Editing Production set
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* ''Non-emptiness.'' The producer has at least one possible course of action. Always holds. |
* ''Non-emptiness.'' The producer has at least one possible course of action. Always holds. |
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* ''[[Closed set|Closure]].'' The production set includes its own boundary. This is a technical property which always holds in practice. |
* ''[[Closed set|Closure]].'' The production set includes its own boundary. This is a technical property which always holds in practice. |
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* ''Separability.'' A production set is separable into inputs and outputs if every field is either non-negative in all elements or non-positive in all elements. This normally holds for individual enterprises but not, for instance, for a national economy. |
* ''Separability.'' A production set is separable into inputs and outputs if every field is either non-negative in all elements or non-positive in all elements. This normally holds for individual enterprises but not, for instance, for a national economy. |
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* ''No free lunch.'' It is impossible to produce something from nothing. Mathematically there is no vector in the production set with at least one positive entry and no negative entries. Always holds. |
* ''No free lunch.'' It is impossible to produce something from nothing. Mathematically there is no vector in the production set with at least one positive entry and no negative entries. Always holds. |
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* ''Possibility of inaction.'' The zero vector belongs to the production set; in other words, it is possible to produce nothing by consuming nothing. This property almost never holds exactly: resources will be needed either to wind up a concern or to maintain it while dormant. The property may be a useful approximation. |
* ''Possibility of inaction.'' The zero vector belongs to the production set; in other words, it is possible to produce nothing by consuming nothing. This property almost never holds exactly: resources will be needed either to wind up a concern or to maintain it while dormant. The property may be a useful approximation. |
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* ''[[Free disposal]].'' If '''''y''''' is an element of production set ''Y'', then so is any vector which consumes more of a given input or produces less of a given output. Mathematically, if '''''e''''' is a vector none of whose entries is negative, and if {{nowrap|'''''y''''' ∈ ''Y''}}, then {{nowrap|'''''y''''' – '''''e''''' ∈ ''Y''.}} This too may be a useful approximation. |
* ''[[Free disposal]].'' If '''''y''''' is an element of production set ''Y'', then so is any vector which consumes more of a given input or produces less of a given output. Mathematically, if '''''e''''' is a vector none of whose entries is negative, and if {{nowrap|'''''y''''' ∈ ''Y''}}, then {{nowrap|'''''y''''' – '''''e''''' ∈ ''Y''.}} This too may be a useful approximation. |
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* ''Single output''. Production is often based on units (e.g. flour mills) which produce a single output from more than one input. A separable production set has a single output if exactly one field contains a positive entry. |
* ''Single output''. Production is often based on units (e.g. flour mills) which produce a single output from more than one input. A separable production set has a single output if exactly one field contains a positive entry. |
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* ''Labour consumption''. Labour is usually an input to all elements of a production set which have any positive output. |
* ''Labour consumption''. Labour is usually an input to all elements of a production set which have any positive output. |
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* ''[[Irreversibility]]''. If {{nowrap|'''''y''''' ∈ ''Y''}} and '''''y'''''≠0 then {{nowrap|(–'''''y''''') ∉ ''Y''.}} Always holds in practice. |
* ''[[Irreversibility]]''. If {{nowrap|'''''y''''' ∈ ''Y''}} and '''''y'''''≠0 then {{nowrap|(–'''''y''''') ∉ ''Y''.}} Always holds in practice. |
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* ''[[Convex set|Convexity]].'' If two vectors lie within the production set, then so do all intermediate points. This often holds as an approximation but cannot apply exactly if inputs or outputs comprise discrete units. |
* ''[[Convex set|Convexity]].'' If two vectors lie within the production set, then so do all intermediate points. This often holds as an approximation but cannot apply exactly if inputs or outputs comprise discrete units. |
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* ''Additivity'' (or ''[[free entry]]''). This property is relevant to the production set of an industry or of an economy but not for instance to a single flour mill. It means that if production vector '''''y''''' is possible, and so is |
* ''Additivity'' (or ''[[free entry]]''). This property is relevant to the production set of an industry or of an economy but not for instance to a single flour mill. It means that if production vector '''''y''''' is possible, and so is <i><b>y'</b></i>, then so too is '''''y'''''+<i><b>y'</b></i>. So if a mill can be built with a view to running in one way, and another mill can be built with a view to running in another way, then both can be built to produce the sum of intended outputs from the sum of intended inputs. Free entry is a postulate of [[perfect competition]]. |
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* ''[[Returns to scale]]'' and ''[[economies of scale]]''. See below. |
* ''[[Returns to scale]]'' and ''[[economies of scale]]''. See below. |
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==Returns to scale== |
==Returns to scale== |
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''Constant returns to scale'' mean that if '''''y''''' is in the production set, then so too is λ'''''y''''' 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. |
''Constant returns to scale'' mean that if '''''y''''' is in the production set, then so too is λ'''''y''''' 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. |
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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 {{nowrap|''F''(λ'''''y''''') > λ''F''('''''y''''')}} for all λ > 1 and {{nowrap|''F''(λ'''''y''''') < λ''F''('''''y''''')}} for all λ<1. A converse condition can be stated for ''decreasing returns to scale''. |
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 {{nowrap|''F''(λ'''''y''''') > λ''F''('''''y''''')}} for all λ > 1 and {{nowrap|''F''(λ'''''y''''') < λ''F''('''''y''''')}} for all λ<1. A converse condition can be stated for ''decreasing returns to scale''. |
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==Limitation== |
==Limitation== |
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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 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)]]). |
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==See also== |
==See also== |