Cardinal Utility Analysis

2.1.1 Cardinal Utility Analysis

Cardinal utility analysis is built on a simple but powerful assumption: utility — the satisfaction a consumer gets from a good — can be measured in numbers. Just as we measure temperature in degrees or weight in kilograms, we can assign a numerical value to the satisfaction a shirt gives us. A consumer might say, "This shirt gives me 50 units of utility." That number is meaningful in itself, and it can be compared with the utility from other goods.

This approach treats utility like a cardinal number — 50 units is exactly twice as much as 25 units. While modern economics often prefers ordinal utility (ranking, not measuring), the cardinal approach remains a clear and intuitive starting point for understanding consumer behaviour, especially the law of demand.

Measures of Utility

Total Utility (TU) is the total satisfaction a consumer gets from consuming a given quantity of a commodity. If a consumer eats 4 bananas, the total utility from those 4 bananas is the sum of the satisfaction each banana provides. More of a commodity generally gives more satisfaction, so total utility rises as consumption increases. We write $TU_n$ to mean the total utility from consuming $n$ units of commodity $x$.

Marginal Utility (MU) is the change in total utility that comes from consuming one additional unit of a commodity. It answers the question: "How much extra satisfaction does the 5th banana give me, compared to having only 4?"

The relationship is straightforward. Suppose 4 bananas give 28 units of total utility, and 5 bananas give 30 units. The 5th banana adds 2 units of utility. So:

In general, for the $n$th unit of a commodity:

where the subscript $n$ refers to the $n$th unit consumed.

Total utility and marginal utility are also linked in the opposite direction. If you know the marginal utility of each unit, you can add them up to get total utility:

This simply says that the total satisfaction from $n$ bananas is the sum of the satisfaction from the first banana, plus the second, and so on up to the $n$th.

A Numerical Example: Table 2.1

The textbook gives a concrete example to show how total and marginal utility behave as consumption increases. The numbers are imaginary but illustrate a real pattern.

Look at the pattern. The first unit gives 12 units of utility. The second adds only 6 — total utility rises to 18. The third adds 4, the fourth adds 2. By the fifth unit, total utility stays at 24 — the marginal utility is zero. The sixth unit actually reduces total utility to 22, giving a negative marginal utility of -2.

The Law of Diminishing Marginal Utility

The pattern in the table is not accidental. It reflects a fundamental observation about human wants: the law of diminishing marginal utility. This law states that as a consumer consumes more and more units of a commodity (while holding consumption of all other commodities constant), the marginal utility from each additional unit declines.

Why does this happen? Because the intensity of a consumer's desire for a good weakens as they get more of it. The first banana when you are hungry is extremely satisfying. The second is still good, but less so. By the fourth or fifth, the desire is nearly or completely satisfied. The law is a statement about human psychology — our wants are satiable.

In the table, notice that $MU_3$ (4 units) is less than $MU_2$ (6 units). Total utility increases, but it does so at a diminishing rate. The rate of change of total utility — which is marginal utility — falls from 12 to 6 to 4 to 2. This is the law in action.

Marginal utility becomes zero at the point where total utility stops rising. In the example, $TU$ is constant at 24 units for the 4th and 5th units, so $MU_5 = 0$. Beyond that, total utility falls and marginal utility turns negative.

Deriving the Demand Curve for a Single Commodity

Cardinal utility analysis gives us a direct way to understand why demand curves slope downward. But first, we need to be clear about what demand and the demand curve mean.

Demand for a commodity is the quantity a consumer is willing and able to buy, given the prices of goods and the consumer's income. It is not just a wish — the consumer must have the purchasing power to back it up. Demand for a good $x$ depends on:

• the price of $x$ itself,
• the prices of other goods (substitutes and complements),
• the consumer's income,
• the consumer's tastes and preferences.

The demand curve is a graph that shows the different quantities of a commodity a consumer is willing to buy at different prices of that same commodity, while holding constant everything else — prices of other goods, income, and tastes.

Figure 2.2 in the textbook shows a hypothetical demand curve for an individual consumer for commodity $x$. The horizontal axis measures quantity, the vertical axis measures price. The curve slopes downward from left to right. At lower prices, the consumer buys more of $x$; at higher prices, she buys less. This negative relationship between price and quantity demanded is called the law of demand.