Chapter 62: Structural Transformation, Dual Economies, and Migration. 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

Inequality rises, then falls

How can moving workers into better jobs first widen and then narrow income dispersion?

With incomes equal inside each sector, all dispersion comes from the gap between sectors. It is largest when the workforce is split evenly and vanishes when everyone is in one sector, so pure composition change traces an inverted U.

$Equation, written in LaTeX: \bar y_1=\frac{2(8)+8(2)}{10}=3.2.$

$Equation, written in LaTeX: V_1=\frac{2(8-3.2)^2+8(2-3.2)^2}{10} =\frac{57.6}{10} =5.76.$

$Equation, written in LaTeX: V_2=\frac{9(8-7.4)^2+(2-7.4)^2}{10} =\frac{32.4}{10} =3.24.$

Scroll sideways for the whole equation

Ten workers earn 2 in the traditional sector or the modern income in the modern sector. The mean income is y bar and V is the population variance of income across the ten workers.

Predict first. As more of the ten workers move into modern jobs, does the variance keep rising?

Your prediction Yes, it keeps rising No, it peaks and then falls

Workers in the modern sector

Modern-sector income

Figure: Inequality rises, then falls. Variance of income against the number of modern workers when modern income is 8 and traditional income is 2. It rises from 0 to a peak of 9.00 at 5 workers and falls back to 0 at 10. The current point, 2 modern workers, has variance 5.76. — Workers in the modern sector: 2, Modern-sector income: 8
Constructed example: the chapter's invented ten-worker economy (incomes 2 and 8; 0, 2, 9 and 10 modern workers); 5 modern workers and modern incomes 6 and 10 are added for comparison.

Modern workers: 2 of 10
Mean income: 3.20
Variance: 5.76
Peak variance for this modern income: 9.00

Mean income = (2 x 8 + 8 x 2) / 10 = 3.20. Variance = (2 x (8 - 3.20)^2 + 8 x (2 - 3.20)^2) / 10 = 57.60 / 10 = 5.76. Inequality is still rising: each extra modern worker widens the spread.

Worked steps

Mean = (2 x 8 + 8 x 2) / 10 = 32 / 10 = 3.20
Modern gap: 8 - 3.20 = 4.80; squared 23.0400
Traditional gap: 2 - 3.20 = -1.20; squared 1.4400
Sum = 2 x 23.0400 + 8 x 1.4400 = 57.60
Variance = 57.60 / 10 = 5.76

Use the idea

Before reading a rise in inequality as a stage on a universal curve, split it into the gap between sectors and the spread inside each sector.

Where the conclusion applies

Equal incomes within each sector and population variance as the measure. Dispersion inside the modern sector, an excluded group or another measure can give a different path.

Check your understanding: With 5 of 10 workers earning 8 and the rest 2, what is the variance?

Mean = (5 x 8 + 5 x 2) / 10 = 5; variance = (5 x 9 + 5 x 9) / 10 = 9, the peak for this case.

Chapter 62 source: section "Kuznets curve".

Demonstration 2 of 4

Where released farm workers go

How much of a productivity gain comes from better farming, and how much from where the freed workers go?

The within-sector term weighs each sector's productivity gain by its starting share; the reallocation term values the change in shares at the new productivities. Releasing farm labor and using it productively are separate achievements.

Equation, written in LaTeX: Y_0=70(4)+30(10)=580,

$Equation, written in LaTeX: Y_1=60(\frac{14}{3})+40(10)=680,$

$Equation, written in LaTeX: 0.70(\frac{14}{3}-4)+0.30(10-10) =\frac{7}{15} \approx0.467.$

Equation, written in LaTeX: Y_1'=280+300+30=610,

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Aggregate productivity P is output per worker across 100 workers, P = sum of s_i p_i over sectors with employment share s_i and output per worker p_i. Farm output stays at 280; manufacturing produces 10 per worker and informal services 3.

Predict first. If the 10 released workers go to informal services instead, does aggregate productivity still rise?

Your prediction Yes, but by less No, it falls

Destination of released workers

Workers released from farming

Figure: Where released farm workers go. Bars of the productivity change split into a within-sector term of 0.467, a reallocation term of 0.533 and a total of 1.000 when 10 released workers go to manufacturing at 10. — Destination of released workers: Manufacturing (10), Workers released from farming: 10
Constructed example: the chapter's invented 100-worker economy (70 farm workers at 4, 30 manufacturing at 10, 10 released, informal services at 3); 5 and 20 released workers are added, with farm output held at 280.

P0 before: 5.8
P1 after: 6.80
Within-sector term: 0.467
Reallocation term: 0.533
Total change: 1.000
Farm output per worker after: 4.6667

10 farm workers are released and go to manufacturing at 10. Y1 = 60 x 4.6667 + 40 x 10 = 280 + 400 = 680, so aggregate productivity moves from 5.8 to 680 / 100 = 6.80. The change splits as 0.467 + 0.533 = 1.000: within sectors plus reallocation. Moving the released workers into higher-productivity jobs adds a positive reallocation term.

Worked steps

Farm output per worker = 280 / 60 = 4.6667
Y1 = 60 x 4.6667 + 40 x 10 = 280 + 400 = 680, so P1 = 680 / 100 = 6.80
Within = 0.70 x (4.6667 - 4) + 0.30 x (10 - 10) = 0.467
Reallocation = 4.6667 x (0.60 - 0.70) + 10 x (0.40 - 0.30) = 0.533
Check: 0.467 + 0.533 = 1.000 = 6.80 - 5.8

Use the idea

When a country's output per worker rises, split the gain into within-sector growth and reallocation before crediting either policy.

Where the conclusion applies

Constant output per worker in manufacturing and informal services however many workers join, and farm output held at 280. Other decompositions place the interaction term differently.

Check your understanding: With 10 released workers going to informal services, what is aggregate productivity?

(280 + 300 + 30) / 100 = 6.1.

Chapter 62 source: section "Structural transformation".

Demonstration 3 of 4

Why concerts get dearer

Why does a service with unchanged productivity grow more expensive relative to goods?

Both sectors pay the same wage, so the service's relative cost equals the ratio of productivities. When the workshop's productivity rises and the concert's does not, the concert gets relatively dearer even though nothing about it changed.

$Equation, written in LaTeX: c_{P,0}=\frac{20}{2}=10.$

$Equation, written in LaTeX: c_{S,0}=8(20)=160.$

$Equation, written in LaTeX: c_{P,1}=\frac{30}{4}=7.50$

$Equation, written in LaTeX: c_{S,1}=8(30)=240.$

$Equation, written in LaTeX: \frac{240}{7.50}=32.$

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c_P is the labor cost of one device (wage divided by devices per hour) and c_S the labor cost of one concert (8 musician-hours times the wage). Before the change the wage is 20 and the workshop makes 2 devices per hour; the venue has 120 seats.

Predict first. The concert still uses the same 8 hours. Does its cost relative to a device change?

Your prediction Yes, it doubles No, it stays at 16

Workshop devices per hour after

Common wage after

Figure: Why concerts get dearer. Two bars of the concert's cost measured in devices: 16 before and 32 after, with the workshop at 4 devices per hour and a common wage of 30. — Workshop devices per hour after: 4, Common wage after: 30
Constructed example: the chapter's invented workshop and ensemble (wage 20 then 30, 2 then 4 devices per hour, 8 hours, 120 seats); 3 devices per hour and a wage of 25 are added.

Device cost after: 7.50
Device cost change: -25%
Concert cost after: 240.00
Concert cost change: 50%
Relative cost: 16 to 32
Labor cost per ticket: 1.33 to 2.00

The device's labor cost moves from 10.00 to 30 / 4 = 7.50 (-25%), and the concert's from 160 to 8 x 30 = 240 (50%). The concert still uses 8 hours, yet its cost in devices rises from 16 to 32 because the workshop now makes 4 devices per hour. The labor cost per ticket goes from 1.33 to 2.00.

Worked steps

Before: device = 20 / 2 = 10.00; concert = 8 x 20 = 160; relative = 160 / 10 = 16
After: device = 30 / 4 = 7.50
After: concert = 8 x 30 = 240
Relative = 240 / 7.50 = 32
Per ticket = 240 / 120 = 2.00, against 160 / 120 = 1.33

Use the idea

When a labor-intensive service gets dearer, compare its productivity growth with the economy's before blaming waste.

Where the conclusion applies

One common wage across sectors, labor as the only cost and fixed hours per concert. Recording, subsidy or changing the service alter the outcome.

Check your understanding: What is the labor cost per ticket after the change?

240 / 120 = 2.00, up from 160 / 120 = 1.33.

Chapter 62 source: section "Baumol cost disease".

Demonstration 4 of 4

Creating jobs that raise unemployment

Can adding formal city jobs leave more people unemployed in the city?

Each new job raises the chance of employment, which draws migrants until the expected city wage falls back to the rural wage. Each job attracts w_u / w_r members of the labor force, so when the city wage is above the rural wage unemployment grows with employment.

Equation, written in LaTeX: p_0w_u=0.5(200)=100=w_r.

Equation, written in LaTeX: 0.6(200)=120>100.

$Equation, written in LaTeX: 100=\frac{120}{L_u}(200), L_u=240.$

Equation, written in LaTeX: U_u=240-120=120.

Scroll sideways for the whole equation

w_r = 100 is the rural wage, w_u the formal city wage, J the number of formal jobs and L_u the urban labor force. Job seekers without a formal job earn zero and moving is free, so migration stops when (J / L_u) w_u = w_r. The baseline has 100 jobs.

Predict first. The city adds 20 formal jobs (100 to 120) at wage 200. Does urban unemployment fall?

Your prediction Yes, it falls by 20 No, it rises by 20

Formal city jobs

Formal city wage

Figure: Creating jobs that raise unemployment. Stacked bars of the urban labor force: baseline 100 employed and 100 unemployed; after the program 120 employed and 120 unemployed at a city wage of 200. — Formal city jobs: 120, Formal city wage: 200
Constructed example: the chapter's invented city (rural wage 100, city wage 200, 100 then 120 jobs); 140 jobs and city wages of 150 and 250 are added for comparison.

Urban labor force: 240
Employed: 120
Unemployed: 120
Migrants: 40
Change in unemployment: +20

Migration stops when the expected city wage equals the rural wage: L_u = 120 x 200 / 100 = 240, of whom 120 are employed and 120 unemployed. 20 new jobs draw 40 migrants, so unemployment rises by 20 even as employment rises by 20.

Worked steps

Baseline: L_u = 100 x 200 / 100 = 200, unemployed 100
On impact: p = 120 / 200 = 0.6000; expected wage 0.6000 x 200 = 120.00 against 100
New equilibrium: L_u = 120 x 200 / 100 = 240
Unemployed = 240 - 120 = 120
Migrants = 240 - 200 = 40

Use the idea

Judge a city job program by employment, unemployment and migrant welfare together, not by the unemployment count alone.

Where the conclusion applies

A fixed formal wage, zero income while searching, costless moves and open competition for jobs. Informal earnings, moving costs or jobs reserved for residents weaken the paradox.

Check your understanding: With 140 jobs at wage 200, what is equilibrium unemployment?

L_u = 140 x 200 / 100 = 280; unemployment = 280 - 140 = 140.

Chapter 62 source: section "Harris-Todaro migration paradox".

Inequality rises, then falls

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Where released farm workers go

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Why concerts get dearer

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Creating jobs that raise unemployment

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies