Unemployment is not a still pond. Each month workers lose jobs and others find them, and the rate in the news is just where those two flows balance. Search and matching models make that balance explicit, and the picture they draw — the Beveridge curve — most often brings intermediate students to an economics tutor in Montreal.
1 · Unemployment is a flow, not a stock
The unemployment rate u is a stock: the share of the labour force without work right now. But that stock is filled and drained without pause. The separation rate s is the fraction of employed workers who lose their job each period; the job-finding rate f is the fraction of the unemployed who find one.
In steady state the stock is constant, so inflow equals outflow: s(1−u) = f u. Solve for the rate:
u = s/(s+f).
Unemployment is high because jobs end fast (large s) or work is hard to find (small f) — a matter of flows, with no wage in sight.
2 · The matching function and market tightness
Where does f come from? A hire needs two sides to meet — an unemployed worker searching and a firm with a vacancy. Write the new hires per period as a matching function m(U, V).
A standard form is m = A·√(U V): Cobb–Douglas with constant returns, where A is matching efficiency. Divide by U and the job-finding rate drops out:
f = m/U = A·√(V/U) = A·√θ,
where θ = V/U is market tightness, the vacancies per unemployed worker. A tighter market raises f, so jobs are easier to find, but it lowers the rate at which each vacancy is filled — a congestion no firm internalises.
3 · The Beveridge curve
Now combine the two ideas. In steady state u = s/(s+f), and f rises with tightness θ = v/u (writing v for the vacancy rate). More vacancies raise f, which lowers u. So across steady states, vacancies and unemployment move in opposite directions. Plot v against u and you trace a downward-sloping curve — the Beveridge curve.
Eliminating θ from the steady-state condition gives its equation: v = s²(1−u)²/(A²u). Two features earn marks. It slopes down — raise u and v falls — and it is convex to the origin, steep at low unemployment and flat at high, because frictions bite hardest when the market is tight. Every point on the curve is a labour market at rest, each at a different tightness.
4 · Movements along the curve versus shifts of it
This is the distinction examiners test hardest: a point can move along a fixed Beveridge curve, or the whole curve can shift.
Along the curve — cyclical. In a downturn firms post fewer vacancies. Tightness θ falls, f falls, and the economy slides down and to the right: more unemployment, fewer vacancies. A boom reverses it. The matching process is unchanged; only the position on the curve moves.
A shift of the curve — structural. If matching itself worsens — the jobless are in the wrong places or hold the wrong skills for the jobs on offer, so efficiency A falls (or separations s rise) — then at every vacancy rate unemployment is higher, and the whole curve shifts outward.
That is the curve’s diagnostic power. High unemployment beside low vacancies looks cyclical, a demand problem. High unemployment beside high vacancies, out on a shifted curve, looks structural — a matching problem stimulus alone will not cure. Tightness itself is set by firms’ vacancy posting, which a fuller model pins down; here it is the lever that slides you along the curve.
Worked example — a mid-sized national labour market
Step 1 — Set up the matching function. Take m = A·√(U V) with A = 0.36 and tightness θ = V/U = 0.25 — one vacancy per four searchers. Then f = A·√θ = 0.36×√0.25 = 0.36×0.5 = 0.18.
Step 2 — Find steady-state unemployment. Each month 2% of the employed lose their job, so s = 0.02. Balancing flows, u = s/(s+f) = 0.02/(0.02+0.18) = 10%.
Step 3 — Locate the point. The vacancy rate is v = θ u = 0.25×0.10 = 2.5%. Call this point B = (10%, 2.5%) — it sits on the Beveridge curve.
Step 4 — A recession hits. Demand falls, vacancies are cut, tightness and job-finding drop: f falls from 0.18 to 0.08. The new steady state is u = 0.02/(0.02+0.08) = 20%, the vacancy rate about 1%. The economy slides down and right along the same curve, from B to C = (20%, 1%) — a movement along.
Step 5 — Contrast a structural shock. Now instead let mismatch worsen, cutting efficiency from A = 0.36 to A = 0.24. The curve shifts out. At the same 2.5% vacancy rate, unemployment climbs from 10% to about 18.5% — point D. Same vacancies, far more joblessness.
Step 6 — Interpret. The two shocks demand different cures. The recession (B → C) is a movement along the curve — a demand shortfall demand policy can lift. The mismatch (B → D) is a shift — a structural fault stimulus will not repair. Same headline unemployment; opposite diagnosis.
Given a jump in unemployment, could you tell a movement along the Beveridge curve from a shift of it — and defend the diagnosis? That distinction, cyclical against structural, is the one examiners test hardest in search and matching. A one-on-one economics tutor works the flow steady state and the matching function with you until u = s/(s+f) is something you rebuild, not remember. Book a trial session.
Practice
Q1. A market has matching function m = A·√(U V) with A = 0.5, tightness θ = 0.09 and separation rate s = 0.05. Find the job-finding rate f and the steady-state unemployment rate u.
Q2. A boom then raises tightness to θ = 0.16, with A and s unchanged. Find the new f and u — a movement along the Beveridge curve, or a shift of it?
Q3. Instead, from θ = 0.09 and s = 0.05, mismatch cuts efficiency to A = 0.25. Find the new f and u — movement, or shift?
Answers. Q1: f = 0.5×√0.09 = 0.5×0.3 = 0.15; u = 0.05/(0.05+0.15) = 25%. Q2: f = 0.5×√0.16 = 0.5×0.4 = 0.20; u = 0.05/(0.05+0.20) = 20%. Only tightness changed — a movement along the curve, up and to the left as unemployment falls and vacancies rise. Q3: f = 0.25×√0.09 = 0.25×0.3 = 0.075; u = 0.05/(0.05+0.075) = 40%. Efficiency A fell, so the curve shifts out: at the same tightness, unemployment is higher.
Key takeaways
- Unemployment is a flow steady state: u = s/(s+f) — high when jobs end fast or are hard to find.
- The job-finding rate comes from a matching function: f = A·√θ. Tightness θ = V/U is the hinge linking vacancies to unemployment.
- The Beveridge curve slopes down and is convex. Every point is a labour market at rest, each at a different tightness.
- Cyclical shocks move you along the curve; structural shocks shift it out. Telling them apart is a diagnosis, not a technicality.
Why Montreal students choose our economics tutoring
- Models built, not memorised: sessions derive the flow steady state and the Beveridge curve from the matching function, so you can reconstruct u = s/(s+f) under exam pressure instead of quoting it.
- The distinction examiners reward, drilled: movement along the curve versus a shift of it — the line that separates a first from a 2:1.
- One-on-one and matched to your course: a tutor works from your own notation and past papers, whether you are preparing for McGill problem sets or another university’s module.
FAQ
Q: What is the Beveridge curve in plain terms?
A: The downward-sloping relationship between the vacancy rate and the unemployment rate. When many jobs sit unfilled while few people are jobless, the economy sits high on it; in a slump it slides low and to the right.
Q: What is the difference between the separation rate and the job-finding rate?
A: The separation rate s is the share of employed workers who lose their job each period; the job-finding rate f is the share of the unemployed who find one. Together they fix u = s/(s+f).
Q: Why does the Beveridge curve slope downward?
A: More vacancies raise tightness, which makes jobs easier to find, which lowers unemployment. So vacancies and unemployment move opposite ways — a negative slope.
Q: How do I tell cyclical unemployment from structural unemployment?
A: A cyclical rise moves along a fixed curve — high unemployment with low vacancies. A structural rise shifts the curve outward — high unemployment alongside high vacancies, because matching has broken down.
Q: Is this the same as the natural rate of unemployment?
A: The flow steady state u = s/(s+f) is one way to define an equilibrium unemployment rate. The Phillips-curve approach reaches the natural rate through inflation expectations — the same idea, a different route.
Book an economics tutor in Montreal or online
Search and matching rewards students who can derive the Beveridge curve, not just draw it — the flow steady state, the matching function, and the movement-versus-shift distinction. One-on-one sessions build that fluency on your own past papers. Tell us your course and exam date, and we will match you with the right tutor this week.
