In any accounting period of length two, a young child named "Junior" receives an allowance $.$ Junior typically spends all of his allowance $$ as soon as he gets it so $c_1,$ consumption in the first half of the accounting period, period $1,$ is $$ while $c_2,$ consumption in the second half of the accounting period, period $2,$ is zero. Luckily Junior is only eight and can find no one to give him a loan, otherwise he would sell his future allowances. Junior's parents, who understand the importance of good money$-$management skills, are worried about his lack of thrift so they decide to induce him to save more by offering to match any interest that he earns on savings. Let $r$ denote the current interest rate that Junior faces which is constant.

a$)$ Sketch Junior's intertemporal budget constraint $($with consumption in the first half of the accounting period $c_1$ on the axis of abscissae and consumption in the second half of the accounting period on the axis of ordinates$)$ before the interest proposal of his parents. Show where Junior consumes in this figure.

b$)$ On the same graph sketch Junior's new intertemporal budget constraint after the interest proposal of his parents. Can you tell whether the plan will work?

Now suppose Junior is already saving some of his allowance, so consumption in the second period $c_2$ is strictly positive. Thus Junior's parents have made their proposal to induce him to save even more than he was before.

c$)$ Using indifference curves and budget constraints illustrate three possible ways Junior could respond to the proposal. Be sure to decompose each case into an income and a substitution effect.