How the saving rates were derived in Azariadis (1996)'s Impatience Trap?

Question

The household maximizes his lifetime utility function according to $$ \max_{c_1, c_2} v(c_1, c_2) := \frac{1}{\beta(c_1)} \log c_1 + \log c_2 - A(c_1, c_2) $$ subject to $$ \beta(c_1) = \begin{cases} \beta_1 > 0 &\text{ and } A(c_1, c_2) = 0 & \text{ if } c_1 < \bar{c} \\ \beta_2 > \beta_1 &\text{ and } A(c_1, c_2) = \frac{\beta_2-\beta_1}{\beta_1\beta_2} \log c_2 & \text{ if } c_1 > \bar{c} \end{cases} $$ Budget constraint is $$ c_1 + \frac{c_2}{R_{t+1}} \leq w_t $$

1. How did he derive the savings as follows $$ s_t = \begin{cases} \frac{\beta_1}{1+\beta_1} w_t & \text{ if } w_t \leq (1+\beta_1) \bar{c}, \\ w_t - \bar{c} & \text{ if } w_t \in \left[ (1+\beta_1)\bar{c}, (1+\beta_2)\bar{c} \right], \\ \frac{\beta_2}{1+\beta_2} w_t & \text{ if } w_t > (1+\beta_2) \bar{c} \end{cases} $$
2. Is there any particular reason why he assumes $A = \frac{\beta_2-\beta_1}{\beta_1\beta_2} \log c_2$ when $c_1 > \bar{c}$?

This is section 3.1 of Azariadis (1996): The Economics of Poverty Traps Part One: Complete Markets in JEG.

Answer 1

The household's utility function is $$ v(c_1, c_2) := \frac{1}{\beta(c_1)} \log c_1 + \log c_2 - A(c_1, c_2) $$ where $$ \beta(c_1) = \begin{cases} \beta_1 > 0 &\text{ and } A(c_1, c_2) = 0 & \text{ if } c_1 < \bar{c} \\ \beta_2 > \beta_1 &\text{ and } A(c_1, c_2) = \frac{\beta_2-\beta_1}{\beta_1\beta_2} \log c_2 & \text{ if } c_1 > \bar{c} \end{cases} $$

We can rewrite the household's utility function as: $$v(c_1, c_2) :=\begin{cases} \frac{1}{\beta_1} \log c_1 + \log c_2 & \text{ if } c_1 < \bar{c} \\ \frac{1}{\beta_2} \log c_1 + \log c_2 - \frac{\beta_2-\beta_1}{\beta_1\beta_2} \log c_2 & \text{ if } c_1 \geq \bar{c} \end{cases}$$

In the paper, it is given that $v$ is a continuous, piecewise differentiable utility function. However, we can see that $v$ as described above is not continuous at $c_1=\bar{c}$. In fact, $v$ is not monotonic either.