Deterministic Preferences

Deterministic Preferences & Utility Functions

Develop a rigorous understanding of preference axioms, ordinal utility representation, and deterministic choice models—the essential building blocks for microeconomic insights and financial mathematics.

Foundational Preference Axioms

Axiom

Reflexivity

For any bundle x, the relation x ⪰ x holds.

Guarantees self-consistency—no option is strictly preferred to itself.

Axiom

Completeness

For any bundles x and y, either x ⪰ y or y ⪰ x (or both).

Ensures decision-makers can compare any two feasible allocations.

Axiom

Transitivity

If x ⪰ y and y ⪰ z, then x ⪰ z.

Eliminates preference cycles and supports consistent optimization.

Monotonicity, Convexity & Continuity

Monotonicity
More of every good is (weakly) preferred to less, aligning with the economic notion of non-satiation in consumption and wealth.
Convexity
Weighted averages of bundles are preferred to extremes, capturing love for diversification and smoothing.
Continuity
Small changes in consumption bundles produce small changes in preferences, enabling utility representation theorems.

Geometry of Preferences

Indifference Curves: Downward sloping due to monotonicity; convex to the origin when preferences exhibit love for diversification.

Marginal Rate of Substitution: Measures the rate at which a decision-maker is willing to trade one good for another while maintaining utility, central to price ratio interpretation.

Utility Level Sets: Show ordinal rankings; any positive monotonic transformation leaves these sets unchanged.

Representative Utility Functions

Example

Cobb–Douglas Utility

U(x_1, x_2) = x_1^{\alpha} x_2^{1-\alpha}

\text{MRS}_{12} = \frac{\partial U / \partial x_1}{\partial U / \partial x_2} = \frac{\alpha}{1-\alpha} \cdot \frac{x_2}{x_1}

Models portfolio allocation between a risk-free asset and a single risky asset with constant relative risk aversion in deterministic settings.

Example

Leontief Preferences

U(x_1, x_2) = \min\{ x_1 / \gamma, x_2 / (1-\gamma) \}

\text{Marginal rate of substitution is undefined at the kink; interior diversification is not optimal.}

Describes perfect complements such as bundled services where diversification is not valuable (e.g., a payments infrastructure requiring both hardware and software).

Example

Linear Utility

U(x_1, x_2) = a x_1 + b x_2

\text{MRS}_{12} = a/b ext{ (constant)}, capturing perfect substitution and risk neutrality.}

Illustrates risk-neutral behavior or perfect substitutes, such as choosing between two equally safe cash accounts with identical yields.

Case Study: Intertemporal Consumption Choice

Define the Opportunity Set

Consider a U.S. household deciding between consumption today (C1) and saving for retirement (C2) with a risk-free Treasury bill.

Specify Preferences

Assume utility u(C1, C2) = C1^{0.4} C2^{0.6}, capturing preference for smooth lifetime consumption.

Maximize Utility

Solve the deterministic optimization problem subject to the intertemporal budget, obtaining the optimal savings rate.

Interpret Results

Demonstrate how monotonicity and convexity deliver interior solutions and how utility shapes consumption smoothing behavior.

First Order Conditions

Solving the Lagrangian for the two-period consumption problem yields the Euler equation that governs optimal savings behavior:

\mathcal{L} = C_1^{0.4} C_2^{0.6} + \lambda \left(w - C_1 - \frac{C_2}{1+r}\right) \\ \Rightarrow \frac{\partial \mathcal{L}}{\partial C_1} = 0.4 C_1^{-0.6} C_2^{0.6} - \lambda = 0, \\ \frac{\partial \mathcal{L}}{\partial C_2} = 0.6 C_1^{0.4} C_2^{-0.4} - \frac{\lambda}{1+r} = 0.

Dividing the first-order conditions yields $\frac{C_2}{C_1} = \left(\frac{0.6}{0.4}\right)(1+r)$ which links optimal consumption growth directly to the interest rate.

Key Takeaways

Axiomatic foundations ensure consistent decision-making and enable mathematical optimization of consumption and investment choices.
Ordinal utilities are unique up to positive monotonic transformations, highlighting that only preference rankings matter under certainty.
Monotonicity and convexity conditions create economically meaningful indifference curves that support diversification and smooth consumption.

Transition to Uncertainty

Deterministic preference theory sets the stage for incorporating uncertainty. When outcomes become probabilistic, we extend utility functions to lotteries and introduce the expected utility operator. The same axiomatic discipline ensures coherent decision-making.

\text{If } \succeq \text{ is complete, transitive, continuous, and strictly monotone on } \mathbb{R}^n_+,\\ \text{then } \exists U: \mathbb{R}^n_+ \to \mathbb{R} \text{ continuous, strictly increasing such that } x \succeq y \iff U(x) \ge U(y).

Expected utility builds upon this representation by adding probability weighting while preserving the core ordering of deterministic bundles.

Explore Expected Utility

Q&A: Deterministic Utility

Why focus on ordinal utility rather than cardinal measures?

Ordinal utility preserves preference rankings without relying on arbitrary units of satisfaction. Financial decisions often require only rankings (e.g., choosing the better portfolio), making ordinal representations sufficient and robust.

How does deterministic utility relate to portfolio theory?

Deterministic utility provides the baseline for mean-variance analysis. By understanding convex preferences and trade-offs under certainty, we extend the framework to uncertainty via expected utility or mean-variance approximations.

Can different utility functions represent the same preferences?

Yes. Any strictly increasing transformation of an ordinal utility function yields the same preference ordering, emphasizing that utility levels lack intrinsic meaning.

Next Steps

Move to uncertainty analysis or solidify insights with targeted practice.

Next Topic: Uncertainty Preferences Practice Deterministic Concepts