A single mother has y $=$ $$400$ in non$-$labour income and w $=$ $$15$ per hour in labour income. She has up to $100$ hours available from which she can decide how much to work. There is no income tax.

$1.$ Write down the budget constraint and draw it on a graph. Make sure to label the axes and intercept correctly. Assume that the mother is already working $40$ hours per month. Show the indifference curve and the optimal point on the graph.

$2.$ Suppose that there is a fixed cost of $$300$ per month for childcare that must be paid as soon as the mother starts working. Show how the fixed cost affects the budget line in question $1.$ Discuss briefly her likely labour supply response. Use indifference curves on your graph to support your answer. Determine the share of substitution and income effects in the total effect. Assume that the government defines a childcare subsidy program such that mothers who are working up to $25$ hours per month receive an equivalent of $80$ percent of their earnings as childcare support. Those who are working between $25$ to $60$ hours per month are fully reimbursed. The subsidy decreases by $50\%$ of earnings for hours beyond $60$ until it has been completely phased out. $($Example: if a mother works $61$ hours. half of the earned income on the last hour is deducted from the full support. That is$,$ she will receive $$292\mathrm{.}5$ in childcare support.$)$

$3.$ Define the equation for the subsidy program for all three parts. $($part $1$: $0$ to $25$ hours; part $2$: $25$ to $60$ hours; part $3$: $60$ to $100$ hours$)$

$4.$ Use your answer in question $4$ and define the budget constraint after this policy. $($Hint: Budget constraint for each interval is equal to the corresponding childcare subsidy plus the budget constraint in question $2)$

$5.$ Draw the new budget constraint. Carefully label all the kinked points, axes and intersection point on the axes.

$6.$ Determine the likely response to the childcare subsidy program in terms of hours of work $($increase$,$ decrease, no change$)$ if she has already been working for: $1.10$ hours $2.40$ hours $3.70$ hours. $($Create a table with current hours and expected change of hour after policy implemented$)$