Precautionary Saving and Liquidity Shortage

Citation: He, G.; Hu, Z Precautionary Saving and Liquidity Shortage Sustainability 2023 , 15 , 2373 https://doi.org/10.3390/su 15032373 Academic Editors: Bruce Morley and Alistair Hunt Received: 30 December 2022 Revised: 24 January 2023 Accepted: 24 January 2023 Published: 28 January 2023 Copyright: © 2023 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/) sustainability Article Precautionary Saving and Liquidity Shortage Guohua He and Zirun Hu * Economics and Management School, Wuhan University, Wuhan 430072, China * Correspondence: huzirun@whu.edu.cn Abstract: Most of the canonical macroeconomic models simulate liquidity anomalies by changing the economic fundamentals or adding massive financial shock to firms’ collateral constraints, but a few facts somehow tell a different story. Instead of relying on the exogenous shocks, we introduce uncertainty into an otherwise classical liquidity framework and try to answer what worsens the aggregate liquidity in the absence of exogenous simulations and what a firm dynamics and financing strategy would be. Our analysis shows that (1) uncertainty induces agents to make decisions under the worst-case scenario and hence generates a unique expectation threshold that drags market (or firms) liquidity from sufficiency to insufficiency even without any shock or economic changes. (2) Precautionary saving occurs before the real liquidity shortage as the expectation shifts, causing firms to secure external financing by raising the equity issuing price and hoarding liquid assets, such as fiat money, against liquidity tightening. (3) To achieve liquidity stability and sustainability, an extra mathematical constraint is supplemented for the uniqueness and the existence of equilibrium under uncertainty. Other properties of firms’ intertemporal allocations, such as the bid-ask spread and return of holding of the illiquid asset, are derived. Moreover, some approaches for further empirical research are discussed Keywords: liquidity shortage; uncertainty; firm dynamics; precautionary saving 1. Introduction Liquidity, as the core factor in financial activities, has received a lot of academic attention, especially since the onset of the subprime crisis in 2008. The severe shortage of liquid assets during the subprime crisis prompted multiple authorities to inject a massive amount of liquidity into the market, in the forms of bailouts, quantitative easing, etc. The instability and unsustainability of liquidity has delivered a huge blow to the global economy and brought liquidity to the forefront of policy debate. To characterize and replicate the crisis theoretically, one of the pivotal studies, Kiyotaki and Moore [ 1 ], incorporates a firm’s liquidity with the standard real business cycle model, opening the gate for macroeconomics to explore firm dynamics and financing strategy along with the change of aggregate liquidity status. Be that as it may, such a perfect model, similar to many other canonical macroeconomic models, blames exogenous shocks for the liquidity anomalies instead of agents’ endogenous decision-making; therefore, it is limited in explaining some specific facts in reality. One robust example is the Knightian shares documented in Bachmann et al. [ 2 ]’s study: firms’ uncertainty/Knightian shares in Greece spiked up shortly after the victory of the Syriza party on 25 January 2015, and peaked when the repayment to the IMF loan was overdue on June 30. Neither fundamental change nor prominent shock was found during this period, but uncertainty was reflected jointly in firm planning and observed risk premia in financial markets. With this in mind, we would like to ask two pivotal questions throughout the paper: What worsens the aggregate liquidity in the absence of exogenous simulation? How does a firm act and cause this effect under uncertainty? These two unresolved issues, according to our perspective, happen to be the primary elements to better understand the volatility of liquidity Sustainability 2023 , 15 , 2373. https://doi.org/10.3390/su 15032373 https://www.mdpi.com/journal/sustainability

Sustainability 2023 , 15 , 2373 2 of 15 A few studies closely related to our research have anatomized the determinants of liquidity both theoretically and empirically. Among the theoretical literature, the early research concerning liquidity is mainly focusing on a firm’s asset structure such as Bolton et al. [ 3 ], but the subprime crisis of 2008 was the turning point and changed the perspective from micro to macro. For example, Kiyotaki and Moore [ 1 ] survey the firm dynamics using the liquidity constraint in external financing, it turns out to be the solid proof of how agent’s decision-making changes the aggregate liquidity status. Del Negro et al. [ 4 ] extend Kiyotaki and Moore [ 1 ]’s work by adding the nominal rigidity to a more comprehensive household sector and further examining the efficiency of quantitative easing around the zero lower bound. In contrast, Guerron-Quintana et al. [ 5 ] abstract Kiyotaki and Moore [ 1 ]’s setup to discuss the hysteresis/superhysteresis of the economic growth in the USA after the asset bubble burst. Similarly, Brunnermeier and Pedersen [ 6 ] point out that the optimistic belief accelerates the liquidity expansion and raises the associated asset risk, while the pessimistic belief is the final push to collapse; together, these two opposite components constitute financial cycles. Some other related examples in the literature include but are not limited to Shi [ 7 , 8 ], Ajello [ 9 ], Brunnermeier and Oehmke [ 10 ], Lorenzoni [ 11 ], Holmstrom and Tirole [ 12 ], Gertler et al. [ 13 ], Gertler and Karadi [ 14 ], and Geanakoplos [ 15 ]. For the empirical literature, Acharya and Merrouche [ 16 ] document that the liquidity demand of large settlement banks in the UK experienced a 30% increase before the subprime crisis and that strong precautionary nature cause the liquidity demand to rise on days of high payment activity and for banks with greater credit risk. Ashcraft et al. [ 17 ] show similar evidence but with the US data. Chiu et al. [ 18 ] examine the relationship between funding liquidity and equity liquidity during the subprime crisis by using the index and ETFs; the empirical results show that a higher degree of funding of illiquidity leads to an increase in bid–ask spread and a reduction in both market depth and net buying imbalance. In a more broad scope, Belke et al. [ 19 ] apply the liquidity shock to the open economy and emphasize that a global liquidity shock leads to a rise in consumer and global house prices, where the latter reaction is more pronounced. Some studies survey liquidity risk and its association with other factors. Cao and Petrasek [ 20 ] think that abnormal stock returns during liquidity crises are strongly negatively related to liquidity risk and that the degree of informational asymmetry and the ownership structure of the firm are the main reasons The strategy under liquidity risk is also shown by Cao et al. [ 21 , 22 ], in which the authors explore the hedge-fund and mutual-fund managers’ timing abilities by examining whether they can time market liquidity through adjusting their portfolios’ market exposures as aggregate liquidity conditions change. In addition, Wegener et al. [ 23 ] examine the yields of traditional Pfandbriefe and Jumbo Pfandbriefe with different maturities where the yield spreads between these two types of German covered bonds can be considered as pure liquidity premia. The authors find that the yields are fractionally co-integrated before and after the crisis, but the degree of integration of the spread increases strongly during the crisis. Except for these macro level studies, a few specific types of research concerning firm level liquidity and firm dynamics are also worth noting. Under the typical agency problem, Lambrecht and Myers [ 24 , 25 ] show that managers tend to smooth payouts in consideration of smoothing the rents they draw from the firm, smoothing can either be performed by lending (capital expansion) or borrowing (leverage). The basic reason for this behavior is that managers are risk averse and the authors provide a few demonstrations of changing managers’ preferences. Similarly, Hoang and Hoxha [ 26 , 27 ] provide a more solid proof of smoothing behavior using the empirical results of the US, China, and Taiwan; the authors find that firms use debt and investment to smooth a large fraction of shocks to the net income to keep payouts less variable Two comments can be presented from these related examples in the literature. First, the recent empirical results are putting forward evidence about the precautionary hoarding of liquidity in a way that most of the canonical macroeconomic models cannot explain. For example, the system in the macroeconomic model is staying at an efficient equilibrium (or

Sustainability 2023 , 15 , 2373 3 of 15 steady state) if no perturbation occurs, implying agents will not suddenly and willingly choose to hoard liquidity for precaution in the absence of exogenous shocks. However, Bachmann et al. [ 2 ] and Acharya and Merrouche [ 16 ] depict a contradicted picture. Although a few studies in macro finance begin to emphasize the expectation-driven element, the associated modeling, often known as the sunspot, can still be regarded as an exogenous shock, such as the bank run settings in Gertler et al. [ 28 ]. Second, tractability could still be a main issue for macroeconomic models, especially the macro finance models, while concerning precautionary hoarding/saving. Using Maxted [ 29 ] as an example, the diagnostic expectation, an extrapolative expectation that overreacts to noise and may lead to a precautionary behavior, is introduced to He and Krishnamurthy [ 30 ]’s macro finance model, but the basic solution to the model is numerical computation. Despite the numerical method being a good and efficient way to solve the precautionary saving puzzle, the analytical solution is always indispensable for putting the mechanism in a more concrete perspective and helping us better understand the reason why firms adopt these strategy rules/choices. These two comments are in line with the two unresolved questions that were asked at the beginning This paper contributes to macro-finance theory; the questions of our topic are responded to with a modified liquidity framework where the investment chance is ambiguous. Specifically, we replace the exogenous shock in Kiyotaki and Moore [ 1 ] with the endogenous uncertainty and emphasize that the model’s economy is without shocks and fundamentals changes. Meanwhile, we solve the modified liquidity framework analytically and collect all the necessary conditions to judge and characterize the firm dynamics and individual financing decisions and further anatomize why these factors can cause precautionary saving and liquidity shortage. We discover a few broad conclusions upon our specific settings: (1) Firms shift their expectations due to the existence of uncertainty and the model endogenously generates a unique expectation threshold that drags the aggregate liquidity from sufficiency to insufficiency, implying the market could spontaneously experience a liquidity shortage under uncertainty. This theoretical result matches Bachmann et al. [ 2 ]’s evidence that risk premia rose in Greece before the true debt crisis unfolded (2) The shifted expectation induces firms to secure external financing by raising the equity issuing price and hoarding liquid assets, such as fiat money, as precautionary saving against liquidity tightening, which determines firms’ precautionary saving behaviors prior to the real liquidity shortage. In echoing Acharya and Merrouche [ 16 ] and Ashcraft et al. [ 17 ], this mechanism explains the spikes of liquidity demand in the UK and the US before the subprime crisis to some extent. (3) Our theory also finds that the model generates a disequilibrium under some certain feasible conditions, where the aggregate liquidity is no longer stable and sustainable. To avoid this instability and unsustainability, an extra constraint is supplemented. Other properties about firms’ intertemporal allocation, such as the bid-ask spread and return of holding of the illiquid asset, are derived 2. Model Setup To explore how precautionary saving and liquidity shortage rises in the absence of exogenous shocks, we introduce the uncertainty to Kiyotaki and Moore [ 1 ]. Basic framework . Assume an infinite-horizon discrete-time economy with firms and workers, the only commodity is produced by firms, and the inputs of this production include social capital and workers’ labor. As for capital, it can be separated into two types according to the degree of liquidity: one is fully liquid fiat money, while the other is equity that is issued by firms for the sake of external financing. In summary, the firms’ utility is provided by: E t ∞ ∑ s = t βs − t u ( c s ) where E t is the expectation operator, β is the objective discounter with 0 < β < 1, and the single period utility has the form of u ( c s ) = ln c s .

Sustainability 2023 , 15 , 2373 4 of 15 Moreover, the production function of firms is provided by: y t = A t k γ t ` 1 − γ t where y t , k t , and ` t are, respectively, the output, capital, and labor, A t denotes for the industrial technology, and γ is input ratio between capital and labor. Obviously, due to the the capital held by firms, the firm’s profit can be expressed as: y t − w t ` t = r t k t Intuitively, w t ` t is the aggregate wage that firms need to pay to the labor force, while r t k t is the aggregate investment revenue on production. One thing should be further noted is that capital depreciates over time: k t + 1 = λ k t + i t , where we assume the depreciation rate is 1 − λ > 0 and i t is the new investment in capital per period In normal economic exchanges, firms are allowed to implement external financing for issuing new equities or reselling old ones, the borrowing constraint is valid if a firm decides to issue a certain amount of equities to the market: prepay the 1 − θ ratio of capital for each unit of external financing as collateral and θ ∈ ( 0, 1 ) . Moreover, the equity is supposed to be partially liquid, so that only a φ ∈ ( 0, 1 ) ratio of old equities can be resold in each period. In contrast, the fully liquid asset in our economy is fiat money that has a total amount of M and without any profit revenue by itself. Assume the asset portfolio of each firm at time t consists of money m t (which can be 0 in equilibrium), the newly issued equities n f t , holdings of other firm’s old equities n o t , and the total net worth n t , then the balance sheet of an arbitrary firm is provided by n t = n o t + k t − n f t and the corresponding liquidity constraints are: n t + 1 ≥ ( 1 − θ ) i t + ( 1 − φ ) λ n t , (1) m t + 1 ≥ 0 (2) To obtain more of an insight, ( 1 ) can be treated as the liquidity constraint throughout asset accumulation: the LHS of this equation is the firm’s net worth stock (saving or income of each period), while the RHS is the prepay plus unsold equities (expenditure of each period); the LHS should surpass the RHS. Moreover, ( 1 ) somehow reflects the market liquidity status depending on whether the equation is binding or not, that is, the excessive liquidity in each firm disappeared whenever the equation was binding, meaning that the firm has no extra asset to perform anything else. Likewise, the money holding m t can be positive when firms face a liquidity shortage. To characterize these traits in detail, let the prices for equity and money as q t , p t , then the cash flow of each firm can be expressed as: c t + i t + q t ( n t + 1 − i t − λ n t ) + p t ( m t + 1 − m t ) = r t n t , (3) where n t + 1 − i t − λ n t and m t + 1 − m t denote the change of assets Unlike the firms, workers have the utility such that E t ∞ ∑ s = t βs − t U c 0 s − ϕ 1 + v ( ` 0 s ) 1 + v , where we use the prime symbol as the superscript to denote workers’ variables, ϕ > 0 is the weight of labor and 1 + v > 0 is the reverse of the elasticity of the labor supply. Compare the settings of the firms, the budget constraint of an arbitrary worker is provided by c 0 t + q t ( n 0 t + 1 − λ n 0 t ) + p t ( m 0 t + 1 − m 0 t ) = w t ` 0 t + r t n 0 t .

Sustainability 2023 , 15 , 2373 5 of 15 Moreover, a worker becomes redundant at both the liquidity equilibrium and illiquidity equilibrium according to Kiyotaki and Moore [ 1 ]; thus, the rest of the paper no longer focuses on the behavior of workers Uncertainty/Ambiguity . To introduce uncertainty to the liquidity framework, we assume that an arbitrary firm in each period has the chance of π to restock its capital, i.e., invest its capital with i t . Meanwhile, the agent has imperfect information about investment probability but infers it using the liquidity market status, implying that the investment probability π each firm faces depends on the finite state space and varyies across time. To a single firm, specify π ( j ) t = Pr ( π t + 1 = j | n t ) as the conditional probability that the equity is n t at t and the investment probability is j at t + 1. Similarly, specify π ( j ) t t = Pr ( π t = j | n t ) as the conditional probability that the equity is n t and investment probability is j at t . Using the Bayesian rule, it follows that π ( j ) t + 1 t + 1 = f ( n t + 1 − n t , j ) π ( j ) t ∑ i f ( n t + 1 − n t , i ) π ( i ) t where f ( n t + 1 − n t , j ) = 1 √ 2 πσ exp − ( n j , t + 1 − n j , t − µ ) 2 2 σ 2 is the density of the net worth difference when it is at the state of j . Moreover, the transition path of the investment probability, π ( j ) t t → π ( i ) t , satisfies the Markov process such that π ( i ) t + 1 = N ∑ j = 1 Π ji π ( j ) t + 1 t + 1 , thus the posterior probability is provided by π ( i ) t + 1 = B i ( n t + 1 − n t , π t ) = ∑ N j = 1 Π ji f ( n t + 1 − n t , j ) π ( j ) t ∑ i f ( n t + 1 − n t , i ) π ( i ) t Moreover, incorporating these settings with the firm’s utility function yields the utility under uncertainty such that: U t = ln c t + β 1 − η ln E π exp ( 1 − η ) E U t + 1 , (4) where E π is the expectation over the distribution of π and η is the ambiguity aversion parameter. Note that, given η → 1, the utility becomes U t = ln c t as the second term approaches 0; while, given η → ∞ , the utility is: U t = ln c t + β min π E U t + 1 , (5) that is, the classical ambiguity function where the agent makes their decision under the worst case scenario. To seek the analytical solutions to our model, we only focus on the case of ( 5 ) throughout the theoretical analysis Equilibrium conditions . According to Kiyotaki and Moore [ 1 ], the basic liquidity framework has two types of equilibrium, one is the liquidity equilibrium (LE) where fiat money serves no purpose, that is, p t = 0, q t = 1. The other one is the illiquidity equilibrium (IE) where firms need to preserve money to survive the time of liquidity shortage; it follows that p t > 0, q t > 1, as money becomes valuable. We begin with the LE conditions. First, note that the aggregate capital in the economy equals the summation of firms’ saving/net worth K t + 1 − λ K t = I t = r t K t − C t ,

Sustainability 2023 , 15 , 2373 6 of 15 combining it with ( 1 )–( 3 ) under the condition of p t = 0, q t = 1, a little computation yields r t n t − c t > ( 1 − θ ) i t − φ t λ n t ⇒ π ( r t K t − C t ) = π I t > ( 1 − θ ) I t − φ t λπ K t ⇒ π ( 1 − λ ) K > ( 1 − θ )( 1 − λ ) K − φλπ K ⇒ π ( 1 − λ ) > ( 1 − θ )( 1 − λ ) − φλπ , (6) which is the core of the LE condition and also the criterion of the first best efficiency. To obtain the feasible policy recommendations for reality, we summarize the properties of ( 6 ) using Remark 1 . Remark 1. The first best efficiency can be achieved by: (1) given φ ∗ ≥ 0 > − 1 − λ λ for θ = 1 or φ ∗ > ( 1 − λ )( 1 − π ) πλ for θ = 0 ; (2) given θ ∗ ≥ 0 > 1 − π 1 − λ for φ = 1 or θ ∗ > 1 − π for φ = 0 Intuitively, Remark 1 characterizes the first best efficiency from both the resellablity constraint and the prepay collateral. For instance, given θ ∗ as the policy parameter, then, for any θ ∗ ∈ [ 0, 1 ] , the LE will be realized whenever φ is sufficiently big (say φ = 1). On the contrary, the firm may not resell enough old equities to back up its financing need if φ is sufficiently small, thus the only way to realize the LE is to lower the prepay/collateral ratio to 1 − θ ∗ < π . The reasoning and logic are very much the same at the perspective that φ ∗ is the policy parameter. To visualize these two efficiency boundaries, Figure 1 presents a numerical illustration 标注 1 ( 第一最优边界 ) 为令均衡达到第一最优，需满足两类边界条件 (1) 给定 θ = 1 存在 ϕ ∗ ≥ 0 > − 1 − λ λ ，或给定 θ = 0 存在 ϕ ∗ > (1 − λ )(1 − π ) πλ ； (2) 给定 ϕ = 1 存在 θ ∗ ≥ 0 > 1 − π 1 − λ ，或给定 ϕ = 0 存在 θ ∗ > 1 − π 。 标注 1 分别从两方面刻画了第一最优的边界，具有一定的调控政策内涵：其一，给定融资垫头 比例 θ ∗ 为可控参数，当 ϕ 足够大时，如 ϕ = 1 ，此时任意 θ ∗ ∈ [0 , 1] 都可保证经济流动性充裕，故此 时融资垫头并不会改变经济的效率配置；而当 ϕ 较小时，如极端情形 ϕ = 0 ，厂商无法通过出售所 持有的外部权益来支付新进投资的融资垫头，则为保证经济效率此时应降低垫头，满足 1 − θ ∗ < π 。 因此，融资垫头比例 θ ∗ 可视为政府投资补助，当市场流动性紧缺时，政府应通过结构性投资帮助 厂商缓解垫头支付压力，从而撬动市场的流动性派生 10 ；而当经济中流动性泛滥时，政府可以通过 增大厂商外部融资垫头或税收的方式削弱厂商的外部资金获取，从而收缩金融流动性。这一性质可 由图 2 -(a) 刻画。其二，与上述分析类似，给定权益变现粘性 ϕ ∗ 为可变参数，当融资垫头要求足 够小时，如 θ = 1 ，市场中任意流动性系数 ϕ ∗ 皆可使流动性保持充裕，此时权益变现粘性可忽略； 而当垫头要求足够大时，如 θ = 0 ，厂商新进投资需求无法通过外部融资满足，需要全额自行支付， 即内部融资。那么此刻为保证流动性充裕需使权益变现粘性满足 ϕ ∗ > (1 − λ )(1 − π ) πλ 。因此， ϕ ∗ 可视为 央行的流动性注入，即央行可根据金融流动性的实际状态进行权益或其他金融资产购买，这在次贷 危机与欧债危机后皆存在实际应用，相关性质与 Del Negro et al. ( 2017 ) 中的观点一致。此边界条 件可由图 2 -(b) 进行刻画。 0 0 5 1 0 0 3 0 6 θ ϕ ∗ (a) 垫头补贴 0 0 5 1 − 1 0 1 ϕ θ ∗ (b) 流动性补助 图 2: 流动性充足的必要条件 ( λ = 0 975 , π = 0 05 ) 下面求解货币均衡一阶条件。此时拥有投资机会厂商 ( 下文称之为投资型厂商 ) 的流动性约束 绷紧，式 ( 5 ) 变为 n i t +1 = (1 − θ ) i t + (1 − ϕ t ) λn t (11) 10 这一派生机制可理解为，给定厂商需支付垫头 1 − θ ，政府此时的投资补助 ∆ θ 可令垫头支付变为 1 − θ − ∆ θ ，则厂商将额外获得 ∆ θ 的流动性。一方面，若厂商将 ∆ θ 纳入新进投资 i t ，其下一期的净资产将额外增加，即 n t +1 (∆ θ ) > n t +1 ，进而有 ϕ t +1 n t +1 (∆ θ ) > ϕ t +1 n t +1 ，意味着经济中未来流动性增加；另一方面，若厂商将 ∆ θ 用于当期外部权益购买，则加总流动性直接增加。 7 Figure 1. Efficiency boundaries We next discuss the IE conditions. The firm has the chance to invest but will face the binding liquidity constraint such that n i t + 1 = ( 1 − θ ) i t + ( 1 − φ t ) λ n t ; inserting this constraint into the cash flow, Equation ( 3 ) provides the new cash flow equation under IE: c i t + q R t n i t + 1 = r t n t + [ φ t q t + ( 1 − φ t ) q R t ] λ n t + p t m t (7) where q R t = 1 − θ q t 1 − θ is the replacement cost and q R t < 1 for q t > 1. According to Kiyotaki and Moore [ 1 ] and the log form of the firm’s utility, the consumption c i t and investment i t can be derived as

Sustainability 2023 , 15 , 2373 7 of 15 c i t = ( 1 − β ) r t n t + [ φ t q t + ( 1 − φ t ) q R t ] λ n t + p t m t i t = ( r t + λφ t q t ) n t + p t m t − c i t 1 − θ q t For these firms that do not have the chance to invest, the liquidity constraint is not necessary and the investment equals 0, i t = 0. Therefore the cash flow equation for this type of firm is provided by: c s t + q t n s t + 1 + p t m s t + 1 = r t n t + q t λ n t + p t m t , (8) and the solution of consumption is c s t = ( 1 − β )( r t n t + q t λ n t + p t m t ) . Moreover, note that c i t < c s t due to q > 1 > q R These two cash flow equations can solved for the firms’ intertemporal decisions using the Euler equations. In doing so, ( 4 ) is further specified as the expected utility containing both types of firms (or say a firm with two different states) such that: U t = ln c t + β 1 − η ln ∑ j π ( j ) t exp ( 1 − η ) E U t + 1 ( c i t + 1 ) + ∑ j ( 1 − π ( j ) t ) exp ( 1 − η ) E U t + 1 ( c s t + 1 ) Combining the last equation with ( 7 ) and ( 8 ) and using the first order condition with respect to n t and m t provides the Euler equations as follows: u 0 ( c t ) = E t p t + 1 p t β π ( j ) t exp ( 1 − η ) E U t + 1 ( c i t + 1 ) u 0 ( c i t + 1 ) ∑ j π ( j ) t exp ( 1 − η ) E U t + 1 ( c i t + 1 ) + ∑ j ( 1 − π ( j ) t ) exp ( 1 − η ) E U t + 1 ( c s t + 1 ) + E t p t + 1 p t β ( 1 − π ( j ) t ) exp ( 1 − η ) E U t + 1 ( c s t + 1 ) u 0 ( c s t + 1 ) ∑ j π ( j ) t exp ( 1 − η ) E U t + 1 ( c i t + 1 ) + ∑ j ( 1 − π ( j ) t ) exp ( 1 − η ) E U t + 1 ( c s t + 1 ) , u 0 ( c t ) = E t β R s t + 1 z }| { ( r t + 1 + λ q t + 1 ) / q t · π ( j ) t exp ( 1 − η ) E U t + 1 ( c i t + 1 ) u 0 ( c i t + 1 ) ∑ j π ( j ) t exp ( 1 − η ) E U t + 1 ( c i t + 1 ) + ∑ j ( 1 − π ( j ) t ) exp ( 1 − η ) E U t + 1 ( c s t + 1 ) + E t β R i t + 1 z }| { ( r t + 1 + φ t + 1 λ q t + 1 + ( 1 − φ t + 1 ) λ q R t + 1 ) / q t · ( 1 − π ( j ) t ) exp ( 1 − η ) E U t + 1 ( c s t + 1 ) u 0 ( c s t + 1 ) ∑ j π ( j ) t exp ( 1 − η ) E U t + 1 ( c i t + 1 ) + ∑ j ( 1 − π ( j ) t ) exp ( 1 − η ) E U t + 1 ( c s t + 1 ) where R i t + 1 , R s t + 1 are the return rates of two different types of firms, with and without investment chance, respectively, for holding the equity from t to t + 1. Moreover, the aggregate capital and market clearing are simply provided by: ( 1 − θ q t ) I t = π β [( r t + λφ t q t ) K t + p t M ] − ( 1 − β )( 1 − φ t ) λ q R t K t , (9) r t K t = I t + ( 1 − β ) r t + ( 1 − π + πφ t ) λ q t + π ( 1 − φ t ) λ q R t K t + p t M (10) 3. Theoretical Analysis at Equilibrium To facilitate the following analysis, it is useful to rewrite the liquidity constraint. In doing so, note that the original liquidity constraint has the form of ( 1 − λ ) θ + λπφ ≤ ( 1 − λ )( 1 − π ) and the binding equation yields φ = ( 1 − λ )( 1 − π − θ ) λπ ,

Sustainability 2023 , 15 , 2373 8 of 15 then the superbinding equation for undersupplied liquidity can be expressed by φ 0 = φ − ∆ One notable point is that the wedge ∆ > 0 separates these two types of liquidity constraints, so that ∆ can also be considered as the gap between the liqudity demand and supply Moreover, given φ ∈ [ 0, 1 ] , the parameter condition for investment chance π is ( 1 − θ )( 1 − λ ) ≤ π ≤ ( 1 − θ ) We now focus on the firm’s behavior and how uncertainty affects the equilibrium. The aggregate condition ( 9 ) and ( 10 ) are solved for the risk free rate r t and real money balances l = pM / K such that: πβ r = ( 1 − β + πβ ) κ + ( 1 − β )[ λπβ ( 1 − π ) η − b θ ] q , (11) π l = ( 1 − π ) κ − [ λπ − λπ ( β + π − πβ ) η + b θ ] q (12) Note that fiat money becomes valuable whenever l > 0, corresponding to the binding liquidity constraint, indicating firms that have the investment chance may face liquidity shortage. Thus, using Equation ( 12 ) and l > 0, we can obtain the upper bound of the equity equilibrium price such that 1 < q < ( 1 − π ) κ λπ − λπ ( β + π − πβ ) η + b θ ≡ b q , ∀ l > 0, where κ , η , b θ are η = π − ( 1 − λ )( 1 − θ ) ( 1 − θ ) λπ , κ = 1 − λ + ( 1 − β )[ π − ( 1 − λ )( 1 − θ )] 1 − θ , b θ = θ 1 − θ π The following content will show that the real money balances and the upper bound of the equity price can change from l = 0, q = 1 to l > 0, q ∈ ( 1, b q ) when firms are under uncertainty/ambiguity. Before anatomizing this effect, we need to propose a pivotal Lemma Lemma 1. There is a unique expectation threshold that changes the status of the liquidity constraint Proof. According to the claim of classical uncertainty theory, agents always make decisions under the worst case scenario, so we set π = π − ξ , ξ > 0 as the investment chance for the worst case. Given the specification of liquidity demand φ , it is useful to define liquidity supply ¯ φ such that ¯ φ > φ ≡ ( 1 − θ − π )( 1 − λ ) λπ > 0 and ¯ φ < 1. Moreover, the derivatives of φ to π further show that ∂φ ∂π = − ( 1 − λ ) λπ + λ ( 1 − θ − π )( 1 − λ ) ( λπ ) 2 < 0, ∂ 2 φ ∂π 2 = 2 λ 2 π ( 1 − λ ) λπ + λ ( 1 − θ − π )( 1 − λ ) ( λπ ) 4 > 0, indicating φ is strictly decreasing in π . Meanwhile lim π → 0 φ = + ∞ , lim π → ( 1 − θ )( 1 − λ ) φ = 1, saying that φ > ¯ φ holds for a unique π > 0, corresponding to a unique ξ , that is, a unique expectation threshold Intuitively, what Lemma 1 tries to convey is that firms tend to expand the size of external financing at the current period when they expect the future investment chance will

Sustainability 2023 , 15 , 2373 9 of 15 drop and that this is the only best response according to their profit maximization. On the other hand, the liquidity market faced an intensified demand rise in the short run due to firms’ “precautionary saving”, resulting in liquidity shortage. Figure 2 provides a proper illustration of this effect 动性强约束，并且 ( 19 ) 大于 0 可推知存在均衡权益价格上限 b q 为 1 < q < (1 − π ) κ λπ − λπ ( β + π − πβ ) η + b θ ≡ b q, ∀ l > 0 当 l = 0 时，经济为无货币均衡，效率达到最优。此外，上式中 κ, η, b θ 为简化系数，下文可以看到 依据流动性约束强弱的不同其具体形式也不尽相同。 此外，由于投资机会不确定性对于流动性约束状态的影响以及方向较为隐性，这令后续分析的 阐述稍显复杂。因此，这里一个较为简便的解决办法便是构建真实不确定性与流动性缺口 ∆ 的联 系。为此，当投资机会 π 不确定时，厂商将以其最差预期 π − ξ 为基础进行决策，而此时我们发现 流动性约束可由弱约束转变为强约束，进而改变厂商对外部权益与货币持有的事前配置。这一性质 归纳为以下引理。 引理 1 ( 不确定性与缺口 ) 当流动性供给水平不变时，存在唯一的投资机会预期临界使流动性约束 在等式与不等式间转换。 证明 ：见附录。 直观而言，引理 1 是指当厂商普遍预期市场投资机会下降时，它们认为未来经济中通过外部融 资的数量也将减小，因此为保证日后的生产效益与可能出现的融资需求，厂商在当期扩大了外部融 资额度，进而在流动性供给相对固定的情况之下导致流动性需求提高，造成流动性相对紧缺，进而 有流动性约束从弱约束变为强约束。这一性质也可由图 3 进行阐述 14 ，图中 ϕ 为流动性需求而 ¯ ϕ 为 流动性供给，可见当厂商预期 π 下降时其流动性加总需求增加并超过已有的流动性供给。此外，由 引理 1 的证明可知投资机会的不确定性越大，其最差情形 π − ξ 越小，而流动性供需缺口 ∆ 越大， 这也是引理 1 的一个重要意义。 0 0 2 0 4 0 6 0 8 0 0 2 0 4 0 6 0 8 1 π ϕ / ¯ ϕ ϕ ¯ ϕ = 0 2 图 3: 不确定性下流动性供需的变化 得益于上述两项定义与引理 1 ，下面我们对厂商的融资行为进行解析。为此，基于欧拉方程中 14 为更清晰说明这一逻辑，基于 Kiyotaki & Moore ( 2012 ) 的参数条件可见当最差投资机会 π − ξ = 0 0203 时 ϕ = 1 ，证明给定任意 流动性供给 ¯ ϕ ∈ (0 , 1) ，存在模糊区间系数 ξ = 0 0297 总能使经济中流动性转为强约束。 10 Figure 2. Liquidity shortage We next explore firms’ financing strategy under uncertainty. The specification of R s and R i in the Euler equations provides: R s = r + λ q q , R i = r + φλ q + ( 1 − φ ) λ q R q , where R s is the return rate of the firms that do not have the investment chance, while R i denotes the return rate of the alternate. Note that for these firms that do not have the investment chance, R s can be regarded as the interest rate of saving; this is because this type of firm can only buy the equities from the market rather than issue the new equities by themselves (see [ 1 ]). In contrast, R i can be regarded as the interest rate of investing. Moreover, recall that q R < 1 if q > 1 under liquidity shortage, which also implies R s > 1 > R i by the last two expressions, suggesting the saving firms are more profitable than the investing firms. In contrast, R s = 1 = R i should hold if model’s economy is free of uncertainty. Therefore, the key to exploring firms’ behaviors under uncertainty is to anatomize how R s and R i change against the expectation threshold in Lemma 1 . Reserve prices . In order to obtain enough external financing to offset the loss under uncertainty, it is reasonable for firms to set a minimum reserve price q (ask price) for their newly issued equities to meet the financing demand; the ask price is rooted in the expression of R i . Similarly, to maximize the return from saving, firms can also set a maximum reserve price q (bid price) on buying equities; this factor is also rooted in R s Theoretically, these reserve prices help connect uncertainty with two return rates R i , R s ; the following proposition characterizes this mechanism in detail Proposition 1. The ask price q increases in the extent of uncertainty, while the bid price q is the opposite Proof. Substituting the superbinding liquidity constraint into R i < 1 provides the ask price as follows: q ≡ 1 − β ( 1 − π ) π ( 1 − β ) + β ( 1 − λ )( 1 − θ ) + λπ ( 1 − β ) ∆ + βπ − β ( 1 − λ )( 1 − θ ) + λπβ ∆ β π − ( 1 − π )( 1 − λ )( 1 − θ ) + λ ∆ πθ + ∆ π ( 1 − θ ) λ − ( 1 − β ) β ( 1 − π )[ π − ( 1 − λ )( 1 − θ )] − πθ + ∆ λπβ ( 1 − π ) − ∆ πθλ ! .

Sustainability 2023 , 15 , 2373 10 of 15 Likewise, the substitution of the superbinding liquidity constraint into R i < 1 provides the bid price: q ≡ 1 − ( 1 − π ) β ( 1 − λ )( 1 − θ ) + ( 1 − β )[ π − ( 1 − λ )( 1 − θ )] + λπ ( 1 − β ) ∆ ( 1 − λ )( 1 − θ ) πβ − ( 1 − β ) β ( 1 − π )[ π − ( 1 − λ )( 1 − θ )] − θπ + λπβ ( 1 − π ) ∆ − λπθ ∆ On the analytical solution of q and q , a little computation shows their properties concerning the expectation threshold π = π − ξ : the derivative of q to π is provided by ∂ q ∂π ∝ − ( 1 − β ) π 2 − ( β − π )( 1 − λ )( 1 − θ ) − ( 1 − β ) λπ 2 ∆ < 0 This result proves that the ask price decreases to the extent of the uncertainty. However, the derivative of the bid price is a slightly complicated due to the indeterminacy of the sign: ∂ q ∂π ∝ ( 1 − β )( 1 − θ ) + λ ( 1 − β )( 1 − θ ) ∆ ( 1 − λ )( 1 − θ ) − π 2 − λβ ( 1 − β ) π 2 ∆ To solve this problem formally, we first let ∆ → 0, which provides ∂ q / ∂π ≥ 0 ⇔ π ≤ p ( 1 − λ )( 1 − θ ) , thus a similar condition π ∈ [( 1 − λ )( 1 − θ ) , p ( 1 − λ )( 1 − θ ) / [ 1 + λβ ( 1 − β ) ∆ ]] ⇔ ∂ q / ∂π ≥ 0 holds for 0 < ∆ < 1 < 1 − ( 1 − λ )( 1 − θ ) λβ ( 1 − β )( 1 − λ )( 1 − θ ) Based on this analytical result and following all the parameterization and π = 0.05 in Kiyotaki and Moore [ 1 ], it follows that ∂ q / ∂π > 0, ∀ π = π − ξ < 0.05, i.e., the bid price is increasing in the extent of uncertainty There are two important aspects of Proposition 1 worth noting. First, the reason we insert the superbinding liquidity constraint into inequality R i < 1 and R s > 1 is that the binding constraint also contains a scenario where the liquidity supply meets the demand perfectly so that the liquidity shortage never happens, which is rare in reality and very much deviated from our topic, thus the use of the superbinding condition is necessary. Second, the change of the reverse prices has a strong effect on the return rates. For instance, the ask price q would go up if π → π , inducing all those feasible prices q go up as well due to q = inf q . Meanwhile, the analytical expression of R i indicates that ∂ R i / ∂ q < 0, thus the uncertainty eventually drags down the level of R i . Apply the same reasoning to the bid price, q goes down when π → π , and all the feasible prices q go down as well due to q = sup q . According to the analytical expression of R s , its derivative ∂ R s / ∂ q < 0 indicates that uncertainty raises R s . This mechanism clearly explains how uncertainty affects the return rates and changes their property from R s = 1 = R i to R s > 1 > R i . We use Figure 3 to describe the change of reserve prices over different pairs ( π , ∆ ) , where ∆ is the wedge between the binding and superbinding liquidity constraint and can be regarded as a multiplier to the uncertainty effect 厂商跨期配置的细项，考虑稳态与完全模糊环境，利用两类收益 R s , R i 的表达式 R s = r + λq q , R i = r + ϕλq + (1 − ϕ ) λq R q 其中 R s 无投资机会的储蓄型厂商的跨期回报率， R i 则为获得投资机会的投资型厂商跨期回报率， 前者包含了权益的出售价格要求而后者包含了权益的购买价格要求。由欧拉方程稳态及 Kiyotaki & Moore ( 2012 ) 中断言 3 知 R s > 1 > R i ，意指在不确定环境下融资供给方要求的回报将高于融资需 求方所获得的实际回报 15 。此外，由于融资需求方 ( 投资型企业 ) 依据经济不确定程度内含一个外部 融资总量的水平，即“保底水平”，那么为达到这一“保底水平”我们将 R i 中所包含的权益价格视 为投资型厂商出售权益的价格下限，记为 q ，意指投资型厂商最低以 q 价格出售其权益。与此对应， 记 q 为 R s 中所包含的权益价格，指储蓄型厂商购买权益价格的上限，即投资型厂商最高以 q 价格 购买市面权益。上述两类价格具体化了两类厂商在不确定状态下的融资决策，例如最低可接受价格 q 的变动展示的是融资需求方融资意愿的的变动，而最高可接受价格 q 展示的则是融资供给方的意 愿。基于此，厂商的融资决策性质如下。 命题 1 ( 厂商融资决策 ) 给定权益价格上下限 q 与 q ，其中最低可接受价格 q 随不确定性增大而下 降，而最高接受价格 q 随不确定性增大而上升。 证明： 见附录。 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 1 1 2 1 4 1 6 π (∆ : 0 01 → 0 1) q / q q (∆ = 0 01) q (∆ = 0 1) q (∆ = 0 01) q (∆ = 0 1) 图 4: 不确定性下权益价格上下限的变动 关于 q 的理论性质可由图 4 蓝线进一步验证。具体而言，在不确定性环境下厂商预期未来投 资机会下降（由模糊准则决定） ，因而资本存量减少（每期资本折旧） 、投资机会趋于稀缺性，则对 于单位新进投资所带来的综合收益 R i 势必相对上升。同时， ∂R i / ∂q < 0 又意味着投资厂商预期的 综合收益 R i 上升对应了权益价格 q 同其下限 q = inf { q } 并同下降。两方面因素引致了这一结果， 厂商为筹资一方面选择尽可能多的出售与兑换手中所持有的友商权益，增加了市面上权益的抛售数 15 事实上， Kiyotaki & Moore ( 2012 ) 附录中断言 3 的证明有误， 为此， 可依据其反证思路假设断言 3 (iv) 不正确， 故有 r + ϕλq +(1 − ϕ ) λqR q ≥ 1 。同时， Kiyotaki & Moore ( 2012 ) 中 (A 11) 式可化写为 r q = 1 − λ q +(1 − β ) [ r q (1 − π )+ r q π + λπϕ + λπ (1 − ϕ ) qR q + λ (1 − π )+ l q ] 。 将上述两式合并，得到 r q ≥ 1 − λ q + (1 − β ) [ r q (1 − π ) + π + λ (1 − π ) + l q ] 。而 Kiyotaki & Moore ( 2012 ) 的附录证明式却为 r q ≥ 1 − λ + (1 − β ) [ r q (1 − π ) + π + λ (1 − π ) + l q ] ，该式为反证过程中的关键不等式。其中，推导该不等式需要额外条件为 q < 1 ， 这便违背了整个断言 3 及其证明基础 q > 1 ，故 Kiyotaki & Moore ( 2012 ) 关于欧拉方程的证明有误。 11 Figure 3. Reserve prices change over uncertainty.

Sustainability 2023 , 15 , 2373 11 of 15 One thing should be underlined here is that the reserve prices are only the boundaries of the equity price trading in the market rather than the equilibrium price for market clearing and the equilibrium price is still one of the pivotal elements throughout our analysis Equilibrium price . A simple way to obtain the equilibrium price under uncertainty is to substitute the superbinding condition φ 0 = φ − ∆ into ( 9 ) and ( 10 ) and solve for a new expression of the upper bound of the equilibrium equity price such that b q ≡ ( 1 − π ) ( 1 − β ) π + β ( 1 − λ )( 1 − θ ) + λπ ( 1 − β ) ∆ λπ ( 1 − θ ) − ( β + π − πβ ) π − ( 1 − λ )( 1 − θ ) − λπ ( β + π − πβ ) ∆ + πθ + λπθ ∆ where the new parameters η , κ , b θ are provided by η = π − ( 1 − λ )( 1 − θ ) ( 1 − θ ) λπ + ∆ 1 − θ , κ = 1 − λ + ( 1 − β )[ π − ( 1 − λ )( 1 − θ )] 1 − θ + λπ ( 1 − β ) ∆ 1 − θ , b θ = θ 1 − θ π + λπ θ ∆ 1 − θ b q marks a certain region for the equity price at equilibrium, that is, one can always find equilibrium candidates within the region of b q . However, this region can be significantly altered by uncertainty; this property is summarized by the following Proposition Proposition 2. b q is increasing in the extent of uncertainty Proof. In order to prove the statement in Proposition 2 , it is useful to rewrite the expression of b q to b q = 1 + ϕ 1 1 + ϕ 2 , where ϕ 1 and ϕ 2 are provided by: ϕ 1 = λπ ( 1 − β ) ( 1 − π ) ( 1 − β ) π + β ( 1 − λ )( 1 − θ ) · ∆ , ϕ 2 = λπθ − λπ β + π ( 1 − β ) ( 1 − π ) ( 1 − β ) π + β ( 1 − λ )( 1 − θ ) · ∆ The derivative of ϕ 1 and ϕ 2 to π are ∂ϕ 1 ∂π ∝ λπ ( 1 − β ) 2 + β ( 1 − λ )( 1 − θ ) , ∂ϕ 2 ∂π ∝ λ ( 1 + π )[ θ − β − π ( 1 − β )] − λπ ( 1 − β ) ( 1 − β ) π + β ( 1 − λ )( 1 − θ ) − λπθ − λπβ − λπ 2 ( 1 − β ) ( 1 − π )( 1 − β ) , and one can conclude that ∂ϕ 2 / ∂π < ∂ϕ 1 / ∂π , according to the parameterization in Kiyotaki and Moore [ 1 ], that is, the change of ϕ 2 against π is smaller than the change of ϕ 1 against π , implying that b q is increasing in π and that the upper bound of equilibrium price is increasing in the extent of uncertainty Figure 4 plots the numerical result of Proposition 2 . Recall the primitive result that is derived from ( 11 ) and ( 12 ), the real money balances and equity price are l = 0, q = b q = 1 if the liquidity is sufficient. However, Proposition 2 reveals that uncertainty raises the value of b q and causes it to be greater than 1; this also pushes up the real money balance to positive, i.e., money starts being valuable. Therefore, the firm’s dynamics or the equity price will always shift to the IE if the economy is imbued with uncertainty. Moreover, unlike Kiyotaki and Moore [ 1 ], with the help of aggregate shock, uncertainty spontaneously triggers firms’ precautionary saving behavior and breaks the liquidity balance in the market, which happens to be irrelevant to the fundamental change.

Sustainability 2023 , 15 , 2373 12 of 15 影响为点在曲线上移动，如预期的投资机会 π 下降后导致 b q ( < ) 减小。流动性供需缺口对 b q ( < ) 的 影响体现为曲线的平移，如缺口 ∆ 上升体现为 b q ( < ) 向上平移，也即缺口对均衡权益上限具有乘数 效应。然而这里需要注意的是预期投资机会的变化将改变流动性供需缺口的大小 ( 引理 1 ) ，因此再 考虑两者同时变动后，均衡权益价格上限最终因不确定性增大而上升，即 π 的降低推升 b q ( < ) ，具 体可由图 5 展示。 0 0 2 0 4 0 6 0 8 1 4 1 5 1 6 1 7 π b q ( < ) 图 5: 紧约束下不确定性对均衡权益价格上限的影响 此外，命题 2 与本文开篇所提及的现实事实有紧密的联系。在不确定性环境中，厂商担忧未 来投资机会的减少，为满足未来可能出现的流动性需求将持有无生息的货币，因其完全流动性的性 质可即时弥补厂商生产上的流动性缺口。然而另一方面，厂商持有的其他友商权益虽可提供一定程 度的融资资金，但由于变现存在粘性而不能即时弥补可能出现的流动性缺口，故当期厂商的最优策 略则是尽可能多的扩大其外部融资额度储备资金供未来急用，这体现为厂商提高单位权益的发售价 格，即 b q ( < ) 升高。正因如此，市场中对货币持有以及外部融资的加总需求增大，流动性供给固定 时加剧了市面流动性的紧缩。 至此，本文已解析了均衡权益价格上限 b q ，权益供给价格下限 q ，权益需求价格上限 q 三者的理 论性质，在仅考虑流动性强约束的前提下， b q ( < ) 为货币均衡的临界线，那么从几何上来讲，货币均 衡中 q 的取值不得超过该线，即 q ≤ b q ( < ) ，这是货币均衡的充分条件。另一方面， [ q, q ] 为市场中融 资供求双方的定价区间，只有 q 在该区间内，即 q ∈ [ q, q ] ，权益的均衡价格才得以存在，这是货币 均衡的必要条件。上述两类条件但凡缺一，货币均衡则不存在 19 。基于此，本文发现均衡条件可被 打破。例如当缺口 ∆ 处于某一范围内时 ( 给定 ∆ = 0 01 ) ，权益价格的性质为 q ≤ b q ( < ) / ∈ [ q, q ] ，则 货币均衡不存在。为清晰说明其中的理论含义，这里定义两种类型厂商的欧拉方程函数为 Γ( q ) 与 Ψ( q ) ，表达如下。 Γ( q ) = (1 − π ) β r + λq q − P u ′ ( c s ) Ψ( q ) = πβ P − r + ϕλq + (1 − ϕ ) λq R q u ′ ( c i ) 由上文结论可知，货币均衡存在时权益价格与货币持有量满足 q > 1 , m > 0 ，则货币的价格比为 P = 1 ；而强约束环境中不持有货币时存在条件 q > 1 , m = 0 ，则有 P = 0 ，并且此时欧拉方程函数 需满足等式 u ′ ( c s ) ≡ Γ( q ) + Ψ( q ) | P =0 或 u ′ ( c i ) ≡ Γ( q ) + Ψ( q ) | P =0 。现考虑 b q ( < ) / ∈ [ q, q ] 的情形，假 设厂商现在需从区间 [ q, q ] 中形成均衡权益价格 q ，但又知任意 q > b q ( < ) 将可能导致均衡时厂商不 再持有货币，故在函数 Γ( q ) 与 Ψ( q ) 中 P = 0 。现给定最差的预期投资机会 π ，存在下述两类情形： 19 已由前文验证，强约束条件下不存在无货币均衡，那么货币均衡的不存在也意味着强约束下模型均衡的不存在。 13 Figure 4. The upper bound of equilibrium price under uncertainty Given Propositions 1 and 2 , we have the properties of reserve prices and equilibrium price under uncertainty; the combination of these two parts is worth commenting on Geometrically, for any equilibrium price q , it should be equal or less than b q due to b q being a cutoff of the equilibrium price; it hence has q ≤ b q . Meanwhile, the reserve prices define the interval for equity prices trading between liquidity supply and demand, suggesting that a deal can be made only when q ∈ [ q , q ] . Therefore, the condition for equilibrium to be steady, unique, and existing is to satisfy { q : q ≤ b q } ∩ { q : q ≤ q ≤ q } . However, there exists a specific region that any subset in that region could violate the equilibrium condition. To show this disequilibrium thoroughly, we simplify the Euler equations for two different types of firms as follows: Γ ( q ) = ( 1 − π ) β r + λ q q − P u 0 ( c s ) , Ψ ( q ) = πβ P − r + φλ q + ( 1 − φ ) λ q R q u 0 ( c i ) We now know that q > 1, m > 0 if the equilibrium is IE, then the inflation at steady state is P = 1. In contrast, assume the equilibrium is still IE but that agents do not hold any money, it then has q > 1, m = 0 and the steady state inflation is P = 0. Accordingly, the expected utilities for both types are u 0 ( c s ) ≡ Γ ( q ) + Ψ ( q ) | P = 0 , u 0 ( c i ) ≡ Γ ( q ) + Ψ ( q ) | P = 0 On the basic parameterization of Kiyotaki and Moore [ 1 ], it provides b q < q < q if the wedge is assumed to be ∆ = 0.01. On the one hand, this result clearly violates the equilibrium condition. On the other hand, if firms pick a trade price within interval [ q , q ] and follows q > b q , this would cause investing firms to experience a liquidity shortage but with zero money balance, which matches q > 1, m = 0, P = 0. Given the worst case scenario of π , the two following Euler equations hold: (1) For the saving firms, ( 1 − π ) 1 1 − π − r + λ q q β u 0 ( c s ) = π r + λφ q + λ ( 1 − φ ) q R q β u 0 ( c i ) ; (2) For the investing firms, ( 1 − π ) r + λ q q β u 0 ( c s ) = π 1 π − r + λφ q + λ ( 1 − φ ) q R q β u 0 ( c i ) On the parameterization of θ = 0.19, λ = 0.975, β = 0.99, and π are assumed temporally to be 0.05; a little computation shows that the solution of item (1) is q s ≈ 1.003, while the solution of (2) is q i ≈ 1.212. The buying equilibrium price and selling equilibrium price are not equal. The left panel of Figure 5 provides the disequilibrium with a straightforward

Sustainability 2023 , 15 , 2373 13 of 15 example. To avoid such a disequilibrium, a supplementary constraint is needed, which is essential for the stability and sustainability of the aggregate liquidity 故此时均衡不存在。而图 6 -(b) 中 ∆ = 0 1 ，符合命题 3 中的限制条件。在 ∥ π − 0 05 ∥ E < ε 的范 围内， q < q | Γ=Ψ < b q ( < ) < q 成立，均衡存在且唯一。 0 5 · 10 − 2 0 1 1 1 2 1 4 1 6 π q b q q q (a) ∆ → 0 01 0 5 · 10 − 2 0 1 1 1 2 1 4 1 6 π q b q q q (b) ∆ → 0 1 图 6: 不确定性下均衡的存在与唯一 围绕命题 3 的数值验证同时也说明了一个厂商的融资行为。在厂商对未来产生不确定性后，投 资机会 π 的预期将下移，此时产生市面流动性的供求缺口 ∆ ( 引理 1 ) 。而在命题 3 的数值验证中， 当缺口处于某一阈值内 ( 对应上文 ∆ = 0 01 ) ，权益供求双方的报价出现错配，以至于权益市场无 法出清。相反，当缺口超过该阈值时 ( 对应上文 ∆ = 0 1 ) 21 ，权益供求双方报价匹配，厂商应对不 确定性的融资决策达成统一。对此结果的一个直观解释是，当不确定性较小以至于引致的流动性供 给缺口较小时，两类厂商对未来融资市场的走势存有分歧，以至于出现权益价格错配。 四、 不确定性下厂商及加总流动性的动态探讨 现在我们对有限模糊的动态性质进行探讨。相对于 Kiyotaki & Moore ( 2012 ) 原型中的欧拉方 程，在纳入预期模糊后厂商跨期配置过程中的贴现因子出现了根本性的变化，变为 Λ c i t,t +1 = exp (1 − η ) E U t +1 ( c i t +1 ) P j π ( j ) t exp (1 − η ) E U t +1 ( c i t +1 ) + P j (1 − π ( j ) t ) exp (1 − η ) E U t +1 ( c s t +1 ) | {z } χ ci,t · β u ′ ( c i t +1 ) u ′ ( c t ) Λ c s t,t +1 = exp (1 − η ) E U t +1 ( c s t +1 ) P j π ( j ) t exp (1 − η ) E U t +1 ( c i t +1 ) + P j (1 − π ( j ) t ) exp (1 − η ) E U t +1 ( c s t +1 ) | {z } χ cs,t · β u ′ ( c s t +1 ) u ′ ( c t ) 上两式中模糊项分别标记为 χ c i ,t 与 χ c s ,t 。由 χ c i ,t , χ c s ,t 的形式知，二者范围皆大于 0 且小于等于 1 ，意指当厂商主体存在预期模糊时，随机贴现因子值减小，未来的动态贴现也将减少，减少值为 [1 − χ c i ,t ] βu ′ ( c i t +1 ) 与 [1 − χ c s ,t ] βu ′ ( c s t +1 ) ，这部分值也可视为主体为消除模糊而放弃的收益（获所 付出的成本）。其次，模糊厌恶系数 η 也决定了模糊项值的大小。如当 η = 1 时， χ c i ,t = χ c s ,t = 1 ， 21 本文根据命题 3 的理论条件进行了数值计算，结果表明 ∆ 与 π 存在非线性关系，具体而言 ∆ 随 π 递减且凸。同时，可以证明符合 命题 3 条件的缺口满足关系式 ∆ ≤ (1 − β )(1 − π )/ λπ ，将此式代入强约束后得到条件 (1 − λ ) θ + λπϕ ≥ ( β − λ )(1 − π ) ，这与 Kiyotaki & Moore ( 2012 ) 中假设 2 所得到的必要条件相近，但两式符号不同。此外，笔者发现能够满足 Kiyotaki & Moore ( 2012 ) 假设 2 的参数并不能总能得到其文中的有货币均衡（例如参数仅满足假设 2 的充分条件而不满足必要条件时） ，但因篇幅有限，以及本 文的研究重点并非检验 Kiyotaki & Moore ( 2012 ) 理论的正误，其后背的原因未做进一步讨论。 15 Figure 5. The existence of equilibrium under uncertainty Proposition 3. To guarantee the uniqueness and the existence of equilibrium under uncertainty, the tuple ( π , ∆ ) for the equity prices should satisfy that q ( π , ∆ ) < b q ( π , ∆ ) < q ( π , ∆ ) To properly show the geometric intuition and the insight of Proposition 3 , it is useful to use Figure 5 as the illustration, where wedge ∆ is fixed at two different values and the equity prices are depicted by three lines, blue, green and red, respectively, varying against the expectation of π . On the left panel, there exists a certain region in which the blue line b q is below the green line q when wedge ∆ is fixed at 0.01 and the expectation threshold is less than 0.05. Within this region, any equilibrium price candidate q will not surpass the upper bound b q ( q ≤ b q ) and it is hence not qualified for the market trading price ( q / ∈ [ q , q ] ) This geometric description is corresponding to the numerical result of q s 6 = q i as shown in the above. In contrast, the right panel plots the result that the constraint in Proposition 3 is implemented, where the blue line ( b q ) lies in between the red ( q ) and green ( q ); it can also be verified that q i = q s ≈ 1.063, meaning the existence of equilibrium under uncertainty is assured 4. Conclusions In this paper, we ask two pivotal questions: What worsens the aggregate liquidity in the absence of exogenous simulation? How does a firm act and cause this effect under uncertainty? By responding to these unresolved questions, the exogenous shock in Kiyotaki and Moore [ 1 ] is replaced with endogenous uncertainty to change the model’s perturbation from fundamental changes/shocks to expectation shifting. Meanwhile, the conditions of the firm dynamics and individual financing decisions are solved for anatomizing why these factors can cause precautionary saving and liquidity shortages. With a detailed and thorough analysis, three main conclusions are found for the topic First, given that agents are uncertain about the investment chance π and making decisions under the worst case scenario, the model generates a unique expectation threshold that drags the market’s (or firms’) liquidity status from sufficiency to insufficiency. The main reason is that uncertainty motivates agents to begin precautionary saving, which irresistibly raises the liquidity demand, and, when the liquidity supply is fixed at a certain level in the short run, this mismatch occurs Second, by splitting firms into saving and investing types, the intertemporal return and reserve price of each type are obtained, as the uncertainty is interacting with them directly and explicitly. Use an investing firm as an intuitive example: to have enough external financing to offset the loss under the worst case scenario, it is reasonable for the

Sustainability 2023 , 15 , 2373 14 of 15 investing firm to raise the selling price of equities, including the ask price q . Meanwhile, the investing firm’s return R i is strictly decreasing in the selling price, resulting in a downfall of the firm’s intertemporal return. For the saving firm, the reverse is also true Third, the uncertainty pushes up the upper bound of the equilibrium equity price and the value of fiat money as well, justifying that firms try to secure the external financing by raising the equity issuing price and hoarding liquid assets, such as fiat money, as the precautionary saving against the liquidity shortage. Moreover, an extra mathematical constraint is supplemented for the uniqueness and the existence of the equilibrium under uncertainty, which is also the essential element for the stability and sustainability of the liquidity Our theoretical results match the evidence of the agent’s precautionary behavior and aggregate fluctuations before the financial crisis in Greece, the UK, and the US as Bachmann et al. [ 2 ], Acharya and Merrouche [ 16 ], and Acharya and Merrouche [ 16 ] documented. For the sake of application and further research, we would like to discuss one computational example to implement the empirical test on our theory. Specifically, note that the stochastic discounters in our model are: Λ c i t , t + 1 = exp ( 1 − η ) E U t + 1 ( c i t + 1 ) ∑ j π ( j ) t exp ( 1 − η ) E U t + 1 ( c i t + 1 ) + ∑ j ( 1 − π ( j ) t ) exp ( 1 − η ) E U t + 1 ( c s t + 1 ) | {z } χ ci , t · β u 0 ( c i t + 1 ) u 0 ( c t ) , Λ c s t , t + 1 = exp ( 1 − η ) E U t + 1 ( c s t + 1 ) ∑ j π ( j ) t exp ( 1 − η ) E U t + 1 ( c i t + 1 ) + ∑ j ( 1 − π ( j ) t ) exp ( 1 − η ) E U t + 1 ( c s t + 1 ) | {z } χ cs , t · β u 0 ( c s t + 1 ) u 0 ( c t ) , and the ambiguous term for the investing and saving firm are Λ c i t , t + 1 and Λ c s t , t + 1 , respectively. Mathematically, χ c i , t = χ c s , t = 1 if the ambiguous parameter η = 1, where the model’s economy is free from uncertainty. Meanwhile, these two ambiguous terms χ c i , t , χ c s , t are decreasing in the ambiguous parameter η , saying the bigger η is, the less resources there are for agent to have after discounting. For simplicity, use η ∈ ( 1, ∞ ) as the basic measure for the ambiguity and, with additional data to calibrate the parameter array, { β , γ , φ , π , 1 − λ , 1 − θ , r / q + λ − 1 } using the formal macroeconometric method (for example, the combination of the Kalman filter and MLE for unobservable parameters), the model can be simulated numerically. In contrast to the original equilibrium, one can assume that the agent’s expectations start to shift after period 10 and the model converges to a new equilibrium at period 100 such that X 100 = X 101 = X 102 = · · · . Instead of using the perturbation method (because expectation shifting causes the steady state to not be fixed), projection and function approximation, such as Chebyshev polynomials, are essential for solving the model’s transitional path Author Contributions: Conceptualization, G.H. and Z.H.; methodology, Z.H.; software, Z.H.; validation, Z.H.; formal analysis, Z.H.; data curation, Z.H.; writing—original draft preparation, Z.H.; writing—review and editing, G.H.; visualization, Z.H. All authors have read and agreed to the published version of the manuscript Funding: This research received no external funding Institutional Review Board Statement: Not applicable Informed Consent Statement: Not applicable Data Availability Statement: Data available on request due to restrictions Conflicts of Interest: The authors declare no conflict of interest.

Sustainability 2023 , 15 , 2373 15 of 15 References 1 Kiyotaki, N.; Moore, J. Liquidity, business cycles, and monetary policy J. Political Econ 2019 , 127 , 2926–2966. [ CrossRef ] 2 Bachmann, R.; Carstensen, K.; Lautenbacher, S.; Schneider, M Uncertainty is More than Risk–Survey Evidence on Knightian and Bayesian Firms ; Technical report; University of Notre Dame, Notre Dame, IN, USA, 2020 3 Bolton, P.; Thadden, V. Blocks, liquidity, and corporate control J. Financ 1998 , 53 , 1–25. [ CrossRef ] 4 Del Negro, M.; Eggertsson, G.; Ferrero, A.; Kiyotaki, N. The great escape? A quantitative evaluation of the Fed’s liquidity facilities Am. Econ. Rev 2017 , 107 , 824–857. [ CrossRef ] 5 Guerron-Quintana, P.A.; Hirano, T.; Jinnai, R Recurrent Bubbles, Economic Fluctuations, and Growth ; Technical Report; Bank of Japan: Ch ¯u ¯o, Japan, 2018 6 Brunnermeier, M.K.; Pedersen, L.H. Market liquidity and funding liquidity Rev. Financ. Stud 2008 , 22 , 2201–2238. [ CrossRef ] 7 Shi, S Liquidity Shocks and Asset Prices in the Business Cycle. In The Global Macro Economy and Finance ; Springer: Berlin/Heidelberg, Germany, 2012; pp. 118–130 8 Shi, S. Liquidity, assets and business cycles J. Monet. Econ 2015 , 70 , 116–132. [ CrossRef ] 9 Ajello, A. Financial intermediation, investment dynamics, and business cycle fluctuations Am. Econ. Rev 2016 , 106 , 2256–2303 [ CrossRef ] 10 Brunnermeier, M.K.; Oehmke, M. The maturity rat race J. Financ 2013 , 68 , 483–521. [ CrossRef ] 11 Lorenzoni, G. Inefficient Credit Booms Rev. Econ. Stud 2008 , 75 , 809–833. [ CrossRef ] 12 Holmstrom, B.; Tirole, J. Financial intermediation, loanable funds, and the real sector Q. J. Econ 1997 , 112 , 663–691. [ CrossRef ] 13 Gertler, M.; Kiyotaki, N.; Queralto, A. Financial crises, bank risk exposure and government financial policy J. Monet. Econ 2012 , 59 , S 17–S 34. [ CrossRef ] 14 Gertler, M.; Karadi, P. A model of unconventional monetary policy J. Monet. Econ 2011 , 58 , 17–34. [ CrossRef ] 15 Geanakoplos, J. The leverage cycle Nber Macroecon. Annu 2010 , 24 , 1–66. [ CrossRef ] 16 Acharya, V.V.; Merrouche, O. Precautionary hoarding of liquidity and interbank markets: Evidence from the subprime crisis Rev Financ 2013 , 17 , 107–160. [ CrossRef ] 17 Ashcraft, A.; McAndrews, J.; Skeie, D. Precautionary reserves and the interbank market J. Money Credit. Bank 2011 , 43 , 311–348 [ CrossRef ] 18 Chiu, J.; Chung, H.; Ho, K.Y.; Wang, G.H. Funding liquidity and equity liquidity in the subprime crisis period: Evidence from the ETF market J. Bank. Financ 2012 , 36 , 2660–2671. [ CrossRef ] 19 Belke, A.; Orth, W.; Setzer, R. Sowing the seeds for the subprime crisis: does global liquidity matter for housing and other asset prices? Int. Econ. Econ. Policy 2008 , 5 , 403–424. [ CrossRef ] 20 Cao, C.; Petrasek, L. Liquidity risk in stock returns: An event-study perspective J. Bank. Financ 2014 , 45 , 72–83. [ CrossRef ] 21 Cao, C.; Chen, Y.; Liang, B.; Lo, A.W. Can hedge funds time market liquidity? J. Financ. Econ 2013 , 109 , 493–516. [ CrossRef ] 22 Cao, C.; Simin, T.T.; Wang, Y. Do mutual fund managers time market liquidity? J. Financ. Mark 2013 , 16 , 279–307. [ CrossRef ] 23 Wegener, C.; Basse, T.; Sibbertsen, P.; Nguyen, D.K. Liquidity risk and the covered bond market in times of crisis: empirical evidence from Germany Ann. Oper. Res 2019 , 282 , 407–426. [ CrossRef ] 24 Lambrecht, B.M.; Myers, S.C. A Lintner model of payout and managerial rents J. Financ 2012 , 67 , 1761–1810. [ CrossRef ] 25 Lambrecht, B.M.; Myers, S.C. The dynamics of investment, payout and debt Rev. Financ. Stud 2017 , 30 , 3759–3800. [ CrossRef ] 26 Hoang, E.C.; Hoxha, I. Corporate payout smoothing: A variance decomposition approach J. Empir. Financ 2016 , 35 , 1–13 [ CrossRef ] 27 Hoang, E.C.; Hoxha, I. A tale of two emerging market economies: Evidence from payout smoothing in China and Taiwan Int. J Manag. Financ 2020 . [ CrossRef ] 28 Gertler, M.; Kiyotaki, N.; Prestipino, A. A macroeconomic model with financial panics Rev. Econ. Stud 2020 , 87 , 240–288 [ CrossRef ] 29 Maxted, P.; Harvard University, Cambridge, MA, USA. A macro-finance model with sentiment. 2020 Unpublished working paper 30 He, Z.; Krishnamurthy, A. A macroeconomic framework for quantifying systemic risk Am. Econ. J. Macroecon 2019 , 11 , 1–37 [ CrossRef ] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.