在銀行不具強制執行還款能力的環境下，通常借款人必須提供資產做為擔保品。在模型中，我們加入了資產報酬的不確定性，當還款金額大於擔保品報酬時，借款人會違約同時也拋棄資產，由於擔保品的價值未必能彌補倒帳金額，銀行會事先收取一筆費用彌補未來的違約損失。因此，一個貸款契約包含貸款額度、無風險利率以及某金額的費用，此筆費用會直接從貸款額度扣除，剩下的金額是資產真正融資的現金。當借款人想提高貸款額度時，銀行承受的違約風險愈高，因此事前會收取更高的費用，在貸款額度與收取費用的權衡之下，借款人選出最適契約來極大化預期效用，此契約對應到一組貸款額度與費用，因此，模型內生求得資產流動性。上述機制並未依賴文獻制定借貸上限以控制借款人違約誘因的假設，因而補捉到不同的結果：在流動性需求較高的環境，借款人會借超過無風險借貸上限，得到具違約機率的均衡。另外是資產不確定性對環境的影響，我們發現當資產報酬不確定性愈高，借款人的違約機率增加，銀行進而收取較高的費用，此推升了借款人提高貸款金額的成本，並得到高風險性資產流動性愈低的結論。

In an environment where there is no technology to force debt repayment, borrowers are required to pledge some assets as collateral to secure loans. When asset returns are subject to uncertainty, borrowers may default when the loss of giving up collateral is less than that of debt repayment. This paper introduces the optimal contract problem into a monetary economy with collateral loans.
Specifically, a contract maximizes the borrower''s expected utility by choosing a loan amount, a loan rate, a credit limit and an up-front fee, that satisfies the bank''s participation constraints. After deducting the fee from the loan amount, the remaining balance is the net loan amount that real assets finance from credit markets. The model delivers the following insights. We endogenize a credit limit that does not rely on the mechanism of voluntary repayment. A constrained borrower can lift his credit limit by paying a higher fee, and it is optimal for a constrained borrower to finance a risky loan. Therefore, we generate a monetary equilibrium with endogenous default probability.
The liquidity of the real asset is derived from a trade-off between the loan amount and the collected fee. As the asset becomes riskier, borrowers default more likely, and banks charge a higher fee, which implies that it is more costly for borrowers to raise the credit limit. Consequently, a riskier asset is less liquid.

Contents
1 Introduction............................ 1
1.1 Literature Review .................. 3
2 The Model............................... 5
3 Equilibium ............................. 7
4 Optimal Contract and Equilibrium .......13
4.1 Discussion .........................21
5 Extension ..............................23
6 Conclusion .............................28
7 References .............................30
8 Appendix ...............................32

1 Berentsen, A., G. Camera and C. Waller, 2007. Money, Credit and Banking. Journal of Economic Theory 135, 171-195.

2 Berentsen A., C. Waller, 2011. Outside Versus Inside Bonds: A Modigliani-Miller Type Result for Liquidity Constrained Economies. Journal of Economic Theory,146, 1852-1887.

3 Brunnermeier M., K. Markus, 2009. Market Liquidity and Funding Liquidity. Review of Financial Studies, 22(6), 2201-2238.

4 Chapman, J., J. Chiu and M. Molico, 2011. Central Bank Haircut Policy. Ann Finance 7, 319-348.

5 Chomsisengphet, S., A. Pennington, 2006. The Evolution of the Subprime Mortgage Market.Federal Reserve Bank of St. Louis Review,January/February 88(1), 31-56.

6 Chiu, J., C. Meh and R. Wright, 2011. Innovation and Growth with Financial, and other Frictions.Working Paper.

7 Ferraris, L., M. Watanabe, 2008. Collateral secured loans in a monetary economy.Journal of Economic Theory 143, 405-424.

8 Foote, C., K. Gerardi and P. Willen, 2008. Negative equity and foreclosure: Theory and evidence. Journal of Urban Economics 64, 234-245.

9 Geanakoplos, J. 2003. Liquidity, Default, and Crashes: Endogenous Contracts in Gerneral Equilibrium.Advances in Economics and Econometrics: Theory and Applications. Econometric Society Monographs. Eighth World Conference 2, 170-205

10 Gorton, G., A. Metrick, 2012. Securitized Banking and the Run on Repo.Journal of Financial Economics 104, 425-451.

11 Kiyotaki, N. and J. Moore, 1997. Credit Cycles.Journal of
Political Economy 105, 211-248.

12 Lagos, R., R. Wright, 2005. A Unified Framework for Monetary Theory and Policy Analysis.Journal of Political Economy 113, 463-84.

13 Lester, B., A. Postlewaite and R. Wright, 2012. Information, Liquidity, Asset Prices and Monetary Policy, Review of Economic Studies 79, 1209-1238.

14 Li, Y., Y. Li, 2013. Liquidity and Asset Prices: A New Monetarist Approach.Journal of Monetary Economics 60, 426-438.

15 Rocheteau, G., 2011. A Monetary Approach to Asset Liquidity. Journal of Monetary Economics 58, 191-205.

16 Telykova, I., R. Wright, 2008. A Model of Money and Credit, with Application to the Credit Card Debt Puzzle. The Review of Economic Studies 75, 629-647.

17 Williamson, S., 1987. Financial Intermediation, Business Failures, and Real Business Cycles. Journal of Political Economy 95, 1196-1216.

18 Williamson, S., 1987. Costly Monitoring, Loan Contracts, and Equilibrium Credit Rationing.The Quarterly Journal of Economics 102, 135-146.