Chapter 85: Search, Matching, Unemployment, and Wage Dynamics. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Sorting pays only with complementarity

When does pairing the best technician with the best plant raise total output?

Sorting gains only when skill and quality are complements (increasing differences). With an additive technology the pairing is irrelevant, and with a negative interaction crossing wins.

Equation, written in LaTeX: F(s_H,q_H)+F(s_L,q_L)>F(s_H,q_L)+F(s_L,q_H).

Equation, written in LaTeX: 4(3)+1(2)=14,

Equation, written in LaTeX: 4(2)+1(3)=11.

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s is technician skill (s_H set here, s_L = 1), q plant quality (q_H = 3, q_L = 2) and F(s, q) match output. Sorted pairs high with high; crossed pairs high with low.

Predict first. Under F = s + q, does sorting change total output?

Your prediction Yes, it raises output No, the two assignments tie

Technology F(s,q)

High technician skill

Figure: Sorting pays only with complementarity. Left: two technicians and two plants linked by sorted and crossed pairings. Right: sorted output 14.0 and crossed output 11.0 under F = sq. — Technology F(s,q): sq, High technician skill: 4
Constructed example: the chapter's hypothetical values are s_H = 4, s_L = 1, q_H = 3, q_L = 2 and the three technologies; s_H = 2 and 3 are added.

Sorted output: 14.0
Crossed output: 11.0
Sorting gain: 3.0

Hypothetical market from the chapter, technology F = sq. Sorted output is 4(3) + 1(2) = 14.0 and crossed output 4(2) + 1(3) = 11.0, so sorting adds 3.0 units.

Worked steps

Sorted: 4(3) + 1(2) = 14.0
Crossed: 4(2) + 1(3) = 11.0
Gain = 14.0 - 11.0 = 3.0: sorting adds 3.0 units

Use the idea

Before reading a wage and firm-quality correlation as productive sorting, ask whether the inputs are complements in output.

Where the conclusion applies

Two types on each side, known types and flexible transfers that support the efficient assignment.

Check your understanding: With F = sq and s_H = 2, what is the sorting gain?

Sorted 2(3) + 1(2) = 8, crossed 2(2) + 1(3) = 7, a gain of 1.

Chapter 85 source: section "Assortative matching".

Demonstration 2 of 4

Matching function, finding rates, steady unemployment

How do matching efficiency and vacancies set job-finding, vacancy-filling and steady unemployment?

More vacancies per searcher help workers find jobs but congest the firm side. Higher matching efficiency raises both rates and lowers steady unemployment.

$Equation, written in LaTeX: M(u,v)=A u^\alpha v^{1-\alpha},$

$Equation, written in LaTeX: u^*=\frac{s}{s+f(\theta)}.$

$Equation, written in LaTeX: 0.6\sqrt{800(450)}=360.$

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M is monthly matches, A matching efficiency, U = 800 unemployed and V vacancies; f = M/U, q = M/V, theta = V/U, and s = 0.03 the separation rate.

Predict first. Raising vacancies from 450 to 800: does each vacancy fill faster or slower?

Your prediction Faster Slower

Matching efficiency A

Vacancies V

Figure: Matching function, finding rates, steady unemployment. Left: job-finding rate 0.450 and vacancy-filling rate 0.800. Right: steady unemployment 6.25% against the book case of 6.25 percent. — Matching efficiency A: 0.6, Vacancies V: 450
Constructed example: the chapter's hypothetical values are U = 800, V = 450 and 800, A = 0.6 and 0.4 and s = 0.03; A = 0.7 and V = 600 are added.

Matches M: 360.0
Job-finding rate f: 0.450
Vacancy-filling rate q: 0.800
Tightness theta: 0.5625
Steady unemployment u*: 6.25%

Hypothetical labor force of 10,000 with 800 unemployed. Efficiency 0.6 and 450 vacancies give 360.0 matches, so f = 360.0 / 800 = 0.450 and q = 0.800. With separations at 0.03, steady unemployment is 0.03 / (0.03 + 0.450) = 6.25%.

Worked steps

M = 0.6 sqrt(800 x 450) = 0.6 x 600.00 = 360.0
f = 360.0 / 800 = 0.450; q = 360.0 / 450 = 0.800
theta = 450 / 800 = 0.5625
u* = 0.03 / (0.03 + 0.450) = 0.0625, or 6.25%

Use the idea

Separate a fall in job finding caused by fewer vacancies from one caused by worse matching efficiency; they call for different responses.

Where the conclusion applies

Constant-returns matching with alpha = 0.5, stocks held at the comparison date, and a steady state that is an accounting benchmark rather than a forecast.

Check your understanding: With A = 0.6 and V = 600, what are f and u*?

M = 0.6 sqrt(480,000) = 415.7, f = 0.520 and u* = 0.03 / 0.550 = 5.46%.

Chapter 85 source: section "Search-and-matching unemployment".

Demonstration 3 of 4

Moving along versus shifting the Beveridge curve

How many vacancies sustain steady flows at each unemployment level, and what does worse matching do?

Fewer unemployed need more vacancies to generate the same hiring flow, so the curve slopes down. Lower matching efficiency raises the vacancies needed at every U: the curve shifts out.

$Equation, written in LaTeX: \Delta U=s(L-U)-M.$

$Equation, written in LaTeX: V=\frac{(184/0.8)^2}{400}=132.25.$

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L = 5,000 is the labor force, U unemployment, V vacancies, s = 0.04 the separation rate and M = A sqrt(UV) monthly hires. Steady flows need s(L - U) = M.

Predict first. Halving matching efficiency at U = 400: does required V double or quadruple?

Your prediction Double Quadruple

Unemployed U

Matching efficiency A

Figure: Moving along versus shifting the Beveridge curve. Two downward-sloping curves of required vacancies against unemployment, for A = 0.8 and A = 0.4, with a point at U = 500 and V = 101.25. — Unemployed U: 500, Matching efficiency A: 0.8
Constructed example: the chapter's hypothetical values are L = 5,000, s = 0.04, A = 0.8 and 0.4 and U = 500 and 400; U = 600 is added.

Separations s(L - U): 180
Required vacancies V: 101.25
Hires at V = 101.25: 180.0

Hypothetical labor force of 5,000 with separation rate 0.04. At U = 500, separations are 0.04 x 4,500 = 180, so steady flows need V = (180 / 0.8)^2 / 500 = 101.25 vacancies. Changing U moves along a curve; changing A shifts it.

Worked steps

Separations = 0.04 x (5,000 - 500) = 180
V = (180 / 0.8)^2 / 500 = 225.0^2 / 500 = 101.25
Check: 0.8 sqrt(500 x 101.25) = 180.0 hires

Use the idea

When vacancies and unemployment rise together, suspect a shift in matching efficiency rather than a demand movement along the curve.

Where the conclusion applies

Square-root matching and a fixed separation rate; the curve shows steady flows, not adjustment.

Check your understanding: At U = 600 and A = 0.8, what vacancy stock sustains steady flows?

Separations 0.04 x 4,400 = 176; V = (176 / 0.8)^2 / 600 = 220^2 / 600 = 80.67.

Chapter 85 source: section "Beveridge curve".

Demonstration 4 of 4

Monopsony, wage floors, and mobility

How does a firm facing upward-sloping labor supply set pay, and what does a wage floor do?

Hiring one more worker raises pay for everyone, so the monopsonist stops short and pays below MRP. A floor between the monopsony wage and MRP removes that motive and raises both pay and jobs. A flatter supply curve (more mobility) also raises employment.

Equation, written in LaTeX: ME(L)=w(L)+Lw'(L).

Equation, written in LaTeX: MRP_L=ME(L).

Equation, written in LaTeX: w=8+0.5L,

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w = 8 + bL is the inverse labor supply facing the firm, ME = 8 + 2bL marginal expenditure and MRP = 20 each worker's marginal revenue product. A floor fixes the wage the firm must pay.

Predict first. Does a floor of 16 raise or reduce employment relative to monopsony (slope 0.5)?

Your prediction Raises it Reduces it

Labor-supply slope b

Wage floor

Employment L: 12
Wage w: 14
Markdown MRP - w: 6
Competitive employment: 24

Hypothetical workshop from the chapter with supply w = 8 + 0.50L and MRP 20. Setting ME = 8 + 1.0L equal to 20 gives L = 12 / 1.0 = 12 and w = 8 + 0.50 x 12 = 14. The competitive benchmark is L = 24.

Worked steps

ME = 8 + 1.0L = 20, so L = 12 / 1.0 = 12
w = 8 + 0.50 x 12 = 14
Markdown = 20 - 14 = 6; competitive L = 12 / 0.50 = 24

Use the idea

Judge a wage floor against the employer's markdown and the competitive benchmark, not as always good or always bad for jobs.

Where the conclusion applies

Linear supply, constant MRP and a single common wage; a floor above 20 would end profitable hiring.

Check your understanding: With slope 0.5 and a floor of 18, what are L and w?

At 18, (18 - 8) / 0.5 = 20 workers accept; MRP 20 > 18, so all 20 are hired at w = 18.

Chapter 85 source: section "Monopsony and Labor-Market Power".

Sorting pays only with complementarity

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Matching function, finding rates, steady unemployment

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Moving along versus shifting the Beveridge curve

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Monopsony, wage floors, and mobility

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies