Why Household Responses Differ
A change in the real interest rate changes the reward for shifting resources between the present and the future. A higher rate makes postponing consumption more attractive: saving today can finance more consumption later. But the same rate change also affects a household’s resources. A net saver may earn more on its assets, while a net borrower may face a higher cost of borrowing. These two channels can push consumption in different directions.
The distinction matters when assessing how rate changes affect household decisions. The substitution effect concerns the relative price of consumption across time. The income effect concerns the change in purchasing power created by the household’s existing net position. Separating them helps avoid the mistaken claim that every household saves more whenever the real interest rate rises.
The Two-Period Household Model
Consider a household that receives real income \(y_0\) today and \(y_1\) one period from today. It chooses real consumption \(c_0\) today and \(c_1\) next period. Let \(r\) be the effective annual real interest rate for that period. Assume \(r>-1\), so the gross real return \(1+r\) is positive. The two-period budget constraint is
The left side is the present value of consumption; the right side is the present value of income. This constraint assumes that the household can borrow or lend at the same rate, has no borrowing limit, faces no taxes or transaction costs, and takes its income and the real rate as given. If borrowing is restricted, the household may be unable to choose a point on the full budget line.
Rearranging the budget constraint gives \(c_1=(1+r)(y_0-c_0)+y_1\). In a diagram with \(c_0\) on the horizontal axis and \(c_1\) on the vertical axis, the budget line has slope \(-(1+r)\) and passes through the endowment point \((y_0,y_1)\). A change in \(r\) rotates the line around that point: it does not change the household’s endowment, but it changes the trade-off between present and future consumption.
Substitution and Income Effects
When the real rate rises, giving up one unit of current consumption yields more future consumption. The relative price of current consumption, measured in units of future consumption, rises from \(1+r\). Holding the household’s utility level fixed, this change in relative prices encourages it to reduce \(c_0\) and increase \(c_1\). This is the substitution effect.
The rate change also changes the value of a household’s position relative to its endowment. To see the direction of this change, consider the household’s original chosen bundle and its original saving \(s=y_0-c_0\). At the original rate, the budget constraint implies \(c_1-y_1=(1+r)s\). If the rate rises by a small amount while the chosen bundle is held fixed, the change in the budget surplus evaluated at that bundle is
Thus, for a small rate increase, the household’s position improves at its original bundle if it is a net saver and worsens if it is a net borrower. This is the basis of the income effect: the higher return benefits a saver’s existing net position, whereas a higher borrowing cost harms a borrower’s. If consumption in each period is a normal good, the income effect tends to raise a saver’s consumption and lower a borrower’s consumption. The substitution effect tends to lower current consumption for either type. For a net saver, these effects can oppose each other; the total response of current consumption, and therefore of saving, is not determined by the rate change alone. For a net borrower with normal goods, the two effects both tend to reduce current consumption, though the precise response depends on the model and assumptions.
A Model for Calculating the Response
To calculate a response, preferences must also be specified. One useful illustrative model assumes the household chooses positive consumption to maximize \(U(c_0,c_1)=\ln(c_0)+\beta\ln(c_1)\), where \(\beta>0\) represents the weight placed on future utility. The model assumes the household is able to borrow or lend at rate \(r\) and has no other constraints. Define lifetime resources in present-value terms as \(W=y_0+y_1/(1+r)\). Substituting the budget constraint into utility, the first-order condition is
This condition says that the household chooses consumption so that the marginal utility trade-off between periods matches the market trade-off. With \(\beta=1\), it implies \(c_1=(1+r)c_0\). Substituting that condition into the budget constraint gives \(c_0+c_0=W\), so the optimal choices are
This model lets us calculate a household’s total response. It does not mean the income and substitution effects are always equal, or that these formulas apply to households with different preferences or borrowing constraints.
Worked Example: A Net Borrower Facing a Higher Real Rate
A hypothetical household has real income \(y_0=60\) today and \(y_1=100\) next year, measured in thousands of dollars. Assume one period of one year, an effective annual real rate of \(10\%\) initially, no uncertainty, no borrowing constraint, and utility \(U=\ln(c_0)+\ln(c_1)\). Find consumption and saving initially, then recalculate after the real rate rises to \(20\%\). Round final amounts to the nearest thousandth of a thousand dollars.
Method. Use the log-utility solution with \(\beta=1\), then calculate saving as \(s=y_0-c_0\). This keeps the real rate in the one-year model and identifies borrowing by the sign of saving.
Initial rate, \(r=0.10\). Lifetime resources are \(W=60+100/1.10=60+90.909091=150.909091\). Therefore, \(c_0=W/2=75.454545\) and \(c_1=1.10W/2=1.10(75.454545)=83.000000\). Saving is \(s=60-75.454545=-15.454545\). The household borrows \(15.455\) thousand dollars.
Higher rate, \(r=0.20\). Lifetime resources are \(W=60+100/1.20=60+83.333333=143.333333\). Thus, \(c_0=143.333333/2=71.666667\) and \(c_1=1.20(71.666667)=86.000000\). Saving is \(s=60-71.666667=-11.666667\), so the household still borrows, but by a smaller amount.
Check. At the higher rate, the present value of consumption is \(71.666667+86/1.20=71.666667+71.666667=143.333334\), equal to lifetime resources apart from rounding. Current consumption falls and borrowing declines, consistent with the substitution incentive and, under these preferences, the adverse income effect on a borrower.
Worked Example: A Net Saver and the Ambiguous Total Effect
A different hypothetical household has \(y_0=120\) and \(y_1=80\), in thousands of real dollars. Assume one year, no uncertainty or borrowing constraint, an effective annual real rate that increases from \(10\%\) to \(20\%\), and utility \(U=\ln(c_0)+\ln(c_1)\). Calculate consumption and saving at both rates. Round final amounts to the nearest thousandth of a thousand dollars.
Method. Apply the same log-utility solution, but classify the household separately at each rate. The key point is that the income effect for a net saver can offset part of the substitution effect; the sign of the total change must be calculated rather than assumed.
At \(r=0.10\), lifetime resources are \(W=120+80/1.10=120+72.727273=192.727273\). Then \(c_0=96.363636\), \(c_1=1.10(96.363636)=106.000000\), and \(s=120-96.363636=23.636364\). This household is a net saver.
At \(r=0.20\), lifetime resources are \(W=120+80/1.20=120+66.666667=186.666667\). Then \(c_0=93.333333\), \(c_1=1.20(93.333333)=112.000000\), and \(s=120-93.333333=26.666667\). The household remains a net saver.
Check. At the new rate, \(93.333333+112/1.20=93.333333+93.333333=186.666666\), equal to the calculated lifetime resources apart from rounding. In this example current consumption falls and saving rises, but the saver’s higher return increases its purchasing power and offsets some of the substitution incentive. The outcome follows from the specified preferences and endowments; the net-saver label alone does not guarantee that saving will rise.
Interest Income on a Net Position
A household’s interest cash flow provides a direct way to check whether a rate change helps or hurts its existing financial position. When assets and liabilities reprice by the same amount, the change in interest receipts minus interest payments depends on the net position. This calculation concerns the immediate cash-flow effect on balances held fixed; it is not, by itself, a complete prediction of the household’s consumption choice.
Worked Example: Repricing Assets and Debt
A hypothetical household holds \(30{,}000\) dollars in real-rate-sensitive assets and owes \(50{,}000\) dollars in real-rate-sensitive debt. Both reprice immediately for the next year. The effective annual real rate rises from \(2\%\) to \(4\%\). Assume balances remain fixed for the year, there are no fees or missed payments, and amounts are measured in constant purchasing-power dollars. Find the change in net interest cash flow.
Method. Calculate interest receipts and payments at each rate, then compare their difference. Equivalently, multiply the net position by the rate change.
At \(2\%\), interest receipts are \(30{,}000(0.02)=600\) dollars and interest payments are \(50{,}000(0.02)=1{,}000\) dollars. Net interest cash flow is \(600-1{,}000=-400\) dollars.
At \(4\%\), receipts are \(30{,}000(0.04)=1{,}200\) dollars and payments are \(50{,}000(0.04)=2{,}000\) dollars. Net interest cash flow is \(1{,}200-2{,}000=-800\) dollars. The change is \(-800-(-400)=-400\) dollars.
Check. The household’s net position is \(30{,}000-50{,}000=-20{,}000\) dollars, and the rate change is \(0.04-0.02=0.02\). Thus the change is \((-20{,}000)(0.02)=-400\) dollars, matching the direct calculation. The household is a net borrower, so the higher rate worsens its interest cash flow.
Common Mistakes and Why the Distinction Matters
- Using gross assets instead of the net position. In the repricing example, applying the two-percentage-point change only to the \(30{,}000\) dollars of assets gives a \(600\)-dollar increase in interest receipts, but ignores the \(1{,}000\)-dollar increase in payments. The correct net change is \((30{,}000-50{,}000)(0.02)=-400\) dollars.
- Confusing a percentage-point change with a percentage change. A rate rising from \(2\%\) to \(4\%\) increases by 2 percentage points, or \(0.02\) in decimal form. It does not increase by \(4\%\) of the original rate.
- Assuming that every saver must save more. The higher return creates a substitution incentive to save, but the additional purchasing power of a net saver can encourage more current consumption. For a saver, the total change in saving depends on preferences and circumstances.
- Treating a cash-flow calculation as a full behavioral prediction. The change in interest receipts and payments on fixed balances identifies one income channel. It does not specify how the household adjusts consumption, assets, or debt.
For household analysis, state the real-rate change, identify whether the household is a net saver or net borrower, and keep the substitution and income effects distinct. This framework is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Check Your Understanding
Use the two-period model and the assumptions stated in each question. Answers give a concise check; show the intermediate steps when solving.
- A household has \(y_0=90\) and \(y_1=110\), in real dollars. At an effective annual real rate of \(10\%\), write the present-value budget constraint and give the slope when \(c_1\) is plotted against \(c_0\). (Answer: \(c_0+c_1/1.10=190\); slope \(=-1.10\).)
- A household has \(12{,}000\) dollars in rate-sensitive real assets and \(5{,}000\) dollars in rate-sensitive real debt. Both balances reprice for one year, and the effective annual real rate rises by 1.5 percentage points. Find the change in annual net interest cash flow. (Answer: \(+105\) dollars.)
- Assume \(U=\ln(c_0)+\ln(c_1)\), no uncertainty or borrowing constraint, \(y_0=y_1=80\), and \(r=25\%\) for one year. Find \(c_0\), \(c_1\), and saving, in real dollars. (Answer: \(c_0=72\), \(c_1=90\), \(s=8\).)
- Assume the same log utility, no uncertainty or borrowing constraint, \(y_0=50\), \(y_1=90\), and an effective annual real rate that rises from \(5\%\) to \(10\%\). Find current consumption and saving at each rate, rounding to the nearest cent. (Answer: At \(5\%\), \(c_0=67.86\) and \(s=-17.86\); at \(10\%\), \(c_0=65.91\) and \(s=-15.91\), all in real dollars.)
- Why does a higher real rate not guarantee that a net saver’s current consumption falls by a particular amount, or that saving rises? (Answer: The substitution effect tends to reduce current consumption, but the income effect can raise a saver’s purchasing power and current consumption; the net response depends on preferences and circumstances.)