A simple model where NGDP targeting beats inflation targeting

Tony Yates complains that Market Monetarists don't have a model to justify their support for NGDP targeting. So I decided to build him a little model. It's a cruddy little one-period model, and it's really only a sketch of a model, but any competent grad student should be able to do the math to finish it.

The only important difference from a standard New Keynesian model is that I've ditched Calvo's fairy, because she keeps on insisting that the firms that change prices are a representative sample of all firms. I've replaced her with a coordination game with menu costs, a bit like Ball and Mankiw (pdf). Relative shocks are skewed, so those firms that most want to change prices are not representative of all firms. I think that's much more realistic than the random flight of Calvo's fairy.

The intuition is that an inflation targeting central bank must create a big enough output gap to dissuade the firms that most want to deviate from the inflation target from doing so. An NGDP targeting central bank can allow those firms to deviate from the implicit inflation target, provided it creates an output gap just big enough to keep NGDP on target. The smaller the percentage of firms that want to deviate, the smaller the output gap needed to keep NGDP on target.

[Update: Or maybe this is a more general way of stating the intuition: The IT central bank must create a big enough output gap to make the sample of firms that want to pay the menu cost an unbiased sample, so on average they leave prices the same. An NGDPT central bank can allow a biased sample of firms to change prices, provided it offsets it with a (smaller than under IT) output gap.]

To keep it simple, there are only three firms: call them A for apples, B for bananas, and C for carrots. (I can't have two firms because then I couldn't have skewed shocks. And NGDP targeting would perform even better with more than three firms.)

This is the order of moves:

1. The central bank announces either an inflation target, or an NGDP target.

2. The firms announce their (provisional) initial prices.

3. Nature rolls a 3-sided die, and announces whether it lands on A, B, or C.

4. Nature then flips a coin, and announces heads or tails.

5. The die and coin cause a shock to preferences. "A-tails" means that consumer preferences shift away from apples, towards bananas and carrots equally. "A-heads" means that consumer preferences shift towards apples, away from bananas and carrots equally. For simplicity the preference parameters are rigged so that preference shocks would have no effect on aggregate output if prices were perfectly flexible. Call that level of output Y*.

6. The central bank does whatever it needs to do to ensure its previously announced target will be hit.

7. Each firm chooses whether to pay a menu cost for permission to adjust its price now that it has observed the preference shock and the central bank's response.

The key to understanding the model is to understand that (if the die lands on A) firm A has a bigger incentive to pay the menu cost and change its price than do firms B or C. Because the relative demand shock is twice as big for firm A than it is for firm B or firm C. There may (or may not) be multiple equilibria, because if they observe "A-tails", firms B and C might choose to pay the menu costs and raise their prices if they expect that firm A will refuse to pay the menu cost to cut its price. But we are unlikely to observe such equilibria, because the driver commonly known to have the most to lose in a head-on crash will be more likely to swerve in a game of "chicken".

What will the equilibrium look like?

Let's start with inflation targeting.

All firms set a provisional initial price that hits the inflation target.

Suppose the preference shock is "A-heads".

If menu costs are very small, all three firms will pay the menu cost, firm A will raise its price, firms B and C will cut theirs, and the inflation target is hit and output is at Y* (the flexible price equilibrium).

If menu costs are very large, no firm will pay the menu cost, provided the central bank does not let Y deviate too far from Y*. Inflation stays on target, but output is indeterminate within that range (the central bank can manipulate output within that range without causing inflation to miss the target). [Update: but if firms expected the central bank to manipulate output away from Y*, they would want to set higher or lower initial relative prices, so there is no equilibrium where inflation is on target. So the central bank would commit not to do this.]

If menu costs are medium-sized, only firm A would pay the menu cost and raise its price if the central bank sets Y at Y*. But then inflation would rise above target. But the central bank knows this, and would have to ensure that Y is far enough below Y* to dissuade firm A from paying the menu cost. Inflation targeting causes a recession. (And inflation targeting causes a boom if the coin comes up tails.)

Now let's look at NGDP targeting.

All firms set an initial price that hits the NGDP target at Y*.

Suppose the preference shock is "A-heads".

If menu costs are very small, all three firms will pay the menu cost, firm A will raise its price, firms B and C will cut theirs, and the NGDP target is hit and output is at Y* (the flexible price equilibrium).

If menu costs are very large, no firm will pay the menu cost, provided the central bank does not let Y deviate too far from Y*. And since it targets NGDP, the central bank will not let Y deviate from Y*.

If menu costs are medium-sized, only firm A would pay the menu cost and raise its price if the central bank sets Y at Y*. But then NGDP would rise above target. But the central bank knows this, and would have to ensure that Y is far enough below Y* that the NGDP target is hit. But the recession would be smaller than under inflation targeting. Because under inflation targeting the central bank would have to create a recession big enough to dissuade firm A from paying the menu cost to raise its price. Under NGDP targeting the central bank can let firm A raise its price, provided it cuts Y by one third (because there are three firms) of the percentage that firm A raises its price. (There would be a boom if the coin came up tails, but a smaller boom than under inflation targeting.)

With three firms, NGDP targeting beats inflation targeting, because it leads to smaller output gaps. The larger the number of firms, the better NGDP targeting will perform. In the limit, as the number of firms gets very large, NGDP targeting will perform perfectly, though that is only true in this special model where only one firm gets chosen by the die. But NGDP targeting will outperform inflation targeting provided the shocks are skewed. Real world shocks will nearly always be skewed, to some extent.

Comments

[Nick Edmonds]

Does this result depend to some extent on the binary nature of the game, where firms face either an increase in demand of X or a decrease in demand of X/2 (or the other way round for a tails-up), but nothing in between?

What would happen if instead we have demand shifts involving a continuum of firms experiencing a smooth distribution of changes to demand, so that by however little the CB reduces Y, some proportion of firms will always be changing their pricing decisions?

[Nick Rowe]

Nick E:

If the shock is large (relative to menu cost) than all firms will pay the menu cost, so prices are perfectly flexible, so both NGDPT and IT give the same Y* equilibrium.

If the shock is small (relative to menu cost) then no firm will pay the menu cost, so prices are perfectly fixed, so both NGDP and IT give the same Y* equilibrium (except IT has that weird range of equilibria).

It's when you get the medium-sized shock (relative to menu cost), so that some firms will pay the menu cost at Y* and some won't, that NGDP beats IT, provided the shock is skewed and not symmetric.

With a continuum of shock sizes, and a continuum of menu costs, I think NGDP would *always* beat IT, because some firms will want to pay the menu cost at Y* and some firms won't, and the ones that want to pay the menu cost will be a biased sample of all firms and will want (on average) to change prices in one direction.

The intuition is that the IT central bank must create a big enough output gap to make the sample of firms that want to pay the menu cost an unbiased sample, so on average they leave prices the same. An NGDPT central bank can allow a biased sample of firms to change prices, provided it offsets it with a smaller output gap.

[Nick Edmonds]

I see. Is this just equivalent to saying that movements in (an upwards sloping) AS curve have less impact on Q, when the AD curve is downwards sloping rather than horizontal?

[Nick Rowe]

Nick E: yes. But we also need a model where shifts in SRAS don't always mean changes in Y*. (Or, more generally, a model where the horizontal shifts in SRAS are *bigger* than the changes in Y*, because if they are the same size then we want those changes in Y to happen, and IT delivers just that.)

Plus, we need a model, period. Because guys like Tony (with some justification) won't believe us unless we can model everything formally, with microfoundations. You can't do welfare economics without people who have preferences. We want to show that NGDP targeting makes them happier than IT.

[Market Fiscalist]

One can imagine a situation where all firms are able to adjust output to changes in demand so that prices never need to change. Perhaps labor is the only production cost and totally homogenous so that a preference shock of "A-heads" leads to a redistribution of labor between firms, and increased production of A and decreased production of B and C but no prices change. It is possible that the change in relative demand will lead to an increase in NGDP (perhaps Apples are seen as a good substitute for money as a savings vehicle and increased demand for apples causes money velocity to increase). An NGDP-targetting CB will cause a recession to eliminate the increased NGDP, while a IT one will do nothing..

Won't inflation targeting be optimal in this case ?

[Majromax]

@MF:

Your situation is unrealistic, since you implicitly assume that the marginal cost of production (in labour terms) of each unit of A, B, and C remains constant with output, combined with full labour flexibility. In such a scenario, there is no reason for prices to ever change, and we cannot distinguish fully flexible prices from fully fixed prices.

Additionally, this model obviously lacks investment. In such a model, discussing "savings vehicles" is nonsensical, since there are no net savings.

[Too Much Fed]

"For simplicity the preference parameters are rigged so that preference shocks would have no effect on aggregate output if prices were perfectly flexible. Call that level of output Y*."

Does that Y* assume "full employment"?

[Nick Rowe]

MF: you are right. If we assume labour is the only input, and constant returns production function, and homogenous perfectly mobile labour, and constant-elasticity demand curves for each of the 3 goods, then relative prices would not need to change if there were preference shocks. And inflation targeting would be optimal.

But NGDP targeting would be optimal too. There would be no difference between them.

But then I could change the model slightly to introduce productivity shocks (weather?) instead of preference shocks, and get my old results back.

I haven't specified an AD function for this model. Because it's more about SRAS shocks than AD shocks.

[Market Fiscalist]

I think I agree. In any model where frictions prevent output and relative prices moving to their equilibrium levels when relative demand changes then I cannot think of any scenarios where NGDPT will be worse than IT. Still thinking it through though.

[Nick Rowe]

MF: I can think of a model where IT does better than NGDPT. Bring back Calvo's fairy, and let there be aggregate real shocks that reduce/increase Y*, then Calvo's fairy would ensure that the SRAS curve shifts left/right by the same amount as Y* falls/rises, and IT would create a horizontal AD curve that would ensure that Y falls/rises by the same amount as Y*, while NGDPT would create a downward-sloping AD curve that would ensure that Y falls/rises by less than Y*. Those fluctuations in Y would be desirable, and NGDPT would make those fluctuations too small.

TMF: think of "full employment" as just a (silly) name for Y*.

[Too Much Fed]

I am probably violating the assumptions here but...

Let's say there are 100 entities that eat. They all eat 50 apples, 50 bananas, and 50 carrots per month each. They won't eat any more because it will give them a belly ache.

Now let's say there 5,000 apples, 5,000 bananas, and 5,000 carrots produced per month at full employment. Now let's say there is a big productivity shock so that 6,000 apples, 6,000 bananas, and 6,000 carrots could be produced per month. Instead only 5,000 per month of each will be produced and employment goes down.

I believe preferences between the 3 have not changed.

Can NGDP targeting or price inflation targeting "fix" that scenario?

[louis]

TMF:
Someone figures out a way to juice and ferment apples, bananas, and carrots so people have something to drink along with their food.
Productivity gains in the real world usually lead to GDP growth, not unemployment. Human desires are never satiated.
Sticking to your constraints though, it depends on who manages the inputs of production. If everyone is a farmer with his/her own land, "unemployment" just means "I work fewer hours to satisfy my fixed desires". If there are a handful of firms that coordinate production and employment, I guess that there could be a problem if they lay off workers (depriving them of income needed to consume) instead of cutting hours across the board.

[Too Much Fed]

"Productivity gains in the real world usually lead to GDP growth, not unemployment. Human desires are never satiated."

What about Warren Buffett and Apple?

I'd say productivity gains lead to potential real AS growth, and if an economy is supply constrained, then real GDP growth.

"If there are a handful of firms that coordinate production and employment, I guess that there could be a problem if they lay off workers (depriving them of income needed to consume) instead of cutting hours across the board."

That is what I see happening. I forget the exact phrase Nick calls it, something to do with oligopolies.

The U.S. vehicle market is a good example. Take someone like Mitt Romney. He has/had three vehicles and does not want any more.

There are about the same number of vehicles being produced as 10 years ago with fewer workers. The vehicle firms are trying to jack up the stock price.

[Majromax]

@TMF:

> Can NGDP targeting or price inflation targeting "fix" that scenario?

There's no market-clearing equilibrium solution here even in a barter economy, so we shouldn't expect any sort of monetary policy to fix this.

Your problem is that the marginal utility of consumption is discontinuous, so that going from 49 to 50 carrots provides positive utility but going from 50 to 51 provides none. Since all needs can be satiated in your scenario, this means that your conclusions are counterintuitive.

Your proposed productivity shock is positive in terms of vegetables produced, but the satiated needs mean that we don't actually care. Instead, your productivity shock is actually a negative supply shock to the level of full employment, measured in hours of labour.

Introducing just one infinite need, let's say leisure time, fixes this. The barter market can clear by adjusting the price of vegetables versus the price of leisure, and assuming frictionless markets (hah) and identical agents this means that everyone works less.

NGDP targeting could accommodate this scenario if "leisure consumed" was considered part of GDP, even though it is not directly traded on a market. My hunch is that whether inflation targeting would work equally depends on the relative stickiness of vegetable prices versus wages.

Your broader point about satiated needs is misleading. As long as there is even a single person who wants something that can be produced but they can't afford, the economy is supply-constrained. The vehicle market (for example) isn't the entire market.

[Too Much Fed]

"There's no market-clearing equilibrium solution here even in a barter economy, so we shouldn't expect any sort of monetary policy to fix this."

Agreed. Now we are getting somewhere.

"Your problem is that the marginal utility of consumption is discontinuous, so that going from 49 to 50 carrots provides positive utility but going from 50 to 51 provides none. Since all needs can be satiated in your scenario, this means that your conclusions are counterintuitive."

Counterintuitive, indeed.

"Your proposed productivity shock is positive in terms of vegetables produced, but the satiated needs mean that we don't actually care. Instead, your productivity shock is actually a negative supply shock to the level of full employment, measured in hours of labour."

Why isn't that a negative demand shock to the level of full employment, measured in hours of labor (firms are demanding fewer hours of labor while supply of hours of labor remains the same)?

"Introducing just one infinite need, let's say leisure time, fixes this. The barter market can clear by adjusting the price of vegetables versus the price of leisure, and assuming frictionless markets (hah) and identical agents this means that everyone works less.

NGDP targeting could accommodate this scenario if "leisure consumed" was considered part of GDP, even though it is not directly traded on a market. My hunch is that whether inflation targeting would work equally depends on the relative stickiness of vegetable prices versus wages."

How about "permanent" leisure time, called retirement? I am not sure how "leisure consumed" could be part of GDP.

"Your broader point about satiated needs is misleading. As long as there is even a single person who wants something that can be produced but they can't afford, the economy is supply-constrained. The vehicle market (for example) isn't the entire market."

Wouldn't that be considered demand-constrained? Wouldn't supply-constrained be considered someone can afford something but not enough of it could be produced?

[Too Much Fed]

Majromax, I believe Nick's Y* at full employment actually means Y* at full employment while minimizing the retirement market.

If I'm remembering Bill Mitchell correctly, there are actually more than one Y*'s depending on the size of the retirement market.