The Encyclopedia of Economic Principals

Chapter 95

Insurance, Education, Poverty, and Targeting

Smoothing, screening, tagging and crowd-out in social programs.

Four of the chapter's worked examples, made interactive: the smoothing gain against the duration response in unemployment insurance, ordeals that screen the right or wrong people, tags with errors, and crowd-out arithmetic. All values are the chapter's hypothetical teaching numbers.

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

Smoothing gain versus duration response

Should the unemployment benefit rise, given how much consumption falls and how duration responds?

A higher benefit is worth more when consumption falls a lot and people are risk averse; it costs more when it lengthens unemployment a lot. The formula compares the two at the current benefit.

Equation, written in LaTeX: \varepsilon_{D,b}=\frac{b}{D}\frac{dD}{db}.

Equation, written in LaTeX: \frac{3{,}000-2{,}400}{3{,}000}=0.20.

Scroll sideways for the whole equation

Consumption is 3,000 employed and c_u unemployed; gamma is risk aversion. A 10 percent benefit rise lengthens expected duration from 10 weeks to D_1. The smoothing index is gamma times the proportional drop; the elasticity is the percent change in duration per percent change in benefit.

Predict first. For a liquidity-constrained claimant (consumption 2,100), does the verdict flip at gamma = 2?

Your prediction

Choose an example

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Figure: Smoothing gain versus duration response. Two bars: smoothing index 0.40 and duration elasticity 0.50.
Risk aversion: 2, Unemployed consumption: 2,400, Duration after 10% raise (weeks): 10.5
Constructed example: the book's hypothetical values are consumption 3,000, 2,400 and 2,100, gamma 2 and duration 10, 10.5 and 10.2 weeks; gamma 1 and 3 are added.

Calculated values

Consumption drop
20%
Smoothing index
0.40
Duration elasticity
0.50
Verdict
Do not raise

Hypothetical teaching values, not estimates from a real program. Consumption falls 20% on unemployment, so with risk aversion 2 the smoothing index is 2 x 0.20 = 0.40. A 10 percent benefit rise lengthens duration to 10.5 weeks, an elasticity of 0.50. In this local comparison the behavioral cost exceeds the smoothing gain, pointing against a higher benefit.

Worked steps

  1. Drop = (3,000 - 2,400) / 3,000 = 0.20
  2. Smoothing index = 2 x 0.20 = 0.40
  3. Elasticity = ((10.5 - 10) / 10) / 0.10 = 0.50
  4. 0.40 against 0.50: do not raise

Use the idea

Measure the consumption drop and the duration response for the actual claimants before arguing for or against a benefit change.

Where the conclusion applies

A simplified local comparison that omits prudence, the unemployment share, match quality and equilibrium effects, as the chapter notes.

Check your understanding: With gamma = 1, consumption 2,100 and duration 10.5, which index is larger?
Smoothing 1 x 0.30 = 0.30 against 0.50, so still do not raise.

Chapter 95 source: section "Baily-Chetty optimal unemployment-insurance formula".

Demonstration 2 of 4

Ordeals that screen, or screen the wrong way

When does an enrollment burden keep out the right people?

An ordeal screens only if it is relatively more costly for the people the program does not target. When burdens fall hardest on the needy, it screens the wrong way, and every ordeal destroys recipient value.

Equation, written in LaTeX: B_i-c_i(z)\geq 0.

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B_i is the package's value to a household (180 high need, 80 low need) and c_i(z) the burden of the ordeal z. Each group has 100 households and the package costs the government 120.

Predict first. If high-need households face a burden of 190 and low-need 60, who enrolls?

Your prediction

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Figure: Ordeals that screen, or screen the wrong way. Grouped bars: high need values the package at 180 against a burden of 100; low need at 80 against 100.
Burden on high-need households: 100, Burden on low-need households: 100
Constructed example: the book's hypothetical values are values 180 and 80, package cost 120, 100 households per group and burdens 100, 190 and 60; a low-need burden of 190 is added.

Calculated values

High-need net value
80
Low-need net value
-20
Households enrolled
100
Transfers
12,000
Ordeal cost borne
10,000

Hypothetical teaching values, not estimates from a real program. With burdens of 100 on high-need and 100 on low-need households, net values are 180 - 100 = 80 and 80 - 100 = -20, so the ordeal screens as intended. Transfers are 12,000 and recipients bear 10,000 of ordeal cost.

Worked steps

  1. High need: 180 - 100 = 80; low need: 80 - 100 = -20
  2. Transfers = 100 x 120 = 12,000
  3. Ordeal cost = 100 x 100 = 10,000

Use the idea

Before adding paperwork or pickup requirements, ask who finds them most costly and count the burden recipients bear as a real cost.

Where the conclusion applies

Homogeneous groups, a fixed package value and a burden that is pure waste.

Check your understanding: With a burden of 60 for both groups, who enrolls and what ordeal cost is borne?
Both (120 and 20 are positive); all 200 enroll, transfers 24,000 and ordeal cost 12,000.

Chapter 95 source: section "Ordeal mechanism".

Demonstration 3 of 4

Tags, errors, and target receipts

How do a tag's hit and false-positive rates change cost and what reaches the target group?

A tag concentrates a larger payment on the target group at lower cost, as long as it is accurate. False positives raise cost without helping the target; missed targets lower what reaches them.

Equation, written in LaTeX: 0.75(400)=300

Equation, written in LaTeX: 0.10(600)=60

Equation, written in LaTeX: 540(8{,}000)=4{,}320{,}000

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400 of 1,000 low-income adults are constrained. The tag identifies a share of them (true-positive rate) and wrongly admits a share of the other 600 (false-positive rate); tagged adults get 8,000. The benchmark pays 4,000 to all.

Predict first. Raising the false-positive rate from 10 to 40 percent raises cost by how much?

Your prediction

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Figure: Tags, errors, and target receipts. Grouped bars: income-only costs 4.00m and delivers 1.60m to the target; the tag costs 2.88m and delivers 2.40m.
Share of constrained identified: 0.75, False-positive rate: 10%
Constructed example: the book's hypothetical values are 400 and 600 adults, payments 4,000 and 8,000, a 0.75 hit rate and false-positive rates of 0.10 and 0.40; hit rates 0.50 and 0.90 and a false-positive rate of 0.25 are added.

Calculated values

True positives
300
False positives
60
Tagged
360
Tag cost
2,880,000
Target receipts
2,400,000
Cost change vs income-only
-1,120,000

Hypothetical teaching values, not estimates from a real program. The tag finds 0.75 x 400 = 300 constrained adults and wrongly admits 0.10 x 600 = 60 others. Paying 8,000 to 360 people costs 2,880,000, of which 2,400,000 reaches the target, against 4,000,000 and 1,600,000 under the income-only rule.

Worked steps

  1. True positives = 0.75 x 400 = 300; false positives = 0.10 x 600 = 60
  2. Cost = 360 x 8,000 = 2,880,000
  3. Target receipts = 300 x 8,000 = 2,400,000
  4. Income-only: 1,000 x 4,000 = 4,000,000, of which 400 x 4,000 = 1,600,000 reaches the target

Use the idea

Judge a categorical program by its true-positive and false-positive rates together, and count any resources spent to obtain the tag as waste.

Where the conclusion applies

Fixed group sizes, payments and error rates; no change in behavior to obtain the tag.

Check your understanding: With a true-positive rate of 0.90 and a false-positive rate of 0.25, what are cost and target receipts?
360 + 150 = 510 tagged; cost 4,080,000; target receipts 360 x 8,000 = 2,880,000.

Chapter 95 source: section "Akerlof tagging principle".

Demonstration 4 of 4

Crowd-out arithmetic

How much of a public program's enrollment is new coverage?

Public enrollment mixes newly covered people with people who would have been privately insured. Only the first group raises coverage; the second shifts who pays.

Equation, written in LaTeX: G=300+60+320=680.

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G is policy-caused public enrollment, P the fall in private coverage (children who switch from employer or individual plans) and N = G - P the net coverage gain. Crowd-out is P / G.

Predict first. If employers raise dependent contributions so fewer children switch (180 and 40), does net coverage change?

Your prediction

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Figure: Crowd-out arithmetic. A stacked bar of public enrollment (300 from employer plans, 60 from individual plans, 320 previously uninsured) next to the net gain of 320.
Employer-covered switchers: 300, Individually insured switchers: 60, Uninsured take-up: 320
Constructed example: the book's hypothetical values are switchers 300 and 60 (or 180 and 40) and uninsured take-up 320; a take-up of 200 is added.

Calculated values

Public enrollment G
680
Private loss P
360
Net gain N
320
Crowd-out P / G
0.529
Net gain per entrant
0.471

Hypothetical teaching values, not estimates from a real program. Public enrollment is G = 300 + 60 + 320 = 680, of which 360 left private plans, so net coverage rises by N = 320 and crowd-out is 360 / 680 = 0.529 (52.9%). Public enrollment alone would hide the displaced private coverage.

Worked steps

  1. G = 300 + 60 + 320 = 680
  2. P = 300 + 60 = 360
  3. N = 680 - 360 = 320
  4. Crowd-out = 360 / 680 = 0.529

Use the idea

Report a coverage expansion by its net gain and crowd-out rate, not by enrollment alone.

Where the conclusion applies

The counterfactual coverage of each child is known; switching is the only private-coverage change.

Check your understanding: With 300 and 60 switchers and 200 uninsured enrolling, what is the crowd-out rate?
G = 560, P = 360, N = 200; 360 / 560 = 0.643, or 64.3 percent.

Chapter 95 source: section "Public-insurance crowd-out".