Leaning against the bubble: Central Bank intervention in Walrasian asset markets

Chang, Chia-ling; Ilomäki, Jukka; Laurila, Hannu Article Leaning against the bubble: Central Bank intervention in Walrasian asset markets Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Chang, Chia-ling; Ilomäki, Jukka; Laurila, Hannu (2021) : Leaning against the bubble: Central Bank intervention in Walrasian asset markets, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 9, Iss. 12, pp. 1-12, https://doi.org/10.3390/risks9120214 This Version is available at: https://hdl.handle.net/10419/258296 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article Leaning against the Bubble: Central Bank Intervention in Walrasian Asset Markets Chia-Lin Chang 1,2,3,* , Jukka Ilomäki 4and Hannu Laurila 4   Citation: Chang, Chia-Lin, Jukka Ilomäki, and Hannu Laurila. 2021. Leaning against the Bubble: Central Bank Intervention in Walrasian Asset Markets. Risks 9: 214. https://doi. org/10.3390/risks9120214 Academic Editor: Mogens Steffensen Received: 12 October 2021 Accepted: 18 November 2021 Published: 1 December 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Applied Economics, National Chung Hsing University, Taichung 402, Taiwan 2Department of Finance, National Chung Hsing University, Taichung 402, Taiwan 3Department of Finance, Asia University, Taichung 41354, Taiwan 4Faculty of Management and Business, Tampere University, FI-33014 Tampere, Finland; [email protected] (J.I.); [email protected] (H.L.) *Correspondence: [email protected] Abstract: The paper presents a two-period Walrasian financial market model composed of informed and uninformed rational investors, and noise traders. The rational investors maximize second period consumption utility from the payoffs of trading risk-free holdings to risky assets in the first period. The central bank reacts directly to asset price movements by selling or buying assets to stabilize the market price. It is found that the intervention makes the risky asset’s market price per share less sensitive to information shocks, which presses the market price towards its average price thus reducing price variance. The informed investors’ prediction coefficient remains unaffected, but that of the uninformed investors is magnified, which cancels out the negative effect on shock sensitivity thus keeping the expected value of the risky asset’s dividend constant. Finally, the introduction of the policy rule does not affect rational investors’ risk per share. A general conclusion is that the central bank’s policy can be regarded as an effective automatic stabilizer of financial markets. Keywords: asymmetric information; CARA; expectations; Walrasian financial markets JEL Classification: D82; E58; G11; G14; G32 1. Introduction The intuition behind the standard “leaning against the wind”, or “leaning against the bubble”, as we call the policy herein, is that if there is evidence of a bubble in the financial market, the central bank can deflate the bubble by raising the market interest rate. Thus, in the case of overpricing, improving financial stability needs tightening of monetary policy. In spite of its intuitively appealing idea, the practical advisability of the policy is controversial. For a long time, there was a widespread opinion that central banks should not intervene because of the difficulty in detecting the bubbles, and because the unintended costs may outweigh the intended benefits of the policy. The financial crisis of 2008 resulted in a more favorable attitude towards the policy, but debate on its usefulness is still ongoing (Ciccarone et al. 2020). Dong et al. (2020) found that the efficacy of the leaning against the bubble policy depends on the nature of the shock and the type of the policy. They also noted that in some cases it is optimal to attach a negative reaction coefficient on asset price movements. This notion motivates the need to examine the efficacy of such policy, where the central bank directly intervenes in the financial market as a powerful market operator, selling and buying assets thus affecting the market price. This paper aims to respond to that need. The paper is a theoretical analysis in the Walrasian framework, where perfect price mechanism makes simultaneously all markets clear. In the basic financial market model, there are rational investors and noise traders (see for example Campbell and Kyle 1993; Wang 1993). The rational investors include informed investors who have private information about the value of the risky asset, and uninformed investors who infer the value Risks 2021,9, 214. https://doi.org/10.3390/risks9120214 https://www.mdpi.com/journal/risks Risks 2021,9, 214 2 of 12 from the asset’s market price. The Walrasian model is a bit problematic in the context of asymmetric information, because the model implies that information about market prices is transferred from informed to uninformed agents. Nevertheless, the framework has been regarded as most reasonable in that context, and it has been widely used in articles published in top journals since the 1970s (see e.g., Grossman and Stiglitz 1980;Admati 1985). In our model, the central bank tackles bubbles by controlling asset prices in the market. This is done by altering the bank’s own holdings of the assets. The policy rule reacts automatically to overpricing or underpricing by making the bank sell or buy risky assets in the market so that the price mechanism corrects any irrational deviations from the average asset value. We investigate what happens, when the central bank enters the financial market with its reactive policy rule. The main findings are the following. The introduction of the central bank’s policy reduces the sensitivity of the risky asset’s market price to information shocks, which presses the market price towards its average thus reducing price variance. The informed investors’ prediction coefficient remains unaffected, whereas that of the uninformed investors is magnified, which cancels out the fall in the sensitivity parameters so that the expected value of the risky asset’s dividend remains unaltered. Upon receiving the market information, the rational investors’ risk per share is not affected by the policy. The conclusion is that the central bank’s market intervention makes the financial market more stable. The remainder of the paper proceeds as follows. Section 2gives a brief literature review, and Section 3presents the basic model. Section 4derives the financial market equilibrium, and Section 5incorporates the central bank’s policy rule into the model. Section 6provides the results of the analyses, that is the implications of the implementation of the policy rule. Section 7concludes and discusses the findings. 2. Literature Review Early opponents of the leaning against the bubble policy include, for example, Bernanke and Gertler (2001), who argued that the central bank should only control inflation. Furthermore, Greenspan (2002,2004) and Posen (2006), who argued that the central bank should focus only on inflation and macroeconomic stability, and that “cleaning up the mess” would be the proper action after a random financial market bubble has burst. White (2009) and Conlon (2015) deemed the policy ineffective because the fundamental value of risky assets is uncertain, and Brunnermeier and Schnabel (2016) referred to historical evidence concerning the difficulty in detecting stock market bubbles with confidence. Svensson (2017) showed that the costs of the policy exceed its benefits by a substantial margin. The advocates of the policy include for example Bean (2004), Roubini (2006) and Yellen (2010), who highlighted the importance of central bank’s activity in tackling market bubbles. Mishkin (2017) provided a review of studies concerning central banks’ reactions to market bubbles after the 2007–2009 financial crisis and concluded that the central bank should intervene earlier rather than later, because the costs of cleaning up the mess policy might become very high. In the paper by Mishkin (2017), leaning against the bubble policy is suitable for tackling credit-driven bubbles, while cleaning up is appropriate if the bubble inflates from irrational exuberance, meaning that investors’ average expectations depart from rational expectations (Allen et al. 2006;Bacchetta and Van Wincoop 2008). In the behavioral finance literature (e.g., Shiller 1981;De Bondt and Thaler 1985), irrational exuberance has been connected to investors’ bounded rationality. Hirshleifer (2015) provided a review of this literature. Grossman and Stiglitz (1980) presented a two-period Walrasian financial market model, where trading leads to market equilibrium with fixed supply of assets. In their model, rational informed investors have noisy private information about the value of a risky asset, and rational uninformed investors infer the asset’s value from the market price. In the two-period model of Hellwig (1980), rational investors with noisy private information allocate their wealth between a risk-free and a risky asset in the first period Risks 2021,9, 214 3 of 12 and consume the accumulated wealth in the second period. Admati (1985) included several risky assets in the model. Mendel and Shleifer (2012) elaborated the two-period model by including noise traders, assuming also that the informed investors receive another private signal in the second period. Gali (2014) incorporated a standard leaning against the bubble policy into the analysis. In his overlapping generations model, the market price of the risky asset consists of fundamental value and rational bubble components, and the bubble is passed to the next generation developing with the risk-free interest rate. The conclusion was that the policy does not operate because the bubble component does not have a discount factor. Ilomäki and Laurila (2021) found that leaning against the bubble is effective if uninformed investors obey their “animal spirits” and discount their future anticipations. The Keynesian concept refers to instincts, proclivities and emotions, which bypass strict rationality in rapid decisions. In his thorough cost–benefit analysis, Svensson (2017) showed that the costs of standard leaning against the bubble policy clearly exceed its benefits mainly due to the unintended effects of the policy. Therefore, it should be obvious that a more direct intervention in the financial market would cause fewer unintended consequences. A natural leaning against the bubble policy rule (or reaction function) would be to buy risky assets in exchange of riskless alternatives when the asset price is low and sell it when the price is high. We examine the effects of such a policy rule connected to market price movements around the average market price. 3. The Basic Model The model builds on Grossman and Stiglitz (1980), Mendel and Shleifer (2012), and Ilomäki and Laurila (2018,2020). In the two-period Walrasian financial market model, there is a set [0, 1] of rational constant absolute risk-averse investors, divided into informed investors with the share µ and uninformed investors with the share 1 −µ . Both allocate their investments between risk-free and risky assets in the first period to maximize their consumption utility from the investment payoffs in the second period. The constant absolute risk aversion (CARA) exponential utility function u(c) = −e−c is applied with a constant absolute risk aversion coefficient equal to one. The numeraire risk-free asset pays one unit of consumption in the second period, and the risky asset pays a random dividend e D∼ND,σ2 D in terms of consumption units in the second period. The market price of the risky asset is Pper share, expressed in terms of consumption. The informed investors observe a noisy private signal: e s=he D−Di+ε , where D is the unconditional expectation of the actual dividend value, and ε∼N0, σ2 ε . The uninformed investors form rational expectations on e D from the observation of P. In the market, there is also a measure 1 of correlated noise traders whose net demand of the risky asset is: e N∼N0, σ2 N. The initial allocation of the risky asset is: µaI+(1−µ)aU+aN=A(1) where aI , aU and aN denote the possessions of the informed investors, uninformed investors and noise traders, respectively, and Adenotes the fixed total amount of the shares. All traders have an identical endowment a0 of the risk-free asset which they can trade against the risky asset at a unit price in the first period. The informed investors form their expectation of e D based on their private signal e sas follows: Ehe De si=D+β[e s](2) where β is the informed investors’ statistical prediction coefficient. According to the OLS method, the prediction coefficient is derived as: β=cov (e D,e s) var(e s) . Noting that: cov(e D , ε) = 0, the prediction coefficient reads: Risks 2021,9, 214 4 of 12 β=σ2 D σ2 D+σ2 ε (3) where σ2 D is the variance of the dividend and σ2 D+σ2 ε=σ2 s is the variance of the noisy signal e s . Obviously, 0 < β < 1, which means that the closer β is to zero the vaguer are the predictions. The variance of the informed investors’ prediction error is then calculated as σ2 I=var(e D)var(ε) var(e s)so that: σ2 I=σ2 Dσ2 ε σ2 D+σ2 ε (4) Equation (4) describes the informed investors’ prediction uncertainty upon given information and can be interpreted as the ex post risk of the risky asset per share. The uninformed investors’ expectation on the dividend value per share e D is based on the observed market price Plike: Ehe DPi=D+γP−P where γ is the uninformed investors’ prediction coefficient and P is the unconditional expectation of P, that is the average price of the risky asset. The formation of the market price depends on the informed investors’ private signal e s and the net actions of noise traders e N . Recalling that e s=he D−Di+ε , the expected value of both e s and e N is zero. A tentative assumption is that Pand the two random variables are correlated like: P=P+be s+ce N(5) where P is the constant term and parameters band crefer to the sensitivity of the market price to exogenous shocks in e s and e N , respectively. Thus, the uninformed investors’ expectation of e Dreads: Ehe DPi=D+γhbe s+ce Ni(6) Referring to Equations (5) and (6), the uninformed investors’ prediction coefficient under given information comes from: γ=bcov(e s,e D) var(be s)+var(ce N), which gives: γ=bσ2 D b2σ2 D+σ2 ε+c2σ2 N (7) The exact value of γ depends on band c, which are so far unknown but derivable from the market equilibrium. The variance of the uninformed investors’ statistical prediction error is calculated as: σ2 U=var(e D)var(bε)+var(e D)var(ce N) var(be s)+var(ce N)=σ2 Db2σ2 ε+σ2 Dc2σ2 N b2(σ2 D+σ2 ε)+c2σ2 N , which can be written as: σ2 U=θσ2 D(8) where θ=b2σ2 ε+c2σ2 N b2σ2 D+σ2 ε+c2σ2 N (9) Equation (8) presents the uninformed investors’ uncertainty of prediction under given information, which can be interpreted as the ex post risk per share. Please note that while 0 < θ < 1 clearly holds, the exact value of θ depends again on the sensitivity parameters b and c. Risks 2021,9, 214 5 of 12 4. The Financial Market In the financial market, rational investors optimize their holdings of risky and risk-free assets to maximize expected utility from consumption. An informed investor’s CARA optimization problem reads: Ehue CIe si=−e−xIE[e D|e s]−[a0−(xI−aI)P]+ x2 I 2σ2 I(10) where the informed investor’s consumption in period 2 is e CI=xIe D+ [a0−(xI−aI]P , xI is the net demand for the risky asset per share, a0 is the endowment of the risk-free asset and aI is the initial holdings of the risky asset per share. Ehe De si is the expected dividend per share, and x2 I 2σ2 I is the risk element under the assumption of normally distributed noise in the signal e s. Taking the first-order condition with respect to demand xIproduces: Ehe De si−P−xIσ2 I=0 and solving for xIleads to: xI=Ehe De si−P σ2 I (11) Equation (11) says that the informed investor’s demand of the risky asset equals the gain from investing in the risky asset divided by the variance of the informed investors’ prediction error. An uninformed investor’s CARA optimization problem reads: Ehue CUPi=−e−xUE[e D|P]−[a0−(xU−aU)P]+ x2 U 2σ2 U(12) where the uninformed investor’s consumption in period 2 is: e CU=xUe D+[a0−(xU−aU)]P , xU is the net demand for the risky asset per share, a0 is the endowment of the risk-free asset and aU is the initial holdings of the risky asset per share. The first-order condition with respect to xUis: Ehe DPi−P−xUσ2 U=0 and solving for xUgives: xU=Ehe DPi−P σ2 U (13) By Equation (13), the uninformed investor’s demand of the risky asset equals the gain from investing in the risky asset divided by the variance of the uninformed investors’ prediction error. As the market supply of the risky asset is given in Equation (1), the market clearing condition for the risky asset per share is: µxI+(1−µ)xU+e N=A(14) which, recalling Equations (2), (5), (6), (11) and (13), turns to: µD+βe s−(P+be s+ce N) σ2 I+(1−µ)D+γ(be s+ce N)−(P+be s+ce N) σ2 U+e N=A(15) In a Walrasian equilibrium where demand equals supply, the sum of the constants on the left-hand side of Equation (15) must equal the constant supply Aof the risky asset on the right-hand side, and the sum of the coefficients of e s and e N must be equal to zero. Therefore, P ,band cin Equation (5) can now be solved. To calculate P , pick the constant Risks 2021,9, 214 6 of 12 terms from Equation (15), manipulate, and get: µD−Pσ2 U+(1−µ)σ2 ID−P=σ2 Iσ2 UA . Solving for Pgives: P=D−σ2 Iσ2 U µσ2 U+(1−µ)σ2 I A(16) Equation (16) says that the average price P equals the unconditional expectation on the dividend minus discount, which consists of the risk term multiplied by A. Please note that A> 0 means that D > P . Note also that a very big value of Awould make P negative, which is not reasonable. This is a tolerable artifact of the otherwise most useful CARA utility function (Mendel and Shleifer 2012). On the other hand, A< 0 would make D<P . This is reasonable because the investors would then be short the asset on average, thus requiring a higher Pto compensate for the risk in borrowing shares to sell. Moreover, if information were symmetric, µ = 1 or 0, the average discount would be like: −σ2 IA or −σ2 UA, respectively. To calculate the values of the sensitivity parameters band c, use Equation (15) to pick the coefficients of e sand e N. After manipulation: b=µβσ2 U µσ2 U+(1−µ)(1−γ)σ2 I (17) c=σ2 Iσ2 U µσ2 U+(1−µ)(1−γ)σ2 I (18) The signs of the sensitivity parameters band cneed closer scrutiny. By Equation (8), σ2 U depends on θ , which in turn depends on band cby Equation (9). To investigate the interdependency, divide bin Equation (17) by cin Equation (18) to get b c=µβ σ2 I which, after using Equations (3) and (4), reduces to: b c=µ σ2 ε (19) Taking squares on both sides yields: b2=c2µ2 σ4 ε , and substitution into Equation (9) gives: θ=µ2σ2 ε/σ4 ε+σ2 N µ2((σ2 D+σ2 ε)/σ4 ε)+σ2 N . Multiply by σ4 ε σ4 εand manipulate to write: θ=µ2σ2 ε+σ4 εσ2 N µ2σ2 D+σ2 ε+σ4 εσ2 N (20) Equation (20) confirms that 0 < θ < 1. As it is given by underlying parameters, σ2 U in Equation (8) is also given by them. Please note that without the noise trader effect: θ=σ2 ε σ2 D+σ2 ε . Then, recalling Equations (4) and (8), θσ2 D=σ2 I=σ2 U . Hence, Equation (20) demonstrates that the difference between the informed and uninformed investors’ variance of prediction errors is due to noise trading. Moreover, Equations (17) and (18) show that the signs of band cdepend on γ , and Equation (7): γ=bσ2 D b2(σ2 D+σ2 ε)+c2σ2 N says that γ depends on band c. Write Equation (7) as γb2σ2 D+σ2 ε+γc2σ2 N=bσ2 D , and use Equations (17) and (18) to substitute for band c. This yields, after some manipulation: γµ2β2σ4 Uσ2 D+σ2 ε+γσ4 Iσ4 Uσ2 N µσ2 U+(1−µ)(1−γ)σ2 I =µβσ2 Uσ2 D µσ2 U+(1−µ)(1−γ)σ2 I Multiply through by µσ2 U+(1−µ)(1−γ)σ2 I , use: βσ2 D+σ2 ε=σ2 D from Equation (3) and manipulate to obtain γµ2βσ2 Dσ4 U+µβσ2 Uσ2 D(1−µ)σ2 I+σ4 Iσ4 Uσ2 N=βµσ2 Uσ2 D Risks 2021,9, 214 7 of 12 µσ2 U+(1−µ)σ2 I . Solve for γ to get γ=µβ[µσ2 U+(1−µ)σ2 I] µβ[µσ2 U+(1−µ)σ2 I]+(σ4 Iσ2 Uσ2 N)/σ2 D . Finally, use σ2 D= σ2 U/θfrom Equation (8) and write: γ=µβµσ2 U+(1−µ)σ2 I µβµσ2 U+(1−µ)σ2 I+θσ4 Iσ2 N (21) Equation (21) confirms that 0 <γ< 1 (note that σ2 N→0 makes γ→1 ). Therefore, b and cin Equations (17) and (18) are clearly positive. Plug Equation (21) into (17), manipulate and get: b=βµσ2 Uµβµσ2 U+(1−µ)σ2 I+θσ4 Iσ2 N µσ2 Uµβµσ2 U+(1−µ)σ2 I+θσ4 Iσ2 N+(1−µ)σ2 Iθσ4 Iσ2 N (22) Likewise, plug Equation (21) into (18) and get: c=σ2 Iσ2 Uµβµσ2 U+(1−µ)σ2 I+θσ4 Iσ2 N µσ2 Uµβµσ2 U+(1−µ)σ2 I+θσ4 Iσ2 N+(1−µ)σ2 Iθσ4 Iσ2 N (23) Finally, substitute Equations (16), (22) and (23) into Equation (5), which leads to: P=e D−σ2 Iσ2 U µσ2 U+(1−µ)σ2 I A+µβσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N} µσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N}+(1−µ)σ2 Iθσ4 Iσ2 Ne s +σ2 Iσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N} µσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N}+(1−µ)σ2 Iθσ4 Iσ2 Ne N (24) Equation (24) presents the formation of the market price of the risky asset. Since the sensitivity parameters band care unambiguously positive, the rational investors’ market reactions follow the signs of the shocks in e sand e Nand the market price responds accordingly. 5. The Central Bank To incorporate the central bank into the model, suppose that it launches the leaning against the bubble policy in the financial market. We use the acronym LAB for the policy, which reacts to asset price movements by buying the risky asset (in exchange of the riskless asset) when its price is too low and selling it when the price is too high. The policy rule is attached to the deviations of the market price from the average price of the risky asset. If the market price is higher than the average price, the rule implies a reduction of the bank’s holdings of the risky asset so that the increased market supply induces the market price to fall, and vice versa. The central bank’s policy function reads: xQ=q0−q1P−P(25) where xQ is the central bank’s net demand per share of the risky asset in the first period, q0 is its net holdings of the risky asset when the market price Pequals its average value P , and q1 > 0 is a sensitivity parameter that tells how much the central bank sells or buys in the asset market if Pdeviates one unit per share from P . Please note that q0 and q1 are independent of each other. Upon introduction, rational investors take the policy rule it into account in their market decisions (Sargent and Wallace 1975). The market clearing condition becomes: µxI+(1−µ)xU+e N+xQ=A(26) Recalling Equations (5), (15) and (25), the market clears when: µD+βe s−(P+be s+ce N) σ2 I+(1−µ)D+γ(be s+ce N)−(P+be s+ce N) σ2 U+e N+q0 −q1be s+ce N=A (27) Risks 2021,9, 214 8 of 12 The constant P and the sensitivity parameters band cunder the LAW policy can now be solved by repeating the same procedure as in the policy-free case above. This produces: P=D+σ2 Iσ2 U µσ2 U+(1−µ)σ2 I (q0−A)(28) ˆ b=µβσ2 U µσ2 U+(1−µ)(1−ˆ γ)σ2 I+q1σ2 Iσ2 U (29) ˆ c=σ2 Iσ2 U µσ2 U+(1−µ)(1−ˆ γ)σ2 I+q1σ2 Iσ2 U (30) The uninformed investors’ prediction coefficient is now denoted by ˆ γ . It can be calculated by substituting ˆ b and ˆ c from Equations (29) and (30) into Equation (7): γ= bσ2 D b2(σ2 D+σ2 ε)+c2σ2 N , which yields, after some manipulation, ˆ γµ2β2σ2 Uσ2 D+σ2 ε+σ4 Iσ2 Uσ2 N+ µβσ2 D(1−µ)σ2 I=µβσ2 Dµσ2 U+(1−µ)σ2 I+q1σ2 Iσ2 U . Solving for ˆ γ , and using Equation (8) gives ˆ γ=µβ[µσ2 U+(1−µ)σ2 I+q1σ2 Iσ2 U] µβ[µθβ(σ2 D+σ2 ε)+(1−µ)σ2 I]+θσ4 Iσ2 N . Use Equation (3) in the first term in the denominator to substitute for βσ2 D+σ2 ε=σ2 D and then use Equation (8) to substitute for θσ2 D=σ2 U. This produces: ˆ γ=µβµσ2 U+(1−µ)σ2 I+q1σ2 Iσ2 U µβµσ2 U+(1−µ)σ2 I+θσ4 Iσ2 N (31) which is clearly positive. Plug Equation (31) into Equation (29) and get, after manipulation: ˆ b=µβσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N} µσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N}+(1−µ)σ2 Iθσ4 Iσ2 N+q1σ2 Iσ2 U(µ2βσ2 U+θσ4 Iσ2 N)(32) which is positive. Likewise, plug Equation (30) into Equation (29), manipulate and obtain: ˆ c=σ2 Iσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N} µσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N}+(1−µ)σ2 Iθσ4 Iσ2 N+q1σ2 Iσ2 U(µ2βσ2 U+θσ4 Iσ2 N)(33) which is also positive. Substituting Equations (32) and (33) into Equation (5), the market price formula under the LAB policy reads: P=e D+σ2 Iσ2 U µσ2 U+(1−µ)σ2 I (q0−A) +µβσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N} µσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N}+(1−µ)σ2 Iθσ4 Iσ2 N+q1σ2 Iσ2 U(µ2βσ2 U+θσ4 Iσ2 N)e s +σ2 Iσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N} µσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N}+(1−µ)σ2 Iθσ4 Iσ2 N+q1σ2 Iσ2 U(µ2βσ2 U+θσ4 Iσ2 N)e N (34) 6. Results Proposition 1. The introduction of the LAB policy makes the risky asset’s market price per share less sensitive to information shocks. Proof. To examine the effects of the central bank’s intervention on the price sensitivity parameters band c, compare them without and with the policy. The impact of the intervention on bcan be seen by comparing Equations (22) and (32). Postulating that b>ˆ breads: βµσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N} µσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N}+(1−µ)σ2 Iθσ4 Iσ2 N >µβσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N} µσ2 U{µβ[µσ2 U+(1−µ)σ2 I]+θσ4 Iσ2 N}+(1−µ)σ2 Iθσ4 Iσ2 N+q1σ2 Iσ2 U(µ2βσ2 U+θσ4 Iσ2 N)