# What utility function represents an agent with a time-discount factor?

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?

• How about $u(x)+\delta u(y)$ where $u$ satisfies $\lim_{z\to 0}u'(z)\to\infty$, so that a corner solution on either day is not optimal. Nov 20, 2021 at 21:08

As @HerrK. points out, utility functions that represent intertemporal discounting are generally of the form $$U\left((x_i)_{i=1}^T\right) = u(x_1) + \delta_1 u(x_2) + \delta_2^2 u(x_3) + \dots$$ where $$\delta_i$$ is the discount factor and $$x_i$$ is the consumption in period $$i$$. This is covered in most micro textbooks, e.g., (for two periods) in Varian's Intermediate Microeconomics.
When $$u() = \ln()$$, the function $$U\left((x_i)_{i=1}^T\right)$$ is Cobb-Douglas type. In case any $$x_i$$ converges to $$0$$, utility converges to $$-\infty$$, thus the optimal solution will be in the interior.