In economics the production set is a construct representing the possible inputs and outputs to a production process.
A production vector represents a process as a vector containing an entry for every commodity in the economy. Outputs are represented by positive entries giving the quantities produced and inputs by negative entries giving the quantities consumed.
If the commodities in the economy are (labour, corn, flour, bread) and a mill uses one unit of labour to produce 8 units of flour from 10 units of corn, then its production vector is (–1,–10,8,0). If it needs the same amount of labour to run at half capacity then the production vector (–1,–5,4,0) would also be operationally possible. The set of all operationally possible production vectors is the mill's production set.
If y is a production vector and p is the economy's price vector, then p·y is the value of net output. The mill's owner will normally choose y from the production set to maximise this quantity. p·y is defined as the 'profit' of the vector y, and the mill-owner's behaviour is described as 'profit-maximising'.
The following properties may be predicated of production sets.
If a production set is separable and has a single output, then a function F(y) can be constructed whose value is the maximum quantity of output obtainable for given inputs, and whose domain is the set of input subvectors represented in the production set. This is known as the production function.
If a production set is separable then we may define a "production value function" fp(x) in terms of a price vector p. If x is a monetary quantity, then fp(x) is the maximum monetary value of output obtainable in Y from inputs whose cost is x.
Constant returns to scale mean that if y is in the production set, then so too is λy for any positive λ. Returns might be constant over a region; for instance, so long as λ is not too far from 1 for a given y. There is no entirely satisfactory way to define increasing or decreasing returns to scale for general production sets.
If the production set Y can be represented by a production function F whose argument is the input subvector of a production vector, then increasing returns to scale are available if F(λy) > λF(y) for all λ > 1 and F(λy) < λF(y) for all λ<1. A converse condition can be stated for decreasing returns to scale.
If Y is a separable production set with a production value function fp, then (positive) economies of scale are present if fp(λx) > λfp(x) for all λ > 1 and fp(λx) < λfp(x) for all λ < 1. The opposite condition may be referred to as negative economies (or diseconomies) of scale.
If Y has a single output and prices are positive, then positive economies of scale are equivalent to increasing returns to scale.
As with returns to scale, economies of scale may apply over a region. If a mill is operating below capacity then it will offer positive economies of scale, but as it approaches capacity the economies will become negative. Economies of scale for the firm are important in influencing an industry's tendency to concentrate in the direction of monopoly or disaggregate in the direction of perfect competition.
The components of a production vector are conventionally portrayed as flows (see Stock and flow), whereas more general treatments regard production as combining stocks (e.g. land) and flows (e.g. labour) (see Factors of production). Accordingly, the simple definition of 'profit' as the net value of output does not correspond to its meaning elsewhere in economics (see Profit (economics)).