---
tags: discussion
---
# Macro HW3 1-f

The question:

>(f) Using the model from part (e), show that the Bellman operator maps degree-$(1- \sigma)$ homogeneous functions into themselves. (Hint: the zero function satisfies this property). Then show that the pointwise limit of degree-$(1- \sigma)$ homogeneous functions is also homogeneous of degree-$(1- \sigma)$.

I guess the question akss us to prove that the Bellman operator in (d) ( not (e) ) map **all** H.D.-($1-\sigma$) funcitons into H.D.-($1-\sigma$) funcitons, and if we start with any bounded continuous H.D.-($1-\sigma$) funciton $w$, $\lim_{k \to \infty} T^k (w)$ will also be a H.D.-($1-\sigma$) funciton.

However, this guess is different form the ambiguous question. First, why $0$ function? If we already proved the property hold for any H.D.-($1-\sigma$) funciton, it also holds for $0$ function. On the other hand, the special case of $0$ function holds doesn't ensure the property holds for any H.D.-($1-\sigma$) funciton.

Second, where is "pointwise limit"? The converge of value function is based on sup norm, which is not pointwise. Or the second part asks us to prove the following question:

>Given a seuqence of H.D-($1-\sigma$) functions, $\{f_k\}_{k=1}^\infty$. Suppose that the sequence pointwisely converge to $f$, i.e.
$$
\lim_{k \to \infty} f_k(x) = f(x), \forall x \in X,
$$
then $f$ is also a H.D-$(1-\sigma)$ functions.

## Possilbe soution

### part 1
Given $x \in X$, consider the below Bellman equation,

$$
Tw(x) = \max_{x'} \{\frac{(x-x')^{1-\sigma}}{1-\sigma} + \beta w(x')\}.
$$

Suppose $w$ is a H.D-($1-\sigma$) function. Let $x^*$ be the solution of the above Bellman equation. For $t>0$, we have another problem,

$$
Tw(tx) = \max_{x'} \{\frac{(tx-x')^{1-\sigma}}{1-\sigma} + \beta w(x')\},
$$

and we want to show that the solution is $tx^*$. First, $tx^* < tx$, which is feasible. Second, if $y \neq tx^*$ is the solution, i.e.

$$
\frac{(tx-y)^{1-\sigma}}{1-\sigma} + \beta w(y) > \frac{(tx-tx^*)^{1-\sigma}}{1-\sigma} + \beta w(tx^*),
$$

then

\begin{align*}
&t^{1-\sigma}\frac{(x-\frac{y}{t})^{1-\sigma}}{1-\sigma} + t^{1-\sigma}\beta w(\frac{y}{t})=\frac{(tx-y)^{1-\sigma}}{1-\sigma} + \beta w(y) \\
&> \frac{(tx-tx^*)^{1-\sigma}}{1-\sigma} + \beta w(tx^*) = t^{1-\sigma}\frac{(x-x^*)^{1-\sigma}}{1-\sigma} + t^{1-\sigma}\beta w(x^*),
\end{align*}

with $t^{1-\sigma}>0$, this implies

$$
\frac{(x-\frac{y}{t})^{1-\sigma}}{1-\sigma} + \beta w(\frac{y}{t}) > \frac{(x-x^*)^{1-\sigma}}{1-\sigma} + \beta w(x^*),
$$

which contradicts the hypothesis that $x^*$ is the solution to the first problem. Hence,

$$
Tw(tx) = \frac{(tx-tx^*)^{1-\sigma}}{1-\sigma} + \beta w(tx^*) = t^{1-\sigma} Tw(x),
$$

and the Bellman operator defined in this problem maps any H.D-($1-\sigma$) function into another H.D-($1-\sigma$) function.

### part 2

Given a sequence of H.D-($1-\sigma$) functions, $\{f_k\}_{k=1}^\infty$. Suppose that the sequence pointwisely converges to $f$, i.e.
$$
\lim_{k \to \infty} f_k(x) = f(x), \forall x \in X.
$$

For any $x \in X$ , $t>0$ and $\varepsilon>0$, there exists $n_1, n_2 \in \mathbb{N}$, such that

$$
|f_m(x) - f(x)| < \varepsilon, |f_m(tx) - f(tx)| < \varepsilon,
$$

if $m \ge \max\{n_1, n_2\}$. Since $f_m$ is H.D-($1-\sigma$), $f_m(tx) = t^{1-\sigma} f_m(x)$, and

$$
|f(tx) - t^{1-\sigma} f(x)| < 2 \varepsilon,
$$

by the triangular inequality. Since $\varepsilon$ is arbitrary, $f(tx) = t^{1-\sigma} f(x)$ and $f$ is a H.D-($1-\sigma$) function.