---
tags: macro, hw, group
---
7040 PS4
===
$$
% My definitions
\def\ve{{\varepsilon}}
\def\dd{{\text{ d}}}
\newcommand{\dif}[2]{\frac{d #1}{d #2}} % for derivatives
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} % for partial derivatives
$$

# Q1 Linear solution to the stochastic growth model
## Question
Consider the Brock-Mirman model from Problem set $\#1$ but now with depreciation but no iid shock:
\begin{aligned}
\max_{c_t, k_{t+1}} & \ E_0 \sum_{t=0}^{\infty} \beta^t \log c_t \\
\text{s.t. } & c_t + k_{t+1} - (1 − \delta) k_t \le a_t k_t^{\alpha}
\end{aligned}
with $k_0$ given, and where $\{a_t\}$ is an $AR(1)$ with $\ln a_t = \rho \ln a_{t−1} + \epsilon_t$, and $\epsilon_t$ is an i.i.d. random variable, and $\alpha, \beta \in (0, 1)$.

(a) Write down the Bellman equation and derive the conditions for maximization.

(b) Derive the Euler equation.

(c) Log-linearize the model and cast it in Gensys format

(d) Consider the i.i.d. and full depreciation case (ρ = 0 and δ = 0). Verify that the conditions for existence and uniqueness of rational expectations equilibirum hold. (Hint: with $\rho$ = 0 you can reduce the system to 2x2 rather than 3x3)

(e) Using gensys, solve for the linear rational expectations of the full model in (c). Plot the policy function for c for a few different values of $\delta$ and $\rho$

(f) How do your numerical solutions in (e) differ from the numerical solution derived in Problem Set $\#3$ and the analyical solution in Problem Set # 1? Interpret.

## Answer

# Q2 Transversality Conditions 1
## Question
(a) write the transversality condition for the problem above in its most general (inequality) form.

(b) Given $k_t \ge 0$, argue why one can write the transversality condition in a conventional (equality) form and show what it is.

## Answer

# Q3 Transversality Conditions 2
## Question
Consider the problem of a household with (internal) habits
\begin{aligned}
\max_{c_t, k_{t+1}}\ & E_0 \sum_{t=0}^{\infty} \beta^t \log(c_t - \mu c_{t-1}) \\
\text{s.t. }& c_t + a_{t+1} \le Ra_t + y_t
\end{aligned}

(a) Write the Bellman equation and derive conditions for maximization

(b) Derive the Euler equation

(c) Write the transversality conditions in general (inequality) form

(d) Can the TVC can be written in conventional (equality) form? (if not, is there some variant that would work?)