I want to maximize an objective given by $$\max_{\{q_n,p_n\}} \sum_{n=0}^\infty (\alpha_1 - \beta_1 n) p_n + (\alpha_2 - \beta_2 n) q_n$$
where $\alpha_1 > \beta_1 >0$ and $\alpha_2 > \beta_2 >0$ such that $q_n, p_n\geq 0$ for all $n$,
$$ p_{n+1} +q_{n+1} \leq \lambda p_n + q_n $$ where $\lambda >1$, $$p_0=c<1, q_0 =0$$ $$\sum_{n=0}^{\infty} (p_n+q_n) = 1 $$
My argument is that you can greedily optimize the objective by doing the following
- At $n\geq 1$, if $\lambda (\alpha_1 - \beta_1 n) \geq (\alpha_2 - \beta_2 n)$, then set $p_n = \min \big\{ \lambda p_{n-1} +q_{n-1}, 1- \sum_{i=0}^{n-1} p_i +q_i\big\}$. Otherwise set $q_n = \min \big\{ \lambda p_{n-1} +q_{n-1}, 1- \sum_{i=0}^{n-1} p_i +q_i\big\}$
Is there any idea how can I prove this argument? Mainly, I want to describe the structure of $p_n$ and $q_n$. Mainly, I want to say that there is an $n$ such that $p_i >0$ till $n$, and zero after that, (or vice versa for $q_n$). If my argument is incorrect, is there any suggestion on how I can approach this?
