Ok: here's the idea: monopolistic competition, many identical producers face identical (inverse)demand functions:
Pi/P = M/P - Qi
Pi is the ith producer's nominal price; P is the average price, Qi is the ith producer's quantity. M is the money supply. Each has no variable costs, only fixed, for simplicity. So we know that the optimal relative price - that which maximizes revenues- is simply half of M/Pi, as is the optimal quantity. In a world without menu costs, the price level adjusts so that M/P = 2. Suppose it were less than 2. Then each producer's optimal relative price is less than 1, so all would cut prices until M/P went back to 2 , at which point each would be happy charging what others are charging.
I want to add menu costs and examine two cases. We cut M in half first. Then we double it.
First case: Here we have two equilibria, one where none adjust and one where all adjust, if menu costs lie in the range .25-1. Suppose all others adjust optimally cutting their price in half. Then real money is unchanged at 2. If you also adjust, you make profits of 1- MC (MC for menu cost). If not, you charge a relative price of 2 and sell nothing, for profits of 0. All adjust is a symmetric Nash equilibrium if MC is less than 1.
If all others fail to adjust, each faces demand curves with intercepts of 1. The firm that doesn't adjust charges a relative price of 1 and sells nothing. The firm that adjusts charges a relative price of 1/2 and sells 1/2, earning profits of 1/4 -MC. So if menu costs are greater than 1/4, none adjusting is also a symmetric Nash equilibrium. This case obviously echoes Ball and Romer's much more sophisticated "Sticky prices as Coordination failure", which inspired the whole thing. Now, there's lots of experimental evidence for coordination failure, so let's say with a negative money shock, we get the sticky price equilibrium, with output falling to 0!
The second case, however, the positive money shock, is very different. So double the money supply. If all others adjust, I face a demand curve with an intercept of 2. If I adjust as well, I gets profits of 1- MC. If I don't adjust, I charge a relative price of 1/2 and sell 3/2, for profits of 3/4 . So I - and everyone else - will NOT want to adjust if all others adjust just in case MC is greater than 1/4. In other words, all adjust is not an SNE.
On the other hand, suppose all but you fail to adjust. You face a demand curve with an intercept of 4. If you adjust, your relative price is 2 and you sell 2, for profits of 4 - MC. If you fail to adjust, you charge a relative price of 1 and sell 3, for profits of 3. So for MC less than 1, you would want to adjust if all others fail to adjust, so stickiness is not an SNE either.
In the second case, for the positive money shock, we have partial adjustment. We have a chicken game, with a mixed-strategy SNE in which each adjusts with a certain probability ( or a proportion adjust equal to that same probability).(Is this a possible micro-foundation for a Calvo fairy?)
So: sticky prices for negative shocks; partial price adjustment - and so less of an output increase - with positive shocks.
It's almost like the Old Keynesian idea of a reversed-L- shaped aggregate supply, except here the vertical segment of the L is leaning to the right.
OK so it's silly. But think about the reason for the asymmetry: It comes about because the benefit of adjusting optimally gross of menu costs is greater when demand is greater. For the negative shock, demand is greater when all adjust, so if the range is right we get two equilibria: MC greater than benefit of adjusting when when demand is low (because none adjust); MC smaller than benefit of adjusting when demand is high (because all are adjusting).
For the positive shock, demand is greater when none adjust, so the benefits of adjusting gross of menu costs are greater when none adjusts, and smaller when all adjust - but you. That's why we get the chicken structure.
I know I need pictures, as Nick Rowe told me about my last post. I will work on it.!
Happy New Year!