Linear Models of Production
Larry Blume
Cornell University & The Santa Fe Institute & IHS
Introduction
Linear models of economic activity have a long history, going back
to Soviet economists’ material balance tables and (arguably) two
centuries farther to the physiocrats. The modern analysis of linear
models begins with Kantorovich (1939), Leontief (1936, 1941) and
von Neumann (1937).
These models sit between general and partial equilibrium. They
focus on interdependencies of production sectors, but ignore the
demand side.
Important sources are the papers in Koopmans (1951) Activity
Analysis of Production and Allocation, and Gale (1960) The
Theory of Linear Economic Models.
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Contemporary Applications
Do not think this stuff is old and irrelevant!!
Game Theory
Hart, S. and D. Schmeidler (1989), “The existence of
correlated equilibria”, Mathematics of Operations Research
Myerson, R. (1997), “Dual reduction and elementary games”,
GEB.
Mechanism Design
Vohra, R. (2011), Mechanism Design: A Linear Programming
Approach.
Cai, Y. et al. (2011), “On optimal multidimensional
mechanism design”, ACM SIGecom Exchanges.
Networks
Acemoglu et al. (2011), “The network origins of aggregate
fluctuations”.
Blume et al. (2009), “Trading networks with price-setting
agents”. GEB.
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Why the Model?
What do we want to understand with linear production models?
The allocation of factors
The choice of techniques
Factor prices
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The Linear Model
There are n non-primary commodities. All are produced.
The matrix A = (aij ) describes production activities; aij ≥ 0 is
the the amount of good i used to produce one unit of
activity j output.
Each activity produces a distinct good, and only that good.
There is neither joint production nor by-products.
Production is linearly homogeneous. Doubling the level of an
activity doubles both input requirements and outputs.
There are no cross-activity externalities, so production is the
sum of the output of all activities.
There is a single primary commodity, commodity 0, say labor.
One unit of good j requires a0j units of labor.
In the simple model, only activity j produces good j.
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Notation
x≫y
for all i , xi > yi ;
x >y
for all i , xi ≥ yi and for some j, xj > yj ;
x ≥y
for all i , xi ≥ yi .
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Production without Labor: I
Even with large amounts of labor, the economic system may not
be capable of producing positive net outputs, because factor
demand is more than can be supplied. To investigate this, neglect,
for the moment, the labor constraint; suppose a0j ≡ 0.
To achieve gross output x of goods 1 through n, activity i must be
run at level xi . Then net output will be
y = (I − A)x.
(1)
What levels of net output (for consumption) are feasible to
produce? That is, for which y does equation (1) have a
nonnegative solution?
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Production without Labor: II
Definition: An activity matrix is productive if there exists x ∗ ≥ 0
such that x ∗ ≫ Ax ∗ .
This is sometimes called the Hawkins-Simon condition.
Example of a non-productive matrix:
0 2
A=
2 0
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Production without Labor: III
Theorem 1: If the matrix A is productive, then for any y ≥ 0 the
equation system
(I − A)x = y
has a non-negative solution.
Lemma 1: If A is productive and x ≥ Ax, then x ≥ 0.
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Production without Labor: IV
Proof of Lemma 1: By assumption there is an x̄ ≥ 0 such that
x̄ ≫ Ax̄. Ax̄ ≥ 0 so x̄ ≫ 0.
Suppose x ≥ Ax but x 6≥ 0. Let θ = inf{θ̄ : x + θ̄x̄ ≥ 0}. Let
x ′ = x + θx̄, and note that x ′ ≥ 0 and x ′ will have at least one
coordinate xi′ , say i = 1, which is 0.
x ′ = x + θx̄
≫ Ax + θAx̄ ≥ 0,
and so x1′ > 0, which is a contradiction.
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Production without Labor: V
Corollary 1: I − A has full rank n.
Proof: If (I − A)x = 0, then (I − A)(−x) = 0. The lemma implies
x ≥ 0 and −x ≥ 0, so x = 0.
Proof of the Theorem: Since I − A is invertible, (I − A)x = y has
a unique solution x. Since y ≥ 0, x ≥ Ax and so x ≥ 0.
Corollary 2: The matrix A is productive iff (I − A)−1 is
non-negative.
Proof: ?
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Interpreting the Model
Lemma: If A is productive, then An → 0.
Proof: There is an x ∗ ≫ 0 such that x ∗ ≫ Ax ∗ . So there is a
0 < λ < 1 such that λx ∗ ≫ Ax ∗ .PInduct to see that λn x ∗ ≫ An x ∗ ,
and so An x ∗ → 0. Now An x ∗ = i xi∗ An ei , and each term is
non-negative, so for all i , An ei → 0.
Theorem: If A is productive, then the series (I + A + A2 + · · · )
converges to (I − A)−1 .
Proof: (I − A)(I + · · · + An ) = I − An+1 , so
(I + · · · + An ) − (I − A)−1 = (I − A)−1 An+1 , and
||(I + · · · + An ) − (I − A)−1 || ≤ ||An+1 || · ||(I − A)−1 || → 0.
Interpret this in terms of recursive factor demands.
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The Simple Leontief Model I
Geometry of the PPF:
Suppose wlog that the labor
supply is 1.
y2
The set of feasible net outputs is
{y ≥ 0 : a0 (I − A)−1 y ≤ 1}.
(I − A)−1 y is the level at which
the activities must be run to
produce y , and a0 (I − A)−1 y is
the labor requirement of this
levels vector.
y1
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The Simple Leontief Model II
Prices
a0j ≫ 0.
The stock of labor is 1.
Given price vector p = (p0 , . . . , pn ), profits to activity j per
unit of level is
X
πj = pj − p0 a0j −
pi aij
i
π = p(I − A) − p0 a0
Definition: An equilibrium price vector is a p ≥ 0 such that profits
to each used activity are 0, and the profits to each unused activity
are not positive. That is, for all j,
πj ≤ 0 and xj πj = 0.
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The Simple Leontief Model II
Theorem: If A is productive, there is a price vector p ≫ 0 unique
up to positive scalar multiplication such that at prices p, π = 0.
Proof: Wlog, consider prices such that p0 = 1. If profits are all to
be 0, prices solve p(I − A) = a0 , where a0 is the vector of labor
requirements per unit of output. Since A is productive, I − A is
invertible, and so p = a0 (I − A)−1 .
To see that p is positive, let e i denote the i th unit vector in Rn .
From Theorem 1, (I − A)x = e i has a non-negative solution x i . So
0 < a0 · x i = p(I − A)x i = p · e i = pi .
Interpretation: Intermediate products are valued according to their
labor content.
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The General Leontief Model
Suppose now that there are m > n activities. These activities can
be partitioned into sets T1 , . . . , Tn such that if column aj ∈ Ti the
single (unit) output of activity j is good i . Let B be a matrix such
that bij = 1 if activity j produces good i and 0 otherwise. Then
P = {y : y ≤ (B − A)x, a0 · x ≤ 1}
denotes the set of feasible net outputs.
Definition: A technology is a set t = {(a1 , a01 ), . . . , (an , a0n )} of
activities such that for all j, aj ∈ Tj . Let P(t) denote the set of
net output vectors producible from this technology without using
any other activities, and let (At , at ) denote the simple Leontief
model associated with technology t.
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The Nonsubstitution Theorem
If technology t can produce net output vector y , denote the
necessary labor requirement by λt (y ) = at (I − At )−1 y .
Theorem: Suppose that the general Leontief model is productive.
Then there is a technology τ such that for all y ∈ Rn+ and all
productive technologies t, λτ (y ) ≤ λt (y ).
Interpretation: Technologies can be ranked by “efficiency”. There
is a maximally efficient set of technologies, and this set does not
depend upon the particular net output vector at which the ranking
is evaluated.
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Proof of the Nonsubstitution Theorem
For the general model, let B denote the matrix where bij = 0 if
aj ∈
/ Ti and 1 otherwise. Then for the general model run at
activity level x ∈ Rm
+ , net output is (B − A)x. Consider the
problem of minimizing labor output for production of a given y .
min
a0 · x
s.t. (B − A)x = y
x ≥0
Since A is productive, this program has a solution for all y ≥ 0. In
particular, choose y = (1, . . . , 1). Since a solution exists, a basic
solution exists, and this involves only n columns of A. Since all
goods are produced, the columns describe a technology τ . Since
(I − Aτ )x τ ≫ 0, τ is productive.
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Proof of the Nonsubstitution Theorem
Let e i denote the i th unit vector in Rn .
Lemma For all productive technologies t, λτ (e i ) ≤ λt (e i ).
Proof: Suppose not. Then wlog, there is a technology t such that
λt (e n ) < λτ (e n ). Define x i such that (I − Aτ )x i = e i for
n . Then
1 ≤ i ≤ nP
− 1, and (I − At )x n = eP
n−1
(I −PAτ ) 1 x n + (I − At )x n = n1 e i = e, so
e i ) + λτ (e n ) is feasible for the linear program. Thus
λτ ( n−1
1
n−1
n−1
X
X
i
t n
τ
e i ) + λτ (e n )
e ) + λ (e ) ≥ λ (
λ (
τ
1
1
and so λt (e n ) ≥ λτ (e n ), a contradiction.
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Proof of the Nonsubstitution Theorem
Proof of the Theorem:PSuppose there is a t and a y ≥ 0 such that
λt (y ) < λτ (y ). Then i yi (λt (e i ) − λτ (e i )) < 0, and so for some
i , λt (e i ) < λτ (e i ), a contradiction.
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von Neumann Growth Model
Introduces Activity Analysis
This and Wald (1935) “Über die Eindeutige. . . ” are the first
modern GE models.
The first multisector growth model, and the origin of the
turnpike theorem.
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Closed Model of Production
Closed dynamic model of production.
n goods and m activities. An activity is described by two row
vectors: aij is the amount of good j used by activity i , and bij
is the amount of good j produced by activity i .
Activities consume inputs at time t in order to produce
outputs at time t + 1.
inpt = xt A → xt B = outt+1 .
Assumption 1: Every column of B has a positive entry. (All goods
are produced.)
Assumption 2: Every row of A has a positive entry. (Production
uses commodities.)
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Closed Model of Production
Definition: A feasible expansion rate is a scalar α > 0 such that for
some x > 0, xB ≥ αxA.
The technological expansion problem is to find x > 0 and α which
solves
max
x>0
s.t.
α
xB ≥ αxA.
An optimal solution to this problem is called the technological
expansion rate and the corresponding optimal x is called the
optimal expansion factor.
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Technological Expansion Problem
Theorem: If (A, B) satisfy assumptions 1 and 2, the technological
expansion problem has an optimal solution.
Proof: For each α > 0, consider the inequality system
x(B − αA) ≥ 0.
(2)
The TEP is to find the largest α for which this inequality system
has a semi-positive solution.
Assumption 1 implies that if x ≫ 0, then xB ≫ 0. Thus for small
enough positive α, x(B − αA) ≫ 0.
Assumption 2 guarantees that for α sufficiently large, the row sums
of (B − αA) will be negative. Thus x(B − αA)e < 0, so the
inequality (2) has no solution. The set of feasible expansion factors
is bounded and not empty. Its least upper bound solves the TEP.
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Equilibrium
A feasible interest rate is a scalar β > 0 and commodity prices p
such that Bp ≤ βAp. Discounted profits are (B − βA)p, so
feasible interest rates make discounted profits non-positive.
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Equilibrium
Equilibrium prices, activity levels, expansion and interest rates
should have the following properties:
x(B − αA) ≥ 0, inputs at t + 1 cannot exceed outputs at t;
(B − βA)p ≤ 0, discounted profits must be non-positive;
x(B − αA)p = 0, overproduced goods should have a zero
price;
x(B − βA)p = 0, inefficient process should not be used.
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Economic Expansion Problem
The economic expansion problem is to find the smallest interest
rate for which the economy earns non-positive profits:
max
p>0
β
s.t. Bp ≤ βAp.
The β and p that solve this problem are the economic interest rate
and the optimal price vector.
Not surprisingly, this problem is dual to the technological
expansion problem.
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Solving the EEP
The economic expansion problem has an optimal solution. Since
every good is produced, bi p > 0 for some index i , hence there is a
positive bound on feasible interest rates. Since some commodity is
used in production, there is a p assigning that commodity positive
price, and with this p a large enough β is feasible. The set of
feasible interest rates is not empty and is bounded away from 0; its
greatest lower bound solves the EEP.
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Optimal Expansion Factors
Let α∗ and β ∗ denote the optimal expansion rates
Lemma: If (A,B) satisfies assumptions 1 and 2, β ∗ ≤ α∗ .
Proof: Let C = B − α∗ A. The inequality xC ≫ 0 has no
non-negative solution, for if so we would have x ′ B > α∗ x ′ A, which
contradicts the maximality of α∗ . The Theorem of the Alternative
implies that there is a p > 0 such that Cp ≤ 0. Then Bp ≤ α∗ Ap,
so α∗ is a feasible interest rate.
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von Neumann’s Theorem
Theorem: If (A, B) satisfies assumptions 1 and 2, then the optimal
level and price vectors x and p and α∗ satisfy
1 xB ≥ α∗ xA;
2 if xb j > α∗ xaj , then pj = 0;
1∗ Bp ≤ α∗ Ap;
2∗ if bi p < α∗ Ai p, then xi = 0.
This is an equilibrium existence theorem.
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Proof of von Neumann’s Theorem
Point 1) holds by definition of α∗ . Since α∗ ≥ β ∗ , point 1∗ ) holds
as well. These together imply α∗ xAp ≤ xBp ≤ α∗ Ap; so
x(B − α∗ A)p = 0. Since the terms (bi − α∗ ai )p and x(b j − α∗ aj )
are all non-negative, 2) and 2∗ ) both hold as well.
Question: Do x, p and β ∗ satisfy the equilibrium conditions?
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Duality and Uniqueness
If the technological and economic expansion problems are really
dual, we would expect their values to be identical.
Definition: A subset S ⊂ {1, . . . , n} of commodities is independent
if it is possible to produce any commodity in S without using any
commodity i ∈
/ S. The model (A, B) is irreducible if there are no
proper independent subsets.
By reordering rows and columns, if there is an independent subset
then
T { A1
0
A=
A′
Duality Theorem: If the model (A, B) satisfies assumptions 1 and
2 and is irreducible, then α∗ = β ∗ .
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Proof of the Duality Theorem
We need only show that β ∗ ≤ α∗ . If x and p are optimal, then
xB ≥ α∗ xA and Bp ≤ β ∗ Ap. Thus α∗ xAp ≤ xBp ≤ β ∗ xAp, so we
need only show that xAp > 0.
Let T denote the rows of A1 , and S indices corresponding to the
columns b j such that xj > 0. This set must be independent since it
is all that is being produced. Irreducibility implies that xb j > 0 for
all j, which is to say that xB ≫ 0. Since p > 0, xBp > 0. Finally,
βxAp ≥ xBp > 0, so xAp is positive.
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Bibliography
I
Gale (1960), The Theory of Linear Economic Models.
Hawkins and Simon (1949), “Some conditions of macroeconomic
stability”, Econometrica.
Kantorovich (1939), “Matematicheskie metody organizatsii i
planirovaniia proizvodstva”, Leningrad: State Publishing House.
(Reprinted in Management Science (1960).)
Kemeny, Morgenstern and Thompson (1956), “A generalization of
the von neumann model of an expanding economy”, Econometrica.
Koopmans (1951) Activity Analysis of Production and Allocation.
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Bibliography
II
Leontief (1936), “Quantitative input and output relations in the
economic systems of the United States”, Review of Economics and
Statistics.
Leontief (1941), The Structure of the American Economy, 1919 –
1929.
von Neumann (1937). “Über ein Ökonomisches Gleichunssystem
und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes”,
Ergeb. Math. Kolloquiums, nr. 8. (Reprinted in RES, 1945.)
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