Equal Marginal Utility per Dollar Across Goods

Why Compare Marginal Utility per Dollar?

A consumer with a fixed budget must decide how to divide spending among goods. The marginal utility of a good measures the additional utility from a small increase in its quantity, but quantities of different goods have different prices. The relevant comparison is therefore not just marginal utility: it is the additional utility produced by a dollar spent on each good.

If one good provides more marginal utility per dollar than another, a small shift in spending toward the first good can raise utility. At an interior optimum, where the consumer chooses a positive amount of each good, these marginal returns per dollar must be equal. This is a first-order condition: it identifies a necessary property of a smooth interior optimum, not by itself a guarantee that a bundle is a maximum.

Definition: For a differentiable utility function $U(x,y)$, the marginal utilities are $MU_x=\partial U/\partial x$ and $MU_y=\partial U/\partial y$. If the goods have positive prices $p_x$ and $p_y$, their marginal utilities per dollar are $MU_x/p_x$ and $MU_y/p_y$, respectively.

Deriving the Equal-Marginal-Utility Condition

Suppose a consumer chooses quantities $x$ and $y$ to maximize differentiable utility $U(x,y)$, subject to the budget constraint $p_xx+p_yy=m$. Assume the prices are positive, both quantities are strictly positive, the budget is exhausted, and the choice is an interior optimum. The Lagrange function is

\mathcal{L}(x,y,\lambda)=U(x,y)+\lambda(m-p_xx-p_yy).

At an interior optimum, the first-order conditions set the derivatives of this function with respect to $x$, $y$, and $\lambda$ equal to zero:

\frac{\partial U}{\partial x}-\lambda p_x=0,\qquad \frac{\partial U}{\partial y}-\lambda p_y=0,\qquad m-p_xx-p_yy=0.

The first two conditions say that the marginal utility from each good equals the price of that good multiplied by the same $\lambda$. Dividing each condition by its positive price gives the central result.

\frac{MU_x}{p_x}=\lambda \qquad\text{and}\qquad \frac{MU_y}{p_y}=\lambda, \qquad\text{so}\qquad \frac{MU_x}{p_x}=\frac{MU_y}{p_y}.

Theorem: At a differentiable interior optimum with positive prices and a binding budget, the marginal utility per dollar is equal across the goods: $MU_x/p_x=MU_y/p_y$. This is a necessary first-order condition. It need not apply at a boundary choice, and equality alone does not establish that a bundle maximizes utility.

The common value $\lambda$ is the increase in the optimized Lagrange objective associated with a one-unit increase in the budget, under the usual regularity conditions. The main comparison here does not require interpreting $\lambda$: the first-order conditions show directly that the consumer allocates spending until neither good offers a greater marginal utility per dollar than the other.

Why a Spending Shift Gives This Comparison

The same condition can be derived by considering a small budget-neutral transfer. Increase spending on $x$ by a small amount $\varepsilon$, while reducing spending on $y$ by the same amount. The corresponding changes in quantities are $\varepsilon/p_x$ and $-\varepsilon/p_y$. The first-order change in utility is therefore:

dU =MU_x\frac{\varepsilon}{p_x}-MU_y\frac{\varepsilon}{p_y} =\varepsilon\left(\frac{MU_x}{p_x}-\frac{MU_y}{p_y}\right).

For a positive, sufficiently small $\varepsilon$, if $MU_x/p_x>MU_y/p_y$, shifting spending from $y$ to $x$ raises utility to first order. If the inequality goes the other way, shifting spending from $x$ to $y$ raises utility to first order. If the ratios are equal, the first-order effect of this budget-neutral shift is zero. This does not mean that a large transfer has no effect; higher-order changes in utility may matter.

Calculate marginal utilities.
Differentiate the utility function with respect to each good, then evaluate both derivatives at the bundle being considered.

Divide by the corresponding prices.
Compare $MU_x/p_x$ with $MU_y/p_y$; do not compare marginal utilities alone when prices differ.

Interpret the direction locally.
If one ratio is larger, a sufficiently small shift of spending toward that good raises utility to first order. If they are equal, the first-order comparison is tied.

Worked Applications

Worked Example: Choosing Which Good Receives More Spending

A consumer has utility $U(x,y)=10\ln x+6\ln y$, where $x>0$ and $y>0$. The prices are $p_x=\$2$ and $p_y=\$3$. At the bundle $(x,y)=(4,2)$, the consumer spends the full $\$14$ budget. Determine the direction of a small spending shift that raises utility, then check a specific transfer of $\$0.60$.

Method. Compare marginal utility per dollar at the starting bundle. The budget is $2(4)+3(2)=8+6=\$14$, as stated.

Differentiate the utility function:

MU_x=\frac{10}{x},\qquad MU_y=\frac{6}{y}.

At $(4,2)$, $MU_x=10/4=2.5$, so $MU_x/p_x=2.5/2=1.25$. Also, $MU_y=6/2=3$, so $MU_y/p_y=3/3=1$. Since $1.25>1$, a small transfer of spending from $y$ to $x$ raises utility to first order.

For a $\$0.60$ transfer, the quantity of $x$ increases by $0.60/2=0.30$, and the quantity of $y$ decreases by $0.60/3=0.20$. The new bundle is $(4.30,1.80)$, which costs $2(4.30)+3(1.80)=8.60+5.40=\$14$.

The exact utility change is:

\begin{aligned} \Delta U &=10\ln(4.30/4)+6\ln(1.80/2)\\ &=10\ln(1.075)+6\ln(0.9)\\ &\approx 10(0.0723216)+6(-0.1053605)\\ &\approx 0.723216-0.632163\\ &\approx 0.0911. \end{aligned}

The finite transfer increases utility by approximately $0.0911$ utility units. The first-order comparison correctly identifies the improving direction; its value of $0.60(1.25-1)=0.15$ is an approximation, not the exact utility change.

Common mistake avoided: Comparing $MU_x=2.5$ with $MU_y=3$ would incorrectly favor $y$. The goods have different prices, so compare marginal utilities per dollar instead.

Worked Example: Equal Ratios Do Not Make Every Transfer Neutral

A consumer has utility $U(x,y)=20x-x^2+16y-2y^2$, with prices $p_x=\$2$ and $p_y=\$4$. At the bundle $(8,2)$, check whether the marginal utilities per dollar are equal. Then calculate the effect of transferring $\$2$ of spending from $y$ to $x$.

Method. First apply the first-order comparison at the stated bundle; then evaluate the specified finite transfer directly.

The marginal utilities are $MU_x=20-2x$ and $MU_y=16-4y$. At $(8,2)$, $MU_x=20-16=4$, and $MU_y=16-8=8$. Thus $MU_x/p_x=4/2=2$ and $MU_y/p_y=8/4=2$: the ratios are equal. The starting bundle costs $2(8)+4(2)=16+8=\$24$.

A $\$2$ transfer from $y$ to $x$ increases $x$ by $2/2=1$ and reduces $y$ by $2/4=0.5$. The new bundle is $(9,1.5)$, and its cost is $2(9)+4(1.5)=18+6=\$24$.

At the original bundle, utility is $20(8)-8^2+16(2)-2(2^2)=160-64+32-8=120$. At the new bundle, it is $20(9)-9^2+16(1.5)-2(1.5^2)=180-81+24-4.5=118.5$. Utility therefore changes by $118.5-120=-1.5$, a decrease of $1.5$ utility units.

Equal ratios mean the first-order change from a small spending shift is zero at the starting bundle. They do not mean that a finite transfer leaves utility unchanged. Here the specified transfer decreases utility. The example’s conclusion is about that transfer and does not, by itself, establish a general maximum.

Common mistake avoided: Treating the equality of marginal utility per dollar as a claim that every reallocation is neutral. It is a local first-order condition, not a statement that utility is constant along the whole budget line.

Worked Example: A Utility Transformation Can Hide the Comparison

A consumer’s utility is represented by $U(x,y)=x+2y-3$. At $(1,1)$, the prices are $p_x=\$2$ and $p_y=\$1$. Compare the original marginal utilities per dollar with those from the strictly increasing transformation $V=U^3$.

Method. Calculate the original derivatives and use the chain rule for the transformed utility. The bundle costs $2(1)+1(1)=\$3$, so it is feasible with a $\$3$ budget.

For $U$, $MU_x=1$ and $MU_y=2$. The marginal utilities per dollar are $1/2=0.5$ for $x$ and $2/1=2$ for $y$. The original representation therefore indicates that a small shift from $x$ toward $y$ raises utility to first order.

At the bundle, $U(1,1)=1+2-3=0$. The transformation $g(u)=u^3$ is differentiable and strictly increasing, with derivative $g'(u)=3u^2$. By the chain rule, $V_x=g'(U)U_x$ and $V_y=g'(U)U_y$. Since $g'(0)=0$, both transformed marginal utilities are zero at this bundle. Their ratios are $0/2=0$ and $0/1=0$, so they do not preserve the original strict ordering there.

The underlying preference ordering is unchanged by a strictly increasing transformation. The problem is that the transformed first derivatives provide no useful comparison at this particular utility level. In general, when $g'(U)>0$, both marginal utilities are multiplied by the same positive factor and their ordering and equality are preserved. If $g'(U)=0$, that first-order comparison can lose information.

Common mistake avoided: Assuming every differentiable, strictly increasing transformation has a strictly positive derivative everywhere. It need not; check the derivative at the bundle before relying on transformed marginal utilities.

Conditions, Interpretation, and Exam Traps

The equal-marginal-utility-per-dollar condition depends on its assumptions. The consumer must be choosing an interior bundle, the utility function must be differentiable there, prices must be positive, and the budget must bind. If a good is not consumed, the choice is at a boundary and the interior equalities need not hold. If utility is not differentiable at the choice, the derivative-based first-order conditions also cannot be applied as written. Those cases require methods that account for the relevant boundary or nonsmoothness.

Even when the assumptions hold, equality of the two ratios is a necessary first-order condition, not a complete test of optimality. It is a reliable way to identify whether a small budget-neutral shift has a positive or negative first-order effect. For a specified large transfer, calculate the resulting utility directly if the problem asks whether that transfer helps.

The numerical level of marginal utility depends on the utility representation. If a differentiable transformation $V=g(U)$ has $g'(U)>0$ at the bundle, then $V_x=g'(U)U_x$ and $V_y=g'(U)U_y$. Dividing by prices multiplies both marginal utilities per dollar by the same positive factor, so their ordering and equality are preserved. If $g'(U)=0$, both transformed marginal utilities vanish at that bundle, and their comparison may no longer reveal the original ordering. This qualification matters whenever marginal utilities are calculated using a transformed utility function.

A practical check is to state the direction of the proposed transfer in words before calculating: spending moves from the good with the smaller marginal utility per dollar to the good with the larger one. Then verify that the quantity changes preserve the budget. This helps prevent both a reversed direction and a price-unit error.

Key takeaway: At a smooth interior optimum with a binding budget, $MU_x/p_x=MU_y/p_y$. Away from equality, the larger ratio identifies the good toward which a sufficiently small spending shift raises utility to first order. This condition is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, compare marginal utility per dollar before deciding what a reallocation implies. Assume quantities are positive at the stated bundles.

A consumer has $U(x,y)=5\ln x+4\ln y$, prices $p_x=\$1$, $p_y=\$2$, and bundle $(5,2)$. Are the marginal utilities per dollar equal? (Answer: Yes; each is 1.)
A consumer has $U(x,y)=8\ln x+3\ln y$, prices $p_x=\$2$, $p_y=\$1$, and bundle $(2,3)$. Which direction of a small spending shift raises utility to first order? (Answer: From $y$ to $x$; the ratios are 2 for $x$ and 1 for $y$.)
For $U(x,y)=18x-x^2+10y-y^2$, prices are $p_x=\$3$, $p_y=\$2$, and the bundle is $(6,3)$. Calculate both marginal utilities per dollar and state whether the first-order spending-shift comparison is tied. (Answer: $MU_x/p_x=2$ and $MU_y/p_y=2$; the first-order comparison is tied.)
A consumer has $U(x,y)=\ln x+4\ln y$, prices $p_x=\$2$, $p_y=\$4$, and starting bundle $(1,2)$. Calculate the utility change if $\$4$ is transferred from spending on $y$ to spending on $x$. (Answer: The new bundle is $(3,1)$; $\Delta U=\ln 3+4\ln(1/2)\approx-1.674$ utility units.)
At a bundle, $U_x=3$, $U_y=6$, $p_x=\$2$, and $p_y=\$2$. A differentiable increasing transformation has $g'(U)=0$ at that bundle. What are the transformed marginal utilities per dollar there, and can they rank the goods as the original ratios do? (Answer: Both transformed ratios are 0; no, the first-order comparison loses the original ordering.)

What You'll Learn