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Currently I'm studying the RCK model at my Advanced Macroeconomics I classes. A question arose almost immediately when I saw the utility function of a household in this model.

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The question is: why are we using exp(-ρt) inside this integral? I mean: why this functional form in particular? I love mastering every single aspect of every model I study, that's why I'm asking so. Thank you really much in advance for your kind responses.

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    $\begingroup$ It's just there for bringing future utility to present value in continuous time. $\rho$ is the discount factor. $\endgroup$ – Pedro Cavalcante Oliveira Dec 2 '18 at 22:47
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I guess you already know that $\exp(-\rho t)$ is the the discount factor. In particular, it is obtained by continuously compounding the discount factor

$$\exp(-\rho t)=\lim_{m \to \infty} \left(1+ \frac{\rho}{m}\right)^{mt}$$

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