An economy has $75$ households, all of which have incomes of $$25,000$ each in period $1,50$ of which have incomes of $$40,000$ each in period $2,$ and $25$ of which have incomes of $$20,000$ each in period $2.$ Assume that the price of the good

is $$1$ in both periods. Suppose that each household decides that its consumption in period $1$ will equal $50$ percent of the present value of its income from both periods. a$.$ Find the equilibrium value of the interest rate. b$.$ Now suppose instead that each household will consume $60$ percent of the present value of its income from both periods in period $1.$ Now what is the equilibrium value of the interest rate? c$.$ Finally, suppose that each household$$s period $1$ consumption equals one$-$half the present value of its income from both periods, but everyone$$s income in period $1$ is lower: just $$20,000.$ What is the equilibrium value of the interest rate now?