Individual and aggregate, marginal and total, incentives to cut prices in a recession

The purpose of this post is to lay out the intuition behind my discussion/argument with Steve Randy Waldman. (I'm not sure whether this post will help or hinder my discussion with Steve. But students of New Keynesian macro might find it useful regardless.)

We need to distinguish between the individual and aggregate incentives of cutting prices. (Steve gets this, of course, which is why he is talking about coordination problems). But we also need to distinguish between the marginal incentive to cut price by a small amount, and the total incentive to cut price by a big enough amount to fully offset the fall in demand.

To keep it simple, assume the central bank targets a constant level of NGDP. [Update: yes, by making that assumption I'm deliberately ducking a dirty big question about the aggregate incentive to cut prices, which depends on what the central bank targets.] But it's a stupid central bank, that keeps changing its NGDP target. Once every 10 years, the central bank flips a coin, and if it's heads the central bank doubles NGDP, and if it's tails the central bank halves NGDP. So NGDP (equals PY) follows a random walk. And that's the only shock to the economy.

The economy is composed of symmetric worker/firms. Each agent is a yeoman farmer who produces a variety of fruit that is different from all other agents'. In Bertrand Nash monopolistically competitive equilibrium, they all set the same price (so the relative price P(i)/P equals one) and all sell the same output. Just like in simple New Keynesian models.

The picture for an individual agent looks like this:

The individual agent's profit (or utility) is the green shaded area (minus fixed costs, if any).

The individual agent's Marginal Cost curve reflects the (subjective) cost of his own labour, plus the cost of inputs purchased from other firms, if any.

Start in equilibrium with actual Y equal "potential" Y*, then suppose the coin lands tails. So we know that PY halves.

If all agents halve their prices, nothing real changes, Y stays at Y*, and no agent is worse off. If all agents hold their prices constant, Y halves, and all agents are worse off.

That's the aggregate experiment. Now let's look at the individual experiment.

1. Suppose all other agents hold their prices fixed, but agent i can choose his price P(i). Agent i will be worse off than he was before the recession, regardless of where he chooses to set his price. That's because the demand curve facing agent i has shifted left.

2. Suppose all other agents hold their prices fixed, but agent i can choose to do one of two things: he can choose either to hold his price P(i) fixed (letting Y(i) fall), or else hold his output Y(i) fixed (letting P(i) fall). He is worse off either way (see 1 above), but which of those two options is better? We don't know the answer to this question. It depends. Look at the picture above, then shift the demand curve (and Marginal Revenue curve) left, and compare the profits when it holds P(i) fixed (cuts Y(i)) to the profits when it holds Y(i) fixed (cuts P(i)), and you will see this. It could go either way. [Update: intuition: it loses more revenue when it holds P(i) fixed (since the demand curve must be elastic), but it also loses costs.]

3. Suppose the answer to question 2 above is that he is better off holding P(i) fixed and letting Y(i) fall than holding Y(i) fixed and letting P(i) fall. This does not mean the individual agent maximises profits by holding P(i) fixed. Because there is almost certainly a third option which is better than both those two options.

3a. The most likely third option is to cut his price P(i), but by less than half. But if each agent wants to cut his relative price, even if only by an epsilon, the only Nash equilibrium is if all agents halve their prices, and the economy stays at Y*.

3b. If the Marginal Cost curve is very flat, and if the elasticity of the demand curve falls in a recession (when it shifts left), it is possible that the individual agent will want to raise his relative price in a recession. In this case the Nash equilibrium exists, but is unstable.

3c. If we rig the assumptions exactly right, we can rig it so the individual agent wants to hold his price fixed when the demand curve shifts left in a recession. In this case there is a continuum of Nash equilibria. But to get this result, we really do have to rig the assumptions exactly right, so each individual agent wants to set a relative price of exactly one, regardless of shifts in his demand curve, otherwise we don't get any Nash equilibrium at all. This result is extremely fragile.

3d. If we rig the assumptions right (they don't need to be exactly right) in a non-linear model (e.g. where the demand curve gets more elastic, then less elastic, then more elastic again, as we get a deeper and deeper recession), we could get a finite number of Nash equilibria, some locally stable and some unstable.

3e. If you make the demand curve kinked, at a relative price of exactly one (which takes some rigging of the assumptions), you can also get a continuum of Nash equilibria. Because there's an upper and lower Marginal Revenue curve, and you have no incentive to change relative price if MR^- < MC < MR⁺.

What we've got here is a model that generates a reaction function for the representative firm, where each firm chooses P(i) as a function of P and NGDP:

P(i) = R(P,N)

If it's a linear function we can write it as:

P(i) = bP + (1-b)N/Y*

Which has a unique Nash equilibrium P=N/Y*, regardless of b. But if b > 1 that equilibrium is unstable. And if b=1 there's a continuum of Nash equilibria.

Comments

[Billikin]

OK, you have a really stupid CB that targets NGDP. If I read you right, if this really stupid CB halves its target, but everybody else in the economy continues to do what they have been doing, there is a recession. Well, it seems obvious that not everybody continues to do what they have been doing, or nothing would change. There is a lot left out of the story. Or is there? In previous posts you have suggested that all the CB, stupid or not, has to do is to change its target and everybody changes their behavior. In this case you assume that they do not change their behavior, yet Y halves. What's the whole story?

[Nick Rowe]

Billikin: It is not stupid to target NGDP. It is stupid to change the NGDP target at random every 10 years. IF this causes a recession, then the central bank is stupid. Whole story.

****IF**** everybody holds prices constant, there is a recession. What we are trying to figure out is the conditions under which there are ****individual incentives*** to cut prices. Jeez!

[Billikin]

OK, if everybody holds prices constant, and there is a recession, what are they doing to cause the recession? And why? Thanks.

[Billikin]

Oh, I didn't mean that it was stupid of the CB to target NGDP, just that they were stupid and they did that.

[Steve Waldman]

Nick — As I mentioned in my latest on the previous post, I think we are finding our way towards consensus. I do have to quibble with one statement, though:

"The individual agent's Marginal Cost curve reflects the (subjective) cost of his own labour, plus the cost of inputs purchased from other firms, if any."

This isn't quite right. The marginal cost curve reflects only the variable costs of inputs purchased from other firms if any. The meaning of leverage — operating leverage, financial leverage — is that there are fixed costs that have to be covered regardless of the level of output. It is these fixed costs, and the risk of firm termination if they are not met, that can put a wedge between the marginal cost curve and the effective supply curve of a leveraged firm.

[Too Much Fed]

"To keep it simple, assume the central bank targets a constant level of NGDP."

Steve, could you describe that real quick?

My guess is that Nick and you have different models here.

[Majromax]

> 3a. The most likely third option is to cut his price P(i), but by less than half. But if each agent wants to cut his relative price, even if only by an epsilon, the only Nash equilibrium is if all agents halve their prices, and the economy stays at Y*.

Wait, why is agents halving their prices a Nash equilibrium?

If the no-response optimum is for an agent to cut its prices by ε, then why isn't the Nash equilibrium for everyone to cut prices by ε? You seem to be iterating this game, where after period 1 firms observe others cutting their prices and then adapt their response.

[Nick Rowe]

TMF: stop. Off topic.

Majro: "Wait, why is agents halving their prices a Nash equilibrium?"

If all agents half their prices, then P(i)/P = 1. And since NGDP halved and P halved, then Y stayed the same, so Y=Y* again, so no firm will want to change P(i), so it's a Nash equilibrium.

"If the no-response optimum is for an agent to cut its prices by ε, then why isn't the Nash equilibrium for everyone to cut prices by ε?"

Because each agent knows that other firms will cut prices by e too, so will want to cut by another e.

That's what Nash Equilibrium means. Best response to best response.

[Nick Rowe]

Steve: suppose they maximise Utility, not profits. And suppose U = U(net profits), where net profits equal gross profits (as in my picture) minus fixed cost.

Under certainty (or if they set prices after observing the shock to NGDP) this won't make any difference, as long as U( ) is a monotonic increasing function.

Under uncertainty, it might make a difference. But even here it would only make a difference if they cannot change prices *after* the shock is observed. Which means we are assuming temporarily sticky prices.

Now, you might argue that U( ) is not strictly monotonic. Trouble is, if U is independent of net profits (in some range), we can't really say what firms will do.

You might say there are two shocks, a macro shock to aggregate demand, and a micro shock to relative demand, and firms can change prices after observing the macro shock, but not after observing the micro shock.

But even here my gut says the only way to get sticky prices is with something like a kink in the demand curve, and it's going to be a very fragile knife-edge equilibrium. But I'm not 100% sure. Sometimes my gut is wrong.

[J.V. Dubois]

I am loving this. It reminds me the following chapter from Samuel Bowles Microeconomics, maybe it will help you think about it more (the whole problem is fortunately part of Google Books): https://books.google.sk/books?id=HAiMDU4qv0IC&lpg=PP1&dq=samuel%20bowles%20microeconomics&pg=PA66#v=onepage&q&f=false

[Nick Rowe]

JV: thanks. In some ways, the distinctions here, between individual experiment and aggregate experiment, run right across all aspects of economics. If each individual is trying to do X, and his choice depends on others' choices, what happens when you put them all together?

[Majromax]

> Because each agent knows that other firms will cut prices by e too, so will want to cut by another e.

Ahah. I've also slept on this, and what I realize is that you're assuming too much rationality.

The game you describe here is equivalent to the guess 2/3 of the average game, which asks participants to guess a number between 0 and 100, with the winner being that closest to 2/3 of the average of all the guesses. (Your game is "guess 0.5 plus (1-ε) times (the average - 0.5)", which is a shifted-and-scaled version of this game.)

The Nash equilibrium is for every participant to guess 0 (or 1), however this makes a strong assumption that all other participants are actively rational. In practice, this Nash equilibrium is not realized, and a winning value is often in the range of 16-30.

This is why you can't get price stickiness. If you're assuming infinite rationality (such that all agents know that every other agent is rational and will iterate the game), then you get an impulse response to a shock that shifts the system to its new equilibrium. You need a limited order of rationality like I describe (I know what I'm doing, but everyone else is as dumb as a bag of rocks) to have a finite-time response to the monetary shock. This limited rationality seems to often occur in practice, especially in not-strictly-finance where it is more difficult to cast a real-world business decision in terms of an abstract reaction function.

[Nick Rowe]

Majro: I'm assuming Nash equilibrium. Yes, some game theorists do reject Nash equilibrium, for those reasons.

Another approach is to repeat the game every period, let players learn from experience, and ask whether it converges to Nash, and how long it takes to converge. (But how long is the "period", in macro?). When I reject "unstable" Nash equilibria, I am implicitly saying that those equilibria are unlearnable.