Quasilinear Utility and Income Effects

Introduction

For many preferences, higher income changes the quantities a consumer chooses of several goods. Quasilinear preferences have a distinctive implication: when the consumer can afford an interior bundle, additional income is spent entirely on one good, while the chosen quantity of the other good stays fixed. This is a statement about demand at fixed prices, not a claim that the consumer’s choices never respond to prices.

The good whose quantity is fixed as income changes is called the non-numeraire good. The other good, which enters utility linearly, is the numeraire. The result holds only while the optimum remains interior. If income is too low to afford the interior quantity, a budget-boundary choice may result instead.

Definition: Preferences are quasilinear in goods $x$ and $y$ if utility can be represented as $U(x,y)=v(x)+y$, where $x$ is the non-numeraire good and $y$ is the numeraire. Assume $x\geq 0$, $y\geq 0$, and $v$ is differentiable and strictly concave over the relevant range. For the interior-demand result below, assume $v$ is increasing and that the equation $v'(x)=p_x/p_y$ has a positive solution.

Deriving Interior Demand

Let the consumer have income $m$ and face positive prices $p_x$ and $p_y$. The budget constraint is $p_xx+p_yy\leq m$. Since utility rises with $y$, the budget is fully spent at an optimum: if any income were unspent, the consumer could increase $y$ and raise utility. Thus $y=(m-p_xx)/p_y$, for a feasible $x$ satisfying $0\leq x\leq m/p_x$.

Substitute this expression for $y$ into utility. The choice of $x$ then maximizes $v(x)+(m-p_xx)/p_y$. The income term $m/p_y$ adds the same amount to the objective for every feasible $x$; it does not affect which $x$ maximizes it. For an interior choice, the first-order condition equates marginal utility from $x$ with its price relative to the numeraire:

\frac{d}{dx}\left[v(x)+\frac{m-p_xx}{p_y}\right] =v'(x)-\frac{p_x}{p_y}=0, \qquad v'(x)=\frac{p_x}{p_y}.

Strict concavity makes the objective strictly concave in $x$. Therefore, if the first-order condition has a solution in the feasible interval, that solution is the unique global maximizer, not merely a candidate stationary point. Denote it by $x^*$. It depends on the shape of $v$ and the relative price $p_x/p_y$, but not on income. The corresponding numeraire quantity is determined by the budget.

x^*= \text{the solution to }v'(x^*)=\frac{p_x}{p_y}, \qquad y^*=\frac{m-p_xx^*}{p_y}.

The interior solution requires $x^*>0$ and $y^*>0$, so its income condition is $m>p_xx^*$. At equality, the same quantity of $x$ exhausts the budget and $y^*=0$, which is a boundary bundle, not an interior one. If income is below this threshold, the interior bundle is unaffordable and the constraint $y\geq0$ matters.

Theorem: Under the assumptions above, if $m>p_xx^*$, quasilinear demand for the non-numeraire good is $x^*$, where $v'(x^*)=p_x/p_y$. At fixed prices, changing income while preserving $m>p_xx^*$ changes $y^*$ by $\Delta m/p_y$ and leaves $x^*$ unchanged.

Why Income Does Not Change Interior Demand

The result can be seen directly from the maximization problem. After substituting the binding budget, the part of the objective that varies with $x$ is $v(x)-(p_x/p_y)x$. Income appears only in the separate constant $m/p_y$. Adding a constant to every feasible value of an objective cannot change its maximizer. Once $x^*$ is chosen, extra income can be used to buy more $y$, whose marginal utility is constant at one.

Equivalently, for two income levels $m_1$ and $m_2$, suppose both satisfy $m_j>p_xx^*$. At the same prices, both consumers solve the same first-order condition, so they choose the same $x^*$. Their numeraire quantities differ by $\left[(m_2-p_xx^*)-(m_1-p_xx^*)\right]/p_y=(m_2-m_1)/p_y$. This reasoning does not say that demand for $x$ is independent of its price: a price change alters the right-hand side of the first-order condition.

Worked Example: Interior Demand With Log Utility

A consumer has utility $U(x,y)=18\ln(x+1)+y$, income $m=\$27$, and prices $p_x=\$3$ and $p_y=\$1$. Find the optimal bundle. Then find the bundle if income rises to $\$42$, with prices unchanged.

Method. Substitute the budget into utility and solve the interior first-order condition. Check that each income level can afford the resulting $x$-quantity.

Step 1: Find the candidate quantity of $x$. Here $v(x)=18\ln(x+1)$, so $v'(x)=18/(x+1)$. The relative price is $p_x/p_y=3/1=3$. Therefore,

\frac{18}{x+1}=3 \quad\Longrightarrow\quad 18=3(x+1) \quad\Longrightarrow\quad x^*=5.

Step 2: Check affordability and calculate $y$ at income $\$27$. Spending on $x$ is $p_xx^*=3(5)=\$15$, which is less than $\$27$. Thus $y^*=(27-15)/1=12$.

Step 3: Recalculate at income $\$42$. The first-order condition and prices have not changed, so $x^*=5$. Then $y^*=(42-15)/1=27$.

Check. Both bundles satisfy the budget: $3(5)+1(12)=27$ and $3(5)+1(27)=42$. In each case $x>0$ and $y>0$, so both are strictly interior. The increase of $\$15$ in income raises $y$ by $15$ units and leaves $x$ unchanged. The common mistake this avoids is allocating a fixed share of income to $x$: that rule does not follow from quasilinear utility.

Corner Choices When Income Is Limited

The interior formula cannot be used without checking the budget. If $m<p_xx^*$, the quantity $x^*$ cannot be purchased while keeping $y\geq0$. Under the stated assumptions, the consumer then chooses the largest feasible $x$, namely $m/p_x$, and spends nothing on $y$. To see why, along the budget boundary the derivative of utility with respect to $x$ is $v'(x)-p_x/p_y$. Since $v'$ is decreasing and $x<x^*$, this derivative is positive: increasing $x$ toward $x^*$ raises utility. But with insufficient income, the consumer cannot reach $x^*$, so the best feasible choice is $x=m/p_x,\ y=0$. At $m=p_xx^*$, the same choice has $y=0$.

This corner result presumes that a positive solution $x^*$ exists and that $v'(x)>p_x/p_y$ below it. If instead marginal utility at $x=0$ is no greater than the relative price, the optimum may be $x=0$; the interior-demand equation alone does not rule that out. Always check the relevant boundary conditions as well as the first-order condition.

Worked Example: Detecting an Unaffordable Interior Bundle

A consumer has utility $U(x,y)=16\ln(x+1)+y$, income $m=\$10$, and prices $p_x=\$4$ and $p_y=\$1$. Find the optimal bundle. Then determine the bundle if income increases to $\$20$.

Method. First calculate the unconstrained interior quantity. Then compare its cost with income and, if it is unaffordable, maximize utility over the feasible budget boundary.

Step 1: Calculate the interior candidate. Since $v'(x)=16/(x+1)$ and $p_x/p_y=4$,

\frac{16}{x+1}=4 \quad\Longrightarrow\quad 16=4(x+1) \quad\Longrightarrow\quad x^*=3.

The cost of this quantity is $p_xx^*=4(3)=\$12$, greater than the available $\$10$. It is therefore not feasible with positive $y$.

Step 2: Optimize under the $\$10$ budget. The feasible quantity satisfies $0\leq x\leq10/4=2.5$. Along the budget boundary, $y=10-4x$, and the derivative of utility with respect to $x$ is $16/(x+1)-4$. At the largest feasible $x=2.5$, this derivative is $16/3.5-4=32/7-28/7=4/7>0$. It is also positive for every smaller feasible $x$, because $16/(x+1)$ decreases as $x$ rises. Utility therefore increases all the way to $x=2.5$. The optimum is $x=2.5,\ y=0$.

Step 3: Recalculate with income $\$20$. The interior quantity $x^*=3$ now costs $\$12<\$20$, so it is affordable with positive numeraire consumption. The resulting quantity is $y=(20-12)/1=8$, giving $x=3,\ y=8$.

Check. At income $\$10$, $4(2.5)+1(0)=10$; at income $\$20$, $4(3)+1(8)=20$. The first bundle is on the boundary because $y=0$, while the second is strictly interior because both quantities are positive. The mistake to avoid is reporting $x=3$ at income $\$10$: it satisfies the marginal condition but violates the budget if paired with nonnegative $y$.

Quadratic Utility and the Income Threshold

A second functional form makes the marginal-utility condition especially transparent. The following example also distinguishes spending the full budget from being at a boundary. A bundle can spend all income and still be interior if both goods are positive.

Worked Example: A Positive Interior Bundle at Two Income Levels

A consumer has utility $U(x,y)=30x-x^2+y$, income $m=\$150$, and prices $p_x=\$10$, $p_y=\$1$. Find the optimal bundle, then repeat at income $\$170$.

Method. Use the marginal-utility condition and verify strict concavity, affordability, positivity, and budget exhaustion at both incomes.

Step 1: Solve the marginal condition. Here $v'(x)=30-2x$, and $p_x/p_y=10$. Thus $30-2x=10$, so $2x=20$ and $x^*=10$. Since $v''(x)=-2<0$, utility is strictly concave in $x$, and this stationary point is the unique maximizer whenever it is feasible.

Step 2: Calculate the bundle at income $\$150$. The cost of $x^*$ is $10(10)=\$100$. The remaining $\$50$ buys $y=50/1=50$. Thus the bundle is $(x,y)=(10,50)$.

Step 3: Calculate the bundle at income $\$170$. Prices and the marginal condition are unchanged, so $x^*=10$. The remaining income is $170-100=\$70$, giving $y=70$.

Check. The first bundle uses $10(10)+1(50)=150$, and the second uses $10(10)+1(70)=170$. Both $x$ and $y$ are positive in each bundle, so both are strictly interior. Spending the full budget does not make a bundle a boundary solution; the relevant boundary here would be $y=0$ or $x=0$. The increase in income changes $y$ by 20 and leaves $x$ at 10.

Common Exam Traps and Practical Interpretation

Quasilinear preferences are useful because they separate two effects in the consumer’s calculation. At fixed prices, the marginal condition determines the interior amount of $x$, while the budget determines how much $y$ remains. This clean separation is sometimes called a zero income effect for $x$ on the interior range. It is not a statement that the consumer has no response to prices, nor does it describe the low-income corner.

Confusing the numeraire with a good that has no price. The numeraire still has price $p_y$. Its role is that utility is linear in $y$, making its marginal utility constant.
Forgetting relative prices. With general prices, the condition is $v'(x)=p_x/p_y$, not $v'(x)=p_x$ unless $p_y=1$.
Applying the interior solution without an affordability check. The bundle requires $m>p_xx^*$ for $y>0$. At equality $y=0$, and below it the interior candidate is infeasible.
Calling every budget-exhausting bundle a corner. A budget constraint binds at a standard optimum, including an interior one. A corner in this two-good setting means at least one quantity is zero.
Assuming income-independent demand means price-independent demand. A change in $p_x/p_y$ generally changes the solution to $v'(x)=p_x/p_y$.

In applications, a quasilinear approximation can be helpful when a change in income is expected to be absorbed mainly by a flexible spending category while consumption of a particular good remains stable over a range. It should not be used mechanically: the functional form, price conditions, and available income must fit the question. For an actuarial candidate, the central discipline is to derive the marginal condition, then check whether the implied bundle is feasible and truly interior.

Key takeaway: For $U(x,y)=v(x)+y$, an interior consumer chooses $x^*$ from $v'(x^*)=p_x/p_y$. The quantity is independent of income only while the consumer can afford it with $y>0$; additional income then raises $y$, not $x$. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, identify whether the stated bundle is interior or a corner, and show the marginal condition and budget check before comparing with the answer.

A consumer has $U(x,y)=18\ln(x+1)+y$, $p_x=3$, $p_y=1$, and $m=30$. Find the optimal bundle. (Answer: $x=5,\ y=15$; the bundle is strictly interior.)
A consumer has $U(x,y)=24\ln(x+2)+y$, $p_x=4$, $p_y=2$, and $m=50$. Find the optimal bundle and classify it. (Answer: $x=10,\ y=5$; the bundle is strictly interior.)
A consumer has $U(x,y)=30x-1.5x^2+y$, $p_x=9$, $p_y=3$, and $m=120$. Find the optimal bundle and verify the marginal condition. (Answer: $x=9,\ y=13$; $v'(9)=3=p_x/p_y$, and the bundle is strictly interior.)
A consumer has $U(x,y)=10\ln(x+1)+y$, $p_x=2$, $p_y=1$, and $m=6$. Find the optimum, noting whether the unconstrained interior quantity is affordable. (Answer: the interior candidate is $x=4$, costing $8$; the optimum is the corner $x=3,\ y=0$.)
A consumer has $U(x,y)=25\ln(x+1)+y$, $p_x=5$, $p_y=2$. Find the bundles at incomes $m=60$ and $m=100$, and calculate the change in $y$. (Answer: $x=9$ at both incomes; $y=7.5$ and $y=27.5$, respectively, so $y$ rises by $20$.)

What You'll Learn