Step 1: Set up notation - h = hours of work per week - w = market wage ($ per hour) - C = weekly disposable income (consumption) Workers choose h to trade off consumption C versus leisure. A worker participates (works some positive hours) if the net wage at very low h is at least as large as their reservation wage (the minimum wage that makes giving up some leisure worthwhile). Step 2: Budget constraints under each plan Plan A (flat grant with 100% clawback up to $100): - Benefit B = 100 − earnings, until it hits 0. So for earnings y = w h ≤ 100, B = 100 − w h; for y > 100, B = 0. - Disposable income: - For 0 ≤ h ≤ 100/w: C = y + B = w h + (100 − w h) = 100 (flat, zero slope) - For h ≥ 100/w: C = w h (benefit exhausted; slope = w) Interpretation: Up to h = 100/w, working adds nothing to income (effective net wage = 0). Only after that point does income rise with hours. Plan B (hourly subsidy up to $100 total): - Subsidy S = 2.50 per hour, capped at $100. So S = 2.5 h for h ≤ 40; S = 100 for h ≥ 40. - Disposable income: - For 0 ≤ h ≤ 40: C = (w + 2.5) h (slope = w + 2.5) - For h ≥ 40: C = w h + 100 (slope = w) Interpretation: The first hours of work have a higher-than-market net wage; after 40 hours the subsidy stops but the worker keeps both market earnings and the full $100 already earned through the subsidy. Step 3: Participation (who works some positive hours?) - Key idea: A worker starts working if the net wage at the margin (dC/dh near h = 0) is at least their reservation wage. - Plan A: For 0 ≤ h ≤ 100/w, dC/dh = 0. The first hours of work increase hours but do not increase income at all. That means the net return to the first hour is zero. - Conclusion: Only workers with a reservation wage ≤ 0 (essentially none) will choose h > 0 unless they are willing to jump all the way beyond h = 100/w in one leap to start getting positive returns. That “jump” is unrealistic for most, and the flat segment creates a strong disincentive to start working at all. Expect many to choose h = 0. - Plan B: For small h, dC/dh = w + 2.5 > 0. The very first hour pays extra relative to the market wage. - Conclusion: Any worker whose reservation wage ≤ w + 2.5 will work some positive hours. Since reservation wages are typically positive, and w + 2.5 > 0, this plan strongly encourages participation. Therefore, Plan B yields a larger proportion of low-income workers who work a positive number of hours. Step 4: Visual intuition (ASCII graphs) For illustration, take a representative wage w = $10/hour. Plan A (flat grant with 100% clawback) - Flat income of $100 until h = 100/w = 10 hours, then income rises with slope 10. Income C ($/week) ^ | / | / | / |--------------*---------------- C = 100 up to h = 10 | | | | +--------------+-------------------------> Hours h 10 - Up to 10 hours, the line is flat (net wage = 0). After 10 hours, the line coincides with C = 10 h. Plan B (hourly subsidy of $2.50 up to $100 total) - Up to 40 hours, slope is 12.5 (= 10 + 2.5), then slope falls to 10 while keeping the $100 already earned from the subsidy. Income C ($/week) ^ | / | / | / | * | / | / +---------------------+---------------------> Hours h 40 - From 0 to 40 hours, C = 12.5 h (steeper than market wage). At h = 40, C = 12.5×40 = 40×10 + 100. Past 40 hours, the slope is 10 (C = 10 h + 100). Step 5: Economic intuition in words - Plan A imposes a 100% implicit tax on the first dollars earned: the first hours of work do not raise income, creating a strong disincentive to start working (large non-participation at h = 0). - Plan B is a wage subsidy on the extensive margin: it makes the first hours especially attractive, increasing the share of workers who enter employment. This is the classic participation effect seen in earned income tax credit–style designs. Final answer Plan B leads to a greater proportion of low-income workers working some positive number of hours per week. The initial net wage is zero under Plan A (discouraging participation), while it is w + $2.50 under Plan B (encouraging participation). The budget-constraint graphs above show why: Plan A has a flat segment at $100 (no gain from early hours), whereas Plan B has a steeper-than-market line for the first hours, pulling people into work.