This section provides a brief review of basic concepts and techniques used in the following chapters. For more detailed arguments, see any basic textbooks on microeconomics, including Varian (2014).

Constrained Maximization

Consider the following constrained maximization problem:

where x_{1} and x_{2} are choice variables. u() is the objective function and g() = 0 is the constraint.

The corresponding Lagrange function is given as

where the variable X is called a Lagrange multiplier.

Differentiating the Lagrangian with respect to each of its arguments, the first order conditions lead to

These three equations determine three unknown variables, xj, x_{2}, and X. The Lagrange multiplier at the solution measures the sensitivity of the optimal value of the objective function.

Pareto Optimality

The standard approach to welfare economics is based on the concept of “Pareto optimality,” a necessary condition for an economic optimum. A Pareto optimum is a situation in which no feasible reallocation of outputs and/or inputs in the economy could increase the level of utility of one or more individuals without lowering the level of utility of other individuals. An efficient social state is often called Pareto- optimal.

For example, suppose that there are fixed amounts X, Y of two goods (x, y) and that there are only two agents A and B. For simplicity, also assume that each agent’s utility u_{i} is given respectively as a quasi-linear function. Thus,

where xi is agent i’s consumption of good x and yi is agent i’s consumption of good y. i = A, B. A Pareto optimal allocation under this circumstance is one that maximizes the utility of agent A, while holding agent B’s utility fixed at some given level of U. Thus,

Substituting the constraint into the objective function, we have the unconstrained maximization problem,

The optimality condition is given as

Now, we consider the relationship between the optimality condition (1.7) and competitive equilibrium. At an equilibrium price p*, each consumer adjusts her or his consumption of good x to have

This equation means that the necessary condition for Pareto optimality is satisfied; market equilibrium can produce a Pareto-optimal allocation of resources. This proposition is usually referred to as the first optimality theorem of welfare economics.

First Optimality Theorem Resource allocation is Pareto-optimal if there is perfect competition and no market failure.

The first basic theorem of welfare economics states that a competitive equilibrium is a Pareto optimum; namely, the equilibrium is one for which no utility level can be increased without decreasing some other utility level.

Further, any allocation that is Pareto-optimal must satisfy (1.7), which determines p*. This implies that such a Pareto-optimal allocation would be generated by a competitive equilibrium. Thus, we have the second theorem of welfare economics.

Second Optimality Theorem Any specified Pareto-optimal resource allocation that is technically feasible can be established by a free market and an appropriate pattern of factor ownership.

The second basic theorem of welfare economics states that any Pareto optimum can be realized as a particular competitive equilibrium; namely, for each Pareto optimum there is an associated price system and a system of resource ownership that would attain, as a competitive equilibrium, this solution with differing distributions of utility. The theorem says that every Pareto-efficient allocation can be attained by means of a decentralized market mechanism.

A Dual Approach

Consider a standard utility maximization problem of a consumer:

where x_{i} is her or his consumption of good i, p_{i} is a consumer price of good i, and M is her or his income (i = 1,2). Then, the maximum utility u is a function of M and the price vector p = (p_{1}, p_{2}).

The indirect utility function indicates the maximum utility attainable at given prices and income:

From this equation, we may derive the expenditure function:

where E( ) indicates the minimum money cost at which it is possible to achieve a given utility at given prices.

The expenditure function summarizes the consumer’s optimizing behavior and has the following properties.

(i) E(p,u) is non-decreasing in p.

(ii) E(p,u) is homogeneous of degree one in p.

(iii) E(p,u) is concave in p.

(iv) E(p,u) is continuous in p.

(v) The compensated demand curve is xi (p,и^{0}) = ^{dE}(^{p}^^{,P2,u} ) _