Skip to main content

On default and uniqueness of monetary equilibria


We examine the role that credit risk in the central bank’s monetary operations plays in the determination of the equilibrium price level and allocations. Our model features trade in fiat money, real assets and a monetary authority which injects money into the economy through short-term and long-term loans to agents. Short-term loans are riskless, but long-term loans are collateralized by a portfolio of real assets and are subject to credit risk. The private monetary wealth of individuals is zero, i.e., there is no outside money. When there is no default in equilibrium, there is indeterminacy. Positive default in every state of the world on some long-term loan endogenously creates positive liquid wealth that supports positive interest rates and resolves the aforementioned indeterminacy. Hence, a non-Ricardian policy across loan markets can determine the equilibrium allocations, while it allows the central bank to earn profits from seigniorage in order to compensate for any losses.

This is a preview of subscription content, access via your institution.


  1. Numerous papers use this framework: Tsomocos (2008) applies the argument to show the determinacy of international monetary equilibria, while Giraud and Tsomocos (2010) prove uniqueness under the limit-price exchange process. Espinoza et al. (2008) and Espinoza and Tsomocos (2014) use a similar framework to connect the supply of liquidity by the central bank to the term structure of interest rates and provide an explanation for the term premium. Goodhart et al. (2005, 2006, 2010), Tsenova (2014) and Tsomocos (2003) use the same model of money to analyze the financial stability in a monetary economy.

  2. See Buiter (2002), Cochrane (2001), Sims (1994) and Woodford (1994) for the importance of non-Ricardian policy in determining the equilibrium price level.

  3. The collateral constraint links the monetary value of required collateral to the total value of asset holdings for each asset k. Hence, the amount of collateral is endogenously determined, i.e., the monetary authority neither specifies the quantity of assets to be pledged as collateral nor their nominal price, but only its total value. We are grateful to an anonymous referee for pointing out how an alternative collateral requirement that links nominal loans to real asset holdings would resolve the indeterminacy of monetary equilibria if binding, which is typically the case when default obtains in equilibrium. In such an arrangement, the price of collateral would be determined a priori as in Shubik and Tsomocos (1992) who allow the monetary authority to set the exchange price of fiat money to gold (the durable asset in our framework). On the contrary, we focus on a collateral constraint that specifies the loan-to-value ratio for long-term loans and emphasize the role of default on some loans in supporting positive interest rates on other non-defaultable loans due to capability of rolling over loans. The argument behind our determinacy result relies crucially on the fact that default endogenously creates “outside” money such that positive interest rates can be supported rather than on the fact that default results in binding collateral constraints. Put differently, it is the positive interest rate and not the binding collateral constraint that pins down prices. However, we hasten to emphasize that our argument requires the overlapping of both defaultable and non-defaultable loans, which are the long-term and short-term loans in our framework, respectively, so that to allow for rolling over of debts. Most importantly, the non-defaultable loans need to carry positive interest rates and the demand for them should be positive in equilibrium. Naturally, agents would prefer to first take loans that default, but the borrowing capacity is restricted by the collateral constraint. If the benefits of borrowing at positive interest rates are high enough, i.e., if the “gains-to-trade” hypothesis is satisfied [(see Dubey and Geanakoplos (2003b)], then there will be demand for non-defaultable loans as well. Nevertheless, our argument may not be robust to a specification whereby these additional intra-period loans do not exist. In principle, our result should also obtain with only defaultable and non-defaultable overlapping intertemporal loans.

  4. To simplify the intricate equilibrium equations that arise with incomplete markets, we confine attention to “active” equilibria in which each agent chooses to buy something in every state. See Tsomocos (2008) for a formal treatment of indeterminacy in the presence of inactive markets.


  • Adão, B., Correia, I., Teles, P.: Short and long interest rate targets. J. Monet. Econ. 66, 95–107 (2014)

    Article  Google Scholar 

  • Balasko, Y., Cass, D.: The structure of financial equilibrium with exogenous yields: the case of incomplete markets. Econometrica 57(1), 135–162 (1989)

    Article  Google Scholar 

  • Bloise, G., Drèze, J.H., Polemarchakis, H.: Monetary equilibria over an infinite horizon. Econ. Theory 25, 51–74 (2005)

    Article  Google Scholar 

  • Buiter, W.H.: The fallacy of the fiscal theory of the price level: a critique. Econ. J. 112(481), 459–480 (2002)

    Article  Google Scholar 

  • Cochrane, J.H.: Long-term debt and optimal policy in the fiscal theory of the price level. Econometrica 69(1), 69–116 (2001)

    Article  Google Scholar 

  • Drèze, J.H., Polemarchakis, H.: Monetary equilibria. In: Debreu, G., Neufeind, W., Trockel, W. (eds.) Economic Essays: A Festschrift for Werner Hildenbrand, pp. 83–108. Springer, New York (2000)

    Google Scholar 

  • Dubey, P., Geanakoplos, J.: The value of money in a finite-horizon economy: a role for banks. In: Dasgupta, P., Gale, D., Hart, D., Maskin, E. (eds.) Economic Analysis of Markets and Games, Essays in Honor of Frank Hahn, pp. 407–444. MIT Press, Cambridge, MA (1992)

    Google Scholar 

  • Dubey, P., Geanakoplos, J.: Inside and outside money, gains to trade, and is-lm. Econ. Theory 21(2), 347–397 (2003)

    Article  Google Scholar 

  • Dubey, P., Geanakoplos, J.: Monetary equilibrium with missing markets. J. Math. Econ. 39(5–6), 585–618 (2003)

    Article  Google Scholar 

  • Dubey, P., Geanakoplos, J.: Determinacy with nominal assets and outside money. Econ. Theory 27(1), 79–106 (2006)

    Article  Google Scholar 

  • Espinoza, R.A., Goodhart, C.A.E., Tsomocos, D.P.: State prices, liquidity, and default. Econ. Theory 39(2), 177–194 (2008)

    Article  Google Scholar 

  • Espinoza, R.A., Tsomocos, D.P.: Monetary transaction costs and the term premium. Econ. Theory (2014). doi:10.1007/s00199-014-0817-z

  • Geanakoplos, J., Mas-Colell, A.: Real indeterminacy with financial assets. J. Econ. Theory 47(1), 22–38 (1989)

    Article  Google Scholar 

  • Geanakoplos, J., Zame, W.R.: Collateral equilibrium, I: a basic framework. Econ. Theory 56(3), 443–492 (2013)

    Article  Google Scholar 

  • Giraud, G., Tsomocos, D.P.: Nominal uniqueness and money non-neutrality in the limit-price exchange process. Econ. Theory 45(1–2), 303–348 (2010)

    Article  Google Scholar 

  • Goodhart, C.A.E., Tsomocos, D.P., Vardoulakis, A.P.: Modelling a housing and mortgage crisis. In: Alfaro, R.A. (ed.) Financial Stability, Monetary Policy, and Central Banking. Central Bank of Chile, Chile (2010)

  • Goodhart, C.A.E., Sunirand, P., Tsomocos, D.P.: A risk assessment model for banks. Ann. Finance 1(2), 197–224 (2005)

    Article  Google Scholar 

  • Goodhart, C.A.E., Sunirand, P., Tsomocos, D.P.: A model to analyse financial fragility. Econ. Theory 27(1), 107–142 (2006)

    Article  Google Scholar 

  • Goodhart, C.A.E., Peiris, U., Tsomocos, D.P., Vardoulakis, A.P.: On dividend restrictions and the collapse of the interbank market. Ann. Finance 6(4), 455–473 (2010)

    Article  Google Scholar 

  • Grandmont, J.-M.: Money and Value. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  • Gurley, J.G., Shaw, E.S.: Money in a Theory of Finance. Brookings Institution, Washington, DC (1960)

    Google Scholar 

  • Lin, L., Tsomocos, D.P., Vardoulakis, A.P.: Debt deflation effects of monetary policy. Finance and Economics Discussion Series 2014–37. Board of Governors of the Federal Reserve System (2014)

  • Lucas, R.E., Stokey, N.L.: Money and rates of interest in a cash-in-advance economy. Econometrica 55(3), 491–513 (1987)

    Article  Google Scholar 

  • Magill, M., Quinzii, M.: Real effects of money in general equilibrium. J. Math. Econ. 21(4), 301–342 (1992)

    Article  Google Scholar 

  • Magill, M., Quinzii, M.: Anchoring expectations of inflation. J. Math. Econ. 50, 86–105 (2014)

    Article  Google Scholar 

  • Magill, M., Quinzii, M.: Term structure and forward guidance as instruments of monetary policy. Econ. Theory 56(1), 1–32 (2014)

    Article  Google Scholar 

  • McMahon, M., Peiris, U., Polemarchakis, H.: Perils of quantitative easing. Warwick WP (2014)

  • Peiris, U., Vardoulakis, A.P.: Collateral and the efficiency of monetary policy. Econ. Theory (2015). doi:10.1007/s00199-014-0857-4

  • Peiris, U., Polemarchakis, H.: Monetary policy and quantitative easing in an open economy: prices, exchange rates and risk premia. Warwick WP (2014)

  • Shubik, M., Tsomocos, D.P.: A strategic market game with a mutual bank with fractional reserves and redemption in gold. J. Econ. 55(2), 123–150 (1992)

    Article  Google Scholar 

  • Sims, C.A.: A simple model for the study of the determination of the price level and the interaction of monetary and fiscal policy. Econ. Theory 4(3), 381–399 (1994)

    Article  Google Scholar 

  • Tsenova, T.: International monetary transmission with bank heterogeneity and default risk. Ann. Finance 10(2), 217–241 (2014)

    Article  Google Scholar 

  • Tsomocos, D.P.: Essays on Monetary Theory, Banking and General Economic Equilibrium. Ph.D Thesis, Yale University (1996)

  • Tsomocos, D.P.: Equilibrium analysis, banking and financial instability. J. Math. Econ. 39(5–6), 619–655 (2003)

    Article  Google Scholar 

  • Tsomocos, D.P.: Generic determinacy and money non-neutrality of international monetary equilibria. J. Math. Econ. 44(7–8), 866–887 (2008)

    Article  Google Scholar 

  • Woodford, M.: Monetary policy and price level determinacy in a cash-in-advance economy. Econ. Theory 4(3), 345–380 (1994)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Alexandros P. Vardoulakis.

Additional information

We are grateful to the seminar participants of the XXIII European Workshop on General Equilibrium Theory in Paris and to John Geanakoplos, Charles Goodhart, Udara Peiris, Skander Van den Heuvel and especially Herakles Polemarchakis for helpful comments. All remaining errors are ours. The views expressed in this paper are those of the authors and do not necessarily represent those of Federal Reserve Board of Governors, anyone in the Federal Reserve System, or any of the institutions with which we are affiliated.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lin, L., Tsomocos, D.P. & Vardoulakis, A.P. On default and uniqueness of monetary equilibria. Econ Theory 62, 245–264 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Determinacy
  • Liquid wealth
  • Default
  • Collateral
  • Monetary policy

JEL Classifications

  • D5
  • E4
  • E5