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I want to analyze the changes in the interest rate $r$ when dealing with the permanent income hypothesis. First of all,

$r$ is the interest rate $0<\beta<1$ is the discount factor, $Y_t$ is the labor income for period $t$, $A_t$ is the net asset wealth, $C_t$ is the consumption in the period $t$.

Now the Euler equation comes up :

$$u'(C_t)=\beta(1+r)u'(C_{t+1}) \ \ (*)$$

And If we assume $\beta(1+r)=1$ (We are going to be assuming this all the time when we are doing our analysis (closed economy)). Then:

Consumption in each period equals to each other (derived from the Euler equation) and it is equal to :

$$C = \frac{r}{1+r}(A_0+\sum_{t=0}^{\infty} (\frac{1}{1+r})^t Y_t ) \ \ (**)$$

I'm not gonna go through the details of how the equation $(**)$ is derived, since it will distort the scope of my question here.

Now Clearly It could be observed that when the interest rate $r$ increases, we use the $eq(*)$ to identify the substation effect and $eq(**)$ to identify the income effect. Substitution effect could be identified easily by using the fact that $u''()<0$

However when identifying the income effect there are two cases :

$(1)$ Consumer is saving $(2)$ Consumer is borrowing

When looking at the case $(2)$ I could easily show that consumers that are borrowing will decrease their consumption for all periods if the interest rates increase (Intuition = formula)

However if Consumer is saving:

Then $A_0 >0$ So $$ A_0 \frac{r_2}{1+r_2} > A_0 \frac{r_1}{1+r_1}$$

Where $r_2$ is the new interest rate. However, We need to look at the change that occurs for $\sum_{t=0}^{\infty} (\frac{1}{1+r})^t Y_t$ when the interest rate increases. However this decreases with a higher $r$. Since the change in $A_0 \frac{r}{1+r}$ and $\sum_{t=0}^{\infty} (\frac{1}{1+r})^t Y_t$ act in opposite directions, I couldn't show whether $C$ would decrease or increase. (i.e I couldn't show that $intuition = formula$)

Any help?

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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 < 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 > 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 > 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.

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