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The expectations-augmented Phillips curve
Does the inflation-unemployment trade-off really exist? Short run versus long run.
π = πᵉ − α·(u − u*) + εProblem / motivation
In 1958, A.W. Phillips published a startling empirical relation: in the United Kingdom, over a century, the lower unemployment was, the faster wages rose. Politicians saw a menu in it: trade a little inflation for a little employment.
Draw the axes once and for all: unemployment u on the horizontal axis, inflation π on the vertical one — everything that follows happens in this plane. The 1958 curve SLOPES DOWN in it: on the left, low unemployment and high inflation; on the right, high unemployment and low inflation. Transposed from wages to prices (Samuelson and Solow, 1960), it shaped a decade of economic policy: pick your point on the curve.
Then came the 1970s and the impossible: high inflation AND high unemployment AT THE SAME TIME — stagflation, a point in the top right of the plane, off the curve. Friedman (1968) and Phelps had announced it: the curve forgets inflation EXPECTATIONS — what everyone expects prices to do, and writes into their contracts. Once πᵉ is included, the trade-off survives only in the short run; in the long run, all that remains is a VERTICAL line planted at u*.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
Your worksheet is still empty. Go for it: propose at least one idea.
0 idea(s) proposedFormalization
Why would low unemployment push prices up? Through the price-wage loop of the course “The causes of inflation”: when hands are scarce, employers outbid each other to hire and to keep people — wages accelerate, costs follow, and so do prices. The further u is BELOW u*, the faster the loop turns: there is the term −α·(u − u*), and α measures the strength of that loop (in points of inflation per point of unemployment gap).
Then the starting point: the increases already WRITTEN into contracts. If everyone expects πᵉ = 2%, wages and prices carry 2% before the labour market even plays its part: πᵉ is the base of the equation. Hence: π = πᵉ − α·(u − u*). It reads like a sentence — inflation is what was expected, PLUS the overheating of the labour market (or MINUS its slack) — and, like any equation, it turns over: fix π and deduce the u needed, which will be exercise 4.
The film's engine is still missing: how does πᵉ move? Simplest version: everyone expects tomorrow the inflation observed today — πᵉ(tomorrow) = π(today). (Textbooks call these ADAPTIVE expectations — broadly, the adjustment may be only partial; our full catch-up in one year is the textbook case.) It is this little law, never far away, that will turn any lasting surprise into acceleration (the solving step unrolls it).
Finally, real life joins in: a supply shock (oil, harvests, freight) pushes prices without going through unemployment. We add it at the end of the formula — ε, zero in calm times: π = πᵉ − α·(u − u*) + ε. The complete formula of the header. Click each term:
Solving / calculation
The film, frame by frame. Starting point: an economy “anchored” at 2% — expectations sit firmly on the target (πᵉ = 2%), u* = 5%, α = 0.5, no shock (ε = 0). The central bank — the institution that issues the currency and watches over prices, the ECB in the euro area — then launches a stimulus: policy rate cut, easy credit, demand boosted, hiring (course “The policy rate and transmission”) — u drops below u*.
- 1. At rest (u = u*): inflation reproduces expectations exactly
π = 2 − 0.5 × (5 − 5)π = 2% (= πᵉ) - 2. Stimulus: u falls to 3%; contracts, signed earlier, still expect 2%
π = 2 − 0.5 × (3 − 5)π = 3% (surprise: +1 pt) - 3. Adaptive expectations: πᵉ ← 3% (the law from the formalisation). To HOLD u = 3%, the new expectations must be beaten
π = 3 − 0.5 × (3 − 5)π = 4% (acceleration required) - 4. The central bank gives up (accelerating for ever, nobody wants that): the stimulus stops, the surprise dies out, u climbs back to u* — but expectations stay perched up high
π = 4 − 0.5 × (5 − 5)π = 4% stable, u back at 5%
The film's verdict: the employment gain was TEMPORARY, and the only lasting legacy is inflation settled two points higher. On the graph: we SLID along the short-run curve (frame 2), then JUMPED curve when πᵉ was revised (frame 3), then climbed back onto the vertical at u* — higher than at the start. In the long run, all possible points stack up on that VERTICAL line.
π = πᵉ − α · (u − u*) + εSet expectations, the loop (α), unemployment and the shock: π computes itself — and the “next year” line applies the revision law (πᵉ ← π) to show you where the film takes you if everything is held in place.
Economic interpretation
The same equation describes two worlds depending on the horizon — and moving from one to the other has a price.
As long as πᵉ is frozen, pushing u below u* raises π: the central bank can “buy” employment at the price of an inflation surprise that agents will correct. And the trade-off works both ways: to DISINFLATE — bring π back down — u must be pushed above u*; economists call that cost in output and jobs the “sacrifice ratio”. Volcker's American disinflation (1979-1982) paid for it with a severe recession.
Once πᵉ = π, the term α·(u − u*) vanishes: on the graph, all that is left is the VERTICAL line planted at u* — every level of inflation is possible on it, none of them buys employment durably. Any sustained inflation ends up absorbed by expectations.
The whole film rests on ADAPTIVE expectations (πᵉ follows π with a lag — hence the possible surprise). If agents are RATIONAL — they use all available information, including the central bank's announcements — the surprise disappears the moment it is announced (Lucas critique, limit 1). Hence central bankers' obsession: ANCHORING πᵉ on the target — “anchored” expectations no longer move, even when π stirs; “unanchored”, they turn any shock into a spiral.
Central banks still steer with it: they watch the gap u − u* and expectations, and set their policy rate accordingly (the course “The Taylor rule” formalises that reflex). With one caution: u* cannot be observed (limit 3) — steering by the NAIRU means navigating by a star you only see in hindsight.
Limits / critiques
Exercises
πᵉ = 3%, α = 0.5, u = 4%, u* = 6%. What inflation π (in %)?
Holding unemployment 2 points BELOW u* year after year. What becomes of inflation?
An economy at rest: u = u* = 5%, πᵉ = 2%. An oil shock strikes: ε = +3 points. What inflation π (in %)?
The central bank estimates u* at 5% — but the TRUE u* is 7%. It steers the economy at u = 5%, believing itself “at rest”. What does inflation do?
True or false: in calm times (no supply shock, ε = 0), when u = u*, actual inflation equals expected inflation (π = πᵉ).
πᵉ = 6%, u* = 5%, α = 0.5. The central bank wants to bring inflation down to 4% this very year. To what unemployment rate u must it push the economy (in %)?
A PERFECTLY CREDIBLE central bank announces its disinflation, and rational agents immediately adjust πᵉ onto the new target. What does the disinflation cost in unemployment?
What concrete mechanism links very low unemployment to rising inflation?