Expected Income and Current Demand

Introduction

A household deciding whether to buy a durable good today may consider more than the income it receives today. If it expects a raise next year, it may feel more able to make a purchase now; if it expects a reduction in income, it may postpone purchases. The forecast can therefore affect current demand even before future income is received.

The key is to keep three ideas separate: income received now, income expected later, and the good’s current price. A change in current price changes quantity demanded along a given demand curve. A change in expected future income can instead shift the current demand curve, with the direction depending on how buyers respond to their expected resources.

Definition: Expected future income is a buyer’s current expectation of income to be received in a later period. A change in expected future income is a nonprice change in a determinant of current demand. If other determinants, including the good’s current price and current income, are held fixed, the change can shift the current demand curve.

How Expectations Enter Current Demand

A demand relationship can include current price, current income, and expected future income as separate variables. For example, let $Q_d$ be the quantity demanded per month, $P$ the current price in dollars, $Y_c$ current monthly income in thousands of dollars, and $Y_f^e$ expected future monthly income in thousands of dollars. An illustrative linear demand function is

$$ Q_d = 100 - 2P + 0.5Y_c + Y_f^e. $$

This equation is a particular model, not a universal law. It assumes that the stated coefficients adequately describe buyers over the range being considered and that other demand determinants are held fixed. In this example, a one-unit increase in $Y_f^e$—that is, an increase of $1,000 in expected future monthly income—increases quantity demanded by one unit at each given price. The coefficient is positive, so the model represents a good for which higher expected income raises current demand.

In general, the sign is not guaranteed. For a good buyers treat as normal with respect to expected resources, a higher expected income tends to raise current demand. For a good buyers treat as inferior, the effect may be negative. Buyers may also respond little if the forecast is uncertain, if future income is far away, or if borrowing and saving decisions prevent them from acting on it. A demand function must state or imply the response it assumes.

This is distinct from a change in current income. Earlier material on income changes distinguishes the resulting demand shift from a movement along demand caused by the good’s own price. Here, the focus is on a change in buyers’ beliefs about later income while current income remains unchanged. A forecast is not itself a receipt of income.

Calculating a Shift at Fixed Prices

To analyze an expectation change, hold the current price and current income fixed, and compare the two demand schedules. The reason this comparison works is that the price term and current-income term do not change; only the expected-income term changes.

\Delta Q_d = Q_{d,\text{new}} - Q_{d,\text{old}} = \beta_f \bigl(Y_{f,\text{new}}^e - Y_{f,\text{old}}^e\bigr),

where $\beta_f$ is the coefficient on expected future income in the specified demand function. This result follows by subtracting the old quantity equation from the new one: the unchanged intercept, price term, current-income term, and other fixed terms cancel. If $\beta_f>0$, an increase in expected future income raises quantity demanded at every given price. If the demand function is linear, the resulting curves are parallel because the coefficient on price has not changed.

Worked Example: A Higher Forecast Raises Demand at Each Price

A hypothetical market’s monthly demand for a home energy device is $Q_d=100-2P+0.5Y_c+Y_f^e$. Quantity is measured in devices, price $P$ in dollars, and both income variables in thousands of dollars per month. Current income is $Y_c=4$. Buyers revise expected future monthly income from $Y_f^e=4$ to $Y_f^e=6$. Calculate the old and new quantities demanded at prices of $20 and $25, and identify the demand change.

Method. Substitute each forecast into the demand function while keeping current income fixed. First, with the old forecast:

Q_{d,\text{old}}=100-2P+0.5(4)+4 =100-2P+2+4 =106-2P.

With the revised forecast:

Q_{d,\text{new}}=100-2P+0.5(4)+6 =100-2P+2+6 =108-2P.

At $P=20$, the old quantity is $106-2(20)=106-40=66$ devices per month. The new quantity is $108-40=68$ devices per month.

At $P=25$, the old quantity is $106-2(25)=106-50=56$ devices per month. The new quantity is $108-50=58$ devices per month.

At either price, demand increases by 2 devices per month. The intercept rises by 2 while the price coefficient remains $-2$, so this linear model gives a parallel rightward shift. Check: the expected-income increase is $6-4=2$ thousand dollars per month, and its coefficient is 1; hence the predicted quantity increase is $1(2)=2$, matching both calculations. The common mistake avoided here is treating the expectation change as a price change: price stayed fixed, so this is a shift, not movement along demand.

Using a Probability-Weighted Income Forecast

Expected income may be calculated from multiple possible outcomes. If future income can take values $y_1,\ldots,y_n$, with probabilities $p_1,\ldots,p_n$, its expected value is the probability-weighted average:

E[Y_f] = \sum_{j=1}^{n} p_j y_j, \qquad \sum_{j=1}^{n}p_j=1.

A simple demand model may use this expected value as its income input. That is a modeling assumption: it means the demand response is represented as depending on the mean forecast. It does not imply that every buyer is certain to receive that mean amount, or that two different income distributions with the same mean must affect every buyer identically. Uncertainty, access to credit, and attitudes toward risk can matter in a richer model.

Worked Example: Revising Demand After a Promotion Forecast

A hypothetical group’s quarterly demand for a professional training course is $Q_d=50-P+0.2Y_f^e$, where $P$ is the current fee in dollars and expected annual future income $Y_f^e$ is measured in thousands of dollars. Before new information, the group uses an expected income of $40,000. It then estimates a 75% chance of income of $40,000 and a 25% chance of income of $60,000. Calculate demand at fees of $18 and $22 before and after the forecast revision.

Method. First calculate the revised expected income, using income in thousands:

Y_{f,\text{new}}^e=0.75(40)+0.25(60)=30+15=45.

The revised expected income is $45,000, an increase of $5,000 from the earlier forecast of $40,000. At a fee of $18, old demand is $50-18+0.2(40)=32+8=40$ course places per quarter. New demand is $50-18+0.2(45)=32+9=41$ places.

At a fee of $22, old demand is $50-22+0.2(40)=28+8=36$ places. New demand is $50-22+0.2(45)=28+9=37$ places.

The forecast revision raises demand by one place at either fee. Check: the income input rises by 5, and $0.2(5)=1$, consistent with the computed difference. The calculation assumes the model responds to the expected income value; it does not substitute the highest possible income or treat it as certain. That is the common mistake this method avoids.

Separating Current Income, Expected Income, and Price

A useful way to avoid confusing causes is to change one input at a time, then combine the separate effects if the problem gives a model that allows it. In a linear demand function, the coefficient on each income variable determines its modeled effect, while the price coefficient determines the change in quantity demanded when price changes along a given schedule.

Worked Example: Comparing Three Changes in a Demand Function

A hypothetical monthly demand function for a household appliance is $Q_d=120-4P+2Y_c+Y_f^e$. Quantity is in appliances, price is in dollars, and both current income and expected future income are in thousands of dollars per month. Initially, $Y_c=3.5$ and $Y_f^e=5$. Compare the initial quantity at prices of $15 and $17 with (a) a $2,000 increase in expected future income only, (b) a $2,000 increase in current income only, and (c) both increases together.

Method. At $P=15$, the initial quantity is $120-4(15)+2(3.5)+5=120-60+7+5=72$ appliances. At $P=17$, it is $120-4(17)+7+5=120-68+7+5=64$ appliances. The price increase from $15 to $17 reduces quantity by $4(2)=8$ appliances on the initial demand curve.

(a) With expected future income increased by 2 and current income unchanged, the expected-income term rises from 5 to 7. At $P=15$, demand is $120-60+7+7=74$; at $P=17$, it is $120-68+7+7=66$. This expectation change adds 2 appliances at each price.

(b) With current income increased by 2 and expected future income unchanged, the current-income term rises from $2(3.5)=7$ to $2(5.5)=11$. At $P=15$, demand is $120-60+11+5=76$; at $P=17$, it is $120-68+11+5=68$. This current-income change adds 4 appliances at each price.

(c) With both income changes, at $P=15$ demand is $120-60+11+7=78$; at $P=17$, it is $120-68+11+7=70$. Check: the combined increase over the initial schedule is $4+2=6$ appliances at either price, and the $2 price increase still reduces quantity by 8 on each schedule. The common mistake avoided is treating the income effects as interchangeable: the function assigns different coefficients to current and expected income.

Why the Effect Is Not Automatic

The calculations above show what follows from a stated model, not a universal prediction about every market. Higher expected income can increase current demand if buyers expect to have more resources and the good is one they buy more of as resources rise. They might bring a planned purchase forward, or feel more able to commit to a payment. A lower expected income can work in the opposite direction if buyers become cautious and postpone spending.

The direction may differ for a particular good. Buyers might respond to improved income prospects by switching away from a good they regard as inferior. They might also save most of an anticipated increase rather than spend it on the good under study. And a forecast that is highly uncertain may have a smaller effect than an equally sized change buyers consider reliable. The analyst should therefore identify the relevant assumption about buyer response rather than infer a direction from the word “income” alone.

A second important distinction is between demand and quantity demanded. A change in expected future income changes the demand relationship: at each given current price, the quantity buyers are willing and able to purchase may change. A change in the good’s own current price changes quantity demanded along whichever demand relationship applies. If both price and expectations change, calculate the effects separately or substitute all new values into the specified function.

Key takeaway: A change in expected future income can shift current demand even when current income and price have not changed. The direction and size of the shift depend on how buyers respond and on the stated model. Always hold price fixed when identifying a shift, and keep expected income distinct from income already received.

Common Exam Traps

Calling an expectation a current-income change. A forecast about next year is not income received today. Use the variable the problem specifies.
Calling a demand shift movement along demand. If the current price is unchanged and expected income changes, compare quantities at the same price on the old and new schedules.
Assuming the effect must be positive. A positive response follows only when the relevant good and model have a positive expected-income effect.
Using a possible future income as if it were the expected value. When probabilities are given and the model uses expected income, calculate the probability-weighted average first.
Applying a coefficient without checking units. If income is measured in thousands of dollars, an increase of $2,000 is an input change of 2, not 2,000.

This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, keep current price fixed when deciding whether the change is a shift or movement along demand.

A hypothetical demand function is $Q_d=70-2P+Y_c+0.5Y_f^e$, with incomes in thousands of dollars and quantity per month. Current income is 3. Expected future income rises from 8 to 10. Find demand before and after at $P=12$, and state the change. (Answer: 53 before, 54 after; demand rises by 1 unit per month, a rightward shift.)
A hypothetical quarterly demand function is $Q_d=64-2P+0.5Y_f^e$, with expected annual income in thousands of dollars. The previous forecast was 38. A new forecast assigns probability 0.4 to income of 30 and probability 0.6 to income of 50. Find the new expected income and compare demand at $P=15$. (Answer: new expected income is 42; demand rises from 53 to 55 places per quarter.)
Use $Q_d=90-3P+Y_c+0.5Y_f^e$, with incomes in thousands of dollars. At $P=10$, initial incomes are $Y_c=4$ and $Y_f^e=6$. Current income rises by 1.5 and expected future income rises by 2. Find initial and new demand. (Answer: 67 initially and 69.5 after both changes; the increase is 2.5 units.)
A hypothetical demand function is $Q_d=150-5P+2Y_f^e$, with expected income in thousands of dollars. Expected income rises from 40 to 43. Find the increase in demand at any fixed price. If initial demand at $P=20$ was 130, find the new quantity at that price and the price on the new curve that would yield 130 units. (Answer: demand rises by 6 units; new quantity at price 20 is 136; the price yielding 130 is $21.20.)
A demand function is $Q_d=80-2P+0.4Y_c+0.6Y_f^e$, with incomes in thousands of dollars. At $P=20$, current and expected future incomes are initially 10 and 20. Current income then rises by 5 while expected future income falls by 2. Find the initial and final quantities and the net demand shift at that price. (Answer: 56 initially and 56.8 finally; quantity demanded rises by 0.8 unit, so the combined changes shift demand right.)

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