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(I am not looking for the law of diminishing marginal utility, even though it is related)

Which economic theory or theories describe the following problem? When a resource is made available at 0 monetary cost (but has a waiting time / access cost), and the number of interested people vastly exceeds the resource's capacity (people/time), the queue to access the resource increases until the net marginal utility of lining up for the resource approaches zero.

For example, I noticed a line (queue) for playing a game. It's free to play (in dollars, not wait time), and so the size of the line (wait time) continues to increase, assuming a large enough crowd. At the limit of certain (artificial) conditions left to the imagination, the size of this line (and cost of wait time) will increase until the net marginal utility of each player will approach 0; that is, players will continue to join the line up until the point of no perceived net marginal utility (the benefit of playing is completely offset by the wait time). At this equilibrium between cost and benefit, the marginal utility of everyone in the line approaches 0 over time.

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    $\begingroup$ Hi! Do people not perceive waiting in the line as a cost? Do they not mind spending their time there? Because if they do, and demand vastly outstrips supply, I see no reason for marginal utility of the good to converge to zero. $\endgroup$
    – Giskard
    Jan 21, 2023 at 20:52
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    $\begingroup$ Adding to the point made by Giskard, as well as the cost of time in the queue, there is also the cost of time spent consuming the resource (eg playing the game), and the likelihood that that time could have been used for other (free or non-free activities) that may provide greater net utility. This strengthens the argument that marginal utility will not approach zero. $\endgroup$ Jan 22, 2023 at 13:58
  • $\begingroup$ @Giskard Yes, part of the problem setup is that the increasing line length is perceived as a cost that, at the limit, offsets the utility of playing the game, and so the complete activity (waiting in line + playing the game) converges to 0. $\endgroup$
    – Brian Bien
    Jan 23, 2023 at 11:47
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    $\begingroup$ So it seems like you are calculating net marginal utility (possibly including opportunity costs). Are you asking if the assumption that "consumers do all activities with positive net marginal utility" has a name? $\endgroup$
    – Giskard
    Jan 23, 2023 at 13:43
  • $\begingroup$ That is not the question I'm asking, but perhaps net marginal utility can get me closer to a question that uses definitions properly: my hypothesis (using the example of people queuing up to wait in line for a free resource) is that, at the limit, the line length will reach a size such that the net marginal utility experienced by one more person lining up to wait will become negative. This means the state converges to everyone waiting in line with approximately 0 net marginal utility from being in the line $\endgroup$
    – Brian Bien
    Jan 23, 2023 at 23:11

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This can be captured by general models of consumer choice that are used to also analyze market interactions.

First, spending time in queue is a cost but I think you meant at no monetary cost. Then this can be described by standard textbook model of consumer choice. Suppose you have utility derived from consuming some good $x$ and leisure $l$ and hence utility is given by some $U(x,l)$. Suppose that to get one unit of $x$ you have to waste $h$ hours in queue. $h$ here can be used as a stand in for a line length since $h$ will be anyway just some monotonous increasing function of line length. The price of one hour of leisure is just one hour of your time. Then you face the following constraint $xh + l = H$ where $H$ is your time in hours.

Then the problem can be set up as:

$$\mathcal{L} = U(x,l)- \lambda ( hx + l -H)$$

Which gives you following FOCs:

$$U_x' = \lambda h$$

and

$$U'_l = \lambda$$

Combine the FOC's and you get:

$$U_x' - U_l'h = 0 \implies U_x' = U_l'h $$

Which basically states that optimum consumption of $x$ will occur at point where marginal utility of $x$ is equal to marginal cost of queuing (which is the marginal utility of leisure times the cost of time to queue for one unit of $x$). As mentioned above higher $h$ can be also directly interpreted as longer queue since the longer the queue is the longer you have to wait.

You could also build model with more than just one representative agent like above, but in the end you will be using some model from standard consumer choice theory. Paying with waiting is not fundamentally different from paying with money, when it comes to models of optimal consumer choice.

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    $\begingroup$ +1 A more sophisticated model might also take account of the points in my comment on the question. $\endgroup$ Jan 22, 2023 at 14:00
  • $\begingroup$ I've updated my description a bit (using net marginal utility), hopefully clarifying the wording. The relevant interesting difference w.r.t. paying with waiting instead of money, I suspect, is that the cost (wait time) will increase naturally up until the point of everyone in the line converging toward receiving 0 net marginal utility from the wait, given some simplifying assumptions (including a large enough supply of people). I suppose it is no different from money if a good is priced efficiently; the interesting point here seems to be a natural convergence to 0 marginal net utility. $\endgroup$
    – Brian Bien
    Jan 23, 2023 at 23:25

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