Question

Assume a welfare recipient can earn $90 per month without having their benefits reduced. Beyond $90, benefits are reduced by 58 cents for every dollar of earnings. Clarence can earn $10 per hour working or receive $577 from welfare if she does not work at all.

a. If she works 10 hours, how much are her earnings; how much is her welfare benefit; and how much is her income?

b. After Clarence works a certain number of hours, she does not receive any benefit at all. What is that number of hours?

c. Use your answers to a & b to plot her budget constraint and assuming her indifference curves are convex, show that it is possible to have more than one most - preferred outcome.

Answer 1

Answer 2

Solution:

Given:

• Earnings disregard: $90/month (no benefit reduction).

• Benefit reduction rate: $0.58forevery$1 earned beyond $90.

• Wage rate: $10/hour.

• Base welfare benefit (if no work): $577.

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Part (a): Work 10 hours

1. Total earnings:
   $10\text{hours}\times \mathrm{\$}10\mathrm{/}\text{hour}=\mathrm{\$}100$.

2. Benefit reduction:
   Earnings beyond 100 - $90 = $10 ).
   Reduction = $\mathrm{\$}10\times 0.58=\mathrm{\$}5.80$.

3. Adjusted welfare benefit:
   $\mathrm{\$}577-\mathrm{\$}5.80=\mathrm{\$}571.20$.

4. Total income:
   $\mathrm{\$}100(\text{earnings})+\mathrm{\$}571.20(\text{benefit})=\mathrm{\$}671.20$.

5. 
   

Answer:

• Earnings: $100

• Welfare benefit: $571.20

• Total income: $671.20

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Part (b): Hours when benefits reach zero

1. Let $x$ = hours worked.
   Earnings = $10x$.

2. **Benefit reduction starts after 90 = 10x - 90 ).
   Reduction = $0.58\times (10x-90)$.

3. 
   

4. Set welfare benefit to zero:
   $577-0.58(10x-90)=0$.
   Solve for $x$:
   $0.58(10x-90)=577$
   $5.8x-52.2=577$
   $5.8x=629.2$
   $x=108.5\text{hours}$.

5. 
   

Answer:
Clarence loses all benefits after working 108.5 hours/month.

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Part (c): Budget Constraint & Indifference Curves

1. Budget Constraint:

• Segment 1 (0–9 hours):
  Earnings = $10x$, benefits = $577.
  Total income = $10x+577$ (linear, slope = $10).

• Segment 2 (9–108.5 hours):
  Benefits reduced by $5.80 for every $10 earned beyond $90.
  Total income = $577+10x-0.58(10x-90)$.
  Simplifies to: $629.2+4.2x$ (slope = $4.2).

• Segment 3 (>108.5 hours):
  No benefits, income = $10x$ (slope = $10).

2. Indifference Curves:

• If convex, multiple tangency points can occur (e.g., one where work is minimal, another where work is significant).

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• Example: A kink at 9 hours or 108.5 hours may yield two optimal points.

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Answer:
The budget constraint has a kinked shape, allowing for multiple optimal outcomes due to convex preferences.