Should monetary policy lean against the wind? Simulations based on a DSGE model with an occasionally binding credit constraint

Economic Modelling xxx (xxxx) xxx

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Should monetary policy lean against the wind? Simulations based on a DSGE model with an occasionally binding credit constraint☆ Jan Žáˇcek Institute of Economic Studies, Charles University, Opletalova 26, CZ-110 00 Prague, Czech Republic

A R T I C L E

I N F O

JEL classification: E30 E44 E50 Keywords: Asset prices DSGE Leaning-against-the-wind Monetary policy Non-linearities Taylor rule

A B S T R A C T

In this research paper, we extend the model constructed by Gambacorta and Signoretti (2014) by introducing an occasionally binding credit constraint based on a penalty function approach in line with Brzoza-Brzezina, Kolasa, and Makarski (2015) to study the performance of the Taylor rule augmented with asset prices. First, we compare the properties of the baseline model and its modified version. Then, we use both models to study the performance of the basic and extended Taylor rule. The performance of Taylor rules is examined under the optimisation of a central bank’s loss function and the welfare maximisation of economic agents. The analysis delivers the following results. The model with an occasionally binding credit constraint has more favourable properties regarding the hump-shaped and asymmetric impulse responses compared to an eternally binding credit constraint model. The best rule regarding the lowest value of the central banks’ loss function proves to be the rule augmented with asset prices. The optimal reactions are, however, shock- and model-dependent, and therefore, any rule-like behaviour does not seem to be appropriate. The welfare maximisation under the occasionally binding credit constraint model reveals that reacting to asset prices might not be welfare-improving for both types of economic agents – households and entrepreneurs. This result is in contradiction with the implications achieved under the eternally binding credit constraint model.

1. Introduction The augmented Taylor rules have been a subject of research for several decades since the prominent publication Taylor (1993). The author of that paper outlines a monetary policy rule according to which a central bank should set its policy interest rate. Since then, researchers tried to implement several types of variables into the basic Taylor rule to investigate whether these variables can carry useful information that should be directly reflected in the setting of the policy interest rate. Many models with a variety of sectors and modelling techniques have been introduced to study this phenomenon. For example, Svensson (2000) and Batini et al. (2003) examine the performance of a battery of monetary policy rules in a small open economy context. Batini et al. (2003) illustrate that there is no suitable general rule and that the type of shock drives the optimality of the rule. Schmitt-Grohé and Uribe (2007) investigate the implications of reacting to inflation and output in terms of welfare losses. Chow, Lim, and McNelis (2014) show that the exchange rate rule has an advantage over a simple Taylor rule when

the shocks are driven by foreign circumstances, while the simple Taylor rule is preferable in the case of domestic shocks. Adolfson (2007) documents that a direct response to exchange rate contributes to lower welfare only if the other reaction coefficients are set to be sub-optimal. The recent financial crisis of 2008 has shown that financial markets play a crucial role in macroeconomic fluctuations and that the interconnection between the real and the financial side of an economy is substantial. Moreover, the pre-crisis consensus on aggressive conduct of an inflation-targeting policy as a sufficient guarantor of macroeconomic stability (represented for example by Bernanke and Gertler (2001)) started to be questioned. As a result, the Taylor rule started to be adjusted by various financial variables. Empirical papers devoted to this research agenda can be classified based on the selected variable included in the basic Taylor rule. The first stream of literature focuses on the rules augmented with spreads, the second brings together articles dealing with credit or leverage, while the last stream emphasises the role of asset prices. Majority of research papers from these streams come to the general conclusion that financial variables can bring some

☆ This research paper has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 681228. The author also acknowledges support from Charles University (project PROGRES and SVV project 260 463). E-mail address: [email protected].

https://doi.org/10.1016/j.econmod.2019.09.043 Received 9 April 2019; Received in revised form 23 July 2019; Accepted 28 September 2019 Available online XXX 0264-9993/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: Žáˇcek, J., Should monetary policy lean against the wind? Simulations based on a DSGE model with an occasionally binding credit constraint, Economic Modelling, https://doi.org/10.1016/j.econmod.2019.09.043

J. Žáˇcek

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useful information for the monetary policy purposes. However, it is not clear to what extent central banks should react to the movements in those variables. For example, Borio and Lowe (2004) document significant predictive power of asset prices and credit volume for financial imbalances, and bring evidence for financial stability purposes in the case of the United States. Belke and Klose (2010) test the performance of the augmented Taylor rule with spreads and find that the Fed and the ECB responded systematically to rising spreads before and after the financial crisis. Baxa, Horváth, and Vašíˇcek (2013) find that central banks often decrease policy rates in reaction to heightened financial stress. The findings of Oet and Lyytinen (2017) reveal that financial stability has played an important role in the Fed’s monetary policy and that a policy rule accounting for changes in the financial system fits data better than the standard policy rule. López-Salido et al. (2017), one of the most recent research papers in this field, highlight the importance of asset prices for the monetary policy setting. A comprehensive literature review related to Taylor rules augmented with various financial variables offers Käfer (2016). In the light of empirical findings, the performance of Taylor rules augmented with various financial variables started to be investigated using DSGE models.1 There has been done a considerable amount of work in this field of research. For example, Curdia and Woodford (2010) show in a New Keynesian model with financial intermediation that monetary policy based on a rule reacting to credit spreads seems to be more plausible than policy based on the standard Taylor rule. Gilchrist and Zakrajšek (2011) evaluate the standard Taylor rule augmented with a credit spread using a New Keynesian model with Bernanke et al. (1999) financial accelerator mechanism, and they show that the rule augmented with the credit spread lowers the negative implications of the financial imbalances. Gambacorta and Signoretti (2014) demonstrate that reacting to asset prices brings macroeconomic benefits in terms of lower implied volatilities of inflation and output when the economy is driven by supply-side shocks. Quint and Rabanal (2014) find that the introduction of a macroprudential policy rule could help reduce macroeconomic volatility. Žáˇcek (2019) tests the performance of the Taylor rule accounting for movements in asset prices in a small open economy setting and finds that the augmented rule can be useful when responding to certain domestic shocks. However, the augmented rule is outperformed by the baseline rule reacting to movements in inflation and output gap when facing shocks coming from abroad. In contrast with these studies, Gelain et al. (2013) conclude that responding to credit improve the performance of some model variables, while the volatility of output and inflation increases. Svensson (2017) demonstrates that reacting to financial variables is not desirable and might be even harmful to future economic developments. The recent years have shown, however, that linear models are not able to fully capture the nonlinear dynamics present in the data and economic theory. The linearisation techniques remove second- and higher-order interactions, and also induce certainty equivalence property. Therefore, the models started to be augmented with nonlinear features. These studies include DSGE models with time-varying parameters, such as Tonner et al. (2011), or DSGE models with occasionally binding constraints. There are basically two approaches of how to model the occasionally binding constraint. The first one resides in the introduction of a penalty function, as shown by, for example, Brzoza-Brzezina et al. (2015). The other one employs a first-order piecewise linear approximation technique introduced by Guerrieri and Iacoviello (2015). Since then, the number of research papers devoted to the nonlinearities in DSGE models is growing rapidly. The most recent contributions are, for example, Pietrunti (2017) or Guerrieri and Iacoviello

(2017). The authors of those research papers demonstrate how the linear approximations affect the recommendations for economic policy decision making. We contribute to the existing research threefold. First, we modify a baseline closed economy model with the financial sector constructed by Gambacorta and Signoretti (2014) by introducing the occasionally binding credit constraint via a penalty function approach in line with Brzoza-Brzezina et al. (2015) which helps us to introduce nonlinearity into the model. Then, we compare the properties of the occasionally binding constraint (henceforth OBC) model with the baseline model incorporating an eternally binding constraint (the so-called EBC model). Our results show that the model with the occasionally binding credit constraint has more plausible properties in terms of hump-shaped and asymmetric impulse responses of the model variables compared to the standard model with the eternally binding credit constraint. Second, we extend the analysis performed by Gambacorta and Signoretti (2014) in several aspects. In contrast to Gambacorta and Signoretti (2014), we analyse the performance of the augmented Taylor rule with asset prices in response to an LTV shock (as an addition to technology and markup shocks), and we test the robustness of the results under the Calvopricing scheme (introduced as an alternative to a Rotemberg pricing). Third and the foremost, we employ the OBC model to assess the performance of the Taylor-type rule accounting for asset prices in terms of (a) a simple loss function and (b) the welfare maximisation. In the light of Gambacorta and Signoretti (2014), we define “leaningagainst-the-wind’’ policy as monetary policy following an augmented Taylor rule which accounts for asset prices above the standard components – inflation and output. As pointed by Gambacorta and Signoretti (2014), such analysis extends a list of financial variables to which central banks might want to react. Selected interpretation of “leaning-against-the-wind’’ policy differs from the usual understanding which assumes that policy interest rates are increased only in case of emerging financial imbalances. The view of Gambacorta and Signoretti (2014) is nevertheless supported by Juselius et al. (2017) who argue that an effective “leaning-against-the-wind’’ policy is based on a systematic approach which is ensured by an augmented version of the Taylor rule. The results of the simulations assessing the performance of the Taylor-type rule accounting for asset prices are the following. We find that there is a difference in the performance of the augmented Taylor rule across the specifications of the constraint. While the optimisations under the model with the eternally binding constraint suggest that the Taylor rule accounting for movements in asset prices is welfare-improving, the optimisations under the model with the occasionally binding credit constraint suggest the opposite in certain cases. When assessing the macroeconomic benefits in terms of lower implied volatilities of output and inflation, both models imply that the rule augmented with asset prices can deliver better performance compared to the standard rule reacting to developments in inflation and output. However, the optimal reactions to asset prices significantly differ across the shocks, and therefore, they are shock-dependent. Our robustness checks show the stability of the results to different calibrations of the key parameters of the model and different modelling approaches to nominal rigidities. It is important to stress that we do not aim to provide precise quantitative prescriptions of the optimal weights to be assigned to asset prices in the central bank’s reaction function. We are aware that the model presented by Gambacorta and Signoretti (2014) is subject to several severe assumptions and rules out considerations related to financial stability. Therefore, the results presented in this paper should be perceived as qualitative rather than quantitative, and should be interpreted with caution. The rest of the paper is organised as follows. Section 2 describes the model and introduces the occasionally binding credit constraint. Section 3 discusses calibration, the properties of the models and workings of the occasionally binding credit constraint. Section 4 is devoted

1

In contrast with empirical literature, DSGE models do not offer such flexibility as empirical models which can potentially include any variable that is seen as a suitable indicator. Therefore, the DSGE literature focuses on a narrower set of variables. 2

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to the optimal monetary policy rules, while section 5 studies the robustness of the results, and section 6 concludes.

The optimisation results in the standard set of equations: the consumption-Euler equation ( ) 1 + rtib 1 (3) = 𝛽 E P t cPt cPt+1

2. The model We modify a closed economy DSGE model built by Gambacorta and Signoretti (2014) who employ, among others, a simplified version of the banking sector introduced by Gerali et al. (2010). In particular, we add a non-linear feature to the model in line with Brzoza-Brzezina et al. (2015). In this section, we first describe the original structure of the baseline model without the non-linear feature, and then we modify the model by introducing the occasionally binding credit constraint. We refer to the baseline model as the eternally binding constraint (EBC) model, while the augmented model is referred to as the occasionally binding constraint (OBC) model throughout the text. The model introduced by Gambacorta and Signoretti (2014) characterises an economy that is populated by two types of agents – patient households and impatient entrepreneurs, each of unit mass. Patient households consume, provide labour and make deposits in banks. Impatient entrepreneurs consume, hire labour, produce intermediate goods, and take loans from banks to produce wholesale goods. Production of goods relies on two sources – physical capital and labour input. Nominal rigidity is introduced into the model via the presence of retailers who face adjustment costs à la Rotemberg (1982). The role of banks is to collect deposits from patient households at the policy interest rate, to issue loans to impatient entrepreneurs at the loan interest rate, and to accumulate capital out of retained earnings. It is assumed that the deposit market is perfectly competitive while the loan market is monopolistically competitive. In particular, banks charge a constant mark-up on the retail loan interest rate. Banks also face an exogenous target capital-to-asset ratio, which serves as a device of tightening or loosening loan supply conditions. To close the model, the central bank is assumed to set its policy interest rate according to a Taylor-type rule. Following Gambacorta and Signoretti (2014), we assume that all debts are indexed to current inflation to isolate the role of the financial frictions.2 For clarity, we present and discuss all model assumptions as in the original paper.

and the labour supply decision wPt cPt

(4)

2.2. Entrepreneurs Entrepreneurs maximise their utility max

{bEt ,cEt ,Kt ,lPt }

E0

∞ ∑

𝛽Et log(cEt )

(5)

t =0

by choosing consumption cEt , labour input lPt from patient households, capital Kt , and loans bEt . While making their decision, entrepreneurs face the budget constraint E P P cEt + qkt Kt + (1 + rtbE −1 )b t −1 + w t l t ≤

yt + qkt (1 − 𝛿)Kt−1 + bEt xt

(6)

where 𝛽 E is the entrepreneurs’ discount factor such that 𝛽 E < 𝛽 P which ensures positive financial flows, cEt is consumption of entrepreneurs, rtbE is the net nominal interest rate on loans, qkt is the real price of capital, 𝛿 is the depreciation rate of capital, and xt is the mark-up of the retailers on the wholesale goods ytE produced according to the technology defined by a Cobb-Douglas production function of the form 𝜇

yt = aEt Kt −1 (lPt )(1−𝜇)

(7)

with aEt being exogenous total factor productivity disturbance3 and

𝜇 being a measure of capital input. When taking loans from banks,

entrepreneurs are limited in the borrowing activity by the value of their collateral holdings represented by capital, and therefore, face a borrowing credit constraint à la Iacoviello (2005) ( ) (8) bEt (1 + rtbE ) ≤ mEt Et qkt+1 Kt (1 − 𝛿)

2.1. Patient households

where mEt is the stochastic loan-to-value ratio. The value of mEt characterises the amount of credit that can be granted to entrepreneurs against the discounted value of their capital stock. The existence of the borrowing constraint introduces the balance-sheet channel into the model. The optimisation problem yields the consumption-Euler equation ( ) 1 + rtbE 1 = 𝛽E Et + sEt (1 + rtbE ), (9) cEt cEt+1

Patient households choose consumption cPt , labour supply lPt and deposits dPt to maximise the expected utility [ ] ∞ ∑ (lP )1+𝜙 (1) max E0 𝛽Pt log(cPt ) − t 1+𝜙 {cPt ,dPt ,lPt } t =0 with respect to the budget constraint cPt + dPt ≤ wPt lPt + (1 + rtib−1 )dPt−1 + JtP

= (lPt )𝜙 .

the labour demand condition characterising the inverse relationship between labour input and the real wage

(2)

where allocations spent on consumption and deposits need to be less than or equal to labour income wPt lPt (with wPt being the real wage), gross reimbursement of spending ((1 + rtib−1 )dPt−1 ) with a net nominal policy interest rate (rtib ) which coincides with a net nominal deposit interest rate, and real profits (JtP ) from ownership of retailers. 𝛽 P is the patient households’ discount factor and 𝜙 characterises labour supply aversion.

wPt = (1 − 𝜇)

yt , lPt xt

and the investment-Euler equation ( ) ( ) qkt 𝜇 yt+1 𝛽E k E E k = Et + qt+1 (1 − 𝛿) + st mt qt+1 (1 − 𝛿) cEt cEt+1 Kt xt +1

(10)

(11)

describing the relationship between the price of capital and the expected discounted values of the marginal contributions and the depreciated value of capital in the next period, with sEt being the Lagrange multiplier on the borrowing constraint.

2 As Gambacorta and Signoretti (2014) discuss, this assumption might be restrictive since it eliminates the nominal-debt channel from the model. As it is widely known, changes in the price level redistribute real sources between lenders and borrowers, and thus, the effect of this assumption on macroeconomic developments might be non-negligible. Nevertheless, as the authors show, the results do not change qualitatively when accounting and not accounting for the debt indexation.

3

Except for a shock to monetary policy, we assume that each disturbance follows a stochastic AR(1) process at = (1 − 𝜌a )a + 𝜌a at −1 + 𝜀at , where 𝜌a is the autoregressive coefficient, a is the respective steady-state value and 𝜀at is i.i.d. process with zero mean and variance 𝜎a2 . 3

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2.3. Capital producers

bank capital. The deposit market is assumed to be perfectly competitive while the loan market is characterised by monopolistic competition. Banks are assumed to be composed of two branches – a wholesale branch and a retail branch.4 Furthermore, it is assumed that there exists an exogenous target level (given by a constant 𝜈 b ) of the capital-to-asset position Ktb ∕Bt (that is the inverse of leverage). Deviations from this target are penalised by quadratic adjustment costs (parametrised by 𝜃 ).5 The existence of quadratic adjustments costs implies a feedback loop between the financial and the real side of the economy.6 Moreover, the target level of the capital-to-asset position introduces a bank lending channel into the model. The resulting loan-supply schedule moves procyclically with the policy rate and banks’ capital position.7 Such characteristics of the loan supply curve is supported by Woodford (2010), who discusses several potential reasons. The wholesale branch operates under perfect competition. It collects deposits Dt = dPt from patient households (at the deposit interest rate that coincides with the policy interest rate rtib ) on the liability side and combines them with bank capital Ktb , while on the asset side issues wholesale loans, Bt = bEt , at the wholesale loan rate Rbt . The wholesale activity is burdened with quadratic adjustments costs parametrised by 𝜃 . Aggregate bank capital, Ktb , evolves according to the standard law of motion8

Perfectly competitive capital producers combine undepreciated capital from the previous period with unsold final goods purchased from the retailers as investment goods, It , to produce new stock of capital. Capital is subject to depreciation characterised by the depreciation rate 𝛿 . The production of capital is subject to quadratic adjustment costs parametrised by 𝜓 k . Therefore, the aggregate stock of capital evolves according to ( Kt = (1 − 𝛿)Kt −1 +

1−

𝜓k

(

2

)2 )

It It −1

It .

−1

(12)

Capital producers choose the level of investment to maximise profits max Et {I t }

∞ ∑ 𝜏=0

[

(

ΛPt,t+𝜏 qkt+𝜏

It +𝜏 −

𝜓k

(

2

It +𝜏 It +𝜏−1

)

)2

−1

It +𝜏

]

− It+𝜏

(13)

where ΛPt,t +𝜏 = 𝛽Pt +𝜏 𝜆Pt ∕𝜆Pt+𝜏 is the real stochastic discount factor of patient households. The optimisation returns the Tobin’s Q equation describing the real price of capital [ 1=

qkt

] ) It 1− − 1 It − 𝜓 −1 2 It −1 It −1 It −1 ( )( ( )2 ) 𝜆Et+1 k It +1 It +1 k +𝛽P Et q 𝜓 − 1 . It It 𝜆Et t+1

𝜓k

(

It

)2

(

k

It

Ktb = Ktb−1 (1 − 𝛿 b ) + JtB−1

is the parameter describing the proportion of bank capital where used in banking activity and JtB are aggregate bank profits. The wholesale branch chooses the optimal level of deposits Dt and loans Bt to maximise the discounted sum of real cash flows

(14)

2.4. Retailers

max E0

{Bt ,Dt }

The retailers buy the intermediate goods from entrepreneurs, differentiate them at no cost and sell with the mark-up. They operate in a monopolistically competitive market and face quadratic adjustment costs (à la Rotemberg (1982)) parametrised by 𝜓 P that introduce nominal rigidity into the model and imply a New Keynesian Phillips curve. Retailers choose the price of the product pt (j) to maximise max E0

{pt (j)}

∞ ∑ t =0

[(

ΛP0,t

pt (j) − mct pt

) yt (j) −

𝜓P 2

(

pt (j) −1 pt −1 (j)

)2 ] yt

𝜆Pt

𝜓 P (𝜋t+1 − 1)𝜋t+1

t =0

ΛP0,t

⎡ ⎢(1 + Rb )Bt − (1 + r ib )Dt − K b − 𝜃 t t t ⎢ 2 ⎣

(

Ktb − 𝜈b Bt

)2

⎤ Ktb ⎥ ⎥ ⎦ (19)

The first order subject to the balance sheet constraint Bt = condition returns the equation describing the relationship between the wholesale loan interest rate, Rbt , and the degree of the capital-to-asset position of the bank ( ) ( )2 Ktb Ktb Rbt = rtib − 𝜃 − 𝜈b . (20) Bt Bt

(15)

After rearranging equation (20) by putting the interest rates on the lefthand side, equation states that the marginal benefit of issuing a new loan is equal to the marginal cost for deviating from the target capitalto-asset position. Therefore, banks choose such a level of loans that equalises marginal costs and benefits. The retail branch operates under monopolistic competition. It purchases wholesale loans from the wholesale branch, differentiates them at no cost and sells them to impatient entrepreneurs at the retail loan

where pt is the price level, mct = 1∕xt represents the real marginal costs defined as an inverse of the mark-up on the wholesale goods, y y and 𝜖t = mkt ∕(mkt − 1) is the stochastic demand price elasticity. The optimisation yields a New Keynesian Phillips curve

𝜆Pt+1

∞ ∑

Dt + Ktb .

with respect to the demand coming from the intra-optimisation of households ( )−𝜖t pt (j) yt (j) = yt (16) pt

(1 − 𝜖t ) + 𝜖t mct − 𝜓 P (𝜋t − 1)𝜋t + 𝛽P

(18)

𝛿b

yt +1 =0 yt

4 This assumption serves as a useful modelling device. As Gerali et al. (2010) argue, it helps to describe the role of bank capital in the market for loanable funds. It is important to add that this presumption hinges on the assumption of bank risk neutrality. 5 The target leverage ratio is interpreted by Gambacorta and Signoretti (2014) as a short cut to the regulatory capital requirements. However, this interpretation is misleading, given that regulators would not penalise banks for maintaining higher equity capital than requested. Therefore, we resort to the interpretation of Gerali et al. (2010) who interpret quadratic adjustments costs with the target 𝜈 b as a device to capture “the trade-off that would arise in the decision of how much own resources to hold.” 6 For clarification see Gerali et al. (2010). 7 For a detailed discussion see Section 3 in Gambacorta and Signoretti (2014). 8 Zero-dividend policy is assumed.

(17) describing the behaviour of aggregate inflation in the economy. 2.5. Banks The banking sector is modelled as a simplified version of the banking sector introduced by Gerali et al. (2010). The role of banks is to collect deposits from patient households, issue loans to impatient entrepreneurs, and accumulate bank capital. Each bank needs to obey the balance sheet identity stating that loans are equal to deposits plus 4

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interest rate rtbE . In doing so, the retail branch sets the retail loan interest rate by applying a constant mark-up, 𝜇 bE , on the wholesale loan interest rate Rbt . Therefore, the retail loan rate, rtbE , is given by ( ) ( )2 Ktb Ktb − 𝜈b + 𝜇bE . (21) rtbE = rtib − 𝜃 Bt Bt Bank profits are the sum of net earnings, and thus are given by ( )2 𝜃 Ktb JtB = rtbE Bt − rtib Dt − − 𝜈 b Ktb . 2 Bt

Table 1 Calibration.

(22)

Leverage is defined as the ratio of aggregate loans and bank capital B levt = bt . Kt

The central bank sets its policy interest rate according to the Taylortype rule ib

( ) ib ( 𝜋 )𝜙𝜋 ( y )𝜙y 1−𝜌 t

𝜋

t

y

𝜀rt

(24)

where rib is the steady state of the policy interest rate, 𝜋 and y are the steady states of inflation and output, 𝜌ib reflects monetary policy inertia, 𝜙𝜋 and 𝜙y are weights put on inflation and output respectively, and 𝜀rt is a white noise monetary policy shock with variance 𝜎r2 . The market clearing equation closing the model is given by yt = ct + It + 𝛿 b Ktb−1

+ cPt .

with aggregate consumption ct = the budget constraint of entrepreneurs E P P cEt + qkt Kt + (1 + rtbE −1 )b t −1 + w t l t =

The flow of funds is given by

yt + qkt (1 − 𝛿)Kt−1 + bEt . xt

(26)

2.7. Occasionally binding credit constraint The usual way of dealing with the inequality constraint given by equation (8) when solving the model is to assume its equality. However, such approach is not optimal as shown by Guerrieri and Iacoviello (2015), Brzoza-Brzezina et al. (2015), or Pietrunti (2017), since this assumption discards valuable information. Therefore, we extend the baseline model by introducing a non-linear feature represented by the occasionally binding credit constraint following Brzoza-Brzezina et al. (2015). The inequality constraint described by equation (8) is replaced with a smooth penalty function

ΨEt =

1

𝜂

exp{𝜂ΓEt }

and the investment-Euler equation qkt cEt

𝛽E

(

cEt+1

𝜇 yt+1 Kt xt +1

)

)

+ qkt+1 (1 − 𝛿) + mEt qkt+1 (1 − 𝛿) exp{𝜂ΓEt } .

The implementation of the OBC framework thus modifies the above mentioned equations and manifests to the model through them.

(27)

ΩEt = (1 + rtbE ) exp{𝜂ΓEt }.

3. Calibration and properties of the models 3.1. Calibration We calibrate the model based on the values presented in Gambacorta and Signoretti (2014) who follow the calibration of Gerali et al. (2010) who calibrate the model for the euro area. The calibration of the parameter related to the penalty function is based on the discussion in Brzoza-Brzezina et al. (2015). The calibration of the model is summarised in Table 1.9 The discount factor of households, 𝛽 P , is calibrated to 0.996, which implies an annual value of 2% of the policy interest rate in the steady

(28)

To clarify the role of the penalty function and the parameter 𝜂 , consider the following results in the limit ⎧∞ ⎪ = ⎨1 ⎪ ⎩0

(

= Et

(32)

where 𝜂 defines the curvature of the penalty function and ΓEt = (1 + rtbE )bEt − mEt qkt+1 (1 − 𝛿)Kt . The derivative of the penalty function (27) with respect to bEt returns the penalty function slope

lim exp{𝜂ΓEt } 𝜂 →∞

0.996 0.975 0.05 0.059 100 6 11 4 28.65 0.2 0.35 0.005 0.09 1 2 0.3 0.77 0.95 0.5 0.87 0.006 0.05 0.005 0.002

The results in (30) imply that the penalty function in the limit (infinity) is equivalent to the baseline model with the borrowing constraint (8) since entrepreneurs are penalised by infinity when borrowing (and exceeding their collateral) and zero when they are not. Since the penalty function (27) is introduced into the model as an additive component of the utility function of impatient entrepreneurs, the difference between the EBC model and the OBC model lies in the specification of the optimisation problem of those agents. Maximising (5) with respect to (6), taking into account (7) and (27), and combining the resulting first order conditions returns the new specifications of the entrepreneurs’ consumption-Euler equation ( ) 1 + rtbE 1 = 𝛽 E + ΩEt (31) E t cEt cEt+1

(25) cEt

Value

Discount factor of households Discount factor of entrepreneurs Depreciation rate of physical capital stock Cost for managing the bank’s capital position Penalty function parameter Demand elasticity of substitution Bank capital adjustment cost parameter Investment adjustment costs Price adjustment costs parameter Share of capital used in the production Steady state LTV of entrepreneurs Mark-up on the policy interest rate Target capital-to-assets ratio Inverse of the Frisch elasticity of labour supply Weight put on inflation Weight put on output gap Monetary policy inertia Autoregressive coefficient – technology shock Autoregressive coefficient – mark-up shock Autoregressive coefficient – LTV shock Standard deviation – technology shock Standard deviation – mark-up shock Standard deviation – LTV shock Standard deviation – monetary policy shock

𝜇 bE 𝜈b 𝜙 𝜙𝜋 𝜙y 𝜌ib 𝜌a 𝜌mk 𝜌m 𝜎a 𝜎 mk 𝜎m 𝜎r

2.6. Monetary authority, equilibrium and other definitions

ib

Description

𝛽P 𝛽E 𝛿 𝛿b 𝜂 𝜖 𝜃 𝜓k 𝜓P 𝜇 mE

(23)

(1 + rtib ) = (1 + r ib )(1−𝜌 ) (1 + rtib−1 )𝜌

Parameter

ΓEt > 0, if ΓEt = 0, if ΓEt < 0. if

These partial results imply that in the limit holds { ∞ if ΓEt > 0, lim exp{ΨEt } = 𝜂 →∞ 0 if ΓEt ≤ 0.

(29)

9 To verify the results of the model simulations, we run several robustness checks based on the different selection of the values of the selected number of the parameters of the model. The robustness analysis can be found in section 5.

(30) 5

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Table 2 Business cycle properties of the EBC model. All variables are detrended with the Hoedrick-Prescott filter. Source: Eurostat. Variable

Output Consumption Credit (loans) Investment

Standard deviation

Correlation with output

Data

EBC model

OBC model

Data

EBC model

OBC model

1.18 0.69 1.41 2.67

1.27 1.21 1.57 2.32

1.17 1.11 1.28 2.49

1 0.97 0.75 0.76

1 0.99 0.55 0.79

1 0.98 0.31 0.74

state. The discount factor of entrepreneurs, 𝛽 E , is set at a lower value of 0.975 as in Iacoviello (2005), which ensures that entrepreneurs are less patient agents than households. Following Galí (2008), the inverse of the Frisch elasticity, 𝜙, is equal to 1. The share of capital used in the production, 𝜇 , is set at 0.2, while the deprecation rate of physical capital 𝛿 is set at 0.05. The steady state of the demand price elasticity, 𝜖 , is equal to 6, which implies the steady state value of the mark-up 1.2 (that is 20%), which is a common value found in the literature. The stickiness parameters, 𝜓 k and 𝜓 P , are set at 4 and 28.65 based on the estimates of Gerali et al. (2010). The LTV ratio of entrepreneurs is set at 0.35 which resembles with an average ratio of long-term loans to the shares and other equity of the non-financial corporations (which is close to a value reported in Christensen et al. (2007)). The calibration of the banking sector parameters follows the estimates and the computations of Gerali et al. (2010) as well. The target capital-to-asset ratio, 𝜈 b , the parameter governing the cost for managing the bank’s capital position, 𝛿 b , and the parameter characterising capital adjustment cost, 𝜃 , are equal to 0.09, 0.059 and 11 respectively. The mark-up on the inter-bank interest rate is set at 2% (annualised). The degree of monetary policy inertia is equal to 0.77, and the weights put on inflation and the output gap are set at 2 and 0.3 respectively. Shocks are calibrated outside the model to replicate some features found in the data. As it is thoroughly explained by Brzoza-Brzezina et al. (2015), the parameter 𝜂 is the governing factor behind the workings of the penalty

function and defines its curvature. In case that 𝜂 approaches infinity, the penalty function is steep and becomes the inequality constraint as specified by equation (8). Therefore, the goal should be to set the value of 𝜂 to the highest possible value. However, high values of 𝜂 are implausible for solving the model using standard perturbation techniques as discussed by De Wind (2008). Therefore, we broadly follow Brzoza-Brzezina et al. (2015) and set the value of 𝜂 to 100. As Brzoza-Brzezina et al. (2015) discuss, it is important to define how the penalty function reacts to changes in leverage in normal times. Following that research paper, we show (by manipulating equation (31) in the steady state) that the steady state slope of the penalty function is dependent on the value of the discount factor of entrepreneurs 𝛽 E . Using several steady-state relationships of the model, we rewrite the steady state of ΩE as [ ( )] [ ] 1 1 1 ΩE = E 1 − 𝛽E (1 + r bE ) = E 1 − 𝛽E − 𝜇 bE . (33) c c 𝛽P To have the non-binding credit constraint in normal times, the penalty function should be flat in the neighbourhood of the steady state. This can be ensured by setting the value of 𝛽 E close to (1∕𝛽P − 𝜇bE )−1 . However, low values of ΩE imply that the steady state of the leverage is well below the threshold at which the constraint becomes binding. Therefore, 𝛽 E is set at 0.975 which ensures that the credit constraint is moderately binding in the steady state. To be specific, the steady state lever-

Fig. 1. Properties of the baseline EBC model. The IRFs depict one standard deviation shocks. All variables except for interest rates are expressed as percentage deviations from the steady state. Interest rates are expressed in absolute deviations. 6

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Fig. 2. Workings of the OBC. The IRFs depict negative one standard deviation shock to the LTV ratio. All variables except for interest rates are expressed as percentage deviations from the steady state. Interest rates are expressed in absolute deviations.

age ratio under the OBC model equals to 34.5% which is 0.5 p.p. below the EBC LTV ratio. We solve the model using the second-order approximation.

more efficient. More efficient production process leads to higher output and lower prices, and therefore, to lower inflation. The interest rate on loans drops, which resembles the reaction of the central bank to cut its policy interest rate to increase inflation. An investment activity rises with better technology and easier access to credit that is mirrored in a higher demand for loans and subsequent expansion in bank leverage. Asset prices increase, and consumption rises due to higher wages. A cost-push shock is modelled as an increase in the mark-up on goods. An adverse inflation shock causes output, investment and consumption to fall significantly. The central bank increases its policy interest rate to decrease inflation. The interest rate on loans follows the dynamics of the policy interest rate and rises as well. Financing conditions of the loans become adverse, which leads to a sharp contraction in loans and a decrease in bank leverage. A contractionary monetary policy shock is represented by a one standard deviation shock in the policy interest rate. After the shock hits the economy, output and inflation decrease on impact. A higher policy interest rate is reflected by the increase in the interest rate on loans. Decreasing asset prices together with the increasing interest rate on loans cause a decline in the value of the collateral resulting in a lower volume of loans. Since the balance sheet identity of the commercial banks needs to hold every period, deposits decrease more than loans to meet this criterion.

3.2. Properties of the models The business cycle properties of both models are described using standard deviations and correlations with output as measures of volatility and the co-movement with output, respectively. Table 2 compares the business cycle properties of the models and the ones found in the data. As second to fourth columns of Table 2 suggest, volatilities of the selected variables (except for consumption) simulated by the EBC and OBC models broadly match volatilities found in the data. In the case of consumption, the data show a very low level of volatility compared to volatilities generated by the models. This particular feature can be given by a very simple structure of the households’ sector. Regarding correlations with output, the EBC model does well in replicating these co-movements. As the sixth column of Table 2 shows, correlations with output generated by the model are very similar to those found in the data. On the contrary, the OBC model fails to generate reasonable correlation between loans (credit) and output. However, it is important to mention that the OBC model can generate negative skewness of the simulated data for loans compared to the EBC model which cannot (skewness of −0.22 vs 0.16). This feature is in line with findings of Iacoviello and Neri (2010) and Brzoza-Brzezina et al. (2015) who document negative skewness of financial data for the USA and the euro area, respectively. To explain the basic dynamics of the EBC model, the following paragraphs describe the impulse response functions to three out of four shocks included in the model.10 The impulse responses are displayed in Fig. 1. An improvement in the production process is modelled as a positive one standard deviation technology shock, which makes the production

Before we proceed to the comparison of the impulse responses under the OBC and EBC frameworks, we briefly describe the dynamics of the model under the negative one standard deviation shock to the LTV ratio of entrepreneurs.11 As Fig. 2 shows, a contraction in loan demand appears due to tighter financing conditions which decrease entrepreneurs’ net worth. Following subdued loan demand, investment activities fall, which translates into lower output and consumption

10 We leave the LTV shock for the explanation of the workings of the OBC framework.

11 The LTV shock can be perceived as a result of the behaviour of financial sector institutions which is a standard interpretation in the literature.

3.3. Workings of the OBC

7

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Fig. 3. Simulated data. The nominal policy interest rate is expressed in net quarterly terms; inflation is reported in gross quarterly terms. All other variables are expressed in levels.

Fig. 4. Asymmetric property of the OBC. The IRFs depict negative and positive shocks to the LTV ratio. All variables except for interest rates are expressed as percentage deviations from the steady state. Interest rates are expressed in absolute deviations.

accompanied by lower inflation. Deposits, bank leverage as well as asset prices fall. The central bank responds to decreased inflation by cutting its policy interest rate. We further examine the impulse responses under the OBC framework to show the differences compared to the EBC model. Fig. 2 offers the dynamics of the model under three alternative calibrations of the parameter 𝜂 , and compares it with the EBC baseline setting. As it has been highlighted before, lower values of 𝜂 imply more relaxed conditions. On the other hand, as 𝜂 approaches infinity, the penalty function becomes steeper, and the OBC constraint becomes more binding. Therefore, the EBC model is a special case of the OBC model. As Fig. 2 shows, the impulse responses of the model under the OBC set-up are more moderate. The responses become hump-shaped in most cases, and are smoother as well (which is something to be expected in real data). Com-

pared to the EBC variant in which the strongest response usually comes on impact, the OBC displays more gradual reactions with the strongest impacts after several quarters.12 As the OBC becomes less binding (that is 𝜂 becomes lower and the penalty function becomes more flat), the responses are less pronounced. To further illustrate the difference between the properties of the EBC and OBC models, we use both models to simulate synthetic data. The result of this exercise is plotted in Fig. 3. This result reveals two conclusions. First, the OBC model behaves similarly to the EBC model except

12 For example Assenmacher-Wesche and Gerlach (2008) or Gilchrist and Zakrajšek (2012) identify hump-shaped responses and inertia in reactions of macroeconomic variables to financial shocks.

8

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Table 3 Optimised Taylor-type rules under the loss function – a positive technology shock. Gain is a percentage difference between the loss achieved under the baseline and the augmented rule. 𝛼

Baseline rule

Augmented rule – asset prices

𝜙𝜋

𝜙y

Lt

𝜙𝜋

𝜙y

𝜙qk

Lt

Gain

3 1.2 1.1 1.1

0 0 0 0.1

0.000 0.005 0.024 0.048

3 3 1.1 1.1

0 0 0 0

0 0.6 1 1

0.000 0.005 0.022 0.043

0.00% 2.42% 9.02% 10.45%

3 1.1 1.1 1.1

0 0 0 0.1

0.000 0.005 0.024 0.048

3 3 1.1 1.1

0 0 0 0

0 0.6 1 1

0.000 0.005 0.022 0.043

0.00% 2.01% 9.15% 10.70%

EBC 0 0.1 0.5 1 OBC 0 0.1 0.5 1

Table 4 Optimised Taylor-type rules under the loss function – a positive mark-up shock. Gain is a percentage difference between the loss achieved under the baseline and the augmented rule. 𝛼

Baseline rule

Augmented rule – asset prices

𝜙𝜋

ϕy

Lt

𝜙𝜋

𝜙y

𝜙qk

Lt

Gain

3 3 1.1 1.1

0 0 0.1 0.5

0.001 0.004 0.01 0.016

3 3 1.1 2.8

0 0 0 0

0 0 0.1 1

0.001 0.004 0.01 0.014

0.00% 0.00% 4.39% 16.17%

3 3 1.1 1.1

0 0 0 0.4

0.001 0.004 0.01 0.016

3 3 1.1 1.1

0 0 0 0

0 0 0.1 0.3

0.001 0.004 0.01 0.014

0.00% 0.00% 1.23% 11.54%

EBC 0 0.1 0.5 1 OBC 0 0.1 0.5 1

for two series – deposits and loans, which are shifted downwards from the series simulated by the EBC model. This is nothing surprising since using the penalty function approach we shift the steady state of the LTV ratio (for discussion see section 3.1). Second, the OBC framework produces less volatile variables.

itive and negative shocks. The OBC framework, on the other hand, offers substantial flexibility and the construction of the penalty function should enable the asymmetric responses of the macroeconomic variables. Therefore, we devote this subsection to the examination of the OBC framework under both positive and negative shocks. Fig. 4 shows the state-dependent responses of the main variables to small (one standard deviation) and large (two standard deviations) shocks to the LTV ratio. For one standard deviation shocks, the responses are merely symmetric. This result is not surprising since, as explained by Brzoza-Brzezina et al. (2015), the penalty function is smooth and this property ensures that there are no substantial asymmetric responses for small shocks. On the contrary, if we consider large

3.4. Asymmetries under the OBC As the data show, recessions are often more severe than expansions, and therefore, data exhibit asymmetries. However, under the EBC, the models are not able to replicate asymmetries in the reactions of the macroeconomic variables, and the IRFs are symmetric under pos-

Table 5 Optimised Taylor-type rules under the loss function – a positive LTV shock. Gain is a percentage difference between the loss achieved under the baseline and the augmented rule. 𝛼

Baseline rule

Augmented rule – asset prices

𝜙𝜋

𝜙y

Lt

𝜙𝜋

𝜙y

𝜙qk

Lt

Gain

3 3 3 3

0.6 0.9 1 1

0.000 0.000 0.000 0.001

3 3 3 3

0.1 0.2 0.5 0.7

0.1 0.2 0.2 0.3

0.000 0.000 0.000 0.000

67.73% 40.42% 47.27% 51.35%

3 3 3 3

0.5 0.6 1 1

0.000 0.000 0.000 0.000

3 3 3 3

0 0.1 0 0

0.1 0.1 0.2 0.2

0.000 0.000 0.000 0.000

83.76% 40.93% 32.85% 35.95%

EBC 0 0.1 0.5 1 OBC 0 0.1 0.5 1

9

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Table 6 Optimised Taylor-type rules under the welfare maximisation. The first column indicates the type of shock. Households

Baseline rule

Augmented rule

𝝓𝝅

𝝓y

WP

𝝓𝝅

𝝓y

𝜙qk

WP

𝝃P

EBC Technology Mark-up LTV

1.1 1.1 1.1

0 1 1

−118.69 −118.25 −118.71

3 1.1 1.1

0 1 0

0.7 1 1

−118.69 −118.10 −118.65

0.01% 0.06% 0.02%

OBC Technology Mark-up LTV

1.1 1.1 1.1

1 1 1

−118.71 −118.44 −118.97

1.1 1.1 1.1

0 1 0

1 1 1

−118.69 −118.16 −118.73

0.01% 0.11% 0.09%

Entrepreneurs

Baseline rule

Augmented rule

𝛟𝛑

𝛟y

WE

𝛟𝛑

𝛟y

𝜙qk

WE

𝝃E

EBC Technology Mark-up LTV

1.1 1.1 1.1

1 1 1

−123.81 −123.13 −123.91

2.2 1.1 1.1

0 1 0

1 1 1

−123.81 −122.91 −123.83

0.01% 0.54% 0.20%

OBC Technology Mark-up LTV

3 3 1.1

0 0 0

−123.66 −121.44 −122.90

3 3 1.1

0 0 0

0 0 0

−123.66 −121.44 −122.90

0.00% 0.00% 0.00%

Table 7 Sensitivity analysis – values of the parameters. Parameter

Description

Low

Calibrated

High

𝜓 𝜓P 𝜙 𝜌a 𝜌mk 𝜌m

Investment adjustment costs Price adjustment costs parameter Inverse of the Frisch elasticity of labour supply Autoregressive coefficient – technology shock Autoregressive coefficient – mark-up shock Autoregressive coefficient – LTV shock

1 10 4.5 0.7 0.3 0.7

5 28.65 1 0.95 0.5 0.87

10 50 1.5 0.97 0.8 0.95

k

shocks, the model starts to display clear asymmetries in the responses to LTV easing and tightening. The difference resides not only in the size of the responses but also in their shapes. Compared to a moderate increase in loans in the case of LTV easing, LTV tightening is followed by a more pronounced decline in loans. The negative impact on loans translates into a more severe decrease in consumption, output and investment. The impact on asset prices is also larger in the case of the negative LTV shock compared to the positive one. The explanation behind these substantial asymmetries resides in the construction of the penalty function. Since the LTV ratio is a direct component of the credit constraint, the OBC can generate asymmetric behaviour of the impulse responses. The same will also hold for other shocks that substantially affect the variables in the credit constraint. As we have demonstrated, the model with occasionally binding constraint proves to be more appropriate, since it has more plausible characteristics, such as the hump-shaped and asymmetric impulse responses.

prices. Therefore, a general Taylor-type rule can be described by

(1 + rtib )

ib (1−𝜌ib )

= (1 + r )

⎛(

(1

ib + rtib−1 )𝜌 ⎜ ⎜

⎝

(

𝜋t )𝜙𝜋 yt 𝜋 y

)𝜙 y (

qkt qk

)𝜙

qk

⎞ ⎟ ⎟ ⎠

1−𝜌ib

(34) where 𝜙𝜋 , 𝜙y and 𝜙qk are respective weights that are to be optimised. The optimisation procedures are described in the following subsection. 4.2. Methodology to measure macroeconomic benefits We employ two distinct approaches to measure macroeconomic benefits and to find the optimal weights of the augmented Taylor-type rules. The first approach is a standard minimisation of a quadratic loss function based on the concept of the Taylor curves. We assume that the central bank minimises the weighted sum of unconditional variances of inflation and output L = Var (𝜋) + 𝛼 Var (y)

4. Taylor-type rules and optimal weights

(35)

where Var is variance and 𝛼 is the weight assigned to the output. 𝛼 is allowed to vary within the interval [0, 1] by increments of 0.1. We assume that the primary goal of the central bank is to maintain stable inflation, and therefore, we assign the weight of the magnitude one to inflation in the loss function. For each value of 𝛼 , we compute the optimal weights of the coefficients of the augmented Taylor-type rules based on a grid-search method. The coefficients of the Taylor-type rules

4.1. Taylor-type rules We test the performance of several Taylor-type rules. First, we assume that the central bank implements the baseline rule, which reacts to inflation and a measure of economic slack represented by output. Second, we augment the baseline rule with the financial variable – asset 10

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Economic Modelling xxx (xxxx) xxx

Fig. 5. Robustness check – a technology shock. Gain is a percentage difference between the loss achieved under the baseline rule and the rule augmented with asset prices. Gain is computed based on the optimised coefficients of the rules for each value of 𝛼 .

Fig. 6. Robustness check – a mark-up shock. Gain is a percentage difference between the loss achieved under the baseline rule and the rule augmented with asset prices. Gain is computed based on the optimised coefficients of the rules for each value of 𝛼 .

are restricted to intervals: 𝜙𝜋 ∈ (1, 3], 𝜙y ∈ [0, 1] and 𝜙qk ∈ [0, 1] by increments of 0.1.13 The second approach is based on the maximisation of agents’ welfare in which we follow Schmitt-Grohé and Uribe (2007). We compute agents’ welfare using a second-order approximation of the utility functions of households and entrepreneurs. To find the optimal coefficients, we compute agents’ welfare for each combination of the weights in

the Taylor-type rule. Since there are two agents in the economy (each with the different discount factor and the different form of the utility function), we work with two distinct welfare functions under given Taylor-type rule R. The welfare functions for households (W0P,R ) and entrepreneurs (W0E,R ) are thus described by W0P,R

13 Selection of the intervals is motivated by Schmitt-Grohé and Uribe (2007). Nonetheless, we have performed additional optimisations based on wider intervals. The results are qualitatively the same.

= E0

W0E,R = E0 11

∞ ∑

[

𝛽Pt

]

log(cPt )

t =0

∞ ∑ t =0

𝛽Et log(cEt ).

(lP )1+𝜙 − t , 1+𝜙

(36)

(37)

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Economic Modelling xxx (xxxx) xxx

Fig. 7. Robustness check – a LTV shock. Gain is a percentage difference between the loss achieved under the baseline rule and the rule augmented with asset prices. Gain is computed based on the optimised coefficients of the rules for each value of 𝛼 .

Since the units of welfare have no direct interpretation, we employ consumption equivalent units, which is a common approach in the literature on the welfare maximisation. The explanation of the consumption equivalent units is then how much of the steady-state share of consumption would make the agent under the augmented Taylor-type rule as well off as with the standard baseline rule. Based on the functional forms of the utility functions, the expressions for the consumption equivalents for both agents are

𝜉 P = 1 − exp{(W0P,BR − W0P,AR )(1 − 𝛽 P )},

(38)

𝜉 E = 1 − exp{(W0E,BR − W0E,AR )(1 − 𝛽 E )}

(39)

assigns positive coefficients to asset prices. Moreover, the gain of the augmented rule with asset prices increases with the higher value of the weight 𝛼 . When comparing the optimised coefficients under the EBC and the OBC setting, their values are almost identical. The augmented rule with asset prices seems to deliver plausible outcomes under the mark-up shock as well. The results are, however, not so convincing as in the case of the technology shock. As Table 4 shows, the results are substantially dependent on the chosen value of the weight 𝛼 . When the central bank pays no or slight attention to variation in output, the best response regarding the minimum loss is achieved under the rule reacting only to inflation developments. On the other hand, if the weight 𝛼 reaches 0.5 or exceeds this threshold, the minimum loss is found under the rule augmented with asset prices. Looking at the results under EBC and OBC variants, the rule augmented with asset prices can deliver the best performance under the OBC variant in certain cases. However, the optimised coefficients are substantially different from those found under the EBC setting (see Table 9 in Appendix A). For values of 𝛼 exceeding 0.6, the optimisation under the EBC model delivers the coefficient on inflation 𝜙𝜋 close to 3 and a strong reaction to the developments in asset prices, whereas the optimised coefficient on inflation takes the lowest possible value accompanied by a slight reaction to asset prices for the OBC model. Moreover, the gain achieved under the OBC setting is not as high as in the EBC case. The gain differs in terms of units of p.p. (for example, 5 p.p. for 𝛼 = 1). The last investigated shock is a positive LTV shock (see Table 5). Unlike the previous cases, both rules prescribe a positive weight to output for all values of the weight 𝛼 in the case of the EBC model. Moreover, the gain achieved under the augmented rule is the most apparent one (the gain achieves almost 70% for 𝛼 = 1). The lowest values of the loss function are found under the rule augmented with asset prices that suggests a strong response to inflation and moderate responses to output and asset prices. These conclusions hold for both EBC and OBC variant (even though there are slight differences, especially in the weight assigned to output).

where W0E,BR and W0P,SR are welfares under the baseline rule, and W0E,AR

and W0P,AR are welfares under the augmented rule.

4.3. Optimised coefficients – the quadratic loss function The results of the optimisation routines based on the simple quadratic loss function are summarised in Tables 3–5. Mentioned tables also offer a comparison between optimisations under the EBC and the OBC model. For the sake of brevity, we do not present all results for each value of the weight 𝛼 .14 Table 3 summarises the optimisation under a positive one standard deviation technology shock. The general result is that the augmented rule with asset prices outperforms the baseline rule except for the case when the central bank does not put any weight on the output stabilisation in its loss function. In this particular setting, the best performance is achieved under the rule prescribing the highest possible weight on inflation. This result is in line with the “Jackson Hole consensus” and is common across the similar studies (for example, Bernanke and Gertler (2000) or Bernanke and Gertler (2001)). The importance of the output component in the baseline rule increases with the value of the weight 𝛼 exceeding a threshold of 0.6 (not reported). However, this is not true in the case of the augmented rule. The optimised augmented rule prescribes zero coefficients on output for each value of 𝛼 , while it

14

4.4. Optimised coefficients – the welfare maximisation Table 6 summarises the coefficients of the welfare-maximising rules along with the consumption equivalents. We find that in the case of the

The results for each value of 𝛼 can be found in Appendix A. 12

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Fig. 8. Robustness check – welfare of households. The vertical axis depicts the consumption equivalents.

5. Robustness checks

EBC model, the augmented rule with asset prices makes the agents better off. The rule is welfare-improving in all investigated shocks. These results go hand in hand with the results stemming from the optimisation of the loss function. For almost all shocks, the augmented rule with asset prices prescribes the highest possible value of the coefficient 𝜙qk . This is not, however, true for the OBC model. In this case, the rule performs as welfare-improving only for households and brings no benefit for entrepreneurs at the same time (the coefficient on asset prices takes 0 value in all cases). This particular result suggests that when the constraint is occasionally binding, the role of financial frictions is relaxed and fluctuations in asset prices are not decisive for entrepreneurs. To conclude, the simulations suggest that reacting to financial developments might be beneficial for the central bank in certain cases. Based on the achieved results, reacting to asset prices can help to stabilise volatility of inflation and output, and can also increase welfare of economic agents in response to specific shocks. The optimal reactions are, however, shock- and model-dependent, and any rule-like behaviour does not seem to be appropriate. An explanation behind why the rule with asset prices delivers the best performance in most cases might be hidden in the construction of the model introduced by Gambacorta and Signoretti (2014). Since asset prices significantly affect marginal costs of the production process through the return to capital and investment activities, they are closely linked to inflation. Our conclusions are in line with Curdia and Woodford (2010) or Christiano et al. (2010). The achieved results are also backed with findings in the empirical literature. For example, López-Salido et al. (2017) confirm the importance of financial variables – namely asset prices – for the conduct of monetary policy. The novelty in our paper – the introduction of the occasionally binding credit constraint – reveals that there are substantial differences in the results under the EBC and the OBC model in certain cases. Especially, the rule augmented with asset prices performs as welfareimproving only for households and brings no benefit for entrepreneurs at the same time. It is important to bear in mind that our results stem from the model with no consideration of financial instability. Therefore, the results should be interpreted with caution.

In the previous section we have demonstrated that the augmented rule with asset prices can bring macroeconomic benefits in certain cases and that there are differences between the results when using the model with the occasionally binding credit constraint compared to the baseline model. In this section, we study the implications of the different model settings. First, we compare the achieved results with the simulations using a different modelling approach to the sector of retailers. Second, we test the robustness of the results to changes in the calibration of the selected number of parameters.

5.1. The Calvo-pricing scheme As discussed by Lombardo and Vestin (2008), despite the strong similarities between the two most common approaches to the modelling of nominal rigidities – Calvo- and Rotemberg-pricing – to a first-order approximation, there might be differences in welfare costs at higherorder approximations. The authors show that the two pricing schemes yield quantitatively different social costs, which are heavily dependent on the sources of shocks. Based on the achieved outcomes, the authors argue that the adopted pricing scheme could significantly affect the prescribed policy action. To test our results, we replace the original optimisation problem given by equations (15) and (16) by the following maximisation procedure in which retailers face the Calvo-pricing scheme in the price setting. In each period, retailers receive a signal with an exogenously given probability of (1 − 𝜗) to re-optimise their retail prices. If a firm is not allowed to change its price, it leaves it fixed to its prevailing value and does not apply any inflation indexation. Therefore, retailers choose pt to maximise the reset price ̃

max E0

{̃ pt (j)} 13

∞ ∑ 𝜏=0

[(

𝜗t+𝜏 ΛPt,t+𝜏

̃ pt (j) pt +𝜏

)

− mct+𝜏 yt+𝜏 (j)

] (40)

J. Žáˇcek

Economic Modelling xxx (xxxx) xxx

Fig. 9. Robustness check – welfare of entrepreneurs. The vertical axis depicts the consumption equivalents.

with respect to the demand ( )−𝜖t ̃ pt (j) yt . yt (j) = pt

(41)

The first order condition yields the expression for the reset price ∑∞ 𝜖t yt +𝜏 𝜖t E0 𝜏=0 (𝜗𝛽P )t+𝜏 (𝜆Pt+𝜏 )−1 mct+𝜏 pt+𝜏 𝜖t Ft ̃ pt = = ∑ 𝜖 − 1 ∞ t +𝜏 (𝜆P )−1 p t y 𝜖t − 1 E0 𝜖 − 1 Ht t (𝜗𝛽 ) P 𝜏=0 t +𝜏 t +𝜏 t +𝜏 𝜖

of the weight 𝛼 and compare the results in terms of gain. In the case of welfare-maximisation, we rely on a comparison between the consumption equivalents units for both types of agents. The set of the parameters included in the sensitivity analysis is based on Gambacorta and Signoretti (2014), and it comprises the parameters characterising the rigidity in the model – the inverse of the Frisch elasticity of labour supply (𝜙), the investment adjustment costs parameter (𝜓 k ) and the price stickiness parameter (𝜓 P ).16 We also add the autoregressive coefficients of the shock processes (𝜌a , 𝜌m and 𝜌mk ). The selected values of the parameters to be analysed are summarised in Table 7. The results of the sensitivity analysis related to the central bank’s loss function are presented in Figs. 5–7, while the outcomes of the sensitivity analysis aimed at the welfare maximisation are depicted by Figs. 8 and 9. As Figs. 5–7 suggest, computed gains under the loss function are both qualitatively and quantitatively merely the same in the case of the EBC and OBC models. The only exception seems to be the autoregressive coefficient on the technology shock where gain from the augmented rule with asset prices starts to materialise with a higher value of the weight in the loss function for the OBC specification. Inspection of the individual parameters across different shocks reveals that under the mark-up shock, gain from the augmented rule with asset prices starts to be positive after passing the specific value of the weight 𝛼 . Last but not least, the highest differences between gains of the augmented rule for the selected specifications of the parameters are found under the LTV shock. The differences in gains achieved under the low and the high values of the parameters can be even more than 30 p.p. Looking at the robustness checks under the welfare maximisation (Figs. 8 and 9), the result that the augmented rule with asset prices makes both agents under the EBC model and only households under the OBC model better off is confirmed. This is given by the positive consumption equivalent

(42)

𝜖 −1

where Ft = mct pt t yt (𝜆Pt )−1 + 𝜗𝛽P E0 Ft +1 and Ht = pt t yt (𝜆Pt )−1 + 𝜗𝛽P E0 Ht+1 . We set the Calvo parameter 𝜗 to 0.661 to match the value of the Rotemberg parameter 𝜓 P using the relationship 𝜓 P = 𝜗(𝜖 − 1)∕((1 − 𝜗)(1 − 𝜗𝛽P )). The results of this exercise are summarised in Tables 11–13 capturing the optimisation under the loss function, while Table 14 refers to the optimisation under the welfare maximisation.15 Overall, these additional simulation routines confirm the previous results achieved under the Rotemberg-pricing scheme. However, we can find slight differences. First, in the case of the mark-up shock, the optimised augmented rule under the Calvo-pricing scheme prescribes a stronger response to movements in asset prices accompanied by even higher gains. Second, when comparing the outcomes under the welfare maximisation, we find that gains stemming from the augmented rule are generally lower compared to the Rotemberg-pricing scheme. Moreover, the coefficients on asset prices are smaller. Despite these minor differences, the augmented rule still outperforms the baseline rule. 5.2. Sensitivity of the results to changes in the parameters The following analysis highlights to what extent does the initial calibration of the model affects the results presented in the previous subsections. In the case of the optimisations based on the central bank’s loss function, we re-run the optimisations for each value

16

Gambacorta and Signoretti (2014) test the robustness of the results to changes in the target capital-to-asset ratio (𝜈 B ) and the bank capital adjustment costs parameter (𝜃 ) as well. How it appears in their analysis, these parameters do not influence the results significantly.

All mentioned tables can be found in Appendix A. 14

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Economic Modelling xxx (xxxx) xxx

units for households and entrepreneurs in the case of the EBC setting, and positive and zero consumption equivalent units for households and entrepreneurs under the OBC setting respectively. Bearing this in mind, the sensitivity analyses show that the results presented in this paper are robust to changes in the values of the key parameters of the model and the conclusions outlined in the previous subsections are relevant even under different values of the key parameters of the model.

in response to an LTV shock (in addition to technology and mark-up shocks), and we test the robustness of the results under the different design of nominal rigidities. Third, we employ the OBC model to assess the performance of the Taylor-type rule accounting for movements in asset prices in terms of (a) a simple loss function and (b) the welfare maximisation. The main results of this paper are the following. The OBC model has more favourable properties regarding the hump-shaped and asymmetric impulse responses compared to the baseline EBC model. The best performance in terms of the lowest value of the central banks’ loss function delivers the rule augmented with asset prices, which outperforms the baseline rule. This result is stable across the model specifications, even though there are slight differences. However, there are differences in the welfare maximisations. The simulations under the OBC model reveal that reacting to asset prices might not be welfare-improving for both types of economic agents (households and entrepreneurs) included in the model. Overall, the optimal reactions to asset prices significantly differ across the shocks, and therefore, any rule-like behaviour does not seem to be appropriate. The results seem to be robust across different calibrations of the key parameters of the model as well as to different approaches to nominal rigidities modelling. Our results are in line with the previous work of Curdia and Woodford (2010) or Lambertini et al. (2013). The results, however, reveal some weak points of the models not accounting for non-linear features as documented by different results under the welfare maximisations. It is important to stress that some caution is needed when interpreting the results of this paper. We do not aim to provide precise quantitative prescriptions of the optimal weights to be assigned to asset prices in the central bank’s reaction function. We rather emphasise the qualitative perspective of the achieved results. We are aware that the model of Gambacorta and Signoretti (2014) stands upon several severe assumptions which might influence the simulations, and that there is no place for financial instability in the model, which might be crucial.

6. Conclusions The augmented Taylor rules have been a subject of research for several decades. The recent financial crisis of 2008 has shown that the interconnection between the real and the financial side of an economy is considerable. Therefore, the performance of Taylor rules augmented by various financial variables started to be discussed. Moreover, the recent papers in DSGE modelling show that nonlinearities play a crucial role, and therefore, the models should reflect this characteristic. The contribution of this research paper resides in a modification of the model introduced by Gambacorta and Signoretti (2014) by introducing an occasionally binding credit constraint in line with Brzoza-Brzezina et al. (2015), and in the evaluation of the performance of the Taylor-type rule augmented with asset prices in terms of a simple loss function and the welfare maximisation. In this research paper, we employ the baseline closed economy model following Gambacorta and Signoretti (2014) (the EBC model) and its modification in terms of the occasionally binding credit constraint (OBC) introduced via a penalty function approach in line with Brzoza-Brzezina et al. (2015) to study the performance of the Taylor rule augmented by asset prices. The contribution of this research paper is threefold. First, we compare the properties of both employed models in terms of impulse response functions, simulated data and moments. Second, we extend the analysis performed by Gambacorta and Signoretti (2014) by analysing the performance of the augmented rule Appendix A. Figures and tables

Table 8 Optimised Taylor-type rules under the loss function – a positive technology shock. Gain is a percentage difference between the loss achieved under the baseline and augmented rule.

𝛼

Baseline rule

Augmented rule – asset prices

𝜙𝜋

𝜙y

Lt

𝜙𝜋

𝜙y

𝜙qk

Lt

Gain

3 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0 0 0 0 0.1 0.1 0.1 0.1 0.1

0.000 0.005 0.01 0.015 0.019 0.024 0.029 0.034 0.038 0.043 0.048

3 3 2.6 1.7 1.2 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0 0 0 0 0 0 0 0 0

0 0.6 1 1 1 1 1 1 1 1 1

0.000 0.005 0.009 0.014 0.018 0.022 0.026 0.03 0.034 0.039 0.043

0.00% 2.42% 4.56% 6.26% 7.77% 9.02% 9.59% 9.90% 10.13% 10.31% 10.45%

3 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0 0 0 0 0.1 0.1 0.1 0.1 0.1

0.000 0.005 0.01 0.015 0.019 0.024 0.029 0.034 0.038 0.043 0.048

3 3 2.5 1.6 1.2 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0 0 0 0 0 0 0 0 0

0 0.6 1 1 1 1 1 1 1 1 1

0.000 0.005 0.009 0.014 0.018 0.022 0.026 0.03 0.034 0.039 0.043

0.00% 2.01% 4.30% 6.19% 7.83% 9.15% 9.77% 10.10% 10.35% 10.54% 10.70%

EBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 OBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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Table 9 Optimised Taylor-type rules under the loss function – a positive markup shock. Gain is a percentage difference between the loss achieved under the baseline and augmented rule.

𝛼

Baseline rule

Augmented rule – asset prices

𝜙𝜋

𝜙y

Lt

𝜙𝜋

𝜙y

𝜙qk

Lt

Gain

3 3 1.7 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0 0 0 0.1 0.1 0.2 0.3 0.4 0.5

0.001 0.004 0.006 0.008 0.009 0.01 0.012 0.013 0.014 0.015 0.016

3 3 1.7 1.1 1.4 1.1 3 2.9 3 3 2.8

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0.1 0.1 0.6 0.7 0.9 1 1

0.001 0.004 0.006 0.008 0.009 0.01 0.011 0.012 0.012 0.013 0.014

0.00% 0.00% 0.00% 0.00% 0.76% 4.39% 7.04% 9.85% 12.32% 14.45% 16.17%

3 3 1.9 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0 0 0 0 0.1 0.2 0.2 0.3 0.4

0.001 0.004 0.006 0.007 0.009 0.01 0.011 0.013 0.014 0.015 0.016

3 3 1.9 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0.1 0.1 0.2 0.2 0.3 0.3

0.001 0.004 0.006 0.007 0.009 0.01 0.011 0.012 0.013 0.013 0.014

0.00% 0.00% 0.00% 0.00% 0.00% 1.23% 3.88% 5.78% 8.04% 9.59% 11.54%

EBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 OBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Table 10 Optimised Taylor-type rules under the loss function – a positive LTV shock. Gain is a percentage difference between the loss achieved under the baseline and augmented rule.

𝛼

Baseline rule

Augmented rule – asset prices

𝜙𝜋

𝜙y

Lt

𝜙𝜋

𝜙y

𝜙qk

Lt

Gain

3 3 3 3 3 3 3 3 3 3 3

0.6 0.9 1 1 1 1 1 1 1 1 1

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.001 0.001

3 3 3 3 3 3 3 3 3 3 3

0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.6 0.6 0.7 0.7

0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

67.73% 40.42% 43.99% 45.58% 46.56% 47.27% 47.81% 48.99% 49.94% 50.71% 51.35%

3 3 3 3 3 3 3 3 3 3 3

0.5 0.6 0.8 0.9 1 1 1 1 1 1 1

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

3 3 3 3 3 3 3 3 3 3 3

0 0.1 0.1 0.2 0 0 0 0 0 0 0

0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

83.76% 40.93% 34.94% 32.16% 31.42% 32.85% 33.85% 34.59% 35.15% 35.59% 35.95%

EBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 OBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

16

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Economic Modelling xxx (xxxx) xxx Table 11 Calvo pricing. Optimised Taylor-type rules under the loss function – a positive technology shock. Gain is a percentage difference between the loss achieved under the baseline and augmented rule.

𝛼

Baseline rule

Augmented rule – asset prices

𝜙𝜋

𝜙y

Lt

𝜙𝜋

𝜙y

𝜙qk

Lt

Gain

3 1.1 1.4 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0.1 0.1 0.1 0.2 0.2 0.3 0.4 0.4 0.5

0.000 0.005 0.009 0.014 0.018 0.022 0.026 0.030 0.034 0.037 0.041

3 1.7 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 0.3

0 1 1 1 1 1 1 1 1 1 1

0.000 0.004 0.008 0.012 0.015 0.019 0.023 0.026 0.029 0.033 0.036

0.00% 9.27% 14.26% 14.63% 14.66% 14.77% 14.04% 13.23% 12.83% 12.11% 11.37%

3 1.1 1.4 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0.1 0.1 0.1 0.2 0.2 0.3 0.4 0.4 0.5

0.000 0.005 0.009 0.014 0.018 0.022 0.026 0.03 0.034 0.037 0.041

3 1.7 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 0.3

0 1 1 1 1 1 1 1 1 1 1

0.000 0.004 0.008 0.012 0.015 0.019 0.023 0.026 0.029 0.033 0.036

0.00% 9.68% 14.75% 15.18% 15.24% 14.71% 14.59% 13.76% 13.31% 12.60% 11.81%

EBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 OBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Table 12 Calvo pricing. Optimised Taylor-type rules under the loss function – a positive markup shock. Gain is a percentage difference between the loss achieved under the baseline and augmented rule.

𝛼

Baseline rule

Augmented rule – asset prices

𝜙𝜋

𝜙y

Lt

𝜙𝜋

𝜙y

𝜙qk

Lt

Gain

3 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0.1 0.3 0.6 0.9 1 1 1 1 1

0.002 0.006 0.009 0.011 0.012 0.013 0.014 0.014 0.015 0.016 0.016

3 1.1 3 3 2.7 2.2 1.8 1.6 1.4 1.2 1.1

0 0 0 0 0 0 0 0 0 0 0

0 0 0.5 0.8 1 1 1 1 1 1 1

0.002 0.006 0.008 0.009 0.01 0.011 0.011 0.011 0.012 0.012 0.012

0.00% 0.00% 8.09% 15.17% 18.57% 19.74% 20.28% 21.29% 22.61% 24.10% 25.68%

3 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0.1 0.2 0.5 0.8 1 1 1 1 1

0.002 0.006 0.009 0.011 0.012 0.013 0.014 0.014 0.015 0.015 0.016

3 1.1 1.1 1.1 1.2 1.1 1.1 1.1 1.1 1.1 1.1

0 0 0 0 0 0 0 0 0 0 0

0 0 0.1 0.2 0.4 0.5 0.6 0.7 0.8 1 1

0.002 0.006 0.008 0.01 0.01 0.011 0.011 0.012 0.012 0.012 0.012

0.00% 0.00% 6.15% 12.49% 16.17% 18.07% 18.78% 19.66% 20.89% 22.32% 23.85%

EBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 OBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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Economic Modelling xxx (xxxx) xxx Table 13 Calvo pricing. Optimised Taylor-type rules under the loss function – a positive LTV shock. Gain is a percentage difference between the loss achieved under the baseline and augmented rule.

𝛼

Baseline rule

Augmented rule – asset prices

𝜙𝜋

𝜙y

Lt

𝜙𝜋

𝜙y

𝜙qk

Lt

Gain

3 3 3 3 3 3 3 3 3 3 3

0.3 1 1 1 1 1 1 1 1 1 1

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.001 0.001

3 3 3 3 3 3 3 3 3 3 3

0.1 0.3 0.4 0.6 0.7 0.9 1 1 1 1 1

0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

56.53% 50.02% 49.98% 50.33% 50.85% 51.45% 52.12% 52.65% 53.08% 53.49% 53.82%

3 3 3 3 3 3 3 3 3 3 3

0.3 0.7 1 1 1 1 1 1 1 1 1

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

3 3 3 3 3 3 3 3 3 3 3

0 0.2 0.3 0.2 0.2 0.3 0.3 0.4 0.5 0.6 0.7

0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

73.33% 43.35% 37.90% 36.04% 36.03% 36.09% 36.15% 36.24% 36.35% 36.49% 36.68%

EBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 OBC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Table 14 Calvo pricing. Optimised Taylor-type rules under the welfare maximisation. The first column indicates the type of a shock. Households

Baseline rule

Augmented rule

𝜙𝜋

𝜙y

WP

𝜙𝜋

𝜙y

𝜙qk

WP

𝜉P

1.1 1.1 1.1

0 0.3 1

−118.68 −119.24 −118.71

3 2.9 1.1

0 0 0.1

0.4 0.5 1

−118.68 −119.23 −118.66

0.00% 0.01% 0.02%

Technology Mark-up LTV

1.1 1.1 1.1

0 1 1

−118.71 −119.46 −118.96

2.4 1.1 1.1

0 0 0

1 1 1

−118.69 −119.35 −118.72

0.01% 0.04% 0.09%

Entrepreneurs

Baseline rule

EBC Technology Mark-up LTV OBC

Augmented rule

𝜙𝜋

𝜙y

WE

𝜙𝜋

𝜙y

𝜙qk

WE

𝜉E

1.1 1.1 1.1

0 0 1

−123.81 −124.69 −123.91

3 2.7 1.1

0 0 0

0.8 0.2 1

−123.81 −124.69 −123.84

0.01% 0.00% 0.17%

3 3 1.1

0 0 0

−123.67 −120.52 −122.81

3 3 1.1

0 0 0

0 0 0

−123.67 −120.52 −122.81

0.00% 0.00% 0.00%

EBC Technology Mark-up LTV OBC Technology Mark-up LTV

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