Perfect-Substitutes Utility and Constant MRS

Introduction

Some goods can serve much the same purpose for a consumer. If the consumer is willing to exchange one good for the other at a fixed rate, a linear utility function is a useful representation of those preferences. The fixed rate is reflected in both the utility function’s coefficients and the slope of its indifference curves.

This tutorial analyzes perfect-substitutes utility, calculates marginal utilities and the marginal rate of substitution (MRS), and derives the straight-line indifference curves. Quantities are assumed to be nonnegative and divisible, and the two goods are assumed to be desirable, so each has a positive coefficient in the utility function.

Definition: A linear utility representation of perfect substitutes has the form \(u(x,y)=ax+by\), where \(x\) and \(y\) are quantities of two goods and \(a>0\) and \(b>0\). The coefficients measure the utility added by one more unit of each good. Because these coefficients are constant, the consumer’s willingness to exchange the goods is constant at every bundle.

Marginal Utilities and the Constant MRS

Marginal utility measures how much utility changes when the quantity of one good changes slightly while the quantity of the other good is held fixed. For a differentiable utility function, marginal utilities are its partial derivatives. For \(u(x,y)=ax+by\), differentiating gives \(MU_x=a\) and \(MU_y=b\): neither depends on the bundle.

The MRS of \(x\) for \(y\), written here as \(MRS_{xy}\), is the amount of \(y\) the consumer is willing to give up for a small additional amount of \(x\) while remaining on the same indifference curve. With positive marginal utilities, the MRS is their ratio, \(MU_x/MU_y\). The indifference-curve slope is negative, so it is \(-MRS_{xy}\), not \(MRS_{xy}\).

Result: For \(u(x,y)=ax+by\), with \(a,b>0\), \(MU_x=a\), \(MU_y=b\), and \(MRS_{xy}=a/b\). The indifference-curve slope, when \(y\) is plotted vertically and \(x\) horizontally, is \(-a/b\). Both the MRS and the slope are constant across bundles.

Worked Example: Calculating a Constant Trade-Off

A hypothetical consumer evaluates quantities \(x\) and \(y\) using \(u(x,y)=4x+2y\). Find the marginal utilities, MRS, and indifference-curve slope. Then check whether moving from \((5,7)\) to \((8,1)\) preserves utility.

Method. Read the marginal utilities from the coefficients, take their ratio for the MRS, and compare utility at the two bundles. This avoids confusing the MRS with its reciprocal or omitting the negative sign from the curve’s slope.

Marginal utilities. \(MU_x=4\) utility units per unit of \(x\), and \(MU_y=2\) utility units per unit of \(y\).

MRS and slope. \(MRS_{xy}=MU_x/MU_y=4/2=2\). The consumer gives up 2 units of \(y\) for 1 additional unit of \(x\) along the same indifference curve. The curve’s slope is \(-2\).

Check the bundles. At \((5,7)\), utility is \(4(5)+2(7)=20+14=34\). At \((8,1)\), utility is \(4(8)+2(1)=32+2=34\). The change adds 3 units of \(x\), contributing \(4(3)=12\) utility units, and removes 6 units of \(y\), reducing utility by \(2(6)=12\). The changes offset, as the constant trade-off predicts.

Why the Indifference Curves Are Straight Lines

An indifference curve contains all bundles that give the same utility level, say \(\bar u\). For linear utility, fixing utility at that level gives an equation that can be rearranged into the equation of a line. The calculation also shows why the line’s slope does not depend on the chosen utility level.

ax+by=\bar u \quad\Longrightarrow\quad y=\frac{\bar u}{b}-\frac{a}{b}x

The intercept on the \(y\)-axis is \(\bar u/b\), and the slope is \(-a/b\). If \(x=0\), the consumer needs \(y=\bar u/b\) to reach utility \(\bar u\); if \(y=0\), the consumer needs \(x=\bar u/a\). Because the slope is the same for each \(\bar u\), different utility levels give parallel lines. Higher utility levels correspond to lines farther from the origin when both coefficients are positive.

Derivation: Along an indifference curve, utility does not change. For a small change in the bundle, the total change in utility is \(du=MU_x\,dx+MU_y\,dy\). Setting \(du=0\) gives \(MU_y\,dy=-MU_x\,dx\), so \(dy/dx=-MU_x/MU_y=-a/b\). This establishes why the curve slopes downward and why its slope equals the negative MRS. For a linear utility function, the same trade-off holds for finite changes as well: \(a\Delta x+b\Delta y=0\).

Worked Example: Finding an Indifference Curve and Its Intercepts

Suppose \(u(x,y)=3x+5y\). Find the indifference curve through the bundle \((5,9)\), its intercepts, and two other bundles on that curve.

Method. First calculate the utility of the given bundle, then set the utility function equal to that value and rearrange. This prevents using a utility level that does not pass through the specified bundle.

Utility level. At \((5,9)\), \(u=3(5)+5(9)=15+45=60\). The required curve is therefore \(3x+5y=60\).

Line and intercepts. Solving for \(y\) gives \(5y=60-3x\), then \(y=12-\frac{3}{5}x\). The \(y\)-intercept is 12, found by setting \(x=0\). The \(x\)-intercept is 20, found by setting \(y=0\) and solving \(3x=60\).

Check other bundles. If \(x=5\), then \(y=12-\frac{3}{5}(5)=12-3=9\), recovering the original bundle. If \(x=15\), then \(y=12-\frac{3}{5}(15)=12-9=3\); its utility is \(3(15)+5(3)=45+15=60\). Both bundles lie on the same indifference curve. The slope is \(-3/5\), so the MRS is \(3/5\) unit of \(y\) per unit of \(x\).

Interpreting Coefficients and Utility Representations

The ratio of the coefficients determines the trade-off, but each coefficient also depends on the chosen scale for utility. Utility is an ordinal representation: its numerical values rank bundles, but the numbers are not, by themselves, a measure of how much happier one bundle makes the consumer. Multiplying utility by a positive constant preserves the ranking and leaves the MRS unchanged.

For example, multiplying \(u(x,y)=2x+6y\) by 3 gives \(v(x,y)=6x+18y\). The marginal utilities in \(v\) are three times those in \(u\), but their ratio remains \(2/6=6/18=1/3\). Thus the same indifference curves and trade-off are represented. Adding a constant to utility also preserves rankings and the MRS, though it changes the numerical labels assigned to utility levels.

Worked Example: Checking an Equivalent Utility Scale

A consumer’s preferences are represented by \(u(x,y)=2x+6y\). A second representation is \(w(x,y)=4u(x,y)-8\). For the bundle \((4,3)\), calculate both utility values, the marginal utilities and MRS under each representation, and verify that the same trade-off preserves indifference.

Method. Substitute the bundle into each representation, differentiate to find marginal utilities, and compare their ratios. This avoids treating a change in utility scale as a change in the consumer’s preferences.

Utility values. Under \(u\), the bundle gives \(u(4,3)=2(4)+6(3)=8+18=26\). Under \(w\), \(w(4,3)=4(26)-8=104-8=96\).

Marginal utilities and MRS. For \(u\), \(MU_x=2\) and \(MU_y=6\), so \(MRS_{xy}=2/6=1/3\). Since \(w=4(2x+6y)-8=8x+24y-8\), its marginal utilities are \(MU_x=8\) and \(MU_y=24\); its MRS is \(8/24=1/3\). Both representations imply a slope of \(-1/3\).

Trade-off check. Increasing \(x\) by 3 and reducing \(y\) by 1 changes \(u\) by \(2(3)+6(-1)=6-6=0\). The change in \(w\) is \(8(3)+24(-1)=24-24=0\) as well. The same bundles remain indifferent under both utility scales.

What Perfect Substitutability Does—and Does Not—Tell You

A constant MRS means the represented preferences value each additional unit of \(x\) at the same rate relative to \(y\), regardless of the bundle. It is a strong assumption. In many settings, a consumer’s willingness to exchange goods changes with the quantities already held; a linear utility representation does not capture that changing trade-off.

The line describes preferences, not a complete decision about what a consumer will purchase. A budget constraint and prices would be needed to analyze affordability and choice. Here, the key result is the relationship between the utility coefficients, the MRS, and the slope of an indifference curve.

Common exam trap: For \(u(x,y)=3x+9y\), the MRS of \(x\) for \(y\) is \(3/9=1/3\), not \(9/3=3\). The curve’s slope is \(-1/3\). The reciprocal would describe the reverse trade-off, while dropping the minus sign would give the MRS rather than the slope. Always state which good is being gained, which is being given up, and whether the question asks for the MRS or the signed slope.

The interpretation also depends on units. If \(x\) is measured in dozens and \(y\) in individual units, the numerical MRS is in individual units of \(y\) per dozen units of \(x\). The ratio should be reported with the relevant units, not as a unit-free comparison of the goods.

Check Your Understanding

For each problem, use the linear utility representation and report the trade-off with the correct direction and sign.

For \(u(x,y)=7x+2y\), find both marginal utilities and the MRS of \(x\) for \(y\). If \(x\) increases by 2 units, how much must \(y\) change to keep utility constant? (Answer: \(MU_x=7\), \(MU_y=2\), \(MRS_{xy}=3.5\), and \(y\) must decrease by 7 units.)
For \(u(x,y)=4x+3y\), find the indifference curve through a bundle with utility 48, and give both axis intercepts. (Answer: \(4x+3y=48\), or \(y=16-\frac{4}{3}x\); the intercepts are \(x=12\) and \(y=16\).)
Under \(u(x,y)=5x+2y\), a consumer holds \((3,10)\). If \(x\) rises by 4 units, what change in \(y\) keeps the consumer on the same indifference curve? Verify the utility level. (Answer: \(y\) decreases by 10 units; both bundles have utility 35.)
For \(u(x,y)=3x+9y\), state the MRS of \(x\) for \(y\), the indifference-curve slope, and the change in \(y\) that offsets an increase of 6 units in \(x\). (Answer: \(MRS_{xy}=1/3\), slope \(=-1/3\), and \(y\) decreases by 2 units.)
Let \(u(x,y)=2x+5y\) and \(z=4u-8\). Find the marginal utilities and MRS under \(z\), and state whether the indifference curves change. (Answer: \(MU_x=8\), \(MU_y=20\), \(MRS_{xy}=2/5\); the indifference curves do not change.)

What You'll Learn