# Difference between revisions of "Activity analysis"

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

− | A more general model [[#References|[a5]]] is developed by defining an activity as a pair of non-negative | + | \begin{equation*} x = A v \text { and } y = B v. \end{equation*} |

+ | |||

+ | A special case (simple linear model, [[#References|[a2]]]) assumes that $m = n$ (the number of goods and activities is the same) and $B = I$, the $( n \times n )$-matrix identity (each process produces only one good). | ||

+ | |||

+ | A more general model [[#References|[a5]]] is developed by defining an activity as a pair of non-negative $m$-vectors $( x , y )$ where the output vector $y$ is producible from the input vector $x$. Assumptions are imposed on the structure of the set $\mathcal{J}$ of all activities ($\mathcal{J}$ 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 [[#References|[a7]]]. | A non-linear activity analysis model in which production is subject to random shocks is studied in [[#References|[a7]]]. | ||

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===Production to maximize revenue given resource constraints.=== | ===Production to maximize revenue given resource constraints.=== | ||

− | Consider [[#References|[a2]]] the simple linear model | + | Consider [[#References|[a2]]] the simple linear model $( A , I )$ and assume that the firm can sell a unit of good $j$ at a price $\beta_j > 0$. In the short run the firm has a fixed supply $\mu _ { i } > 0$ of good $i$ that can be used as input. The problem is to choose a non-negative vector $v = ( v _ { j } )$ (interpreted as the firm's production schedule) so as to maximize revenue subject to the constraints of resource availability; more formally, |

− | + | \begin{equation*} \left\{ \begin{array} { l } { \operatorname{max} \ \ \sum _ { j = i } ^ { N } \beta _ { j } v _ { j } } \\ { \text { subject to } \ \ \sum _ { j = 1 } ^ { n } a _ { i j } v _ { j } \leq \mu _ { i } } \\ { v _ { j } \geq 0. } \end{array} \right. \end{equation*} | |

Problems of this type have been solved by [[Linear programming|linear programming]] methods [[#References|[a2]]]. | Problems of this type have been solved by [[Linear programming|linear programming]] methods [[#References|[a2]]]. | ||

===Planning for consumption targets.=== | ===Planning for consumption targets.=== | ||

− | Consider ([[#References|[a2]]], [[#References|[a5]]]) | + | Consider ([[#References|[a2]]], [[#References|[a5]]]) $( A , I )$, now interpreted as an economy with $n$ sectors, each producing a single good. Suppose that the sectors are operated at an intensity vector $v$. The output vector is given by $v$ and the input vector is $A v$. The net output available for consumption is $v - A v = ( I - A ) v$. The problem of attaining a consumption target is posed as follows: Given any non-negative "target" consumption vector $c$, does there exist a non-negative $v$ such that $( I - A ) v = c$? The answer is "yes" if and only if $( I - A ) ^ { - 1 } v$ exists and is non-negative. A number of equivalent conditions are available ([[#References|[a2]]], [[#References|[a5]]]). |

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

− | Consider ([[#References|[a2]]], [[#References|[a5]]], [[#References|[a9]]]) an economy with a technology | + | Consider ([[#References|[a2]]], [[#References|[a5]]], [[#References|[a9]]]) an economy with a technology $\mathcal{J}$. Assume that: |

− | i) | + | i) $\mathcal{J}$ is a closed, convex cone in $\mathbf{R} _ { + } ^ { 2 m }$; |

− | ii) | + | ii) $( 0 , y ) \in \mathcal{J}$ implies that $y = 0$ (impossibility of free production); |

− | iii) | + | iii) $( x , y ) \in \cal{J}$, $x ^ { \prime } \geq x$, $0 \leq y ^ { \prime } \leq y$ imply that $( x ^ { \prime } , y ^ { \prime } ) \in \mathcal{J}$ (free disposal); |

− | iv) for every commodity | + | iv) for every commodity $j$ there is some activity $( x ^ { j } , y ^ { j } ) \in \mathcal{J}$ such that $y _ { j } ^ { j } > 0$ (each good is produced by some activity). In view of i), the last assumption iv) can be equivalently stated as: there is a $( x , y ) \in \cal{J}$ with a strictly positive $y$. To understand von Neumann's problem of balanced growth of all goods, define the rate of expansion $\lambda ( x , y )$ of an activity $( x , y )$, with a positive $x > 0$, as |

− | + | \begin{equation*} \lambda ( x , y ) = \operatorname { sup } \{ \lambda : y \geq \lambda x \}. \end{equation*} | |

− | 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 | + | 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 $( x ^ { * } , y ^ { * } ) \in \mathcal{J}$, and a $\lambda > 0$ such that $y ^ { * } = \lambda ^ { * } x ^ { * }$, $\lambda ^ { * } = \lambda ( x ^ { * } , y ^ { * } )$, and $\lambda ^ { * } \geq \lambda ( x , y )$ for all $( x , y ) \in \cal{J}$ with $x > 0$. Moreover, there is a positive price system $p ^ { * } > 0$ such that $p ^ { * } y \leq \lambda ^ { * } p ^ { * } x$ for all $( x , y ) \in \cal{J}$. One refers to $( x ^ { * } , y ^ { * } , p ^ { * } )$ 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 $( A , I )$, 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.=== | ||

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 [[#References|[a6]]]. | 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 [[#References|[a6]]]. | ||

− | Consider the simple linear model | + | Consider the simple linear model $( A , I )$ and assume that each column of $A$ is positive. Let $y _ { 0 }$ be the strictly positive vector of initial stocks of $m$ goods. A program of resource allocation is a sequence of non-negative vectors $\langle x _ { t } , y _ { t } , c _ { t } \rangle$ satisfying $x _ { t } + c _ { t } = y _ { t }$, $x _ { t } \geq A y _ { t + 1}$ for all $t \geq 0$. A program $\langle x _ { t } , y _ { t } , c _ { t } \rangle$ is efficient if there is no other program $\langle x _ { t } ^ { \prime } , y _ { t } ^ { \prime } , c _ { t } ^ { \prime } \rangle$ from the same initial $y _ { 0 }$ such that $c _ { t } ^ { \prime } \geq c _ { t }$ for all $t$, and $c _ { t } ^ { \prime } > c _ { t }$ for some $t$. In other words, $\langle x _ { t } , y _ { t } , c _ { t } \rangle$ 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 | + | One can show that for any program $\langle x _ { t } , y _ { t } , c _ { t } \rangle$ from a given $y _ { 0 }$, |

− | + | \begin{equation*} \sum _ { t = 0 } ^ { \infty } A ^ { t } c_{ t} \leq y_0; \end{equation*} | |

− | a program is efficient if and only if | + | a program is efficient if and only if $\sum _ { t = 0 } ^ { \infty } A ^ { t } c _ { t } = y _ { 0 }$. |

− | The problem of optimal growth in the simple linear model | + | The problem of optimal growth in the simple linear model $( A , I )$ when one maximizes a welfare function over all programs from the initial stock $y _ { 0 }$ is studied in [[#References|[a1]]]. An account of the pioneering empirical work of W.W. Leontief involving activity analysis model is [[#References|[a4]]]. |

====References==== | ====References==== | ||

− | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> S. Dasgupta, T. Mitra, "Intertemporal optimality in a closed linear model of production" ''J. Econom. Th.'' , '''45''' (1988) pp. 288–315</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> D. Gale, "The theory of linear economic models" , McGraw-Hill (1960)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> T.C. Koopmans, "Three essays on the state of economic science" , McGraw-Hill (1958)</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> W.W. Leontief, "Input-output economics" , Oxford Univ. Press (1986) (Edition: Second)</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> H. Nikaido, "Convex structures and economic theory" , Acad. Press (1968)</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> M. Majumdar, "Efficient programs in infinite dimensional spaces: A complete characterization" ''J. Econom. Th.'' , '''7''' (1974) pp. 355–369</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> M. Majumdar, R. Radner, "Stationary optimal policies with discounting in a stochastic activity analysis model" ''Econometrica'' , '''51''' (1983) pp. 1821–37</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a9]</td> <td valign="top"> J. von Neumann, "A model of general economic equilibrium" ''Rev. Econom. Studies'' , '''13''' (1945-6) pp. 1–9</td></tr></table> |

## Latest revision as of 16:55, 1 July 2020

A linear activity analysis model [a9] is specified by two non-negative $( m \times n )$-matrices $( A , B )$, where $m$ is the number of producible goods (assumed to be completely divisible) and $n$ is the number of possible processes or activities of production. The matrix $A = ( a _ { ij} )$ is the input matrix: $a _ { i j}$ indicates the quantity of good $i$ required as an input if the $j$th activity is used at a unit level of intensity. The matrix $B = ( b _ { i j } )$ is the output matrix: $b _ {ij }$ is the quantity of good $i$ produced when the $j$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 $v = ( v _ { j } )$ denotes the levels of operations, an output vector $y$ is produced by using an input vector $x$ defined as:

\begin{equation*} x = A v \text { and } y = B v. \end{equation*}

A special case (simple linear model, [a2]) assumes that $m = n$ (the number of goods and activities is the same) and $B = I$, the $( n \times n )$-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 $m$-vectors $( x , y )$ where the output vector $y$ is producible from the input vector $x$. Assumptions are imposed on the structure of the set $\mathcal{J}$ of all activities ($\mathcal{J}$ 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 $( A , I )$ and assume that the firm can sell a unit of good $j$ at a price $\beta_j > 0$. In the short run the firm has a fixed supply $\mu _ { i } > 0$ of good $i$ that can be used as input. The problem is to choose a non-negative vector $v = ( v _ { j } )$ (interpreted as the firm's production schedule) so as to maximize revenue subject to the constraints of resource availability; more formally,

\begin{equation*} \left\{ \begin{array} { l } { \operatorname{max} \ \ \sum _ { j = i } ^ { N } \beta _ { j } v _ { j } } \\ { \text { subject to } \ \ \sum _ { j = 1 } ^ { n } a _ { i j } v _ { j } \leq \mu _ { i } } \\ { v _ { j } \geq 0. } \end{array} \right. \end{equation*}

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

### Planning for consumption targets.

Consider ([a2], [a5]) $( A , I )$, now interpreted as an economy with $n$ sectors, each producing a single good. Suppose that the sectors are operated at an intensity vector $v$. The output vector is given by $v$ and the input vector is $A v$. The net output available for consumption is $v - A v = ( I - A ) v$. The problem of attaining a consumption target is posed as follows: Given any non-negative "target" consumption vector $c$, does there exist a non-negative $v$ such that $( I - A ) v = c$? The answer is "yes" if and only if $( I - A ) ^ { - 1 } v$ 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 $\mathcal{J}$. Assume that:

i) $\mathcal{J}$ is a closed, convex cone in $\mathbf{R} _ { + } ^ { 2 m }$;

ii) $( 0 , y ) \in \mathcal{J}$ implies that $y = 0$ (impossibility of free production);

iii) $( x , y ) \in \cal{J}$, $x ^ { \prime } \geq x$, $0 \leq y ^ { \prime } \leq y$ imply that $( x ^ { \prime } , y ^ { \prime } ) \in \mathcal{J}$ (free disposal);

iv) for every commodity $j$ there is some activity $( x ^ { j } , y ^ { j } ) \in \mathcal{J}$ such that $y _ { j } ^ { j } > 0$ (each good is produced by some activity). In view of i), the last assumption iv) can be equivalently stated as: there is a $( x , y ) \in \cal{J}$ with a strictly positive $y$. To understand von Neumann's problem of balanced growth of all goods, define the rate of expansion $\lambda ( x , y )$ of an activity $( x , y )$, with a positive $x > 0$, as

\begin{equation*} \lambda ( x , y ) = \operatorname { sup } \{ \lambda : y \geq \lambda x \}. \end{equation*}

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 $( x ^ { * } , y ^ { * } ) \in \mathcal{J}$, and a $\lambda > 0$ such that $y ^ { * } = \lambda ^ { * } x ^ { * }$, $\lambda ^ { * } = \lambda ( x ^ { * } , y ^ { * } )$, and $\lambda ^ { * } \geq \lambda ( x , y )$ for all $( x , y ) \in \cal{J}$ with $x > 0$. Moreover, there is a positive price system $p ^ { * } > 0$ such that $p ^ { * } y \leq \lambda ^ { * } p ^ { * } x$ for all $( x , y ) \in \cal{J}$. One refers to $( x ^ { * } , y ^ { * } , p ^ { * } )$ 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 $( A , I )$, 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 $( A , I )$ and assume that each column of $A$ is positive. Let $y _ { 0 }$ be the strictly positive vector of initial stocks of $m$ goods. A program of resource allocation is a sequence of non-negative vectors $\langle x _ { t } , y _ { t } , c _ { t } \rangle$ satisfying $x _ { t } + c _ { t } = y _ { t }$, $x _ { t } \geq A y _ { t + 1}$ for all $t \geq 0$. A program $\langle x _ { t } , y _ { t } , c _ { t } \rangle$ is efficient if there is no other program $\langle x _ { t } ^ { \prime } , y _ { t } ^ { \prime } , c _ { t } ^ { \prime } \rangle$ from the same initial $y _ { 0 }$ such that $c _ { t } ^ { \prime } \geq c _ { t }$ for all $t$, and $c _ { t } ^ { \prime } > c _ { t }$ for some $t$. In other words, $\langle x _ { t } , y _ { t } , c _ { t } \rangle$ 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 $\langle x _ { t } , y _ { t } , c _ { t } \rangle$ from a given $y _ { 0 }$,

\begin{equation*} \sum _ { t = 0 } ^ { \infty } A ^ { t } c_{ t} \leq y_0; \end{equation*}

a program is efficient if and only if $\sum _ { t = 0 } ^ { \infty } A ^ { t } c _ { t } = y _ { 0 }$.

The problem of optimal growth in the simple linear model $( A , I )$ when one maximizes a welfare function over all programs from the initial stock $y _ { 0 }$ 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=50090