Technological set

Technological set - a concept used in microeconomics , formalizing the set of all technologically feasible vectors of net output.

Content

Definition

Let the economy have $N$ ${\ displaystyle N}$ good In the process of production of them $n$ ${\ displaystyle n}$ benefits are being spent. Denote the vector of these benefits (costs) $x$ ${\ displaystyle x}$ (vector dimension $n$ ${\ displaystyle n}$ ) Other $m=N-n$ ${\ displaystyle m = Nn}$ goods are produced in the production process (vector dimension - $m$ ${\ displaystyle m}$ ) Denote the vector of these benefits $y$ ${\ displaystyle y}$ . Then the vector $z=(-x,y)$ ${\ displaystyle z = (- x, y)}$ (dimension - $N$ ${\ displaystyle N}$ ) is called the net issue vector. The totality of all technologically feasible vectors of clean issues make up the technological set . This is actually a subset of space. $R^{N}$ ${\ displaystyle R ^ {N}}$ .

Properties

Non-emptiness : the technological set is not empty. Non-emptiness means a fundamental possibility of production.
Admissibility of inaction : the zero vector belongs to the technological set. This formal property means that zero output at zero cost is acceptable.
Closure : the technological set contains its own boundary and the limit of any sequence of technologically feasible vectors of clean issues also belongs to the technological set.
Freedom of spending : if a given vector $z$ ${\ displaystyle z}$ belongs to the technological set, then any vector belongs to it $z'\leqslant z$ ${\ displaystyle z '\ leqslant z}$ . This means that formally the same output can be produced at high cost.
Lack of “cornucopia” : of non-negative vectors of clean output, the technological set contains only the zero vector. This means that non-zero costs are required to produce products in positive quantities.
Irreversibility : for any valid vector $z$ ${\ displaystyle z}$ opposite vector $-z$ ${\ displaystyle -z}$ does not belong to the technological set. That is, it is impossible to produce resources from the output in the same amount in which they are used for the production of these products.
Additivity : the sum of two valid vectors is also a valid vector. That is, a combination of technologies is allowed.
Properties related to returns on scale:
- Non-increasing returns to scale : for anyone $\lambda \in (0;1)$ ${\ displaystyle \ lambda \ in (0; 1)}$ if z belongs to the technological set, then $\lambda z$ ${\ displaystyle \ lambda z}$ also belongs to the technological array.
- Non-decreasing returns to scale : for anyone $\lambda >1$ ${\ displaystyle \ lambda> 1}$ if z belongs to the technological set, then $\lambda z$ ${\ displaystyle \ lambda z}$ also belongs to the technological array.
- Constant returns to scale : simultaneous fulfillment of the two previous properties, i.e. for any positive $\lambda$ ${\ displaystyle \ lambda}$ if $z$ ${\ displaystyle z}$ belongs to the technological set then $\lambda z$ ${\ displaystyle \ lambda z}$ also belongs to the technological array. The constant recoil property means that the technological set is a cone.

8. Bulge : for any two valid vectors $z_{1},z_{2}$ ${\ displaystyle z_ {1}, z_ {2}}$ any vectors are also valid $\alpha z_{1}+(1-\alpha )z_{2}$ ${\ displaystyle \ alpha z_ {1} + (1- \ alpha) z_ {2}}$ where $0<\alpha \leqslant 1$ ${\ displaystyle 0 <\ alpha \ leqslant 1}$ . The property of bulge means the ability to "mix" technology. It, in particular, is satisfied if the technological set has the property of additivity and non-increasing returns to scale. Moreover, in this case, the technological set is a convex cone.

Effective Technology Boundary

Acceptable Technology $z$ ${\ displaystyle z}$ called effective if there is no other different from it acceptable technology $z'\geqslant z$ ${\ displaystyle z '\ geqslant z}$ . Many effective technologies form the effective frontier of the technological set.

If the condition of freedom of expenditure and closure of the technological set is fulfilled, then it is impossible to infinitely increase the production of one good without reducing the output of others. In this case, for any valid technology $z$ ${\ displaystyle z}$ there is an effective technology $z'\geqslant z$ ${\ displaystyle z '\ geqslant z}$ . In this case, instead of the entire technological set, only its effective boundary can be used. Usually, the effective boundary can be defined by some production function.

Production Function

Let's consider single-product technologies $(-x,y)$ ${\ displaystyle (-x, y)}$ where $y$ ${\ displaystyle y}$ - dimension vector $m=1$ ${\ displaystyle m = 1}$ , but $x$ ${\ displaystyle x}$ Is the vector of costs of dimension $n$ ${\ displaystyle n}$ . Consider the set $X$ ${\ displaystyle X}$ including all possible cost vectors $x$ ${\ displaystyle x}$ such that for each $x$ ${\ displaystyle x}$ exist $y$ ${\ displaystyle y}$ such that the net issue vectors $(-x,y)$ ${\ displaystyle (-x, y)}$ belong to the technological multitude.

Numeric function $f(x)$ ${\ displaystyle f (x)}$ on $X$ ${\ displaystyle X}$ called a production function if for each given cost vector $x$ ${\ displaystyle x}$ value $f(x)$ ${\ displaystyle f (x)}$ determines the maximum value of the allowable release $y$ ${\ displaystyle y}$ (such that the net output vector (-x, y) belongs to the technological set).

Any point of the effective boundary of a technological set can be represented as $(-x,f(x))$ ${\ displaystyle (-x, f (x))}$ , and the converse is true if $f(x)$ ${\ displaystyle f (x)}$ is an increasing function (in this case $y=f(x)$ ${\ displaystyle y = f (x)}$ - effective boundary equation). If the technological set has the property of freedom of expenditure and allows the production function to be described, then the technological set is determined on the basis of the inequality $y\leqslant f(x)$ ${\ displaystyle y \ leqslant f (x)}$ .

In order for the technological set to be set using the production function, it is enough that for any $x$ ${\ displaystyle x}$ a bunch of $F(x)$ ${\ displaystyle F (x)}$ allowable issues at a given cost $x$ ${\ displaystyle x}$ , was limited and closed. In particular, this condition is satisfied if the technological set satisfies the properties of isolation, non-increasing returns to scale and lack of cornucopia.

If the technological set is convex, then the production function is concave and continuous on the interior of the set $X$ ${\ displaystyle X}$ . If the condition of freedom of expenditure is satisfied, then $f(x)$ ${\ displaystyle f (x)}$ is a non-decreasing function (in this case, the convexity of the technological set also follows from the concavity of the function). Finally, if the condition for the absence of the cornucopia and the permissibility of inaction are fulfilled simultaneously, then $f(0)=0$ ${\ displaystyle f (0) = 0}$ .

If the production function is differentiable, then local scale elasticity can be determined in the following equivalent ways:

$e(x)={\frac {df(\lambda x)}{d\lambda }}\cdot {\frac {\lambda }{f(x)}}|_{\lambda =1}={\frac {f'(x)x}{f(x)}}$ ${\ displaystyle e (x) = {\ frac {df (\ lambda x)} {d \ lambda}} \ cdot {\ frac {\ lambda} {f (x)}} | _ {\ lambda = 1} = {\ frac {f '(x) x} {f (x)}}}$

Where $f'(x)$ ${\ displaystyle f '(x)}$ Is the gradient vector of the production function.

Having thus determined the elasticity of scale, it can be shown that if a technological set has the property of constant returns to scale, then $e(x)=1$ ${\ displaystyle e (x) = 1}$ if decreasing returns to scale, then $e(x)\leqslant 1$ ${\ displaystyle e (x) \ leqslant 1}$ if increasing returns, then $e(x)\geqslant 1$ ${\ displaystyle e (x) \ geqslant 1}$ .

Manufacturer Task

If a price vector is specified $p$ ${\ displaystyle p}$ then the product $p z$ ${\ displaystyle pz}$ represents the profit of the manufacturer. The manufacturer’s task is to search for such a vector $z$ ${\ displaystyle z}$ so that for a given price vector, the profit is maximum. The set of prices of goods at which this problem has a solution is denoted by $P$ ${\ displaystyle P}$ . It can be shown that for a nonempty, closed technological set with a non-increasing return on scale, the producer's problem has a solution on the set of prices $P$ ${\ displaystyle P}$ giving negative profit in the so-called recessive directions (these are vectors $z$ ${\ displaystyle z}$ technological set for which, for any non-negative $\lambda$ ${\ displaystyle \ lambda}$ vectors $\lambda z$ ${\ displaystyle \ lambda z}$ also belong to the technological array). In particular, if the set of recessive directions coincides with $R_{-}^{N}$ ${\ displaystyle R _ {-} ^ {N}}$ , then the solution exists at any positive prices.

Profit function $\pi (p)$ ${\ displaystyle \ pi (p)}$ defined as $pz(p)$ ${\ displaystyle pz (p)}$ where $z(p)$ ${\ displaystyle z (p)}$ - solution of the manufacturer’s task at given prices (this is the so-called supply function, possibly multi-valued). The profit function is positively homogeneous (first degree), i.e. $\pi (\lambda p)=\lambda \pi (p)$ ${\ displaystyle \ pi (\ lambda p) = \ lambda \ pi (p)}$ and continuous on the inside $P$ ${\ displaystyle P}$ . If the technological set is strictly convex, then the profit function is also continuously differentiable. If the technological set is closed, then the profit function is convex on any convex subset of permissible prices $P$ ${\ displaystyle P}$ .

Function (display) of the sentence $z(p)$ ${\ displaystyle z (p)}$ is positively homogeneous of degree zero. If the technological set is strictly convex, then the supply function is unique on P and continuous on the inside $P$ ${\ displaystyle P}$ . If the sentence function is twice differentiable, then the Jacobi matrix of this function is symmetric and non-negative definite.

If a technological set is represented by a production function, then profit is defined as $pf(x)-wx$ ${\ displaystyle pf (x) -wx}$ where $w$ ${\ displaystyle w}$ - vector of prices for factors of production, $p$ ${\ displaystyle p}$ in this case, the price of products. Then for any internal solution (i.e., belonging to the interior $X$ ${\ displaystyle X}$ ) the manufacturer’s task is the equality of the marginal product of each factor to its relative price, that is, in vector form $f'(x)=w/p$ ${\ displaystyle f '(x) = w / p}$ .

If the profit function is specified $\pi (p)$ ${\ displaystyle \ pi (p)}$ being a twice continuously differentiable, convex and positively homogeneous (first degree) function, we can restore the technological set as a set containing for any non-negative price vector $p$ ${\ displaystyle p}$ net issue vectors $z$ ${\ displaystyle z}$ satisfying the inequality $pz\leqslant \pi (p)$ ${\ displaystyle pz \ leqslant \ pi (p)}$ . It can also be shown that if the offer function is positively homogeneous of degree zero and the matrix of its first derivatives is continuous, symmetric and non-negative definite, then the corresponding profit function satisfies the above requirements (the converse is also true).