Why the MRS-Price Rule Is Easy to Misapply
The marginal rate of substitution (MRS) compares how much of one good a consumer is willing to give up for a small increase in another, holding utility constant. At an interior optimum, this trade-off must match the trade-off imposed by market prices. The rule is simple to state, but errors in the order of the goods, in using the budget constraint, or in interpreting tangency can lead to an incorrect bundle.
The rule applies to a specific setting: a consumer chooses quantities \(x\) and \(y\) to maximize a differentiable utility function \(U(x,y)\), subject to \(p_xx+p_yy\le m\), where \(p_x,p_y>0\) and income \(m>0\). If utility is strictly increasing in both goods, an optimum uses all income. If the optimum is interior, with \(x>0\) and \(y>0\), the first-order tangency condition applies. It does not by itself decide whether a candidate is feasible or whether it is a maximum.
Deriving the Ratio Without Reversing It
A reliable way to avoid a reversed ratio is to derive the rule using the same ordering of goods throughout. On the budget line \(p_xx+p_yy=m\), a small change \((dx,dy)\) that stays on the line satisfies \(p_xdx+p_ydy=0\). Thus the budget-line slope, with \(y\) on the vertical axis and \(x\) on the horizontal axis, is \(dy/dx=-p_x/p_y\).
Along an indifference curve, utility does not change to first order, so \(dU=MU_xdx+MU_ydy=0\). Dividing by \(dx\) and solving for the slope gives \(dy/dx=-MU_x/MU_y\). At an interior tangency, the two slopes are equal. Equating their magnitudes gives \(MU_x/MU_y=p_x/p_y\), which is the stated rule. Dividing both sides by the positive prices and marginal utilities also gives the equal-marginal-utility-per-dollar condition.
The ordering matters. If the MRS is instead defined as \(MRS_{yx}=MU_y/MU_x\), the matching price ratio is \(p_y/p_x\). Both conventions can be used, but combining one convention’s MRS with the other convention’s price ratio reverses the answer.
Write whether the MRS means \(MU_x/MU_y\) or \(MU_y/MU_x\); do not rely on an unstated convention.
Differentiate the specified utility function and form the MRS using the stated ordering.
Use the matching ratio at an interior optimum, then solve together with the budget equation.
Confirm the quantities satisfy nonnegativity and the budget, and establish that the candidate is a maximum rather than just a stationary point.
Worked Examples: Applying and Checking the Rule
Worked Example: Correcting a Reversed Price Ratio
A consumer has utility \(U(x,y)=\sqrt{xy}\), income \(m=24\), and prices \(p_x=2\) and \(p_y=1\). Find the interior optimum and identify the error in setting \(MRS_{xy}=p_y/p_x\).
Method. Use \(MRS_{xy}=MU_x/MU_y\), match it to \(p_x/p_y\), and solve the tangency condition together with the budget. This avoids pairing an MRS in one order with a price ratio in the reverse order.
For positive \(x,y\), the marginal utilities are \(MU_x=\tfrac12\sqrt{y/x}\) and \(MU_y=\tfrac12\sqrt{x/y}\). Therefore
The correct interior condition is \(y/x=p_x/p_y=2/1=2\), so \(y=2x\). The budget is \(2x+y=24\). Substituting \(y=2x\) gives \(2x+2x=24\), hence \(4x=24\), \(x=6\), and \(y=12\). The bundle costs \(2(6)+1(12)=24\), so it is feasible and exhausts income.
As a check, \(MRS_{xy}=12/6=2\), and \(p_x/p_y=2/1=2\). The marginal utilities per dollar also agree: \(MU_x=\tfrac12\sqrt{2}\), so \(MU_x/p_x=\sqrt{2}/4\); \(MU_y=\tfrac12\sqrt{1/2}=\sqrt{2}/4\), so \(MU_y/p_y=\sqrt{2}/4\). The utility is \(\sqrt{72}=6\sqrt{2}\).
Common mistake and correction. Setting \(y/x=p_y/p_x=1/2\) gives \(y=x/2\), and the budget then gives \(2x+x/2=24\), \(x=9.6\), \(y=4.8\). Although this bundle is feasible, it has \(MRS_{xy}=4.8/9.6=0.5\), not \(p_x/p_y=2\), so it fails the interior first-order condition. The ratio must follow the stated MRS ordering.
For this utility, \(U\) is concave but not strictly concave on the positive quadrant: it is homogeneous of degree one, so proportional bundles can give equality in the concavity inequality. Uniqueness here can instead be checked by restricting utility to the budget line: \(y=24-2x\), so \(U^2=x(24-2x)=24x-2x^2\), a strictly concave quadratic with its unique maximum at \(x=6\). Since \(U\ge0\), maximizing \(U^2\) gives the same bundle as maximizing \(U\).
Worked Example: A Tangency Ratio Does Not Determine a Bundle
A consumer has \(U(x,y)=\sqrt{xy}\), prices \(p_x=3\) and \(p_y=2\), and income \(m=30\). A proposed solution reports only \(y=1.5x\). Complete the calculation and diagnose what is missing.
Method. First use the tangency condition to find the relative quantities; then use the budget equation to find their levels. The ratio alone describes a ray of bundles, not one bundle.
As in the previous calculation, \(MRS_{xy}=y/x\). At an interior optimum, \(y/x=p_x/p_y=3/2\), so \(y=1.5x\). Substituting this relationship into the budget gives
The resulting bundle costs \(3(5)+2(7.5)=15+15=30\). Its MRS is \(7.5/5=1.5\), equal to the price ratio \(3/2=1.5\). Its utility is \(\sqrt{5(7.5)}=\sqrt{37.5}\), approximately \(6.124\) to three decimals.
For example, the ratio \(y=1.5x\) also includes \((x,y)=(4,6)\). That bundle costs \(3(4)+2(6)=24\), leaving \(6\) unspent. With strictly increasing utility and positive prices, the consumer can improve utility by purchasing more goods with the remaining income, so this bundle cannot be optimal. The correct solution requires both tangency and budget feasibility.
Common mistake and correction. Reporting only \(y/x=3/2\) treats a relative-quantity condition as a complete choice. Add the budget equation to determine the actual quantities, then verify the expenditure and the MRS-price equality.
Worked Example: Tangency Can Be a Minimum
A consumer chooses nonnegative \(x,y\) with \(x+y=10\) and utility \(U(x,y)=x^2+y^2\). Find the tangency candidate and determine whether it maximizes utility.
Method. Find the stationary tangency point, then evaluate utility over the feasible budget line, including its endpoints. This avoids assuming that every tangency is a maximum.
The prices are both \(1\). The marginal utilities are \(MU_x=2x\) and \(MU_y=2y\). For a positive interior tangency, \(MU_x/MU_y=p_x/p_y=1\), so \(2x/2y=1\), or \(x=y\). Combined with \(x+y=10\), this gives the tangency candidate \((5,5)\), with utility \(5^2+5^2=50\).
To check optimality, substitute \(y=10-x\), where \(0\le x\le10\):
This expression is minimized at \(x=5\), where utility is \(50\), not maximized there. At either endpoint, \((10,0)\) or \((0,10)\), utility is \(100\). In fact, since \(0\le x\le10\), \((x-5)^2\le25\), and the displayed expression is at most \(2(25)+50=100\). Thus both endpoints are global maxima, while the tangency is a minimum along the budget line.
Common mistake and correction. Solving the first-order tangency equation and stopping at \((5,5)\) confuses a necessary stationary condition for an interior optimum with a sufficient test for a maximum. Compare utility on the feasible set or use conditions that establish global optimality. Here the utility function is convex, not concave, so the tangency does not guarantee a maximum.
When the Rule Is Necessary—and What Else Must Be Checked
The MRS-price equality is an interior first-order condition under differentiability and positive prices. It is not a universal formula for every optimum. If one good is consumed in zero quantity, the optimum is at a boundary and the interior tangency equation need not hold. If utility is not differentiable at a candidate, marginal utilities may not exist there. If the budget does not bind, as can occur when preferences are not strictly increasing, spending the full income cannot be assumed. These cases require checking the relevant constraints and the behavior of utility at the candidate.
Even when the candidate is interior and the budget binds, tangency alone only establishes a first-order condition. A sufficient global check is available when \(U\) is concave and the feasible set is convex: any feasible point satisfying the appropriate first-order optimality condition is a global maximizer. Strict concavity of utility is a sufficient condition for uniqueness, but it should not be asserted without checking it. Alternatively, for a two-good problem, substitute the budget equation into utility and examine the resulting one-variable objective, including endpoints.
| Check | Question to ask | What a failure means |
|---|---|---|
| Ratio convention | Does the MRS ordering match the price-ratio ordering? | The tangency equation has been reversed. |
| Budget feasibility | Does the candidate satisfy \(p_xx+p_yy\le m\) and \(x,y\ge0\)? | The candidate is not an available choice. |
| Budget use | If utility increases in both goods, does expenditure equal income? | Unspent income can buy more of a good, so the candidate is not optimal. |
| Optimality | Is the stationary candidate a maximum over the feasible set? | A tangency could be a minimum or another nonmaximizing stationary point. |
For exam-style calculations, a compact but complete solution states the MRS convention, computes the marginal utilities, sets the correctly ordered ratio equal to the price ratio, solves with the budget equation, and checks the result. If the question asks for an optimum rather than only a tangency candidate, also explain why the candidate maximizes utility or compare it with relevant boundary choices. This procedure is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Check Your Understanding
For each problem, state the ratio convention you use and check any proposed optimum against the budget.
- A consumer has \(U(x,y)=\sqrt{xy}\), prices \(p_x=4\), \(p_y=1\), and income \(m=40\). Find the interior optimum using \(MRS_{xy}=MU_x/MU_y\). (Answer: \(x=5,\ y=20\).)
- A consumer has \(U(x,y)=xy\), prices \(p_x=3\), \(p_y=1\), and income \(m=24\). Use the interior tangency condition and budget equation to find the candidate, then verify that utility along the budget line is maximized there. (Answer: \(x=4,\ y=12\), with utility \(48\), the maximum on the feasible budget line.)
- A consumer has \(U(x,y)=\sqrt{x}+\sqrt{y}\), prices \(p_x=1\), \(p_y=4\), and income \(m=20\). Find the interior optimum and verify its expenditure. (Answer: \(x=16,\ y=1\), with expenditure \(16+4=20\).)
- With \(x+y=10\) and \(U(x,y)=x^2+y^2\), a candidate solution reports the equal-price tangency \((5,5)\). Is it a maximum, and what is the maximum utility? (Answer: No; \((5,5)\) is a minimum with utility \(50\), while \((10,0)\) and \((0,10)\) maximize utility at \(100\).)
- A consumer has \(U(x,y)=xy\), prices \(p_x=2\), \(p_y=5\), and income \(m=30\). Find the interior candidate from the MRS-price condition and budget equation, and check the ratio. (Answer: \(x=7.5,\ y=3\); \(MRS_{xy}=3/7.5=0.4=p_x/p_y\), and expenditure is \(15+15=30\).)