# Difference between revisions of "Activity analysis"

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012060.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012060.png" /></td> </tr></table> | ||

− | One can show that under assumptions i)–iv) there is a maximal rate of expansion that the economy can attain. More formally, there are an activity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012061.png" />, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012062.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012064.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012065.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012066.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012067.png" />. Moreover, there is a positive price system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012069.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012070.png" />. One refers to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012071.png" /> as a von Neumann equilibrium. It has played a prominent role in characterizing "turnpike" properties of a class of finite horizon planning models [[#References|[a8]]]. For the simple linear model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012072.png" />, the existence of a von Neumann equilibrium can be studied by using the Perron–Frobenius | + | One can show that under assumptions i)–iv) there is a maximal rate of expansion that the economy can attain. More formally, there are an activity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012061.png" />, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012062.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012064.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012065.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012066.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012067.png" />. Moreover, there is a positive price system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012069.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012070.png" />. One refers to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012071.png" /> as a von Neumann equilibrium. It has played a prominent role in characterizing "turnpike" properties of a class of finite horizon planning models [[#References|[a8]]]. For the simple linear model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120120/a12012072.png" />, the existence of a von Neumann equilibrium can be studied by using the [[Perron–Frobenius theorem]] on the dominant characteristic root of a positive matrix [[#References|[a5]]] (cf. also [[Frobenius matrix|Frobenius matrix]]). |

===Efficient allocation of resources.=== | ===Efficient allocation of resources.=== |

## Revision as of 12:28, 23 November 2014

A linear activity analysis model [a9] is specified by two non-negative -matrices , where is the number of producible goods (assumed to be completely divisible) and is the number of possible processes or activities of production. The matrix is the input matrix: indicates the quantity of good required as an input if the th activity is used at a unit level of intensity. The matrix is the output matrix: is the quantity of good produced when the th activity is operated at unit intensity. The processes can be interpreted as "different ways of doing things" in a firm ([a2], [a3]) or different "sectors" of an economy ([a3], [a4]). It is assumed that these can be operated at any non-negative level. If a non-negative column vector denotes the levels of operations, an output vector is produced by using an input vector defined as:

A special case (simple linear model, [a2]) assumes that (the number of goods and activities is the same) and , the -matrix identity (each process produces only one good).

A more general model [a5] is developed by defining an activity as a pair of non-negative -vectors where the output vector is producible from the input vector . Assumptions are imposed on the structure of the set of all activities ( is called the technology or production possibility set of the firm or the economy).

A non-linear activity analysis model in which production is subject to random shocks is studied in [a7].

Some theoretical applications are sketched below.

## Contents

### Production to maximize revenue given resource constraints.

Consider [a2] the simple linear model and assume that the firm can sell a unit of good at a price . In the short run the firm has a fixed supply of good that can be used as input. The problem is to choose a non-negative vector (interpreted as the firm's production schedule) so as to maximize revenue subject to the constraints of resource availability; more formally,

Problems of this type have been solved by linear programming methods [a2].

### Planning for consumption targets.

Consider ([a2], [a5]) , now interpreted as an economy with sectors, each producing a single good. Suppose that the sectors are operated at an intensity vector . The output vector is given by and the input vector is . The net output available for consumption is . The problem of attaining a consumption target is posed as follows: Given any non-negative "target" consumption vector , does there exist a non-negative such that ? The answer is "yes" if and only if exists and is non-negative. A number of equivalent conditions are available ([a2], [a5]).

### Balanced growth at a maximal rate and the von Neumann equilibrium.

Consider ([a2], [a5], [a9]) an economy with a technology . Assume that:

i) is a closed, convex cone in ;

ii) implies that (impossibility of free production);

iii) , , imply that (free disposal);

iv) for every commodity there is some activity such that (each good is produced by some activity). In view of i), the last assumption iv) can be equivalently stated as: there is a with a strictly positive . To understand von Neumann's problem of balanced growth of all goods, define the rate of expansion of an activity , with a positive , as

One can show that under assumptions i)–iv) there is a maximal rate of expansion that the economy can attain. More formally, there are an activity , and a such that , , and for all with . Moreover, there is a positive price system such that for all . One refers to as a von Neumann equilibrium. It has played a prominent role in characterizing "turnpike" properties of a class of finite horizon planning models [a8]. For the simple linear model , the existence of a von Neumann equilibrium can be studied by using the Perron–Frobenius theorem on the dominant characteristic root of a positive matrix [a5] (cf. also Frobenius matrix).

### Efficient allocation of resources.

The problem of attaining an efficient allocation of resources by using a decentralized system of decision making in which prices are used to coordinate individual decisions has long been of interest to economists (theorists as well as policy makers). Although the problem was first posed and solved in a static model (somewhat artificial, since "production" takes time), the subtleties that are involved were exposed in the context of an infinite horizon economy [a6].

Consider the simple linear model and assume that each column of is positive. Let be the strictly positive vector of initial stocks of goods. A program of resource allocation is a sequence of non-negative vectors satisfying , for all . A program is efficient if there is no other program from the same initial such that for all , and for some . In other words, is efficient if there is no other program from the same initial stock that generates at least as much consumption of all goods in all periods, and strictly more of some good in some period.

One can show that for any program from a given ,

a program is efficient if and only if .

The problem of optimal growth in the simple linear model when one maximizes a welfare function over all programs from the initial stock is studied in [a1]. An account of the pioneering empirical work of W.W. Leontief involving activity analysis model is [a4].

#### References

[a1] | S. Dasgupta, T. Mitra, "Intertemporal optimality in a closed linear model of production" J. Econom. Th. , 45 (1988) pp. 288–315 |

[a2] | D. Gale, "The theory of linear economic models" , McGraw-Hill (1960) |

[a3] | T.C. Koopmans, "Three essays on the state of economic science" , McGraw-Hill (1958) |

[a4] | W.W. Leontief, "Input-output economics" , Oxford Univ. Press (1986) (Edition: Second) |

[a5] | H. Nikaido, "Convex structures and economic theory" , Acad. Press (1968) |

[a6] | M. Majumdar, "Efficient programs in infinite dimensional spaces: A complete characterization" J. Econom. Th. , 7 (1974) pp. 355–369 |

[a7] | M. Majumdar, R. Radner, "Stationary optimal policies with discounting in a stochastic activity analysis model" Econometrica , 51 (1983) pp. 1821–37 |

[a8] | R. Radner, "Paths of economic growth that are optimal with regard only to final states: A turnpike theorem" Rev. Econom. Studies , 28 (1961) pp. 98–104 |

[a9] | J. von Neumann, "A model of general economic equilibrium" Rev. Econom. Studies , 13 (1945-6) pp. 1–9 |

**How to Cite This Entry:**

Activity analysis.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Activity_analysis&oldid=15443