1-λof the assets of the financial intermediary. Hence lenders will only supply funds if the gainsfrom stealing are lower than the continuation value of the financial intermediary. This gives rise tothe following constraint:Vj,t≥λ(qktskj,t+qbtsbj,t)⇒νktqktskj,t+νbtqbtsbj,t+ηtnj,t≥λ(qktskj,t+qbtsbj,t)(5)The optimization problem can now be formulated in the following way:max{qktskj,t,qbtsbj,t}Vj,t,s.t.Vj,t≥λ(qktskj,t+qbtsbj,t)From the first order conditions we find thatνbt=νkt.Hence the leverage constraint (5) can berewritten in the following way:νkt(qktskj,t+qbtsbj,t) +ηtnj,t≥λ(qktskj,t+qbtsbj,t)⇒qktskj,t+qbtsbj,t≤φtnj,t,φt=ηtλ-νkt(6)whereφtdenotes the ratio of assets to net worth, which can be seen as the leverage constraint ofthe financial intermediary.The intuition for the leverage constraint is straightforward: a highershadow value of assetsνktimplies a higher value from an additional unit of assets, which raises thecontinuation value of the financial intermediary, thereby making it less likely that the banker willsteal. A higher shadow value of net worthηtimplies a higher expected profit from an additionalunit of net worth, while a higher fractionλimplies that the banker can steal a larger fractionof assets, which induces the household to provide less funds to the banker, resulting in a lowerleverage ratio everything else equal. Substitution of the conjectured solution into the right handside of the Bellman equation gives the following expression for the continuation value of the financialintermediary:Vj,t=EthΩt+1nj,t+1i,Ωt+1=βΛt,t+1(1-θ) +θ[ηt+1+νkt+1φt+1]Ωt+1can be thought of as a stochastic discount factor that incorporates the financial friction. Now9

substitute the expression for next period’s net worth into the expression above:Vj,t=EthΩt+1nj,t+1i=EthΩt+1(1 +rkt+1)qktskj,t+ (1 +rbt+1)qbtsbj,t-(1 +rdt+1)dj,t+ngj,t+1-˜ngj,t+1i=EthΩt+1(rkt+1-rdt+1)qktskj,t+(rbt+1-rdt+1)qbtsbj,t+(1 +rdt+1+τnt+1-˜τnt+1)nj,ti(7)After combining the conjectured solution with (4), we find the following first order conditions:ηt=EthΩt+1(1 +rdt+1+τnt+1-˜τnt+1)i(8)νkt=EthΩt+1(rkt+1-rdt+1)i(9)νbt=νkt=EthΩt+1(rbt+1-rdt+1)i(10)Ωt+1=βΛt,t+1(1-θ) +θ[ηt+1+νkt+1φt+1]2.2.1Financial sector supportWe assume that support provided to an individual intermediary, if provided, will be proportionalto the intermediary’s net worth in the previous period. Hence individual financial support is givenby:ngj,t=τntnj,t-1,ζ≤0,l≥0τnt=ζ(ξt-l-ξ)Repayment of the support is parametrized proportionally to the sector’s net worth in the periodpreceding the pay back period:˜ngj,t= ˜τntnj,t-1where ˜τntis a scaling factor that is obviously time dependent and incorporates the return paid bythe sector to the government over the support funds.2.2.2Aggregation of financial variablesIntegrating the individual balance sheets of the financial intermediaries yields the aggregate balancesheet of the financial sector:pt=nt+dt(11)Aggregation over the asset side of the balance sheet gives the composition of the aggregated financialsystem:pt=qktskt+qbtsbt(12)φtdoes not depend on firm specific factors, so we can aggregate the leverage constraint (6) acrossfinancial intermediaries to link sector wide assets and net worth:pt=qktskt+qbtsbt=φtnt(13)10

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