01 · The case

Do we have discriminatory pay?

Group A earns £22 per hour and Group B earns £16. They work in comparable jobs and have the same educational qualifications.

What we know: A earns £6 per hour more than B.
What we need: an explanation for the difference, rather than a verdict based on the gap alone.

Make a first assessment

Does this £6 gap demonstrate discriminatory pay?

Choose an answer. We will test an experience-based explanation and then examine what remains. Nothing is recorded.

02 · The first hypothesis

Could four years explain everything?

Suppose experience is the only difference that matters. To explain £6 with four extra years, experience would need to earn £1.50 per year.

This is the wage schedule required for experience alone to close the gap, not a schedule measured from data.

Try the experience explanation.

B's experience8 years

A, 12 years£22

B, hypothetical pay£16

Increasing experience closes the gap if both groups earn £1.50 per extra year. We have not yet tested that.

03 · Check the evidence

What do workers actually earn?

We need more than the two group averages. Compare estimated wages at two experience levels within each group, holding education and job type fixed.

These are illustrative predictions from separate fitted pay schedules for comparable, equally educated workers. Actual estimation would require many worker observations and appropriate controls.

Do both groups earn the same return?

The earlier £1.50 return was a hypothesis. We must check each group's pay schedule separately.

04 · Test the experience explanation

How much experience would B really need?

Using B's measured £0.50 return, give B the four years needed to match A's experience. Then keep going until B reaches A's £22 wage.

B starts at 8 years and £16. Matching A's 12 years raises B to only £18.

Move B along its own wage schedule.

B's years of experience8

A · 12 years£22

B · current wage£16

First try 12 years. Does experience alone close the wage gap?

05 · Change one thing at a time

What if B received A's pay rule?

B still has eight years of experience and the same qualifications. We change only the estimated pay schedule from B's to A's.

B's fitted pay rule£12 + £0.50 × years

At 8 years: £12 + £4 = £16

A's fitted pay rule£13 + £0.75 × years

At 8 years: click to reveal the prediction.

This is a counterfactual: B's characteristics stay unchanged while we apply A's fitted pay structure. The difference is £3 at B's eight years; it was £4 at 12 years because the fitted returns differ. Neither difference proves unfair treatment.

Hold experience at eight years.

B under B's pay rule£16

B under A's pay rule—

Same worker characteristics. Different estimated pay rule. What changes?

06 · Build the decomposition

Where does the £6 gap come from?

Start at B's £16. First apply A's pay schedule at B's eight years. Then increase experience to A's 12 years using A's pay rule.

B: 8 years, B pay rule£16

→

B: 8 years, A pay rule?

→

A: 12 years, A pay rule£22

Pay-schedule component—

Characteristics component—

Keep one thing fixed at each step. First switch the wage rule at eight years. Then change experience using A's reference wage rule.

Reveal each step of the bridge.

B earns £16. What if B's eight years received the pay rule estimated for A?

07 · Now formalise the result

The equation adds up each difference.

We found £6 = £3 + £3. The symbol k simply means: examine each characteristic, then add its contribution.

Observed gap = Characteristics + Pay-schedule difference£6 = £3 + £3

Δw = ∑_k(X̄_A,k − X̄_B,k) · β̂_A,k+∑_kX̄_B,k · (β̂_A,k − β̂_B,k)

Explained: characteristicsUnexplained: pay schedules

Read it left to right: for each k, multiply the relevant difference by its estimated coefficient β̂. Then ∑_k means add all those contributions. No transpose is needed.

Click a component to highlight its sum.

Choose either component. Its £3 and corresponding ∑_k expression will light up, with the actual arithmetic.

How to read the notation

ΔwObserved average wage gap: £22 − £16

A, BThe two groups; subscripts tell you which group

kOne characteristic at a time, such as education or experience; also base pay

X̄_A,kA's average amount of characteristic k

β̂_A,kA's estimated pay coefficient for k

∑_kAdd the contributions from all characteristics

Our worked case

Education: identical and held constant

Experience: A = 12 years; B = 8 years

Base pay + experience: A = £13 + £0.75/year; B = £12 + £0.50/year

Education is absorbed into base pay here; in a full model its measured level and estimated return can be separate k terms.

08 · Return to the question

Do we have discriminatory pay?

We have split the observed £6 gap into two statistical components. But which parts, if any, represent unfair treatment?

Characteristics£3

A has four extra years of experience, valued using A's fitted return.

Pay-schedule difference£3

At the same observed characteristics, fitted wages still differ.

The crucial distinction: 'Explained' does not mean fair. 'Unexplained' does not mean discrimination. The decomposition identifies where to investigate, not who is guilty.

Two claims. Do they follow?

1 · Because £3 is explained by experience, that part of the gap must be fair.

2 · The remaining £3 proves discriminatory pay.

Answer both claims, then reveal why we must be careful about what is fair and what counts as discrimination.