3. Many states offer child-care grants for low-income single mothers (e.g. New York), as opposed to (or in addition to) standard cash grants. The primary purpose of this type of policy is to invoke single mothers, who likely need to be compensated with a high wage to offset childcare costs, into the labor force. First, consider the daily labor supply decision of Sarah, who has preferences dictated by U(C, L) = 2C2/3 L1/3, faces a wage rate of w = $8 and earns non-labor income of V = $200. (a) Sarah's marginal rate of substitution is given by MRSL,C C/2L. Compute her reserva- [2 pts] tion wage. Will she work at the market wage rate? (b) What is Sarah's optimal bundle of leisure hours and consumption dollars? [2 pts] Now, consider Sarah's twin sister Tara, who has a child. Like Sarah, Tara has non-labor income = of V $200, a wage rate of w = $8, and preferences represented by U(C, L) 2C2/3 1/3 However, Tara faces a "fixed cost" to participating in the labor force. If she does not participate (choosing h=0), she does not have to pay for childcare. If she chooses to participate and work h0 hours, she must pay Pc = $100 each day for childcare. (c) What is her non-labor income if she does not participate in the labor force? What is her [2 pts] effective non-labor income if she does participate in the work force and has to pay for child care? (d) On the graph below, plot Tara's budget constraint, and label it B. (Hint: Think carefully [2 pts] about how much consumption income she has, less her childcare costs, at 0 labor hours, 1 labor hour, 0.01 labor hours, etc.). Consumption 800 700 600 500 400 300 200 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Leisure Figure 4: Tara's' Budget Set (e) Using indifference curves (you don't need to use her preferences, just general, convex in- [2 pts] difference curves), illustrate why Tara would likely prefer to stay out of the work force, as opposed to entering the work force and incurring the childcare costs. (f) Now, suppose the government offers a $100 childcare payment to Tara which she receives [2 pts] if, and only if, she works h > 0 hours. Plot her new budget constraint, and label this constraint B. Briefly explain how this effectively gives her the same labor force incentives as Sarah; in other words, explain why this would lead her to return to the labor force.