Inflation
How prices rise in the model, and why they rise faster or more slowly from one country to another. This page covers equations E3, E4 and E5.
In plain words
People expect a certain level of inflation, and their expectation blends the central bank's target with the inflation they have lived through in recent years (E3). Today's inflation moves towards this expectation, and rises a little more when the economy is running above its capacity (E4). The published figure, for its part, compares today's prices with those of a year ago (E5).
In the game
Every month, the dashboard publishes inflation over twelve months. No order sets it directly, but the player brings it down by raising the policy rate, which slows activity (E1), and pushes it up with a more generous budget. In an advanced country, the effect is slow, since in France two extra points on the rate for five years lower inflation by only 0.13 points (§7.3). The studio knows that this effect is too weak. In these countries, the rate acts on prices through the value of the currency as much as through demand, and since this exchange-rate channel is missing from the game, the effect of the rate on prices there is half as strong as the bottom of the range found by studies.
The basics
Inflation is the general rise in prices. With 2% inflation a year, a basket that costs 100 euros today costs 102 a year later. Most central banks aim for low and stable inflation, often around 2% in advanced countries.
Part of inflation comes from what people expect. If everyone expects prices to rise by 5%, wages and prices are negotiated on that basis, and the rise really does happen. Trust in the central bank therefore matters a great deal, because if people believe it will hold to its target, the target is what they expect, and inflation stays calm.
The other part comes from tension in the economy. Prices rise when people want to buy more than the country can produce, and more slowly when activity is weak.
Finally, the figure you see in the newspapers compares prices with those of twelve months earlier, which makes it react with a delay. When inflation drops suddenly, it takes about a year to follow.
E3. Inflation expectations
In plain words
The inflation people expect is an average of the central bank's target and the inflation they have lived through, and the more credible the central bank, the more this average leans towards the target. The memory of past inflation fades slowly and loses half its weight in 474 days.
Take an advanced country whose target is 2% and where people have seen 6% inflation. If the central bank is fully credible, 77% of the expectation rests on the target and 23% on the memory, which gives 0.77 × 2 + 0.23 × 6 = 2.92%. If it has lost half its credibility, only 38.5% rests on the target and the expectation rises to 4.46%. With no credibility at all, people simply expect the 6% they have lived through.
In the formula:
| Symbol | What it is | In the example |
|---|---|---|
| \(\pi^e\) | the inflation people expect | 2.92% |
| \(\theta\) | the share placed on the target when the central bank is fully credible | 0.77 |
| \(\psi\) | the central bank's credibility, from 0 to 1 (E17) | 1 |
| \(\bar\pi\) | the central bank's target | 2% |
| \(\pi^a\) | the memory of past inflation | 6% |
| \(\pi^{vu}\) | the inflation people see every day, price shocks included | |
| \(s_a\) | the share of the gap that the memory catches up each day |
Show the full formula and its sourcesThe exact form at the daily step, every parameter with its source, and what the equation leaves out.
ψ is the central bank's credibility, between 0 and 1 (E17); πvu is the inflation people see, price shocks included (E13).
| Parameter | Value | Status | Source |
|---|---|---|---|
expectations_anchoring θ | 0.77 / 0.57 / 0.32 | the anchoring of a fully credible central bank. Advanced: Ball and Mazumder 2019, table 4, United States after 1998 (0.773; before 1998: 0.067); emerging: IMF, WEO of October 2018, annex 3.2, with the constraint a + b = 1 (0.566). This is the weight of 3-year expectations in the Phillips curve, taken as the anchored share on the assumption that these expectations sit on the target; low-income: no published estimate, midpoint of the sourced range 0.07 to 0.57 | |
inflation_memory_persistence | 0.875 per quarter | Ball and Mazumder 2019: decay of the exponential average of past inflation (half-life: 474 days) |
Why. The more credible a central bank is, the more expectations stay settled on its target, and the less an inflation shock feeds on itself. This is what changed in the United States at the end of the 1990s (θ from 0.07 to 0.77), and what separates advanced and emerging countries today (World Bank 2019; Kose et al. 2019). Anchoring is not fixed. Credibility modulates it, as the stock of credibility modulates the weight of target-oriented expectations in the IMF's semi-structural models (E17). What it simplifies: the memory retains the inflation people see, price shocks included, with the same decay whatever the credibility; in Carvalho et al. (2023), the gain of the memory itself changes with anchoring.
E4. The Phillips curve
In plain words
Today's inflation moves towards what people expect, and rises a little higher when the economy is running above its potential. Each point of output gap then adds 0.09 points of inflation in an advanced country, and 0.16 points in an emerging country. This catching up takes time, and inflation covers half the distance in 30 days.
When inflation is already high, shopkeepers change their prices more often. A price stays in place for 4.5 months on average at zero inflation, but changes almost every month above 100% a year, and the same overheating then passes through to prices far more strongly. The studio has made this effect depend on how often prices change because, with a fixed effect, a country starting from 220% inflation would take a century to bring it back down.
Back to the previous example, with expectations at 2.92% and an economy running 2% above its potential. The push is 0.09 × 2 = 0.18 points, and inflation heads towards 3.10%. In an emerging country where inflation is around 50%, the curve is 4.25 times steeper, so the same overheating pushes inflation up there by 1.36 points instead of 0.32.
In the formula:
| Symbol | What it is | In the example |
|---|---|---|
| \(\pi^{*}\) | the inflation that today's inflation is heading towards | 3.10% |
| \(\pi\) | today's inflation | |
| \(\pi^e\) | expected inflation (E3) | 2.92% |
| \(\kappa\) | the slope, that is, what one point of output gap adds to inflation | 0.09 |
| \(\tilde y\) | the output gap, as a % of potential (E1) | +2 |
| \(R\) | how many times steeper the curve is than at low inflation, 1 below 14% | 1 |
| \(\lambda\) | how many times a month a price changes | |
| \(s_\pi\) | the share of the gap closed each day |
Show the full formula and its sourcesThe exact form at the daily step, every parameter with its source, and what the equation leaves out.
λ is the monthly frequency of price changes at trend inflation πa (the memory of E3); R says how much steeper the curve is than at low inflation, where κ was estimated: 1 below 14%, 4.3 at 50%, 10.3 at 100% and beyond. The inflation memory (E3) follows at the same pace, and its quarterly persistence becomes 0.875λ(π)·4.5.
| Parameter | Value | Status | Source |
|---|---|---|---|
phillips_slope κ | 0.09 / 0.16 / 0.16 points of annual inflation per point of output gap | advanced: BIS, working paper 748 (Jašová, Moessner and Takáts 2018), 0.022 per quarter non-annualised after 2008, i.e. 0.09 annualised; cross-checked by Ball and Mazumder 2019 (0.236 per point of unemployment × Okun 0.40 = 0.094). Emerging: 0.040 × 4 = 0.16 (BIS 748), and 0.159 (IMF, WEO 2018). Low-income: no estimate, value of the emerging countries | |
inflation_half_life_days | 30 days | the relation, quarterly in the studies, plays out within its quarter: a half-life of 30 days closes 88% of the gap in 91 days | |
price_duration_months | 4.5 months | Alvarez, Beraja, Gonzalez-Rozada and Neumeyer 2019: duration of a price at zero inflation, in aggregate (Argentina, 1988-1997) | |
price_frequency_threshold | 14% per year | same source: below this threshold, the frequency of price changes barely moves; 12.5% in Mexico (Gagnon 2009) | |
price_frequency_elasticity | 0.53 | same source, table I: 0.53 for the simple pooled estimator, 0.48 to 0.68 depending on the estimator, 2/3 in theory | |
price_frequency_cap | 100% per year | beyond this, the slope stops rising: prices change almost every month, and a linear relation between the output gap and inflation no longer makes sense (at 500%, the Calvo slope would be 170 times that of low inflation, and the slightest gap would sweep everything away) | |
calvo_discount | 0.99 per quarter | the usual discount factor of Calvo-pricing models, which translate a frequency into a slope |
Why. An economy running above its potential pushes prices up. The slope is small today in advanced countries (Blanchard 2016: about 0.2 per point of unemployment since the late 1980s, against 0.7 in the mid-1970s; Hazell et al. 2022 find it smaller still).
Steeper at high inflation. When inflation is high, prices change more often. They change once every 4.5 months at zero inflation and almost every month above 100% (Alvarez et al. 2019). As a result, the same demand passes more into prices than into output. Ball, Mankiw and Romer (1988) find this across 43 countries, where the share of a demand shock that goes to output falls from 0.62 at zero inflation to 0.14 at 10%. The slope estimated at low inflation therefore does not hold for Argentina. Translating the frequency into a slope through the Calvo model is a choice. It is a cautious one, since the data come from a world with menu costs; and with a frequency that rises with inflation, the slope rises too (Ascari and Sbordone 2014). Without this, a country starting from 220% inflation would take a century to disinflate.
What it simplifies: no import prices, no oil, no exchange rate. For low-income countries, where three quarters of the variance of inflation come from global shocks (World Bank 2019), this is the heaviest simplification of the core. And even when steeper, the curve is not enough to end a high inflation within a few years. High inflations end with a change of regime that re-anchors expectations at once (Sargent 1982; Fischer, Sahay and Végh 2002), which the model does not have (§8.2).
E5. The price level and published inflation
In plain words
Every day, the price level rises by one day's worth of inflation, so that after a year of days, it has risen by exactly the annual inflation. Temporary price shocks, on food or energy, are added on top (E13). The published figure compares this level with that of a year earlier. It is the figure the player sees, and it is the one that counts for popularity and credibility.
Today's inflation held at 3% all year gives a published figure of 3%. If it falls sharply from 6% to 3%, the published figure does not follow it straight away and falls little by little over a year, as the months at 6% drop out of the comparison.
In the formula:
| Symbol | What it is | |
|---|---|---|
| \(\tilde P\) | the price level driven by today's inflation | |
| \(x\) | the temporary gap that price shocks add to it, in % | |
| \(P\) | the published price level | |
| \(\pi^{12}\) | published inflation, over twelve months |
Show the full formula and its sourcesThe exact form at the daily step and what the equation leaves out.
P̃ is the price level driven by the Phillips curve, x the temporary deviation that price shocks add to it (E13), and P the published level. π12 is the inflation that statistical institutes publish, over a rolling twelve months. It is the inflation the player sees, and the one that enters popularity and credibility. What it simplifies: a single price, no index per product.