\[ \begin{aligned} \max_{c(t),k(t)} & \int_0^\infty e^{-\delta t} u(c(t)) dt \\ \text{ s.t. } & \frac{dk}{dt} = f(k(t)) - \phi k(t) - c(t) \\ & k(0) = k_0 \\ & 0 \leq c(t) \leq f(k(t)) \end{aligned} \]

\[ \begin{aligned} \max_{k(t),i(t),l(t)} & \int_0^T e^{-r t} \left[p f(k(t),l(t)) - qi(t) - wl(t) - c(i(t),k(t)) \right] dt \\ \text{ s.t. } & \frac{dk}{dt} = i(t) - \delta k(t) \\ & k(0) = k_0 \\ & k(t) \geq 0 \\ & l(t) \geq 0 \end{aligned} \]

\[ \begin{aligned} \max_{q(\theta),T(\theta)} & \int_{\theta_l}^{\theta_h} \left[T(\theta) - cq(\theta)\right] f_\theta(\theta) d\theta \notag \\ & \text{s.t.} \notag \\ &\theta \nu\left(q(\theta)\right) - T(\theta) \geq 0 \forall \theta \label{pc} \\ &\theta \nu\left(q(\theta)\right) - T(\theta) \geq \max_{\tilde{\theta}} \theta \nu\left(q(\tilde{\theta}) \right) - T(\tilde{\theta}) \forall \theta \label{ic} \end{aligned} \]

\[ \begin{aligned} \max_{\ell,t} & \int_{w_l}^{w_h} G\left(u(w\ell(w) - t(w\ell(w)),\ell(w)) \right) f(w) dw \\ & \text{ s.t. } \\ & \int_{w_l}^{w_h} t(w) f(w) dw \geq g \\ & \ell(w) \in \argmax_{\tilde{\ell}} u(w\tilde{\ell} - t(w\tilde{\ell}), \tilde{\ell} ) \end{aligned} \]

\[ \begin{aligned} \max_{x(t),y(t)} & \int_0^T F(x(t),y(t),t) dt \\ & \text{ s.t.} \\ & \frac{d y}{dt} = g(x(t),y(t),t) \forall t \in [0,T] \\ & y(0) = y_0 \end{aligned} \]

• Divide \([0,T]\) into \(J\) intervals of length \(\Delta=T/J\)
• \(\int_0^T F(x(t),y(t), t) dt \approx \sum_{j=1}^J F(x(\Delta j), y(\Delta j), \Delta j) \Delta\)
  • Let \(x_j = x(\Delta j)\), \(y_j = y(\Delta j)\)
• \(\frac{dy}{dt}(\Delta j) \approx \frac{y_j - y_{j-1}}{\Delta}\)f(x

Discretized problem: \[ \begin{aligned} \max_{x_1, ..., x_J, y_1, ..., y_J} & \sum_{j=1}^J \Delta F(x_j, y_j,\Delta j)\Delta \\ & \text{ s.t.} \\ & y_j - y_{j-1} = \Delta g(x_j,y_j,\Delta j) \text{ for} j = 1,...,J \end{aligned} \]

• Discrete first order conditions \[ \begin{aligned} [ x_j ]: && \frac{\partial F}{\partial x} + \lambda_j \frac{\partial g}{\partial x} = & 0 \\ [y_j]: && \frac{\partial F}{\partial y} + \lambda_j \frac{\partial g}{\partial y} = & -\frac{\lambda_{j+1} - \lambda_j}{\Delta} \\ [\lambda_j]: && g(x_j,y_j,\Delta j) = & \frac{y_j - y_{j-1}}{\Delta}. \end{aligned} \]

• Limit as \(\Delta \to 0\) \[ \begin{aligned} [ x_j ]: && \frac{\partial F}{\partial x} + \lambda(t) \frac{\partial g}{\partial x} = & 0 \\ [y_j]: && \frac{\partial F}{\partial y} + \lambda(t) \frac{\partial g}{\partial y} = & -\frac{d\lambda}{dt} \\ [\lambda_j]: && g(x_j,y_j,\Delta j) = & \frac{dy}{dt} \end{aligned} \]

• To analyze maximization in \(\R^n\), we looked at \(f(x^* + v)\) for small \(v\)
• We can do the same here, look at e.g. \[ \int_0^T F(x^*(t) + v(t),y^*(t),t)dt \]
• \(v\) is now a function instead of vector in \(\R^n\)
  • Like vectors in \(\R^n\), functions can be added and multiplied by scalars

• \(Q(x,y) \equiv \int_0^T F(x(t),y(t),t)dt\)
• Directional derivative, for \(v:[0,T] \to \mathbb{R}\) and \(w:[0,T] \to \mathbb{R}\), \[ \begin{aligned} dQ(x,y;v,w) = & \lim_{\alpha \to 0} \frac{Q(x+\alpha v, y + \alpha w) - Q(x,y)}{\alpha} \\ = & \frac{d}{d\alpha} Q(x+\alpha v, y + \alpha w) \vert_{\alpha=0} \end{aligned} \]

\[ \begin{aligned} L(x,y,\lambda,\mu_0) = & \int_0^T F(x(t),y(t),t) dt - \\ & - \int_0^T \lambda(t)\left( \frac{dy}{dt} - g(x(t),y(t),t) \right) dt - \\ & - \mu_0 (y(0) - y_0) \end{aligned} \]

• Problem: \[ \begin{aligned} \max_{x,y} & \int_0^T F(x(t),y(t),t) dt \\ & \text{ s.t.} \\ & \frac{d y}{dt} = g(x(t),y(t),t) \forall t \in [0,T] \\ & y(0) = y_0 \end{aligned} \]
• Hamiltonian \[ H(x,y,\lambda,t) = F(x(t),y(t),t) + \lambda(t) g(x(t),y(t),t) \]

• If \(x^*\) and \(y^*\) are a local constrained maximizers, then there exists \(\lambda^*(t)\) such that \[ \begin{aligned} [ x ]: && 0 = & \frac{\partial H}{\partial x}(x^*,y^*,\lambda^*,t) \\ [ y ]: && -\frac{d\lambda}{dt}(t) = & \frac{\partial H}{\partial y}(x^*,y^*,\lambda^*,t) \\ [ \lambda ]: && \frac{dy}{dt}(t) = & \frac{\partial H}{\partial \lambda}(x^*,y^*,\lambda^*,t) \end{aligned} \]

\[ \begin{aligned} \max_{x,y} & \int_0^T p(t) y(t) - s(t) x(t) - c(t) x(t)^2 dt \\ \text{s.t.} & \frac{dy}{dt} = x(t) \\ & y(0) = y_0. \end{aligned} \]

• \(y(t)\) units of housing
• \(p(t)\) rental price
• \(x(t)\) housing investment
• \(s(t)\) price of housing, \(c(t)x(t)^2\) adjustment cost

\[ \begin{aligned} \max_{s,k} & \int_0^T (1-s(t)) k(t) dt \\ \text{ s.t. } & \frac{dk}{dt} = s(t)k(t) \\ & k(0) = k_0 \\ & k(t) \geq 0 \\ & 0 \leq s(t) \leq 1 \end{aligned} \]

\[ \begin{aligned} \max_{x,y} & \;\; p y_T - \int_0^T \left( c x(t)^2 + s y(t) \right) dt \\ \text{ s.t. } & \frac{dy}{dt} = x(t) \\ & y(T) = y_T \\ & y(0) = 0 \\ & x(t) \geq 0 \end{aligned} \]

\[ \begin{aligned} \max_{c(t),k(t)} & \int_0^\infty e^{-\delta t} u(c(t)) dt \\ \text{ s.t. } & \frac{dk}{dt} = f(k(t)) - \phi k(t) - c(t) \\ & k(0) = k_0 \\ & 0 \leq c(t) \leq f(k(t)) \end{aligned} \]

• Constraint \[ \frac{dk}{dt} = f(k(t)) - \phi k(t) - c(t) \]
• First order conditions \(\Rightarrow\) \[ \frac{dc}{dt} = -\frac{u'(c)}{u''(c)} \left(f'(k(t)) - \phi - \delta \right) \]
• Given \(k(0)\) and a choice of \(c(0)\), these equations determine \(c(t)\) and \(k(t)\)
• Steady state: \(dk/dt = dc/dt= 0\) \[ f'(k^{ss}) - \phi - \delta = 0 \] \[ f(k^{ss}) - \phi k^{ss} - c^{ss} = 0 \]
• Stable path: given \(k(0) = k_0\), there is unique \(c(0)\) such that \(c(t)\) and \(k(t)\) reach the steady state

\[ \begin{aligned} \max_{q(\theta),T(\theta)} & \int_{\theta_l}^{\theta_h} \left[T(\theta) - cq(\theta)\right] f_\theta(\theta) d\theta \\ & \text{s.t.} \\ &\theta \nu\left(q(\theta)\right) - T(\theta) \geq 0 \forall \theta \\ &\theta \nu\left(q(\theta)\right) - T(\theta) \geq \max_{\tilde{\theta}} \theta \nu\left(q(\tilde{\theta}) \right) - T(\tilde{\theta}) \forall \theta \end{aligned} \]

\[ \begin{aligned} \max_{q(\theta),T(\theta)} & \int_{\theta_l}^{\theta_h} \left[T(\theta) - cq(\theta)\right] f_\theta(\theta) d\theta \\ & \text{s.t.} \\ &\theta_l \nu\left(q(\theta_l)\right) - T(\theta_l) \geq 0 \\ & \theta \nu'(q(\theta))q'(\theta) - T'(\theta) = 0 \\ & \frac{dq(\theta)}{d\theta} \geq 0 \end{aligned} \]