# How is patience modelled?

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.

• The title of your question does not quite fit the text, which seems to instead be, "Here is a simple model of the Coase conjecture that I wrote down. Does my model show that the Coase conjecture is false?" – Kenny LJ Mar 29 '19 at 3:04

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.