But I know it will work. We will (almost) all set our watches one hour slow, and we will (almost) all start doing (almost) everything one hour later, by the sun, compared to what we would have done if we hadn't changed our watches. But why?

This weekend the government will tell us all to put our watches back one hour. They want us all to do everything one hour later. It's hard to understand how it will work. Do they think we will all forget we've set our watches one hour slow? What's more, they can't even force us to change our watches.

I've heard stories about people who set their watches 10 minutes fast, so they won't be late for meetings. It's hard to understand how it could work. Do they forget they set their watches 10 minutes fast? Because if they remember, they should be able to figure out they've got an extra 10 minutes, so there's still plenty of time to grab a quick coffee before the meeting starts. If it works, they must be fooling themselves.

A purely nominal change (whether we say than solar noon is called "12.00" or "13.00") will have a big real effect. Even though we all know that it's just a nominal change. Nobody is fooled by the government changing our watches without us seeing them do it. Nobody suffers from "time illusion". We know it's not really earlier. In fact, it's the people who don't hear that the time has changed who are likely to still get up at the same real time, by the sun.

And the real effects of this nominal change will last, at least until the Spring, when the government will tell us to change our watches forward again.

I'm not the first economist who has drawn the analogy between Daylight Savings Time and money. Milton Friedman used the same analogy to explain why it was easier to adjust the real exchange rate through adjusting the nominal exchange rate than through adjusting millions of nominal prices.

Milton Friedman's analogy is even more apt this time around. The US changed the date at which Daylight Savings Time would end. After some discussion, because the new US date didn't work so well for us up North, Canada decided it was more important to stay in synch with the US than to stay in synch with our own dawn and dusk. Canada and the US may not be an optimal currency area, but it's been decided that we are an optimal time area. No currency union, but we do have a time union. It's the equivalent of fixed exchange rates. When the US devalues its time, we devalue our time too.

Why does money have real effects? It's just bits of paper. It's not real. We are still stuck on David Hume's puzzle. If we double the number of bits of paper each one should be worth half as much. It should be a purely nominal change. Nothing real should change. If we switch from meauring turkeys in pounds to measuring them in kilograms, the price per unit weight should be divided by 2.2, but the same turkey should cost exactly the same in pounds or kilograms, and we should buy exactly the same number of turkeys as before.

Metrification was a nominal change that had negiligible real effects, as far as I know. Daylight Savings Time is a nominal change that has real effects. Some monetary changes, like currency reforms where we knock a couple of zeroes off the old currency and call it the new currency, are like metrification, where nothing real changes. And maybe all monetary changes are like metrification in the long run. But some monetary changes are like Daylight savings Time, and have real effects, at least in the short run.

If we understood Daylight Savings Time better, and how it works, we might understand monetary policy better.

First, let's think about the individual's choice problem. When I choose to do something depends on two things: the sun; and everyone else's chosen time. Let t be the time I choose, let T be the time everybody else chooses, and let S be the time the sun chooses. So my reaction function is t = R(T,S). Assume that dR/dT>0, dR/dS>0, and dR/dT+dR/dS=1. What this means is that if everybody else gets up one hour later, I will get up (say) 45 minutes later. And if the sun gets up one hour later, I will get up (say) 15 minutes later. But if everybody else and the sun all get up one hour later, I will get up one hour later too.

For example, t = 0.75T + 0.25S could be a simple linear reaction function.

Now, assume everyone else has the same reaction function as me. We solve for the symmetric Nash equilibrium, where t = T. The answer is T = S. Everyone follows the sun, and so I'll follow the sun too.

OK. That theory failed miserably. In equilibrium it doesn't matter what our watches say, and so Daylight Savings Time makes no difference to what we do. Which is not what happens. And that theory is the exact counterpart of the individual firm's profit maximising price-setting equation where the price an individual firm sets depends on the prices other firms set and on the money supply. We can't get non-neutrality that way.

We need some sort of discontinuity in the reaction function. If we are one minute late for the meeting, it's almost as bad as being 10 minutes late. Otherwise, if the sun gets up one hour later, and everybody arrives at the meeting one minute after they expect everyone else to arrive, the meeting starts one minute late, and then the next day two minutes late, and when everyone figues out what's happening, and extrapolates to the new Nash Equilibrium, the meeting is an hour late, just like the sun.

What sort of loss function would gives us the linear reaction function like t = 0.75T + 0.25S ? It would have to be a quadratic loss function. Each individual wishes to minimise Loss = a(t-T)^2 + b(t-S)^2. The first order condition for a minimum is dLoss/dt = 2a(t-T) + 2b(t-S) = 0. And we solve for t as t = (a/(a+b))T + (b/(a+b))S.

That loss function was smooth. The indifference curves are roughly circular (ellipsoid?). Suppose we changed the loss function to something that has a kink in it. Something like this: Loss = aABS(t-T) + b(t-S)^2 . The marginal cost of moving away from the sun starts at zero, when you are following the sun exactly, and steadily increases as you move further and further away from the sun. But the marginal cost of moving away from everyone else is a constant. It doesn't start at zero. Being even one minute late for the meeting is costly, and being two minutes late is twice as costly. So, unless everyone else is a long way away from the sun, you will want to show up exactly on time for the meeting. It's only when they hold the meeting well before dawn that you decide to sleep in, forget everybody else, and just follow the sun. The reaction function is discontinuous.

Now we are getting somewhere. If we solve for the symmetric Nash equilibrium we find there's a range of equilibria around the sun. Any time, as long as it's not too far away from the sun, is an equilibrium time to hold the meeting. Everybody shows up when they expect everybody else to show up. And if the government tells, or merely suggests, that we all put our watches back one hour, and if everyone expects that everyone else will follow the government's suggestion, we all go to the meeting one hour later by the sun. Because everyone hears the government's suggestion, and knows that everyone else hears it too, and so on, it acts as a focal point to coordinate our expectations about when meetings will take place. But if the government suggests we all set our watches back two hours, or maybe three or four, we decide to ignore it, and know that everyone else will too.

Back to monetary policy. What we need is some sort of loss function with a kink in it. So the losses to a firm that raises or lowers its price, relative to other firms' prices, are significant even if it raises or lowers its price by just a penny. The marginal costs of having the wrong relative price don't start at zero. So firms won't cut their prices unless they expect other firms to cut theirs too, even if all prices are too high relative to the sun, I mean the money supply. The losses from small deviations of price from the money supply must be second order of smalls, but losses from small deviations of price from everyone else's price must be first order of smalls.

The Beatles.