1. **Stating the problem:** We need to distinguish between cardinal utility theory and ordinal utility theory using diagrams and highlight two weaknesses of each approach. 2. **Cardinal Utility Theory:** This theory assumes that utility can be measured and expressed in numerical values. It implies that the difference in utility values is meaningful. - **Diagram:** Imagine a utility function $U(x)$ where $x$ is quantity of a good. The utility values are on a cardinal scale, e.g., $U(1) = 10$, $U(2) = 20$, meaning the utility of 2 units is twice that of 1 unit. - **Formula:** Utility is a function $U: X \to \mathbb{R}$ where differences $U(x_2) - U(x_1)$ are meaningful. - **Weaknesses:** 1. Measuring utility numerically is often unrealistic because utility is subjective. 2. Assumes interpersonal comparisons of utility are possible, which is controversial. 3. **Ordinal Utility Theory:** This theory only ranks preferences without assigning numerical values. It states that only the order of preferences matters, not the magnitude. - **Diagram:** A preference ordering where $x_2$ is preferred to $x_1$, but no numerical utility values are assigned. - **Formula:** Utility function $U$ represents preferences such that $x_2 \succ x_1$ if and only if $U(x_2) > U(x_1)$, but differences $U(x_2) - U(x_1)$ are not meaningful. - **Weaknesses:** 1. Cannot measure intensity of preferences, only order. 2. Does not allow for interpersonal utility comparisons. 4. **Summary:** Cardinal utility provides measurable utility but is often unrealistic; ordinal utility is more realistic but less informative about preference strength.