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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