In the two-period model, suppose a household's income in the first period is $40,000, income in the second period is $30,000, and the real interest rate is 25 percent. Draw a diagram showing the budget constraint. Now, suppose the real interest rate rises to 30 percent. Draw the new budget constraint. For the budget constraints you have drawn, be sure to show the values of the intercepts on each axis. If the household decides that its consumption in period 1 should always equal its consumption in period 2, determine whether the household is worse off or better off because of the decline in the real interest rate. Show your work.
What will be an ideal response?
Initial situation: intercept for first-period consumption axis = 40,000 + 30,000/1.25 = 64,000; intercept for second-period consumption axis = 10,000 + 40,000 + 30,000 = 80,000.
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New situation: intercept for first-period consumption axis = 40,000 + 30,000/1.3 = 63,077; intercept for second-period consumption axis = 12,000 + 40,000 + 30,000 = 82,000.
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Budget constraint rotates clockwise, from AB to CD.
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For consumption to be the same in each period, we must find the point where C1 = C2 and which is on the budget constraint, which has the equation: C2 = 80,000 ? 1.25 C1. In the initial situation, we would have: C1 = C2 = 80,000 ? 1.25 C1, so 2.25 C1 = 80,000, so: C1 = 35,556. So, the household saves 40,000 ? 35,556 = 4,444. Note that then C2 = 30,000 + (4,444 × 1.25) = 35,555. (The discrepancy is just rounding error.)
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In the new situation, the budget constraint is: C2 = 82,000 ? 1.3 C1, so C1 = C2 = 82,000 ? 1.3 C1, so 2.3 C1 = 82,000, so: C1 = 35,652. So, the household saves 40,000 ? 35,652 = 4,348. Note that then C2 = 30,000 + (4,348 × 1.3) = 35,652.
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In both cases, the household saves, so the increase in the real interest rate makes it better off. Note, as well, that the household's consumption is higher in both periods with the higher interest rate.
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