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| Original file line number | Diff line number | Diff line change |
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| # Brunnermeier Sannikov "A Macroeconomic Model with a Financial Sector" (2014) | ||
| # The idfference with Ditella is that Ψ appears in the last algebratic equation, making it much much harder to solve for it | ||
| # using non linear solver | ||
| using EconPDEs | ||
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| Base.@kwdef mutable struct BrunnermeierSannikov | ||
| # See Table H Numerical examples | ||
| ρE::Float64 = 0.06 | ||
| ρH::Float64 = 0.06 | ||
| γ::Float64 = 2.0 | ||
| ψ::Float64 = 0.5 | ||
| δ::Float64 = 0.05 | ||
| σ::Float64 = 0.1 | ||
| κ::Float64 = 10.0 | ||
| a::Float64 = 0.11 | ||
| alow::Float64 = 0.1 | ||
| χlow::Float64 = 1.0 | ||
| firstregion::Bool = true | ||
| q::Float64 = 1.0 | ||
| qx::Float64 = 0.0 | ||
| end | ||
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| function (m::BrunnermeierSannikov)(state::NamedTuple, y::NamedTuple, Δx, xmin) | ||
| (; ρE, ρH, γ, ψ, δ, σ, κ, a, alow, χlow) = m | ||
| (; x) = state | ||
| (; pE, pEx_up, pEx_down, pExx, pH, pHx_up, pHx_down, pHxx) = y | ||
| # χ is the fraction of risk held by experts. | ||
| # ψ is the fraction of (real) capital held by experts | ||
| # so χ * ψ / x corresponds to total risk held by experts / their networth | ||
| # σE = κE / γ + (1 - γ) / (γ * (ψ - 1)) * σpE | ||
| # implies κE = γ * σE - (1 - γ) / ((ψ - 1)) * σpE | ||
| # so κE - κH = γ * (σE -σH) - (1 - γ) / ((ψ - 1)) * (σpE -σpH) | ||
| # = γ * (χ * ψ / x - (1-χ * ψ) / (1-x) - (1 - γ) / (γ * (ψ - 1)) * (σpE - σpH) | ||
| # . Specifically, we suppose that experts must retain at least a fraction χbar ∈ (0, 1] of equity. (i | ||
| # drift and volatility of state variable ν | ||
| #@show x, p, ψ | ||
| pE = max(pE, 1e-3) | ||
| pH = max(pH, 1e-3) | ||
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| χ = max(χlow, x) | ||
| # market clear givges | ||
| # x / pE + (1-x) / pH = (ψ * a + (1-ψ) * alow - i) / q | ||
| # this implies ψ | ||
| pEx, pHx = pEx_up, pHx_up | ||
| iter = 0 | ||
| @label start | ||
| q = m.q | ||
| if x ≈ xmin | ||
| m.firstregion = true | ||
| # poin toutside the gri, so zero | ||
| Ψ = 0.0 | ||
| q = (Ψ * a + (1 - Ψ) * alow + 1 / κ) / ((x / pE + (1 - x) / pH) + 1 / κ) | ||
| end | ||
| if m.firstregion | ||
| # in the second region (a - alow) / q = (κE - κH) * (σ + σq) | ||
| # here q corresponds to value in the previous x grif | ||
| i = (q - 1) / κ | ||
| Ψ = max(((x / pE + (1 - x) / pH) * q + i - alow) / (a - alow), (1e-10 + x) / χ) | ||
| K = q / (a - alow) * χ * σ^2 * (γ / (x * (1 - x)) - (1 - γ) / (ψ - 1) * (pEx / pE - pHx / pH)) | ||
| X =χ * Ψ - x | ||
| qx = (1 - sqrt(max(eps(), X * K))) / X * q | ||
| qx = max(min(qx, 4.0), 0.0) | ||
| q = q + qx * Δx | ||
| #@assert abs((1 - qx / q * X)^2 - K * X) < 1e-8 | ||
| if Ψ > 1.0 | ||
| m.firstregion = false | ||
| Ψ = 1 | ||
| q = (a + 1 / κ) / ((x / pE + (1 - x) / pH) + 1 / κ) | ||
| qx = - (a + 1 / κ) / ((x / pE + (1 - x) / pH) + 1 / κ)^2 * (1 / pE - 1 / pH - x / pE^2 * pEx - (1-x) / pH^2 * pHx) | ||
| end | ||
| else | ||
| # in the second region Ψ = 1 | ||
| Ψ = 1 | ||
| # We get q through market clearing equation after plugging i = (p - 1) / κ | ||
| q = (a + 1 / κ) / ((x / pE + (1 - x) / pH) + 1 / κ) | ||
| qx = - (a + 1 / κ) / ((x / pE + (1 - x) / pH) + 1 / κ)^2 * (1 / pE - 1 / pH - x / pE^2 * pEx - (1-x) / pH^2 * pHx) | ||
| i = (q - 1) / κ | ||
| @assert abs((a - i) / q - (x / pE + (1 - x) / pH)) < 1e-8 | ||
| end | ||
| #@show m.firstregion, x, qx | ||
| #@assert x isa Float64 | ||
| qxx = (qx - m.qx) / Δx | ||
| m.q = q | ||
| m.qx = qx | ||
| q = max(1e-3, q) | ||
| i = (q - 1) / κ | ||
| Φ = log(q) / κ | ||
| σx = (χ * Ψ - x) * σ / (1 - (χ * Ψ - x) * qx / q) | ||
| σpE = pEx / pE * σx | ||
| σpH = pHx / pH * σx | ||
| σq = qx / q * σx | ||
| σE = Ψ * χ / x * (σ + σq) | ||
| σH = (1 - Ψ * χ) / (1 - x) * (σ + σq) | ||
| κE = γ * σE - (1 - γ) / (ψ - 1) * σpE | ||
| κH = γ * σH - (1 - γ) / (ψ - 1) * σpH | ||
| μx = x * ((a - i) / q - 1 / pE) + σx * (κE - σ - σq) + x * (σ + σq) * (1 - χ) * (κE - κH) | ||
| #μx = x * (1 - x) * (κE * σE - κH * σH + 1 / pH - 1 / pE - (σE - σH) * (σ + σq)) | ||
| if (iter == 0) && (μx < 0) | ||
| iter += 1 | ||
| pEx, pHx = pEx_down, pHx_down | ||
| @goto start | ||
| end | ||
| μpE = pEx / pE * μx + 0.5 * pExx / pE * σx^2 | ||
| μpH = pHx / pH * μx + 0.5 * pHxx / pH * σx^2 | ||
| μq = qx / q * μx + 0.5 * qxx / q * σx^2 | ||
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| r = min(max((a - i) / q + Φ - δ + μq + σ * σq - (χ * κE + (1 - χ) * κH) * (σ + σq), -0.1), 0.1) | ||
| #if ψ < 1 | ||
| # @assert r ≈ (alow - i) / p + Φ - δ + μp + σ * σp - κH * (σ + σp) | ||
| #end | ||
| # Market Pricing | ||
| pEt = - pE * (1 / pE - ψ * ρE + (ψ - 1) * (r + κE * σE) + μpE - (ψ - 1) * γ / 2 * σE^2 + (2 - ψ - γ) / (2 * (ψ - 1)) * σpE^2 + (1 - γ) * σpE * σE) | ||
| pHt = - pH * (1 / pH - ψ * ρH + (ψ - 1) * (r + κH * σH) + μpH - (ψ - 1) * γ / 2 * σH^2 + (2 - ψ - γ) / (2 * (ψ - 1)) * σpH^2 + (1 - γ) * σpH * σH) | ||
| (; pEt, pHt), (; x, r, μx, σx, Ψ, κ, κE, κH, σE, σH, σq, pEt, pHt, q, qx, qxx, μq) | ||
| end | ||
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| # He Krishnamurthy "Intermediary Asset Pricing" AER (2013) | ||
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| using EconPDEs | ||
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| Base.@kwdef mutable struct HeKrishnamurthy | ||
| # See Table 2—Parameters and Targets | ||
| m::Float64 = 4 # intermediation multiplier | ||
| λ::Float64 = 0.6 # Debt/assets of intermediary sector | ||
| g::Float64 = 0.02 # dividend growth | ||
| σ::Float64 = 0.09 # dividend volatility | ||
| ρ::Float64 = 0.04 # time discount rate | ||
| γ::Float64 = 2.0 # risk aversion specialist | ||
| l::Float64 = 1.84 # labor income ratio | ||
| end | ||
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| # Description of the model | ||
| # Households cannot invest directly in the risky asset market: can only invest in bonds and intermediaries. Households unwilling to invest more than m * w_t in the equity of intermediary, where w_t is networth intermediary (constant on outside equity financing) | ||
| # Specialists have the knowledge to invest in the risky assets, and can invest in the risky asset on behalf of the households | ||
| # Specialists accept household funds and allocate between risky and riskless. networth of specialist is w and wealth allocated by households is H. | ||
| # Denote αI ratio of risky asset holdings to total capital w + H | ||
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| # households have log utility. Assume that they die continuously so that we can treat financial wealth as the thigns to maximize in t + dt (instead of total wealth), so that we can treat labor income easily. | ||
| # We make assumptions so that the household sector chooses to keep a minimum of λ w^h in short-term deb issued by intermediary. Formally, we assume that a fraction λ can only ever invest in riskless bond while remaining can invest in intermediation market. but both are poooled at the end of the period. | ||
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| # Denote pS the wealth-to-consumption ratio of specialists | ||
| function (m::HeKrishnamurthy)(state::NamedTuple, y::NamedTuple) | ||
| (; m, λ, g, σ, ρ, γ, l) = m | ||
| (; pS, pSx_up, pSx_down, pSxx) = y | ||
| (; x) = state | ||
| iter = 0 | ||
| pSx = pSx_up | ||
| @label start | ||
| # market clearing for goods gives x / pS + (1 - x) * ρ = (1 + l) / p | ||
| # This allows to obtain p (and its derivatives) in terms of pS (and its derivatives) | ||
| p = (1 + l) / (x / pS + (1 - x) * ρ) | ||
| px = - (1 + l) / (x / pS + (1 - x) * ρ)^2 * (1 / pS - ρ - x / pS^2 * pSx) | ||
| pxx = - (1 + l) / (x / pS + (1 - x) * ρ)^2 * (- 1 / pS^2 * pSx - 1 / pS^2 * pSx - x / pS^2 * pSxx + 2 * x / pS^3 * pSx^2) + 2 * (1 + l) / (x / pS + (1 - x) * ρ)^3 * (1 / pS - ρ - x / pS^2 * pSx)^2 | ||
| @assert x / pS + (1 - x) * ρ ≈ (1 + l) / p | ||
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| # market clearing for equity gives ((1-x) * (1-λ) * αH + x) * αI = 1 | ||
| # If unconstrained, assumption is that αI = 1.0 | ||
| αH = 1 | ||
| αI = 1 / (1 - λ * (1 - x)) | ||
| # total equity of households cannot be higher than m * networth of specialists | ||
| if (1 - x) * (1 - λ) * αH > x * m | ||
| # constrained case | ||
| αH = x * m / ((1 - x) * (1 - λ)) | ||
| αI = 1 / (x * (1 + m)) | ||
| end | ||
| @assert ((1-x) * (1-λ) * αH + x) * αI ≈ 1.0 | ||
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| # we have the usual feeback loop between asset prices and wealth | ||
| # σx = x * (αI - 1) * (σ + σp) | ||
| σx = x * (αI - 1) * σ / (1 - x * (αI - 1) * px / p) | ||
| σpS = pSx / pS * σx | ||
| σp = px / p * σx | ||
| @assert σx ≈ x * (αI - 1) * (σ + σp) | ||
| σR = σ + σp | ||
| κ = γ * (αI * σR - σpS) | ||
| # as usual, μx = x * (1 - x) * (growth rate of wealth of specialists minus growth rate of wealth of households) | ||
| μx = x * (1 - x) * ((αI - (1 - λ) * αI * αH) * κ * σR - 1 / pS + ρ - l / ((1 - x) * p)) | ||
| if (iter == 0) && (μx < 0) | ||
| iter += 1 | ||
| pSx = pSx_down | ||
| @goto start | ||
| end | ||
| μpS = pSx / pS * μx + 0.5 * pSxx / pS * σx^2 | ||
| μp = px / p * μx + 0.5 * pxx / p * σx^2 | ||
| r = 1 / p + g + μp + σ * σp - κ * σR | ||
| @assert x * (r + αI * κ * σR - 1 / pS) + (1 - x) * (r + (1 - λ) * αI * αH * κ * σR - ρ + l / ((1-x) * p)) ≈ g + μp + σ * σp | ||
| σCS = κ / γ | ||
| μCS = (r - ρ) / γ + (1 + 1 / γ) / (2 * γ) * κ^2 | ||
| pSt = - (1 / pS + μCS + μpS + σCS * σpS - r - κ * (σCS + σpS)) | ||
| μR = r + κ * σR | ||
| (; pSt), (; pS, p, r, κ, σp, μR, σR, αI, μx, σx, x) | ||
| end | ||
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| m = HeKrishnamurthy() | ||
| xn = 1000 | ||
| stategrid = OrderedDict(:x => range(0, 1, length = xn+2)[2:(end-1)].^1.5) | ||
| yend = OrderedDict(:pS => ones(length(stategrid[:x]))) | ||
| y, residual_norm, a = pdesolve(m, stategrid, yend) | ||
| @assert residual_norm <= 1e-5 | ||
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