Permanent Income Hypothesis - Changes in r
If you impose the condition $ \beta (1 + r) = 1 $ and then move around $ r $, you're assuming that consumer preferences are changing. In other words, you have introduced a third effect: increased interest rates lead to consumers decrease their preference for consuming in the future. In other words, there is no substitution effect in your model, there is only an income effect. The result you derived for consumption is fine, and the reason you see an ambiguity is because the consumer not borrowing today doesn't mean the consumer will not borrow at some point in the future, and if the consumer does want to borrow, the higher interest rates will induce them to borrow and consume less. This happens when the consumer expects their income to rise in the future. To illustrate, assume that $ Y_t = Y (1 + \gamma)^t $ for some $ \gamma &lt; r $ (which could change the NPG condition you used to derive this equation, but whatever) so we get

$$ C = \frac{r}{1+r} A_0 + \frac{r}{r- \gamma} Y $$

If $ \gamma \leq 0 $ and $ A_0 &gt; 0 $, i.e if the consumer expects his future income to stay constant or decline, then the impact of a rise in $ r $ is unambiguous: it leads to higher consumption in all periods. However, if the consumer expects his income to rise, i.e $ \gamma &gt; 0 $, then the two effects work in opposite directions: higher rates mean that the consumer borrows less, and this decreases $ C $. The net effect is ambiguous, and depends on the values of the variables in the model.