3.7 Appendix: Proofs
3.7.5 Proofs for Applications
Putting (3.38) and (3.39) together, we get Tg(α∗) = Eg[min(c(h), L)1{e<K1}]
Eg[min(K2, e)1{e<K1}] ≥ Ef[min(c(h), L)1{e<K1}]
Ef[min(K2, e)1{e<K1}] =Tf(α∗) =α∗ which concludes the proof.
Proof of Corollary 3.4.1. Comparing the income-increasing payment of the contribution-based system
α∗min(K2, e)
to the income-constant payment of the premium-based system E[min(L, c(h))]
yields the existence of a threshold e∗ ∈ [0,e] such that for all¯ e < e∗ we have α∗min(K2, e) <
E[min(L, c(h))], and for all e > e∗ we have α∗min(K2, e) > E[min(L, c(h))]. As health benefits are equal in both system customers with incomee > e∗enjoy a higher utility and customers with incomee < e∗ enjoy a lower utility in the premium-based system.
We now argue thate∗ ∈(0,¯e). Firstly, observe that α∗ = E[min(L, c(h))1P U(α∗)]
E[min(K2, e)1P U(α∗)] ≥ E[min(L, c(h))]
E[min(K2, e)] ,
where the equality follows from α∗ being a fixed point of T(·), and the inequality follows from (3.15). Therefore, we can conclude that
α∗min(K2,e)¯ ≥ E[min(L, c(h))]
E[min(K2, e)] min(K2,¯e)>E[min(L, c(h))].
Secondly, note that
α∗min(K2,0) = 0<E[min(L, c(h)], which concludes the proof.
Proof of Proposition 3.4.2. Fix a payment p(h, e) for each customer type. Given this set of payments, welfare is
W(p(h, e)) =E[u(min(c(h), L)−c(h) +e−p(h, e))]. (3.42)
Proof of (i). SetK1 = ¯e. Recall that the payment in the contribution-based system isα∗min(K2, e), whereas it isA∗ =E[min(L, c(h))] in the premium-based system. AsK1= ¯e, PU insures all cus-tomers,1P U = 1 and budget-balancing of PU implies
α∗E[min(K2, e)] =E[min(L, c(h))] =A∗. (3.43) Note that the result still holds if customers are allowed to randomize, i.e., if β(h, e)∈ [0,1] denotes the probability that customer-(h, e) chooses PM1.
To save on notation define ψ(h, e) = min(c(h), L)−c(h) +e and note that ψ(·,·) is increasing in both arguments. Consider the welfare difference between the premium-based system and the contribution-based system, with (3.43)
E[u(ψ(h, e)−α∗ E[min(K2, e)])]−E[u(ψ(h, e)−α∗min(K2, e))]
<E
u0(ψ(h, e)−α∗min(K2, e)) (α∗min(K2, e)−α∗ E[min(K2, e)])
, (3.44) where the inequality follows from strict concavity ofu(·) . Observe that
1. u0(ψ(h, e)−α∗min(K2, e)) is decreasing in (h, e) becauseu0(·) is decreasing andψ(h, e)− α∗min(K2, e) is increasing in (h, e) asα∗ ≤1.
2. α∗min(K2, e)−α∗ E[min(K2, e)] is weakly increasing in (h, e).
Hence, the FKG inequality implies that (3.44) is bounded from above by the constant E
u0(ψ(h, e)−α∗min(K2, e))
E[α∗(min(K2, e)−E[min(K2, e)])] = 0, where the last equality follows from
E[α∗(min(K2, e)−E[min(K2, e)])] = 0. (3.45) Therefore, the contribution-based system with K1 = ¯e gives the population a strictly higher welfare than the premium-based system. Recall that welfare is increasing inK1. Thus, for suffi-ciently highK1 the contribution-based system is welfare-dominant.
Proof of (ii). Consider the income redistribution scheme that is defined by the transfer τ(e) to agent with income e, where
τ(e) =α∗E[min(K2, e)]−α∗min(K2, e).
By definition the premium-based system together with this income redistribution scheme gives the population the same welfare as the welfare-optimal contribution-based system, i.e., the system with K1 = ¯e. Furthermore, (3.45) implies that the income redistribution scheme is budget-balanced.
Proofs for Welfare-Optimal Payments
Proof of Proposition 3.4.3. LetA:=E[min(c(h), L)] be the aggregate health benefits of the population. Formally, we consider the problem
max
p(h,e) E[u(min(c(h), L)−c(h) +e−p(h, e))], (3.46)
s.t. A≤E[p(h, e)]. (3.47)
The Lagrangian
E[u(min(c(h), L)−c(h) +e−p(h, e)) +λ(p(h, e)−A)]
yields the first-order condition
u0(min(c(h), L)−c(h) +e−p(h, e)) =λ. (3.48) Note thatu0(·) is strictly decreasing. Solving forp(h, e) and inserting into the constraint, (3.47), gives
A=E
−u0−1(λ) + min(c(h), L)−c(h) +e . Using the definition of A, we obtain
λ=u0(E[e−c(h)]). (3.49)
Equating (3.48) and (3.49) yields
u0(min(c(h), L)−c(h) +e−p(h, e)) =u0(E[e−c(h)]). (3.50) Again exploiting thatu0(·) is strictly decreasing and after rearranging terms we obtain
popt(h, e) = min(c(h), L) +e−c(h)−E[e−c(h)].
Proof of Proposition 3.4.4. We start by rewriting (3.46) to account for the fact that the payment may not depend on h. For clarity we spell out all expectations explicitly.
max
p(e)
Z
E
Z
H
u(min(c(h), L)−c(h) +e−p(e))f(h|e) dh f(e) de, (3.51) s.t. A≤
Z
E
p(e)f(e) de. (3.52)
The Lagrangian for the problem is Z
E
Z
H
u(min(c(h), L)−c(h) +e−p(e))f(h|e) dh+λ(p(e)−A)f(e) de.
Using Leibniz’s integral rule we obtain the first-order condition Z
H
u0(min(c(h), L)−c(h) +e−p(e))f(h|e) dh−λ= 0. (3.53) (3.53) definesp as an implicit function ofe. Denote the left side of (3.53) byG(e, p). Then
∂G(e, p)
∂p =
Z
H
−u00(min(c(h), L)−c(h) +e−p)f(h|e) dh >0, (3.54) where the last inequality follows from strict concavity ofu(·). Furthermore
−∂G(e, p)
∂e = ∂G(e, p)
∂p +
Z
H
−u0(min(c(h), L)−c(h) +e−p) ∂f(h|e)
∂e dh. (3.55)
Rewrite the second term on the right side of inequality (3.55) as Z
H
−u0(min(c(h), L)−c(h) +e−p) ∂logf(h|e)
∂e f(h|e) dh.
Observe that:
1. By affiliation ∂log∂ef(h|e) is increasing inh. Indeed, we have 0≤ ∂2logf(h, e)
∂e ∂h = ∂2log(f(h|e)f(e))
∂e ∂h = ∂
∂h
∂logf(h|e)
∂e
.
2. −u0(min(c(h), L)−c(h) +e−p) is increasing inh because min(c(h), L)−c(h) is increasing and −u0(·) is increasing by concavity.
Neglecting the argument of −u0(·) for convenience and applying the FKG inequality we get Z
H
−u0(·) ∂logf(h|e)
∂e f(h|e) dh≥ Z
H
−u0(·)f(h|e) dh Z
H
∂logf(h|e)
∂e f(h|e) dh. (3.56) Rewriting the second term on the right hand side of inequality (3.56) and using Lebesgue’s dominated convergence theorem we see that
Z
H
∂logf(h|e)
∂e f(h|e) dh= Z
H
∂f(h|e)
∂e dh= ∂
∂e Z
H
f(h|e) dh
= 0.
sinceR
Hf(h|e) dh= 1. Consequently, note that Z
H
−u0(·) ∂logf(h|e)
∂e f(h|e) dh≥0. (3.57)
Applying the implicit function theorem, we conclude that dˆpopt
de = −∂G(e,p)∂e
∂G(e,p)
∂p
=
∂G(e,p)
∂p
∂G(e,p)
∂p
+ R
H−u0(·) ∂log∂ef(h|e) f(h|e) dh
∂G(e,p)
∂p
≥1,
where the inequality follows from (3.54) and (3.57).
Bibliography
Basel Committee on Banking Supervision and International Association of Deposit Insurers . Core principles for effective deposit insurance systems, 2009.
Viral V. Acharya, Sreedhar T. Bharath, Anand Srinivasan, et al. Understanding the recovery rates on defaulted securities. Centre for Economic Policy Research, 2003.
Tobias Adrian and Adam B. Ashcraft. Shadow banking regulation. FRB of New York Staff Report, (559), 2012.
Michalis Attalides, Antonis Polemitis, and Stavros Zenios. Cyprus: Six reasons for staying within the euro and one question about whether it can. 2015.
Jennie Bai, Arvind Krishnamurthy, and Charles-Henri Weymuller. Measuring liquidity mismatch in the banking sector. Available at SSRN 2343043, 2014.
Pedro Pita Barros. Cream-skimming, incentives for efficiency and payment system. Journal of Health Economics, 22(3):419–443, 2003.
BCBS Basel III. The liquidity coverage ratio and liquidity risk monitoring tools. Bank for International Settlements, 2013.
Timothy Besley. Publicly provided disaster insurance for health and the control of moral hazard.
Journal of Public Economics, 39(2):141–156, 1989.
˚Ake Blomqvist and Henrik Horn. Public health insurance and optimal income taxation. Journal of Public Economics, 24(3):353–371, 1984.
˚Ake Blomqvist and Per-Olov Johansson. Economic efficiency and mixed public/private insurance.
Journal of Public Economics, 66(3):505–516, 1997.
Arturo Bris, Ivo Welch, and Ning Zhu. The costs of bankruptcy: Chapter 7 liquidation versus chapter 11 reorganization. The Journal of Finance, 61(3):1253–1303, 2006.
Markus K. Brunnermeier and Lasse Heje Pedersen. Market liquidity and funding liquidity.Review of Financial studies, 22(6):2201–2238, 2009.
Markus K. Brunnermeier, Gary Gorton, and Arvind Krishnamurthy. Risk topography. InNBER Macroeconomics Annual 2011, Volume 26, pages 149–176. University of Chicago Press, 2011.
Markus K. Brunnermeier, Gary B. Gorton, and Arvind Krishnamurthy. Liquidity mismatch measurement. Available at SSRN, 2014.
John Bryant. A model of reserves, bank runs, and deposit insurance. Journal of banking &
finance, 4(4):335–344, 1980.
Jeremy I. Bulow, John D. Geanakoplos, and Paul D. Klemperer. Multimarket oligopoly: Strategic substitutes and complements. The Journal of Political Economy, pages 488–511, 1985.
Hans Carlsson and Eric Van Damme. Global games and equilibrium selection. Econometrica:
Journal of the Econometric Society, pages 989–1018, 1993.
Varadarajan V. Chari and Ravi Jagannathan. Banking panics, information, and rational expec-tations equilibrium. Journal of Finance, pages 749–761, 1988.
Kalman J. Cohen. Improving the capital adequacy formula. Journal of Bank Research, 1(1):
13–16, 1970.
Basel Committee et al. Basel iii: A global regulatory framework for more resilient banks and banking systems. Basel Committee on Banking Supervision, Basel, 2010.
Helmuth Cremer and Pierre Pestieau. Redistributive taxation and social insurance.International Tax and Public Finance, 3(3):281–295, 1996.
Tri Vi Dang, Gary Gorton, and Bengt Holmstr¨om. Haircuts and repo chains. Technical report, Working paper, 2013.
Sergei A. Davydenko. Insolvency, illiquidity, and the risk of default. Technical report, Working Paper. University of Toronto, 2012.
Sergei A. Davydenko and Julian R. Franks. Do bankruptcy codes matter? a study of defaults in france, germany, and the uk. The Journal of Finance, 63(2):565–608, 2008.
Angus S. Deaton and Christina H. Paxson. Aging and inequality in income and health. American Economic Review, pages 248–253, 1998.
Douglas W. Diamond and Philip H. Dybvig. Bank runs, deposit insurance, and liquidity. The journal of political economy, pages 401–419, 1983.
Thomas M. Eisenbach. Rollover risk as market discipline: A two-sided inefficiency. SSRN 1970451, 2013.
European Central Bank. Ela procedures.
Hans Fehr, Heinrich Jess, and BfA Berlin. Health premiums or health contributions? an evalua-tion of health care reform opevalua-tions in germany. Schmollers Jahrbuch, 126(1):21–57, 2006.
Nathan Foley-Fisher, Borghan Narajabad, Stephane Verani, et al. Self-fulfilling runs: Evidence from the us life insurance industry. Technical report, Board of Governors of the Federal Reserve System (US), 2015.
Thierry Foucault, Marco Pagano, and Ailsa R¨oell. Market Liquidity: Theory, Evidence, and Policy. Oxford University Press, 2013.
Julian R. Franks and Walter N. Torous. A comparison of financial recontracting in distressed exchanges and chapter 11 reorganizations.Journal of financial economics, 35(3):349–370, 1994.
Frederick T. Furlong and Michael C. Keeley. Capital regulation and bank risk-taking: A note.
Journal of banking & finance, 13(6):883–891, 1989.
Lawrence G. Goldberg and Sylvia C. Hudgins. Response of uninsured depositors to impending s&l failures: evidence of depositor discipline.The Quarterly Review of Economics and Finance, 36(3):311–325, 1996.
Lawrence G. Goldberg and Sylvia C. Hudgins. Depositor discipline and changing strategies for regulating thrift institutions. Journal of Financial Economics, 63(2):263–274, 2002.
Itay Goldstein and Ady Pauzner. Demand-deposit contracts and the probability of bank runs.
The Journal of Finance, 60(3):1293–1327, 2005.
Gary Gorton and Andrew Metrick. Securitized banking and the run on repo.Journal of Financial economics, 104(3):425–451, 2012.
Gary Gorton, Andrew Metrick, Andrei Shleifer, and Daniel K. Tarullo. Regulating the shadow banking system [with comments and discussion].Brookings Papers on Economic Activity, pages 261–312, 2010.
Gary B. Gorton and Andrew Metrick. Haircuts. Technical report, National Bureau of Economic Research, 2009.
Zhiguo He and Wei Xiong. Dynamic debt runs. Review of Financial Studies, page hhs004, 2012.
Karl Hinrichs. The impact of german health insurance reforms on redistribution and the culture of solidarity. Journal of Health Politics, Policy and Law, 20(3):653–687, 1995.
Bengt Holmstrom. Understanding the role of debt in the financial system. 2015.
Rajkamal Iyer and Manju Puri. Understanding bank runs: the importance of depositor-bank relationships and networks. Technical report, National Bureau of Economic Research, 2008.
Charles J. Jacklin and Sudipto Bhattacharya. Distinguishing panics and information-based bank runs: Welfare and policy implications.The Journal of Political Economy, pages 568–592, 1988.
Klaus Jacobs and Sabine Schulze. Systemwettbewerb zwischen gesetzlicher und privater kranken-versicherung: Idealbild oder schim¨are. GGW, H, 1:7–18, 2004.
Larry Karp, In Ho Lee, and Robin Mason. A global game with strategic substitutes and comple-ments. Games and Economic Behavior, 60(1):155–175, 2007.
Alexander Kemnitz. A simple model of health insurance competition.German Economic Review, 14(4):432–448, 2013.
Daesik Kim and Anthony M. Santomero. Risk in banking and capital regulation. The Journal of Finance, 43(5):1219–1233, 1988.
Philipp K¨onig, Tijmen R. Dani ¨Els, and Bank NV De Nederlendesche. Illiquidity risk and liquidity regulation. 2014.
Margaret Meyer and Bruno Strulovici. Beyond correlation: Measuring interdependence through complementarities. 2015.
Robert J. Milgrom, Paul R.and Weber. A theory of auctions and competitive bidding. Econo-metrica: Journal of the Econometric Society, pages 1089–1122, 1982.
James A. Mirrlees. An exploration in the theory of optimum income taxation. The review of economic studies, 38(2):175–208, 1971.
Stephen Morris and Hyun Song Shin. Unique equilibrium in a model of self-fulfilling currency attacks. American Economic Review, pages 587–597, 1998.
Stephen Morris and Hyun Song Shin. Global games: theory and applications. 2001.
Stephen Morris and Hyun Song Shin. Illiquidity component of credit risk. Work. Pap., Princet.
Univ, 2009.
Werner Neudeck and Konrad Podczeck. Adverse selection and regulation in health insurance markets. Journal of Health Economics, 15(4):387–408, 1996.
Mark V. Pauly. Is cream-skimming a problem for the competitive medical market? Journal of Health Economics, 3(1):87–95, 1984.
Alessandro Petretto. Optimal social health insurance with supplementary private insurance.
Journal of Health Economics, 18(6):727–745, 1999.
Andrew Postlewaite and Xavier Vives. Bank runs as an equilibrium phenomenon. The Journal of Political Economy, pages 485–491, 1987.
Jean-Charles Rochet. Incentives, redistribution and social insurance.The Geneva Papers on Risk and Insurance Theory, 16(2):143–165, 1991.
Jean-Charles Rochet and Xavier Vives. Coordination failures and the lender of last resort: was bagehot right after all? Journal of the European Economic Association, 2(6):1116–1147, 2004.
Jean-Charles Rochet et al. Liquidity regulation and the lender of last resort. Financial Stability Review, 11:45–52, 2008.
Michael Rothschild and Joseph Stiglitz. Equilibrium in competitive insurance markets: An essay on the economics of imperfect information. In Foundations of Insurance Economics, pages 355–375. Springer, 1976.
Linda M. Schilling. Capital structure, liquidity and miscoordination on runs. Liquidity and Miscoordination on Runs (October 14, 2015), 2015.
Moshe Shaked and J. George Shanthikumar. Stochastic orders. Springer Science & Business Media, 2007.
Hyun Song Shin. Reflections on northern rock: the bank run that heralded the global financial crisis. The Journal of Economic Perspectives, pages 101–120, 2009.
Michal Szkup. Debt overhang and rollover risk! 2015.
Sarah Thomson and Elias Mossialos. Choice of public or private health insurance: learning from the experience of germany and the netherlands. Journal of European Social Policy, 16(4):
315–327, 2006.
Karin S. Thorburn. Bankruptcy auctions: costs, debt recovery, and firm survival. Journal of financial economics, 58(3):337–368, 2000.
Fabrice Tourre. Debt runs and the value of liquidity reserves. 2015.
Xavier Vives. Strategic complementarity, fragility, and regulation. Review of Financial Studies, 27(12):3547–3592, 2014.
Wissenschaftlicher Beirat beim BMF. Nachhaltige Finanzierung der Renten- und Krankenver-sicherung, 2004.
Markus Worz and Reinhard Busse. Analysing the impact of health-care system change in the eu member states-germany. Health Economics-Chichester, 14(1):S133, 2005.