Example 3.8. The corresponding continuous-time version of the consumer's problem in Example 3.1 is J(Ao)=max/u(c)e-pt s.t. A=rA A(0)=A,A(∞)≥0 Characterize the optimal consumption path. I Example 3.9. Many other types of constraints can also be handled similarly. For ex- ample, consider the follo J(o, tr, to)=max/f[r(t),u(t), t]dt st.i(t)=g{(1),u(t),t, o h[r(t), u(t), t]dt=c, (to) 3. Phase Diagram ee Chiang(1984, 1992)and Sydsaeter et al(2005, P. 244) 3.1. The Linear System a phase diagram is often used to analyze the solution from a dynamic system. From an optimal control problem, the Euler equation and the motion equation generally lead to two equations of the form ∫(x,y,t) Phase diagrams handle the autonomous case with f= f(, y)and g=g(a, y) 1. For simple f and g, we may be able to solve this equation system directly 2. For complicated f and g, we may use a phase diagram to illustrate the solution pathExample 3.8. The corresponding continuous-time version of the consumer’s problem in Example 3.1 is J(A0) ≡ maxc ] ∞ 0 u(c) e−ρt dt s.t. A˙ = rA − c A(0) = A0, A(∞) ≥ 0. Characterize the optimal consumption path.  Example 3.9. Many other types of constraints can also be handled similarly. For ex￾ample, consider the following problem J(x0, xT , t0) ≡ maxu ] T t0 f[x(t), u(t), t] dt s.t. x˙(t) = g[x(t), u(t), t], U T t0 h[x(t), u(t), t] dt = c, x(t0) = x0, x(T) = xT .  3. Phase Diagram See Chiang (1984, 1992) and Sydsaeter et al (2005, p.244). 3.1. The Linear System A phase diagram is often used to analyze the solution from a dynamic system. From an optimal control problem, the Euler equation and the motion equation generally lead to two equations of the form: ⎧ ⎪⎨ ⎪⎩ x˙ = f(x, y, t), y˙ = g(x, y, t). Phase diagrams handle the autonomous case with f = f(x, y) and g = g(x, y). 1. For simple f and g, we may be able to solve this equation system directly. 2. For complicated f and g, we may use a phase diagram to illustrate the solution path. 3—7