The basic welfare economics problem is to find the theoretical maximum of a social welfare function, subject to various constraints such as the state of technology in production, available natural resources, national infrastructure, and behavoural constraints such as consumer utility maximization and producer profit maximization. In the simplest possible economy this can be done by simultaneously solving seven equations. This simple economy would have only two consumers (consumer 1 and consumer 2), only two products (product X and product Y), and only two factors of production going into these products (labour (L) and capital (K)). The model can be stated as:
maximize social welfare: W=f(U1 U2) subject to the following set of constraints:
K = Kx + Ky (The amount of capital used in the production of goods X and Y)
L = Lx + Ly (The amount of labour used in the production of goods X and Y)
X = X(Kx Lx) (The production function for product X)
Y = Y(Ky Ly) (The production function for product Y)
U1 = U1(X1 Y1) (The preferences of consumer 1)
U2 = U2(X2 Y2) (The preferences of consumer 2)
The solution to this problem yields a Pareto optimum. In a more realistic example of millions of consumers, millions of products, and several factors of production, the math gets more complicated.
Also, finding a solution to an abstract function does not directly yield a policy recommendation! In other words, solving an equation does not solve social problems. However, a model like the one above can be viewed as an argument that solving a social problem (like achieving a Pareto-optimal distribution of wealth) is at least theoretically possible.


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