Individual Demand From Utility Maximization

From Consumer Choice to Individual Demand

A consumer’s choice depends on preferences, prices, and income. Individual demand records the quantities the consumer chooses when those conditions are specified. This connects utility maximization to demand: instead of asking only which bundle is best at one set of prices, we can ask how the chosen quantity changes when one price changes and the other determinants are held fixed. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

The key distinction is between a choice at particular conditions and a demand function across conditions. A solution to one utility-maximization problem gives a bundle. To obtain demand as a function of a price, solve the choice problem while treating that price as a variable and keeping income, preferences, and all other prices fixed.

Definition: Given a preference relation represented by utility $u(x_1,\ldots,x_n)$, a price vector $p=(p_1,\ldots,p_n)$ with positive prices, and income $m\geq 0$, the budget set is the set of nonnegative bundles affordable at those conditions: $$ B(p,m)=\left\{(x_1,\ldots,x_n): x_i\geq 0\text{ for each }i,\quad \sum_{i=1}^n p_i x_i\leq m\right\}. $$ The Marshallian demand correspondence is the set of utility-maximizing bundles in that budget set: $$ x(p,m)\in\operatorname*{arg\,max}_{x\in B(p,m)}u(x). $$ If there is exactly one maximizing bundle for each specified price vector and income, the correspondence is a Marshallian demand function. Its $i$th component, $x_i(p,m)$, is the consumer’s chosen quantity of good $i$.

The word “correspondence” matters: preferences and prices do not always produce a unique maximizing bundle. When several bundles tie for the highest utility, demand is set-valued unless an additional selection rule is specified. Also, the utility representation need not measure happiness in an absolute sense. The maximization uses the ranking of bundles, so any strictly increasing transformation of utility represents the same preferences and yields the same maximizing choices.

With positive prices, the budget set is bounded and closed. If utility is continuous, a maximum exists on this set. If preferences are also locally nonsatiated, a maximizing consumer spends all available income, so the budget constraint binds. Without local nonsatiation, an optimum need not exhaust income; the inequality in the budget-set definition remains important.

How a Price Change Generates Demand

For good $x$, write its demand as $x(p_x,p_y,m)$ when there are two goods. To study how its own price relates to the chosen quantity, vary $p_x$ while holding $p_y$, $m$, and the preference representation fixed. At every value of $p_x$, solve the consumer’s maximization problem again. The resulting choices define demand in terms of that price. The choices of other goods may change too, because the consumer is still choosing a whole bundle.

A useful way to organize the calculation is to distinguish the optimization problem from the demand result. First specify the budget set and find its maximizing bundle at arbitrary prices. Then substitute particular prices and income to calculate choices, or compare choices across price values. For a smooth interior optimum, first-order conditions can help find the bundle; they must be accompanied by appropriate checks that the solution is feasible and is indeed optimal.

Specify the choice problem.
Write preferences, nonnegativity restrictions, prices, and income. The consumer chooses only from bundles in the budget set.

Find the maximizing bundle.
Use the method suited to the problem, such as first-order conditions for an interior choice or a comparison of feasible boundary choices.

Check the solution.
Verify feasibility and optimality. If several bundles attain the maximum, report the full set of choices rather than asserting a unique quantity.

Vary one price.
Keep income, preferences, and the other prices fixed, and solve again. The selected quantity of the good at each price is its Marshallian demand.

Worked Examples

Worked Example: Finding an Interior Choice

A consumer has utility $u(x,y)=2\sqrt{x}+\sqrt{y}$, where $x,y\geq 0$. The prices are $p_x=2$ and $p_y=1$, and income is $m=18$. Find the Marshallian choice. State why the method gives a global maximum, not just a candidate.

Method. Use the budget constraint and the interior first-order condition, then check feasibility and optimality. The budget set is $2x+y\leq18$. Utility is strictly increasing in both goods, so the consumer spends all income at an optimum: $2x+y=18$.

At an interior optimum, the marginal rate of substitution $MU_x/MU_y$ equals $p_x/p_y$. Here $MU_x=1/\sqrt{x}$ and $MU_y=1/(2\sqrt{y})$, so $MU_x/MU_y=2\sqrt{y/x}$. Thus the condition is $2\sqrt{y/x}=2/1=2$, which gives $\sqrt{y/x}=1$ and $y=x$.

Substitute $y=x$ into the budget equation: $2x+x=18$, so $3x=18$, $x=6$, and $y=6$. The bundle is feasible and has strictly positive quantities. The utility function is strictly concave on the nonnegative domain, so a feasible interior stationary point is the unique global maximum. Utility there is $2\sqrt{6}+\sqrt{6}=3\sqrt{6}$, approximately $7.348$.

Check. Expenditure is $2(6)+1(6)=18$, exactly the income. The first-order condition holds because the marginal-utility ratio is $2\sqrt{6/6}=2=p_x/p_y$. A common mistake is to reverse the price ratio; the correction is to use $MU_x/MU_y=p_x/p_y$ with the marginal utilities in the same order as the goods.

Worked Example: Varying One Price to Obtain Demand

Use $u(x,y)=2\sqrt{x}+\sqrt{y}$ again, now with income $m=18$ and the price of $y$ fixed at $p_y=1$. Find the demand for $x$ as a function of $p_x>0$, and calculate the chosen quantities when $p_x=1,2,$ and $3$.

Method. Solve the interior problem with $p_x$ left as a variable. At an interior optimum the marginal-utility ratio gives $2\sqrt{y/x}=p_x$, so $y=p_x^2x/4$. Since utility is strictly increasing, the budget binds: $p_xx+y=18$. Substitution gives $(p_x+p_x^2/4)x=18$. Therefore $x=18/(p_x+p_x^2/4)$, and $y=p_x^2x/4$.

For $p_x=1$, the denominator for $x$ is $1+1/4=5/4$. Thus $x=18/(5/4)=72/5=14.4$, and $y=14.4/4=3.6$. Expenditure is $1(14.4)+1(3.6)=18$.

For $p_x=2$, the denominator is $2+4/4=3$. Thus $x=18/3=6$, and $y=4(6)/4=6$. Expenditure is $2(6)+1(6)=18$.

For $p_x=3$, the denominator is $3+9/4=21/4$. Thus $x=18/(21/4)=24/7$, and $y=9(24/7)/4=54/7$. Expenditure is $3(24/7)+54/7=126/7=18$. To three decimal places, these choices are respectively $x=14.400, 6.000, 3.429$ and $y=3.600, 6.000, 7.714$.

The derived demand for $x$, with the stated income, preferences, and $y$-price fixed, is $x(p_x)=18/(p_x+p_x^2/4)$. The decreasing choices across these three prices are a result for this specified problem, not a claim that every individual demand function must have the same shape. The strict concavity and positive candidate quantities ensure the interior solution is the unique global maximum at each of these prices.

Check. In every case the quantities are nonnegative, the budget binds, and the marginal-utility ratio equals the price ratio. The method avoids the mistake of changing income or the other good’s price along with $p_x$; doing so would compare different demand conditions.

Worked Example: Boundary Choice and Multiple Demands

A consumer has utility $u(x,y)=x+2y$, income $m=12$, and prices $p_x=2$, $p_y=3$. Find the maximizing bundle. Then suppose only $p_y$ changes to $4$. Describe demand for $y$ at the new price.

Method. Compare utility gained per dollar. At the original prices, one dollar spent on $x$ buys $1/2$ unit and yields utility $1/2$; one dollar spent on $y$ buys $1/3$ unit and yields utility $2/3$. Because $y$ yields more utility per dollar, the consumer spends all income on $y$: $x=0$, $y=12/3=4$. The resulting utility is $0+2(4)=8$, and expenditure is $3(4)=12$.

When $p_y=4$, both goods yield $1/2$ unit of utility per dollar: $1/2$ for $x$ and $2/4=1/2$ for $y$. Every bundle on the budget line $2x+4y=12$ is optimal. Dividing that equation by $2$ gives $x+2y=6$, which is exactly the utility of every such budget-line bundle. For $y$, the possible maximizing quantities range from $0$ to $3$, with $x=6-2y$.

Check. At the new prices, the endpoints $(6,0)$ and $(0,3)$ both cost $12$ and both give utility $6$; every feasible mixture between them does too. Demand is therefore a correspondence, not one unique quantity. A common mistake is to claim the utility equals expenditure: here the budget equation implies $x+2y=6$, while expenditure is $2x+4y=12$.

Why the Demand Definition Matters

The definition keeps three ideas separate. Preferences determine how bundles are ranked; the budget set determines which bundles are affordable; and utility maximization selects the best affordable bundle or bundles. A change in one price changes the budget set and can therefore change the selected bundle even when preferences and income are unchanged. This is the basis for describing a chosen quantity as a function of price.

The condition $MU_x/MU_y=p_x/p_y$ is useful only for an interior choice where the relevant derivatives exist. It is not a universal demand formula. At a boundary, the consumer may choose zero of a good, so equality of the marginal-utility ratio and price ratio need not hold. With ties, as in the last example, the maximizing choice may not be unique. The earlier material on boundary optimization and consumer choice helps identify which case applies before interpreting a first-order condition.

Common exam trap: reversing the MRS-price ratio. In the first example, the correct condition is $2\sqrt{y/x}=p_x/p_y=2$, giving $y=x$. If the price ratio is incorrectly reversed, the equation becomes $2\sqrt{y/x}=1/2$, giving $y=x/16$. Combining that incorrect relation with the budget gives $2x+x/16=18$, so $x=96/11$ and $y=6/11$. Although this bundle exhausts income, it is not optimal: its marginal-utility ratio is $2\sqrt{(6/11)/(96/11)}=1/2$, which does not equal the actual price ratio $2$. Budget feasibility alone does not establish utility maximization.

When reporting an individual demand result, state the good, the conditions held fixed, and whether the maximizing quantity is unique. This makes clear what the result means and prevents a one-time bundle calculation from being mistaken for a demand function.

Check Your Understanding

For each problem, identify the maximizing choice or demand set and verify it against the stated budget and preferences.

A consumer has $u(x,y)=2\sqrt{x}+\sqrt{y}$, income $10$, and prices $p_x=p_y=1$. Find the interior maximizing bundle. (Answer: $x=8,\ y=2$.)
A consumer has $u(x,y)=3x+y$, income $8$, and prices $p_x=2,\ p_y=1$. Find the maximizing bundle. (Answer: $x=4,\ y=0$.)
A consumer has $u(x,y)=x+2y$, income $20$, and prices $p_x=2,\ p_y=4$. Describe all maximizing bundles and the range of demand for $y$. (Answer: Every bundle satisfying $x+2y=10$ is optimal; $0\leq y\leq5$.)
A consumer has $u(x,y)=2\sqrt{x}+\sqrt{y}$, income $21$, $p_y=1$, and $p_x=3$. Find the maximizing quantities and check total expenditure. (Answer: $x=4,\ y=9$; expenditure is $3(4)+9=21$.)
For $u(x,y)=2\sqrt{x}+\sqrt{y}$, income $12$, and $p_y=2$, compare demand for $x$ when $p_x=1$ and when $p_x=2$. (Answer: At $p_x=1$, $(x,y)=(32/3,2/3)$; at $p_x=2$, $(x,y)=(24/5,6/5)$.)

What You'll Learn