Social Welfare Functions

This scene merges two people's happiness into one score. It picks the best among all possible happiness pairs.

One slider changes how the merging works. Count only the sum, and the bigger total wins. Hate inequality more, and the score changes. At the extreme, only the worse-off person counts. The best point slides toward the equal diagonal. The formula below fixes the merging rule.

For example, two 🧑 friends split 15 🍬 candies. One friend gets 9 and the other gets 6. Judged by the sum alone, this split scores best. Care more about fairness, and they split 7.5 each. The worse-off friend's happiness sets the score.

W_\rho = \left(\frac{u_1^\rho + u_2^\rho}{2}\right)^{1/\rho}

Part of these learning paths

General Equilibrium and Welfare

Examples & Exercises

On the utility possibility frontier $(u_1/12)^2 + (u_2/10)^2 = 1$ , find the utilitarian optimum (maximize $u_1 + u_2$ ).

(u_1^*, u_2^*) \approx (9.22,\ 6.40)

\rho = 1

— utilitarian optimum

(9.22, 6.40)

Show proof

Lagrangian: $1 = \lambda \cdot 2u_1/144$ , $1 = \lambda \cdot 2u_2/100$ give $u_1/144 = u_2/100$ , i.e. $u_2 = 0.6944\, u_1$ .
Substituting into the frontier: $u_1^2/144 + (0.6944\,u_1)^2/100 = 1$ gives $u_1^2 = 84.98$ , so $u_1 \approx 9.22$ , $u_2 \approx 6.40$ .

On the same frontier, find the Rawlsian (maximin) optimum

and compare it with the utilitarian optimum.

(u_1^R, u_2^R) \approx (7.68,\ 7.68) \ne (9.22,\ 6.40)

\rho \to -\infty

— Rawls point

(7.68, 7.68)

Show proof

Maximizing $\min(u_1, u_2)$ means the frontier meets the 45-degree line: with $u_1 = u_2 = u$ , $u^2/144 + u^2/100 = 1$ gives $u^2 = 59.02$ , $u \approx 7.68$ .
The Rawls point $(7.68, 7.68)$ differs from the utilitarian point $(9.22, 6.40)$ — the choice of social welfare function (inequality aversion $\rho$ ) changes the "optimal" distribution.

17. Social Choice and Welfare