A Worked Interior Choice with Square-Root Utility

Why Work Through This Utility Function?

An optimality condition is most useful when you can turn it into an actual bundle and then check that bundle against the budget. For the utility function $U(x,y)=\sqrt{x}+\sqrt{y}$, the marginal utilities have a simple form, but the quantities demanded move in an initially surprising direction: the consumer buys more of a good when its price is lower. This tutorial derives the quantities, explains why the solution is an interior maximum, and shows how to verify a calculation.

The Choice Model and Its Conditions

Model: A consumer chooses quantities $x$ and $y$ to maximize $U(x,y)=\sqrt{x}+\sqrt{y}$, subject to $p_xx+p_yy\leq m$. Quantities must be nonnegative, prices $p_x$ and $p_y$ are strictly positive, and income $m$ is strictly positive. The goods are divisible, and the consumer may spend no more than the available income.

Each additional unit of either good raises utility, so any bundle leaving some income unspent can be improved by using part of it to buy more of a good. Therefore the budget binds at the optimum: $p_xx+p_yy=m$. In this model, the optimal bundle is also interior: both quantities are positive. At a bundle with one quantity equal to zero, a small purchase of that good adds utility in proportion to the square root of the purchase, while reducing the positive quantity of the other good by the same spending amount reduces utility only in proportion to that amount. For sufficiently small purchases, the gain is larger than the loss.

An interior choice can be found using the equal-marginal-utility-per-dollar condition from the earlier material on consumer choice. The marginal utility of a good measures the extra utility from a small increase in its quantity. Dividing marginal utility by price gives the marginal utility obtained per dollar spent on that good.

Deriving the Optimal Quantities

Differentiate utility with respect to each quantity. For positive $x$ and $y$, the marginal utilities are:

MU_x=\frac{1}{2\sqrt{x}}, \qquad MU_y=\frac{1}{2\sqrt{y}}.

At an interior optimum, the two marginal utilities per dollar must be equal. Otherwise, moving a small amount of spending from the good with lower marginal utility per dollar to the good with higher marginal utility per dollar would raise utility.

\frac{MU_x}{p_x}=\frac{MU_y}{p_y} \quad\Longrightarrow\quad \frac{1}{2p_x\sqrt{x}}=\frac{1}{2p_y\sqrt{y}} \quad\Longrightarrow\quad p_y\sqrt{y}=p_x\sqrt{x}.

The last equality implies $y=(p_x^2/p_y^2)x$. Substituting this relationship into the binding budget gives an equation in one unknown:

p_xx+p_y\left(\frac{p_x^2}{p_y^2}x\right)=m \quad\Longrightarrow\quad x\left(p_x+\frac{p_x^2}{p_y}\right)=m \quad\Longrightarrow\quad x^*=\frac{mp_y}{p_x(p_x+p_y)}.

Substituting $x^*$ into the relationship between $x$ and $y$ gives the second quantity:

y^*=\frac{p_x^2}{p_y^2}x^* =\frac{mp_x}{p_y(p_x+p_y)}.

Result: For positive prices and income, the utility-maximizing bundle is $$ x^*=\frac{mp_y}{p_x(p_x+p_y)}, \qquad y^*=\frac{mp_x}{p_y(p_x+p_y)}. $$ The formula gives quantities, not expenditure amounts. The corresponding expenditures are $p_xx^*=mp_y/(p_x+p_y)$ and $p_yy^*=mp_x/(p_x+p_y)$.

These formulas satisfy the budget exactly:

p_xx^*+p_yy^* =\frac{mp_y}{p_x+p_y}+\frac{mp_x}{p_x+p_y} =m.

They also satisfy the marginal condition. Taking square roots of the two quantities gives $\sqrt{x^*}=\sqrt{mp_y/[p_x(p_x+p_y)]}$ and $\sqrt{y^*}=\sqrt{mp_x/[p_y(p_x+p_y)]}$. Their ratio is $\sqrt{x^*/y^*}=p_y/p_x$, so $p_x\sqrt{x^*}=p_y\sqrt{y^*}$, as required.

The condition identifies the interior candidate; concavity explains why it is the global solution. The square-root function is strictly concave on nonnegative quantities. Thus their sum is strictly concave, and the set of bundles satisfying the linear budget is convex. A strictly concave objective has at most one maximum on that feasible set. Since the formulas give a positive, affordable bundle satisfying the interior condition, that bundle is the unique maximum.

Worked Applications

Worked Example: A Lower-Priced Second Good

A consumer has $100$ dollars to spend on goods $x$ and $y$. Their prices are $p_x=\$1$ and $p_y=\$4$ per unit, and utility is $U(x,y)=\sqrt{x}+\sqrt{y}$. Find the optimal bundle, check its cost, and verify the marginal condition.

Method. Apply the derived quantity formulas with $m=100$, $p_x=1$, and $p_y=4$:

x^*=\frac{100(4)}{1(1+4)}=\frac{400}{5}=80, \qquad y^*=\frac{100(1)}{4(1+4)}=\frac{100}{20}=5.

Affordability check. The consumer spends $1(80)=\$80$ on $x$ and $4(5)=\$20$ on $y$. Total spending is $\$80+\$20=\$100$, equal to income.

Marginal-condition check. The marginal utility per dollar for $x$ is $1/[2(1)\sqrt{80}]=1/(8\sqrt{5})$. For $y$, it is $1/[2(4)\sqrt{5}]=1/(8\sqrt{5})$. The values are equal. Utility at the bundle is $\sqrt{80}+\sqrt{5}=4\sqrt{5}+\sqrt{5}=5\sqrt{5}$. The bundle is positive, affordable, and satisfies the optimality condition.

Worked Example: Verifying a Bundle With Unequal Prices

A consumer has $240$ dollars and faces prices $p_x=\$3$ and $p_y=\$12$ per unit. With utility $U(x,y)=\sqrt{x}+\sqrt{y}$, find the optimal quantities and show that neither the budget nor the marginal condition is being misread.

Method. Substitute the stated income and prices into the two formulas:

x^*=\frac{240(12)}{3(3+12)}=\frac{2880}{45}=64, \qquad y^*=\frac{240(3)}{12(3+12)}=\frac{720}{180}=4.

Budget check. The expenditure on $x$ is $3(64)=\$192$. The expenditure on $y$ is $12(4)=\$48$. Total expenditure is $\$192+\$48=\$240$, so the bundle is affordable and uses the full budget.

Marginal-condition check. For $x$, marginal utility per dollar is $1/[2(8)(3)]=1/48$. For $y$, it is $1/[2(2)(12)]=1/48$. The two rates match. Utility is $\sqrt{64}+\sqrt{4}=8+2=10$. In particular, the consumer buys more units of the cheaper good, but still spends more dollars on $x$ than on $y$ because the chosen quantities and the prices both determine expenditure.

Worked Example: Checking the Price Ratio and the Full Calculation

A consumer spends up to $100$ dollars on two goods with prices $p_x=\$4$ and $p_y=\$1$ per unit. Their utility is $U(x,y)=\sqrt{x}+\sqrt{y}$. Find the maximizing bundle, calculate its utility, and verify both affordability and equal marginal utility per dollar.

Method. Use the quantity formulas with $m=100$, $p_x=4$, and $p_y=1$:

x^*=\frac{100(1)}{4(4+1)}=\frac{100}{20}=5, \qquad y^*=\frac{100(4)}{1(4+1)}=\frac{400}{5}=80.

Affordability check. Spending on $x$ is $4(5)=\$20$, and spending on $y$ is $1(80)=\$80$. The total is $\$20+\$80=\$100$, exactly the available income.

Marginal-condition check. For $x$, marginal utility per dollar is $1/[2\sqrt{5}(4)]=1/(8\sqrt{5})$. For $y$, it is $1/[2\sqrt{80}(1)]=1/(8\sqrt{5})$, since $\sqrt{80}=4\sqrt{5}$. Utility is $\sqrt{5}+\sqrt{80}=5\sqrt{5}$. The expenditure shares are not the same as the quantity shares: the cheaper good has many more units, while the more expensive good costs more per unit.

Common Mistake: Reversing the Expenditure Formula

The quantities contain the other good’s price in the numerator: $x^*$ contains $p_y$, while $y^*$ contains $p_x$. A frequent mistake is to reverse this relationship, perhaps by assigning the larger expenditure share to the good with the higher own price. That does not generally equalize marginal utility per dollar.

For the first worked example, the incorrect assignment would spend $\$20$ on $x$ and $\$80$ on $y$. At prices $\$1$ and $\$4$, that produces the affordable bundle $(x,y)=(20,20)$. But its marginal utilities per dollar are:

\frac{MU_x}{p_x}=\frac{1}{2\sqrt{20}}, \qquad \frac{MU_y}{p_y}=\frac{1}{8\sqrt{20}}.

They are not equal: the marginal utility per dollar for $x$ is four times that for $y$. The correction is to use the formulas for quantities, then multiply each quantity by its own price to check spending. For the correct bundle $(80,5)$, the two marginal utilities per dollar are both $1/(8\sqrt{5})$, and the expenditures add to the full $\$100$.

Key takeaway: For square-root utility with positive prices and income, derive the bundle from equal marginal utility per dollar and the binding budget. Then check the result in two separate ways: total expenditure must equal income, and the two marginal utilities per dollar must match.

Check Your Understanding

For each problem, use the square-root utility model and verify the final bundle against the budget and the marginal condition where requested.

A consumer has income $60$, with prices $p_x=2$ and $p_y=3$. Find the maximizing quantities. (Answer: $x^*=18,\ y^*=8$.)
Income is $100$, and the prices are $p_x=1$ and $p_y=9$. Find the maximizing quantities and the resulting utility. (Answer: $x^*=90,\ y^*=10/9,\ U^*=10\sqrt{10}/3$.)
A consumer has income $140$, with $p_x=5$ and $p_y=2$. Find the bundle and state each good’s expenditure. (Answer: $x^*=8,\ y^*=50$; expenditures are $40$ on $x$ and $100$ on $y$.)
Income is $180$, and prices are $p_x=3$ and $p_y=12$. A proposed bundle is $(30,5)$. Check whether it uses the full budget and whether it satisfies the equal-marginal-utility-per-dollar condition; then give the optimum. (Answer: The proposed bundle costs $150$ and is not optimal; the optimum is $(48,3)$, costing $144+36=180$.)
A consumer has income $260$, with prices $p_x=4$ and $p_y=9$. Find the maximizing quantities and verify the total expenditure. (Answer: $x^*=45,\ y^*=80/9$; expenditure is $180+80=260$.)

What You'll Learn