The Encyclopedia of Economic Principals

Chapter 80

Cities, Agglomeration, Spatial Equilibrium, and Regional Multipliers

Follow a local shock into wages, rents, gaps and jobs.

Four of the chapter's worked examples, made interactive: wages and rents after an amenity or productivity shock, a regional gap that shrinks or compounds, the local jobs that follow an exporter, and a public works multiplier with leakages. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

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

Rosen-Roback wages and rents

How do a park and a productivity gain split between wages and rents?

Both shocks make the place more attractive, so rent rises either way. An amenity lets employers pay less because workers accept lower wages to live there; productivity makes firms compete for workers and wages rise. The sign of the wage change identifies the shock.

Equation, written in LaTeX: \Delta V=\Delta w-0.5\Delta r+10\Delta a,

Equation, written in LaTeX: \Delta\Pi=8\Delta A-\Delta w-0.25\Delta r.

Equation, written in LaTeX: -0.25\Delta r-0.5\Delta r+10=0.

Equation, written in LaTeX: 8-0.5\Delta r-0.25\Delta r=0,

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dw and dr are changes in the annual wage and rent, in thousands of dollars. a is a local amenity and A local productivity. Mobile households must stay indifferent (dV = 0) and firms must keep zero profit (dPi = 0).

Predict first. Does a productivity shock raise or lower wages?

Your prediction

Choose an example

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Figure: Rosen-Roback wages and rents. Wage change against rent change: the household and firm lines shift and cross at dr = 13.333 and dw = -3.333 thousand dollars; dashed lines show the starting position at the origin.
Shock: Amenity (park)
Constructed example: the chapter's hypothetical linearization (coefficients 0.5, 10, 8 and 0.25; one-unit amenity and productivity shocks); the combined shock is added for comparison.

Calculated values

Rent change dr ($000)
13.333
Wage change dw ($000)
-3.333
Wage
falls

Hypothetical teaching numbers, not regional data. The amenity lowers the household line by 10. Adding the two conditions gives 10 = 0.75 dr, so rent rises by 13.333 thousand and the wage falls by 3.333 thousand (dw = 0 - 0.25 x 13.333 = -3.333).

Worked steps

  1. Households: dw - 0.5 dr + 10 x 1 = 0; firms: 8 x 0 - dw - 0.25 dr = 0
  2. Add the two: 0 + 10 - 0.75 dr = 0
  3. dr = 10 / 0.75 = 13.333
  4. dw = 0 - 0.25 x 13.333 = -3.333

Use the idea

Read wages and rents together: rising rents with falling wages point to amenities, rising rents with rising wages to productivity.

Where the conclusion applies

A local linear approximation, perfectly mobile households and firms, and fixed coefficients.

Check your understanding: With both shocks of one unit, what are dr and dw?
8 + 10 = 0.75 dr gives dr = 24 and dw = 8 - 0.25 x 24 = 2.

Chapter 80 source: section "Rosen-Roback spatial equilibrium".

Demonstration 2 of 4

Circular cumulative causation

When does a regional income gap feed on itself?

Each year carries a fraction s of the gap forward. With s below one the gap dies away; above one it compounds. Policy can change s by cutting one link in the loop.

Equation, written in LaTeX: g_1=0.40(\$10\text{ million})=\$4\text{ million},

Equation, written in LaTeX: g_2=0.40(\$4\text{ million})=\$1.6\text{ million}.

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g is the income gap between two regions in millions of dollars, starting at 10. s is net reinforcement: the share of each year's gap carried into the next, after offsets.

Predict first. Which value of s makes the gap grow?

Your prediction

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Figure: Circular cumulative causation. Bars of the regional income gap from year 0 to 5 at s = 0.40: 10, 4.00, 1.60, and 0.10 in year 5.
Net reinforcement s: 0.40 (base)
Constructed example: the chapter's hypothetical regions (gap 10 million, s = 0.40, 1.10 and 0.60); the path to year 5 is added for comparison.

Calculated values

Net reinforcement s
0.40
Gap after one year
4.00
Gap after two years
1.60
Gap after five years
0.10
Gap
shrinks

Hypothetical teaching numbers, not regional data. Net reinforcement is s = 0.40 (demand and migration carry 0.60, offset by 0.20). The gap moves from 10 to 0.40 x 10 = 4.00 and then 0.40 x 4.00 = 1.60 million, so it shrinks each year and reaches 0.10 million after five years.

Worked steps

  1. g1 = 0.40 x 10 = 4.00
  2. g2 = 0.40 x 4.00 = 1.60
  3. g5 = 10 x 0.40^5 = 0.10

Use the idea

List the feedback channels in a regional decline and ask which one policy could cut to bring s below one.

Where the conclusion applies

A constant reinforcement coefficient and no further shocks.

Check your understanding: Under the guarantee (s = 0.60), what are g1 and g2?
0.60 x 10 = 6 and 0.60 x 6 = 3.6 million.

Chapter 80 source: section "Circular cumulative causation".

Demonstration 3 of 4

Local employment multiplier

How many local service jobs follow 100 new export jobs?

The local multiplier depends on how much income stays and is spent locally and on whether local suppliers expand output or raise prices. Housing limits and inelastic supply both shrink it.

Equation, written in LaTeX: 85(\$80{,}000)(0.30)=\$2.04\text{ million}.

Equation, written in LaTeX: \frac{0.75(\$2.04\text{ million})}{\$60{,}000}=25.5.

Equation, written in LaTeX: \frac{0.50(\$1.44\text{ million})}{\$60{,}000}=12

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An exporter adds 100 jobs paying 80,000 dollars. Resident workers spend 30 percent locally; a local service job needs 60,000 dollars of revenue, and only the share met by more quantity (not higher prices) creates jobs. m is local jobs per direct job.

Predict first. Which constraint cuts the first round more: housing (60 residents) or a weaker quantity response (0.50)?

Your prediction

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Figure: Local employment multiplier. Left: payroll 8 million and local demand 2.04 million. Right: 100 direct jobs and 25.5 first-round local jobs plus 6 from re-spending.
Resident workers: 85, Share of demand met by more quantity: 0.75
Constructed example: the chapter's hypothetical exporter (100 jobs at 80,000, 85 or 60 resident workers, local share 0.30, 60,000 per job, quantity share 0.75 or 0.50, re-spending 6 or 3); the two mixed cases are added for comparison.

Calculated values

Local demand ($ million)
2.04
First-round jobs
25.5
Re-spending jobs
6
Multiplier m
0.315

Hypothetical teaching numbers, not regional data. 85 resident workers spend 30 percent locally: 85 x 80,000 x 0.30 = 2.04 million. With 75 percent met by more output, the first round is 0.75 x 2,040,000 / 60,000 = 25.5 jobs. With the book's 6 re-spending jobs, m = (25.5 + 6) / 100 = 0.315.

Worked steps

  1. Demand = 85 x 80,000 x 0.30 = 2,040,000
  2. First round = 0.75 x 2,040,000 / 60,000 = 25.5
  3. m = (25.5 + 6) / 100 = 0.315

Use the idea

Before quoting a jobs multiplier for a new plant, ask where its workers will live and how easily local services can expand.

Where the conclusion applies

Fixed local spending share and revenue per job; re-spending rounds are taken from the book only for its two cases.

Check your understanding: With 60 residents and a 0.75 quantity share, what is the first round?
60 x 80,000 x 0.30 = 1.44 million; 0.75 x 1.44 million / 60,000 = 18 jobs (a constructed mixed case).

Chapter 80 source: section "Local employment multiplier".

Demonstration 4 of 4

Public works multiplier with leakages and crowding out

How much output does 10 million of repairs add once taxes, imports and crowding out are counted?

Each round of spending is a fraction of the one before; taxes and imports shrink that fraction, so the total is smaller. Crowding out and the project's own value are separate ledgers.

Equation, written in LaTeX: k_G=\frac{1}{1-0.60}=2.5.

Equation, written in LaTeX: k_G=\frac{1}{1-0.60(1-0.20)+0.15}=\frac{1}{0.67}\approx1.493.

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G = 10 million dollars of repairs. c is the marginal propensity to consume, t the tax rate and m the import share of income. k_G is the spending multiplier. 3 million of private output is crowded out and the repaired asset yields services worth 12 million.

Predict first. Does raising c to 0.75 matter more with or without leakages?

Your prediction

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Figure: Public works multiplier with leakages and crowding out. Bars of output added in each spending round, starting at 10 million and shrinking by a factor 0.60; the first eight rounds are shown, and the full series sums to 25.00 million.
Marginal propensity to consume c: 0.6, Taxes and imports: None
Constructed example: the chapter's hypothetical repairs (10 million, c = 0.60, t = 0.20, m = 0.15, crowd-out 3, services 12); c = 0.50 and 0.75 and the crowd-out in the no-leakage case are added for comparison.

Calculated values

Multiplier k
2.500
Output gain ($ million)
25.00
Net of crowd-out ($ million)
22.00
Measured multiplier
2.200
Project surplus ($ million)
2.00

Hypothetical teaching numbers, not regional data. The denominator is 1 - 0.60 = 0.40, so k = 2.500 and 10 million of repairs adds 2.500 x 10 = 25.00 million in total. Subtracting the 3 million of crowded-out private output leaves 22.00 million, a measured multiplier of 2.200. The project's own value is 12 - 10 = 2 million, a separate question.

Worked steps

  1. 1 - 0.60 = 0.40
  2. k = 1 / 0.40 = 2.500
  3. Output = 2.500 x 10 = 25.00
  4. Net of crowd-out = 25.00 - 3 = 22.00; measured 22.00 / 10 = 2.200
  5. Project surplus = 12 - 10 = 2, a separate ledger

Use the idea

Report three numbers for a public works plan: gross demand propagation, net measured output and the project's lifetime value.

Where the conclusion applies

Idle resources, a fixed propensity to consume and fixed leakage rates; the 3 million crowd-out is the book's figure for the leakage case, applied to every state for comparison.

Check your understanding: At c = 0.75 with leakages, what are k and output?
1 / (1 - 0.75 x 0.80 + 0.15) = 1 / 0.55 = 1.818; output 18.18 million; net of 3 crowd-out 15.18.

Chapter 80 source: section "Fiscal Multiplier and Public Works".