Quasilinear Utility and Constant Marginal Utility

Introduction

A utility function may treat two goods differently: utility can change at a constant rate with one good, while the rate of change with the other depends on how much of that good is consumed. This pattern is called quasilinear utility. It is useful to recognize because it gives a clear result about marginal utility and a distinctive pattern for the marginal rate of substitution (MRS).

The key is to distinguish two claims. In a quasilinear utility representation, the marginal utility of the good that enters linearly is constant. The MRS, however, is generally not constant: it depends on the quantity of the nonlinear good. This tutorial develops those results using marginal utilities and indifference curves, concepts covered in the earlier material on utility and MRS. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: A two-good utility function is quasilinear in \(y\) if it can be written as \(U(x,y)=v(x)+cy\), where \(v\) is a function of \(x\) and \(c\) is a positive constant. The good \(y\) enters linearly; the good \(x\) may enter nonlinearly. The common normalized form is \(U(x,y)=v(x)+y\), which sets \(c=1\). The goods and function are considered on a domain where the utility function and the derivatives being used are defined.

Marginal Utilities and MRS

For a differentiable quasilinear utility function \(U(x,y)=v(x)+cy\), the marginal utility of a good is the partial derivative of utility with respect to that good, holding the other good fixed. Differentiating gives:

MU_x=\frac{\partial U}{\partial x}=v'(x), \qquad MU_y=\frac{\partial U}{\partial y}=c.

Thus \(MU_y\) is constant: an additional unit of \(y\) changes utility by \(c\), regardless of the current quantities of either good. By contrast, \(MU_x=v'(x)\) can vary with \(x\). The term “quasilinear” does not require \(v\) to be concave, although a concave \(v\) is common in applications. Nor does the definition by itself say that the marginal utility of \(x\) is diminishing; that depends on the specific function \(v\).

Using the convention that \(MRS_{xy}\) is the amount of \(y\) a consumer would give up for an additional unit of \(x\) while staying on the same indifference curve, the marginal-utility formula is \(MRS_{xy}=MU_x/MU_y\), when \(MU_y\ne 0\). Consequently:

MRS_{xy}=\frac{v'(x)}{c}.

The MRS depends on \(x\), not on \(y\). This does not mean that the MRS is necessarily constant: it will change as \(x\) changes whenever \(v'(x)\) changes. In this convention, the slope of an indifference curve is \(-MRS_{xy}\).

Why the MRS Has This Form

Along an indifference curve, utility stays fixed. A small change in the quantities therefore has zero total change in utility. The total differential of \(U\) is \(dU=v'(x)dx+c\,dy\). Setting it to zero and solving for the slope gives the result:

Hold utility fixed.
On an indifference curve, \(dU=0\), so \(v'(x)dx+c\,dy=0\).

Find the indifference-curve slope.
Divide by \(dx\), for a local change with \(dx\ne 0\), to obtain \(\frac{dy}{dx}=-\frac{v'(x)}{c}\).

Apply the MRS convention.
The amount of \(y\) given up per unit increase in \(x\) is the negative of the slope, so \(MRS_{xy}=-\frac{dy}{dx}=\frac{v'(x)}{c}\).

The same conclusion follows by solving the utility-level equation \(v(x)+cy=\bar U\) for \(y\): \(y=(\bar U-v(x))/c\). Its derivative with respect to \(x\) is \(-v'(x)/c\), matching the slope just derived. This calculation applies where the function is differentiable and \(c>0\); at a point where a derivative does not exist, the derivative-based MRS may not be defined.

Worked Applications

Worked Example: Square-Root Utility in One Good

Suppose a consumer’s preferences are represented on \(x\ge 0,\ y\ge 0\) by \(U(x,y)=6\sqrt{x}+2y\). Find the utility, both marginal utilities, and \(MRS_{xy}\) at \((x,y)=(9,4)\). Then find the quantity of \(y\) that keeps utility unchanged if \(x\) rises from 9 to 16.

Method. Identify the linear term and differentiate each term with respect to its own good. The coefficient of \(y\) is \(c=2\), so \(MU_y=2\). Since \(v(x)=6\sqrt{x}\), for \(x>0\):

MU_x=6\left(\frac{1}{2\sqrt{x}}\right)=\frac{3}{\sqrt{x}}.

At \(x=9\), \(\sqrt{9}=3\), so \(MU_x=3/3=1\). At the stated bundle, utility is \(U(9,4)=6(3)+2(4)=18+8=26\). The MRS is \(MU_x/MU_y=1/2=0.5\): locally, the consumer would give up 0.5 units of \(y\) for one additional unit of \(x\) at this bundle.

To keep utility at 26 after \(x\) rises to 16, solve \(6\sqrt{16}+2y'=26\). Since \(\sqrt{16}=4\), this becomes \(24+2y'=26\), so \(2y'=2\) and \(y'=1\). The bundle \((16,1)\) gives \(U=24+2=26\), which checks the result. The quantity of \(y\) falls from 4 to 1.

Common mistake avoided: Treating the MRS at \(x=9\) as the exact amount of \(y\) needed for a finite change from \(x=9\) to \(x=16\). The MRS is a local trade-off; the utility-level equation gives the exact finite compensation.

Worked Example: Solving an Indifference-Curve Equation

On the domain \(x>0,\ y\ge0\), let \(U(x,y)=10\ln(x)+3y\). A consumer has bundle \((4,6)\). Calculate utility, marginal utilities, and \(MRS_{xy}\) at that bundle. Then find \(y\) at \(x=9\) that gives the same utility. Give the final quantity of \(y\) to three decimal places.

Method. Differentiate the utility function: \(MU_x=10/x\) and \(MU_y=3\). At \(x=4\), \(MU_x=10/4=2.5\), so \(MRS_{xy}=2.5/3=5/6\), or approximately 0.8333. The initial utility is \(10\ln(4)+3(6)=10\ln(4)+18\).

Let \(y'\) be the amount of \(y\) at \(x=9\) that preserves this utility. Equate the two utility levels:

10\ln(9)+3y'=18+10\ln(4).

Rearrange to isolate \(y'\):

3y'=18+10\ln(4)-10\ln(9) =18+10\ln(4/9).

Using \(\ln(4/9)=\ln(4)-\ln(9)\approx-0.810930\), we get \(10\ln(4/9)\approx-8.10930\), and hence \(3y'\approx18-8.10930=9.89070\). Therefore \(y'\approx9.89070/3=3.29690\), or \(3.297\) to three decimal places. This is nonnegative, as required by the stated domain. Substituting the unrounded value into \(10\ln(9)+3y'\) recovers \(18+10\ln(4)\), so the utility-level equation is satisfied.

Common mistake avoided: Losing the negative sign in \(\ln(4/9)\), or evaluating \(18+10\ln(4/9)\) as a negative number. Since \(4/9<1\), its logarithm is negative, but the sum is approximately \(9.891\), not negative. The division by 3 is also necessary to obtain \(y'\).

Worked Example: Linear Utility in the First Good

On \(x\ge0,\ y>0\), consider \(U(x,y)=5x+8\sqrt{y}\). Find the utility, marginal utilities, and \(MRS_{xy}\) at \((2,16)\). Then calculate how much \(x\) must increase if \(y\) falls from 16 to 9 and utility is to remain unchanged.

Method. Here \(x\) is the linearly entering good, with constant marginal utility \(MU_x=5\). For \(y>0\), \(MU_y=8/(2\sqrt{y})=4/\sqrt{y}\). At \(y=16\), \(MU_y=4/4=1\). The utility at the initial bundle is \(U(2,16)=5(2)+8(4)=10+32=42\), and \(MRS_{xy}=5/1=5\).

At \(y=9\), the nonlinear term is \(8\sqrt{9}=24\). To keep utility equal to 42, solve \(5x'+24=42\). Thus \(5x'=18\), and \(x'=18/5=3.6\). The required increase is \(3.6-2=1.6\) units of \(x\). Checking, \(U(3.6,9)=5(3.6)+24=18+24=42\).

Common mistake avoided: Assuming that because \(MU_x\) is constant, the MRS must be constant too. At \(y=16\), MRS is 5; at \(y=9\), it is \(5/(4/3)=15/4=3.75\). The MRS changes because \(MU_y\) changes with \(y\).

What Is Constant—and What Is Not

These examples show why it is important to name the good whose marginal utility is constant. For \(U(x,y)=v(x)+cy\), it is \(MU_y=c\). If instead \(U(x,y)=ax+g(y)\), where \(a>0\) is constant, then \(MU_x=a\), and the MRS is \(MRS_{xy}=a/g'(y)\) wherever \(g'(y)\ne0\). In either orientation, quasilinearity means that one good enters linearly; it does not mean that both marginal utilities or the MRS are constant.

A function linear in both goods, such as \(U(x,y)=ax+cy\), is a special case in which the MRS is constant as well. That result follows because both marginal utilities are constants, so their ratio is constant. In a general quasilinear function, the nonlinear good’s marginal utility can vary, and that changes the MRS.

Utility representations also require care in interpretation. Marginal utility and the algebraic form of a utility function are properties of the chosen representation, not direct measurements of a consumer’s satisfaction. The quasilinear form is useful because it makes the marginal utility and MRS calculations transparent. A different increasing transformation can represent the same preference ordering while changing the numerical marginal utilities or no longer displaying the same quasilinear form.

Key takeaway: For \(U(x,y)=v(x)+cy\), \(MU_y=c\) is constant, \(MU_x=v'(x)\), and \(MRS_{xy}=v'(x)/c\). The MRS is independent of \(y\), but it need not be constant as \(x\) changes.

Check Your Understanding

For each problem, use the stated utility representation and show enough work to check the marginal utilities, MRS, or utility-preserving change.

Let \(U(x,y)=7\sqrt{x}+2y\), with \(x\ge0,\ y\ge0\). At \((9,3)\), find utility, \(MU_x\), \(MU_y\), and \(MRS_{xy}\). (Answer: \(U=27,\ MU_x=7/6,\ MU_y=2,\ MRS_{xy}=7/12\).)
Let \(U(x,y)=4\ln(x+2)+5y\), with \(x\ge0,\ y\ge0\). Starting from \((2,1)\), find the quantity of \(y\) at \(x=6\) that preserves utility. Give the final value to four decimal places. (Answer: \(y=1-(4/5)\ln 2\approx0.4455\).)
Let \(U(x,y)=3x+6\sqrt{y}\), with \(x\ge0,\ y>0\). If \(y\) falls from 9 to 4, how much must \(x\) rise to preserve utility? (Answer: \(2\) units of \(x\).)
Let \(U(x,y)=12\ln(x)+4y\), with \(x>0,\ y\ge0\). From \((1,6)\), find the value of \(y\) at \(x=4\) that preserves utility, and find \(MRS_{xy}\) at both endpoints. Give \(y\) to four decimal places. (Answer: \(y=6-3\ln 4\approx1.8411;\ MRS_{xy}=3\) at \(x=1\) and \(0.75\) at \(x=4\).)
For \(U(x,y)=2\sqrt{x}+y\), with \(x>0,\ y\ge0\), compare \(MRS_{xy}\) at \(x=1\) and \(x=4\). Does the constant marginal utility of \(y\) imply a constant MRS? (Answer: \(MRS_{xy}=1\) at \(x=1\) and \(0.5\) at \(x=4\); no, the MRS varies with \(x\).)

What You'll Learn