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I have the following dynamic programming problem:

$$\max_{\{x_t,y_{t+1}\}_{t=0}^\infty}\sum_{t=0}^\infty \beta^tu(F(x_t)-y_{t+1})\;\;\;\;\;\text{s.t}\;\;\;\;y_{t+1}\in\Gamma(x_t)$$

where $\Gamma(x)=\{y\in\mathbb{R}_{+}\mid 0\le y\le F(x)\}$, and the functions $u$ and $F$ have the following properties:

  1. $u:\mathbb{R}_{+}\to\mathbb{R}$ is continuous, strictly increasing, and strictly concave;
  2. $F:\mathbb{R}_{+}\to\mathbb{R}$ is continuous, strictly increasing, and concave;
  3. $F(0)=0$;
  4. And there exists $\bar{x}>0$ such that:
  • If $x\in[0,\bar{x}]$ then $F(x)\in[x,\bar{x}]$.

  • If $x>\bar{x}$ then $F(x)<x$.

I've got to show that the function $G(x,y)\equiv u(F(x)-y)$ is bounded over the set $A\equiv\{(x,y)\in \mathbb{R}_{+}\times\mathbb{R}_{+}\mid y\in\Gamma(x)\}$.

What I have tried: Let $(x,y)\in A$, then $y\in\Gamma(x)$. First, by definition the functions $u$ and $F$ clearly $G$ is bounded below. Now in order to show that $G$ is bounded above, I noticed that the set of images $G(x,\Gamma(x))$ is compact in the second argument because $G$ is continuous and $\Gamma(x)$ is compact, i.e. given $x$ we have that the function $G(x,\cdot)$ is bounded in $\Gamma(x)$.

Now, trying to show that $G$ is bounded above in $x$ (i.e. in the first argument) has proven to be more involved, I'm trying to use the fourth property shown above about the existence of $\bar{x}$, where we have two cases, if $x\in[0, \bar{x}]$ then clearly $G(x,y)\le u(F(\bar{x})-y)\le u(F(\bar{x}))$. But when considering $x>\bar{x}$ we have $F(x)<x$, thus $G(x,y)<u(x-y)$ which does not show that $G$ is bounded since the "bound" still depends on $x$ which is in a unbounded space $(\bar{x},\infty)$.

I think there must be a trick with $F(x)<x$ but I have not been able to see it, any help would be appreciated.

Thanks!

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The function $u(F(x)-y)$ is not necessarily bounded on $A$. For example, if $u(x) = F(x) = \sqrt{x}$ then: $$ u(F(x)-y) = \sqrt{\sqrt{x}-y}, $$ Taking $y = 0$, this gives $u(F(x)) = \sqrt{\sqrt{x}}$, which is clearly unbounded in $x$.

The problem you are looking at is the following: $$ \max_{(x_t, y_t)_{t = 0}^\infty} \sum_{t = 0}^\infty \beta^t u(F(x_t) - y_{t+1}) \text{ s.t. } y_{t+1} \in [0, F(x_t)]. $$ Notice that this problem is not well defined. In particular, given that $u$ is strictly increasing, the best choice for $y_{t+1}$ is to set it equal to zero at every instance $t$. This gives the payoff function $$ \sum_{t = 0}^\infty \beta^t u(F(x_t)), $$ which is clearly increasing in $x_t$ (as both $u$ and $F$ are strictly increasing). As such, the best thing to do is to set $x_t$ as large as possible (so equal to $+ \infty$). This means that your maximisation problem has no solution.

Now, assume that you have a better behaving model, for example, $y_t = x_t$. This then gives the problem: $$ \max_{(x_t)_{t = 0}^\infty} \sum_{t = 0}^\infty \beta^t u(F(x_t) - x_{t+1}) \text{ s.t. } x_{t+1} \in [0, F(x_t)]. $$ The function $u(F(x)-y)$ is still unbounded, but now you can show that the objective function $\sum_{t=0}^\infty \beta^t u(F(x_t) - x_{t+1})$ is bounded for every feasible path. Let $\partial u(0)$ be a supdifferential (derivative) for $u$ at zero. Then, using concavity of $u$, we have: $$ \begin{align*} \sum_{t = 0}^n \beta^t u(F(x_t) - x_{t+1}) &\le \sum_{t = 0}^n \beta^t \left(u(0) + \partial u(0)[F(x_t) - x_{t+1}]\right),\\ &= u(0) \sum_{t = 0}^n \beta^t + \partial u(0) \underbrace{\sum_{t = 0}^n \beta^t [F(x_t) - x_{t+1}]}_{=A}. \end{align*} $$ Now, $F(x) \le \overline{x} + x$ (if $x \le \overline{x}$ then $F(x) \le \overline{x}$ and if $x \ge \overline{x}$, then $F(x) \le x$). So, $$ \begin{align*} A &\le \sum_{t = 0}^n \beta^t [\overline{x} + x_t - x_{t+1}],\\ &= \overline{x} \sum_{t = 0}^n \beta^t + \underbrace{\sum_{t = 0}^n \beta^t (x_t - x_{t+1})}_{=B} \end{align*} $$ Then, $$ \begin{align*} B &= \sum_{t = 0}^n \beta^t (x_t - x_{t+1}),\\ &=x_0 \underbrace{- x_1 + \beta x_1}_{\le 0} \underbrace{- \beta x_2 + \beta^2 x_2}_{\le 0} \underbrace{- \beta^2 x_3 + \,\,}_{\le 0} \ldots \underbrace{\,\, + \beta^n x_n}_{\le 0} - \beta^n x_{n+1},\\ &\le x_0 - \beta^n x_{n+1} \le x_0 \end{align*} $$ Putting it all together, we obtain the bound: $$ \begin{align*} \sum_{t = 0}^n \beta^t u(F(x_t) - x_{t+1}) & \le (u(0) + \partial u(0) \overline{x}) \sum_{t = 0}^n \beta^t + \partial u(0) x_0 \end{align*} $$ Taking the limit $t \to \infty$, the right hand side converges to $\frac{u(0) + \partial u(0) \overline{x}}{1 - \beta} + \partial u(0) x_0$ (a fixed number).

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