Let's break down and **assess the statement** step-by-step, and then **draw/describe the budget sets and indifference curves** for the three cases. ## 1. Understanding the Policy and Statement - **Cash welfare program**: Suppose the program guarantees a certain amount of cash benefit ($G$) to someone with zero earnings. - **Benefit reduction rate (tax rate)**: For every dollar earned, benefits are reduced by a certain percentage. Initially, it's **50%** (for every $1 earned, benefits drop by $0.50). The proposal is to lower this to **25%**. - **Guarantee unchanged**: The maximum benefit (at zero earnings) stays the same. **Statement to assess:** > If I reduce the benefit reduction rate for cash welfare from 50% to 25%, and leave the guarantee unchanged, it will save me money because some people on the program will work more so I will pay less. Is this statement correct? Let's analyze. --- ## 2. Budget Set Before and After the Change Let: - $w$ = wage rate (per hour) - $h$ = hours worked - $E = wh$ = earnings from work - $G$ = guarantee (maximum benefit) - $r$ = benefit reduction rate (0.5 or 0.25) - $T$ = total income = earnings + welfare ### **A. Before Change: Benefit reduction rate = 50%** - For $E = 0$, benefits = $G$, total income = $G$ - For $0 < E < \text{breakeven}$, benefits = $G - rE$, total income = $E + (G - rE) = G + (1 - r)E$ - For $E \geq$ breakeven, benefits = 0, total income = $E$ **Breakeven point:** $G - rE = 0 \implies E = G/r$ So, the budget line is: - Slope = $1 - r$ up to $E = G/r$ - After $E = G/r$, slope = 1 **Example:** Suppose $G = \$200$ - At $r = 0.5$, breakeven at $E = 200/0.5 = \$400$ ### **B. After Change: Benefit reduction rate = 25%** - Slope up to $E = G/0.25 = 4G$ - Slope = $1 - 0.25 = 0.75$ up to $E = 4G$ - After $E = 4G$, slope = 1 --- ## 3. Budget Set Diagrams ### (i) **People not working at all** - At $h = 0$, total income = $G$ - Can increase hours and earnings, but for each dollar earned, lose only $0.25$ in benefits (instead of $0.50$) - The budget line is flatter (less steep) with the lower reduction rate, so the "kink" is farther out. ### (ii) **People working and getting benefits** - Their budget line between $h = 0$ and the breakeven point has a higher slope (they get to keep more of each extra dollar they earn). - With the lower reduction rate, this segment is longer and the slope is less steep (i.e., higher reward for working). ### (iii) **People near the breakeven point** - Before: Breakeven at $E = G/0.5 = 2G$ - After: Breakeven at $E = G/0.25 = 4G$ - The range over which the benefit is positive is wider, covering higher earnings. #### **Visual summary:** - The budget set *expands* when the reduction rate drops. - The "kink" moves to the right (higher earnings). - The slope before the kink increases (from 0.5 to 0.75). --- ## 4. Indifference Curves - Indifference curves are upward sloping ("bowed" toward the origin), representing combinations of leisure and income that yield the same utility. - The optimal point is where the highest possible indifference curve is tangent to the budget constraint. ### **For each group:** - **Not working at all:** Initially at $(0, G)$. With the higher slope, some may be induced to work a little. - **Working, getting benefits:** May work more since the implicit tax is lower (they get to keep more of each extra dollar earned). - **Near breakeven:** The region where benefits are available is now larger, might "bunch" just below new, higher breakeven. --- ## 5. **Assessment of the Statement** The claim is: > Lowering the benefit reduction rate will *save money* because people will work more, so I will pay them less. Let's analyze: ### **A. Effect on work incentives** - Lowering the reduction rate **does** increase the incentive to work (since you lose less in benefits per dollar earned). - Some people will work more. ### **B. Effect on total benefit expenditure** - For a given level of earnings, benefits are *reduced more slowly* with a lower reduction rate. - The breakeven point (where benefits go to zero) is pushed out: people keep some benefits at higher earnings than before. - Unless the increase in earnings is very large, **most people will still receive more in benefits than before** for the same amount of work. ### **C. Fiscal impact:** - **People who continue not working:** Still get $G$ (no change). - **People who work a bit more:** Will get *higher* benefits than before at the same earnings, unless they work a *lot* more. - **People near the old breakeven:** Can earn more and still get some benefits. Many will continue to receive partial benefits, so the welfare rolls may expand. **In practice, the lower reduction rate is likely to cost more, unless the increase in earnings is dramatic.** ### **Numerical Example** Suppose: - $G = \$200$ - Old reduction rate: 50% - New reduction rate: 25% - Suppose someone earns $100$ **Old benefit:** $200 - 0.5 \times 100 = \$150$ Total income: $100 + 150 = \$250$ **New benefit:** $200 - 0.25 \times 100 = \$175$ Total income: $100 + 175 = \$275$ So at the same earnings, they get **more** in benefits, and have a higher total income. Even if they increase earnings, unless they go all the way past the new breakeven, they get more. --- ## **Final Answer** ### **Assessment:** > **The statement is incorrect.** > > Lowering the benefit reduction rate from 50% to 25%, while keeping the guarantee unchanged, will **cost more money**, not less. This is because people retain more benefits at any given level of earnings, and the breakeven point (where benefits go to zero) is pushed out, so more people will qualify for partial benefits. While some people may work more, the increase in earnings is unlikely to be large enough to offset the higher outlays from the reduced benefit reduction rate. --- ### **Budget Sets and Indifference Curves:** - **Not working at all:** - Budget set starts at $(0, G)$. - With lower reduction rate, the budget line is less steep, and the "kink" occurs at higher earnings. - **Working and getting benefits:** - Old: Slope of budget line = 0.5, kink at $E = 2G$ - New: Slope = 0.75, kink at $E = 4G$ - Indifference curves may shift outward, as people can reach higher utility by working more. - **Near breakeven:** - Old: Lose all benefits at $E = 2G$ - New: Lose all benefits at $E = 4G$ - Many may now qualify for partial benefits at higher earnings. **(See below for a sketch of the budget set)** ``` Vertical axis: Total income Horizontal axis: Earnings | | / | / | / | / | / Total income | / | / | / | / |/ ----------------+-------------------> Earnings /| / | / | / | / | ``` - The steeper segment (lower slope) is the old 50% reduction rate. - The flatter, longer segment is the new 25% reduction rate. - Indifference curves are tangent to the budget set at the optimal choice. --- ### **Summary Table** | Group | Old Budget Set (50%) | New Budget Set (25%) | Effect | |-----------------------------|----------------------|----------------------|-------------------------------| | Not working | $G$ | $G$ | No change | | Working, getting benefits | Slope 0.5, kink at $E=2G$ | Slope 0.75, kink at $E=4G$ | Keep more of each $ earned | | Near breakeven | Lose benefits at $E=2G$ | Lose at $E=4G$ | Benefits phase out more slowly| **Bottom line:** **Costs rise** with a lower reduction rate, because benefits phase out more slowly and more people qualify for partial benefits, even though some will work more. --- ### **References** - Mankiw, N. Gregory, *Principles of Economics*, section on welfare and labor supply. - Blank, Rebecca, "It Takes a Nation: A New Agenda for Fighting Poverty," on welfare program design.