Neo-classic theory of the producer

The neo-classic theory of the producer is during Théorie of the consumer (micro-economics). It is about an economic modeling of the behavior of a Economic agent as a producer of goods and services. According to the neo-classic framework , one considers as producer an agent which transforms inputs ( inputs ) into outgoing products ( outputs ) according to a Fonction of production. It is also supposed that these agents do not encounter any problem of Demande and that the level of the production is conditioned by the Offre (tallies of pure competition and perfect).

Primal program and dual program

Technology and together of production

Together of production

Definitions

The producing neo-classic is a “block box”: one is interested only in what enters and to what leaves, without préocupper of the exact mode of transformation of the inputs into outgoing. In all that follows, one thus will be interested in the only list of the outgoing Nets $y=\; (y\_1,\dots ,\; y\_n)$, where the factors of negative sign correspond to the inputs, the others with the outgoing ones.

The plane of production is thus defined as the list of the outputs Nets of the producer.

One can thus define a Relation of order partial on the unit of the plans of production: a plan of $y\_a$ production is better than a $y\_b$ plan, noted $y\_a>y\_b$ if and only if the absolute value of the inputs of $y\_a$ is lower (of which at least an input strictly), either the absolute value of the outputs of $y\_a$ is higher (idem), or both.

An effective plan of production is a plan of production such as it does not exist any other vector of outputs Nets better than that of this plan of production.

The together of production is all the realizable plans of productions by a given producer.

Properties

It is considered that the economic whole of productions $Y$ checks the properties following:

• the whole of production is not-vacuum: it is possible at least nothing to produce, $0\; \backslash \; in\; Y$

• monotonicity
• divisibility : $\backslash \; forall\; (there,\; \backslash \; lambda),\; \backslash \; lambda\; \backslash \; in\; \backslash \; Rightarrow\; \backslash \; lambda\; there\; \backslash \; in\; Y$
• additivity : $\backslash \; forall\; (there,\; y\text{'})\; \backslash \; in\; Y^2,\; (y+y\text{'})\; \backslash \; in\; Y$. This condition implies in particular that there is no difficulty with the aggregation of the whole of production.
• convexity : $\backslash \; forall\; (there,\; y\text{'},\; \backslash \; alpha)\; \backslash \; in\; Y^2\; \backslash \; times,\; (\backslash \; alpha\; there\; +\; (1\; \backslash \; alpha)\; y\text{'})\; \backslash \; in\; Y$

Under these assumptions, one can define the function of production like the border of the whole of production.

Functions of productions

See also Function of production

Definition and properties

By convention, one notes $y$ the outputs and $x$ the inputs, in the form of positive quantities. This form of notation makes it possible to define functions of production associated with a plan of production $y=F\; (X)$.

According to what then precedes, if the whole of production Y is convex, F is concave .

It is almost always supposed that the neo-classic functions of production are subjected to decreasing returns at the margin , which are the equivalent for the producer of the taste for the diversity of the consumer:

Rate and elasticity of substitution

Isoquantes of production

The isoquantes of production are the whole of the combinations of inputs making it possible to obtain the same level of output.

If the functions of production are concave, the isoquantes of production are convex.

Marginal rate of technical substitution

The marginal rate of technical substitution (TMST) of Y for X is the positive relationship between quantity there factors Y which it is possible to give up and quantity X of X which it is possible to substitute to him to maintain constant the level of production. TMST = marginal productivity of X/productivité marginal of Y

By analogy with the marginal Rate of substitution of the consumer, one defines the rate of substitution technique' like: $TST=\; \backslash \; frac\; \{\backslash \; partial\; partial\; F\; \backslash \; x\_1\}\; \{\backslash \; partial\; F/\backslash \; partial\; x\_2\}$

The TSMT is the opposite of the slope of the isoquantes of production (see supra ).

Outputs of scale

detailed Article: Outputs of scale

A function of production presents increasing Rendements of scale

• if $F\; (\backslash \; lambda\; X)\; >\; \backslash \; lambda\; F\; (X)$;
• decreasing if ;
• constant if $F\; (\backslash \; lambda\; X)\; =\; \backslash \; lambda\; F\; (X)$;

Elasticity of substitution

The elasticity of substitution $\backslash \; sigma$ is a measurement of the curve of the isoquantes:

$\backslash \; sigma=\; \backslash \; frac\; \{\backslash \; frac\; \{\backslash \; Delta\; x\_2/x\_1\}\; \{x\_2/x\_1\}\}\; \{\backslash \; frac\; \{\backslash \; Delta\; TST\}\; \{TST\}\}\; =\; \backslash \; frac\; \{\backslash \; frac\; \{D\; (x\_2/x\_1)\}\{x\_2/x\_1\}\}\; \{\backslash \; frac\; \{dTST\}\; \{TST\}\}\; =\; \backslash \; frac\; \{dln\; (x\_1/x\_2)\}\{dln\; (TST)\}$

Primal program: the maximization of the profit

It is supposed that the producers are taking price , and more generally one places within the framework of the pure competition and perfect.

In first approach, one considers that the producer seeks to measure his profit $\backslash \; Pi=p.y-c.x$, where p and C are respectively the price of the outputs and the cost of the inputs. The whole of the production costs taken into account is very extensive, since it include the hiring of the buildings, the remuneration of the contractor, the hiring of the capital, etc

The program of the producer is thus written:

The resolution of this program is done by means of a Lagrangien:

$\backslash \; mathcal\; \{L\}\; =p.y-c.x-\; \backslash \; lambda\; (y-F\; (X))$

who gives the conditions to the first order:

One draws the requirement from it from optimality:

$TST=\; \backslash \; frac\; \{\backslash \; frac\; \{\backslash \; partial\; F\; (X)\}\{\backslash \; partial\; x\_i\}\}\; \{\backslash \; frac\; \{\backslash \; partial\; F\; (X)\}\{\backslash \; partial\; x\_j\}\}\; =\; \backslash \; frac\; \{c\_i\}\; \{c\_j\}$

It also comes:

$\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; x\_i\}\; =\; \backslash \; frac\; \{c\_i\}\; \{p\}$

what means that the marginal productivity is equal to the real costs.

One can then finish the resolution of the program to obtain the offer of the firm $y\; (p,\; c)$, the requests for factors and the profit. The Lemme of Hotelling highlights a relation between the first and the last of these factors: $y\; (p,\; c)=\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; pi\}\; \{\backslash \; partial\; p\}\; (p,\; c)$ One shows this result by the Théorème of the envelope.

Under the assumptions above, the function of offer is homogeneous of degree zero compared to $p$ and $c$, and increasing compared to $p$.

For the same reasons, the requests for factors are also homogeneous of degree zero in $p$ and $c$.

Dual program: the minimization of the costs

One can see the problem of the producer like the maximization of the profit, or conversely like a minimization of his production costs. The stake is then to show that the two programs lead to the same result in terms of level of production and request of the factors, therefore of profit.

See too

Internal bonds

• Microéconomie
• Function of production
• Theory of the consumer (micro-economics)
• Outputs of scale

External bonds

• Theory of the producer
• Behavior of the consumer and the producer, concept of surplus and analyzes public policies
• the theory of production and costs

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