The Encyclopedia of Economic Principals

Chapter 84

Labor Supply, Participation, and Household Insurance

Search costs, care hours, wage insurance and self-selection.

Four of the chapter's worked examples, made interactive: when a job seeker stops searching, where the child penalty comes from, a steady wage as insurance, and how self-selection moves sector averages. All numbers are the chapter's hypothetical values.

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

When does a job seeker stop searching?

When is search worth its cost, and what happens to measured unemployment when searchers give up?

Search continues while expected gain covers its cost. Weak prospects lower p, and the person moves from unemployment to nonparticipation, which shrinks both the numerator and the labor force of the unemployment rate.

Equation, written in LaTeX: pJ\geq k.

Equation, written in LaTeX: pJ=0.10(\$2{,}000)=\$200,

Equation, written in LaTeX: u=\frac{U}{E+U}.

Scroll sideways for the whole equation

p is the perceived job-finding probability, J = $2,000 the gain from an acceptable offer and k the cost of qualifying search. E is employment and U active unemployment.

Predict first. If p falls to 0.04, does Jordan keep searching at a cost of $120?

Your prediction

Choose an example

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Figure: When does a job seeker stop searching? Left: expected gain $200 beside search cost $120. Right: measured unemployment 10.0% with all searching and 8.2% after twenty stop.
Perceived job-finding probability: 0.1, Search cost ($): 120
Constructed example: the chapter's hypothetical values are J = $2,000, costs $120 and $70, p = 0.10, 0.07 and 0.04, and the town of 900 employed and 100 searching with 20 discouraged; p = 0.15 and a cost of $170 are added.

Calculated values

Expected gain pJ
$200
Search cost k
$120
Decision
Search
Town rate, all search
10.0%
Town rate, 20 discouraged
8.2%

Hypothetical numbers from the chapter. With a perceived job-finding probability of 0.10, Jordan's expected gain is 0.10 x 2,000 = 200 dollars against a search cost of $120, so Jordan searches. In the town, twenty discouraged searchers cut measured unemployment from 10.0% to 8.2% although nobody was hired.

Worked steps

  1. pJ = 0.10 x 2,000 = 200
  2. Compare: 200 > 120, so Jordan searches
  3. All searching: u = 100 / (900 + 100) = 10.0%
  4. Twenty discouraged: u = 80 / (900 + 80) = 0.0816, or 8.2%

Use the idea

Read a falling unemployment rate alongside participation: exits from search can lower the rate with no hiring at all.

Where the conclusion applies

A fixed gain from an offer and a fixed search cost over the decision horizon; the town's flows are taken as given, with no hiring.

Check your understanding: With p = 0.07 and a search cost of $170, does Jordan search?
pJ = 0.07 x 2,000 = $140, below $170, so no; at a cost of $120 he would.

Chapter 84 source: section "Discouraged-worker effect".

Demonstration 2 of 4

Where the child penalty comes from

How do uncovered care hours, and who covers them, turn into a gendered earnings penalty?

The binding care constraint removes paid hours from whoever covers the gap. Formal care that fills the gap removes the hours channel; an equal split keeps the income loss but removes the gender gap.

Equation, written in LaTeX: t_m+t_f+t_{paid}\geq C.

Equation, written in LaTeX: \frac{\$1{,}200-\$600}{\$1{,}200}=50\%.

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C = 50 is the child's weekly care need, t_paid formal care, t_m and t_f the parents' care hours. Relatives give 10 hours; each parent earns $30 an hour for up to 40 hours.

Predict first. With an equal split and 20 formal hours, is the gendered penalty zero even though both lose income?

Your prediction

Choose an example

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Figure: Where the child penalty comes from. Left: stacked weekly hours, mother 20 paid and 20 care, father 40 paid and 0 care. Right: penalties 50.0% and 0.0%.
Formal childcare hours: 20, Who covers uncovered hours: Mother only
Constructed example: the chapter's hypothetical values are the $30 wage, 40 hours, 50 care hours, 10 from relatives, 20 or 40 formal hours and the equal split; 30 formal hours are added.

Calculated values

Uncovered care hours
20
Mother paid hours
20
Mother weekly earnings
$600
Mother penalty
50.0%
Father weekly earnings
$1,200
Father penalty
0.0%

Hypothetical family from the chapter. With 20 formal hours, 50 - 10 - 20 = 20 care hours remain uncovered, covered by the mother. The mother earns $600 (penalty 50.0%) and the father $1,200 (penalty 0.0%), so the gendered gap is 50.0%.

Worked steps

  1. Uncovered = 50 - 10 - 20 = 20 hours, covered by the mother
  2. Mother: 40 - 20 = 20 hours x $30 = $600
  3. Father: 40 - 0 = 40 hours x $30 = $1,200
  4. Mother penalty = (1,200 - 600) / 1,200 = 50.0%; father = (1,200 - 1,200) / 1,200 = 0.0%

Use the idea

When a parent's earnings fall after a birth, ask how many care hours are uncovered and how they are divided before attributing the loss to pay discrimination.

Where the conclusion applies

Fixed wage and care need, uncovered care taken only from paid hours, all else constant; the chapter's later $27 wage shows a separate dynamic channel not shown here.

Check your understanding: With 30 formal hours and the mother covering the rest, what is her penalty?
Uncovered = 50 - 10 - 30 = 10; she works 30 hours and earns $900, so the penalty is 300 / 1,200 = 25%.

Chapter 84 source: section "Child penalty".

Demonstration 3 of 4

A steady wage as insurance

When does a worker prefer a fixed wage to pay that follows revenue, and what does the firm earn?

Concave utility makes a sure wage worth more than a risky pair with a slightly higher mean. A diversified firm can absorb the swing, earning in booms what it loses in slumps.

Equation, written in LaTeX: w_s^{spot}=p_sF'(n_s).

Equation, written in LaTeX: \frac{\sqrt{40}+\sqrt{16}}{2}\approx5.16.

Equation, written in LaTeX: \sqrt{27}\approx5.20,

Scroll sideways for the whole equation

Spot pay equals revenue per worker in each equally likely state: 40 in the boom and the slump value set here. The worker's utility is sqrt(w); the firm offers one fixed wage in both states.

Predict first. If the slump deepens to $8, does the fixed $27 contract still give the firm a nonnegative average margin?

Your prediction

Choose an example

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Figure: A steady wage as insurance. Left: square-root utility with the spot chord at expected utility 5.16 and the fixed wage point at 5.20. Right: firm margins 13.0, -11.0 and average 1.0.
Slump revenue per worker ($): 16, Fixed contract wage ($): 27
Constructed example: the chapter's hypothetical values are revenue 40 and 16 (8 in the deep slump) and the $27 wage; a slump of 24 and wages of 26 and 28 are added.

Calculated values

Spot expected utility
5.16
Fixed-wage utility
5.20
Firm margin, boom
13.0
Firm margin, slump
-11.0
Firm average margin
1.0

Hypothetical two-state economy from the chapter. Spot pay of 40 or 16 gives expected utility 5.16; a fixed $27 gives 5.20, so the worker prefers the fixed wage. The firm's margins are 40 - 27 = 13.0 in the boom and 16 - 27 = -11.0 in the slump, so the firm earns $1.00 on average for bearing risk.

Worked steps

  1. Spot EU = (sqrt(40) + sqrt(16)) / 2 = (6.3246 + 4.0000) / 2 = 5.1623
  2. Fixed wage: sqrt(27) = 5.1962, so the worker prefers the fixed wage
  3. Margins: 40 - 27 = 13.0; 16 - 27 = -11.0
  4. Average margin = (13.0 + (-11.0)) / 2 = 1.0

Use the idea

Look for the insurance signature: stable pay in bad states offset by restraint in good states within a continuing relationship.

Where the conclusion applies

Two equally likely states, a risk-neutral firm, fixed employment and no other contracting frictions; the chapter's layoffs in the deep slump are not modelled.

Check your understanding: At a slump of 16 and a wage of 28, does the worker prefer the contract, and what is the firm's average margin?
sqrt(28) = 5.29 > 5.16, so yes; margins are +12 and -12, an average of 0.

Chapter 84 source: section "Implicit-contract smoothing".

Demonstration 4 of 4

Who chooses coding, and what the average hides

How can a sector's observed mean pay fall when every incumbent's pay rises?

Workers sort on comparative payoffs, so observed sector means mix price changes with changes in who is in the sector. An entrant with lower coding pay drags the mean down.

Equation, written in LaTeX: j_i^*=\arg\max_j\{p_jq_{ij}-c_{ij}\}.

Equation, written in LaTeX: \frac{70+60}{2}=65,

Equation, written in LaTeX: \frac{77+66+44}{3}\approx62.3,

Scroll sideways for the whole equation

Each worker picks the sector j with the highest net payoff p_j q_ij - c_ij. Potential pay (coding, sales) in $000: Ana (70, 50), Ben (60, 58), Cara (48, 65), Dev (40, 55). Only the chosen sector's pay is observed.

Predict first. When coding pay rises 10 percent and Dev faces the $12,000 licence, does the observed coding mean rise or fall?

Your prediction

Choose an example

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Figure: Who chooses coding, and what the average hides. Left: the four workers placed by net sales pay and coding pay, coders in teal. Right: observed means, coding 65.00 and sales 60.00.
Coding pay increase: 0%, Dev's sales licence cost ($000): 0
Constructed example: the chapter's hypothetical values are the four workers' pay, the 10 percent raise and Dev's 12 licence; a 20 percent raise is added.

Calculated values

Coders
Ana and Ben
Sellers
Cara and Dev
Observed coding mean
65.00
Observed sales mean
60.00

Hypothetical workers from the chapter. With coding pay up 0% and a sales licence cost of 0 for Dev, Ana and Ben code and Cara and Dev sell. The observed coding mean is (70.0 + 60.0) / 2 = 65.00 and the sales mean (65 + 55) / 2 = 60.00 (in $000). A change in who enters moves the averages even when no incumbent's pay falls.

Worked steps

  1. Coding pay x 1.00: Ana 70.0, Ben 60.0, Cara 48.0, Dev 40.0
  2. Dev compares coding 40.0 with net sales 55 - 0 = 55
  3. Coding mean = (70.0 + 60.0) / 2 = 65.00
  4. Sales mean = (65 + 55) / 2 = 60.00

Use the idea

Do not read a cross-sector wage gap as the gain anyone would get from switching; track who enters and leaves.

Where the conclusion applies

Fixed potential pay, equal entry costs except Dev's licence, and ties resolved toward coding.

Check your understanding: With a 20 percent coding raise and no licence, who codes and what is the coding mean?
Coding pay 84, 72, 57.6 and 48 against sales 50, 58, 65 and 55: Ana and Ben code; mean (84 + 72) / 2 = 78.

Chapter 84 source: section "Roy self-selection model".