Interest rate Effects on labour supply

Question

I was analysing, free of specificities, the effects of an increase in the real interest rate on labour supply in a two period setting. The obvious effect on leisure is the intertemporal substitution effect as leisure in the present becomes more expensive than leisure in the future. Nevertheless, all the books I have looked at do not mention any income effect from this. Shouldn't the latter one exist as well? The increase in interest rates, if one is a net lender, allows one to increase leisure in both periods.

Now looking at the typical optimization FOC:

$$\frac{\frac{\partial U}{\partial l}}{\frac{\partial U}{\partial l'}}=\frac{(1+r)w}{w'}$$

(1+r) increases, thus, the ratio needs to increase. Nevertheless, that can happen by increasing both levels of leisure, but by different amounts. Am I reasoning incorrectly?

{Please disregard from mentioning all other intertemporal and cross effects, ex. consumption and leisure as I am aware of them}

Answer 1

Yes, there is an income effect. It's unclear why books wouldn't mention that, perhaps it was not a complete treatment.

Clearly, for individuals with zero assets, the income effect is zero, and goes to +- infinity as assets go to plus/minus infinity. Hence, @ChinG's presumption, that the substitution effect would always dominate, is not clearly true for me. That might be the case under some particular assumptions onto assets and preferences, but not in general.

Unless books clearly express that there is no income effect [in which case you're free to update the question], I'd assume that it's just not a complete exposition.

Answer 2

Yes, there is an income effect, as the individual needs to work less in order to obtain the same amount in the numerator (relative to before the increase in exchange rate). The assumption is that the substitution effect dominates the income effect here, regardless of whether the individual is a borrower or a lender.