An economy has 100 households.The forty rich households each have incomes of $50,000 in period 1 and $75,000 in period 2.The sixty poor households each have incomes of $20,000 in period 1 and $25,000 in period 2.Assume that the price of the good is $1 in both periods.Also assume that the households borrow from each other.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.The equilibrium real interest rate is about

Question

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In period 1, the 40 rich households have a total income of $2 million and the 60 poor households have a total income of $1.2 million. Each household decides to consume 50 percent of the present value of its income, which is equal to the income in period 1 plus half of the income in period 2 (because the discount rate is 50 percent).

So, the total consumption in period 1 is $1.2 million (from the poor households) plus $1.5 million (from the rich households who consume $37,500 each) for a total of $2.7 million.

Total output in period 1 is $1.6 million (since there are 100 households consuming $16,000 each), so there is excess demand for the good. The interest rate will need to rise to encourage some households to postpone consumption to period 2, which will increase the supply of the good in period 1.

In equilibrium, the interest rate will be such that the present value of the total consumption in both periods equals the total income in both periods. The present value of the consumption in period 1 is $2.7 million, and the present value of the consumption in period 2 is $1.5 million (from the rich households) plus $0.6 million (from the poor households who consume $10,000 each) for a total of $2.1 million. The total present value of consumption is $4.8 million.

The total present value of income is $3 million (from the rich households who earn $125,000 each) plus $1.44 million (from the poor households who earn $22,500 each) for a total of $4.44 million.

Therefore, the equilibrium real interest rate is (4.8/4.44-1) = 40%.

An Economy Has 100 Households