A Little Economic Analysis of Waiting as Deadweight Loss (after Stokey & Zeckhauser)

'''DEADWEIGHT LOSS''': inefficiency – price that someone is paying but no one is receiving.

Imagine a golf course with a capacity of 150 golfers per day in a community with well over this many golfers. Experience has shown that the relationship between the number of people who want to golf on a given day and how long they have to wait to tee up looks like this:

What do we read in this graph? The obvious situation that the more golfers there are, the longer the wait.

But it makes sense to make a further assumption: that people have '''different levels of willingness to wait''' to play. Some will wait for hours, others will decide not to play unless they can play within 15 minutes. This can be represented like this:

The equation for this will look generically like this:

(1)
$$N = f(W)$$

where N is the number who want to play and W is the expected wait time.

Let's ignore that flat part of the graph for now. The "Y" intercept (actually the N-intercept here) is, say 235 (based on empirical data probably — e.g., how many folks show up before word goes out that the course is crowded).

and let's say the slope is $-\frac{5}{2}$. That is, for each extra 2 minutes of expected wait, 5 fewer people are interested in waiting.

And so we have $N = -\frac{5}{2} W + 235$

We would like, though, to express W as a function of N ($W = f(N)$) since the original graph had N on the x-axis and W on the y-axis. We can do this by solving for W.

(2)
\begin{align} N = -\frac{5}{2} W + 235 \end{align}
(3)
$$2N = -5W + 470$$
(4)
\begin{align} -\frac{2}{5} N = W - \frac{470}{5} \end{align}
(5)
\begin{align} -\frac{2}{5} N + \frac{470}{5} = W \end{align}
(6)
\begin{align} W = -\frac{2}{5} N + 94 \end{align}

These can be combined in a single diagram

But let's stop for a second and figure out what the new version of the W and N line means. For any given wait time, we saw above, there is a group of people who want to play. In that group are very patient people who really really love to golf and who will wait a long time. But what is the boundary of a given group? It's formed by the person in the group with the least patience. The person who is only just willing to wait that given amount of wait time.

This means that for any given wait time, the corresponding point on the straight NW line tells us the patience level of the least patient person willing to wait that long.

To be sure we see what's going on, let's imagine two situations. First, what happens when 135 people want to play golf? Alternatively, what happens if only 30 want to play?

What's going on where the lines intersect?

Remember, the X-axis is how many people play golf. And the downward diagonal line represents how long they'd be willing to wait. We can think of this as "what playing a round of golf is worth to them." So we can read from the diagram what they actually paid in waiting time and what they "got" in terms of what they'd have been willing to pay.

Assume that people's time is worth $3 per hour. Since the vertical scale represents waiting time, we can relabel it in dollars at this going rate. At the equilibrium point, the average wait is 44 minutes. All told, the 125 golfers pay for this average wait even though some of them are so hard core that they would be willing to pay more. There is a sense in which these golfers got a bargain at only waiting 44 minutes. But this "payment" is burned money. Everyone who golfs pays the fee but no one receives the payment. What if we tried to direct this payment to keep it "in the system"? The town council looks at charging$2 per round of golf. What's the effect?

If time is worth \$3 per hour this is like throwing an 40 minute wait at people before anyone even shows up to play. What effect does this have on the number of people who wish to play?

Note: fee shading in the above diagram should go up to next line. See book page 86.