Consider an agent who has a fixed budget, and should decide how to split it between consumption today and consumption tomorrow. For simplicity, suppose there is no interest, and no borrowing/lending, and the price is the same in both days. In my studies, I encountered two ways to model the agent's decision problem:

1.  Consider each daily consumption as a different good, then model the agent's preferences using a utility function. For example, if the agent prefers consumption today to consumption tomorrow, but still wants to consume something tomorrow, then the utility function can be a Cobb-Douglas utility function, e.g. $x^{0.8} y^{0.2}$ where $x, y$ are the consumptions today and tomorrow respectively. Then, the agent maximizes his utility given the budget constraints, just like in a usual consumer's choice problem.

2. Assume that the agent has a certain discount factor $\delta$, which determines how much the agent prefers to consume today over tomorrow. How can this be modeled using a utility function? Initially I thought that the corresponding utility function would be $x + \delta y$. But then, the optimal solution (when $\delta<1$) is to consume everything today, and starve tomorrow. This does not make much sense. 

My question: what utility function corresponds to a discount factor of $\delta$? Can it be represented, for example, by a Cobb-Douglas function?