Inverse problems in models of distribution of resources A. A. - PowerPoint PPT Presentation

Inverse problems in models of distribution of resources A. A. Shananin

Contents 1. Introduction. New problems of mathematical economics under conditions of globalization. 2. The Houthakker–Johansen model of distribution of resources with substitution of production factors at the micro-level. 3. Aggregation and the inverse problem statement. The Bernstein theorems on characterisation of the Radon transform of non-negative measures with support in a cone. 4. A model of distribution of resources with substitution of production factors at the micro-level. New problems of integral geometry. 5. The problem of estimation of elasticity of substitution of production factors at the micro-level and its relation to study of combinatorial structures.

The Houthakker–Johansen model ◮ x = ( x 1 , . . . , x n ) – technology; ◮ µ ( dx ) – non-negative measure describing the distribution of powers over technologies; ◮ l = ( l 1 , . . . , l n ) – vector of available production factors; ◮ u ( x ) – technology loading coefficient; ◮ F ( l ) – production function, relating amounts of product to amounts of resources used in production process

The problem of distribution of resources in the Houthakker–Johansen model �  u ( x ) µ ( dx ) → max ,      R n   +   � (1) xu ( x ) µ ( dx ) ≤ l ,    R n   +    0 ≤ u ( x ) ≤ 1 . 

The generalized Neumann–Pearson lemma 1. If l ≥ 0 then the problem (1) has a solution. 2. If u 0 ( x ) is a solution to problem (1) then there exist Lagrange multipliers p 0 ≥ 0, p = ( p 1 , . . . , p n ) ≥ 0, not simultaneously equal to zero, such that � 0 for almost all x w.r.t. µ such that p 0 < px ; u 0 ( x ) = 1 for almost all x w.r.t. µ such that p 0 > px ; � � � p j l j − x j u 0 ( x ) µ ( dx ) = 0 , j = 1 , . . . , n . R n + � 3. If p 0 > 0, p = ( p 1 , . . . , p n ) ≥ 0, l = + x θ ( p 0 − px ) µ ( dx ) R n then u ( x ) = θ ( p 0 − px ) is a solution to (1).

Duality of production and profit functions ◮ Profit function � Π( p , p 0 ) = ( p 0 − px ) + µ ( dx ) , (2) R n + ◮ Production function F ( l ) is concave, non-decreasing and continuous on R n + . Π( p , p 0 ) = sup ( p 0 F ( l ) − pl ) , l ≥ 0 F ( l ) = 1 p ≥ 0 (Π( p , p 0 ) + pl ) . inf p 0

Aggregation ◮ Let F 0 ( X 0 ) be a positively homogeneous, concave, positive, continuous on R n + utility function; ◮ Let q 0 ( p ) be the price index: pX 0 q 0 ( p ) = inf F 0 ( X 0 ) , { X 0 ≥ 0 | F 0 ( X 0 ) > 0 } pX 0 F 0 ( X 0 ) = inf q 0 ( p ) . { p ≥ 0 | q 0 ( p ) > 0 } ◮ X j = ( X j 1 , . . . , X j m ) — amounts of products of other industries used in j -th industry; ◮ l j = ( l j 1 , . . . , l j n ) — amounts of raw resources used in j -th industry; ◮ F j ( X j , l j ) — production function of j -th industry.

The problem of distribution of resources (the nonlinear input-output model) l = ( l 1 , . . . , l n ) — total amounts of available raw resources.  F 0 ( X 0 ) → max ,    m   �  F j ( X j , l j ) ≥ X i j , j = 1 , . . . , m ,      i = 0    m (3) l j ≤ l , �    j = 1    X 0 ≥ 0 , X 1 ≥ 0 , . . . , X m ≥ 0 ,       l 1 ≥ 0 , . . . , l m ≥ 0 .  

Equilibrum market mechanisms Let l > 0. Then vectors X 0 , . . . , X m , l 1 , . . . , l m satisfying the restrictions of problem (3) solve this problem if and only if there exist p 0 > 0, p = ( p 1 , . . . , p m ) ≥ 0, s = ( s 1 , . . . , s n ) ≥ 0 such that ◮ X 0 ∈ Arg max { p 0 F 0 (˜ X ) − p ˜ X | ˜ X ≥ 0 } ; X , ˜ X − s ˜ X ≥ 0 , ˜ ◮ ( X j , l j ) ∈ Arg max { p j F j (˜ l ) − p ˜ l | ˜ l ≥ 0 } , j = 1, . . . , m ; F j ( X j , l j ) − � m ◮ p j � i = 0 X i � = 0 , j = 1 , . . . , m ; j l j − � m ◮ s j � i = 0 l i � = 0 , j = 1 , . . . , m . j

Aggregated macro-description X , ˜ X − s ˜ p j F j (˜ l ) − p ˜ ◮ Π j ( s , p ) = sup ˜ � � l – profit function X ≥ 0 , ˜ l ≥ 0 of j -th industry; ◮ F A ( l ) – aggregated profuction function. Variational principle (dual problem) �� m ◮ Π A ( s , p 0 ) = max � j = 1 Π j ( s , p ) | p ≥ 0 , s ≥ 0 , q 0 ( p ) ≥ p 0 ; ◮ Π A ( s , p 0 ) – aggregated profit function.

Inverse problem statement Π A ( s , p 0 ) = sup p 0 F A ( l ) − sl � � , l ≥ 0 F A ( l ) = 1 Π A ( s , p 0 ) + sl � � inf . p 0 s ≥ 0 Find a non-negative measure µ A ( dx ) supported in R n + and such that � Π A ( s , p 0 ) = � � p 0 − sx + µ A ( dx ) . R n +

Relation to integral geometry problems ∂ 2 � Π A ( s , p 0 ) = µ A ( dx ) , ∂ p 2 0 sx = p 0 + ∞ � ∂ Π A ( s , τ ) � � � e − sx µ ( dx ) = e − τ d τ . ∂τ R n 0 + Theorem (G. M. Henkin, A. A. Shananin) Suppose that a measure µ ( dx ) satisfies the conditions ◮ � + e − A | x | | µ | ( dx ) < ∞ for some A > 0 , ( 4 ) R n ◮ � + ( p 0 − sx ) + µ ( dx ) = 0 for all p 0 > 0 , s ∈ K, where K is an R n open cone in R n + . Then µ ( dx ) ≡ 0 .

Characterization theorem (G. M. Henkin, A. A. Shananin) A function Π( s , p 0 ) can be represented in the form � ( s , p 0 ) ∈ R n + 1 Π( s , p 0 ) = ( p 0 − sx ) + µ ( dx ) , , + R n + where µ ( dx ) is a non-negative measure supported in R n + and satisfying condition (4) if and only if 1. Π( s , p 0 ) is a positively homogeneous convex function on R n + 1 + ∂ 2 and for fixed s ∈ R n + the measure ∂τ 2 Π( s , τ ) decays exponentially as τ → + ∞ ; � + ∞ � ∂ Π( s ,τ ) e − τ d τ � belongs to C ∞ ( R n 2. function G ( s ) = + ) and ∂τ 0 for some open cone Γ ⊂ int R n + and some s ∈ Γ for all λ > 0, ξ 1 , . . . , ξ k ∈ Γ , k ≥ 1 the following inequality holds: ∂ ( − 1 ) k D ξ 1 · · · D ξ k G ( λ s ) ≥ 0 , � D ξ = ξ = ( ξ 1 , . . . , ξ n ) . ξ j , ∂ s j j

Example 1 Let n = 2, let F CES be a CES production function: � − γ/ρ , α 1 l − ρ 1 + α 2 l − ρ � F CES ( l 1 , l 2 ) = α 1 , α 2 > 0 , ρ ≥ 1 , 0 < γ < 1 . 2 Then the profit function equals ρ ρ 1 1 1 � γ ( 1 + ρ ) γ 1 − γ ( 1 − γ ) p ρ ( 1 − γ ) . 1 − γ � 1 + ρ 1 + ρ 1 + ρ 1 + ρ Π CES ( s 1 , s 2 , p 0 ) = γ + α α s s 0 1 1 2 2 For ρ > − 1 there exists a distribution of powers over technologies corresponding to these functions.

Example 2 ◮ Let m = 2, n = 2, F 0 ( X 0 1 , X 0 2 ) = min ( X 0 1 , X 0 2 ) , µ 1 ( dx ) = k 0 δ ( x − z ) , z = ( z 1 , z 2 ) , y j = ( y j 1 , y j µ 2 ( dx ) = k 1 δ ( x − y 1 ) + k 2 δ ( x − y 2 ) , 2 ) , j = 1 , 2 , where k 1 + k 2 > k 0 , y 1 1 > y 2 1 , y 1 2 > y 2 2 . Then � � � � � Π A ( s , p 0 )= max p 0 − s ( z + y 1 ) p 0 − s ( z + y 2 ) ( k 0 − k 2 ) + + + min ( k 0 , k 2 ) + , � � � � � p 0 − s ( z + y 1 ) p 0 − s ( z + y 2 ) min ( k 0 , k 1 ) + +( k 0 − k 2 ) + . + + | sy 2 ≤ sy 1 � + | sy 1 ≤ sy 2 � ◮ Denote K 1 = � s ∈ R 2 � s ∈ R 2 , K 2 = . � Π A ( s , p 0 ) = max Then Π j ( s , p 0 ) , Π j ( s , p 0 ) = ( p 0 − sx ) + µ j ( dx ) , j R 2 + Π A ( s , p 0 ) = Π j ( s , p 0 ) R n + = ∪ n for s ∈ K j ; j = 1 K j , � + ∞ � ∂ Π j ( s , τ ) � G ( s ) = max G j ( s ) , G j ( s ) = e − τ d τ . ∂τ j 0

Stable correspondances (A. V. Karzanov, A. A. Shananin) Let X = { x 1 , . . . , x m } ⊂ R n + , Y = { y 1 , . . . , y m } ⊂ R n + and C ⊂ R n + be a cone. Definition. A bijection γ : X → Y is called a C -stable correspondance if for any x i , x j ∈ X , p ∈ C the inequality px i < px j implies p γ ( x i ) ≤ p γ ( x j ) . Theorem A bijection γ : X → Y is a C-stable correspondance if and only if for any x i , x j ∈ X, x i � = x j ◮ if x j − x i ∈ C ∗ then γ ( x j ) − γ ( x i ) ∈ C ∗ ; ◮ if x j − x i �∈ C ∗ , x i − x j �∈ C ∗ then there exist such λ ≥ 0 , µ ≥ 0 , λ + µ > 0 that λ ( x j − x i ) = µ ( γ ( x j ) − γ ( x i )) .

A model of industry with substitution of production factors at the micro-level. Let f ( u ) be a positively homogeneous of first order, concave, continuous function on R n + , positive on int R n + . A technology is given by a vector x = ( x 1 , . . . , x n ) . � � u 1 x 1 , . . . , u n A production function at the micro-level: f . x n Examples: ◮ The Leontieff function with constant proportions f ( u ) = min ( u 1 , . . . , u n ) corresponds to the production function at the micro-level in the Houthakker–Johansen model. ◮ CES-function � − 1 /ρ = u 1 ⊕ ρ · · · ⊕ ρ u n , u − ρ + · · · + u − ρ � f ( u ) = ρ ≥ − 1 . n 1

The problem of distribution of resources in presence of substitution of production factors at the micro-level  � � � u 1 ( x ) , . . . , u n ( x ) �� min 1 , f µ ( dx ) → max u ,   x 1 x n    R n   +   � (5) u j ( x ) µ ( dx ) ≤ l j , j = 1 , . . . , n ,    R n   +    � � u ( x ) = u 1 ( x ) , . . . , u n ( x ) ≥ 0 .  pu Put q ( p ) = inf f ( u ) , { u ≥ 0 | f ( u ) > 0 } � � p ◦ x = ( p 1 x 1 , . . . , p n x n ) , π ( x , p , p 0 ) = p 0 − q ( p ◦ x ) + .