steady state at time t. 7 The percentage point deviation of nominal interest rate around zero ination steady state value of R isR t .
Accordingly, the NKPC adjusted for the cost channel, is derived using equations (4) and (6) as follows: It is clear from equation (7), that when J = 1, the NKPC boils down to Ravenna and Walsh (2006) and, when J = 0, it turns to the standard NKPC. Iterating this equation forward yields the following: 7 Equilibrium exible price output is discussed in detail below.
This equation shows that current ination not only depends on the current and future path of the output gap but also on the current and future path of the nominal interest rates.
The latter inuences current ination directly due to the inclusion of the cost channel in the model.
Log linearising the Euler equation given by (1) yields the well-known dynamic IS equation: where u t is an exogenous demand disturbance term.
Since I am comparing the results with JTW, to be compatible with their model, I introduce natural rate of interest (r n t ) as dened by JTW.
8 Accordingly, the dynamic IS equation becomes this: wherer n t is the percentage point deviation of the net natural interest rate around its zero ination steady state value of r n . At the zero ination steady state, nominal interest rate is equal to natural rate of interest rate.
9 Accordingly, at the zero ination steady state, the following result holds: R = 1 + r n = 1 β .
8 JTW denes the natural interest rate as follows: is the potential output and g t is a disturbance that uctuates independently of changes in the real interest rate. 9 At the zero ination steady state, I assume the potential growth in the economy to be zero and that there will be no disturbances to the natural rate of interest. Accordingly, the natural interest rate at zero ination steady state is equal to 1 β − 1. From the Euler equation given by equation (1), it is easy to nd the zero ination steady state value of the net nominal interest rate is also equal to 1 β − 1.

Production Sector
Optimal Monetary Policy at the ZLB in a Cost Channel Economy

Aggregate Resource Constraint
The economy I consider in this model is a simple economy. It abstracts from aggregate demand components such as investments, government purchases or net exports. Accordingly, aggregate resource constraint of the economy is given by this: where Y t is the aggregate production.

Flexible Price Equilibrium
The model developed above is characterised by three distortions. The rst of them is the presence of market power in the goods market due to the monopolistic competition of the rms. The second is due to price rigidity. These two distortions are basic in the standard new Keynesian model. The third distortion is specic to this study, and it is due to the cost channel. In the following section, I relax the price rigidity assumption and examine the equilibrium output under exible prices.
Suppose that all rms adjust prices optimally in each period, i.e. prices are fully exible.
When prices are fully exible, all rms charge the same price. Each rm sets its price equal to a markup, δ(= θ θ−1 > 1) over its nominal marginal cost, which is constant over time. Hence, it follows that the real marginal cost will also be constant and equal to the the inverse of the optimal markup chosen by rms.
10 Let superscript f denote the exible price equilibrium values of relevant variables. Accordingly: Hence: 10 See Walsh (2010, p334-335) for a detailed description of the exible price mechanism.
Households equate the real wage to the marginal rate of substitution between leisure and consumption. From equation (2): Combining Equation (9) and (10) together with production function and resource constraint yields the following: Hence: This shows that the equilibrium exible price output is distorted by monetary policy as the nominal interest rate is an argument in the equation and the presence of market power in the goods market. With regard to the distortions by monetary policy, for example, an increase in nominal interest rate decreases labour demand, which in turn reduces the equilibrium level of exible price output. This distortion is directly due to the inclusion of the cost channel in the model.
The steady state value of the exible price output is given as follows: where R f is the steady state value of the exible price nominal interest rate.

Monetary Authority
Optimal Monetary Policy at the ZLB in a Cost Channel Economy The steady state value of the exible price output is also distorted by monetary policy and monopolistic competition. If J = 0, then by construction the cost channel is eliminated and the distortion is also eliminated. On the other hand, if the nominal interest rate is zero (or R f = 1), the distortion brought in by the cost channel is eliminated. Distortion due to monopolistic competition can be eliminated by setting δ = 1.
Next, I specify the objective of the monetary authority and her problem.

Monetary Authority
The monetary authority has one monetary instrument, which is the short-term nominal interest rate. It attempts to minimise the loss function: where λ is a positive parameter representing the weight assigned to output stability. This loss function has been derived using second-order Taylor
The problem is as follows: subject to This problem cannot be solved by applying standard solution methods for rational expectations models because of the complications brought in by the non-linear constraint in equation (13). To make the analysis more tractable, I consider the agents with perfect foresight under both discretion and commitment policies in the following sections.
In this analysis, following JTW, it has been considered that the economy is in a liquidity trap following a large negative demand shock to the natural interest rate. The natural rate of interest follows an AR(1) process following the shock and converges to steady state value in and after period one. The AR(1) process is as follows: r n t = ρ t 0 + r n for t = 0, 1, 2, 3..., where 0 is the large negative shock that occurs in the time t = 0, and ρ is the persistence of the shock (0 < ρ < 1).
The optimisation problem under each commitment and discretionary policies is considered in the following sections.

Optimisation under Discretion
Under discretion, the central bank treats the optimisation problem as a sequential optimisation problem. Accordingly, the central bank makes whatever decision is optimal in each period without committing to future actions. The central bank chooses (x t , π t ) in order to minimise the objective function given by equation (12) where µ t , δ t and ν t are Lagrangian multipliers.
Under discretion, the central bank optimises in each period. Accordingly, the Karush-Kuhn-Tucker (KKT) conditions of the problem are the following: At steady state, dene x t = x, π t = π, r n t = r n , δ t = δ, µ t = µ, ν t = ν,R t =R andr n t = 0.
Also dene R ss as the value of the gross nominal interest rate relevant to the particular steady state.
11 Accordingly, the KKT conditions become the following: Potentially there can be two steady states in the system, an interior solution and a corner solution. First, I will consider the interior solution. In this case, the nominal interest rate is strictly positive, i.e. R ss > 1. According to the KKT conditions, ν = 0. Substituting these into the above steady state conditions and solving the linear system of equations yields: π = 0, x = 0, R ss = 1 + r n = R, µ = 0, δ = 0 and ν = 0.
As required by the KKT conditions, ν is strictly negative at the corner solution for the following values of J: Therefore, there exists a second steady state at the ZLB when J is suciently small. JTW show that there is a second steady state under discretion for a no-cost channel economy, i.e.
when J = 0. For the baseline parametrisation values set at the calibration section below, maximum value of J to have a second steady state is 0.9. This steady state does not minimise the central bank loss function since both ination and the output gap have been deviated from zero.
The Friedman rule [Friedman (1969)] of zero nominal interest rate is not optimal in this model. One reason for this dierent conclusion is the absence of any explicit role for money in the utility approximation of equation (11), as showed by Walsh (2010, p355). Another one is, as mentioned by JTW, the central bank loss functions dened in these types of optimisation studies do not include the existence of shoe-leather cost. Friedman argues that distortions due to shoe-leather costs are proportional to nominal interest rates, therefore, these distortions can be eliminated by setting nominal interest rate to zero.

Optimisation under Commitment
Under commitment, the central bank optimises the system and commits to a current and future policy plan. I assume full credibility of the central bank. Accordingly, the central bank species the desired levels of ination and the output gap for all possible dates and the states 3.4 Optimisation Problem Optimal Monetary Policy at the ZLB in a Cost Channel Economy of nature. The central bank is assumed to choose a state contingent sequence {x t, π t } ∞ t=0 which minimises its objective function given by equation (12) subject to the adjusted NKPC, the dynamic IS curve and the ZLB constraint. Accordingly, the KKT conditions are as follows: Since lagged values of the Lagrange multipliers are appearing in the KKT conditions, it is clear that the KKT conditions are history dependent. Accordingly, the optimal choice of ination, the output gap and the nominal interest rate depend on the past values of the endogenous variables. If the central bank deviates from its policy plan (a credibility loss), the outcome is dierent.
This shows that under each policy, the interior solution converges to the same steady state with zero ination and output gap minimising the central bank loss function. Now turn to the corner solution under commitment. The solution for ν is as follows: The sign of ν, which is the Lagrangian multiplier of the ZLB constraint is strictly positive.
This contradicts the KKT conditions. Therefore, under commitment there does not exist a second steady state at the ZLB. and not binding thereafter (i.e. t ≥ T d + 1).
Consider the dynamic path where t ≥ T d + 1. Here, ν t = 0. Accordingly, the KKT conditions under discretion can be stated as follows: From equations (16)(18): Combining this result with equations (19) and (20) yields: When J = 0, i.e. when rms do not borrow externally to nance the wage bill, equation (22) is identical to JTW, which is given by π t+1 = β −1 1 + κ 2 (σ+η) 2 λ π t . Since τ is always greater than unity as shown by JTW, this dierence equation has a bounded solution which 3.5

Optimal Path under Discretion
Optimal Monetary Policy at the ZLB in a Cost Channel Economy is given by π t = 0.
However, when J > 0, τ is not necessarily greater than unity. The value depends on the parametrisation. To have an idea about the value of τ , I plot the value of the coecient with dierent values of J given in Figure 1. 12 It shows that the value of τ is greater than unity for smaller values of J (when J < 0.59), while it is less than unity for higher values of J. Optimal Monetary Policy at the ZLB in a Cost Channel Economy interest rate follows a stochastic AR(1) process; however, he did not consider a cost channel economy.
Further, the fact that |τ | < 1 means that there are multiple equilibria under discretionary policy. This leads to the equilibrium policy path selection. In the following simulation exercise, I consider the policy in which the economy returns to the zero ination steady state on or before the 100 th quarter. First, the weight on output in the loss function, (λ) has been set at 0.25 in the baseline calibration. However, underlying theory implies a much smaller value for λ. 13 In most monetary policy literature, including Ravenna and Walsh (2006), employs a large value for λ considering the empirical relevance. Accordingly, I chose a large value for λ. Second, as mentioned before, the zero ination steady state value of the natural interest rate is the value that has been calculated under the assumption that there is no growth in the potential output. Accordingly, Steady state value of the natural rate of interest is set at 1 β − 1. These values are based on a time period equal to three months (one quarter).

Simulation
Optimal Monetary Policy at the ZLB in a Cost Channel Economy

.2 Simulation
In the baseline simulation, I consider the initial shock to the economy of the size of 0 = −0.05, which is equivalent to around a 19 per cent drop in the annualised natural interest rate. In these simulations, I consider three values for J. 14 They are two extreme values J = 0 and J = 1, and a more empirically relevant value of J = 0.6. The dynamic path of the exogenous natural interest rate due to the large negative demand shock is depicted in   In this section, the welfare losses under optimal policy are considered. I consider welfare losses in two ways. First, the more natural measurement of welfare loss in these kind of models, i.e. by evaluating the central bank's objective function given in equation (11). However, the welfare units found in that way do not have a proper interpretation. Therefore, I also consider the consumption equivalent welfare loss. The gure shows a well-known result in the optimal monetary policy literature at the ZLB: that welfare loss under commitment policy is less than under discretion. The reason for this is that the use of expected ination is unavailable under the discretionary policy, as there is no incentive to implement promised ination ex-post. The ZLB, therefore, generates signicant additional welfare losses under discretionary policy.
With regard to the cost channel, welfare loss under both discretion and commitment is high compared to no-cost channel economies (compare the cases when J = 0 and J = 1 in Figure 5). The negative impact of cost channel on welfare under discretionary policy is substantially high compared to its impact under commitment policy. In cost channel economies, under commitment welfare loss increases by only 9 per cent, compared to a 95 per cent increase under discretion. Demirel (2013) in a dierent context also found that, in a cost channel economy, a switch from discretion to commitment in monetary policy yields greater welfare gains relative to a no-cost channel economy.

4.3
Welfare Losses Optimal Monetary Policy at the ZLB in a Cost Channel Economy 16 Method of calculation of consumption equivalent welfare is based on Adam and Billi (2007). Adam and Billi (2007, Page 748) show that the utility equivalent percentage loss of consumption in the steady state is given by: p = 100 * 1 t+i + λy 2 t+1 ) and ζ is elasticity of a rms' real marginal cost. In addition to the baseline parameterisation given in Table 11, following Adam and Billi (2007), I set θ = 7.66 and ζ = 0.47 for this calculation.   The results are given in Table 3. The table shows that when the prices are relatively exible (when ω takes relatively smaller values) and also when they are relatively rigid (when ω takes relatively larger values), there is no dierence between a cost channel economy and a no-cost channel economy.
17 Recall T d (T c ) is the time, which denotes that the ZLB is binding under discretionary (commitment) policy Price Rigidity ω = 0.3 ω = 0.75 ω = 0.9 I also considered the sensitivity of results with regard to discount factor. Results do not change to various values of β, a result that was found by JTW for a no-cost channel economies. It is conrmed here for the cost channel economies too.

Conclusion
In this study, I incorporated the cost channel of monetary policy into an otherwise standard new Keynesian model and evaluated the optimal monetary policy at the zero lower bound on nominal interest rates. The novelty of the study is that this is the rst time a new Keynesian type study has been performed to analyse the optimal monetary policy at the ZLB with the cost channel. I considered that the economy was initially in a recession with a liquidity trap following a large negative demand shock. The solution methodology was dierent to the standard new Keynesian model as the ZLB brings non-linearity into the model. I followed the JTW solution methodology in a perfect foresight environment, which solves the problem considering that the economy is already at the ZLB.
The study revealed some important results in the conduct of the optimal monetary policy in a cost channel economy at the ZLB. First, the discretionary policy requires central banks to keep interest rates at the zero lower bound for longer in a cost channel economy. This is because, in cost channel economies, the deation is high and persistent due to a large negative demand shock compared to no-cost channel economies. Further, cost channel economies Optimal Monetary Policy at the ZLB in a Cost Channel Economy introduces a policy trade-o between ination and output gap. This result contradicts the nding by JTW that short-term interest rates follow a one-to-one exogenous natural rate of interest following a negative demand shock in a no-cost channel economy.
Under the commitment policy with a fully credible monetary authority, the simulation exercise has shown that the central bank is able to terminate the zero interest rate regime earlier in a cost channel economy than otherwise. This result is in contrast to the results found under discretionary policy. The reason for that is, in a cost channel economy, the private sector has inated inationary expectations. This is because the cost channel increases future cost of production, and in turn, ination when the central bank starts tightening monetary policy on a future known date.
Welfare losses are calculated using the central bank's objective function. It was revealed that the cost channel generates substantially high welfare losses, both under discretionary and commitment policies. Accordingly, abstracting the cost channel in these types of models can lead to under estimation of welfare losses.
The robustness of the results was examined using a sensitivity analysis. The basic results found in the baseline parametrisation were conrmed in the sensitivity analysis. It was found that the dierence between a cost channel economy and a no-cost channel economy is marginal with regard to the timing of the termination of zero interest rates when prices are relatively exible or relatively rigid.

Appendix
Appendix A Log linearising real marginal cost Taking log of (5) and substituting A t = Yt Nt yield: For simplication purposes denote (23) as follows: where S t = Wt Pt Yt Nt At steady state (24): Log linearised equation given by taking the dierence of (24) and (25): Now considerŝ t :ŝ t =ŵ t −p t +ŷ t −n t , Using (2) ,ŝ t = ηn t + σĉ t . Dene x t as output gap lead to: Substituting this result in (26) yields: ψ = (σ + η)x t + JR.