welfare economics Welfare economics is a field of economics that applies microeconomic techniques to evaluate the overall well-being (welfare) of a society. The principles of welfare economics are often used to inform public economics, which focuses on the ...

, a utility–possibility frontier (or utility possibilities curve), is a widely used concept analogous to the better-known

production–possibility frontier In microeconomics, a production–possibility frontier (PPF), production possibility curve (PPC), or production possibility boundary (PPB) is a graphical representation showing all the possible quantities of outputs that can be Production (econom ...

. The graph shows the maximum amount of one person's utility given each level of utility attained by all others in society. The utility–possibility frontier (UPF) is the upper frontier of the utility possibilities set, which is the set of utility levels of agents possible for a given amount of output, and thus the utility levels possible in a given consumer

Edgeworth box In economics, an Edgeworth box, sometimes referred to as an Edgeworth-Bowley box, is a graphical representation of a market with just two commodities, ''X'' and ''Y'', and two consumers. The dimensions of the box are the total quantities Ω''x'' an ...

. The slope of the UPF is the trade-off of utilities between two individuals. The absolute value of the slope of the utility-possibility frontier showcases the utility gain of one individual at the expense of utility loss of another individual, through a marginal change in outputs. Therefore, it can be said that the frontier is the utility maximisation by consumers given an economies' endowment and technology. This means that points on the curve are, by definition,

Pareto efficient In welfare economics, a Pareto improvement formalizes the idea of an outcome being "better in every possible way". A change is called a Pareto improvement if it leaves at least one person in society better off without leaving anyone else worse ...

, which are represented by ''E, F'' and ''G'' in the image to the right. Meanwhile the points that do not lie on this curve are not

, as shown by point ''H''. The utility possibility frontier also represents a social optimum, as any point on the curve is a maximisation of the given

social welfare function In welfare economics and social choice theory, a social welfare function—also called a social ordering, ranking, utility, or choice function—is a function that ranks a set of social states by their desirability. Each person's preferences ...

. However, based on the extent of society’s preferences for an equal distribution of real income, a point off the curve may be preferred. All points on or below the utility–possibility frontier are attainable by society; all points above it are not attainable. The utility–possibility frontier is derived from the

contract curve In microeconomics, the contract curve or Pareto set is the set of points representing final allocations of two goods between two people that could occur as a result of mutually beneficial trading between those people given their initial allocati ...

. In a competitive economy, any allocation over the utility–possibility frontier is a Pareto optimum, as the UPF is a representation of the Pareto contract curve in a different dimension (utilities rather than goods). The UPF is the set of points which, for a given level of utility of person 1, utility of person 2 is maximized (subject to resource availability). Because all points along the UPF represent different real income distributions, all being Pareto efficient, it is difficult to determine which utility combination is preferable to society. Usually, the

, which incorporates the deservedness of the two individuals and states how society’s well-being relates to that of the two individuals, is required to maximize social welfare. It is assumed that the value of social welfare changes as the individual utility of any member of society changes. To maximize social welfare, a point on the UPF would be chosen that also falls on the highest

indifference curve In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...

for society. The shape of the utility possibility curve is often represented as being concave to the origin, as

cardinal utility In economics, a cardinal utility expresses not only which of two outcomes is preferred, but also the intensity of preferences, i.e. ''how much'' better or worse one outcome is compared to another. In consumer choice theory, economists originally ...

is often assumed. Cardinal utility implies that consumers can rank their preferences over goods (utility in this case).

Constructing a Utility-Possibility Frontier

To graphically construct a utility possibility-frontier, the utility levels of consumers are plotted for every Pareto efficient allocation. The frontier is traced from the varying levels of the allocation The process to derive the slope of the Utility Possibility Frontier. The two individuals have a utility function of U_A (X, Y) and U_B (X, Y), where X and Y represent two goods. At optimality, MRS^A_XY = MRS^B_XY As stated above, the slope of the Utility Possibility Frontier maps the effect of a marginal change in utility. U^A' = U^A_X (-dX) + U^B_Y dY U^B' = U^B_X dX + U^B_Y dY Since, MRS^N_XY = U^N_X / U^N_Y MRS^A_XY = U^A_X / U^A_Y = (U^A' / U^A_Y - dY) / (-dX) MRS^B_XY = U^B_X / U^B_Y = (U^B' / U^B_Y - dY) / dX Since, the MRS^A = MRS^B along the UPF At Optimality, along the UPF Thus, as the MRSs must be equal along the UPF, which implies, after some rearrangement, that: U^B' / U^A' = - U^B_Y/U^A_Y The slope, as seen above, is equivalent to the absolute value of the marginal utilities of y.

References

Theodore Bergstrom lecture notes
Welfare economics {{economics-stub