Abstract
We present a dynamic general equilibrium model of production economies with adverse selection in the financial market to study the interaction between funding liquidity and market liquidity and its impact on business cycles. Entrepreneurs can take on short-term collateralized debt and trade long-term assets to finance investment. Funding liquidity can erode market liquidity. High funding liquidity discourages firms from selling their good long-term assets since these good assets have to subsidize lemons when there is information asymmetry. This can cause a liquidity dry-up in the market for long-term assets and even a market breakdown, resulting in a financial crisis. Multiple equilibria can coexist. Credit booms combined with changes in beliefs can cause equilibrium regime shifts, leading to an economic crisis or expansion.
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Adjemian, S., Bastani, H., Juillard, M., Karamé, F., Mihoubi, F., Perendia, G., Pfeifer, J., Ratto, M., Villemot, S.: Dynare: Reference Manual, Version 4. Dynare working papers, 1, CEPREMAP (2011)
Akerlof, G.: The market for lemons: quality uncertainty and the market mechanism. Q. J. Econ. 84(3), 488–500 (1970)
Benhabib, J., Dong, F., Wang, P.: Adverse Selection and Self-Fulfilling Business Cycles (No. w20642). National Bureau of Economic Research (2014)
Bernanke, B., Gertler, M., Gilchrist, S.: Financial accelerator in a quantitative business cycle framework. In: Taylor, J., Woodford, M. (eds.) Handbook of Macroeconomics. Elsevier Science, Amsterdam (1999)
Bigio, S.: Endogenous liquidity and the business cycle. Am. Econ. Rev. 105(6), 1883–1927 (2015)
Brunnermeier, M.K., Pedersen, L.H.: Market liquidity and funding liquidity. Review of Financial Studies 22(6), 2201–2238 (2009)
Carlstrom, C.T., Fuerst, T.S.: Agency costs, net worth, and business fluctuations: a computable general equilibrium analysis. Am. Econ. Rev. 87, 893–910 (1997)
Eisfeldt, A.: Endogenous liquidity in asset markets. J. Finance 59(1), 1–30 (2004)
Gorton, G., Ordoñez, G.: Collateral crises. Am. Econ. Rev. 104(2), 343–378 (2014)
Gorton, G., Ordoñez, G.: Crises and Productivity in Good Booms and in Bad Booms. Working paper, University of Pennsylvania (2015)
Gourinchas, P.-O., Obstfeld, M.: Stories of the twentieth century for the twenty-first. Am. Econ. J. Macroecon. 4, 226–265 (2012)
Guerrieri, V., Shimer, R.: Dynamic adverse selection: a theory of illiquidity, fire sales, and flight to quality. Am. Econ. Rev. 104, 1875–1908 (2014)
Guerrieri, V., Shimer, R., Wright, R.: Adverse selection in competitive search equilibrium. Econometrica 78, 1823–1862 (2010)
House, C.L., Masatlioglu, Y.: Managing markets for toxic assets. J. Monet. Econ. 70, 84–99 (2015)
Ikeda, D., Phan, T.: Toxic asset bubbles. Econ. Theor. 61(2), 241–271 (2016)
Kaminsky, G.L., Reinhart, C.M.: The twin crises: the causes of banking and balance-of-payments problems. Am. Econ. Rev. 89, 473–500 (1999)
Kindleberger, C.P.: Manias, Panics, and Crashes: A History of Financial Crises. Basic Books, New York (1978)
Kiyotaki, N., Moore, J.: Liquidity, Business Cycles and Monetary Policy. Working paper, Princeton University (2008)
Kiyotaki, N., Moore, J.: Credit cycles. J. Polit. Econ. 105(2), 211–248 (1997)
Kurlat, P.: Lemons markets and the transmission of aggregate shocks. Am. Econ. Rev. 103(4), 1463–1489 (2013)
Lagos, R., Rocheteau, G., Wright, R.: Liquidity: a new monetarist perspective. J. Econ. Lit. 55, 371–440 (2017)
Lester, B., Postlewaite, A., Wright, R.: Liquidity, information, asset prices and monetary policy. Rev. Econ. Stud. 79, 1209–38 (2012)
Li, S., Whited, T.M.: Capital Reallocation and Adverse Selection. Working paper, University of Rochester (2014)
Li, Y., Rocheteau, G., Weill, P.-O.: Liquidity and the threat of fraudulent assets. J. Polit. Econ. 120, 815–846 (2012)
Malherbe, F.: Self-fulfilling liquidity dry-ups. J. Finance 69(2), 947–970 (2014)
McKinnon, R.I., Pill, H.: Credible economic liberalizations and overborrowing. Am. Econ. Rev. 87, 189–93 (1997)
Mendoza, E.G., Terrones, M.E.: An Anatomy of Credit Booms and Their Demise. Working paper no. 18379, National Bureau of Economic Research (2012)
Miao, J.: Introduction to economic theory of bubbles. J. Math. Econ. 53, 130–136 (2014)
Miao, J., Wang, P., Zhou, J.: Asset bubbles, collateral, and policy analysis. J. Monet. Econ. 76, S57–S70 (2015)
Miao, J., Wang, P., Lifang, X.: Stock market bubbles and unemployment. Econ. Theor. 61(2), 273–307 (2016)
Minsky, H.P.: The financial instability hypothesis: an interpretation of Keynes and an alternative to “standard” theory. Neb. J. Econ. Bus. 16(1), 5–16 (1977)
Reinhart, C.M., Rogoff, K.: This Time is Different: Eight Centuries of Financial Folly. Princeton University Press, Princeton (2009)
Santos, M.S., Woodford, M.: Rational asset pricing bubbles. Econometrica 65(1), 19–57 (1997)
Schularick, M., Taylor, A.: Credit booms gone bust: monetary policy, leverage cycles and financial cycles, 1870–2008. Am. Econ. Rev. 102(2), 1029–1061 (2012)
Tomura, H.: Asset illiquidity and market shutdowns in competitive equilibrium. Rev. Econ. Dyn. 15(3), 283–294 (2012)
Williamson, S., Wright, R.: Barter and monetary exchange under private information. Am. Econ. Rev. 84, 104–123 (1994)
Zhao, B.: Rational housing bubble. Econ. Theor. 60(1), 141–201 (2015)
Acknowledgements
We are grateful to an anonymous referee for helpful suggestions to improve the paper. We have benefitted from comments by Hengjie Ai, Wei Cui, Zhen Huo, Nicolas Jacquet, Yang Jiao, Benjamin Moll, Nicola Pavoni, Vincenzo Quadrini, Jose-Victor Rios-Rull, Cheng Wang, Yi Wen, Randy Wright, Jianhuan Xu, Tao Zha, Shenghao Zhu, Fabrizio Zilibotti, as well as participants at the NYU Conference on Multiple Equilibria and Financial Crises, the Fourth HKUST International Workshop on Macroeconomics, the 2015 AFR Summer Institute of Economics and Finance at Zhejiang University, the Stockman Conference at University of Rochester, Tsinghua University, Shanghai University of Finance and Economics, Singapore Management University, Renmin University of China, and Peking University.
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Appendices
Appendix
A Proofs
Proof of Proposition 1
We first consider an entrepreneur’s decision problem. For ease of notation, we suppress the subscript j. Let \(V_{t}\left( k_{t},\varepsilon _{t},h_{t}^{g},h_{t}^{l},b_{t}\right) \) denote the value function, where we have suppressed the aggregate state variables. Then \(V_{t}\) satisfies the following Bellman equation
subject to the constraints described in Sect. 3, where the conditional expectation is taken with respect to \(\varepsilon _{t+1}\). Conjecture that the value function \(V_{t}\) takes the following form:
where \(q_{t}(\varepsilon _{t}),\) \(\phi _{t}^{g}(\varepsilon _{t}),\) \(\phi _{t}^{l}(\varepsilon _{t}),\) and \(\phi _{t}^{b}(\varepsilon _{t})\) are to be determined.
Then \(V_{t+1}\) is also linear and we can write
where we define
We use the flow-of-funds constraint and other constraints in Sect. 3 to derive
If \(p_{t}^{g}>P_{t}^{g},\) then all entrepreneurs would purchase as much good assets as possible. If \(p_{t}^{g}<P_{t}^{g},\) then no entrepreneurs would purchase any good asset. In both cases a competitive equilibrium could not exist. Thus we must have \(p_{t}^{g}=P_{t}^{g}\) and \(p_{t}^{l}=P_{t}^{l}\). If \(p_{t}^{b}>1/R_{\textit{ft}},\) then all entrepreneurs would prefer to buy bonds and an equilibrium could not exist. If \(p_{t}^{b}<1/R_{\textit{ft}},\) then all entrepreneurs would borrow until the borrowing constraint binds. In this case all entrepreneurs would also want to purchase as much financial assets as possible in order to take leverage. But this would not constitute an equilibrium. Thus \(1/R_{\textit{ft}}=p_{t}^{b}.\)
We can simplify the last equality to derive
Let \(\varepsilon _{t}^{*}=1/Q_{t}\). Since \(i_{t}\ge 0,\) it is optimal to make as much investment as possible if and only if \(\varepsilon _{t}\ge \varepsilon _{t}^{*}.\)
By the flow-of-funds constraint and the borrowing constraint,
Since a firm with \(\varepsilon _{t}>\varepsilon _{t}^{*}\) wants to invest using as many resources as possible, it will not purchase any asset and will sell all its assets; that is
Moreover, it will borrow as much as possible up to the borrowing limit. A firm with \(\varepsilon _{t}<\varepsilon _{t}^{*}\) will not invest. Since \(p_{t}^{b}=1/R_{\textit{ft}}\), \(p_{t}^{l}=P_{t}^{l},\) and \(p_{t}^{g}=P_{t}^{g},\) the firm is indifferent between saving and borrowing and is indifferent between buying and selling assets. We then obtain the optimal investment rule
Thus we can derive aggregate investment and the law of motion for capital in Eqs. (20) and (21), where we have used the market-clearing condition for bonds, i.e., \(\int b_{\textit{jt}}\mathrm{d}j=0\).
Substituting the decision rules back into (A.1) and using the conjectured value function, we can derive
Matching coefficients yields
Using the preceding definition of \(Q_{t},\) \(p_{t}^{g},p_{t}^{l}\) and \( p_{t}^{b}\) and noting that \(p_{t}^{g}=P_{t}^{g}\), \(p_{t}^{l}=P_{t}^{l},\) \( p_{t}^{b}=1/R_{\textit{ft}},\) we can derive their asset pricing equations given in Proposition 1.
Since firms with \(\varepsilon _{t}\le \varepsilon _{t}^{*}\) are indifferent between buying and selling assets, we allow them to purchase assets so that asset markets can clear
Without loss of generality, we can set individual purchasing choice as
Moreover firms with \(\varepsilon _{t}<\varepsilon _{t}^{*}\) are indifferent between saving and borrowing. \(\square \)
Proof of Lemma 1
It is straightforward to check that \(\beta \left[ 1+\int _{\varepsilon ^{*}}^{\varepsilon _{\max }}\left( \frac{\varepsilon }{\varepsilon ^{*}} -1\right) \mathrm{d}F\left( \varepsilon \right) \right] \) decreases with \(\varepsilon ^{*}\). Since
where the second inequality comes from Assumption 1, it follows from the intermediate value theorem that there exists a unique solution, denoted by \( \varepsilon _{b}^{*}\in \left( \varepsilon _{\min },\varepsilon _{\max }\right) ,\) to the equation \(\beta \left[ 1+\int _{\varepsilon ^{*}}^{\varepsilon _{\max }}\left( \frac{\varepsilon }{\varepsilon ^{*}} -1\right) \mathrm{d}F\left( \varepsilon \right) \right] =1\). \(\square \)
Proof of Proposition 2:
Equation (17) in the steady state gives (26). For \( P^{g}>0,\) we need
If follows from Eq. (18) that \(P^{l}=0\). By Lemma 1, the condition above is equivalent to \(\varepsilon ^{*}>\varepsilon _{b}^{*}\). Using \(Q=1/\varepsilon ^{*}\) and Eq. (16), we can derive the steady-state capital stock in Eq. (27). Using Eq. (20) in the steady state yields
Substituting (26) for \(P^{g}\) and (27) for \(K=K\left( \varepsilon ^{*}\right) \) into the equation above gives an equation for \( \varepsilon ^{*},\) (28). We need the following lemma to complete the proof.
Lemma 3
For a sufficiently small \(\mu \), \(K\left( \varepsilon ^{*}\right) \) increases with \(\varepsilon ^{*}\) on \(\left( \varepsilon _{\min },\varepsilon _{\max }\right) .\)
Proof
Let
We can compute that
For a sufficiently small \(\mu \in \left( \varepsilon _{\min },\varepsilon _{\max }\right) ,\) \(h^{\prime }\left( \varepsilon ^{*}\right) <0.\) Thus by (27),
increases with \(\varepsilon ^{*}.\) \(\square \)
Simple algebra shows that the expression
increases with \(\varepsilon ^{*}\) on \(\left( \varepsilon _{b}^{*},\varepsilon _{\max }\right) .\) Let \(D\left( \varepsilon ^{*}\right) \) denote the expression on the left-hand side of (28). Then since \(D\left( \varepsilon ^{*}\right) \) is the product of the preceding expression and \(K\left( \varepsilon ^{*}\right) \), it increases with \( \varepsilon ^{*}.\)
We can check that
decreases with \(\varepsilon ^{*}\) on \(\left( \varepsilon _{b}^{*},\varepsilon _{\max }\right) .\) As \(\varepsilon ^{*}\) decreases to \( \varepsilon _{b}^{*},\ S\left( \varepsilon ^{*}\right) \) approaches infinity since \(R_{f}\left( \varepsilon _{b}^{*}\right) =1\) by Lemma 1, but \(D\left( \varepsilon _{b}^{*}\right) \) is finite. As \(\varepsilon ^{*}\) increases to \(\varepsilon _{\max },\) \(D\left( \varepsilon ^{*}\right) \) approaches infinity, but the limit of \(S\left( \varepsilon ^{*}\right) \) is finite. By the intermediate value theorem, there is a unique solution to \(\varepsilon ^{*}\in \left( \varepsilon _{b}^{*},\varepsilon _{\max }\right) \) in Eq. (28).
Differentiating the expressions on the two sides of Eq. (28) yields
We then have
Since \(\frac{\partial D\left( \varepsilon ^{*}\right) }{\partial \varepsilon ^{*}}>0\) and \(\frac{\partial S\left( \varepsilon ^{*}\right) }{\partial \varepsilon ^{*}}<0\) for small \(\mu ,\) we have \( \frac{\partial \varepsilon ^{*}}{\partial \mu }>0.\) Since \(K\left( \varepsilon ^{*}\right) \) and \(R_{f}\left( \varepsilon ^{*}\right) \) increase with \(\varepsilon ^{*}\), Y increases with \(\mu \) and \(P^{g}\) decreases with \(\mu .\) \(\square \)
Proof of Proposition 3:
We can write down an entrepreneur’s decision problem by dynamic programming as in (A.1) subject to the constraints given in Sect. 2. We suppress the subscript j throughout the proof. Conjecture that the value function takes the form as in (A.2). Then we have
where \(Q_{t},\) \(p_{t}^{g},\) \(P_{t}^{l},\) and \(p_{t}^{b}\) are defined as in (A.3) and (A.4).
Using the flow-of-funds constraint and the preceding equation, we can derive
By a similar argument to the proof of Proposition 1, for the entrepreneur’s optimal decisions to be compatible with a competitive equilibrium, we must have
Thus we have
Since \(i_{t}\ge 0,\) it is optimal for the firm to make real investment if and only if \(\varepsilon _{t}\ge 1/Q_{t}=\varepsilon _{t}^{*}.\) When making the investment, the firm will invest as much as possible. By the flow-of-funds constraint (6) and the borrowing constraint (7), we have
To leave the maximum resource for investing, the firm will not purchase any asset; that is, \(x_{t}=0.\) The borrowing constraint must also bind when \( \varepsilon _{t}>1/Q_{t}=\varepsilon _{t}^{*}.\) Thus we obtain the investment rule
Substituting this investment rule into the right-hand side of the Bellman equation in (A.1), we can derive that for \(\varepsilon _{t}>\varepsilon _{t}^{*},\)
where in the last equality we have used the fact that \(s_{t}^{l}=h_{t}^{l}\) since \(Q_{t}\varepsilon _{t}P_{t}\ge P_{t}\ge p_{t}^{l}\) and that \( s_{t}^{g}=h_{t}^{g}\) if \(Q_{t}P_{t}\varepsilon _{t}\ge p_{t}^{g}\) and \( s_{t}^{g}=0,\) otherwise.
If \(\varepsilon _{t}\le \varepsilon _{t}^{*}\), then \(i_{t}=0\) and we have
where the second equality follows from the fact that \(s_{t}^{g}=0\) since \( P_{t}\le p_{t}^{g}\) and that \(s_{t}^{l}=h_{t}^{l}\) since \(P_{t}\ge p_{t}^{l}.\)
We now combine the preceding two cases for all \(\varepsilon _{t}\in \left[ \varepsilon _{\min },\varepsilon _{\max }\right] \). If
then
If
then
Let \(\varepsilon _{t}^{**}\equiv \min \left( \frac{p_{t}^{g}}{ P_{t}Q_{t}},\varepsilon ^{\max }\right) .\) Then for any \(\varepsilon _{t}\in \left( \varepsilon ^{\min },\varepsilon ^{\max }\right) \), we can write
Substituting the preceding equation into the Bellman equation and using (A.2), we match coefficients to derive that for any \(\varepsilon _{t}\in \left( \varepsilon _{\min },\varepsilon _{\max }\right) \),
Substituting these equations into the previous definitions of \(Q_{t},\) \( p_{t}^{g},\) \(p_{t}^{l},\) and \(p_{t}^{b},\) we obtain their asset pricing equations as in the proposition. \(\square \)
Proof of Proposition 4:
In a frozen equilibrium, \(P_{t}=0\) for all t. No firms want to sell any good assets since the holding value \(p_{t}^{g}>0.\) In a frozen equilibrium, the market for long-term assets breaks down. We conjecture that the value function \(V_{t}\) takes the following form:
Then we can write
where we define \(Q_{t},\) \(p_{t}^{g},\) \(p_{t}^{l},\) and \(p_{t}^{b}\) as before. The Bellman equation is given by
subject to (5), \(b_{t+1}/R_{\textit{ft}}\ge -\mu _{t}k_{t},\) and
Using the flow-of-funds constraint, we can compute the objective function in (A.5) as
where we have used the fact that \(h_{t+1}^{g}=h_{t}^{g}=h_{0}^{g}\) and \( h_{t}^{l}=h_{t}^{l}=h_{0}^{l}\) for all t. We then obtain the investment rule in the proposition. Substituting this investment rule back into (A.5) and matching coefficients, we obtain
Using the definitions of \(Q_{t},p_{t}^{g},p_{t}^{l},\) and \(p_{t}^{b},\) we can derive (16), (17), and \(p_{t}^{l}=\beta E_{t}\left( p_{t+1}^{l}\right) .\) By the transversality condition, we deduce that \(p_{t}^{l}=0\) for all t. \(\square \)
Proof of Proposition 5:
By (36), \(Q_{t}P_{t}\varepsilon _{t}\le Q_{t}P_{t}\varepsilon _{\max }<p^{g}.\) No firms want to sell the good assets so that \(\varepsilon _{t}^{**}=\varepsilon _{\max }\) and \(\varTheta _{t}=0\). Thus \( P_{t}=p_{t}^{l}\) by (31). We then use (29) to derive (33) and use (30) to derive (32).
We use Proposition 3 to derive Eq. (44) for aggregate investment. We then obtain the law of motion for aggregate capital in Eq. (34). Using (3), (4), and the labor market-clearing condition \( N_{t}=1\), we can derive that
In addition,
By the decision rule in Proposition 3 and the market-clearing condition for financial assets,
we can derive
Since \(x_{t}\) is indeterminate at the individual firm level, we can set
for all j. \(\square \)
Proof of Proposition 6:
By Lemma 1, there exists a unique cutoff value \(\varepsilon _{b}^{*}\in \left( \varepsilon _{\min },\varepsilon _{\max }\right) \) to Eq. (24). In the bubbly lemon steady state, \(P>0\) and hence Eq. (32) is equivalent to Eq. (24). This implies that \(\varepsilon _{b}^{*}\) is the investment threshold in the bubbly lemon steady state. By (24) and (33), we can derive \(p^{g}\) as in (40). Using Eqs. (16) and (22), we can show that the steady-state capital stock is equal to \(K\left( \varepsilon _{b}^{*}\right) \) where \(K\left( \cdot \right) \) is given in (27). Using Eqs. (22) and (34), we can solve for P as in (41). We need to verify that the condition
holds in the steady state. But this is equivalent to (39). \(\square \)
Proof of Proposition 7:
In a pooling equilibrium the restriction in (45) must hold. Firms with \(\varepsilon _{\textit{jt}}\ge \varepsilon _{t}^{**}\) sell their good assets. By Proposition 3 and the market-clearing conditions for assets, we can compute \(\varTheta _{t}\) as in the proposition. Using the decision rule for investment in Proposition 3 and aggregating individual decision rules, we obtain (43) and (44). \(\square \)
Proof of Lemma 2:
Under Assumption 1, Lemma 1 establishes the existence of a unique solution \(\varepsilon _{b}^{*}\) to Eq. (24). Since \(\beta \left( 1+\int _{\varepsilon ^{*}}^{\varepsilon _{\max }}\left( \frac{\varepsilon }{\varepsilon ^{*}}-1\right) \mathrm{d}F\left( \varepsilon \right) \right) \) decreases with \(\varepsilon ^{*},\) it follows that
for \(\varepsilon ^{**}>\varepsilon _{b}^{*}.\) Thus we deuce that
Since the expression on the right-hand side of Eq. (50) decreases continuously with \(\varepsilon ^{*}, \) it follows from the intermediate value theorem that there exists a unique solution to \(\varepsilon ^{*}\) in \(\left( \varepsilon _{b}^{*},\varepsilon ^{**}\right) \) in Eq. (50). \(\square \)
Proof of Proposition 8:
Following the strategy used in the context, we know pooling equilibrium can be supported if and only if
where \(\overline{c}^{P}\left( \pi \right) =\underset{\varepsilon ^{**}\in \left[ \varepsilon _{b}^{*},\varepsilon _{\max }\right] }{\max } \varGamma \left( \varepsilon ^{**},\pi \right) \), and
As in the proof of Proposition 2 and Lemma 3, for a sufficiently small \(\mu , \) the expression
increases with \(\varepsilon ^{*}.\) Thus the numerator of the expression for \(\varGamma \) given above satisfies
for any \(\varepsilon ^{*}\ge \varepsilon _{b}^{*}.\) In addition, it follows from Lemma 1 that
for \(\varepsilon ^{**}>\varepsilon _{b}^{*}\) so that the denominator of the expression for \(\varGamma \) given above is also positive. We deduce that
for all \(\varepsilon ^{**}\in \left( \varepsilon _{b}^{*},\varepsilon _{\max }\right) .\) Since
and other limits are finite, we have
By the intermediate value theorem, there exists a solution to \(\varepsilon ^{**}\) in \(\left( \varepsilon _{b}^{*},\varepsilon _{\max }\right) \) in Eq. (52).
We can verify that
where the first equality uses the fact that \(\varPhi \left( \varepsilon _{\max }\right) =\varepsilon _{b}^{*}\) and the second uses the definition of \( \underline{c}^{B}\left( \pi \right) \) by Eq. (38). Therefore we know that \(\overline{c}^{p}\left( \pi \right) >\underline{c}^{B}\left( \pi \right) \).
The steady-state capital stock \(K\left( \varepsilon _{p}^{*}\right) \) is derived from Eq. (16) using \(\varepsilon _{p}^{*}=1/Q. \) \(\square \)
Proof of Proposition 9:
We apply Proposition 4. Aggregation leads to the equations for aggregate capital and investment in the proposition. \(\square \)
Proof of Proposition 10:
By Eq. (16), we can derive the steady-state capital stock \(K\left( \varepsilon ^{*}\right) \) defined in (27). We need to solve for \(\varepsilon ^{*}.\) By (22) and (56), we can derive Eq. (58). As in the proof of Proposition 8, we know that the right-hand side of (58) strictly increases with \(\varepsilon ^{*}\). In addition, we can show that
and
Therefore there exists a unique solution \(\varepsilon ^{*}\in \left( \varepsilon _{\min },\varepsilon _{\max }\right) \) to Eq. (58) for any \(c>0\).
Proof of Proposition 11:
By Lemma 3, for a sufficiently small \(\mu \), \(K\left( \varepsilon ^{*}\right) \) increases with \(\varepsilon ^{*}\). To prove \(K\left( \varepsilon _{p}^{*}\right)>K\left( \varepsilon _{b}^{*}\right) >K\left( \varepsilon _{a}^{*}\right) \), we only need to show that \( \varepsilon _{p}^{*}>\varepsilon _{b}^{*}>\varepsilon _{a}^{*}\) when \(\mu \) is small enough. By Lemma 2 and Proposition 8, \(\varepsilon _{p}^{*}>\varepsilon _{b}^{*}.\)
By definition,
By (37) and (38), \(\varepsilon _{b}^{*}\) satisfies the equation
By Proposition 10, \(\varepsilon _{a}^{*}\) satisfies
As shown in Proposition 6, a bubbly lemon steady-state equilibrium can be supported if \(c<\bar{c}^{B}\left( \pi \right) \). Therefore Eqs. (A.6) and (A.7) jointly imply
As in the proof of Proposition 2, \(D\left( \varepsilon ^{*}\right) \) strictly increases with \(\varepsilon ^{*}\) for a sufficiently small \(\mu \). Then Eq. (A.8) implies that \(\varepsilon _{b}^{*}>\varepsilon _{a}^{*}.\) \(\square \)
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Dong, F., Miao, J. & Wang, P. The perils of credit booms. Econ Theory 66, 819–861 (2018). https://doi.org/10.1007/s00199-017-1076-6
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DOI: https://doi.org/10.1007/s00199-017-1076-6