最適傳統與非傳統貨幣政策 - 政大學術集成

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(1)國立政治大學社會科學院經濟學系研究所碩士論文. 最適傳統與非傳統貨幣政策 Optimal Conventional and Unconventional Monetary Policy. 指導教授：黃俞寧研究生: 蘇醒文. 博士撰. 中華民國 106 年 7 月.

(2) 中文摘要在面臨 2008 年金融危機後嚴峻之需求面衝擊下，多國央行紛紛調降利率，甚至調降至零利率，以期望國家能盡速脫離經濟衰退。在利率調降的過程中，由於受零利率下限的限制，單純藉由傳統貨幣政策之調控，仍可能無法使經濟從嚴峻的需求面衝擊下恢復。因此，非傳統貨幣政策也在美國、日本、英國、歐元區等國家被央行所採行做為對抗經濟衰退的政策工具。在此背景下，我們想了解如何實施最適非傳統貨幣政策。在最適貨幣政策理論發展中，最適傳統貨幣政策發展較早，因此也較為完備。而在最適非傳統貨幣政策方面，前瞻指引(forward guidance)這項政策工具已被引入最適理論中來進行討論，但量化寬鬆(quantitative easing)直至今日還未被引入最適理論中來討論。因此，本文建構一個封閉的新興凱因斯模型(New Keynesian)，開創性的將量化寬鬆引入最適貨幣政策理論中，來討論最適傳統與非傳統貨幣政策，也藉此與過去文獻比較施行量化寬鬆與未施行量化寬鬆之影響。. I.

(3) Abstract After financial crisis occurred in 2008, countries, such as United States of America, United Kingdom, Japan and the euro area, have been mired in slow economic growth, compared with long-term, and have confronted with severe negative demand shocks which can’t be offset by zero interest-rate policy. Under the circumstance where zero lower bound was present, unconventional monetary policy was implemented by central bank of each country to end up the economic recession prospectively. As a result, we are interested in how to implement optimal unconventional monetary policy. The literatures regarding implementing conventional optimal monetary policy under demand shock were developed maturely. Forward guidance, one of unconventional monetary policy, was already developed in field of optimal monetary policy under impacted by demand shock, but quantitative easing has not been conducted in the theory of optimal monetary policy so far. Thus, we construct a close-economy by New Keynesian model to conduct quantitative easing into the theory of optimal monetary policy to discuss the optimal conventional and unconventional monetary policy, and to compare the difference corresponding to the previous literatures which without quantitative easing.. II.

(4) Contents 1. Introduction .......................................................................................................... 1. 2. Model .................................................................................................................... 3 2.1. 3. 4. 5. Households ............................................................................................... 3 2.1.1. Saver ................................................................................................ 5. 2.1.2. Borrower ......................................................................................... 6. 2.2. Financial intermediaries ........................................................................... 7. 2.3. Firms ......................................................................................................... 8. 2.4. Government .............................................................................................. 9. 2.5. Conventional and unconventional monetary policy ............................... 11. 2.6. Market clearing conditions ..................................................................... 12. Analytical solution.............................................................................................. 12 3.1. Efficient allocation.................................................................................. 12. 3.2. Approximate equilibrium ....................................................................... 13. 3.3. Analytical method of policy analysis: discretion and commitment ....... 16 3.3.1. Optimal policy under discretion .................................................... 16. 3.3.2. Optimal policy under commitment ............................................... 20. Numerical solution ............................................................................................. 25 4.1. Parameters .............................................................................................. 25. 4.2. Numerical method of policy analysis: discretion and commitment ....... 26 4.2.1. Optimal policy under discretion .................................................... 26. 4.2.2. Optimal policy under commitment ............................................... 27. Conclusion .......................................................................................................... 29. References .................................................................................................................... 30 Appendix ...................................................................................................................... 33. III.

(5) 1. Introduction After financial crisis occurred in 2008, countries, such as United States of. America, United Kingdom, Japan and the euro area, have been mired in slow economic growth, compared with long-term, and have confronted with severe negative demand shocks which can’t be offset by zero interest-rate policy. Under the circumstance where zero lower bound was present, unconventional monetary policy was implemented by central bank of every country to end up the economic recession prospectively. After that, economists have paid much attention to the studies on unconventional monetary policies which involve the forward guidance and quantitative easing, such as Williams (2011, 2012), Campbell et al. (2012), Vissing-Jorgensen (2011) and Woodford (2012). Williams (2011, 2012) mentioned that authority administers unconventional monetary policy including two kinds of tools, forward guidance and quantitative easing (QE), to cause the long-term interest rates to drop to assist the nation’s economic recovery. Campbell et al. (2012) examined how public statements of FOMC intentions, forward guidance, can substitute for lower rates at the zero bound. Krishnamurthy and Vissing-Jorgensen (2011) and Woodford (2012) empirically evidenced that quantitative easing effects the interest rate on national debts. On the other hand, credit default swap (CDS) of nation’s debt was ascended by rising sovereign risk which had been fueled by higher level of nation’s debts after the crisis. Several literatures, such as De Bruyckere, Gerhardt, Schepens and Vennet (2013) and Alter and Beyer (2014), evidenced that spillover effects of sovereign risk exist in real economy. Corsetti, Kuester, Meier and Muller (2013, 2014) illustrated that rising sovereign risk not only decreases the price of nation’s debts but increases financing cost of private sector because of risk spilled over from sovereign debt market to financial market as sovereign risk channel. Therefore, implementing the quantitative easing can reduce the rising sovereign risk, 1.

(6) and then to alleviate the increasing financing cost. While these studies have shown that unconventional monetary policies do generate significant effects during the financial crisis, how to implement these policies optimally can be crucial for the monetary authority. In recent decades, the theory of optimal policy has been evolved from ad hoc to micro-founded by series of studies, such as Clarida, Galí and Gertler (1999) and Woodford (2003). The works on studies above have exhibited how a quadratic loss function, which represents an approximation to the representative agent’s welfare function associated with a given policy, is derived from a New Keynesian model. Under this framework, the performance of a given policy could be evaluated formally by this quantitative criterion. Referring to Kydland and Prescott (1977), a central bank implements monetary policy under commitment or discretion. The difference between them is that the policy under commitment will follow an assurance about rules in future, but the policy under discretion will be re-optimized period-by-period. Under commitment or discretion, the authority optimizes monetary policy by considering social welfare measured by loss function and the real economy represented by New Keynesian Phillips Curve and dynamic IS curve. Furthermore, the literatures regarding implementing policies optimally under impacted by demand shock can respectively discuss in optimal conventional monetary policy and optimal unconventional monetary policy. First, the literatures regarding optimal conventional monetary policy have unanimously shown that the positive demand shock can be offset by an interest-rate policy, but the response to a negative demand shock is generally distinguished into two cases: slightly demand shock and severe demand shock. Clarida et al. (1999) and Woodford (2003) expressed that when an economy was impacted by a slightly negative demand shock, central bank can offset the impact on output gap and inflation by tools of monetary policy, and the economy will be stabilized. On the other hand, Eggertsson and Woodford (2003) and Jung et al. (2005) expressed both output gap and inflation will fluctuate when the 2.

(7) economy was impacted by severe negative demand shock which makes the central bank encounter the zero lower bound. Second, several literatures regarding optimal unconventional monetary policy have studied forward guidance. Eggertsson and Woodford (2003) and Jung, Teranishi and Watanabe (2005) discussed that when an economy impacted by a demand shock, a central bank optimally implements forward guidance can encourage the economy to escape from zero interest rate sooner than without implementing forward guidance. However, in contrast to the optimal implementation of forward guidance which has been studied extensively in the literature, optimal quantitative easing involving purchases of government debts by the central bank has not been studied yet. Therefore, the primary aim of this study is to examine the optimal quantitative easing. Under this background, this study conducts the unconventional monetary policy in line with. the theory of optimal monetary policy to discuss two topics. First, we. discuss the problem of optimal unconventional monetary policy by a closed-economy of standard New Keynesian model. Difference from the previous literatures, we conduct both forward guidance and quantitative easing into our analysis. Thus, we can show the effects of implementing quantitative easing. Second, in our analysis, we discuss the optimal level of asset purchased while central bank implements quantitative easing, and the timing to suspend the policy. This study is divided into five chapters. Chapter 1 is the introduction. Chapter 2 outlines a New Keynesian model with two types of households and sovereign risk channel. Chapter 3 and Chapter 4 show the analytical solution and numerical solution respectively. The last chapter is the conclusion.. 2. Model. 2.1 Households Following Bhattarai et al (2015) and Benigno et al (2016), imagine a 3.

(8) closed-economy model which is endowed with two representative households indexed by m ∈ {s, b} , the notation s and b denote “saver” and “borrower” respectively. The population of household m is measured by ς m , and total population of. 1 . Both representative economy is normalized to one which denote ς s + ς b = households have the expected-utility form as. ∞. (. Et ∑ ξt + k β k U ( ctm+ k ) − ∫ V ( ltm+ k ( h ) ) dh k =0. 1. 0. ). (1). ,. where Et is the standard expectation operator; 0 < β < 1 is a discount factor; ξt m. denotes an exogenous preference shock; lt. (h). is labor supply of household m to. firm h , and ctm is consumption bundle of household m which has the form as. θ. θ −1  1  θ −1 ctm ≡  ∫ ctm (h) θ dh  0  . ,. (2). where ctm (h) is the consumption of good h which belongs to a continuum of goods produced in the interval [ 0, 1] , and θ is the intertemporal elasticity of substitution. ( ). between goods. We define U ctm. (. and V ltm ( h ). U ( ctm ) = 1 − exp(−ϖ ctm ) and V ( ltm ( h ) ). where. ). as. (l ( h )) = m t. 1+ ϕ. 1+ϕ. ,. (3). ϖ > 0 and ϕ denotes inverse of the labor supply elasticity. Following. Benigno et al (2016), the reason which we choose an exponential utility in consumption is that it makes the mathematical processes in following chapters simpler. The difference between the budget constraints of two types of household is that saver 4.

(9) holds deposits and long-term government bonds, but borrower is credited from the financial intermediaries. 2.1.1 Saver Saver makes decisions to maximize expected utility, Eq. (1), subjected to the budget constraint as. s g g , out Pc t t + d t + Pt bt. = Rt dt −1 + (1 − ϑt ) Pt g Rtg btg−,1out + ∫ wts ( h ) lts ( h ) dh + ∫ π t ( h ) dh − tts + trt def 1. 1. 0. 0. ,. (4). where Pt is a corresponding price index of consumption bundle, wts (h) is the nominal wage of saver paid by firm h , π t ( h ) is the profit of firm h by producing goods, tts is the lump-sum tax on saver and ϑt is a random variable of the sovereign default rate. The variable dt −1 represents the amount of deposits saved at period t − 1 which will be repaid with the interest rate on deposits Rt by financial intermediaries at the end of the period. Moreover, Rt is also a monetary policy tool controlled by monetary authority. In line with Woodford (2001), the long-term bond Btg−1 issued by government is perpetual bond with the price Pt g at time t and will pay an exponentially decaying coupon rate ι s at time t + s + 1 , and the (gross) yields of long-term bond is Rtg= (1 + ιts Pt g ) / Pt g . btg−,1out represents the long-term bond held by saver. Following Corsetti et al. (2013, 2014), government debt is not riskless. In each period, government may be obligated to execute its debt contract, in which case def def ϑt = 0 ; or it may partially default, in which case ϑt = ϑ , with ϑ ∈ (0, 1) . We. assume that trt def is a lump-sum transfer that compensates to bondholders when the sovereign default occurs, and the transfer is written as. trt def = ϑt Pt g Rtg btg−,1out .. (5) 5.

(10) Every period, the saver optimizes utility by choosing consumption, deposits, long-term bond and labor supply. The labor-consumption tradeoff, Euler equation derived by optimizing deposits and Euler equation derived by optimizing long-term bond are given as. (l ( h )). ϕ. wts ( h ) , = Pt ϖ exp(−ϖ cts ) s t. . (6). . Rt +1   , Π t +1 . . (1 − ϑt ) Pt +g1Rtg+1 . −ϖ cts ) β Et xt +1 exp(−ϖ cts+1 ) xt exp(=. −ϖ cts ) β Et xt +1 exp(−ϖ cts+1 ) xt exp(=. Pt g P t +1. . (7).  , . (8). Pt +1 / Pt represents the inflation. where P t +1 = 2.1.2 Borrower Borrower makes decisions to maximize expected utility, Eq. (1), subject to the following budget constraint. b b b b fi b , Pc t t + Rt bt −1 =bt + ∫ wt ( h ) lt ( h ) dh + ∫ π t ( h ) dh + d t − tt 1. 1. 0. 0. (9). where wtb (h) is the nominal wage of borrower paid by firm h , bt −1 is the amount of loans borrowed at period t − 1 which should be repaid with the interest rate on loans Rtb to financial intermediaries at the end of the period, δ t fi is the transfers from the. financial intermediary sector , and ttb is the lump-sum tax on borrower. 1. 1. The transfer qt. fi. will be discussed in the section of financial intermediaries. 6.

(11) Every period, the borrower optimizes utility by choosing consumption, loans, and labor supply. The labor-consumption tradeoff and Euler equation derived by optimizing loans are given as. (l ( h )) = w ( h ) P ϖ exp ( −ϖ c ) ϕ. b t. b t. b t. (10). ,. t.  Rtb+1  b xt exp(= −ϖ c ) bx Et  t +1 exp(−ϖ ct +1 )  . Π t +1   b t. (11). 2.2 Financial intermediaries We construct perfectly competitive financial intermediaries by following Cúrdia and Woodford (2016). In each period, a portion of loans χ t are not repaid (due to, say, fraud), so cause the financial (credit) friction and the spread of nominal interest rate between deposits and loans exists. Even though the degree of risk on fraudulent loans and non-fraudulent loans aren’t equivalent, financial intermediaries require the same interest rate on loans, Rtb , on all loans, (1 + χ t ) Bt , because the loans can’t be distinguished into fraudulent or non-fraudulent at initiation. Moreover, we assume that deposits Dt are repaid from the payments of the non-fraudulent loans, so Rt Dt = Rtb Bt , where Dt = ς s dt and Bt = ς bbt . As a result, the cash flow in period. t. of financial intermediaries is given as Dt − Bt − χ t Bt . We set 1 + ωt ≡ Rtb / Rt which represents the spread between interest rates on loans and deposits. Then, substituting Dt= (1 + ωt ) Bt and choosing Bt to maximize the profits yield the first-order. condition as. χ t = ωt. (12). .. 7.

(12) The first-order condition represents that the existence of the default risk affect the credit spread which rsises the cost of borrowing. Furthermore, following the assumption made by Corsetti et al. (2013), χ t depends on sovereign risk to capture the risk spilled over from sovereign debt market to financial market. That is. α.  Pg Rg  = χ t ψ t  t +g1 t +1  − 1 ,  Pt Rt +1 . (13). where parameter ψ t is a financial friction shock, and α measures the strength of the spillover from the sovereign risk premium to the private risk premium. At last, the transfers, ∆ tfi = ς bδ t fi , from financial intermediaries to borrower, includes loans which are not repaid. 2 Hence ∆ tfi = χ t Bt .. 2.3 Firms We assume a continuum of firms indexed by h ∈ [0, 1] in a monopolistically competitive goods market. In line with Benigno et al. (2016), each firm produces differentiated goods, but the technology of production is identical Cobb-Douglas form as. Y= L= t ( h) t ( h). ( L ( h) ) ( L ( h) ) s t. ςs. b t. ςb. (14). ,. where Yt (h) is the goods produced by firm h , and Lst (h) and Lbt (h) are the labor demand for saver and borrower respectively. Thus, the problem of cost minimization, subject to technology constraint, derives the marginal cost as. 2. The setting of the transfers ∆ t. fi. is following Cúrdia and Woodford (2016). The earning from. fraud is assumed to be a lump-sum transfers to borrower. Thus, borrower has additional income, in each period. 8. χt bt ,.

(13) MCt (h= ) W= (ς s ) −ς s (ς b ) −ς b (Wt s )ς s (Wt b )ς b . t. (15). Following the staggered prices proposed by Calvo (1983), each firm may reset its price Pt (h) with probability 1− Θ in every period. Therefore, the proportion of producers who reset their price is 1− Θ . Furthermore, a firm chooses Pˆt (h) to be. t. the resetting price to maximize its profit at time. ∞. as. (. k Et ∑ ( Θ ) Λ t + k , t (1 + t ) Pˆt (h)Yt + k , t (h) − Wt + k ( h ) Yt + k , t (h) k =0. ). (16). subject to the sequence of demand constraint. −θ.  P ( h)  Yt + k , t (h) =  t  Yt + k ,  Pt + k . (17). β k ( (U c ,t +1 / Pt ) / (U c ,t / Pt + k ) ) is a stochastic discount factor, and 1 + τ where Λ t + k , t = is a proportional subsidy. In equilibrium, all firms choose Pˆt (h) = Pˆt . The optimal condition is given as. ∞. Pˆt = µ. Et ∑ ( Θβ ) exp ( −ϖ Yt + k ) Pt + k θ Wt + k ( z ) Pt + k −1Yt + k k. k =0. ∞. Et ∑ ( Θβ ) exp ( −ϖ Yt + k ) Pt + k k. k =0. , θ −1. Yt + k. where µ = θ / (1 + τ )(θ − 1) .. 2.4 Government In line with Corsetti et al (2014), the government’s budget constraint is. 9. (18).

(14) 1. Pt g Btg + Tt s + Tt b =(1 − ϑt ) Pt g Rtg Btg−1 + t ∫ Pt (h)Yt (h)dh + TRtdef , 0. (19). where Tt s = ς s tts , Tt b = ς bttb and TRtdef = ς s trt def . We simply assume that government expenditure is zero, and sovereign bonds are issued fixedly ( Pt g Btg ≡ P g B g ). In each period, the total amount of the sovereign bonds issued by government is Pt g Btg , and the total amount of lump-sum taxies collected from saver and borrower are Tt s and Tt b respectively. Pt g Btg , Tt s and Tt b are used to finance the bond repayment 1. (1 − ϑt ) Pt g Rtg Btg−1 , the subsidy t ∫ Pt (h)Yt (h)dh on good producers and the transfer 0. def t. TR. for sovereign default. In line with Corsetti et al (2014), the compensation of. sovereign default, the transfer TRtdef , is set in such way that a sovereign default does not alter the actual debt level. Formally, we set. TRtdef = ϑt Pt g Rtg Btg−,1out . 3. (20). After plugged sovereign default compensation, Eq. (20), into government budget constraint, Eq. (19), the government budget constraint is rewritten as. 1. + Tt b Pt g Rtg Btg−1 + t ∫ Pt (h)Yt (h)dh . Pt g Btg + Tt s = 0. (21). Furthermore, we assume that the probability of sovereign default, ptdef , in each period follows a cumulative distribution function of beta distribution with parameters. 3. The setup in Corsetti et al (2014) is in line with Yeyati and Panizza (2011). The costs of. sovereign default materialize in the run-up to defaults rather than at the time when the default actually arise. 10.

(15) α b , bb and Ξ tg ,max as g. g. def t. p.  Pt g Btg ,out  1 , = Fbeta  ab g , g  . g B B   4 PY Ξ B ,max t  . (22). where Btg , out = ς s btg , out represents the total amount of the sovereign bond outstanding held by saver, and Ξ tg ,max is the upper bound of debt-to-GDP ratio. Thus the sovereign default rate is. ϑ. def t. ϑ def = 0. with probability. ptdef. with probability 1 − ptdef. .. (23). 2.5 Conventional and unconventional monetary policy The monetary policies can be distinguished into conventional monetary policy and unconventional monetary policy. The conventional monetary policy indicates that the monetary authority will consider the situation of economy to set the current short-term interest rate, Rt in period. t . The unconventional monetary policy. indicates that the monetary authority responses situation of economy by two instruments, forward guidance and quantitative easing. Forward guidance is the policy announcing the future short-term interest rate in current period. Therefore, the monetary authority sets {Rk }tk+= 1t++n1 in period. t , where. n is positive integer.. Quantitative easing is the policy that the monetary authority purchases the long-term bonds from market to affect interest rate on private sector. Here, we employ ΩQE to t represent the total amount of long-term bonds purchased by monetary authority. In a nutshell, the monetary authority’s conventional monetary policy instrument is current short-term interest rate, and unconventional monetary policy instruments are forward 11.

(16) guidance (future short-term interest rate) and the quantitative easing.. 2.6 Market clearing conditions The clearing conditions of goods market and labor market are. Yt = Lt =. ∑C. m=s , b. ∑. m=s , b. m t. ,. (24). Lmt ,. (25). where Ctm = ς m ctm and Lmt = ς mltm , Then, clearing conditions of the financial intermediation and sovereign bond market are. Dt= (1 + χ t ) Bt ,. (26). ΩQE + Pt g Btg ,out , Pt g Btg = t. (27). where Dt = ς s dt , Bt = ς bbt and Btg ,out = ς s btg ,out .. 3. Analytical solution To discuss the optimal monetary policy, we characterize that the monetary. authority chooses the target variables and the policy instrument to minimize the loss function subject to dynamic IS (DIS) curve and New Keynesian Phillips Curve (NKPC), and thus we can discuss this problem under discretion and commitment respectively.. 3.1 Efficient allocation In line with Bhattarai et al. (2015), the efficient allocation can be described as the optimal problem of social planner as. 12.

(17) 1+ϕ   1 ( Yt ( z ) ) dz  max ∑ xt β t 1 − exp(ϖ Ct ) − ∫ 0   1+ ϕ t =0   ∞. (28). subject to the technological and resource technology constraint which is.  1 Ct =  ∫ Yt ( z ) 0 . θ −1 θ.  dz  . θ θ −1. (29). .. In period t , the first-order condition is given as. 1. −1. ϖ Yt θ Yt (h) θ exp ( −ϖ Yt ) = Yt (h)ϕ ,. (30). and it is easy to show that Yt (h) = Yt under efficiency. As a result, we do log-linearization on Eq. (30) to obtain the efficient level of aggregate output. Yt E = 0 .. (31). 3.2 Approximate equilibrium We do analyses by the log-linearized model, and the detailed derivations are attached in appendix. 4 After log-linearizing on Euler equations of both households, Eq. (7) and Eq. (11), we use weighted combination by population on two types of household’s linearized Euler equation to derive the DIS curve of whole economy as. 4. The detailed derivations of DIS curve and Loss function are attached in the Appendix A and. Appendix C respectively, and the derivation of NKPC is following Galí (2015). 13.

(18) (.  QE ) − r n − Π  X t X t +1 − σ −1 it + (ς bψ t − ς bαϒ 0 Ω = t t t +1. ). ,. (32). where. ′ ϑ ϒ 0 ≡ ( Fbeta. def. /Ξ. B g ,max.  ΩQE  5 QE  ) , Ωt ≡  t  .  PY . The parameter σ denotes the coefficient of relative risk aversion. 6 The variable it +1 denotes the (net) nominal interest rate on deposits which is also a policy instrument controlled by monetary authority. The variable X t ≡ Yt − Yt E represents the log deviation output gap between actual output and efficient output, and rt +n1 is the natural rate and represents the demand shock which is derived from preference shock. rt +n1 =r + (ξt − ξt +1 ) . 7 The parameter Fbeta ′ denotes the probability density function of the beta distribution at steady-state. Furthermore, the method of deriving the New Keynesian Phillips Curve (NKPC) is commonly same as economic textbooks. We use the first order condition obtained from goods producer to derive the NKPC as.  = βΠ  + κ X , Π t t +1 t. (33). where. 5. The variable with tilde represents the log-deviation from steady-state, xt ≡ log( xt / x ) .. 6. In line with Benigno et al (2016), the coefficient of relative risk aversion is defined as. σ ≡ 1/ (ϖ Y ) . 7. The setting about rt +1 also appeares in Woodford (2003) and Galí (2015). n. 14.

(19) κ=. (1 − Θ )(1 − Θβ )(σ + ϕ ) Θ (1 + θϕ ). .. Finally, following Woodford (2003) and Bhattarai et al. (2015), we consider a social planner maximizing the social welfare of economy as. = Wt. ∞. ∑ bV ∑ k. k 0= m s, b =. m. U ( c m ) − 1V ( l m (h) ) dh  . t +k ∫0 t +k  . (34). Then we take second-order Taylor expansion on Eq. (34) around the steady-state to obtain the approximation of welfare as. ∞ 1 Wt ≈ − U cY ∑ β k Lt + k + t.i. p. + O (|| ξ 3 ||) , 2 k =0. (35). (. where t.i. p. represents the terms independent of monetary policy, O || ξ 3 ||. ). contains the terms which order are higher than two, and Lt is the loss function which form is. ( ). = Lt λX X t. 2.  )2 + λ R ( c R )2 , + λΠ ( Π t t C. (36). where. λX= σ + ϕ , λc R =. (s + ϕ ) ς sς bs (1 + ϕ ). , λΠ =. 15. (1 + ϕ ) Θθ 2 . (1 − Θ )(1 − βΘ ).

(20) The variable ctR represents the relative consumption of two types of household. 8 This term has been featured in the literatures discussing welfare-theoretic loss function, such as Benigno (2009), Fiore and Tristani (2013), and Cúrdia and Woodford (2016). Considering the quantitative easing policy, the loss function, Eq. (38), can be rewritten as. Td Td  −1   2 2 QE    Lt λX X t + λΠ Π t + λc R σ  αϒ 0 ∑ Ω k − ∑ψ k   = k t= k t  =   . 2. (37). .. 3.3 Analytical method of policy analysis: discretion and commitment We assume a circumstance that the economy has been impacted by a severe negative demand shock, and the shock can’t be offset by zero interest-rate policy. Under the circumstance, the monetary authority responses the shock by conventional instruments and unconventional instruments. Considering the assumptions, we analyze the optimal monetary policy under discretion and commitment respectively. 3.3.1 Optimal policy under discretion Under discretionary policy, the monetary authority will not commit any action of monetary policy in future. Thus, the optimal problem of monetary authority is one-period problem. The monetary authority minimizes the loss in welfare subject to DIS curve and NKPC. The Lagrangian function which represents the optimal problem is. = Γ. (. 2, t. 8. ). U cY 1  ˆ QE ) − Π  − r n  Lt + φ1, t  X t − X t +1 + it + (ς bψ t − ς bαϒ 0 Ω t t +1 t  σ 2    −bΠ  − κ X +φ Π. (. t. t +1. t. ). The detailed derivation of the relative consumption is attached in the Appendix B. 16. ,. (38).

(21) where φ1, t and φ2, t denote the Lagrange multipliers. 9 After differentiating the  QE , φ  , X , Ω Lagrangian function Γ , the first order conditions of variables Π 1, t t +1 t t. and φ2, t are given as.  + φ =0 , U cY λΠ Π 2, t t. (39). ( ). 0, U cY λY Yt + φ1, t − κφ2, t = U c Y λc R (σ. ). −1 2. (40). Td Td   −1 QE ˆ  αϒ 0 ∑ Ω k − ∑ψ k   ϒ 0 − φ1, t ς bσ αϒ 0 =0 , = k t= k t   . (. (41). ). ˆ QE ) − r n − Π  , X t − X t +1 + σ −1 it + ς b (ψ t − αϒ 0 Ω t t t +1 = 0. (42).  −βΠ  − κ X =0 . Π t t +1 t. (43). And the Kuhn-Tucker conditions regarding the non-negativity constraint on the nominal interest rate it are. it φ1, t = 0 ,. (44). φ1, t ≥ 0 ,. (45). it ≥ 0 .. (46). {Π t }t = −∞ = {φ1, t }t = −∞ = {φ2, t }t = −∞ = 0 We assume the initial conditions as { X t }t = −∞ = −1. −1. −1. −1. and {= it }t−=1−∞ {= rt n }t−=1−∞ r , and we solve the optimal problem under discretion in line with Jung et al. (2005).. 9. The expectation operator, Et , not appear here because the future path of the natural rate of. interest is perfectly foreseen. 17.

(22) Observing the Lagrangian function as above, we obtain ∂Γ / ∂it = 2σ φ1,t by −1. differentiating the variable it . Because the parameter σ is positive, the sign of ∂Γ / ∂it is dependent on Lagrange multiplier φ1, t . We discuss two cases in here. First,. if the nominal interest rate it is positive, which indicates the non-negativity constraint is not binding, ∂Γ / ∂it and the Lagrange multiplier φ1, t equals zero. This means the monetary authority can perfectly offset the fluctuation caused by demand shock through controlling nominal interest rate. On the other hand, if the nominal interest rate it equals zero, which indicates the non-negativity constraint is binding, ∂Γ / ∂it and the Lagrange multiplier φ1, t is positive. This means the monetary. authority can’t offset the fluctuation caused by demand shock through controlling nominal interest rate. In other words, the zero lower bound is present. We solve the optimal problem by following Jung et al. (2005). Given that the non-negativity constraint on nominal interest rate is not binding at steady-state, we assume the economy is impacted by a severe negative demand shock which causes the monetary authority encountering the zero lower bound from periods 0 to T d . This implies the non-negativity constraint is binding from period 0 to period T d . Thus, we derive the dynamic path by segmenting periods in two parts, periods 0 to T d and periods T d + 1 to infinite. The constraint is binding in periods 0 to T d , and the constraint is not binding in periods T d + 1 to infinite. First, we characterize the path of variables from periods T d + 1 to infinite. Because the constraint is not binding, we substitute φ1, t = 0 into first-order conditions to obtain a unique bounded solution as. zt = 0 ,. (47). n = it rt= r , t = T d +1, .... (48).  X ]′ . We where zt represents a vector including inflation and output gap, zt ≡ [Π t t 18.

(23) can reform the Eq. (41) to derive the quantitative easing as. ˆ QE = ϒ (φ − φ ) + ϒ ψ , Ω t 1 1, t 1, t +1 2 t. where ϒ1 ≡. (49). ςb 1 1 and ϒ 2 ≡ . −1 U c Y λc αϒ 0σ αϒ 0 R. Second, in periods from 0 to T d , we substitute it = 0 into first-order conditions and use Eqs. (39), (40), (42), (43) and (49) to obtain a system of difference equations as. zt= Azt − a (rt n + ϒ 4ψ t ) , t = 0, ... , T d , +1. (50). where.   β −1 − β −1κ   A ≡  σ −1 ( β −1 + ( β −1 − 1)U c Y λΠ ϒ3κ ) 1 + σ −1 (U c Y λX ϒ3 +β −1 (κ + U c Y λΠ ϒ3κ 2 ) )  , −  1 + σ −1U c Y λX ϒ3 1 + σ −1U c Y λX ϒ3   0     , ϒ ≡ ς αϒ ϒ and ϒ ≡ ς (1 − αϒ ϒ ) . a≡ σ −1 3 b 0 1 4 b 0 2  1 + σ −1U c Y λX ϒ3 . After combined the terminal condition zT d +1 ≡ [0 0]′ with Eq. (50), the system of difference equation has a unique bounded solution which can be written as. zt =. Td. ∑A k =t. − ( k −t +1). a(rkn+1 − ϒ 4ψ k ). n d , rk +1 < 0 , ψ k > 0 , t = 0,..., T .. 19. (51).

(24) According the Eq. (47) and (51), the dynamic path of inflation and output gap for all periods are given as.  T − ( k −t +1) n a (rk +1 − ϒ 4ψ k )  ∑A zt =  k =t 0  d. , t = 0,..., T d. (52). .. = , t T + 1,... d. ˆ QE can be rewritten by combining Eqs. (39), (40), (41) The quantitative easing Ω t and (51) as. T T  ˆ QE = Ω −U cY ϒ1u  ∑ ( A− ( k −t +1) arkn+1 ) − ∑ ( A− ( k −t +1) arkn+1 )  t k = t +1  k= t  d. d. T T  + U cY ϒ1u  ∑ ( A− ( k −t +1) a ϒ 4ψ k ) − ∑ ( A− ( k −t +1) aϒ 4ψ k )  + ϒ 2ψ t k = t +1  k= t  d. d. for t = 0,..., T , d. where u ≡ [κλΠ. ,. (53). λX ] .. 3.3.2 Optimal policy under commitment Under discretionary policy, the monetary authority commits the actions of future monetary policy. Thus, the expecting inflation of private sector is affected by monetary authority. The monetary authority minimizes the loss in welfare subject to DIS curve and NKPC. The Lagrangian function which represents the optimal problem starting from t = 0 is. ( X − X.  U cY Lt + φ1, t  Γ =∑ b  2 t =0  + φ2, t  ∞. t. (. )).  ˆ QE ) − r n − Π  + σ −1 it +1 + (ς bψ t − ς bαϒ 0 Ω t t t +1   ,   − bΠ  − κ X Π t t +1 t  t. t +1. (. ). (54). 20.

(25) After differentiated the Lagrange function Γ , the first order conditions of  , X , ΩQE , φ , φ are given as variables Π 1,t 2,t t t t.  − β −1σ −1φ U c Y λΠ Π 0 , t 1, t −1 + φ2, t − φ2, t −1 =. (55). U cY λX X t + φ1, t − β −1φ1, t −1 − φ2, tκ = 0 , U c Y λc R (σ. ). −1 2. (56). Td Td   QE −1 ˆ  αϒ 0 ∑ Ω k − ∑ cψ , k   αϒ 0 − φ1, t ς bσ αϒ 0 =0 , k t= k t =   . ). (. (57). ˆ QE ) − r n − Π  X t − X t +1 + σ −1 it +1 + (ς bψ t − ς bαϒ 0 Ω t t t +1 = 0 ,. (58).  − βΠ  − κ X =0 . Π t t +1 t. (59). And the Kuhn-Tucker conditions regarding the non-negativity constraint on the nominal interest rate, it , are. it φ1, t = 0 ,. (60). φ1, t ≥ 0 ,. (61). it ≥ 0 .. (62). The difference between the first-order conditions in here and the first-order conditions in section 3.3.1 is that the lagged Lagrange multipliers, φ1, t −1 and φ2, t −1 , here can capture the loss in expecting inflation and output gap changed in period t − 1 . We.  }−1 = {Π {φ1, t }t−=1−∞ = {φ2, t }t−=1−∞ = 0 and assume the initial conditions as { X t }t−=1−∞ = t t = −∞ {= it }t−=1−∞ {= rt n }t−=1−∞ r to solve the optimal problem under commitment.. As Eq. (57), we reform the Eq. (57) to derive the quantitative easing as. 21.

(26) ˆ QE = ϒ (φ − φ ) + ϒ ψ . Ω 1 1,t 1,t +1 2 t t. (63). In line with Jung et al. (2005), given that the non-negativity constraint on nominal interest rate is not binding at steady-state, it implies φ= φ= 0 in 1, −1 2, −1 period −1 . We assume the economy is impacted by a severe negative demand shock starting from period 0 which causes the non-negativity constraint is binding, and we guess the constraint is binding until period T c . Thus, the constraint is binding in periods 0 to T c , and the constraint is not binding in periods T c + 1 to infinite. In section 3.3.1 we segment the periods in two part, periods 0 to T d and periods after T d , to solve the dynamic path, but in here, we derive the dynamic path by segmenting periods in three parts which are periods 0 to T c , period T c + 1 and periods T c + 2 to infinite. First, we characterize the path of variables from the periods T c + 2 to infinite. Because the constraint is not binding from periods T c + 1 to infinite, we substitute. {φ1,t }=t∞ T c +1 = 0 into first-order conditions to obtain. zt = kφ2, t −1 ,. (64). φ2,t = µ1φ2,t −1 ,. (65). it += rt +n1 + ϒ5φ2,t −1 , 1. (66). where.  (1 − µ1 )   , k ≡  k µ   U cY λX 1 . 22.

(27)  κσ ϒ5 ≡ 1 −  U cY λ X.   µ1 (1 − µ1 ) . . The parameter µ1 is a real eigenvalue of the system difference equations of Eqs. (55) and (59) under {φ1,t }=t∞ T c +1 = 0 , satisfying 0 < µ1 < 1 . Second, in period T c + 1 , we substitute φ1,T c +1 = 0 into first-order conditions to obtain a system of difference equations as.  zT c +1  −1 −1 = φ  M QzT c + 2 + M J φT c , c  2,T +1 . (67). where.  1 M ≡ U c Y λΠ  0. −κ 0 U cY λX. β 0   1  , Q ≡ 0  0 −κ . 0 0  0   −1 1  . 0  , J ≡ ( βσ )  β −1 0  0 . We need the terminal condition, value of zT c + 2 , and initial condition, value of φT c , to complete the above solution. Finally, the constraint is binding in periods 0 to T c . Thus, we substitute {it }Tt = 0 = 0 into first-order conditions to obtain c. ˆ − a (r n − ϒ ψ ) − a ϒ (φ − φ ) , zt= Az +1 4 k 1,t +1 t c k +1 c 3 1,t = φt Cφt −1 − Dzt. , t = 0,1, , T c ,. (68) (69). Where 23.

(28)  β −1 −κβ −1  Aˆ ≡   , −1 1 + κ ( βσ ) −1   −( βσ )  β −1 + κ ( βσ ) −1 κ  C≡  , ( βσ ) −1 1  U Y λ κ D≡ c Π  U c Y λΠ. U cY λX   , 0 . , φt = [φ1,t φ2,t ]′ and a ≡ [0 σ −1 ]′ ..  , X , φ1,t and φ2,t can be determined by given The unique dynamic path of Π t t. Eqs. (55)-(62) and initial conditions. The path of the nominal interest rate from periods T c + 1 to infinite can be derived by combing the path of inflation and output gap with Eq. (58), and the path of nominal interest rate from periods 0 to T c are known, {it }Tt =0 = 0 . If these conditions are not verified, checking the Kuhn-Tucker c. conditions in each period, then the procedure is repeated for a new value for T c . 10 While the conditions were verified, we can derive the equilibrium path of the quantitative easing by using Eqs. (55), (56), (57) and path of zt .  QE }∞ can be derived by The equilibrium path of quantitative easing {Ω t t =0. substituting the equilibrium path of the variables φ1,t and φ2,t into Eq. (63). Given the complexity model, we adopt numerical methods to solve the equilibrium path by those dynamic equations in next section.. 10. See, e.g., Siddhartha and Betty (2015) for a relevant analysis algorithm. 24.

(29) 4. Numerical solution In this section, we use numerical method to present the solution of optimal. monetary policy under discretion and commitment.. 4.1 Parameters We adopt the parameters commonly used in the existing literature or calibrated to match the US data. Following Benigno et al. (2016) as observing US data, we set the subjective discount factor β equals 0.9963. Following Corsetti et al. (2013), the annual long-term yield in US at the steady state is 4.3274% in line with the averaged daily treasury long-term rate data, between 2000 and 2015. The ratio of debt-to-GDP at steady-state is 60% in line with the US averages over the last 20 years. The sovereign default rate ϑ def is setting at 0.5. The upper bound of debt-to-GDP ratio Ξ tg ,max equals 1.64 which can be obtained from the no-arbitrage conditions of saver. and the default probability, Eqs. (7), (8) and (22). Strength of the spillover effect α is set to 0.55 which consists with the finding of Harjes (2011). About the household sector, we assume the population of two types of households are equal, ς s = ς b = 0.5. We set the coefficient of relative risk aversion σ = 1 by following Galí (2015), and the inverse of the labor supply elasticity ϕ = 1/1.9 by following Corsetti et al. (2013). These imply the parameter ϖ is equal to 1.9255. About the firm sector, we set the elasticity of substitution θ = 9 by following Galí (2015), and the probability of price fixed Θ = 0.66 by following Eggertsson et al. (2014). Table 1 shows all the parameter values reported in our study.. Table 1 Parameters. ςs ςb β α. ΞB. 0.5 0.5 0.9963 0.55. g,. σ. ϕ. θ 25. max. 1.64 1 1/1.9 9.

(30) ϑ def. Θ. 0.55. 0.66. 4.2 Numerical method of policy analysis: discretion and commitment We showed the analytical solution of the optimal policy in section 3. In this subsection, we show the numerical analysis based on the benchmark parameter values attached in Table 1. The path of endogenous variables are simulated that economy was impacted by adverse demand shocks including the natural rate shock and the financial friction shock, equal to -0.25 percent and 1 percent respectively, from periods t = 0 to t = 7 . Furthermore, under both discretion and commitment, we compare two cases: First, the monetary authority implements three tools of monetary policy including the current nominal (deposit) interest rate, forward guidance (future nominal interest rate) and quantitative easing when the economy was impacted by the adverse demand shocks. Second, the monetary authority implements only two tools of monetary policy including the current nominal (deposit) interest rate as forward guidance (future nominal interest rate) as the cases discussed in Jung et al. (2005) and Galí (2015). We assme the first (second) case is called OMP (OMP without QE) for short. 4.2.1 Optimal policy under discretion When monetary authority implements policy under discretion, forward guidance is absent in the policy tools, only current nominal interest rate or quantitative easing can be adopted. Thus, in first case, OMP case, we consider that the monetary authority is able to use the current nominal interest rate and quantitative easing to be the policy tools. In second case, OMP without QE case, we consider that the monetary authority is only able to use the current nominal interest rate to be the policy tool. We simulate these two cases to compare the difference of the effect between the policy with quantitative easing and without. The simulated dynamic path of the variables, shown in Figure 1, include inflation, output gap, nominal interest rate, sovereign debt purchases, spread between interest rates on loans and deposits rate, and natural rate. 26.

(31) Fig.1 OMP case vs. OMP without QE case under discretion. Figure 1 has shown that the quantitative easing reduces the interest spread through the sovereign risk channel, and the variations of inflation and output gap are shrunk after implementing the quantitative easing. 4.2.2 Optimal policy under commitment When monetary authority implements policy under commitment, forward guidance exists in the policy tools. Thus, in first case, OMP case, we consider that the monetary authority is able to use the current nominal interest rate, forward guidance and quantitative easing to be the policy tools. In second case, OMP without QE case, we consider that the monetary authority is able to use the current nominal interest rate and forward guidance to be the policy tool. We simulate these two cases to compare the difference of the effect between with policy with quantitative easing and without. The simulated dynamic path of the variables, shown in Figure 2, include inflation, output gap, nominal interest rate, sovereign debt purchases, spread between interest rate on loans and deposits and natural rate. 27.

(32) Fig.2 OMP case vs. OMP without QE case under commitment. The previous literatures discussing the optimal monetary policy without quantitative easing, such as Jung et al. (2005) and Galí (2015), have several features. Frist, the period of implementing zero lower bound under commitment is not less than under discretion, T c ≥ T d . Second, the path of inflation and output gap are negative during initial period and then becomes positive. This caused by a looser monetary policy promised by monetary authority, presenting in Figure 2 which shown that the period of remaining at zero lower bound is longer than the period of facing negative shocks. Figure 2 has also shown that the quantitative easing reduces the interest spread through the sovereign risk channel, the variations of inflation and output gap are shrunk after implementing the quantitative easing, and the period of remaining at zero lower bound in OMP case is shorter than OMP without QE case.. 28.

(33) 5. Conclusion In this study, we discuss the issue of optimal conventional and unconventional. monetary policy, and the effect of implementing quantitative easing policy by monetary authority. The main results are as follows. First, we show the analytical solution of the amount of asset total purchased under a given demand shock. Second, we present that the quantitative easing reduces the impact of demand shock to real economy, different from previous literatures, the OMP without QE case. Third, under commitment, implementing qualitative easing makes the economy escape from zero lower bound of nominal interest rate sooner than without qualitative easing under commitment, which means the period that nominal interest rate at zero is shorter. Regarding the research in the future, we suggest to conduct a different channel on unconventional monetary policy to compare the result with the sovereign risk in this study. Furthermore, other policy rules could also be considered in optimal unconventional monetary policy.. 29.

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(37) Appendix Appendix A. Derivation of the dynamic IS curve. The Euler equations of two types of households, Eq. (7) and Eq. (11), are derived from their utilities.. We do log-linearize on both Euler equations to obtain. ( ( R. ) )). s  − (ξ − ξ ) c= cts+1 − s −1 Rt − Π t t +1 t t +1 b c= ctb+1 − s −1 t. b t.  − (ξ − ξ −Π t +1 t t +1. (A1). .. By using the weighted combination by population on the two types of households’s linearized Euler equation. = ∑ ς mctm. ∑ς. = m s ,= b m s ,b. m.  m 1 m     ct +1 − s Rt − Π t +1 − (ξt − ξt +1 )  ,. (. ). (A2). the dynamic IS curve is derived out as.  − (ξ − ξ )) . = X t X t +1 − s −1 (ς s Rt + ς b Rtb − Π t +1 t t +1. (A3). Note that X = Yt − Yt E , while Yt E equals 0 which means X t = Yt . t Combing the log-linearized Eqs. (12), (13) and (22), the spread of interest rate is written as.  QE . Rtb+1 − Rt +1= ψ t − αϒ 0 Ω t. (A4). Finally, combing the Eqs. (A3) and (A4) the dynamic IS curve is written as 33.

(38) (.  QE ) − r n − Π  X t X t +1 − σ −1 it + (ς bψ t − ς bαϒ 0 Ω = t t t +1. ). ,. (A5). where rt n =r + (ξt − ξt +1 ) is natural rate and represents a demand shock. 11. Appendix B. Derivation of the relative consumption. The relative consumption is assumed as. ctR =. cts ctb. (B1). .. Combing the log-linearized Eq. (7), (11) and (B1) with Eq.(A4), the relative consumption is written as. T T   QE − ∑ψ  .  σ  αϒ 0 ∑ Ω c= k k = k t= k t   d. R t. d. −1. Appendix C. (B2). Derivation of the loss function. The utility of the households is given as. Ut = =. ∑V. m=s , b. m. (U (c ) − ∫ V (l (h))dh ) m t. 1. 0. m t. U (c ) − ∑ ∑ VV ∫ V (l. m m t = m s ,= b m s, b. 1. m 0. m t. ,. (C1). (h))dh. and we take the second-order Taylor expansion on two terms of Eq. (C1) respectively. 11. Note that we set Rt +1= it − ρ . 34.

(39) to obtain. ∑ς. U (c ) ≈. ∑ς. m m t = m s ,= b m s, b. 1−s m 2   U cY  ctm + ct )  + t.i. p. + O (|| ξ 3 ||) , ( 2  . . 1+ ϕ. 1+ ϕ.  ss ( − 1) . ∑ VVV ∫ V (l (h))dh ≈ V l  Y + 2 Var (Y (h)) + 2 (Y ) +  2(1 + ϕ )  ( 1. m=s , b. m 0. m t. l. (C2). m. . t. t. +t.i. p + O (|| ξ ||. 3. 2. t. ). . . s b.  (ctR ) 2 )   . (C3). By combing the Eqs. (36), (C1), (C2) and (C3), we derived the loss function from welfare function as ∞. Wt = ∑ b t + kU t + k k =0. ∞   s + ϕ ) ς sς bs R 2 1 + ϕ ) Θθ 2 ( ( 1 t 2   )2   ≈ − U cY ∑ bs (ct ) + (Π  ( + ϕ ) (Yt ) + t  2 (1 − Θ )(1 − bΘ ) (1 + ϕ ) t = t0   .. + t.i. p. + O (|| ξ 3 ||). ∞ 1 = − U cY ∑ b t Lt + t.i. p. + O (|| ξ 3 ||) 2 t = t0. The loss function Lt is denoted as. ( ). = Lt λX X t. 2.  )2 + λ R ( c R )2 , + λΠ ( Π t t c. (C4). where. λX= σ + ϕ , λΠ =. (1 + ϕ ) Θθ 2 (1 − Θ )(1 − βΘ ). , λc R =. (s + ϕ ) ς sς bs (1 + ϕ ). .. Finally, substituting the Eq. (B2) into Eq. (C5), the loss function is rewritten as. 35.

(40) 2. Td Td  −1   2 2 QE    + λ R σ  αϒ ∑ Ω − ∑ψ   . Lt λX X t + λΠ Π = 0 t k k c k t= k t  =   . 36. (C5).

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