Financial intermediation and efficient risk sharing in two-period lived OLG models

Ritschel, Paul; Wenzelburger, Jan Article — Published Version Financial intermediation and efficient risk sharing in twoperiod lived OLG models Economic Theory Bulletin Provided in Cooperation with: Springer Nature Suggested Citation: Ritschel, Paul; Wenzelburger, Jan (2024) : Financial intermediation and efficient risk sharing in two-period lived OLG models, Economic Theory Bulletin, ISSN 2196-1093, Springer International Publishing, Cham, Vol. 12, Iss. 1, pp. 57-78, https://doi.org/10.1007/s40505-024-00263-z This Version is available at: https://hdl.handle.net/10419/315867 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Economic Theory Bulletin (2024) 12:57–78 https://doi.org/10.1007/s40505-024-00263-z RESEARCH ARTICLE Financial intermediation and efficient risk sharing in two-period lived OLG models Paul Ritschel1·Jan Wenzelburger1 Received: 18 November 2023 / Accepted: 26 February 2024 / Published online: 4 April 2024 © The Author(s) 2024 Abstract This article investigates a two-period lived overlapping-generations (OLG) model that incorporates financial intermediation. A risk-neutral bank offers loan and deposit contracts that insure risk-averse agents against idiosyncratic income shocks. Agents prefer financial intermediation to capital markets if it provides efficient risk sharing. The analysis demonstrates that in any two-period lived OLG model in which productive capital is increasing in investment levels, financial intermediation, when implemented for the purpose of efficient risk sharing, cannot instigate business cycles or complex dynamics. The resulting dynamics is monotonic and qualitatively indistinguishable from the dynamics of the classical OLG model by Diamond (Am Econ Rev 55(5):1126–1150, 1965). Business cycles may only occur if banks offer inefficient contracts. Efficient contracts will, in general, not induce dynamically efficient growth paths. Keywords Financial intermediation ·Overlapping generations ·Risk sharing · Business cycles ·Loan contracts JEL Classification D53 ·E32 ·E44 ·G21 ·O41 1 Introduction Understanding the pace and patterns of economic growth is one of the central topics in macroeconomics. The empirical evidence that a well-functioning financial system is vital for economic development is plentiful, e.g. see Levine (1997) or the excellent Paul Ritschel and Jan Wenzelburger contributed equally to this work. BPaul Ritschel [email protected] Jan Wenzelburger jan.wenzelbur[email protected] 1Faculty of Business Studies and Economics, University of Kaiserslautern-Landau, Gottlieb-Daimler-Str. 42, 67663 Kaiserslautern, Rhineland-Palatinate, Germany 123 58 P. Ritschel, J. Wenzelburger reviews of the empirical literature by Levine (2005) and Aziakpono (2011). Over the past four decades, relatively few theoretical contributions have incorporated financial intermediation into growth models. These try to link the promotion of economic growth to the fundamental functions that financial intermediaries carry out in an economy, cf. Pagano (1993). The seminal contribution by Greenwood and Jovanovic (1990), for example, highlights how risk sharing and the informational advantage of financial intermediaries encourage high-yield investments and economic growth. Bencivenga and Smith (1991), to mention another important contribution, extends the Diamond and Dybvig (1983) view by demonstrating that liquidity provision induces savings behaviour of agents that enhances capital accumulation. The literature on business cycles in OLG models with financial intermediation is relatively scarce. Williamson (1987) demonstrates that indivisibilities in investment projects may be a cause for business cycles. Smith (1998) finds that monopolistic financial intermediaries can increase the severity of existing business cycles. Azariadis and Smith (1998) show that bank-loan financed capital investments may generate business cycles if there is an adverse selection problem regarding the ability of the borrowers to honour their debt. Banerji et al. (2004) consider an OLG model in which loan and deposit contracts enable risk-averse agents to completely insure against idiosyncratic income shocks. They argue that risk sharing may expose the economy to endogenous fluctuations in the form of real-sector business cycles and conclude that the promotion of economic growth by financial intermediaries comes at the cost of the full variety of complex dynamics. Finally, the stochastic OLG model with financial intermediation developed in Gersbach and Wenzelburger (2003,2008,2012) exhibits persistent business cycles. Macroeconomic productivity shocks trigger the failure of individual production projects. This risk cannot be diversified away so that the model, unlike that in Banerji et al. (2004), has aggregate uncertainty. This article addresses the extent to which efficient risk sharing can induce endogenous business cycles in two-period lived OLG models, which, in the absence of financial intermediation, are known to admit only monotonic growth. Following on from Banerji et al. (2004) by allowing for the standard class of intertemporal preferences used in that literature, we find that a collective bank can implement the efficient allocation by offering suitable loan and deposit contracts. These contracts provide complete risk sharing and must enlarge the disposable income in order to be accepted by agents. This income effect, which in Banerji et al. (2004) is deemed responsible for causing endogenous fluctuations, is a consequence of a mere incentive problem and it turns out that it does not alter the qualitative dynamics of the economy. The key feature of our model is that productive capital and thus capital income is increasing in investment levels. We demonstrate that financial intermediation which implements the efficient allocation and diversifies away idiosyncratic risk does not generate business cycles or even complex dynamics as the dynamics of the economy is always monotonic. In our framework, agents’ incentive compatibility and participation constraints are explicit. Our analysis reveals that even though the bank maximises agents’ welfare, incentive problems remain so that the acceptance of an efficient contract is not as straightforward as one would expect. Contrary to Banerji et al. (2004), who argue that financial intermediation may generate a complex backwards dynamics, we will focus 123 Financial intermediation and efficient risk sharing... 59 on the forward dynamics of the economy. The reason is that the usefulness of the backwards dynamics for a forward-time interpretation is limited and that the analysis is often restricted to a limited range of model parameters in the neighbourhood of a steady-state solution, e.g. see Grandmont (1989) and Medio and Raines (2006). The remainder of this article is organised as follows. The next section lays out the basic model and all essential assumptions. In Sect.3, we formulate the decision problems of both agents and the bank. We then introduce our notion of an efficient contract and establish its existence and uniqueness. Section 4is dedicated to the dynamics induced by efficient contracts and contains our main results. Section5concludes. 2 Model prerequisites We consider a two-period lived OLG model with discrete time t=0,1,...,∞. There is a single perishable good that can be consumed and invested. At the beginning of each period t, a new generation comprising a unit-mass continuum of homogeneous agents is born. Agents are risk-averse and live for two periods. Their intertemporal preferences over consumption are represented by a life-cycle utility function U(c1,c2):= u(c1)+v(c2), where c1,c2≥0 denote youthful and old-age consumption, respectively. Assumption 1 (Preferences) The utility functions u,v :R+→Rare twice continuously differentiable, strictly increasing, strictly concave, and satisfy the Inada conditions. A young agent may become an entrepreneur by undertaking a risky production project, which may either be successful or fail. The likelihood of success depends on the amount of capital invested and is determined by a success function p :R+→(0,1] that stipulates the success probability p(I)of the capital investment I≥0. The uncertainty about the outcome of a project resolves one period after capital has been invested. A project generates a verifiable gross rate of return >0 if successful and zero if it fails.1Invoking the law of large numbers, the productive capital stock of the economy is (I):= p(I)I. The properties of the success function pare of central importance to our analysis and are stated in terms of properties of . Assumption 2 (Productive capital) The function :R+→R+, defined by (I)= p(I)I, is twice continuously differentiable, strictly increasing, and concave. 1As an alternative interpretation, one may think of the entrepreneur as an investor who invests capital into a firm. A failed project is then equivalent to the default of a firm. 123 60 P. Ritschel, J. Wenzelburger The important assumption for our results is that productive capital (I)is strictly increasing in the investment level I. The concavity of implies that the success probability p(I)is non-increasing in I.2Such a choice features the economic intuition that large-scale projects are more likely to fail. Example 1 The function (I)=κI 1+I, where 0 <κ≤1 is some constant, satisfies Assumption 2. The corresponding success probability p(I)=κ 1+Iis decreasing in I. The production sector of the economy is perfectly competitive. A neoclassical technology with constant returns to scale transforms labour N≥0 and real capital K≥ 0 into output. Capital depreciates fully during production. We denote by k:= K/N the capital-labour ratio and by f:R+→R+the production function of the representative firm in intensive form. Assumption 3 (Technology) The production function f:R+→R+is thrice continuously differentiable, strictly increasing, strictly concave, and satisfies the Inada conditions. Moreover, it holds that f(k)k f(k)>−1 and f(k)k f(k)>−2 for all k∈R+. The last two properties imposed on fin Assumption 3imply that capital income f(k)kis strictly increasing and strictly concave in the capital-labour ratio k. These properties facilitate the existence and uniqueness of an efficient loan contract and are satisfied by many standard production functions in the literature.3 The young generation constitutes the workforce of the economy. Each young agent supplies one unit of labour inelastically to a perfectly competitive labour market. Labour and capital are paid their marginal products. The old generation is retired and receives capital income only. Given a capital investment I, the productive capital stock of the subsequent period is k=(I)and is paid its marginal product =f((I)). The capital income of the old generation thus becomes g(I):= f((I)) (I). (1) The following properties of gare essential for our results. Lemma 1 (Capital income) Let Assumptions 2and 3be satisfied. Then capital income of the old generation g :R+→R+, defined by (1), is strictly increasing and strictly concave with g(0)=0. In the proof of Lemma 1, it is shown that under the hypotheses of Assumption 3, concavity of productive capital is a sufficient condition for strict concavity of capital income g. 2Concavity of implies (I)I (I)=p(I)I p(I)+1≤1sothatp(I)≤0 for all I>0. 3Assumption 3is fulfilled by the Cobb–Douglas production function and by a wide range of parameterisations of the CES production function. 123 Financial intermediation and efficient risk sharing... 61 3 Financial intermediation To transfer resources to the second period of their lives, young agents may invest part of their wage income into a production project which exposes them to the idiosyncratic risk of an old-age income shock. Agents may form an endogenous coalition in the form of a risk-neutral collective bank as a device to share this risk.4The bank offers young agents a loan contract (Bt,It,Rt), where Bt≥0 is the size of the loan, It≥0is the capital investment into the project, and Rt≥0 is the gross interest rate on loans. Agents who accept a loan contract are protected by limited liability as they do not have to repay the loan in case their project fails.5To finance its loans, the bank raises deposits from young agents by offering a risk-free gross rate rt≥0 on deposits. 3.1 Decision problems Each young agent must decide whether to invest into a project by accepting a loan contract or to undertake the project without funding from the bank instead. Independently of her investment decision, however, a young agent is also allowed to deposit part of her wage income at the bank. The decision problem of a young agent is thus the following. Suppose the bank offers the loan contract (Bt,It,Rt)and the deposit rate rton savings in period t. Consider first the case in which the agent accepts the loan contract. Given her wage income wt, the agent must decide on how much to consume and how much to save for retirement. By accepting the loan contract, her disposable income becomes wd t:= wt+Bt−It, so that youthful consumption is c1=wd t−D,(2) where D≥0 is the amount saved and deposited at the bank. Old-age consumption is c2g≥0 if the project is successful and c2b≥0 if the project fails. Since agents have limited liability, the constraint for old-age consumption reads c2g=rtD+π(It)−RtBt c2b=rtD,(3) where rtDare the proceeds from the deposits, π(It):= f((It))Itis the revenue from a successful project, and RtBtis the loan repayment obligation. The objective of a young agent is to maximise her expected utility of lifetime consumption. Inserting the budget constraints (2) and (3), the agent’s objective function 4For further details on financial-intermediary coalitions, we refer to the paper by Boyd and Prescott (1986) and, for an overview, to Freixas and Rochet (2008). 5We tacitly assume that the bank possesses a monitoring technology that enables it to observe agents’ investment behaviour and enforce the contract. A stipulated investment is a particular form of monitoring. For details we refer to the seminal contribution by Holmström and Tirole (1997) on the role of monitoring in settings with limited liability. 123 62 P. Ritschel, J. Wenzelburger becomes max 0≤D≤wd t u(wd t−D)+p(It)v(rtD+π(It)−RtBt)+(1−p(It)) v(rtD). (4) Given a loan contract (Bt,It,Rt), a deposit rate rt, and a wage rate wt, a solution to (4) is given by the agent’s savings function SR5 +→R+, which is defined by S(wt,Bt,It,Rt,rt):= argmax 0≤D≤wd t u(wd t−D)+p(It)v(rtD +π(It)−RtBt)+(1−p(It)) v(rtD). Inserting Sinto the objective function in (4) yields the value function for Problem (4), which is denoted by V(wt,Bt,It,Rt,rt). (5) Consider now the case in which the agent rejects the loan contract and undertakes the project without funding from the bank. To do so, she will invest the amount IA into the project and deposit the amount DAat the bank in order to safeguard old-age consumption against the failure of the project. Formally, the corresponding decision problem reads max IA,DAu(wt−DA−IA)+p(IA)v(rtDA+π(IA)) +(1−p(IA)) v(rtDA) s.t. IA,DA≥0 and IA+DA≤wt. (6) The value function associated with Problem (6) is well defined and stipulates the agent’s reservation utility, which for any given wtand rtis denoted by Ures(wt,rt). The decision problem of the bank is the following. Since the bank is collectively owned, it offers a loan contract (Bt,It,Rt)and a deposit rate rtso as to maximise the agent’s expected utility Vgiven in (5). Using the law of large numbers, the bank correctly anticipates that for any given It, the loan default rate is 1−p(It). Therefore, the two feasibility constraints of the bank are the profit constraint p(It)RtBt−rtDt≥0,(PrC) stating that bank profits must be non-negative, and the resource constraint Dt≥Bt,(RC) noting that the bank has no equity. Both the loan and the deposit contract have to be compatible with the young agent’s savings behaviour. Since the amount saved is at the 123 Financial intermediation and efficient risk sharing... 63 discretion of the agent, the bank has to fulfil the incentive compatibility constraint Dt=S(wt,Bt,It,Rt,rt)(IC) in order to obtain the amount of deposits required in (RC). Finally, since an agent may decide to invest without funding from the bank, the loan contract must be designed in such a way that the agent prefers the loan contract to undertaking the project without the bank. Formally, this participation constraint reads V(wt,Bt,It,Rt,rt)≥Ures(wt,rt), (PC) that is, the expected utility of accepting both the loan and the deposit contract is at least as high as the reservation utility. Given the wage rate wt, the decision problem of the bank thus takes the form max B,I,R,r≥0V(wt,B,I,R,r) s.t. p(I)RB −rS(wt,B,I,R,r)≥0,S(wt,B,I,R,r)≥B, and V(wt,B,I,R,r)≥Ures(wt,r). (7) 3.2 Efficient allocations We next establish the efficient allocation that a myopic social planner would implement. Given the wage rate wt, the planner’s objective in period tis to maximise the welfare of the generation born in t.6Applying the law of large numbers, the mass of successful agents in the subsequent period is p(I), while the mass of failed agents is 1−p(I). Capital income in the subsequent period is g(I), independently of the state of nature. The social planner’s maximisation problem thus becomes max I,c1,c2g,c2bu(c1)+p(I)v(c2g)+(1−p(I)) v(c2b) s.t. I,c1,c2g,c2b≥0,c1+I≤wt, and p(I)c2g+(1−p(I)) c2b≤g(I). (8) The solution (I t,c1 t,c2g t+1,c2b t+1)to Problem (8) will be referred to as efficient allocation. It is provided by the following proposition. Proposition 1 (Efficient allocation) Let Assumptions 1–3be satisfied and wt>0be given. Then Problem (8)admits a uniquely determined solution (I t,c1 t,c2g t+1,c2b t+1), where the efficient consumption plan is c1 t=wt−I t,c2g t+1=c2b t+1=g(I t) 6Since our research question is not concerned with intergenerational externalities as, for example, in Ennis and Keister (2003), a myopic social planner is justified. 123 64 P. Ritschel, J. Wenzelburger and the efficient investment level 0<I t<w tsolves max 0≤I≤wt u(wt−I)+v(g(I)). (9) Proposition 1states that if resources are allocated efficiently, the young generation consumes its wage income less the efficient investment level, while the old generation consumes aggregate capital income, which is perfectly smoothed out across both possible states of nature. Observe that the uniqueness of efficient allocations hinges on the concavity of the objective function in (9). This in turn is guaranteed, since u and vare strictly concave by Assumption 1and gis concave by Lemma 1. 3.3 Efficient contracts The natural question now is whether financial intermediation that offers loan and deposit contracts in line with Problem (7) can implement the efficient allocation determined in Proposition 1. In situations in which agents’ private actions are difficult to control, the arising incentive constraints make it questionable whether an efficient outcome can be achieved, cf. Myerson (1979). To address this problem, we will next define an efficient contract as a contract that implements the efficient allocation and is optimal for both agents and the bank. Definition 1 (Efficient contract) Given a wage rate wt, a loan contract (Bt,It,Rt) together with a deposit rate rtis called an efficient contract (in period t)ifthefollowing holds: (i) The quadruple (Bt,It,Rt,rt)solves the bank’s problem (7). (ii) The allocation induced by (Bt,It,Rt,rt)is efficient in the sense of Proposition 1. Observe that by definition, any efficient contract must be incentive compatible. With an efficient contract, agents are fully insured and old-age consumption is independent of the success of the project because the bank completely diversifies away idiosyncratic risk. Incentive compatibility of the efficient deposit rate rtimplies in particular that agents decide at their own discretion to save the amount of funds required for complete risk sharing. Our next proposition establishes the existence of a unique efficient contract.7 Proposition 2 (Existence of an efficient contract) Let Assumptions 1–3be satisfied and wt>0be given. Then there exists a uniquely determined efficient contract (Bt,It,Rt,rt), which is given by the following equations: (i) The investment level satisfies 0<It<w tand solves max 0≤I≤wt u(wt−I)+v(g(I)). (10) 7Note again that the uniqueness of the efficient investment level hinges on the concavity of the objective function in (10) for which the (strict) concavity of u,v,andgis a sufficient condition. 123 Financial intermediation and efficient risk sharing... 71 c1=wt−I. The first-order conditions are therefore −u(wt−I)+p(I)[v(c2g)−v(c2b)]+λ[g(I)−p(I)(c2g−c2b)]=0(A1) p(I)v(c2g)−λp(I)=0(A2) (1−p(I)) v(c2b)−λ(1−p(I)) =0(A3) and the complementary slackness condition is λ[g(I)−p(I)c2g−(1−p(I)) c2b]=0.(A4) Assumption 1implies that in an optimum, the constraint on the consumption plan must hold with equality. Since g(0)=0, the Inada conditions on vimply that I>0 because otherwise c2g=c2b=0by(A4). Therefore, 0 <p(I)<1. Conditions (A2) and (A3) then imply v(c2g)=v(c2b)=λ>0. Since v <0, it follows from (A4) that a social optimum requires c2g=c2b=g(I). (A5) Moreover, (A1) reduces to −u(wt−I)+v(g(I)) g(I)=0.(A6) Observe that (A6) is the first-order condition of the maximisation problem max 0≤I≤wt u(wt−I)+v(g(I)). (A7) The objective function in (A7) is either already a continuous function or can be transformed into a continuous function on the compact interval [0,w t]using the exponential function. Hence, a solution I tto (A7) exists. It follows that any solution (I t,c2g t+1,c2b t+1)to the first-order conditions (A1)–(A3) must satisfy (A5) with I t being a maximiser of Problem (A7). In other words, the social planner’s problem (8) reduces to Problem (A7), so that any maximiser I tof (A7) together with (A5) and c1 t=wt−I tis a maximiser of (8). The concavity of gimplies that the objective function in (A7) is strictly concave so that the social optimum is uniquely determined.  Proof of Proposition 2The proof comprises four steps. Step 1 (Relaxed problem). We establish the existence and uniqueness of a solution to Problem (7) without the participation constraint. The Lagrangian of Problem (7) without the participation constraint is L(B,I,R,r,λ 1,λ 2):= u(wt+B−I−D)+p(I)v(rD+π(I)−RB) +((1−p(I)) v(rD)+λ1(p(I)RB −rD)+λ2(D−B), (A8) 123 72 P. Ritschel, J. Wenzelburger where λ1,λ 2≥0 are the Lagrange multipliers and D=S(wt,B,I,R,r)to simplify notation. The four first-order conditions for a solution (Bt,It,Rt,rt)are: 0=p(It)Rtv(rtDt+π(It)−RtBt)−λ1+λ2−u(wt+Bt−It−Dt) +(λ1rt−λ2)∂S ∂B(wt,Bt,It,Rt,rt)(A9) 0=h(It)p(It)v(rtDt+π(It)−RtBt)+p(It)v(rtDt+π(It)−RtBt)−v(rtDt) +λ1p(It)RtBt−u(wt+Bt−It−Dt)−(λ1rt−λ2)∂S ∂I(wt,Bt,It,Rt,rt) (A10) 0=λ1−vrtDt+π(It)−RtBtp(It)Bt−(λ1rt−λ2)∂S ∂R(wt,Bt,It,Rt,rt) (A11) 0=p(It)vrtDt+π(It)−RtBt+[1−p(It)]v(rtDt)−λ1Dt −(λ1rt−λ2)∂S ∂r(wt,Bt,It,Rt,rt), (A12) where Dt=S(wt,Bt,It,Rt,rt). The two complementary slackness conditions are: λ1[p(It)RtBt−rtDt]=0 (A13) λ2(Dt−Bt)=0.(A14) Assume that λ1rt−λ2=0. We will show below with (A34) that in an optimum, this identity must hold. As a consequence, all terms involving derivatives of Sin the first-order conditions (A9)–(A12) are zero. Since p>0, (A11) is equivalent to λ1−v(rtDt+π(It)−RtBt)Bt=0.(A15) By Assumption 1,v>0 so that two cases can occur in (A15). First, Bt>0 and λ1=v(rtDt+π(It)−RtBt)>0.(A16) Second, Bt=0. Case 1. Since Bt>0 and λ1>0, (A13) requires p(It)RtBt=rtDt,(A17) stating that the profit constraint is binding. Inserting (A16), it follows that (A9) holds with λ2=u(wt+Bt−It−Dt)>0.(A18) Since λ2>0, (A14) implies that the resource constraint is binding, Bt=Dt>0.(A19) 123 Financial intermediation and efficient risk sharing... 73 Using (A19), it follows from (A17) that rt=p(It)Rt.(A20) Inserting (A16)into(A12) yields [1−p(It)]Dtv(rtDt)−v(rtDt+π(It)−RtBt)=0.(A21) (A21) has two possible solutions. First, since 0 <p(I)<1forI>0, It=0isa solution whenever p(0)=1. In this case, (A20) implies Rt=rtand thus RtBt=rtDt so that the attained utility level is u(wt)+v(0). By Assumption 1, this level cannot be optimal. Since v <0, the second solution to (A21)is RtBt=π(It). (A22) It follows from (A16) and (A18) that λ1=v(rtDt)>0 and λ2=u(wt−It)>0.(A23) Combining (A20) with (A22) yields Bt=g(It) rt .(A24) Since Dt=Bt,(A24) implies rtDt=g(It). (A25) Therefore, λ1=v(g(It)). (A26) Inserting (A19), (A22), (A25), and (A26), Condition (A10) reduces to −u(wt−It)+v(g(It)) g(It)=0.(A27) Condition (A27) determines the optimal investment level It. Observe that (A27)isthe first-order condition for the maximisation problem max 0≤I≤wt u(wt−I)+v(g(I)). (A28) Equations (A22) and (A25) imply that any utility-maximising consumption plan of the relaxed problem (A8) has to satisfy c1 t=wt−Itand c2g t+1=c2b t+1=rtDt=g(It). (A29) Hence, any solution to (A8) is already determined by a solution Itto Problem (A28). Since Problem (A28) coincides with Problem (10), existence and uniqueness of 0 < It<w tobtain from the same arguments as presented in the proof of Proposition 1. 123 74 P. Ritschel, J. Wenzelburger Case 2. If Bt=0, then the profit constraint (PrC) implies that rtDt=0. The strict concavity of vyields u(wt−I−D)+p(I)v(π(I)) +(1−p(I)) v(0)<u(wt−I)+v(g(I)) (A30) for all I,D>0 with I+D≤wt. Note that the r.h.s. of (A30) is the objective function of Problem (A28), which assumes its maximum in 0 <It<w twith Itbeing determined by (A27). Hence, Bt=0 cannot be optimal. It follows that Bt>0 is optimal and that the optimal solution to the relaxed problem (A8) is uniquely determined by (A27) together with (A29). Step 2 (Incentive compatibility). In Step 1, the optimal deposit rate rthas not yet been determined. Given the loan contract (Bt,It,Rt)determined in Step 1, the incentive constraint (IC) implies that deposits Dt=S(wt,Bt,It,Rt,rt)must satisfy the first-order condition u(wt+Bt−It−Dt)=p(It)v(rtDt+π(It)−RtBt)+(1−p(It)) v(rtDt)rt. (A31) Inserting (A19), (A22), and (A25), Condition (A31) simplifies to −u(wt−It)+v(g(It)) rt=0.(A32) A comparison of (A27) with (A32) shows that the optimal deposit rate is rt=g(It). (A33) Using (A23), (A26), and (A32), it follows that rtsatisfies λ1rt−λ2=v(g(It)) rt−u(wt−It)=0,(A34) thus justifying the assumption made at the outset of the proof. Finally, inserting (A33) into (A24) and (A20) yields Bt=g(It) g(It)and Rt=g(It) p(It). Step 3 (Efficiency). To see that the contract (Bt,It,Rt,rt)computed above implements the efficient allocation, recall that the first-order conditions (A6) and (A27) coincide, so that It=I tis the efficient investment level. It follows from (A22) and (A25) that c2g t+1=c2b t+1=rtS(wt,It,Bt,Rt,rt)=g(It)=g(I t)=c2g t+1=c2b t+1. Thus, (Bt,It,Rt,rt)implements the efficient allocation. 123 Financial intermediation and efficient risk sharing... 75 Step 4 (Participation constraint). We prove that the relaxed problem without the participation constraint (A8) has the same solution as Problem (7) by showing that agents will accept the efficient contract (Bt,It,Rt,rt)computed above. An agent who rejects the efficient loan contract may save and invest with idiosyncratic risk, solving Problem (6). By the strict concavity of v, the objective function in (6) satisfies u(wt−IA−DA)+p(IA)v(rtDA+π(IA)) +[1−p(IA)]v(rtDA) ≤u(wt−IA−DA)+v(rtDA+g(IA)) (A35) for all IA,DA≥0 with IA+DA≤wt. Replacing the objective function in Problem (6) with the r.h.s. of Inequality (A35), an auxiliary problem obtains. We will next establish that the uniquely determined maximiser of this auxiliary problem is (IA t=It,DA t=0), where Itis the efficient investment level, and show that agents will be worse off rejecting the efficient contract. Observe first that the auxiliary objective function is strictly concave if gis strictly concave. The Inada conditions on uand vimply that any solution (IA t,DA t)to Problem (6) must satisfy 0 <IA t+DA t<w t. Thus, there are three possible solutions to the auxiliary problem. Case 1: IA t=0, DA t>0. The resulting first-order conditions in this case read −u(wt−DA)+v(rtDA)g(0)+λ1=0 (A36) −u(wt−DA)+v(rtDA)rt=0.(A37) The Inada conditions imply that a solution 0 <DA t<w tto (A37) exists. Inserting (A37)into(A36), we see that (IA t=0,DA t)is a possible maximum if λ1=v(rtDA t)[rt−g(0)]≥0. However, since rt=g(It)and g <0, it follows that λ1must be negative. Hence, (IA t=0,DA t)does not satisfy the first-order conditions. Case 2: IA t>0, DA t=0. The corresponding first-order conditions are −u(wt−IA)+v(g(IA)) g(IA)=0 (A38) −u(wt−IA)+v(g(IA)) rt+λ2=0.(A39) As shown in the proof of Proposition 1, the unique solution to (A38) is the efficient investment level IA t=It. Since rt=g(It),itfollowsthat(IA t=It,DA t=0) together with λ2=0 solves the first-order conditions. Case 3: IA t>0, DA t>0. The resulting first-order conditions are −u(wt−IA−DA)+v(rtDA+g(IA)) g(IA)=0(A40) −u(wt−IA−DA)+v(rtDA+g(IA)) rt=0.(A41) 123 76 P. Ritschel, J. Wenzelburger A comparison of (A40) with (A41) shows that any solution IA trequires rt=g(IA t). Since rt=g(It)and g <0, it follows that IA t=It. A comparison with Case 2 shows that (IA t=It,DA t=0)solves (A40) and (A41). Since both equations are strictly decreasing in D, no solution with positive savings (IA t=It,DA t>0)exists. These considerations show that the maximum of the auxiliary problem obtains in (IA t=It,DA t=0)and achieves the utility level u(wt−It)+v(g(It)). Inequality (A35) implies Ures(wt,rt)≤u(wt−It)+v(g(It)) =V(wt,Bt,It,Rt,rt), showing that agents are indeed willing to accept the efficient contract (Bt,It,Rt,rt).  Proof of Theorem 3Endogenous fluctuations are ruled out if G>0. Differentiating (18) yields G(k)=I(w(k))I(w(k)) w(k). (A42) By Assumption 2,>0. Moreover, w>0 by Assumption 3. We next show that I>0 such that G>0 holds. The investment function Iis defined by the first-order condition −u(w −I(w)) +vg(I(w))g(I(w)) =0.(A43) Differentiating (A43) yields I(w) =u(w −I(w)) u(w −I(w)) +vg(I(w))g(I(w))2+vg(I(w))g(I(w)). (A44) By Assumption 1,u <0 such that the numerator in (A44) is strictly negative. Since 0<I(w) < w is a maximiser, the second-order condition for the objective function in (A28) is satisfied, implying that the denominator in (A44) is strictly negative. Thus, we conclude that (A44) is strictly positive so that G>0.  Proof of Proposition 4(i) Steady states of Gare determined by solutions k≥0to k! =I(w(k)).(A45) Note that (0)=0. If w(0)=0, then I(w(0)) =0 and, consequently, k=0solves (A45). On the contrary, if w(0)>0, then the Inada conditions stated in Assumption 1 imply 0 <I(w(0)) < w(0). Since >0, it follows that I(w(0))>0, showing that k=0 cannot solve (A45). Hence, k=0solves(A45) if and only if w(0)=0. (ii) It follows from the definition of and Proposition 2that 0≤G(k)=I(w(k))≤f(k)for all k≥0.(A46) If w(0)>0, then G(0)>0, so that in the local neighbourhood of zero, we have G(k)>k. On the other hand, if w(0)=0, then G(0)=0. In this case, it follows from the property limk→0G(k)>1 that in the local neighbourhood of zero, 123 Financial intermediation and efficient risk sharing... 77 G(k)>kholds. Inequality (A46), the strict concavity of f, and the Inada condition limk→∞ f(k)=0 then imply that there exists at least one k>0 that solves (A45). Of these solutions, the largest one must satisfy 0 <G(k)<1 and thus be asymptotically stable.  Proof of Lemma 2Assumptions 2and 3imply that I→ f((I)) is strictly increasing and strictly concave. Hence, a solution IGto max I≥0f((I)) −I(A47) is unique, if it exists. Observe that f((I)) −I≤f(I)−Ifor all I>0. It follows from Assumption 3that the function I→ f(I)−Ihas a unique maximum. 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