# Interest rate Effects on labour supply

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}