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A Model with Explicit Solution Jesús Fernández-Villaverde University of Pennsylvania December 2, 2012 Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 1 / 34 Motivation Brunnermeier and Sannikov, 2012. Full equilibrium dynamics of an economy with …nancial frictions: 1 Nonlinearity: model will respond very di¤erently to small and large shocks. 2 Volatility paradox: lower values of exogenous risk may lead to higher levels of endogenous risk. Features: 1 Continuous time. 2 We have more productive but less patient agents borrowing from less productive but more patient agents. Financial frictions di¢ cult the ‡ow of funds between both groups. Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 2 / 34 Preferences Continuum of in…nitely lived, risk-neutral agents: 1 Experts, I = [0, 1]: E0 Z e ρt dct where ct is cumulative consumption until time t. We impose dct 0. 2 Households, J = [0, 1]: E0 Z e rt dc t where c t is cumulative consumption until time t. We do not impose dc t 0 (negative consumption can be thought as additional labor e¤ort): hence r is risk-free rate. Assumption : r < ρ. Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 3 / 34 Technology I Experts with e¢ ciency units of capital kt produce output: yt = akt Experts can invest: dkt = (φ (ιt ) δ) kt dt + σkt dZt where 1 2 3 4 ιt is investment rate per unit of capital. φ (ιt ) is an investment technology with adjustment costs (φ (0) = 0, φ0 ( ) = 0, and φ00 ( ) < 0). We do not impose ιt > 0. Concavity of φ ( ) imposes large costs to disinvestment. dZt is a Brownian motion. Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 4 / 34 Technology II Households with e¢ ciency units of capital k t produce output: y t = ak t where a < a. Households can invest: dk t = (φ (ιt ) δ) k t dt + σk t dZt where δ > δ. We take output as numeraire. Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 5 / 34 Price of Capital Capital can be traded at price qt , which evolves as: dqt = µqt qt dt + σqt qt dZt Boundaries for price of capital: q = max ι q = max ι Jesús Fernández-Villaverde (PENN) r r a ι φ (ι) + δ a ι φ (ι) + δ A Model with Explicit Solution December 2, 2012 6 / 34 Returns I Thus, the value of the capital hold by an expert generates: 1 Capital gains: d (kt qt ) = φ ( ιt ) kt qt 2 δ + µqt + σσqt dt + σ + σqt dZt Dividend a ιt qt 3 Total return: drtk = a ιt qt δ + µqt + σσqt dt + σ + σqt dZt + φ ( ιt ) Similarly, return for a household: dr kt = a ιt qt Jesús Fernández-Villaverde (PENN) + ( φ ( ιt ) δ + µqt + σσqt ) dt + (σ + σqt ) dZt A Model with Explicit Solution December 2, 2012 7 / 34 Returns II Two components of risk on returns: 1 2 σdZt is exogenous risk caused by stochastic process for capital e¢ ciency. σqt dZt is endogenous risk caused by …nancial frictions. Without them, σqt = 0 because qt = q. Even if experts are risk-neutral with respect to consumption, they exhibit risk-averse behavior because the return to investment is time-varying. Experts su¤er losses when they want to buy more capital: its price is lower. Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 8 / 34 Budget Constraint Because of an agency problem, experts must retain 100 percent of equity and …nance the rest of their investment with risk-free debt. Experts: dnt = xt drtk + (1 nt xt ) rdt dct nt where nt 0 is net wealth, a fraction xt 0 invested in capital and a fraction 1 xt in the risk-free asset. In general, xt > 1. The solvency constrain: nt 0. Households: with x t dnt = x t dr kt + (1 nt x t ) rdt dc t nt 0. Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 9 / 34 Equilibrium I De…nition For any initial endowments of capital Z I k0i di + n Z J k0i , k j0 ; i 2 I, j 2 J o such that: k i0 dj = K0 an equilibrium is described by a group of stochastic processes on the …ltered probability space de…ned by the Brownian motion n o fZt , t 0g: the price j i process for capital fqt g, net worths nt , nt 0 , capital holdings n o n o kti , k jt 0 , investment decisions ιit , ιjt , and consumption choices n o 0, dc jt of individual agents i 2 I, j 2 J such that: dcti 1 2 Given prices, all experts and households maximize. Initial net worths satisfy n0i = k0i q0 and ni0 = k i0 q0 for all i 2 I, j 2 J. Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 10 / 34 Equilibrium II De…nition 3. Markets clear: Z I dcti di + dKt = Z I Z Z J dc it dj = φ ιit Jesús Fernández-Villaverde (PENN) I kti di + Z I δ kti di + Z J a Z J k it dj = Kt ιit kti di + φ ιit A Model with Explicit Solution Z J δ k jt dj a ιjt k jt dj dt dt + σKt dZt December 2, 2012 11 / 34 Excess Returns I First, to maximize experts return with respect to ιt , from drtk = a ιt qt + ( φ ( ιt ) set: φ 0 ( ιt ) = δ + µqt + σσqt ) dt + (σ + σqt ) dZt 1 ) ι t = ι ( qt ) qt Similarly, for a household: φ 0 ( ιt ) = 1 ) ι t = ι ( qt ) qt Thus: ι t = ι t = ι ( qt ) Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 12 / 34 Excess Returns II Now, de…ne expected excess returns: Et drtk dt Et dr kt dt r = r = Jesús Fernández-Villaverde (PENN) a a ι ( qt ) + φ (ι (qt )) qt ι (qt ) + φ (ι (qt )) qt A Model with Explicit Solution δ + µqt + σσqt r δ + µqt + σσqt r December 2, 2012 13 / 34 Optimal Strategies I Consider a feasible strategy for the experts fxt , d ζ t g with payo¤: θ t nt = Et Z ∞ e ρ (s t ) t dcs where dct = nt d ζ t Then, for a price process for capital fqt g that generates a …nite maximal payo¤, a feasible strategy is optimal if and only if: max s.t. Jesús Fernández-Villaverde (PENN) ζt 0 xbt 0,d b nt d b ζ t + Et (θ t nt ) dnt =b xt drtk + (1 b xt ) rdt nt Et e ρt θ t nt ! 0 A Model with Explicit Solution db ζt December 2, 2012 14 / 34 Optimal Strategies II Moreover, for d θt = µtθ dt + σtθ dZt θt a feasible strategy is optimal if and only if: 1 2 3 θt 1 and d ζ t > 0 only when θ t = 1. µtθ = ρ r . Either: xt > 0 and or xt = 0 and 4 Ee ρt θ t nt Et drtk dt Et drtk dt r= σ + σqt σtθ (desire to leverage) r σ + σqt σtθ (‡ight to quality) ! 0. Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 15 / 34 Optimal Strategies III For the households, Et dr kt dt with equality if 1 1 ψt = Kt r Z J 0 k it dj > 0 It can be veri…ed that, in equilibrium, ψt qt Kt > Nt = Jesús Fernández-Villaverde (PENN) Z I A Model with Explicit Solution nti di December 2, 2012 16 / 34 Wealth Distribution I Aggregate wealth: Nt = qt Kt Z I nti di Nt = Z J nit dj Hence, experts’s wealth share: ηt = Jesús Fernández-Villaverde (PENN) Nt 2 [0, 1] qt Kt A Model with Explicit Solution December 2, 2012 17 / 34 Wealth Distribution II Law of motion: d ηt ηt ψt = + ηt 2 dr k rdt (σ + σqt ) dt ηt a ι ( qt ) + (1 ψt ) (δ δ) dt d ζ t qt If ψt > 0 ) xt = 0 and we have r= Et drtk + (σ + σqt ) σtθ ) rdt = Et drtk + (σ + σqt ) σtθ dt dt Then: dr k rdt = dr k Et drtk = (σ + σqt ) dZt = (σ + σqt ) dZt Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution (σ + σqt ) σtθ dt (σ + σqt ) σtθ dt σtθ dt December 2, 2012 18 / 34 Wealth Distribution III We can substitute d ηt ηt ψt ηt (σ + σqt ) dZt σtθ dt (σ + σqt ) dt ηt a ι ( qt ) + + (1 ψt ) (δ δ) dt d ζ t qt ! ψt η t q q θ η t ( σ + σt ) σ + σt + σt dt = + a qι(tqt ) + (1 ψt ) (δ δ) ψ ηt + t (σ + σqt ) dZt d ζ t ηt = Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 19 / 34 Wealth Distribution IV By de…ning µnt = ψt ηt ηt (σ + σqt ) σ + σqt + σtθ + η σt = ψt ηt ηt ι ( qt ) + (1 qt a ψt ) ( δ δ) (σ + σqt ) we get d ηt η = µnt dt + σt dZt ηt Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution d ζt December 2, 2012 20 / 34 Markov Equilibria Search for functions: qt θt ψt = q (η t ) = θ (η t ) = ψ (η t ) Then, once we know η t , we can get qt , θ t , ψt , and from d ηt η = µnt dt + σt dZt ηt d ζt we get d η t . Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 21 / 34 A Proposition I Let us suppose that we know η, q (η ) , q 0 (η ) , θ (η ) , θ 0 (η ) . 1 Find ψ 2 η, η + q 0 (η ) q (η ) such that: a a +δ q (η ) δ + σtθ (σ + σqt ) = 0 where: η Jesús Fernández-Villaverde (PENN) σt = σqt = σtθ = 1 η1 (ψ η) a (ψ η) q 0 (η ) q (η ) q 0 (η ) η σ η q (η ) t θ 0 (η ) η σ η θ (η ) t A Model with Explicit Solution December 2, 2012 22 / 34 A Proposition II 2 If ψ > 1, set ψ = 1 and recalculate σt , σqt , σtθ . 3 Compute: η µnt = µqt = r η σt a σ + σqt + σtθ + ι (q (η )) q (η ) a ι (q (η )) + (1 q (η ) φ (q (η )) + δ µtθ = ρ q 00 (η ) = 2 (µqt q σtθ (σ + σqt ) q 0 (η ) µnt η ) (η ) η 2 η2 θ 0 (η ) µnt η 2 µtθ θ (η ) η 2 σt Jesús Fernández-Villaverde (PENN) δ) r σt θ 00 (η ) = σσqt ψ) (δ A Model with Explicit Solution η2 December 2, 2012 23 / 34 A Proposition III 4 Use boundary conditions q 0 (η ) = 0, q (0) = q θ (η ) = 1, θ 0 (η ) = 0 lim θ (η ) = ∞ where η is the re‡ecting boundary when experts consume. Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 24 / 34 An Algorithm Set q (0) = q, θ (0) = 1, and θ 0 (0) =small number. Set qL = 0 and qH =large number. qL Guess q 0 (0) = qH + and solve for q (η ) and θ (η ) until the …rst of the 2 three conditions holds: 1 2 3 q (η ) = q. θ 0 (η ) = 0. q 0 (η ) = 0. If q 0 (η ) = 0, set qL = q 0 (0), otherwise qH = q 0 (0). Iterate until convergence. Check that q 0 (η ) and θ 0 (η ) reach 0 at the same point η . Normalize θ (η ) = Jesús Fernández-Villaverde (PENN) θ (η ) θ (η ) to match boundary condition. A Model with Explicit Solution December 2, 2012 25 / 34 Calibration Parameters ρ 0.06 r 0.05 a 0.1 a 0.05 δ 0.03 δ 0.05 σ 0.4 pφ (ι) 0.1 1 + 20ι This implies q = 0.5858 and q = 1.3101. Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 26 / 34 Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 27 / 34 Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 28 / 34 Three Ine¢ ciencies 1 Capital misallocation: for low η t , households manage part of the capital (ψ < 1). 2 Under investment: ι (qt ) < ι (q ) . 3 Consumption distortions: experts should only consume at time 0. Note: ine¢ ciencies get worse for low η t . Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 29 / 34 Adverse Feedback Loop Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 30 / 34 Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 31 / 34 Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 32 / 34 Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 33 / 34 Jesús Fernández-Villaverde (PENN) A Model with Explicit Solution December 2, 2012 34 / 34