Skip to content
We use only essential cookies to keep you signed in and remember your preferences. See our Cookie Policy.
TestStop
MocksPracticeResourcesIndiaBlogPricing
Sign in
Resources › Inflation and the Phillips curve › 12. The Delhi School set
Inflation and the Phillips curve · Lesson 12 of 12

The hard problems: the Delhi School set

The Delhi School entrance asks about wage-setting and price-setting in letters. The wage rule is given as an equation, the mark-up is a symbol, and the question is the Phillips relation it implies, the natural rate it gives, or how that moves when productivity or the mark-up does. Four techniques cover all seventeen questions, and the Natural-Rate Workbench is preloaded with the set-up of each.

← Inflation and the Phillips curve moduleLesson 12 of 12 · about 60 minutes
1Measuring inflation2Measuring unemployment3Causes4Costs5The original curve6Okun's law7Expectations8The natural rate9Stagflation10Disinflation11In the data12Delhi School
A question to hold on to
The Phillips curve relation for an economy is πt − πet = 0.1 − 3ut, where πet = πt−1. Suppose inflation in year (t − 1) is 2 percent. In year t the central bank decides to keep the unemployment rate at 3 percent forever. Find the rate of inflation for years t, t + 1, t + 2 and t + 3.
Delhi School of Economics entrance, 2021Solved at the end ↓
In this guide
Lesson · about 60 minutes
  1. How the Delhi School asks
  2. Which technique? A drill
  3. Part 1: building the supply relation
  4. Part 2: natural-rate numbers and comparative statics
  5. The set-ups in the Workbench
  6. Part 3: the NAIRU and the accelerationist path
  7. Part 4: definitions that need care
  8. The opening question, solved
  9. The lesson on one screen
  10. The Delhi School questions
01
The approach

How the Delhi School asks

Four habits separate these questions from the ones in earlier lessons.

  1. Read how productivity enters the wage rule. The sign of its effect on the natural rate depends on it (Lesson 8): none when the claim scales with A, a rise when the claim rises but price-setting does not, a fall when the claim rises less than one for one.
  2. Keep inflation as a ratio. The set-ups give prices, so write 1 + π = P/P₋₁ before you simplify, and expected inflation as Pᵉ/P₋₁ − 1. The relation is 1 + π = (1 + πᵉ) times the wage-setting factor.
  3. The natural rate is where π = πᵉ. Put the wage-setting factor equal to 1 and solve for u. Whatever else changes, that equation is the whole of it.
  4. Units. The relation in 2021 gives u as a fraction (0.03), not 3. Neat numbers are a hint that you have the set-up right.
02
Practice

Which technique? A drill

Before solving a problem, name the technique. Seven problems from the set, each asking only for the method. The four options are:

  1. Write the price level from the wage rule and the mark-up, then divide by last year's price
  2. Set π equal to πᵉ (the wage-setting factor equal to 1) and solve for u
  3. See how the natural rate moves when one parameter moves
  4. Iterate the relation year by year
Pause and predict
W = Pᵉ F(u, A) and P = (1 + μ)W. Which equation links inflation to expected inflation?
Pause and predict
A 20% mark-up and a wage claim W = P(1 − u). What is the natural rate?
Pause and predict
Does a rise in productivity raise or lower the natural rate in this set-up?
Pause and predict
π(t) − π(t−1) = 0.1 − 3u(t) and unemployment is held at 3%. What is inflation in the next four years?
Pause and predict
The union sets W = Pᵉ(Z − αu), firms mark up by m and each worker makes one unit. What is the aggregate supply curve?
Pause and predict
F = A/u − 1 with static expectations. What is the NAIRU?
Pause and predict
Productivity grows at rate γ in the general form. What must happen to the NAIRU?
03
Part 1

Part 1: building the supply relation

The general set-up
1+πt​=(1+πte​)(1+μ)F(ut​,At​)
Wage-setting W = Pᵉ F(u, A) with F falling in u and rising in productivity A, and price-setting P = (1 + μ)W. Divide by last year's price P₋₁.
DSE 2012 Q51 · hard
Wt = PetF(ut, At), Pt = (1 + μ)Wt. Which equation relates expected to actual inflation?
  1. 1
    Build the price.
    Pt = (1 + μ)PetF(ut, At).
  2. 2
    Divide by last year's price.
    Pt/Pt−1 = 1 + πt and Pet/Pt−1 = 1 + πet, so 1 + πt = (1 + πet)(1 + μ)F.
Answer
(B): πt = (1 + πet)(1 + μ)F(ut, At) − 1.
DSE 2012 Q52 · hard
With productivity constant at Ā, which equation defines the natural rate?
  1. 1
    The natural rate is where π = πᵉ.
    Then 1 + π = 1 + πᵉ, so (1 + μ)F(u, Ā) = 1.
  2. 2
    Solve for F.
    F(u, Ā) = 1/(1 + μ).
Answer
(B): F(u, Ā) = 1/(1 + μ).
DSE 2018 Q37 · hard
The union sets W = Pᵉ(Z − αu), the producer sets P = (1 + m)W, and each employed worker produces one unit of output Y, with labour force L. Find the aggregate supply function.
  1. 1
    Employment is output.
    Y = L(1 − u), so u = 1 − Y/L.
  2. 2
    Substitute.
    W = Pᵉ(Z − α + αY/L).
  3. 3
    Mark up.
    P = Pᵉ(1 + m)(Z − α + αY/L).
Answer
(D): P = Pᵉ(1 + m)(Z − α + αY/L). Option A and C have a minus sign in front and B has Z − 1 where it should have Z − α.
04
Part 2

Part 2: natural-rate numbers and comparative statics

Put the wage-setting factor equal to 1, solve for u, and then ask how the answer moves.

DSE 2020 Q06 · standard
Firms charge a 25% mark-up over unit wage cost and workers demand W/P = 1 − u. What is the natural rate?
  1. 1
    Price-setting.
    W/P = 1/1.25 = 0.8.
  2. 2
    Meet wage-setting.
    1 − u = 0.8, so u = 20%.
Answer
(A) 20%.
DSE 2019 Q33 · standard
The mark-up is 20% and wage-setting is W = P(1 − u). What is the natural rate?
  1. 1
    Price-setting.
    W/P = 1/1.2 = 0.8333.
  2. 2
    Solve.
    1 − u = 0.8333, so u = 1/6 = 16.7%, closest to 17%.
Answer
(D) 17%.
DSE 2019 Q24 · standard
Price-setting gives W/P = A/(1 + m) and wage-setting gives W/P = A(1 − u). What is the effect of an increase in technology A on unemployment?
  1. 1
    Equate.
    A(1 − u) = A/(1 + m); A cancels, so 1 − u = 1/(1 + m) whatever A is.
Answer
(B) zero.
DSE 2020 Q08 · hard
Y = AL, firms mark up the effective unit wage cost W/A by μ, and workers’ expected real wage is W/Pᵉ = Aα(1 − u)1−α, 0 < α < 1. If expected and actual prices are equal, what does a rise in A do to unemployment?
  1. 1
    Price-setting.
    P = (1 + μ)W/A, so W/P = A/(1 + μ).
  2. 2
    Equate with wage-setting.
    A/(1 + μ) = Aα(1 − u)1−α, so (1 − u)1−α = A1−α/(1 + μ).
  3. 3
    Solve.
    1 − u = A/(1 + μ)1/(1−α). It rises with A, so u falls.
Answer
(B) negative. Only part of a productivity gain reaches the wage claim, so firms can afford more than workers claim.
DSE 2012 Q53 · hard
A one-shot increase in the average productivity of workers, ceteris paribus, leads to?
  1. 1
    The natural rate solves F(u, A) = 1/(1 + μ).
    F rises with A at every u.
  2. 2
    To restore F = 1/(1 + μ).
    F falls with u, so u must rise.
Answer
(A) an increase in the natural rate of unemployment. This differs from 2019 Q24 and 2020 Q08 only in how A enters the claim.
DSE 2012 Q54 · hard
A one-shot increase in the producer’s mark-up, ceteris paribus, leads to?
  1. 1
    The target moves.
    F(u, A) = 1/(1 + μ) falls when μ rises.
  2. 2
    Lower F needs higher u.
    F falls with u, so u rises.
Answer
(A) an increase in the natural rate.
05
Interactive

The set-ups in the Workbench

Each button loads the set-up of one of the questions above. Press 2019 Q24 and raise A, then 2012 Q51 to Q56 and raise A, then 2020 Q08 and raise A, and watch the natural rate stay, rise and fall. The same tool stands alone at the Workbench page.

Pick the wage-setting rule, then move the mark-up, productivity and the workers’ claim. Price-setting (red) is the real wage firms allow, wage-setting (blue) is the real wage workers claim at each unemployment rate, and they meet at the natural rate (gold). The second plot holds unemployment at a target you choose, measured from the natural rate, and follows inflation for ten years.

W/P = A(1 + z − u). The claim scales with productivity, as does the real wage firms allow, so productivity cancels.

0.10.20.30.40.50.60.511.5unemployment rate, ureal wage, W/P
Wage-settingPrice-settingNatural rate
24681051015202530yearinflation, %
Inflation2% anchor
The natural rate

20%. uₙ = z + m/(1 + m) = 0 + 0.25/1.25 = 20%. The real wage there is 0.8.

Holding unemployment at 18%

That is 2 points below the natural rate. After ten years inflation is 30.57%. With θ = 1 it climbs every year and does not settle.

The Phillips relation this rule implies

1 + π = (1 + πᵉ) × (1 + m)(1 + z − u) = (1 + πᵉ) × 1.25 × (1 − u). Near the natural rate, π ≈ πᵉ − β(u − uₙ) with β = 1.25 (inflation lost per point of unemployment above uₙ).

What moves the natural rate
ChangeNew uₙNatural rate
Productivity A up 10%20%no change
Mark-up m up 5 points23.1%rises by 3.08 points
Workers' claim z up 0.0525%rises by 5 points
Checks, recomputed live

At uₙ with πᵉ = 2%, π = 2% ✓. At uₙ with πᵉ = 10%, π = 10% ✓. The wage-setting factor at uₙ is 1, so uₙ does not depend on expected inflation.

06
Part 3

Part 3: the NAIRU and the accelerationist path

DSE 2007 Q06 · standard
NAIRU implies?
  1. 1
    What the name says.
    The non-accelerating inflation rate of unemployment is the rate at which inflation does not accelerate: it is constant. It is not zero unemployment and it does not fix the growth of GDP.
Answer
(B) inflation is constant.
DSE 2012 Q55 · hard
Expectations are static, πet = πt−1, F(u, A) = A/u − 1 and A is constant at Ā. What is the NAIRU?
  1. 1
    Stable inflation.
    πt = πt−1 = πet, so (1 + μ)F = 1.
  2. 2
    Substitute F.
    (1 + μ)(Ā/u − 1) = 1, so Ā/u = 1 + 1/(1 + μ) = (2 + μ)/(1 + μ).
  3. 3
    Solve.
    u = Ā(1 + μ)/(2 + μ).
Answer
(C) u = ((1 + μ)/(2 + μ))Ā.
DSE 2012 Q56 · hard
Productivity now grows at a constant rate γ. To maintain non-accelerating inflation, the rate of unemployment has to increase at the rate?
  1. 1
    The NAIRU is proportional to A.
    u = A(1 + μ)/(2 + μ), and the factor (1 + μ)/(2 + μ) is constant.
  2. 2
    So it grows with A.
    If A grows at γ, so does u.
Answer
(A) γ.
DSE 2021 Q36 · hard (the opening question)
πt − πet = 0.1 − 3ut, πet = πt−1, πt−1 = 2%, u held at 3%. Find π for t to t + 3.
  1. 1
    The gap each year.
    π − πᵉ = 0.1 − 3 × 0.03 = 0.01, one percentage point.
  2. 2
    Iterate.
    πt = 2 + 1 = 3, πt+1 = 4, πt+2 = 5, πt+3 = 6.
  3. 3
    Why it climbs.
    The natural rate here is 0.1/3 = 3.33%, so 3% is below it, and with expectations equal to last year’s inflation, inflation rises by the gap every year (Lesson 7).
Answer
(C) 3%, 4%, 5% and 6%.
07
Part 4

Part 4: definitions that need care

DSE 2011 Q51 · standard
Pt = (1 + a)WtNt/Yt: a fixed mark-up over unit labour cost. With π inflation, w wage inflation and λ productivity growth (productivity = Y/N), which is true?
  1. 1
    Rewrite with productivity.
    P = (1 + a)W/(Y/N): the price is the mark-up times the wage over productivity.
  2. 2
    Take growth rates.
    The mark-up is constant, so π = w − λ (exactly, 1 + π = (1 + w)/(1 + λ)).
Answer
(C) π = w − λ (Lesson 5).
DSE 2019 Q07 · standard
The efficiency wage theory argues that?
  1. 1
    The idea.
    Firms pay more than the market-clearing wage so that workers work harder or stay (Lesson 8). A wage above the clearing level is a higher real wage than the one at which the market clears.
Answer
(B) firms choose to pay a higher wage than the classical equilibrium wage, so the real wage is higher than the market-clearing one.
DSE 2006 Q44 · hard
A negative supply shock (for example an oil price rise) shifts the Phillips curve and ____ the natural rate of unemployment. If the government wants to keep the economy at the original rate of unemployment, it must have ____ inflation.
  1. 1
    The key's reading.
    The shock lifts the curve, π = πᵉ − β(u − uₙ) + ε, and the natural rate is the structural rate where π = πᵉ without the shock, so it does not change.
  2. 2
    Hold unemployment there.
    At u = uₙ the equation gives π = πᵉ + ε. If a continuing shock is accommodated and πᵉ is last year’s inflation, inflation rises by ε every year: ever increasing (Lesson 9).
Answer
(D) does not change, ever increasing.
A convention to know

Lesson 9 also described an oil price rise as a rise in the mark-up, which raises the natural rate (option C’s first blank). The key to this question takes the first reading: the shock shifts the curve and the natural rate is unchanged. Which reading a question uses is usually clear from whether it says the natural rate changes.

DSE 2010 Q03 · hard
If an economy is experiencing hyperinflation, then?
  1. 1
    Seigniorage as inflation rises.
    Revenue is the inflation tax rate times real money balances. In Lesson 4 money demand falls as inflation rises, and revenue peaks at an inflation rate of 1/α and then falls.
  2. 2
    Hyperinflation is past the peak.
    The fall in real balances outweighs the higher tax rate, so seigniorage goes down.
Answer
(B) government seigniorage goes down. The key reads hyperinflation as inflation beyond the revenue-maximising rate.
08
Solving the opening question

The opening question, solved

It was solved in Part 3 (DSE 2021 Q36). The relation gives a gap of one point a year at 3% unemployment, because the natural rate is 3.33%, so inflation is 3%, 4%, 5% and 6%. The other options start from the wrong place or step by the wrong amount: A lowers inflation in year t, B leaves year t at 2%, and D goes from 2% to 4%. Year t already has inflation of 3%, because expected inflation is the 2% of the year before, and each later year adds exactly one point.

09
Revise

The lesson on one screen

The lesson on one screen
General set-up
W = Pᵉ F(u, A), P = (1 + μ)W gives 1 + π = (1 + πᵉ)(1 + μ)F.
Natural rate
π = πᵉ: (1 + μ)F = 1, so F = 1/(1 + μ).
Linear forms
W/P = 1 − u with a mark-up m: uₙ = m/(1 + m). A cancels from W/P = A(1 − u).
Productivity
None, a rise or a fall: read how A enters the claim.
NAIRU, F = A/u − 1
u = A(1 + μ)/(2 + μ); it grows at the rate of A.
Union rule
P = Pᵉ(1 + m)(Z − α + αY/L).
Accelerationist
π − πᵉ = 0.1 − 3u with πᵉ = π₋₁: a point a year at 3%.
10
Real questions

The Delhi School questions

Part 1 · new in this lesson

Building the supply relation from wage-setting and price-setting

Write the price level from a wage rule and a mark-up, find the Phillips relation it implies, the natural rate from it, and the aggregate supply curve of a union economy.

3 questionsPractise →
Part 2 · from Lesson 8

Natural-rate numbers and comparative statics

A mark-up and a wage rule give a number; then what a rise in productivity or in the mark-up does to it, in three different wage-setting forms.

6 questionsPractise →
Part 3 · from Lesson 7

The NAIRU, productivity growth and the accelerationist path

Inflation that does not accelerate, how the NAIRU must move when productivity grows, and a year-by-year path when unemployment is held below the natural rate.

4 questionsPractise →
Part 4 · from Lessons 3 and 4 and 8 and 9

Definitions and set-ups that need care

A mark-up over unit labour cost in growth rates, efficiency wages, a supply shock and the natural rate, and seigniorage in a hyperinflation.

4 questionsPractise →
All 17 DSE questions, moderate before hard within each part: practise the full set →

✦These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.

← Lesson 11The Phillips curve in the data: the US, and India

Read next

All resources →
MacroeconomicsNatural-Rate Workbench: wage-setting, price-setting, NAIRU and the Phillips relation in one screenPick one of four wage-setting rules and see the natural rate with its working, the Phillips relation it implies, what moves it and the path of inflation when unemployment is held away from it, with the identities checked live.Interactive diagram · 15 minMacroeconomicsThe natural rate of unemployment: wage-setting, price-setting, efficiency wages and hysteresisPrice-setting and the mark-up, wage-setting and bargaining, why they meet at one unemployment rate, what moves it, the three ways productivity can enter, efficiency wages, insiders and outsiders, hysteresis, and the step to the Phillips curve. With the Natural-Rate Workbench.Complete guide · 55 minMacroeconomicsIS–LM 10: the Delhi School problemsThe DSE's harder questions on the module: IS–LM in letters, supply from the labour market and open-economy models; the wage-setting and price-setting ones are in the inflation module.Complete guide · 50 min