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7

One way to see (intuitively) the connection between the left hand sides is to write the discrete case as: $$\frac{u'(c(t + \tau))}{u'(c(t))},$$ for $\tau = 1$. Now if we generalise this to a setting where $\tau$ is now a variable in $\mathbb{R}$, this becomes a function of $\tau$. Taking the derivative with respect to $\tau$ and evaluating at $\tau = 0$, ...

6

You cannot completely ignore the RHS. Starting with $$\frac{U'(c_{t+1})}{U'(c_{t})}=RHS,$$ replace $t+1$ by $t+\Delta t$ to get $$\frac{U'(c(t+\Delta t))}{U'(c(t))}=RHS_{\Delta t},$$ where $RHS_{\Delta t}$ is the modified version of $RHS$ which contains terms depending on $\Delta t$, e.g. the modified discount factor. Expanding around $c(t)$ (and neglecting ...

5

As already commented, the equation you probably meant is $$\rho V(k)= \sup_c \{\, u(c) + V'(k) ( f(k) -\delta k -c ) \,\}.$$ I have never seen this equation called the HJB equation (probably missing a basic reference on my part). I'll call it "dynamic programming PDE". What you're really asking is the connection between two approaches to solve ...

4

First, the Present value hamiltonian equals: $${\cal P} = e^{-rt} \frac{c(t)^{1-\theta}}{1 - \theta} + \lambda(t)(Ak(t)^\alpha - \delta k(t) - c(t))$$ This gives the first order conditions:  \begin{align*} &\frac{\partial{\cal I}}{\partial c} = e^{-rt} c(t)^{-\theta} - \lambda(t) = 0 \tag{1.1}\\ &\dot \lambda(t) = - \lambda(t) (\alpha A (k_t)^{\...

1

Answer to Q1: If we re-write FONC as a function of $a$: $u'(c(a))-V_{a} =0$ Differentiate wrt $a$ (as in Walde 2010): $u''(c(a)) c'(a) -V_{aa} =0$ We know from SOSC that $u''(c(a))<0$. If we assume consumption is increasing in wealth $c'(a)>0$, then $V_{aa}<0$ $\left[\begin{array}{l} \rho V(a_{t}) = \frac{\gamma (V_a)^{1-\frac{1}{\gamma}}}{1-\... 1 Since you mention Walton, here is something from his notes page 111 Definition 4: The Hamiltonian and is defined$H(t,x,\rho) := min _v \{c(t,x,v) - \rho v\}$Notice the analogue to the Hamiltonian you have written.$v$in here is the control (your$c$),$c(t,x,v)$in here is the$u(c)$function you have and$-\rho = \partial_xL(t,x)$in here where$L\$ is ...

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