How is patience modelled?

Question

How is consumer patience modelled? My first thought was "discount factors, but the problem is that it doesn't seem to capture the intuitive notion of what being patient means.

For example, the following sounds very reasonable (see Coase Conjecture):

A monopolist selling a durable good loses market power if consumers become increasingly patient.

But if we model patience using discount factors, this result does not hold. For example, say consumer utility across two periods is $$U(c_1, c_2) = c_1 + \beta c_2$$ where $c_1, c_2$ is consumption in the two periods and $\beta$ is a discount factor. If $\beta = 1$, then the consumers are very patient, because they place equal value on the consumption in both periods.

However, that means that utility is $c_1 + c_2$. This is strictly greater than $c_1 + \beta c_2$ for $\beta < 1$. Therefore, the utility of the consumer is strictly higher across any level of consumption $\{(c_1, c_2) \in \mathbb{R}^2\}$.

Clearly that would mean that the firm gets more market power, because their consumer base is now strictly better off from buying their products than they were before.

Answer 1

There is a fundamental flaw in your reasoning and that is that you are comparing the utility values of two different consumers.

Remember that utility functions don't actually mean anything and they're just an invention to explain consumer behaviour. When you change from $c_1+\beta c_2$ to $c_1+c_2$ you are changing the consumer and you can't compare these utility values anymore. For example I can also change the utility function to $c_1+c_2+123456789$ and my consumers have higher utility but it doesn't mean anything about market power. Never compare utility values across consumers.

Next, with regards to the opening question about patience, patience can mean many different things depending on the model. Usually it has to do with discounting, but discounting isn't always a simple factor (e.g. hyperbolic discounting). It has an intuitive meaning of "valuing the future more highly relative to the present" but the formality changes by model. Also, if you want to talk about arbitrarily patient agents, you probably want infinite time horizon.

Finally, with regards to the Coase conjecture, Stokey proved it in 1981 in a very important result in mechanism design. Reading that will probably give some insight into the formality of patience.