must satisfy the Euler equation aHt,n(,2(O)=Hn1,(),(),t∈(0,T) If the terminal value u(T) is free, the transversality condition is HnT,u'(T),i(T)]=0. (3.10) If the initial value u(to) is free, the transversality condition is Hilto, u(to), i*(to)=0 (311) If the terminal condition is u(T)20, the transversality conditions are (T)HaT,(T),i'(T)=0,Ha[,u'(T),t(T)≤0. (312) Conversely, if H(t, u, i) is concave in(u, i), then any u*E A satisfying(3.9)and the initial and terminal conditions is a solution of (3.8). Here. T can be either finite or infinite By Kamien-Schwartz(1991, p 43), the Legendre condition is Hilt, u(t),i'(tIso It is a second order necessary condition, but not sufficient even locally Two boundary conditions are needed to pin down the two arbitrary constants in the general solution. When a boundary condition is missing, a transversality condition replaces it Cxample 3.6. Solve the following agency model max y-s( f(y, ady (313) s.t.u(s(y)lf(y, a) c(a)+i Example 3. 7. Characterize the solution of the following model in asymmetric informa- max/u[=(0), 0]+v(=(0), 0)) dF(e) st.i(6)≥0.must satisfy the Euler equation: d dtHu˙ [t, u∗ (t), u˙ ∗ (t)] = Hu[t, u∗ (t), u˙ ∗ (t)], t ∈ (0, T). (3.9) If the terminal value u(T) is free, the transversality condition is Hu˙ [T, u∗ (T), u˙ ∗ (T)] = 0. (3.10) If the initial value u(t0) is free, the transversality condition is Hu˙ [t0, u∗ (t0), u˙ ∗ (t0)] = 0. (3.11) If the terminal condition is u(T) ≥ 0, the transversality conditions are u∗ (T)Hu˙ [T, u∗ (T), u˙ ∗ (T)] = 0, Hu˙ [T, u∗ (T), u˙ ∗ (T)] ≤ 0. (3.12) Conversely, if H(t, u, u˙) is concave in (u, u˙), then any u∗ ∈ A satisfying (3.9) and the initial and terminal conditions is a solution of (3.8).  Here, T can be either finite or infinite. By Kamien—Schwartz (1991, p.43), the Legendre condition is Hu˙u˙ [t, u∗ (t), u˙ ∗ (t)] ≤ 0. It is a second order necessary condition, but not sufficient even locally. Two boundary conditions are needed to pin down the two arbitrary constants in the general solution. When a boundary condition is missing, a transversality condition replaces it. Example 3.6. Solve the following agency model: max a, s(·) ] v [y − s(y)] f(y, a)dy (3.13) s.t. ] u [s(y)] f(y, a)dy ≥ c(a)+¯u.  Example 3.7. Characterize the solution of the following model in asymmetric informa￾tion: max x(·) ] ¯θ θ {u [x (θ), θ] + v [x (θ), θ]} dF (θ) s.t. x˙ (θ) ≥ 0.  3—5