Why Discrete Marginal Utility Matters
A utility table often reports utility only at selected bundles or quantity levels. In that setting, the change in utility from one listed quantity to another is observable, but a derivative may not be available: the quantities could be discrete, or the utility function may not be specified between table entries. The practical task is to calculate the utility gain over the stated interval and, when useful, express it as an average gain per unit.
This is related to marginal utility as a partial derivative, introduced in the earlier material on marginal utility, but it is not the same calculation. A derivative describes a local rate of change for a smooth utility function. A difference calculated from a table describes a finite change across an interval. Keeping the interval and the units explicit prevents a common error: reporting a total utility change as though it were a per-unit marginal utility.
Reading Utility Tables Carefully
For adjacent rows in a table, subtract the earlier utility from the later utility. If the quantity rises by more than one unit, divide the utility change by the quantity change when the question asks for an average per-unit amount. The total utility change and the average discrete marginal utility answer different questions, so it is useful to report both when there could be ambiguity.
For example, suppose utility is 23 at quantity 3 and 38 at quantity 6. The total utility change is \(38-23=15\). The quantity change is \(6-3=3\), so the average utility change is \(15/3=5\) utility units per unit of quantity. It does not follow that each of the three individual units added exactly 5 utility units; the table only establishes the average over the interval.
Worked Example: Average Marginal Utility Across Unequal Table Intervals
A consumer's utility from weekly servings of a product is recorded as follows. Calculate the total utility change and average discrete marginal utility from 1 to 3 servings, and from 3 to 6 servings. Identify what can and cannot be inferred about each individual serving.
| Servings \(q\) | Utility \(U(q)\) |
|---|---|
| 0 | 0 |
| 1 | 9 |
| 3 | 23 |
| 6 | 38 |
Method. For each interval, subtract the starting utility from the ending utility. Then divide by the number of additional servings to obtain the average utility change per serving.
From 1 to 3 servings, the quantity change is \(3-1=2\), and the utility change is \(23-9=14\). Therefore the average discrete marginal utility is \(14/2=7\) utility units per serving.
From 3 to 6 servings, the quantity change is \(6-3=3\), and the utility change is \(38-23=15\). The average is \(15/3=5\) utility units per serving.
The total utility changes are 14 and 15, respectively; the average per-serving changes are 7 and 5. The table does not show the utility at quantity 2, 4, or 5, so it does not tell us the exact utility gain from each of those individual servings.
Check. The interval averages reconcile with the total changes: \(2(7)=14\) and \(3(5)=15\). The common mistake avoided here is dividing by the ending quantity rather than by the number of units added.
Finite Differences and Derivatives
When a smooth utility function is given, a finite difference can be calculated even if the quantity is treated as continuous. The finite difference gives the exact change over a specified interval. Its average per-unit value can be compared with the derivative, but it should not be labeled the derivative unless the interval shrinks to zero.
Record the initial quantity \(q_0\), the final quantity \(q_1\), and the interval length \(q_1-q_0\).
Evaluate the utility function at both endpoints and subtract: \(U(q_1)-U(q_0)\).
The average discrete marginal utility is the total change divided by \(q_1-q_0\). A derivative instead describes a local rate at a point.
The connection to a derivative follows directly from its definition. For a differentiable one-variable utility function, take a finite increment \(h\ne 0\). The average utility change per unit over that increment is \([U(q+h)-U(q)]/h\). As the increment approaches zero, this difference quotient approaches the derivative \(U'(q)\):
Thus a finite difference is not an approximation merely because it is calculated from a function: it is the exact change over its interval. The difference quotient is the exact average rate over that interval. It approximates the derivative at \(q\) when \(h\) is small, but the two need not be equal. If \(U\) is continuous on the interval and differentiable inside it, the mean value theorem says the average rate equals \(U'(c)\) for at least one point \(c\) between the endpoints. That point need not be the starting quantity.
Worked Example: Comparing a Finite Change With a Derivative
Let utility for a quantity \(q\) be \(U(q)=30q-q^2\), for \(q\geq 0\). Calculate the total utility change and average discrete marginal utility as quantity rises from 4 to 7. Then compare the result with the derivative at the start and end of the interval.
Method. Evaluate the function at the endpoints for the finite change. Separately differentiate the function to find the local marginal utility at each endpoint.
At \(q=4\), \(U(4)=30(4)-4^2=120-16=104\). At \(q=7\), \(U(7)=30(7)-7^2=210-49=161\). The total utility change is \(161-104=57\), and the quantity change is \(7-4=3\). The average discrete marginal utility is \(57/3=19\) utility units per unit of quantity.
The derivative is \(U'(q)=30-2q\). At the starting quantity, \(U'(4)=30-8=22\); at the ending quantity, \(U'(7)=30-14=16\). Neither endpoint derivative is 19. The average rate of 19 describes the whole interval, whereas 22 and 16 are local rates at particular quantities.
As a check, the one-unit utility changes are \(U(5)-U(4)=125-104=21\), \(U(6)-U(5)=144-125=19\), and \(U(7)-U(6)=161-144=17\). Their sum is \(21+19+17=57\), and their average is \(57/3=19\). The common mistake avoided is using the derivative at \(q=4\) as though it were the exact average utility change across the full three-unit interval.
Changes in More Than One Good
A utility table or function may involve several goods. If several quantities change at once, the total utility change is the difference between utility at the final bundle and utility at the initial bundle. To describe the contribution associated with each good, specify a path: change one quantity while holding the others fixed, then change the next. The component changes along that path add to the total, but their values can depend on the order because the other quantities may affect utility.
Worked Example: Decomposing a Joint Change Along a Stated Path
Suppose utility is \(U(x,y)=2x+xy+y\). A bundle changes from \((1,1)\) to \((3,2)\). Calculate the total utility change, decompose it by first changing \(x\) and then \(y\), and compare with the path that changes \(y\) first.
Method. Evaluate utility at the initial bundle, final bundle, and each intermediate bundle on the specified path. Each path's component changes must sum to the same total change.
The initial utility is \(U(1,1)=2(1)+(1)(1)+1=4\). The final utility is \(U(3,2)=2(3)+(3)(2)+2=6+6+2=14\). The total change is \(14-4=10\).
On the path that changes \(x\) first, the intermediate bundle is \((3,1)\). Its utility is \(U(3,1)=2(3)+(3)(1)+1=6+3+1=10\). Changing \(x\) therefore adds \(10-4=6\) utility units while \(y=1\), across two units of \(x\); the average is \(6/2=3\) per unit of \(x\). Changing \(y\) next adds \(U(3,2)-U(3,1)=14-10=4\). The components sum to \(6+4=10\).
On the path that changes \(y\) first, the intermediate bundle is \((1,2)\). Its utility is \(U(1,2)=2(1)+(1)(2)+2=6\). The \(y\) change adds \(6-4=2\), and then the \(x\) change adds \(14-6=8\). Again, \(2+8=10\).
The total change is 10 on either path, but the component changes differ. Dividing the total change of 10 by the two-unit change in \(x\) gives 5, but that ratio is not the discrete marginal utility of \(x\): \(y\) changed too. The \(x\)-only average on the first path is 3, and on the second path it is \(8/2=4\), because \(x\) changes while \(y\) is held at a different level. The common mistake avoided is assigning the entire joint utility change to the change in only one good.
Utility Representation and Numerical Differences
Utility functions represent preferences: bundles with higher utility are ranked above bundles with lower utility. If utility is ordinal, any strictly increasing transformation preserves those rankings. However, a nonlinear increasing transformation can change utility differences and therefore change the numerical value of discrete marginal utility. Numerical differences are meaningful only relative to the particular utility representation being used; they are not, by themselves, comparable measures of satisfaction across people or across different representations.
For example, if \(U(q)=q\) for nonnegative \(q\), then the average utility change from 1 to 3 is \((3-1)/(3-1)=1\). The strictly increasing transformation \(V(q)=U(q)^2=q^2\) gives the same ranking of these quantities, but its average change over the same interval is \((9-1)/(3-1)=4\). Both representations rank 3 above 1; their numerical marginal utility values differ. A positive change in a good that raises utility remains positive under a strictly increasing transformation, but its size is not preserved.
Common Errors to Avoid
- Confusing a total change with a per-unit average. For a change from \(q_0\) to \(q_1\), the total change is \(U(q_1)-U(q_0)\); divide by \(q_1-q_0\) only when an average per-unit rate is requested.
- Using the ending quantity as the denominator. The denominator is the quantity added, \(q_1-q_0\), not \(q_1\).
- Calling a finite difference a derivative. A difference quotient is an average rate over a nonzero interval. A derivative is its limit as the interval shrinks to zero.
- Attributing a joint change to one good. If other quantities also change, the total utility difference does not isolate a one-good marginal utility.
- Comparing numerical utility changes across representations. Ordinally equivalent utility functions can preserve rankings while assigning different numerical differences.
Check Your Understanding
For each problem, report the requested finite change or average and keep the interval explicit.
- A utility table gives \(U(0)=3\), \(U(2)=15\), and \(U(5)=30\). Find the total utility change and average discrete marginal utility from 0 to 2, and from 2 to 5. (Answer: From 0 to 2, total change \(12\) and average \(6\) per unit; from 2 to 5, total change \(15\) and average \(5\) per unit.)
- For \(U(q)=18q-q^2\), calculate the total utility change and average discrete marginal utility as \(q\) increases from 3 to 5. (Answer: \(U(3)=45\), \(U(5)=65\), total change \(20\), and average \(10\) per unit.)
- Let \(U(x,y)=3x+2xy+y\). Starting at \((1,2)\), change \(x\) to 3 first and then \(y\) to 3. Calculate each component change and the total. (Answer: Initial utility \(9\), intermediate utility \(23\), final utility \(30\); the \(x\) change adds \(14\), the \(y\) change adds \(7\), and the total change is \(21\).)
- At quantities 1 and 4, a utility representation gives values \(U(1)=8\) and \(U(4)=20\). Another representation is \(V=3U+5\). Find the average utility change per unit under each representation and state whether the rankings change. (Answer: Under \(U\), \((20-8)/3=4\) per unit; under \(V\), the endpoint values are 29 and 65, so the average is \(36/3=12\) per unit; rankings do not change because \(V\) is strictly increasing in \(U\).)
- For \(U(q)=q^2+2q\), calculate the average discrete marginal utility from \(q=4\) to \(q=5\), then compare it with the derivative at \(q=4\). (Answer: \(U(4)=24\), \(U(5)=35\), so the finite average is \(11\) per unit; \(U'(q)=2q+2\), so \(U'(4)=10\).)