By Ilmanen A.
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Extra resources for A Framework for Analyzing Yield Curve Trades. Understanding the Yield Curve: Part 6
19) Sequential Precision of Predictions in Models of Economic Growth 29 The necessary optimality conditions of the maximum principle are expressed in the Hamiltonian system of equations ⎧ 1 ⎪ k˙ = f (k, u) − − λk, ⎪ ⎪ ⎪ ψ 1 ⎪ ⎪ ⎪ ⎪ ∂f (k, u) ⎪ ⎪ ˙ , ⎪ ⎨ ψ1 = ψ1 δ + λ − ∂k u ⎪ ⎪ ⎪ , u˙ = υu 1 − ⎪ ⎪ ρ ⎪ ⎪ ⎪ ⎪ ⎪ u ⎪ ⎩ ψ˙ 2 = ψ2 δ − υ + 2υ ρ − ψ1 (20) ∂f (k, u) . ∂u To resolve peculiarities of the system we introduce variables z1 = ψ1 k, z2 = ψ2 u, (21) which can be interpreted as costs of capital and useful work.
Then the national income is defined by the formula Y (t) = C(t) + I (t) = (1 − s(t))Y (t) + s(t)Y (t). (3) It is assumed that the model characterizes growth in an aggregative closed economy. A. M. 1 Dynamics of Capital and Labor The capital stock is accumulated according to equation ˙ = s(t)Y (t) − μK(t). K(t) (4) Here parameter μ > 0 is the rate of capital depreciation. It is assumed that the labor input grows exponentially ˙ L(t) = n, L(t) (5) with a constant growth rate n > 0. Then dynamics of capital per worker is described by equation ˙ = s(t)y(t) − λk(t), k(t) (6) where λ = μ + n is capital decay, and n is capital dilution.
T) with respect to ξ ∈ G(t). The defect (109) (t, ξ ) from the last formula can be rewritten as (t, ξ ) = 1 a (t) + y (t), ξ − x ∗ (t) − y(t)x ∗ (t) ) ν˜ (t) + H(t, ξ, ∂ξ S(t, ξ )). (110) The component a(t) of the dual variable S(t, ξ ) will be chosen from the HamiltonJacobi equation (t, x ∗ (t)) = 0 and we obtain a (t) = y(t)T x ∗ (t) − H(t, x ∗ (t), y(t))˜ν (t). (111) With the Lagrangian L(t, ξ ) = − (t, ξ ) + λ(t)(x0 − ξ ) + μ(t)(ξ − Ceαt ) the Karush-Kuhn-Tucker conditions for the problem (109) are (112) 20 S.
A Framework for Analyzing Yield Curve Trades. Understanding the Yield Curve: Part 6 by Ilmanen A.