Why Quasilinear Utility Simplifies Consumer Choice
In many consumer-choice problems, income affects how much of each good a person buys. Quasilinear utility is a useful special case: it separates the utility from one good from the utility supplied by another good that enters utility linearly. At an interior optimum, the consumer can first determine the quantity of the nonlinear good from its marginal utility and relative price, then use the budget to calculate the quantity of the linear good.
The key is not to mistake this simplification for a general rule about preferences. The result depends on the utility form and on having a feasible interior solution. If the calculated quantity of the linear good is negative, for example, the interior calculation is not a feasible bundle.
The linear good has constant marginal utility: \(MU_y=1\). The nonlinear good has marginal utility \(MU_x=v'(x)\), which changes as its quantity changes. At an interior optimum, marginal utility per dollar is equal across the goods. Equivalently, the marginal rate of substitution, \(MU_x/MU_y\), equals the price ratio \(p_x/p_y\). Since \(MU_y=1\), the condition becomes \(v'(x)=p_x/p_y\).
Deriving the Interior Choice
The budget binds because utility rises with \(y\): if the consumer left some income unspent, spending a little more on \(y\) would increase utility. Thus, for an interior bundle, \(p_xx+p_yy=m\), or \(y=(m-p_xx)/p_y\). Substitute this expression into utility. The consumer’s problem becomes choosing \(x\) to maximize a one-variable expression.
Differentiating this expression with respect to \(x\) gives \(v'(x)-p_x/p_y\). At an interior maximum, that derivative is zero. Strict concavity, \(v''(x)<0\), makes the substituted objective strictly concave, so any feasible solution to the first-order condition is its unique maximum. These steps justify the result and also show why the income term does not affect the condition for \(x\).
Calculate \(p_x/p_y\), the marginal utility per dollar required at an interior optimum.
Set \(v'(x)=p_x/p_y\) and solve for \(x^*\). Strict concavity supports a unique maximizing quantity.
Calculate \(y^*=(m-p_xx^*)/p_y\). Check that both quantities are positive and that spending equals income.
Worked Applications
Worked Example: Logarithmic Utility and the Budget Check
A consumer has utility \(U(x,y)=10\ln(1+x)+y\), income \(m=30\), and prices \(p_x=2\) and \(p_y=1\). Assume \(x,y\geq 0\). Find the interior optimum, verify its feasibility, and check the marginal condition.
Method. The nonlinear part is \(v(x)=10\ln(1+x)\), so \(v'(x)=10/(1+x)\). The relative price is \(p_x/p_y=2/1=2\). The interior condition is \(10/(1+x)=2\), so \(10=2(1+x)\), \(10=2+2x\), and \(x^*=4\).
Use the budget. The spending on \(x\) is \(p_xx^*=2(4)=8\). The amount left for \(y\) is \(30-8=22\), and because \(p_y=1\), \(y^*=22/1=22\). Both quantities are positive, so this candidate is interior.
Check. Total spending is \(2(4)+1(22)=8+22=30=m\). Also, \(v'(4)=10/(1+4)=2=p_x/p_y\). The candidate satisfies both the budget and the interior marginal condition.
Common mistake avoided. Inverting the price ratio would set \(10/(1+x)=1/2\), producing \(x=19\). That would cost \(2(19)=38\), already more than the income of 30, so the resulting bundle cannot be the consumer’s choice.
Worked Example: Square-Root Utility With a Nonunit Price for the Linear Good
Suppose \(U(x,y)=9\sqrt{x}+y\), income is \(m=60\), and the prices are \(p_x=3\) and \(p_y=2\). Assume nonnegative consumption and look for an interior solution. Find the bundle and verify the condition.
Method. Here \(v(x)=9\sqrt{x}\), so for \(x>0\), \(v'(x)=9/(2\sqrt{x})=4.5/\sqrt{x}\). The relative price is \(p_x/p_y=3/2=1.5\). Set marginal utility equal to that price ratio: \(4.5/\sqrt{x}=1.5\). Dividing both sides by 1.5 gives \(3/\sqrt{x}=1\), so \(\sqrt{x}=3\) and \(x^*=9\).
Use the budget. The expenditure on \(x\) is \(3(9)=27\), leaving \(60-27=33\) for \(y\). Since each unit of \(y\) costs 2, \(y^*=33/2=16.5\).
Check. Both quantities are positive. Spending is \(3(9)+2(16.5)=27+33=60\). The marginal utility at \(x=9\) is \(4.5/\sqrt{9}=4.5/3=1.5\), equal to \(p_x/p_y=1.5\). The interior bundle is \(x^*=9,\ y^*=16.5\).
Common mistake avoided. The budget amount left after buying \(x\) is 33 dollars, not 33 units of \(y\). Dividing by \(p_y=2\) is necessary to convert the remaining dollars into units.
Income and the Linear Good
The first-order condition \(v'(x^*)=p_x/p_y\) contains prices but not income. Therefore, with prices fixed, a change in income does not change \(x^*\) as long as the consumer continues to have a feasible interior solution. The budget then assigns the change in income to \(y\): an additional unit of income increases \(y^*\) by \(1/p_y\). This is a property of the quasilinear form, not a general claim that income never affects demand.
The condition has an economic interpretation. If \(v'(x)\) exceeds \(p_x/p_y\), an additional small amount of \(x\), paid for by reducing \(y\), raises utility. If \(v'(x)\) is below \(p_x/p_y\), reducing \(x\) and using the released spending to buy \(y\) raises utility. At an interior optimum, neither small reallocation improves utility, so the two marginal utilities per dollar are equal.
Worked Example: Income Changes While Prices Stay Fixed
A consumer has \(U(x,y)=24x-x^2+y\), with \(x,y\geq 0\), and faces prices \(p_x=8\), \(p_y=2\). Compare the interior optimum at income \(m=100\) with the optimum at income \(m=120\). Assume \(v(x)=24x-x^2\) is the nonlinear component.
Method. Differentiate the nonlinear component: \(v'(x)=24-2x\). The relative price is \(8/2=4\). The first-order condition is \(24-2x=4\), so \(2x=20\) and \(x^*=10\). The second derivative is \(v''(x)=-2<0\), consistent with strict concavity.
At income 100. Spending on \(x\) is \(8(10)=80\), leaving \(100-80=20\) dollars for \(y\). Thus \(y^*=20/2=10\). Total spending is \(80+2(10)=100\).
At income 120. The prices have not changed, so the marginal condition still gives \(x^*=10\). Spending on \(x\) remains 80, leaving \(120-80=40\) dollars for \(y\). Thus \(y^*=40/2=20\). Total spending is \(80+2(20)=120\).
Check and interpretation. In each bundle, \(v'(10)=24-20=4=p_x/p_y\). Increasing income by 20 raises \(y\) by \(20/2=10\) units while leaving \(x\) unchanged. The calculation is valid at both incomes because the resulting quantities of both goods are positive.
Feasibility, Scope, and Exam Traps
The interior equations are a conditional result, not a complete solution method for every quasilinear problem. Always calculate \(y^*\) after solving for \(x^*\). If \(m-p_xx^*\leq 0\), then the proposed bundle has no positive quantity of the linear good, and the interior assumptions fail. Similarly, a candidate \(x^*\leq 0\) is not an interior quantity of \(x\). In those cases, the interior formula alone is insufficient to identify the optimum; it should not be reported as an interior bundle.
A frequent exam error is to use the reciprocal price ratio. The correct equality is \(MU_x/MU_y=p_x/p_y\). Here \(MU_y=1\), so \(v'(x)=p_x/p_y\). A useful check is to ask whether the units make sense: \(v'(x)\) is marginal utility per unit of \(x\), while \(p_x/p_y\) is the cost of one unit of \(x\) measured in units of \(y\). Reversing the ratio reverses the relative cost.
Another error is to treat the budget as a direct formula for \(x\). The budget determines the tradeoff between the goods, but the marginal condition selects \(x^*\). Only then does the budget determine \(y^*\). This two-part calculation is the central practical advantage of quasilinear utility.
Check Your Understanding
For each problem, use the interior condition first, then use the budget and verify the result.
- A consumer has \(U(x,y)=8\ln(1+x)+y\), income \(m=24\), \(p_x=2\), and \(p_y=1\). Find the interior bundle. (Answer: \(x^*=3,\ y^*=18\).)
- A consumer has \(U(x,y)=10\sqrt{x}+y\), income \(m=40\), \(p_x=5\), and \(p_y=2\). Find the interior bundle and verify the budget. (Answer: \(x^*=4,\ y^*=10\); spending is \(5(4)+2(10)=40\).)
- Let \(U(x,y)=30\ln(1+x)+y\), \(p_x=5\), \(p_y=1\), and \(m=70\). Find the interior bundle and check the marginal condition. (Answer: \(x^*=5,\ y^*=45\); \(30/(1+5)=5=p_x/p_y\).)
- Let \(U(x,y)=16x-x^2+y\), \(p_x=4\), \(p_y=2\), and \(m=60\). Find the interior bundle and verify strict concavity and budget balance. (Answer: \(x^*=7,\ y^*=16\); \(v''(x)=-2<0\), and spending is \(4(7)+2(16)=60\).)
- A consumer has \(U(x,y)=12\sqrt{x}+y\), with \(p_x=2\) and \(p_y=1\). Compare the interior bundles at incomes \(m=25\) and \(m=35\). (Answer: \(x^*=9\) at both incomes; \(y^*=7\) when \(m=25\), and \(y^*=17\) when \(m=35\).)