Question

A welfare program for low-income people offers a family a basic grant of \(\$ 6,000\) per year. This grant is reduced by \(\$ 0.75\) for each \(\$ 1\) of other income the family has. a. How much in welfare benefits does the family receive if it has no other income? If the head of the family earns \(\$ 2,000\) per year? How about \(\$ 4,000\) per year? b. At what level of earnings does the welfare grant become zero? c. Assume the head of this family can earn \(\$ 4\) per hour and that the family has no other income. What is the annual budget constraint for this family if it does not participate in the welfare program? That is, how are consumption ( \(c\) ) and hours of leisure ( \(h\) ) related? What is the budget constraint if the family opts to participate in the welfare program? (Remember, the welfare grant can only be positive. e. Graph your results from parts (c) and (d). f. Suppose the government changes the rules of the welfare program to permit families to keep 50 percent of what they earn. How would this change your answers to parts (d) and (e)? g. Using your results from part (f), can you predict whether the head of this family will work more or less under the new rules described in part (f)?

Short answer

Answer: For a non-participant in the welfare program, the relationship between consumption (c) and hours of leisure (h) can be expressed as: \(c = 4 * (8760 - h)\), where 8760 is the total number of hours in a year and the head of the family can earn $4 per hour.

Step-by-step solution

a. Welfare benefits with varying incomes

First, we are given that the grant is \(6,000 per year. We are then told that the grant is reduced by \)0.75 for every $1 of other income the family has. Let's calculate the welfare benefits for different incomes mentioned: 1) No other income: the grant stays at $6,000 (full grant, no reductions). 2) The head of the family earns $2,000 per year: Reduction due to income: 2000 * 0.75 = $1,500. Total grant: 6000 - 1500 = $4,500. 3) The head of the family earns $4,000 per year: Reduction due to income: 4000 * 0.75 = $3,000. Total grant: 6000 - 3000 = $3,000.

b. Level of earnings for zero welfare grant

Let the earnings of the family be \(e\). We need to find the point where the grant would reduce to zero. We are given that for every dollar, a reduction of 0.75 dollars happens. We have: \(e * 0.75 = 6000\) Solving for \(e\), we get: \(e = 6000 / 0.75 = 8000\) The earnings level at which the grant becomes zero is $8,000.

c. Annual budget constraint for non-participant

Given that the head of the family can earn $4 per hour, and there's no other income or welfare, we can determine the annual budget constraint. We need to first express income (I) in terms of hours worked (w): \(I = 4 * w\) Here we define c = consumption and h = hours of leisure in a year. As there are 8760 hours per year, the relationship between consumption and hours of leisure for a non-participant can be given as: \(c = 4 * (8760 - h)\)

d. Annual budget constraint for participant

While participating in the welfare program, the family receives an adjusted grant based on their income. We can modify the equation for consumption and hours of leisure as: \(c = 4 * (8760 - h) + 6000 - 0.75 * 4 * (8760 - h)\) Simplifying the equation: \(c = 4 * (1 - 0.75) * (8760 - h) + 6000\) Adding the non-negative welfare grant condition: \(c = 4 * (1 - 0.75) * (8760 - h) + 6000\); grant >= 0 \(c = 1 * (8760 - h) + 6000\); grant >= 0

e. Graphing the results

The results from parts (c) and (d) can be graphed by plotting the consumption (c) in the vertical axis and hours of leisure (h) in the horizontal axis. The non-participant budget constraint shows a steeper slope (-4) and starts from \(c = 4 * 8760 = 35040\), while the participant's budget constraint has a less steep slope (-1) and starts from \(c = 6000\).

f. Change in answers for parts (d) and (e) due to rule change

Now, the government allows families to keep 50% of what they earn. We should redo the participant's budget constraint calculation with the new grant reduction rate. New Reduction rate = \(1 - 0.5 = 0.5 \) Modified equation for participant’s budget constraint: \(c = 4 * (8780 - h) + 6000 - 0.5 * 4 * (8760 - h)\) Simplifying the equation: \(c = 4 * (1 - 0.5) * (8760 - h) + 6000\) The graph would still show consumption in the vertical axis and hours of leisure in the horizontal axis, but now the slope will be different, and the change in the welfare program will affect the benefits for different levels of earnings in part (d).

g. Predicting work hours change under new rules

Using the results from part (f), we can determine if the head of the family will work more or less due to the new rules. Due to the lower reduction rate in welfare benefits as the family earns more, it is likely that they will choose to work more under the new rules, as the opportunity cost of working is reduced. This will depend on the individual's preferences and their desired consumption and leisure balance.

Key concepts

Welfare Economics

Welfare economics is a branch of microeconomic theory that focuses on the optimal allocation of resources and goods, aiming to understand how an economy can be organized in a way that improves social welfare. Social welfare in this context refers to the overall well-being and economic prosperity of a society, taking into consideration factors like income distribution and the efficiency of resource allocation. In welfare programs, such as the one described in the exercise, the government typically provides financial assistance to low-income families. The impact of such programs is assessed not only by measuring how they aid the recipients directly, but also by how they affect work incentives and economic behavior.

For instance, in the textbook problem, we see a common concern with welfare programs: the trade-off between providing financial aid and creating disincentives to work, known as the tapering effect. This is where a recipient's welfare benefits decrease as they earn more income, which may reduce their motivation to work. Economists study how the design of these programs can be improved to balance assistance and incentive, aiming to achieve an overall increase in social welfare.

Income Effect

The income effect in economics describes how changes in a person's income can affect their purchasing behavior. It's a critical component in understanding how welfare programs influence work incentives. When a family receives a welfare grant, it effectively increases their income, allowing them to afford more goods and services, or in some cases, to enjoy more leisure time without needing to work additional hours. However, as income rises from other sources, like employment, the reduction of benefits due to welfare programs can temper this effect.

For example, in the exercise, we see that as the head of the family earns more money, their welfare grant is reduced. The family experiences an income effect in both their actual income from work and their income from welfare. When they earn \(2,000, they still receive a part of the grant, but as their income from working increases to \)4,000, the grant decreases further. If the welfare grant changes, such as in the proposed modification to allow families to keep 50 percent of their earnings, the income effect will also adjust, potentially encouraging more work hours, as there's a lower reduction in benefits.

Budget Constraint

A budget constraint represents the concept in microeconomic theory that consumers can only purchase what their budgets allow. It's a fundamental principle that underpins consumer choice theory and illustrates the trade-off between different goods and services, including the trade-off between leisure and labor. In terms of welfare economics, a budget constraint can also depict the feasible combinations of income and government assistance families can obtain.

During the exercise, we analyze two different scenarios: one in which the family does not participate in the welfare program and another where they do. When not participating, the family's budget constraint is simply the product of their wage rate and the number of hours worked, as there's no grant to consider. However, participation introduces a new element. The constraint now includes the grant, which adjusts downwards as work hours, and thus income, increase. Graphically, these constraints can be represented by lines on a graph that illustrate the trade-off between leisure and consumed goods, or income, showing how the slope becomes less steep when the family is part of the welfare program, indicating the welfare grant's effect on the budget constraint.