Calvo (staggered) contracts
Based on Wikipedia: Calvo (staggered) contracts
The Stubborn Stickiness of Prices
Imagine you run a coffee shop. Every morning, hundreds of customers stream through your doors expecting to pay roughly what they paid yesterday. One day, you hear that the cost of coffee beans has doubled due to a drought in Brazil. Economic theory might suggest you should immediately raise your prices. But will you?
Probably not.
You'd have to print new menus. Update the digital signs. Retrain your staff. Deal with confused and possibly angry customers. Maybe wait to see if your competitors raise their prices first. All of these invisible frictions add up to something economists call "nominal rigidity" — the stubborn tendency of prices to stay put even when the underlying economic conditions have shifted.
This seemingly mundane observation about sticky prices turns out to be one of the most important ideas in modern macroeconomics. And in 1983, an Argentine economist named Guillermo Calvo proposed an elegant mathematical way to think about it that has since become the standard approach in economic models around the world.
The Calvo Lottery
Calvo's insight was beautifully simple. Instead of trying to model all the complex reasons why a particular firm might or might not change its price — the menu printing, the competitor watching, the customer psychology — he proposed treating price changes like a lottery.
In each period (say, each month or each quarter), every firm in the economy gets to spin a wheel. With some fixed probability, call it "h," the firm gets the chance to reset its price to whatever it wants. With probability one minus h, the firm is stuck with its old price for another period. The key insight is that this probability stays constant regardless of how long it's been since the firm last changed its price.
This might seem almost too simple. In the real world, surely a price that has been stuck for two years is more likely to change than one set last week? We'll return to this criticism later. But Calvo's genius was recognizing that this simplification — what mathematicians call a "constant hazard rate" — made the problem tractable while still capturing the essential economic intuition.
Living with Uncertainty
What makes Calvo contracts psychologically interesting is the uncertainty they create. When a firm sets a price, it doesn't know how long that price will last. It might get to change it next month. Or it might be stuck with it for years.
The mathematics here follow what statisticians call a geometric distribution. If your probability of changing price in any given period is twenty-five percent, then on average your price will last four periods. But — and this is the counterintuitive part — no matter how long your current price has already been in place, the expected additional duration is still four more periods.
This is exactly like flipping a biased coin. If you've flipped tails ten times in a row, the coin doesn't remember. Your chance of heads on the next flip is exactly what it always was. Economists sometimes call this the "memoryless property."
When Shocks Hit the Economy
The real power of the Calvo model becomes apparent when we think about how the economy responds to surprises. Suppose the central bank unexpectedly raises interest rates, or oil prices suddenly spike, or a pandemic shuts down half the world's supply chains. In classical economic models, all prices would instantly adjust to the new reality. But that's not what happens in the real world, and the Calvo model explains why.
When a shock hits, only a fraction of firms — those lucky enough to win the price-change lottery — can respond immediately. Everyone else is stuck with prices set before the shock even happened.
The next period, more firms get their chance to adjust. But there are still plenty of firms operating with outdated prices. The period after that, the same story. The adjustment spreads through the economy like ripples in a pond, gradually diminishing but never quite disappearing.
If twenty-five percent of firms can change prices each period, then after one period, seventy-five percent of prices are still at their pre-shock levels. After two periods, about fifty-six percent haven't adjusted. After four periods, about thirty-two percent. The proportion declines, but only gradually.
Contrast with Taylor Contracts
There's an older approach, proposed by the economist John Taylor, that handles price stickiness differently. In Taylor contracts, every firm is locked into its price for a fixed period — say, exactly four quarters — after which it must reset. It's like having an annual price review that happens on a predictable schedule.
The difference matters. Under Taylor contracts, after exactly four periods, literally every price in the economy has been reset. The pre-shock prices are completely gone. Under Calvo contracts, there's always some fraction of prices that were set arbitrarily long ago. The old prices fade away but never quite vanish.
Which is more realistic? Different industries probably behave differently. Some prices (like gasoline) change constantly. Others (like magazine subscriptions or gym memberships) tend to be reviewed on fixed schedules. The Calvo approach has become more popular in academic models partly because it's mathematically more convenient, but both capture important aspects of how prices actually behave.
The Phillips Curve Gets a Makeover
One of the most important applications of the Calvo model came in 1995, when economist John Roberts used it to derive what's now called the "New Keynesian Phillips Curve." To understand why this matters, we need a brief detour into the history of macroeconomic thought.
The original Phillips Curve, discovered by economist A.W. Phillips in 1958, documented an apparent trade-off between inflation and unemployment in British data. When unemployment was low, inflation tended to be high, and vice versa. For a while, policymakers thought they could exploit this trade-off — accepting a bit more inflation in exchange for lower unemployment.
This thinking fell apart in the 1970s when many countries experienced "stagflation" — high inflation and high unemployment simultaneously. Economists realized they had been missing something crucial: expectations. If people expect prices to rise, they'll demand higher wages, which will push prices up further, creating a self-fulfilling prophecy.
The New Keynesian Phillips Curve incorporates these expectations explicitly, and the Calvo model provides the microeconomic foundation. The basic equation says that current inflation depends on two things: expected future inflation and current economic activity (usually measured by output or unemployment).
Looking Forward, Not Backward
Here's what makes the New Keynesian Phillips Curve special: it's fundamentally forward-looking. When firms get the chance to set their prices, they don't just think about conditions today. They think about what conditions might be like in all the future periods during which their price might be stuck.
If a firm expects high inflation next year, it will set a higher price today, even if current conditions don't warrant it. After all, it might be stuck with this price for a while. This forward-looking behavior is what makes monetary policy work in these models — by managing inflation expectations, central banks can influence current behavior.
Think about what this means for a coffee shop owner trying to set prices during uncertain times. They're not just thinking about today's coffee bean costs. They're thinking about where costs might be six months from now, when they might finally get another chance to update their menu.
Measuring Stickiness
How rigid are prices really? Economists have tried to measure this in several ways, and the results are surprisingly complex.
One approach is to measure the average age of current prices. Go around to every firm in the economy and ask: how long has your current price been in effect? Average those numbers up.
With Calvo pricing, if twenty-five percent of firms change prices each period, the average age of prices in the economy will be about four periods. Most prices are relatively fresh, but there's a long tail of older prices pulling the average up.
But there's a subtle problem with this measure called "interruption bias." When we observe prices at any given moment, we're catching them in the middle of their lifespan. We can see how old they are, but we don't know how much longer they'll last.
Completed Length Versus Current Age
A different measure asks: when a price finally does change, how long will it have lasted in total? This "completed length" turns out to be longer than the average age — in fact, quite a bit longer.
The mathematics work out so that the average completed length equals twice the average age minus one. If the average age of prices is four periods, the average completed length is seven periods. This seems paradoxical at first, but it reflects the fact that longer-lasting prices are more likely to be observed at any given time, just as you're more likely to encounter a slow-moving car on the highway than a fast one.
These measurement issues matter enormously for policy. If a central bank is trying to estimate how quickly its interest rate changes will feed through into prices, it needs to know just how sticky prices really are. Getting this wrong can mean the difference between a well-timed intervention and a policy disaster.
The Trouble with Calvo
For all its elegance, the basic Calvo model has a significant problem: it doesn't fit the data very well.
When economists look at actual inflation data, they find that past inflation predicts current inflation better than the pure forward-looking New Keynesian Phillips Curve would suggest. Inflation seems to have momentum — it's harder to stop than the basic model implies.
This has led researchers to develop what's called the "hybrid" New Keynesian Phillips Curve, which includes both forward-looking expectations and backward-looking momentum. Current inflation depends not just on expected future inflation and current economic activity, but also on what inflation was last period.
Indexation to the Rescue
One way to generate this backward-looking behavior within a Calvo framework is called indexation. The idea is that even when firms can't fully reset their prices, they can make automatic adjustments based on recent inflation.
Think of it like a rental contract with a clause saying "rent increases by the rate of inflation each year." The landlord doesn't have to renegotiate — the adjustment happens automatically. With indexation, the Calvo probability now refers to the chance of fully resetting the price versus having it adjust mechanically based on past inflation.
This modification has become popular among New Keynesian researchers because it preserves the mathematical tractability of the Calvo framework while generating inflation dynamics that look more like real-world data.
When Age Matters
A more fundamental modification challenges the constant hazard rate itself. In 1999, economist Alexander Wolman proposed allowing the probability of price change to depend on how old the current price is.
This makes intuitive sense. A price set yesterday is probably still appropriate for current conditions. A price set three years ago is increasingly likely to be out of date. The pressure to change should build over time.
Alternatively, some prices might actually become "stickier" with age. If a firm has successfully maintained a price point for years, customers come to expect it, and changing becomes increasingly costly.
The generalized Calvo model with duration-dependent hazard rates can capture both possibilities. Mathematically it's more complex, but it opens up a much richer set of dynamics that can better match how different industries actually behave.
Why Any of This Matters
You might reasonably ask: why spend so much intellectual effort on the mundane fact that prices don't change instantly?
The answer is that this mundane fact has profound implications for economic policy. In a world where prices adjusted instantly to any change in economic conditions, monetary policy would be essentially powerless. The central bank could print more money, but prices would immediately rise to compensate, leaving real economic activity unchanged.
Price stickiness is what gives central banks their leverage over the real economy. When the Federal Reserve raises interest rates, it takes time for all those sticky prices to adjust. In the meantime, the higher rates affect real borrowing costs, real spending decisions, and real employment.
The specific form of price stickiness matters too. How long do prices stay stuck? Do they eventually catch up all at once, or gradually? Do firms look forward or backward when setting prices? These questions determine how quickly and effectively monetary policy transmits through the economy.
The Ongoing Debate
Four decades after Calvo's original paper, economists are still arguing about the best way to model price dynamics. The basic Calvo contract remains the workhorse assumption in most New Keynesian models, valued for its mathematical elegance. But researchers continue to propose modifications, extensions, and alternatives.
Some have explored state-dependent pricing, where the probability of changing price depends on economic conditions rather than just time. Others have built models with menu costs — explicit fixed costs of changing prices — and solved for when firms optimally choose to bear those costs.
The quest continues for a model that is both tractable enough to use in large-scale policy analysis and realistic enough to capture the complex ways that prices actually evolve in modern economies.
From Abstract to Everyday
The next time you notice that your favorite restaurant hasn't changed its prices despite rising food costs, or that your gym membership stays fixed year after year, you're observing the phenomenon that Guillermo Calvo formalized forty years ago.
These small frictions — the reluctance to print new menus, the fear of alienating customers, the desire to wait and see what competitors do — aggregate up into something that shapes the behavior of entire economies. They determine how quickly shocks propagate, how effective monetary policy can be, and why inflation, once started, can be so difficult to stop.
What began as a clever mathematical trick to make economic models tractable has become central to how we understand the macroeconomy. The Calvo contract sits quietly at the heart of the models that central bankers use to make decisions affecting billions of lives. Not bad for an idea inspired by the simple observation that changing prices is harder than it looks.