The Interior Tangency Rule for Two Goods

Why Interior Tangency Matters

When a consumer chooses between two goods, an affordable bundle must satisfy the budget constraint. But affordability alone does not show that the bundle is best. At a differentiable interior optimum, the consumer’s willingness to trade one good for the other must match the tradeoff imposed by market prices. This is the interior tangency rule.

The rule is useful both for solving choice problems and for checking proposed answers. It is a necessary condition under the stated assumptions, not a universal recipe: it does not cover corner choices or kinks, and tangency alone does not guarantee that a candidate is a global maximum.

Definition: Let \(x\) and \(y\) denote quantities of two goods, with differentiable utility \(U(x,y)\). Their marginal utilities are \(MU_x=\partial U/\partial x\) and \(MU_y=\partial U/\partial y\). Using the standard convention here, the marginal rate of substitution of \(x\) for \(y\) is \(MRS_{xy}=MU_x/MU_y\), when \(MU_y\ne 0\). It measures the number of units of \(y\) the consumer is locally willing to give up for one more unit of \(x\), holding utility constant.

Let the goods’ positive prices be \(p_x\) and \(p_y\), and let income be \(m\). The budget set consists of bundles satisfying \(p_xx+p_yy\le m\). An interior bundle has \(x>0\) and \(y>0\). Under the differentiability and positive-marginal-utility assumptions used for the rule, an interior utility-maximizing bundle must use the full budget and satisfy \(MRS_{xy}=p_x/p_y\).

Deriving the Tangency Condition

The condition comes from comparing the slopes of two curves through the chosen bundle. On an indifference curve, utility is constant. A small change in quantities therefore satisfies \(dU=MU_x\,dx+MU_y\,dy=0\). If \(MU_y\ne0\), rearranging gives the indifference-curve slope. The budget line has a fixed slope because its equation is \(p_xx+p_yy=m\).

\left.\frac{dy}{dx}\right|_{U\ \text{constant}} =-\frac{MU_x}{MU_y}, \qquad \left.\frac{dy}{dx}\right|_{\text{budget}} =-\frac{p_x}{p_y}.

At a differentiable interior optimum, the indifference curve and budget line are tangent: their slopes are equal. Equating the two negative slopes gives the rule below. In economic terms, if the slopes differ, a small reallocation along the budget line can move the consumer to a higher indifference curve.

Theorem: If a consumer has a differentiable utility function, faces positive prices, chooses a differentiable interior optimum, and has \(MU_y\ne0\) there, the tangency condition is

\frac{MU_x}{MU_y}=MRS_{xy}=\frac{p_x}{p_y}.

This is a necessary first-order condition under these assumptions. The budget condition must be checked as well. If the consumer is nonsatiated and can increase utility by increasing consumption, a maximizing bundle exhausts income, so \(p_xx+p_yy=m\). The rule does not apply as stated if the optimum is on a boundary, the indifference curve has a kink, or the relevant derivatives do not exist. Nor does solving the two equations alone establish that the resulting candidate is a global maximum; that requires additional analysis of preferences and feasible alternatives.

Testing a Proposed Bundle

A proposed interior choice can be checked in two separate stages: calculate its expenditure, then compare the marginal-utility ratio with the price ratio. Both tests matter. A bundle that satisfies tangency but leaves money unspent is not the full-budget choice under nonsatiation; a bundle that exhausts the budget but fails tangency is not an interior optimum.

Worked Example: Checking Budget Exhaustion and Tangency

A consumer has utility \(U(x,y)=\ln x+2\ln y\), faces prices \(p_x=3\) and \(p_y=2\), and has income \(m=22\). Test the proposed bundle \((x,y)=(4,5)\) for budget exhaustion and interior tangency.

Method. First calculate expenditure. Then find both marginal utilities and compare their ratio with the price ratio. This avoids mistaking a budget-feasible bundle for an optimum.

Budget check. Expenditure is \(3(4)+2(5)=12+10=22\), equal to income. The bundle exhausts the budget, and both quantities are positive.

Marginal utilities. Differentiating gives \(MU_x=1/x\) and \(MU_y=2/y\). At the proposed bundle, \(MU_x=1/4\) and \(MU_y=2/5\). Thus the inverse marginal-utility ratio is \(MU_y/MU_x=(2/5)/(1/4)=8/5=1.6\). With the convention \(MRS_{xy}=MU_x/MU_y\), the MRS is its reciprocal: \(MRS_{xy}=(1/4)/(2/5)=5/8=0.625\).

Tangency check. The price ratio is \(p_x/p_y=3/2=1.5\), not \(0.625\). The bundle therefore exhausts the budget but does not satisfy the interior tangency condition. It is not an interior optimum under the stated conditions. The reciprocal calculation \(1.6\) is correct for \(MU_y/MU_x\), but it must not be labeled \(MRS_{xy}\) under the convention used here.

Check. The budget test passes, while the tangency test fails; keeping those conclusions separate prevents an affordability check from being mistaken for an optimality check.

Solving the Two Conditions Together

To find an interior tangency candidate, write down the marginal utilities, set their ratio equal to the price ratio, and combine that equation with the full-budget equation. Keeping the two equations visible helps avoid reversing the price ratio or using the wrong reciprocal of the marginal-utility ratio.

State the model.
Record the utility function, positive prices, income, and the assumption that the solution sought is interior and differentiable.

Calculate marginal utilities.
Differentiate utility with respect to each good, then form \(MU_x/MU_y\).

Impose tangency and the budget.
Set \(MU_x/MU_y=p_x/p_y\) and \(p_xx+p_yy=m\).

Solve and verify.
Check that the solution has positive quantities, spends the stated income, and gives the same marginal-utility and price ratios.

Worked Example: Finding an Interior Tangency Candidate

Suppose \(U(x,y)=\ln x+3\ln y\), prices are \(p_x=2\) and \(p_y=3\), and income is \(m=40\). Find the interior bundle satisfying tangency and budget exhaustion, and verify both conditions.

Method. Use the tangency equation to express one quantity in terms of the other, then substitute into the budget equation. The common mistake is to compare \(MU_x/MU_y\) with \(p_y/p_x\) rather than \(p_x/p_y\).

Step 1: Marginal utilities. Differentiation gives \(MU_x=1/x\) and \(MU_y=3/y\). Therefore \(MRS_{xy}=(1/x)/(3/y)=y/(3x)\).

Step 2: Tangency. The price ratio is \(p_x/p_y=2/3\). Set \(y/(3x)=2/3\). Multiplying both sides by \(3x\) gives \(y=2x\).

Step 3: Budget exhaustion. Substitute \(y=2x\) into \(2x+3y=40\): \(2x+3(2x)=40\), so \(8x=40\) and \(x=5\). Then \(y=2(5)=10\).

Step 4: Verification. The quantities are positive. Expenditure is \(2(5)+3(10)=10+30=40\), exactly the income. At \((5,10)\), \(MU_x=1/5\), \(MU_y=3/10\), and \(MRS_{xy}=(1/5)/(3/10)=2/3\), equal to \(p_x/p_y=2/3\).

The bundle \((5,10)\) is the interior tangency candidate. This calculation verifies the necessary conditions; a separate argument would be needed to establish that a candidate is a global maximum.

Checking Tangency with Nonlinear Marginal Utilities

The condition is not limited to utility functions whose marginal utilities are simple reciprocal expressions. The same procedure applies whenever the utility function is differentiable and the marginal-utility ratio can be calculated at an interior bundle. The derivatives must be evaluated at the candidate, not at an average or at some other bundle.

Worked Example: A Quadratic Utility Function

A consumer has \(U(x,y)=12x-x^2+8y-y^2\), faces \(p_x=2\) and \(p_y=1\), and has income \(m=12\). Find the interior bundle satisfying the tangency and full-budget equations, then verify that the marginal utilities are positive there.

Method. Differentiate each utility term, equate the marginal-utility ratio to the price ratio, and solve with the budget. Checking marginal utilities at the end confirms that the candidate lies in the increasing part of this utility function.

Step 1: Marginal utilities. The derivatives are \(MU_x=12-2x\) and \(MU_y=8-2y\). Thus the tangency equation is \((12-2x)/(8-2y)=2/1=2\).

Step 2: Rearrange tangency. Multiplying by \(8-2y\) gives \(12-2x=16-4y\). Rearranging yields \(2y-x=2\), or \(y=(x+2)/2\).

Step 3: Use the budget. The full-budget equation is \(2x+y=12\). Substitution gives \(2x+(x+2)/2=12\). Multiplying by 2 gives \(4x+x+2=24\), so \(5x=22\) and \(x=22/5=4.4\). Then \(y=(4.4+2)/2=3.2=16/5\).

Step 4: Verify. Expenditure is \(2(4.4)+3.2=8.8+3.2=12\), and both quantities are positive. The marginal utilities are \(MU_x=12-2(4.4)=3.2\) and \(MU_y=8-2(3.2)=1.6\). Their ratio is \(3.2/1.6=2\), equal to \(p_x/p_y=2/1=2\). Both marginal utilities are positive at this candidate.

The calculated bundle is an interior tangency candidate that exhausts the budget. As in the preceding example, those checks establish the stated conditions, not global optimality by themselves.

Assumptions, Interpretation, and Common Traps

Tangency describes a local relationship between the consumer’s indifference curve and budget line. At an interior optimum, a small movement along the budget line must not produce a first-order utility gain in either direction. If the curves cross with different slopes instead of touching, a nearby affordable reallocation can improve utility. This slope interpretation also explains why the price ratio must use the same order of goods as the marginal-utility ratio.

The standard formula \(MRS_{xy}=p_x/p_y\) uses \(MRS_{xy}=MU_x/MU_y\). If a problem instead defines its MRS as \(MU_y/MU_x\), the corresponding price ratio is \(p_y/p_x\). Both conventions can express the same tangency; mixing one convention’s MRS with the other convention’s price ratio reverses the comparison.

A differentiable, strictly increasing transformation of utility preserves the consumer’s preference ranking. If \(V=g(U)\), then the transformed marginal utilities are \(V_x=g'(U)MU_x\) and \(V_y=g'(U)MU_y\). Their ratio agrees with \(MU_x/MU_y\) when \(g'(U)>0\). Strictly increasing differentiable transformations can have \(g'(U)=0\) at some utility levels, however; there the transformed marginal utilities may both be zero and their ratio may be undefined. Preference preservation alone does not ensure the transformed marginal-utility ratio is defined at every point.

Common exam trap: Do not conclude “optimal” from either budget exhaustion or tangency alone. Check both equations, use a consistent MRS convention, and confirm that the bundle is interior and differentiable. A boundary solution or a kink may be optimal even though the interior tangency equation does not apply.

A practical check is to substitute the proposed quantities back into the original budget and tangency equations. This catches arithmetic errors, a reversed price ratio, and solutions that violate the assumed interior conditions. If either quantity is zero, or if the utility function has a kink there, stop before applying the interior rule and analyze the relevant boundary or kink separately.

Check Your Understanding

For each problem, state whether the bundle satisfies the specified conditions or solve for the interior tangency candidate. Use \(MRS_{xy}=MU_x/MU_y\).

A consumer has \(U(x,y)=\ln x+\ln y\), prices \(p_x=5\) and \(p_y=2\), and income \(m=24\). Test the bundle \((4,2)\) for budget exhaustion and tangency. (Answer: Expenditure is \(5(4)+2(2)=24\), so the budget is exhausted; \(MRS_{xy}=(1/4)/(1/2)=1/2\), while \(p_x/p_y=5/2\), so it is not tangent.)
For \(U(x,y)=2\ln x+\ln y\), let \(p_x=3\), \(p_y=2\), and \(m=30\). Find the interior bundle satisfying tangency and budget exhaustion. (Answer: \(MRS_{xy}=2y/x=3/2\), so \(x=4y/3\); the budget gives \(3(4y/3)+2y=30\), hence \(y=5\) and \(x=20/3\).)
For \(U(x,y)=10x-x^2+6y-y^2\), let \(p_x=1\), \(p_y=2\), and \(m=8\). Solve the two interior conditions and check marginal utilities. (Answer: Tangency gives \((10-2x)/(6-2y)=1/2\), hence \(y=2x-7\); the budget \(x+2y=8\) gives \(x=22/5=4.4\), \(y=9/5=1.8\), with \(MU_x=1.2\) and \(MU_y=2.4\), whose ratio is \(1/2\).)
Let \(U(x,y)=x+y\) and \(V(x,y)=(U(x,y)-2)^3\). At \((1,1)\), explain why the preferences are represented by the same ranking but the ratio of transformed marginal utilities is undefined. (Answer: The cube is strictly increasing, so it preserves the ranking; at \(U=2\), its derivative is zero, so \(V_x=V_y=0\) and \(V_x/V_y\) is undefined.)

What You'll Learn