Short answers, pulled from the story.
Equity in economics refers to a condition of fairness where economic processes and their outcomes do not unduly favor or disadvantage any particular group or individual. It is closely tied to taxation policies, welfare economics, and public finance, and is often measured using the Gini index. Economists acknowledge that formulating a universally accepted definition is difficult because it requires interpersonal comparisons of utility.
Horizontal equity means treating people alike when they are in the same or similar economic situations, such as applying the same tax burden to comparable incomes. Vertical equity means treating people differently according to differences in their income, wealth, or ability to pay, typically supporting progressive taxation where higher earners contribute a greater proportion.
Hammond's equity principle holds that a progressive transfer, redistributing income from a richer person to a poorer one without altering their relative positions on the utility scale, constitutes a welfare improvement for society. It is grounded in the idea that a utility gain for a worse-off person offset by a utility loss for a better-off person is a net social gain.
Sen's weak equity axiom states that if person i has a lower level of welfare than person j at every level of individual income, then the optimal distribution of a given total income must give person i a higher income than person j. The axiom prioritizes the needs of the most disadvantaged when allocating resources.
Inter-nation equity is concerned with the allocation of national gain and loss in the international context, aiming to ensure that each country receives an equitable share of tax revenues from cross-border transactions. It has been an important consideration in debates about how taxing rights should be divided between source and residence countries.
Equitability in fair division means every person's subjective valuation of their own share of some goods is the same. The surplus procedure achieves a more complex variant called proportional equitability. For more than two people, a division cannot always be both equitable and envy-free, meaning the two fairness criteria can mathematically conflict.