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Federal Reserve Bank of Chicago
Business Networks, Production Chains,
and Productivity: A Theory of InputOutput Architecture
Ezra Oberfield
WP 2011-12
Business Networks, Production Chains, and Productivity:
A Theory of Input-Output Architecture∗
Ezra Oberfield
Federal Reserve Bank of Chicago
ezraoberfield@gmail.com
November 27, 2011
Abstract
This paper studies an analytically tractable model of the formation and evolution of chains of
production. Over time, entrepreneurs accumulate techniques to produce their good using goods
produced by other entrepreneurs and labor as inputs. The value of a technique depends on both
the productivity embodied in the technique and the cost of the particular input; when producing,
each entrepreneur selects the technique that delivers the best combination. The collection of
known production techniques form a dynamic network of potential chains of production: the
input-output architecture of the economy. Aggregate productivity depends on whether the
lower cost firms are the important suppliers of inputs. When the share of intermediate goods
in production is high, the lower cost firms are selected as suppliers more frequently. This raises
aggregate productivity and also increases the concentration of sales of intermediate goods.
Keywords: Networks, Productivity, Supply Chains, Ideas, Diffusion
JEL Codes: O31, O33, O47
∗
Preliminary and Incomplete, please do not quote without permission of the author. I appreciate the comments of
Amanda Agan, Fernando Alvarez, Gadi Barlevy, Marco Bassetto, Jarda Borovicka, Jeff Campbell, Thomas Chaney,
Aspen Gorry, Joe Kaboski, Sam Kortum, Alejandro Justiniano, Robert Lucas, Devesh Raval, Rob Shimer, Nancy
Stokey, Nico Trachter, and Andy Zuppann, as well as various seminar participants. The views expressed herein are
those of the author and not necessarily those of the Federal Reserve Bank of Chicago or the Federal Reserve System.
All mistakes are my own.
Why do some firms have a lower cost of production? One possibility is that a firm is technically
efficient: it is good at transforming inputs into output. Alternatively, a firm could have access to
a supplier that provides particularly cheap inputs. When inputs are cheap, output is inexpensive
to produce even if production is not efficient. Having a low cost of production depends not just
on what the firm knows, but on who it knows. If finding a low-cost supplier is important, much of
what we think of as aggregate productivity is comprised of relationships that link entrepreneurs.
This paper develops a model of the formation and evolution of input-output links. In the model,
entrepreneurs search for the most cost-effective techniques to produce their goods and for new uses
for their goods. It is based on the premise that there may be multiple ways to produce a good,
each using a different set of inputs.1 When one entrepreneur meets another, she may develop a new
technique for producing her good using the other entrepreneurs’s good as an input. When deciding
which technique to use, an entrepreneur cares both about the technical efficiency of the technique
and about the cost of the inputs. If she can charge a lower price, other entrepreneurs are more
likely to use her good as an input. As new techniques are discovered and as suppliers adjust their
prices, entrepreneurs substitute across methods of production.
The collection of known production techniques form a dynamic network comprising the set of
supply chains available to make each final product. When an entrepreneur discovers a new, more
cost effective technique, the cost savings diffuse through the network. How large those effects
are and how quickly these more efficient techniques are developed hinge on both the density and
structure of the network.2
1
For example, an electricity producer may have a production technique that uses oil as an input and another less
productive technique that uses coal.
2
Consider an entrepreneur who discovers an extremely cost effective technique. If the firm has many potential
customers (i.e., there are many other firms that have techniques that use the entrepreneur’s good as an input), those
lower prices will ripple down many supply chains. If, however, the firm has few potential customers, the lower prices
may not have much of an immediate impact.
1
I first study a planner’s problem in an economy in which new techniques arrive randomly to
all entrepreneurs at a uniform rate. For this baseline case, there is a closed form expression for
aggregate productivity, relating features of the network (and parameters that shape the network)
to aggregate output. An important determinant of aggregate productivity is whether the lower cost
producers are able to become the important suppliers of the economy. To become an important
supplier, a firm must have many potential customers who know how to use the firm’s good and to
be able to charge those potential customers a relatively low price. When intermediate goods are
more important in production (relative to labor), the ability to charge a low price becomes more
important in winning customers, and the lowest cost firms are more likely to become superstar
suppliers. This increases aggregate productivity and also increases the market concentration in
sales of intermediate goods.
To better understand what determines the emergence of superstar suppliers, I study a simple
extension that allows for heterogeneous rates of link formation across different subgroups. The
particular configuration determines how productivity and demand covary and hence how frequently
the lowest cost firms are selected to supply intermediate goods. Concentration of link formation
among a subset of firms, e.g., in a city, leads to positive correlation of productivity and demand
across subgroups and hence higher aggregate output.
A final extension allows for production functions that use multiple inputs. Raising the number
of inputs makes the cross sectional distribution of marginal cost more concentrated and dampens
the impact of superstar suppliers.
I then study equilibria of a particular market structure with monopolistic competition in sales
of final goods and bilateral two part pricing between all potential input-output pairs. I consider
the set of pairwise stable equilibria: arrangements for which no there are no profitable unilateral or
2
pairwise deviations. For each of these equilibria, all firms set the marginal price equal to marginal
cost (along a non-negative fixed fee) so production chains avoid double marginalization. There
are many equilibria that decentralize the planner’s optimal allocation but differ in how profits are
divided across firms. There is also always at least one equilibrium in which no production occurs
(although there is a sense in which that equilibrium is fragile).
This paper relates to several disparate literatures. Most directly connected is the literature on
social networks, especially as applied to firms. Atalay et al. (2011) document cross-sectional facts
about input-output links, while Chaney (2011) discusses the network structure of international
shipments of goods. Both explain their empirical findings using extensions of the preferential
attachment models of Barabasi and Albert (1999) and Jackson and Rogers (2007). Relative to
this literature, this paper makes three contributions. First, this paper endogenizes the formation of
observed input-output links in the economy. Each entrepreneur selects a supplier from among many
potential suppliers, and these choices are mediated by prices. In the preferential attachment model,
the formation of links is mechanical: no choices are made. The endogeneity of network formation is
advantageous in that it opens the door to the analysis of policy experiments that change incentives
and thus the link formation process. Second, in this model the network structure determines
not just the distribution of input-output links but also both firm level and aggregate productivity,
providing a more clear connection between the network structure and economic variables of interest
such as aggregate output and welfare. Third, this paper gives a natural alternative explanation for
a skewed cross-sectional distribution of input-output links.
The model emphasizes technological interdependence in the spirit of Rosenberg (1979)3 . In
3
“The social payoff of an innovation can rarely be identified in isolation. The growing productivity of industrial
economies is the complex outcome of large numbers of interlocking, mutually reinforcing technologies, the individual components of which are of very limited economic consequence by themselves. The smallest relevant unit of
observation is seldom a single innovation but, more typically, an interrelated clustering of innovations.”
3
trying to make sense of the black box that is aggregate productivity, this paper provides a structure
that connects features of technological relationships to an aggregate production function and to
cross sectional characteristics of firms.
The structure of the model is also related to the work of Kortum (1997), Eaton and Kortum
(2002), Alvarez et al. (2008), and Lucas (2009) who study flows of ideas. The most closely connected
is Lucas (2009), who studies a model in which individuals learn ideas from other individuals over
time.
The idea that network structure determines how shocks propagate through an economy has been
incorporated into the real business cycle literature, centered around the model of Long and Plosser
(1983). Recently, the discussion has focused on whether shocks to particularly well connected
sectors can account for aggregate fluctuations.4 These models typically assume that each sector
has a representative firm that produces using a Cobb-Douglas production function, using inputs
from all other sectors.5 Like these models, this paper is concerned with how cost savings spill
over to other firms through input-output links. However, while that literature has focused on the
consequences of sectoral shocks taking the input-output structure as given, this paper studies the
endogenous formation and evolution of the input-output structure.
This paper proceeds as follows: Section 1 describes the basic technology, setting up and solving
a social planner’s problem. In this section there are simple formulas relating the density of the
network to aggregate output. Section 2 describes the size distribution. Section 3 discusses market
structures that could be overlaid on the economic environment. Section 4 generalizes the model
4
See Horvath (1998), Dupor (1999), Carvalho (2007), Acemoglu et al. (2010), and Foerster et al. (2008). This
literature has focused on the sectoral level both because the most fine input-output data is at that level and because
solving these models involves inverting matrices, which becomes computationally intensive as the number of nodes
in the network grows large.
5
Jones (2008) uses a similar model to argue the input-output structure can help us understand cross country
income differences. In that setup misallocation in one sector raises input prices in other sectors, and the magnitude
of the overall effect depends on the input-output structure.
4
presented in the first section to multiple types to allow for more interesting network configurations,
while Section 5 studies how particular configurations relate to aggregate output. Section 6 extends
the model to allow for multiple inputs. Section 7 concludes.
1
The Baseline Model
1.1
Economic Environment
There is a unit mass of infinitely-lived firms, and each firm is associated with producing a particular
good. Each good is used for final consumption and potentially can be used as an intermediate input
by other firms. A representative consumer has Dixit-Stiglitz preferences over the goods and supplies
labor inelastically (both of these can easily be relaxed). There is no storage technology.
Over time, each firm accumulates production techniques. For a firm, a technique is method
of producing its good using some other firm’s good as an intermediate input. Each technique
φ = {j, i, z} consists of three components: (i) the good that is produced, j; (ii) the good used
as an input, i; and (iii) a production technology associated with using that input, indexed by the
productivity parameter z:
y=
αα (1
1
zxα L1−α
− α)1−α
where y is the quantity of output of good j produced, x is the quantity of good i used as an input,
L is labor.
Time is continuous and firms discover and lose access to techniques randomly over time. Each
firm discovers new techniques at the arrival rate µ̃(t). When firm j discovers a new technique,
φ = {j, i, z} the identity of the supplier i is random and uniformly distributed across all firms in
5
the economy.6 The productivity parameter, z, of a technique is drawn from a fixed distribution
with CDF H(·) and is constant over time. It is assumed that the support of H is bounded below
by some z0 > 0 and that
R∞
z0
log(z)dH(z) < ∞. Each existing technique becomes infeasible at rate
δ.7
Given a menu of techniques and input prices, each firm produces using the technique that
delivers the best combination of input cost and productivity. Since the production functions exhibit
constant returns to scale, generically using a single technique is optimal.8
z5
A
z3
z1
z4
D
z5
A
z7
z2
C
B
z6
z3
z1
z7
z2
E
C
z8
B
z4
D
z6
E
z8
F
z9
G
F
(a) The Set of Techniques
z9
G
(b) Techniques that are Used
Figure 1: A Graphical Representation of the Input-Output Structure
Figure 1a shows the set of techniques at a point in time. Each node is a firm, and an edge
that connect two nodes corresponds to a technique. The direction indicates which firm produces
the output, and which firm produces the output. Each edge has a weight associated with the
productivity of the technique. In Figure 1b the solid arrows are techniques that are currently
used, while the dashed arrows correspond to techniques that are not.
At a point in time, the state of the economy can be summarized by the set of available techniques,
Φ(t). Figure 1 gives a visual representation of this set as a weighted, directed graph. In the figure,
6
Section 4 relaxes this uniformity.
δ plays a minor role in the analysis and setting δ = 0 would change little. It is included (i) for generality and (ii)
so that when µ̃(t) is constant there is a well defined steady state.
8
In principle a firm could produce using more than one technique. In the analysis I will assume without loss that
each firm uses only a single technique.
7
6
firms are represented by nodes. Each technique is represented by an edge or link connecting two
nodes. Each edge has a direction that corresponds to the flow of goods for the technique, indicating
which firm provides the intermediate input and which produces the output. In addition, each edge
has a weight (a number) corresponding to the productivity of the technique.
Figure 1a shows the set of all available techniques. Note that several firms, C, D, and E, each
have knowledge of multiple techniques that use different producers. In equilibrium they will each
select a single technique with which to produce. Figure 1b gives an example of these selections.
Over time the network will evolve as new techniques are discovered and some old techniques
become infeasible. While the productivity of a technique is constant over time, the attractiveness
of a technique varies with the cost of the associated inputs.
1.2
A Planner’s Problem
Throughout Section 1 I will focus on a planner’s problem in order to build intuition about the
economic environment and to describe solution techniques without getting bogged down with the
details of a particular market structure. There are many market structures that could be layered
on top of the technological environment. In Section 3 I discuss a particular market structure that
decentralizes the planner’s solution.
Consider the problem of a planner that takes the network of techniques as given. At each point
in time the planner makes production decisions and allocates labor to maximize the instantaneous
utility of the representative agent. To increase readability time subscripts will be suppressed when
unnecessary. Let yj0 be production of good j for final consumption. Given the set of existing
techniques, Φ, we can define two subsets relevant for each firm: let ΦSj be the set of techniques
available to produce good j (potential suppliers for firm j) and let ΦB
j be the set of techniques that
7
would use good j as an input (potential buyers of good j). For a technique φ = {j, i, z}, define the
following quantities:
• z(φ) is the productivity parameter associated with the technique.
• s(φ) is the identity of the supplier, in this case i.
• b(φ) is the identity of the buyer, in this case j.
The planner selects a technique for firm j to use, φ∗j , from among the available techniques to
produce good j, ΦSj . With that, the planner chooses how much of good j to produce, yj , a quantity
of labor, Lj , and the quantity of inputs of good s(φj ∗) to use.
n
o
Formally, the planner chooses an allocation φ∗j , yj0 , yj , xj , Lj
j∈J
to maximize final consump-
tion:
Z
ε
(yj0 ) ε−1
max
ε
ε−1
j∈J
subject to: (i) technological constraints
yj ≤
αα (1
1
z(φ∗j )xαj L1−α
,
j
− α)1−α
∀j ∈ J
(ii) goods feasibility constraints
X
yj0 +
xb(φ) ≤ yj
∗
φ∈ΦB
j |φ=φb(φ)
and (iii) a labor resource constraint
Z
Lj ≤ L
j∈J
8
∀j ∈ J
The left hand side of the second constraint for good j consists of the uses of the good: output
for final consumption and for use as an intermediate input in other firms’ production.
Let M Cj be the marginal social cost of producing good j (the multiplier on the goods feasibility
constraint for j), and let w be the marginal social cost of labor (the multiplier on the labor resource
constraint). The first order necessary conditions from this problem imply that for each j,
M Cj
1
= min
S
w
z(φ)
φ∈Φj
For each technique φ,
1
z(φ)
M Cs(φ)
w
α
M Cs(φ)
w
α
(1)
gives the marginal social cost of producing good j using that
technique, with the planner using the one that delivers the lowest marginal social cost.
It will be convenient to define qj ≡
1
M Cj
as a measure of the efficiency of producing good j. If
we choose units of utility so that w = 1, we can rewrite equation (1) as
α
qj = max z(φ)qs(φ)
φ∈ΦS
j
1.3
(2)
The Supply Chain Interpretation
Given the structure of the network, we can back out the production technology used by the social
planner to produce each good. We consider here how labor is allocated across the different stages
of production in the supply chain for each good.
For a given good, say j, the planner uses a particular chain of techniques to produce good j.
Let {φ0 φ1 φ2 ...} denote the chain of techniques for good j, with φ0 being furthest downstream (the
technique actually used by firm j).
Let L̄j be total amount of labor used in production of good j for final consumption (given
9
constant returns to scale in production, this is well defined). Also let Lkj be the labor used in the
kth to last stage of production of good j for final consumption, so that L0j is the labor used by firm
j to produce for final consumption (as opposed to for production of the good for intermediate use).
Aggregating across stages gives L̄j =
P∞
k
k=0 Lj .
The first order conditions imply that Lk+1
= αLkj : the labor used at each stage is a constant
j
fraction of the labor used in the subsequent stage. We can therefore write
L̄j =
∞
X
k=0
Lkj =
∞
X
αk L0j =
k=0
We also have the first order condition yj0 =
1
0
1−α qj Lj ,
yj0 = qj L̄j
1
L0
1−α j
so that
(3)
In a sense equation (3) should not be surprising; qj was defined to be the ratio of the marginal
social cost of producing good j and the marginal social cost of labor, or the efficiency with which
the planner can produce good j in units of labor.
We can derive a more basic interpretation of equation (3) in terms of more fundamental objects
in the model. With similar notation, we can define qjk to be the efficiency of the kth to last firm
in the chain of production for good j, and z(φk ) to be the productivity parameter of the technique
α
in the kth step. This means that along the supply chain for good j, qjk = z(φk ) qjk+1 . By
10
definition, qj0 = qj , so we make repeated substitutions to get
qj
= qj0 = z(φ0 ) qj1
=
∞
Y
z(φk )α
α
α α
= ...
= z(φ0 ) z(φ1 ) qj2
k
k=0
The planner therefore faces a production function describing the social cost of producing good final
j:
"
yj0 =
∞
Y
#
αk
z(φk )
L̄j
(4)
k=0
The efficiency with which the planner can produce good j depends on the productivity of each
technique at each step in the supply chain, with the techniques furthest downstream weighted
more heavily.
1.4
The Allocation of Labor and Welfare
We now use several more first order conditions from the planners problem to arrive at an expression
R
1
ε−1 ε−1
for total final consumption. Define Q ≡
q
, a standard productivity aggregator for
J j
economies with Dixit-Stiglitz preferences. The first order conditions with respect to each yj0 imply
ε
yj0
qj
=
. We can now use the labor resource constraint and equation (3) to write
0
Q
Y
Z
L=
Z
L̄j =
J
J
yj0
=
qj
Z
J
Y 0 Q−ε qjε−1 = Y 0 /Q
(5)
or more conveniently
Y 0 = QL
(6)
To solve for aggregate output, we need to characterize firms’ efficiencies. One could use equa11
tion (2) to create an operator on {qj }j∈J and look for a fixed point.9 However, with a continuum of
firms, this is neither computationally feasible nor would it be particularly illuminating. We proceed
to impose more structure on the set of techniques motivated by the dynamics of the model.
1.5
The Cross Sectional Distribution of Efficiency
To this point, the treatment of the network of techniques has been quite general. In this section
we impose the probabilistic structure of the dynamic model in order to more clearly characterize
the solution to the planner’s problem. In particular, we will set up a fixed point problem for the
cross sectional distribution of efficiency.
There are two important elements for characterizing the distribution. The first is how techniques
are distributed across firms. The second is the distribution of efficiency delivered by each technique.
We characterize each in turn.
1.5.1
The Distribution of Techniques
Here we describe how techniques are distributed across firms and how this distribution evolves.
Roughly, given the history of Poisson arrival rates of new techniques and decay rates of existing
techniques, the fraction of firms with n techniques at time t can be fully described by a Poisson
distribution with a mean λ̃(t).
More formally, let ω(n, t) be the fraction of firms with access to n techniques at time t. Over
9
Taking logs of both sides gives an operator T , where the jth element of T ({log qj }j∈J ) is maxφ∈ΦB {log z(φ) +
j
α log qs(φ) }. T satisfies monotonicity and discounting. If the support of z were bounded above, the operator would be
a contraction on the appropriate bounded function space. If the efficiencies {qj }j∈J can take values on the extended
real line, then the mapping will not be a contraction. Indeed, there may be multiple fixed points, as discussed below.
12
time, this evolves according to the following law of motion:
ω̇(n, t) = µ̃(t)ω(n − 1, t) + (n + 1)δω(n + 1, t) − µ̃(t)ω(n, t) − nδω(n, t)
(7)
ω(n, t) increases when a firm with n − 1 techniques discovers a new one and when a firm with
n + 1 techniques loses one of their n + 1 techniques. Similarly ω(n, t) decrease when a firm with n
techniques either gains a new technique or loses one of its n techniques.
If at some t0 the distribution of n is given by a Poisson distribution with mean λ̃ (t0 ), then a
solution to equation (7) is such that at any time t > t0 , the distribution of n will also follow a
Poisson distribution with mean λ̃(t), where λ̃(t) satisfies the differential equation:10
˙
λ̃(t) = µ̃(t) − δ λ̃(t)
(8)
To interpret this, it helps to take the limit as t0 → −∞ (and imposing that limt0 →−∞ λ̃ (t0 ) is
bounded) giving
Z
t
λ̃(t) =
e−(t−τ )δ µ̃(τ )dτ
−∞
This is closely related to the fact that the sum of independent Poisson random variables is also
a Poisson random variable. The takeaway from this is that regardless of history of arrival rates
({µ̃(τ )}τ ≤t ), the cross-sectional distribution of techniques ω(n, t) at a given point in time can be
summarized by a single number, λ̃(t). Since λ̃(t) is also the average number of techniques per firm,
I will refer to it as the density of techniques in the network.
10
In fact, for an arbitrary initial distribution, the distribution of links will converge asymptotically to Poisson
distribution.
13
1.5.2
The Cross Sectional Distribution of Efficiency
Let F (q) be the fraction of firms with efficiency no greater than q given the decisions of the planner.
This is an endogenous object that will need to be solved for. The strategy is to use the fact that
each potential supplier has efficiency no greater than q with probability F (q) to set up a fixed point
problem for F .
If firm j discovers a single technique φ, there are two parts that determine how useful it is:
a productivity parameter, z, drawn from an exogenous distribution H(z), and the efficiency of
the supplier, qi . Recall from equation (2) that if firm j produces using technique φ = {j, i, z}, j
will produce at efficiency qj = z(φ)qiα . Let G(q) be the cumulative distribution of the efficiency
delivered by a single random technique. Given equation (2), we can write G(q) as
∞
Z
G(q) =
0
To interpret this, note that for each z, F
1
q α
dH(z)
F
z
1
q α
z
(9)
is the portion of potential suppliers that, in
combination with that z, would leave the firm with efficiency no greater than q.
We now ask, what is the probability that, given all of its techniques, a firm has efficiency no
greater than than q? We can write this as
Pr (qj ≤ q) =
=
∞
X
n=0
∞
X
Pr (All n draws are ≤ q) ω (n)
G (q)n
n=0
= e−λ̃[1−G(q)]
14
λ̃n e−λ̃
n!
To interpret this last expression, if λ̃ [1 − G(q)] is a parameter of a Poisson distribution (the arrival
rate of techniques that would provide efficiency better than q), then e−λ̃[1−G(q)] is the probability
that no such techniques arrived.
When the number of firms is large, a standard abuse of the law of large numbers gives Pr (qj ≤ q) =
F (q).11 We can substitute the expression for G(q) from equation (9) to get a fixed point problem
for the distribution of efficiency F (q):
F (q) = e
−λ̃
R∞
1−F
0
1
( zq ) α
dH(z)
(10)
This recursive equation is the key to solving the planner’s problem.
1.6
Properties of the Planner’s Solution
Consider the space F̄ of non-decreasing functions f : R+ 7→ [0, 1]. Consider the operator T on this
space defined as
T f (q) ≡ e
−λ̃
R∞
0
1
dH(z)
1−f ( zq ) α
With probability one the planner’s solution will be a fixed point of the operator T on F̄.
The qualitative behavior of network depends on whether the average number of techniques λ̃ is
greater or less than 1. The more interesting case in which the average number of links exceeds one
will be the focus of this paper, but for completeness I will discuss both.
11
The proof of Proposition 1 uses such a law of large numbers for a continuum of random variables described by
Uhlig (1996). To use this one must verify that firms’ efficiencies are pairwise uncorrelated. In the present context this
is not immediately obvious: it is possible that two firms’ supply chains overlap or that one is in the other’s supply
chain. However, by assumption the network is sufficiently sparse that with high probability the supply chains will
not overlap: there is a continuum of firms but only a countable number of those are in any of given firm’s potential
supply chains. Therefore, for any two firms, the probability that their supply chains overlap is zero. The law of large
numbers then implies that equation (10) holds with probability one.
15
If firm j does not have access to any techniques, it cannot produce. Similarly, if firm j has
techniques but its suppliers do not, then those suppliers will not be able to produce and consequently
neither will firm j. Continuing with this logic, if a supply chain is finite, it is not viable.
Consider the probability that a single firm will have access to at least one supply chain that
continues indefinitely; only firms with access to such a supply chain will have positive efficiency.
As shown in Appendix B the probability that a firm has no such chains is the smallest root ρ of
ρ = e−λ̃(1−ρ) .12 For λ̃ ≤ 1, the probability of no such chains is one, while for λ̃ > 1, the probability
is strictly less than one.13
Few Techniques: λ̃ ≤ 1
If λ̃ ≤ 1, there are so few techniques available that the probability that any individual firm has
access to a viable supply chain is zero. In this case one can show that T is a contraction, with the
unique solution f = 1: all firms produce with efficiency 0.
Many Techniques: λ̃ > 1
As λ̃ crosses the critical value of 1, one can show that there are multiple fixed points of the
operator T on F̄ (see Appendix C). Recall that the functional equation equation (10) was constructed from necessary (but not sufficient) conditions to the planner’s problem, so one must check
which of these solutions to equation (10) actually solves the planner’s problem.
There are two solutions in which F (q) is constant for all q, both of which stem from the fact
that equation (10) is formulated as a recursive equation. The first is F (q) = 1 for all q, which again
corresponds to zero efficiency (infinite marginal social cost) for all goods. However, the rationale
12
There is a direct analogy between this problem and the Galton-Watson problem. Consider a world in which
every individual has a random number of children drawn from some fixed distribution. The Galton-Watson problem
is: What is the asymptotic probability of extinction?
13
The starkly different behavior of the network when λ̃ crosses 1 is called a phase transition. Such a phase transition
is a typical property of random graphs, a result associated with the Erdos-Renyi Theorem. See Kelly (1997) and
Kelly (2005) for examples in which this kind of phase transition is given an economic interpretation.
16
is different than when λ̃ ≤ 1; here, the logic is recursive. If the marginal social cost of every input
is infinite, then the marginal social cost of each output must be infinite as well. The allocation
that arises from this solution is feasible, but we will show that it is dominated by another feasible
allocation and is therefore not the solution to the planner’s problem.
There is a second constant solution, F (q) = ρ ∈ (0, 1), which follows a similar recursive logic.
This fixed point implies infinite marginal social cost for those firms that cannot produce, and zero
marginal social cost for all other firms. In other words, any firm with a viable supply chain has
infinite efficiency. The rationale is similar: if inputs have zero marginal social cost, output has zero
social cost. Unfortunately, this leads to an infeasible allocation, and is therefore not a solution to
the planners problem either.14
There is always a third fixed point. In Appendix C a subset F ⊂ F̄ is constructed along with
a partial ordering. The subset does not contain either of the two constant fixed points. We can
show that there is a fixed point of T using the Tarski fixed point theorem, which also provides an
algorithm to numerically solve for such a fixed point. Further we can show that the fixed point is
unique. See Appendix C for a more complete statement of the theorem and a proof.
Proposition 1 There exists a unique fixed point of T on F, F sp . With probability one, F sp is the
CDF of the cross sectional distribution of efficiencies in the solution to the planner’s problem and
aggregate productivity is Q =
R∞
0
1
q ε−1 dF sp (q) ε−1 .
Multiple solutions to first order necessary conditions of the planner’s problem is actually a
feature of most models in which a portion of output is used simultaneously as input, such as
14
These two constant solutions have further economic meaning. Given the distribution of productivity draws H(·)
with support [z, z̄] with 0 ≤ z ≤ z̄ ≤ ∞, let q and q̄ be the lowest and highest possible efficiencies among firms that
are able to produce. q (q̄) is derived from the supply chain in which every technique has the worst (best) possible
1
1
productivity draw, so that q = z 1−α (q̄ = z̄ 1−α ). If λ̃ > 1 the solution to the planners problem must have F (q̄) = 1
and F (q) = ρ, the two constant solutions to equation (10).
17
a standard growth model with roundabout production. If the same good enters a technological
constraint as both an input and an output, the Lagrange multiplier on that good will be on
both sides of a first order condition. Consequently the first order condition will be satisfied if the
Lagrange multiplier takes the value of zero or infinity. One can usually sidestep this issue by finding
an alternative way to describe the production technology, e.g., solving for final output as a function
of primary inputs. Much of the work in the proof of Proposition 1 is in finding and characterizing
such an alternative description of production possibilities.
1.7
A Parametric Assumption
I now describe a special case that proves to be analytically tractable. Assume that the productivity
−ζ
parameter embodied in a technique is drawn from a Pareto distribution, H(z) = 1 − zz0
,
with the restriction that ζ > ε − 1 so that final output is finite. In addition, parameterize the
arrival rate (and initial condition) of techniques so that µ̃(t) = µ(t)z0−ζ for all t which implies that
λ̃(t) = λ(t)z0−ζ .
With these assumptions we can compare economies with different values of z0 (holding {λ(t)}
fixed). In an economy with a lower z0 , there will be two differences: and (ii) techniques have
stochastically lower productivity and (ii) each firm discovers new techniques more frequently. In
fact, the parameterization is such that varying z0 has no impact on the average number of techniques
with productivity above any threshold ẑ, λ̃ (1 − H(ẑ)): the two effects cancel exactly. The only
difference is that with a lower z0 there are additional relatively unproductive techniques.
We then look at the limit of a sequence of economies as z0 → 0. This adds many relatively
unproductive techniques (low z) to the economy without changing the number of productive techniques (high z). In the limit, the measure of firms without access to any techniques goes to zero.
18
In this special case, we can show that every solution F (·) to equation (10) follows a Frechet
distribution. To see this, note that we can use the change of variables x = (q/z)1/α to write
∞
Z
1 − G(q) =
z0
Z
=
1
q α
H (z) 1 − F
dz
z
0
q
z0
1
α
H0
0
q
(1 − F (x)) qαx−α−1 dx
xα
Using the functional form H 0 (z) = ζz0ζ z −ζ−1 , we can then write
λ̃ [1 − G(q)] =
λz0−ζ
Z
= q
1
α
ζz0ζ
0
−ζ
q
z0
Z
λ
q
z0
q −ζ−1
(1 − F (x)) qαx−α−1 dx
xα
1
α
αζxαζ−1 (1 − F (x)) dx
0
For any F (·), as z0 → 0, this expression will clearly go to q −ζ multiplied by a constant. Label this
−ζ
constant θ, so that equation (10) can be written as F (q) = e−θq , the cumulative distribution of a
Frechet random variable. Note that the exponent ζ is the same as that of the Pareto distribution
H. This means that the distribution of efficiencies F inherits the tail behavior of the distribution
of productivity draws, H.
We next solve for θ, which was defined to satisfy
Z
θ=λ
∞
αζxαζ−1 (1 − F (x)) dx
0
Integrating by parts gives
Z
θ=λ
∞
xαζ F 0 (x)dx
0
19
Plugging in the functional form F (q) = e−θq
−ζ
and making the substitution s = θx−ζ gives
Z
θ = λ
∞
θα s−α e−s ds
0
so that θ satisfies15
θ = Γ(1 − α)λθα
(11)
where Γ(·) is the gamma function.
With this, we can compute Q, the relevant measure of welfare:
Z
Q=
∞
q
0
ε−1
1
1
ε−1
ε − 1 ε−1
1/ζ
dF (q)
=θ Γ 1−
ζ
Putting these together, we get an expression for final consumption:
1
1 1
1 1
ε − 1 ε−1
Γ(1 − α) 1−α ζ λ 1−α ζ L
Y =Γ 1−
ζ
(12)
There are several immediate implications. First, aggregate output is increasing in the density of
the network, λ. In a more densely populated network, firms on average have a larger set of supply
chains to choose from, and hence are more likely to have lower cost.
Second, the share of intermediate goods in production α plays two roles. First, as in other models
with roundabout production, it determines the extent to which lower prices of input cost feed back
into lower cost of production. This is the exponent
1
1−α
that appears in several places. Second,
α determines the likelihood that the lowest cost producers are selected as the actual suppliers of
15
I write equation (11) in this form rather than solving directly for θ in order to emphasize the fact that the
equation has three non-negative roots, two of which are zero and infinity. In addition, in later sections it will be
easier to see parallels with the analogous expressions when this equation when written in this form.
20
inputs. Recall that the efficiency delivered by a single technique is zq α , where z is the productivity
embedded in the technique and q is the efficiency of supplier. α determines the relative impact
of each of these factors on the cost effectiveness of a technique, and consequently in the selection
of supplier. When α is high, the cost of the inputs matters more, which means the most efficient
producers are selected as suppliers more frequently. In other words, superstar suppliers will be
much more relevant for aggregate production when α is closer to one. Mathematically, this shows
up in the term Γ(1 − α).16 Summarizing, when α is high, each supplier is able to pass through cost
savings to its customers at a higher rate and the most efficient firms are selected to be suppliers
more frequently, so that their high efficiency can be passed through to a larger share of customers.
This will be discussed in greater detail in Section 2
1.8
Dynamics
Over time, some firms discover new techniques while others substitute across techniques in response
to changes in cost along their supply chains. For aggregate quantities, however, all of these changes
can be summarized by changes in the density of the network, λ. All relevant aggregate dynamics
can be summarized by two equations
1
1
Y = κλ 1−α ζ L
and
λ̇(t) = µ(t) − δλ(t)
If the (normalized) arrival rate of new techniques µ(t) is constant over time (and if δ > 0), then
16
Γ(x) is decreasing on (0, 1). Γ(1) = 1 and limx→0 Γ(x) = ∞.
21
there is a steady state with
λss =
µ
δ
Alternatively if the arrival rate of new techniques is growing over time, say µ(t) = µeγt , then
there is a balanced growth path (for any δ ≥ 0) with
λ̇
=γ
λ
2
Ẏ
1 1
=
γ
Y
1−αζ
and
Size Distribution
Here I discuss two dimensions of the cross sectional distribution of size. First, I give expressions
for the conditional and unconditional distributions of the number of customers (other firms that
purchase intermediate goods). Second, I describe the cross sectional distribution of employment.
2.1
Number of Customers
Consider a single draw of a technique that uses firm i’s good as an input. Given the firm’s efficiency, qi , we can compute the probability that the technique is the potential buyer’s best available
technique. To do this, we first characterize the following object: For a potential customer that has
drawn a technique that uses i, what is the probability that it has no other techniques better that
deliver efficiency better than q?
Given the Poisson distribution over the number of techniques, a firm will have n − 1 other
techniques with probability
e−λ̃ λ̃n
.
n!(1−e−λ̃ )
The CDF of efficiency delivered by each of these techniques
is G(q). We can therefore write the probability that the potential buyer has no other technique
22
that delivers better than q, conditioning on having at least one technique, as:
P∞
n=1
e−λ̃ λ̃n
n−1
n! G(q)
1 − e−λ̃
=
"
1
G(q)(1 − e−λ̃ )
∞ −λ̃ n
X
e λ̃
n!
n=0
#
−λ̃
n
G(q) − e
=
F (q) − e−λ̃
G(q)(1 − e−λ̃ )
Among techniques that use i as a supplier, the fraction that deliver efficiency less than q
is H qqα , with density q1α H 0 qqα . We can now characterize the probability that a particular
i
i
i
technique is the potential buyer’s best technique:
Z
∞
Pr (φ is used|qi ) =
0
1 0
H
qiα
q̃
qiα
F (q̃) − e−λ̃
G(q̃)(1 − e−λ̃ )
dq̃
How many known techniques use a given firm as a potential supplier? In other words, how
many potential customers does a given supplier have? Across all firms, the distribution over the
number of potential customers follows a Poisson law with mean λ̃. Each one of those techniques has
an equal chance of being the potential buyer’s best technique, so the distribution over the number
of actual customers will also be a Poisson, with parameter:
Z
λ̃
0
∞
1 0
H
qiα
q̃
qiα
F (q̃) − e−λ̃
G(q̃)(1 − e−λ̃ )
dq̃
Using the functional form for H and taking the limit as z0 → 0 yields
λ
qiαζ
θ
So among firms with efficiency q, the distribution over the number of customers is a Poisson
23
αζ
distribution with parameter λ q θ . One can see that the distribution among high efficiency suppliers
first order stochastically dominates the distribution among low efficiency suppliers: high efficiency
firms get more customers.
Figure 2 shows the average number of customers at each percentile in the efficiency grouping
for different values of α. As expected, higher efficiency firms are able to attract more customers
than low efficiency firms.
It is notable that curve in Figure 2 depends on only one parameter, α (the share of inputs in
production). This is because buyers choose to use the technique that gives the best combination of
efficiency (z) and input cost (q). α determines the relative importance of these two factors. Recall
that the efficiency associated with a single technique is zq α . If α is large then the the share of
inputs is higher, and the cost of inputs becomes relatively more important. An increase in α makes
the techniques using high efficiency suppliers even more cost effective. In contrast, when α is low,
more weight is put on the idiosyncratic productivity associated with the technique. Because the
productivity draws are drawn from the same distribution regardless of the efficiency of the supplier,
lowering α increases the odds that a low efficiency firm will be able to attract customers; the low
efficiency becomes less relevant to its customers.
When α is higher, more weight is put on the cost of inputs, so the advantage of the high efficiency
suppliers is even larger. The shift is evident in Figure 2: with higher α, the high efficiency firms
are capturing a much larger share of the customers, at the expense of the low efficiency firms.17
We next look at the unconditional distribution over the number of customers among all firms. To
find the mass of suppliers with n customers, we simply integrate over suppliers of each efficiency.
The resulting formula is given in the following proposition, which also describes the tail of the
17
It is easy to show a single crossing property: Curves for different values of α in Figure 2 cross exactly once.
24
Avg Num Customers
2.5
Α = .25
2.0
Α = .5
1.5
1.0
0.5
q Percentile
0.0
0.2
0.4
0.6
0.8
1.0
Figure 2: Conditional Distribution of Customers
The mean number of actual customers for each percentile in the efficiency distribution.
distribution.
Proposition 2 Let pn be the mass of firms with n customers. Then
Z
pn =
0
∞
w−α
Γ(1−α)
n
−
e
w−α
Γ(1−α)
n!
e−w dw
The counter cumulative distribution has a tail index of 1/α.
It is notable that the distribution of number of customers has only a single parameter, α. In
particular the density of the network, λ does not enter this equation. While a higher λ raises the
number of potential customers for any firm, it also raises the number of alternatives available to
each of those potential customers.
Figure 3 shows the distribution of customers for different values of α. When α is high, more
weight is put on the cost of inputs, so the distribution is more skewed. The tail is thicker, and but
there are also more firms without any customers. In contrast, when α is low, the middle of the
25
pmf
0.8
pmf
10-7
Α = .9
Α = .7
Α = .5
Α = .3
Α = .1
0.6
10-17
10-27
0.4
Α = .9
Α = .7
Α = .5
Α = .3
Α = .1
10-37
0.2
10-47
1
2
3
4
5
Customers
2
(a) Number of Customers
5
10
20
50
Customers
(b) Number of Customers, Log-Log
Figure 3: The Distribution of Customers
Figure 3a gives the mass of firms with n customers for several different values of the input share
α. Figure 3b also gives the mass of firms with n customers, but on a log-log plot, to better show
the tail of the distribution.
distribution is thicker.
2.2
Comparison to Preferential Attachment
The preferential attachment model of Barabasi and Albert (1999) was designed to match the
scale free nature of the distribution of links in several real world networks. For example, relative
to a network in which links are formed randomly and with uniform probability, there are too
many websites with many incoming weblinks and also too many with very few incoming weblinks.
Roughly, the preferential attachment model explains this as follows: There is an initial network and
over time new links are formed and new firms are born. The probability that a new link involves
a particular firm is increasing in the number of links that firm already has.
This paper gives an alternative explanation for skewed cross-sectional distribution of inputoutput links. Because entrepreneurs differ in productivity (some have more efficient supply chains
than others), they will also differ in their ability to attract customers. While the distribution of
potential customers is the same as one would expect in a uniformly random network, the distribution
26
of actual customers is not. This is precisely because of endogenous selection: some of those firms
with many potential customers are able to offer lower prices and win over a larger fraction of those
potential customers.18
2.3
Distribution of Employment
In characterizing the cross sectional distribution of employment, we first derive a convenient fact.
Let B (q|qi ) be the the CDF of the efficiency of customers of suppliers with efficiency qi . In other
words, B (q|qi ) is the distribution efficiency among firms whose best technique uses a supplier with
efficiency qi . We will show that in the limit as z0 → 0, B (q|qi ) = F (q). We can solve for this
distribution of customers’ actual efficiency with an application of Bayes rule:
B 0 (q|qi ) =
1
0
qiα H
R∞
0
1
0
qiα H
q
qiα
q̃
qiα
F (q)−e−λ̃
G(q)(1−e−λ̃ )
F (q̃)−e−λ̃
dq̃
G(q̃)(1−e−λ̃ )
The numerator is the density of efficiency delivered by techniques that use suppliers with qi ,
q
1
0
H
α
q
q α , multiplied by the probability that a such a technique is the potential customer’s best
i
i
technique. The denominator is the probability that a technique from i is the customer’s best
technique. We can use the functional forms for H and take a limit as z0 → 0:
0
lim B (q|qi ) =
z0 →0
lim R ∞
z0 →0
ζz0ζ qiαζ q −ζ−1 e−θq
z0 qiα
ζz0ζ qiαζ q̃ −ζ−1 e−θq̃−ζ dq̃
= ζθq −ζ−1 e−θq
=
−ζ
−ζ
lim F 0 (q)
z0 →0
18
There is an analogous comparison for the world wide web. The preferential attachment model says the
www.nytimes.com has many incoming links because it happened to have many such links initially, and consequently
many other websites linked to it as well. In contrast, a model with endogenous network formation would give the
explanation that www.nytimes.com offers better content than other sites, and therefore other sites choose to link to
it more frequently.
27
This implies that knowing the efficiency of a supplier gives no information about the identity
of its customer. That is, there are no systematic differences between the customers of low and high
efficiency suppliers.
This has two implications. First, it gives insight into the determinants of size. Aside from
more sales of final goods, high efficiency firms will on average be larger because they have more
customers, not because their customers are any bigger.
Second we can treat the characteristics of customers as independent, identically distributed
random variables. This will be helpful in several ways. Of particular use here is the fact that we
can treat the size of a customer as an IID random variable.
Distribution of Employment
We are interested in adding together the labor used to make goods for final consumption and
for intermediate use for each customer. Since the latter can be treated as independent random
variables, it is easiest to work with characteristic functions of the relevant distributions.
Let χ(s) be the characteristic function associated with the cross sectional distribution of employment. To get at it, we will create a fixed point problem. Roughly, the information that is used
includes:
• Given a firm’s efficiency, the quantity of labor used for production of final output.
• Given a firm’s efficiency, the distribution over the number of actual customers.
• The fact that the quantity of labor used by a customer is independent of the efficiency of the
supplier.
• The fact that if a customer uses L units of labor, the supplier will use αL units of labor to
make the inputs for that customer.
28
As described in Appendix D.2, the characteristic function solves the equation:
Z
χ(s) =
0
∞
− ε−1
−α
ζ
t
t
exp is
[1 − χ(αs)] − t dt
(1 − α) L −
Γ 1 − ε−1
Γ(1 − α)
(13)
ζ
There are several things to note. First, as L changes, the distribution of labor scales uniformly.19
Second the distribution depends on only two parameters, α and
ε−1
ζ .
The share of intermediates
matters for the same reason as before, it determines the skew of the distribution of number of
customers.
ε−1
ζ
is a composite of two parameters, the elasticity of substitution in final consumption,
and ζ, the tail index of both H(·) (the Pareto distribution from which productivity shocks are
drawn) and F (·) (the cross sectional distribution of efficiency). In combination, these parameters
determine the tail index of the distribution of final consumption. When ζ is small, the efficiency
distribution has a thicker tail, inducing a thicker tail in the distribution of final consumption. When
ε is high, consumers are more willing to substitute toward low cost goods, also thickening the tail
of final consumption.
Equation (13) can be used to solve for χ(s) numerically.20 We can consequently use standard
methods to back out the distribution of employment form its characteristic function.
Figure 4 shows the distribution for the parameters α =
ε−1
ζ
= 1/2 and L = 1. One can see that
this density is quite skewed, with the mode well below the mean of 1.
19
In fact, one
could write the characteristic function associated with the fraction of labor used by each firm as
χ̂ (s) = χ L1 s . This satisfies
Z ∞
− ε−1
t ζ
t−α
(1 − α) −
χ̂ (s) =
exp is
[1 − χ̂ (αs)] − t dt
Γ 1 − ε−1
Γ (1 − α)
0
ζ
20
While equation (13) is a functional equation, it resembles a difference equation. One can solve this using a reverse
shooting algorithm, starting near the point χ(0) = 1 and interpolating.
29
PDF
1 - CDF HLogL
1.000
1.5
0.500
1.0
0.100
0.050
0.5
0.010
0.005
Employment
1
2
3
4
5
0.05 0.10
(a) Density of Employment
0.50 1.00
5.0010.00
Employment HLogL
(b) Right CDF of Employment, Log-Log
Figure 4: The Distribution of Employment
Figure 4a gives the density of employment with α = ε−1
= 1/2. Figure 4b gives right CDF on
ζ
a log-log plot to show the shape of the tail of the distribution.
3
Market Structure and Decentralized Equilibria
We will consider a market structure in which there is monopolistic competition across final goods
but bilateral two part tariffs for intermediate goods. In addition we will consider contracting
terms that are “pairwise stable.”21 This equilibrium concept is reminiscent of Nash equilibrium,
except there is no explicit description of a formal game. We will characterize the set of contracting
arrangements for which the are no profitable unilateral or mutually beneficial pairwise deviations.
As an equilibrium concept, pairwise stability captures the dynamic spirit of the model. As new
techniques are found, some of the cost savings are passed on to downstream customers in the form
of lower prices. An important aspect of this is that when the terms of a contract are no longer
optimal, those terms can be renegotiated to reflect the change in cost structure in the supply chain.
While the equilibrium definition is a static one, the motivation is inherently dynamic, and pairwise
21
As an equilibrium concept, pairwise stability is used frequently in the networks literature (Jackson (2008) provides
an excellent survey). For many applications in this literature, payoffs are a function of the particular links that are
formed, and the idea is to find a network of links for which no pair of nodes wish to change whether or not they
are connected (see Jackson (2003)). Here, the usual concept is extended to include contractual terms of trade in any
input output relationship.
30
deviations capture the spirit of the renegotiation that would take place over time.22
One result will be that for each technique that is used in equilibrium, marginal price equals
marginal cost. While the space of possible contracts is fairly large, it seems reasonable to focus on
a space of contracts that allows for this possibility. Anything other than this kind of two part tariff
would lead to double marginalization and leave surplus on the table. Input-output relationships
are generally long-lived, so it would be surprising if the contracting terms remained inefficient. In
addition, the informational demands of these contracts (in equilibrium) are not large. Firms do
not need to know that much about the environment to get the terms of the contract right.23 While
these concerns are outside the model, they motivate the focus on two part pricing.
Monopolistic competition across final goods leads to a uniform markup of
ε
ε−1 .
This gives
surplus for each final good that is divided across firms in that supply chain. We will show that
allowing for two part tariffs decentralizes the planner’s solution: in this decentralization, the inputs,
outputs, and labor used by each firm are the same as the the efficient allocation. Each firm will
choose the technique that gives the best combination of cost/productivity. There are many ways
to divide the surplus from sales of final goods across supply chains, but this is not relevant for the
allocation of resources. Since the marginal input price equals marginal cost, the quantity supplied
will be efficient, and since labor is supplied inelastically the monopoly markup on final goods is not
22
Another a solution concept, Nash bargaining, is tricky precisely because of renegotiation: there is no well defined
outside option. If a supplier i and buyer j are bargaining over the contract terms for a technique, it is natural that
if the bargaining breaks down j will use its next best supplier. However, if this happens, does j renegotiate with
its current contracts with its customers? More generally, the Nash bargaining solution requires well defined outside
options, and to do this would require a well defined order of negotiation and renegotiation across pairs of firms.
23
Another consideration is that allowing for bilateral two part tariffs leads to a characterization of equilibrium that
is tractable. Consider an alternative in which firms set a single price for all customers. A difficulty in solving for the
optimal prices in this world is that demand curves facing firms are not continuous let alone differentiable, as lowering
a price a little may allow a supplier to beat out a competitor and give a spike in quantity demanded (or may allow
the buyer to lower its price enough to beat out its competitor, giving that buyer and consequently the supplier a
spike in quantity demanded). If two part tariffs are not available, solving for the optimal prices (and consequently
the allocation) is quite challenging.
31
distortionary.24
3.1
Pairwise Stable Equilibrium
Here we define a pairwise stable equilibrium. This has two stages. First an “arrangement” determines which techniques are used and pricing for each of those techniques. Second, firms set prices
of final goods and select their input mix. In defining an arrangement, care is taken to ensure enough
structure so that payoffs after deviations are well defined.
Definition 1 An arrangement consists of the following:
(i) For each technique φ ∈ Φ (t), define the triple {p(φ), τ (φ), A(φ)} ∈ R+ × R× {0, 1}.
p(φ) and τ (φ) are the price and fixed part of a two part tariff. A(φ) = 1 implies that the
supplier associated with the technique, s(φ), is willing to supply goods at those terms, whereas
A(φ) = 0 indicates she is not.
(ii) For each firm, j, a ordering of techniques. Formally, this is a bijective function Oj : ΦSj →
n
o
1, ..., ΦB
.
j
Let φ∗j ≡ arg min{φ∈ΦS |A(φ)=1} Oj (φ) be the technique actually used by firm j in the arrangej
ment. This is jointly determined by the ordering Oj and the whether or not each of j’s potential supn
o
B |φ = φ∗
pliers are willing to accept the contracting terms of the arrangement. ΦB∗
≡
φ
∈
Φ
j
j
b(φ)
is the set of customers that use firm j’s good as an input in the arrangement.
Given an arrangement, firms choose a price and quantity for final output along with an input
P
mix to maximize profit. p0j yj0 is revenue from final output, φ∈ΦB∗ [p(φ)x(φ) + τ (φ)] is revenue from
j
output for use as an intermediate input, and τ φ∗j + p φ∗j x φ∗j is the cost of intermediates
24
If labor were supplied elastically, the only differences between the planner’s allocation and the decentralized
equilibrium would be that less labor would be supplied because of the monopoly markups and all production would
scale down in proportion to the decrease in aggregate labor.
32
inputs used in production. Profit for firm j is
πj = max p0j yj0 +
p0j ,yj0 ,L,x
X
[p(φ)x(φ) + τ (φ)] − τ φ∗j − p φ∗j x − wL
φ∈ΦB∗
j
subject to the technological constraints and a downward sloping demand curve for final output:
X
yj0 +
φ∈ΦB∗
j
yj0
Given an arrangement,
α 1−α
1
z φB
x φB
L
j
j
1−α
αα (1 − α)
≤ D p0j
x(φ) ≤
n
o
{p(φ), τ (φ), A(φ)}φ∈ΦB , Oj
j
j∈J
, consider two types of deviations:
A unilateral deviation for firm j is an alternative order Ôj and/or an alternative acceptance
Â(φ) for each φ ∈ ΦB
j .
For a technique φ, a pairwise deviation is an alternative two part tariff p̂(φ), τ̂ (φ), an alternative acceptance for the supplier s(φ) about whether to accept the contract, Â(φ), and an alternative
ordering for the buyer, Ôb(φ) .
Either of these types of deviations delivers an alternative arrangement, and hence alternative
payoffs.
Definition 2 A pairwise stable equilibrium is an arrangement
o
n
, and a wage w such that
firms choices, p0j , yj0 , xj , Lj
n
o
{p(φ), τ (φ), A(φ)}φ∈ΦB , Oj
j
j∈J
,
j∈J
• Given wages and profits, the final consumers maximize utility.
• For each j ∈ J,
n
o
p0j , yj0 , xj , Lj maximize firm j’s profit given the arrangement, wages, and
the final demand.
33
• Labor and final goods markets clear
• There are no unilateral deviations that would increase a firm’s profit.
• There are no pairwise deviations that would increase each firms’ profit.
Let qjpw be the efficiency of firm j, equal to the wage divided by marginal cost. In Appendix E
we prove the following:
Proposition 3 In any pairwise stable equilibrium:
(i) For each technique that is actually used, price is equal to marginal cost, p φ∗j =
the fixed fee is nonnegative, τ φ∗j ≥ 0;
α
pw
(ii) qjpw = maxφ∈ΦS z(φ) qs(φ)
w
,
qjpw
and
j
These conditions are the same as the necessary conditions to the planners problem. In fact,
there are many pairwise stable equilibria that decentralize the planner’s solution. In each of these,
marginal costs are the same as in the planner’s solution, but the various equilibria differ in the fixed
part of the tariffs. In each of these equilibria, the markup in the sales of goods to the final consumer
generates a profit for each good. This profit is then divided up across the supply chain to produce
that good according to the fixed fees.25 Normally the markups would distort the consumptionleisure margin, but since labor is supplied inelastically, the markup has no effect on the allocation
of goods and labor. Even if labor were elastic, all of these equilibria would generate the same
aggregate production function as the planner would choose.
There are additional equilibria which do not decentralize the planner’s problem. For example,
there is always an equilibrium in which all firms set a price of infinity (this corresponds to one of
25
There is an upper limit to the fees that can be charged in equilibrium: they cannot be so high that the buyer
would find it beneficial to use an alternative supplier. Short of that limit, any nonnegative fee will do.
34
constant fixed points of equation (10)).
There is a sense in which that equilibrium is fragile (along with any others that might exist).
There is a slightly perturbed economic environment no such equilibrium exists. More specifically,
consider adding to each firm’s techniques a production function that uses only labor, y = qL, with
q common to all firms. In this environment we can provide a welfare theorem: every pairwise stable
equilibrium decentralizes the planner’s solution.
Proposition 4 If q > 0, then with probability one F sp = F pw and Qsp = Qpw .
See Appendix E for proof. For the remainder of the paper, we will focus on equilibria that
decentralize the planner’s allocation.
3.2
An Alternative Interpretation of the Model
An alternative interpretation of the model is that the technology embodied in a technique is nonrival and freely available for others to replicate. In this interpretation, each good is produced by an
island of identical firms, and labor is perfectly mobile across islands. The state of technology can
still be represented as a network, but each node is an island of firms producing a single good rather
than an individual entrepreneur. Again, the network represents the input-output architecture of
the economy, but among islands of firms. In this interpretation perfect competition might be a
more natural benchmark with all prices are set at marginal cost. It is straightforward to show that
competitive equilibrium allocations correspond to those of pairwise stable equilibria.
35
4
Asymmetric Networks
The preceding analysis studied a very specific type of network, and leaves open the question of how
alternative network configurations would affect aggregate productivity. For example, if many firms
are in the center of a city and others are in the outskirts, how would aggregate productivity respond
to increased concentration? If some entrepreneurs were particularly active in finding new techniques
(and others particularly inactive), how would this change patterns of diffusion of productivity gains
from newly discovered techniques?
To get at these, we first extend the previous setup to allow for more interesting network configurations. There are multiple types of firms, indexed by k ∈ K. The only structural difference
between firms of different types is how frequently they are involved with new techniques that are
discovered. Let Mt be the mass of each type k firms and, abusing notation, let K be the number
of types (in addition to the set of types).
The social planner’s problem is exactly the same as in Section 1.2 and all first order conditions
carry over. Instead of characterizing the distribution of efficiency across all firms, it will be convenient to characterize the distribution among firms of each type. Let Fk (q) be the fraction of type
k firms with efficiency less than q. We proceed to characterize these distributions by setting up a
fixed point problem.
We next define several objects that have analogs in Section 1. ωk (n; k 0 ) is the fraction type k
firms that have access to n techniques with suppliers of type k 0 . At a given point in time this follows
a Poisson distribution with mean λ̃k (k 0 ). Let Gk (q; k 0 ) be the distribution of efficiency provided by
36
a single technique drawn by a type k firm with supplier type k 0 . Gk (q; k 0 ) is then:
0
∞
Z
Gk (q; k ) =
1
q
α
F
k0
z
0
dH(z)
Given Gk (q; k 0 ), we can compute Pr (qj ≤ q|j is type k), the probability that, given all of its
draws of techniques, a firm of type k has efficiency no greater than q:
∞
Y X
Pr (qj ≤ q|j is type k) =
ωk (n; k 0 )Gk (q; k 0 )n = e−
P
k0
λ̃k (k0 )[1−Gk (q;k0 )]
k0 ∈K n=0
The same abuse of the law of large numbers gives Fk (q) = Pr (qj ≤ q|j is type k), giving the fixed
point problem, K functional equations for the K unknown functions {Fk (·)}k∈K :
Fk (q) = e
4.1
−
P
k0 ∈K
1
R
λ̃k (k0 ) 1− 0∞ Fk0 ( zq ) α dH(z)
(14)
A Parametric Assumption
We will use the same functional forms as in the one type model, H(z) = 1 −
−ζ
z
z0
and λ̃k (k 0 ) =
λk (k 0 )z0−ζ . We will then look at the equilibrium of the limiting economy as z0 → 0. With a similar
argument, we will show that any set of solutions to equation (14) must follow Frechet distributions.
As before we can write
λ̃k (k 0 ) 1 − Gk (q; k 0 ) = q −ζ λk (k 0 )
Z
0
37
q
z0
1
α
αζxαζ−1 (1 − Fk0 (x)) dx
Substituting into equation (14) for each k 0 we get
− log Fk (q) = q
−ζ
X
Z
0
λk (k )
q
z0
1
α
αζxαζ−1 (1 − Fk0 (x)) dx
0
k0 ∈K
For any set of {Fk (·)}k∈K , as z0 → 0, this expression goes to q −ζ multiplied by a constant. For
−ζ
each k, label this constant θk so that Fk (q) = e−θk q .
We next solve for {θk }k∈K , which are defined to satisfy
θk =
X
0
Z
λk (k )
∞
αζx
αζ−1
(1 − Fk0 (x)) dx =
0
k0 ∈K
X
k0 ∈K
0
Z
λk (k )
0
∞
xαζ Fk0 0 (x)dx
Plugging in the functional form for Fk0 (q) gives
θk = Γ(1 − α)
X
λk (k 0 )θkα0
(15)
k0 ∈K
Notice also that, as before, for any {λk (k 0 )}k,k0 ∈K there are three solutions to equation (15): θk =
0, ∀k ∈ K, θk = ∞, ∀k ∈ K, and a third solution that is the solution to the planner’s problem.
4.2
Aggregate Output
Given the distribution of efficiency across firms, total output will be Y 0 = QL where Qε−1 =
ε−1
J qj
R
(again, the analysis in the one type economy carries over). It will be convenient to define
Qk ≡
R∞
0
q ε−1 dFk (q)
1
ε−1
to be a productivity aggregator among firms of type k. We can then
write
Qε−1 =
X
k∈K
38
Mk Qε−1
k
With the functional forms, we can write (as before):
Qk =
1/ζ
θk Γ
1
ε − 1 ε−1
1−
ζ
The productivity aggregator for the whole economy can then be written as
ε−1
Q
ε−1
ε−1 X
=Γ 1−
Mk θ k ζ
ζ
k∈K
5
Superstars and Productivity Spillovers
The purpose of this section is to demonstrate how changing the configuration of the network
affects aggregate output. We consider examples in which there are two types of firms, indexed by
k ∈ {A, B}. The total mass of firms is MA + MB = 1. Again, the types of firms differ only in how
frequently they are involved with new techniques that are discovered. For a firm of type k let λk
be the (cumulative) arrival rate of techniques, so that λk =
P
k0
λk (k 0 ).
To focus on the influence of the configuration, I will hold the total number of techniques in the
network constant but vary their distribution. In other words, I will let {λk (k 0 )} and {Mk } vary
subject to
λ = MA (λA (A) + λA (B)) + MB (λB (A) + λB (B))
holding λ constant.
The heterogeneity across types is parameterized as follows: Let ρs ≡
λA
λB
=
λA (A)+λA (B)
λB (A)+λB (B)
(s is for
“supplier”). This is a measure of how much more frequently type A firms discover new techniques.
When ρs is larger, type A has a larger advantage in finding potential suppliers.
39
Similarly, let ρb ≡
(λA (A)+λB (A))/MA
(λA (B)+λB (B))/MB
(b is for buyer). This is a measure of how much more
frequently type A firms find potential buyers. When ρb is larger, type A has a larger advantage
in finding potential buyers. If ρb = 1 then the probability of being a supplier is uniform across all
firms. If ρb > 1, then a new technique is relatively more likely to use a type A firm than would be
suggested by MA and MB .
Given the values of ρs and ρb (along with the assumption that
λA (B)
λA (A)
=
λB (B)
λB (A)
which can easily
be abandoned) we can solve for the implied values of {λA (A), λA (B), λB (A), λB (B)}:
λA (A) =
λA (B) =
λB (A) =
λB (B) =
ρb MA
ρb MA + MB
MB
ρb MA + MB
ρb MA
ρb MA + MB
MA
ρb MA + MB
ρs
λ
ρ s MA + M B
ρs
λ
ρ s MA + M B
1
λ
ρ s MA + M B
1
λ
ρ s MA + M B
To compute aggregate output, we need an expression for the productivity aggregator Q =
R
1
ε−1 ε−1
. Given the values of {λk (k 0 )}k,k0 ∈K we can solve for the values of θA and θB using
q
J j
equation (15). With this, we can then solve for the productivity aggregator Q:
ε−1
ε−1
ζ
ε−1
Qε−1 = MA QA
+ MB Qε−1
+ MB θBζ
B ∝ MA θ A
where QA and QB are the productivity aggregators among firms of each type. We are interested
in how Q varies as the structure of the network changes.
Specialization
We first consider an environment in which firms specialize in activities. Type A firms have an
40
advantage in finding new production techniques (ρs > 1) but type B firms have an advantage in
finding potential buyers (ρb < 1). One can think of type A firms as specializing in R&D and type
B firms as specializing in marketing.
Figure 5 shows aggregate output relative to a uniform network (ρs = ρb = 1) with the same
number of total techniques. It is evident that when firms specialize, Aggregate output is quite a
bit lower than a uniformly random network.
Because the R&D firms discover more techniques, they are more productive. However, the firms
specializing in marketing are acquiring a larger share of potential customers. The cost savings found
by the R&D firms get less of a chance to be passed on to other firms: the wrong firms are supplying
the inputs.
Relative Output
1.00
Relative Output
2
3
4
5
6
Ρs ,1Ρb
20
40
60
80
100
Ρs ,1Ρb
0.9
M A = .4
0.95
M A = .4
M A = .5
M A = .5
0.8
M A = .6
M A = .6
0.7
0.90
0.6
0.85
0.5
0.4
0.80
(a) Small Scale
(b) Large Scale
Figure 5: Specialization: Varying ρs > 1, ρb < 1
Figure 5 shows aggregate output relative to a uniform network for various values of activity
(Figure 5a and Figure 5b show the same graph at different scales). When ρs is larger, type A
firms discover techniques more frequently. When ρb < 1, type B firms find potential buyers
more frequently.
Hubs
We first consider the case in which type A firms are hubs: they are more likely to discover new
techniques and they are relatively more likely to be the supplier when other firms get new techniques.
41
This scenario is motivated by a partially urbanized economy in which urban entrepreneurs interact
with each other frequently, while rural entrepreneurs are less active. Formally we set ρs = ρb and
vary this common number for various values of MA . Figure 6 shows aggregate output relative to a
uniform network (ρs = ρb = 1) with the same number of techniques per firm. When the network is
close to uniform, aggregate output rises with the increased concentration of techniques among type
A firms. However this relationship is non-monotonic; as ρs , ρb → ∞, relative productivity slowly
falls back to 1.
Aggregate output is higher than in the uniform network because the more productive firms
are becoming the important suppliers in the economy. When ρs > 1, type A firms draw many
techniques and are therefore more likely to have high efficiency. While type B firms draw a smaller
number of techniques, the techniques they do draw are likely to have type A firms as suppliers,
meaning that the type B firms are likely to end up with a low marginal cost. When ρb , ρs → ∞,
the impact disappears, because all type B firms are essentially disconnected from the network
and are no longer able to benefit from the high efficiency of the type A firms. Type A firms are
extremely productive (there are more techniques among them) but fewer firms are producing, and
consequently less gains from variety. These effects exactly offset, and aggregate output is the same
as the uniform case.
For intermediate values, the increase in aggregate output in response to concentration is larger
when MA is small. This happens because more techniques are concentrated within the type A
firms, so productivity among those firms is high. Type B firms are likely to have techniques that
use type A firms, so that they are increasingly able to benefit from the high productivity among
type A that is due to the increased concentration. This is most stark as MA → 0, in which case
the peak of the curve rises unboundedly. More formally, for a given MA , set ρb = ρs =
42
1/2
MA .
Then
Relative Output
Relative Output
1.07
1.08
M A = .4
1.06
M A = .4
M A = .5
M A = .5
M A = .6
1.05
M A = .6
1.06
1.04
1.04
1.03
1.02
1.02
1.01
1.00
1
2
3
4
5
6
Ρs ,Ρb
(a) Small Scale
20
40
60
80
100
Ρs ,Ρb
(b) Large Scale
Figure 6: Type A are Hubs: Varying Both ρs and ρb
Figure 6 shows aggregate output relative to a uniform network for various values of activity
(Figure 6a and Figure 6b show the same graph at different scales). When the common value of
ρp and ρs is larger, a larger share techniques involve type A firms.
limMA →0 Q = ∞.26
Advantage in Finding Potential Suppliers
We next examine the case in which ρs varies but ρb = 1, shown in Figure 7. In this case
type A firms are more likely to discover new techniques, but do not have any advantage in finding
potential buyers. Here we can see that total output is smaller than the uniform case (and the drop
is persistent: as ρs → ∞, relative productivity stays depressed below 1). Here, while type A firms
are more productive as a group than they would be in the uniform case, the type B firms are less
able to take advantage of this, because the techniques they find aren’t especially concentrated on
the type A firms. As a result, type B firms are less productive in than they would be in the uniform
case, so much so that this dominates the increased productivity among type A firms.
The intuition for why aggregate productivity is lower than the uniform case can be found in
26
An even more stark example (which is easier to analyze by hand) is ρs =
1/2
,
MA
1/2
MB
MB
MA
and ρb → ∞. This leads to
(normalizing λ = 1), λA (A) =
λB (A) =
, λA (B) = λB (B) = 0. Here, no matter how few type A firms there
are, half of all total techniques are drawn by those firms. However, all techniques use type A firms as inputs. Again,
we have limMA →0 Q = ∞.
43
equation (15) relating the average efficiencies of each type: θk = Γ(1 − α)
P
k0 ∈K
λk (k 0 )θkα0 . Note
that θk is a linear combination of concave functions of each of the θk0 s. Thus θk would be higher if
the θk0 s were closer together.
Relative Output
1.00
Relative Output
2
3
4
0.98
5
6
Ρs
20
40
60
0.95
M A = .4
100
M A = .4
M A = .5
0.96
80
M A = .5
0.90
M A = .6
M A = .6
0.85
0.94
0.80
0.92
0.75
0.90
(a) Small Scale
(b) Large Scale
Figure 7: Type A discover techniques more frequently: Varying ρs with ρb = 1
Figure 7 shows aggregate output relative to a uniform network for various values of activity
(Figure 7a and Figure 7b show the same graph at different scales). When ρs is larger, type A
firms discover techniques more frequently.
Advantage in Finding Potential Buyers
Lastly we examine the the case in which ρs = 1 but ρb varies, so that A has an advantage in
finding potential buyers, but not in finding potential suppliers. As shown in Figure 8, aggregate
productivity is exactly the same as in the uniform case. Considering this and the previous case, we
can infer the different roles of ρb and ρs . ρs generates productivity differences across the different
types, as drawing more techniques leads to (on average) higher productivity. ρb determines how
much of these productivity differences spill over to the other types. In this case where ρs = 1, there
are no productivity differences to spill over, so varying ρb makes no differences. Compare this to
ρs 6= 1, in which case varying ρb can make a big difference.
44
Ρs
Relative Output
1.0
0.8
M A = .4
M A = .5
0.6
M A = .6
0.4
0.2
20
40
60
80
100
Ρb
Figure 8: Type A discover potential buyers more frequently: ρs = 1, Varying ρb
Figure 8 shows aggregate output relative to a uniform network for various values of activity.
When ρb is larger, type A firms are more likely to find potential buyers. This is not a misprint;
all three lines take the constant value 1.
6
Multiple Inputs
In the preceding analysis, each firm used a single input in production. While this is assumption
makes the analysis (and visualization) of the model economy considerably easier, it is obviously at
odds with production in the real world. In this section we extend the analysis so that firms use
multiple inputs. While there are several ways to model what happens when firms discover new
production methods, I will focus on a version that I consider to be both natural and tractable.
A firm needs to accomplish a fixed set of tasks. Over time, firms discover new techniques to
accomplishing each task, and each of these techniques is a sub-production function that requires a
particular input. In this way, a firm can improve production by finding techniques that are more
cost effective in accomplishing individual tasks.27
The central assumption is that each firm uses a production function that combines N tasks into
a unit of output. To accomplish each task, the firm uses a technique. For the technique used for
27
An alternative modeling strategy is that a new technique requires an entirely new input bundle. In that model,
a firm would change suppliers all at once, rather than one at a time.
45
the nth task, let zn be the cost of inputs and qn be the efficiency of the supplier associated with
that task. Then the production is such that the efficiency of the firm is
q=
N
Y
1
(zn qnα ) N
n=1
The production function that justifies this can be written in two ways that are equivalent as long
as the firm minimizes cost:
y =
N
Y
n=1
y =
zn xαn Ln1−α
1−α 1−α α α
N
1
(1 − α)1−α
!1
N
N
N
Y
N
α − α Y
N
n=1
N
1
N
zn
N
Y
1
N
xn
!α
L1−α
n=1
n=1
The difference between the two is that in the first version labor is explicitly allocated separately to
each task. The important feature is that α is the share of intermediate inputs.
Let F (q) be the fraction of firms with efficiency no greater than q. We will use the same method
as before to solve to define a fixed point problem for F .
For a firm, let vn be the efficiency of a firm in accomplishing the nth task. Also, the number
of techniques a firm has to accomplish a given task is follows a Poisson distribution with mean
λ̃ (which is the same for each task). Let K(v) be probability that vn < v.28 Then by the same
analysis as before, we can write
−λ̃
K(v) = e
R ∞h
0
1−F
28
( vz )
1/α
i
dH(z)
The analysis can easily be extended so that λ̃ varied by task, which would imply that the CDFs of each vn to be
different (i.e. Kn (·)).
46
Since q =
1
QN
N
n=1 vn , we can write F (q) as
Z
∞
F (q) =
∞
Z
...
0
K
0
!
qN
QN
n=2 vn
dK (v2 ) dK (v3 ) ...dK (vN )
We now impose the parametric assumptions H(z) = 1 − (z/z0 )−ζ and λ̃ = λz0−ζ . As shown in
Appendix F, in the limiting economy as z0 → 0, the fraction of tasks with techniques delivering
efficiency no greater than v converges to a Frechet distribution
K(v) = e−θv
−ζ
α N
θ = λθα Γ 1 −
N
The expression for F is messier, but we can derive an expression for aggregate productivity Q =
R∞
0
q ε−1 dF (q)
1
ε−1
:
(ε − 1) /ζ
Q=Γ 1−
N
N
ε−1
1
1
α N (1−α)ζ
λ (1−α)ζ Γ 1 −
N
This generalizes the formula for aggregate productivity with a single input given by equa
N
ε−1
α N
tion (12).Since Γ 1 − N
and Γ 1 − (ε−1)/ζ
are decreasing in N , aggregate productivity is
N
declining in the number of inputs, as shown in Figure 9.
Aggregate productivity declines with the number of inputs as a consequence because the tail of
superstar suppliers shrinks. This happens because of two mutually reinforcing mechanisms. First,
the distribution of efficiency F (q) becomes more concentrated. With a single input, a firm would
end up with a high efficiency with one great technique. With more inputs, a firm that gets lucky
enough to draw a great technique for one of its tasks may only have low efficiency techniques for its
47
Q
1.3
1.2
1.1
1.0
2
4
6
8
10
N
Figure 9: Aggregate Productivity by Number of Inputs
Figure 9 shows aggregate productivity Q the as the number of inputs changes. The parameters
are α = 1/2, ζ = 4, and = 4, with λ set so that the limiting aggregate productivity when the
number of inputs grows large is 1. The purple line is the asymptotic aggregate productivity as
the number of inputs grows large.
other tasks. The more inputs there are, the harder it is for a firm to have a high overall efficiency,
as the the law of large numbers (across tasks) begins to kick in.
On top of this, when efficiencies are more concentrated among potential suppliers, it becomes
harder to draw a high efficiency technique for a single task. The efficiency provided by a technique
is zq α , and less dispersion in q implies that relatively high efficiency techniques are drawn less
frequently.
A stark feature of the model that drives this result is that all tasks enter symmetrically into
production. If instead production were dominated by relatively few core tasks, then the distribution
of efficiencies would be less concentrated and superstar suppliers would be more relevant.
Alternatively, if tasks are substitutes then the high efficiency tasks will have a large cost share.
In the limit as the elasticity of substitution goes to infinity, the N input model would converge to
48
the single input model.
7
Conclusion
This paper has described a tractable model of the formation and evolution of chains of production.
The model aggregates easily, with a simple formula connecting features of the network to aggregate
output. A key feature driving aggregate productivity is whether the low cost producers are able
to become the important suppliers of the economy. To become an important supplier, a firm must
have many potential customers who know how to use the firm’s good and to be able to charge
those potential customers a low price. When intermediate goods are more important in production
(relative to labor), the ability to charge a low price becomes more important in winning customers,
and the lowest cost firms are more likely to become superstar suppliers. This increases aggregate
productivity and also increases the market concentration in sales of intermediate goods.
With more interesting network configurations, the model can be solved almost as easily. With
multiple types, aggregate productivity depends on how productivity and demand covary across
groups. Holding the total quantity of techniques fixed, concentration of link formation among a
subset of firms, e.g., in a city, leads to positive correlation and higher aggregate output. In contrast,
negative correlation, e.g., if some firms specialize in R&D whereas others specialize in marketing,
can lead to significantly lower output.
49
Appendix
A
Notation
A chain of techniques is a sequence of techniques (finite of infinite) φ0 φ1 φ2 ... with the s(φk+1 ) =
b(φk ). Given the set of all existing techniques, Φ, we can define several objects.
Let C(j) be the set of chains of techniques with the property φ0 φ1 φ2 ... with the additional
property that b(φ0 ) = j. These are the distinct supply chains available that could be used to
produce good j. For example, if firm j1 has access to a single technique φ1 with supplier j2 , and
j2 has access to two techniques φ2 and φ3 , then C contains the three distinct chains φ1 , φ1 φ2 , and
φ1 φ3 .
Let CN (j) ⊆ C(j) be the set of chains of techniques to produce good j with exactly N techniques. Let bN (j) ≡ |CN (j)| be the number of such distinct supply chains of length N .
Lastly, let C∞ (j) ⊆ C(j) be the set of infinite chains of techniques to produce good j. These
∞ (j) is the set of chains of techniques of
are all of the viable supply chains to produce good j. CN
length N that form the beginning of an infinite supply chains. For example, φ0 φ1 φ2 ... ∈ C ∞ (j),
then φ0 φ1 is in C2∞ (j).
B
The Number of Supply Chains
For completeness, we show the derivation of two results from the theory of branching processes
that will be used in this paper (see for example Athreya and Ney (1972)).
−λ̃ k
Let p(k) be the probability that a firm has exactly k techniques, in this case equal to e k!λ̃ ,
and PN (l, k) be the probability that, in total, l different firms have among them k supply chains
of length N . Note that PN (1, k) is the probability a firm has exactly k supply chains of length N
(i.e., the probability
bN = k).
Pthat
∞
Define ϕ(x) = k=0 p(k)xk to be the probability generating function for the random variable
b1 . In this case ϕ(x) = e−λ̃(1−x) . Also, for each N , let ϕN (·) be the probability generating function
associated with bN . If ϕ(N ) is the N -fold composition of ϕ then we have the convenient result:
Claim 1 ϕN (x) = ϕ(N ) (x)
Proof. We proceed by induction. By definition, the statement is true for N = 1. Noting that
P
∞
k
l
k=0 P1 (l, k)x = ϕ(x) , we have
ϕN +1 (x) =
∞
X
l=0
=
∞
X
PN +1 (1, l)xl =
∞ X
∞
X
PN (1, k)P1 (k, l)xl =
l=0 k=0
∞
X
k=0
PN (1, k)ϕ(x)k = ϕN (ϕ(x))
k=0
We immediately have the following:
Claim 2 For any x, E xbN = ϕ(N ) (x).
50
PN (1, k)
∞
X
l=0
P1 (k, l)xl
P
k
(N ) (x)
Proof. E xbN = ∞
k=0 PN (1, k)x = ϕN (x) = ϕ
We next study the probability that a single firm has no chains that continue indefinitely. This
can be characterized as follows:
Claim 3 The probability that a single firm has no chains that continue indefinitely is the smallest
root, ρ, of y = ϕ(y).
Proof. The probability that a firm has no chains greater than length N is given by PN (1, 0), or
equivalently ϕ(N ) (0). Then the probability that a single firm has no chains that continue indefinitely
is limN →∞ ϕN (0).
Next note that that ϕ is increasing and convex, ϕ(1) = 1, and ϕ(0) ≥ 0. This implies that in
the range [0, 1], the equation ϕ(y) = y has a either a unique root at y = 1 or two roots, y = 1 and
a second in (0, 1).
Let ρ be the smallest root. Noting that for y ∈ [0, ρ), y < ϕ(y) < ρ, while for y ∈ (ρ, 1) (if such
y exist), ρ < ϕ(y) < y. Together these imply that if y ∈ [0, 1), the sequence {ϕN (y)} is monotone
and bounded, and therefore has a limit. We have ϕN +1 (0) = ϕ(ϕN (0)). Taking limits of both
sides (and noting that ϕ is continuous) gives limN →∞ ϕN +1 (0) = ϕ(limN →∞ ϕN (0)). Therefore
the limit must be a root of y = ϕ(y), and therefore must be ρ. In other words, it must be that
limN →∞ ϕN (0) = ρ.
Claim 4 If λ̃ ≤ 1 then with probability 1 the firm has no chains that continue indefinitely. If λ̃ > 1
then there is a strictly positive probability the firm has a chain that continues indefinitely.
Proof. In this case, we have ϕ(x) = e−λ̃(1−x) . If λ̃ ≤ 1 then the smallest root of y = ϕ(y) is y = 1.
If λ̃ > 1 the smallest root is strictly less than 1.
C
Existence and Characterization of the Solution to the Planner’s
Problem
The strategy is as follows: We define a sequence of random variables {XN }N ∈N with the property
that the maximum feasible efficiency of a firm is given by the limit of this sequence, if such a limit
exists. We then show that XN converges to a random variable X sp in Lε−1 . Next we show that the
distribution of X sp is given by the unique fixed point of T in F, a subset of F̄ (and of course that
such a fixed point exists). Letting F sp be this least fixed point, we apply the law of large numbers
for continuum of random variables (Uhlig (1996)) to argue that the cross-sectional distribution of
efficiencies (in Lε−1 ) is given by F sp and that aggregate productivity is simply kX sp kε−1 .
C.1
A Unique Fixed Point
We begin by defining three functions, f¯, f 1 , and f , in F̄. To do so, we define several objects
that will parameterize these functions. Let ρ ∈ (0, 1) be the smallest root of ρ = e−λ̃(1−ρ) . In
the definition of f¯, q2 and β are defined as follows: Let β > ε − 1, let q2 be defined so that
n
o
1
R∞
α
q21−α > max 1, λ̃ β exp z0 log(z)dH(z) . In the definition of f , q0 = z01−α . The three functions
51
are:
(
f¯(q) ≡
ρ,
1 − (1 − ρ)
f 1 (q) ≡
ρ, q < 1
1, q ≥ 1
ρ, q < q0
1, q ≥ q0
f (q) ≡
q < q2
β
q
q2
, q ≥ q2
Consider the partial order on F̄, the set of right continuous, weakly increasing functions f :
R+ → [0, 1] given by the binary relation : f1 f2 ⇔ f1 (q) ≤ f2 (q)∀q ≥ 0. Clearly f¯ f 1 f .
Let F ⊂ F̄ be the subset of set of nondecreasing functions f : R+ → [0, 1] that satisfy f¯ f f .
Lemma 1 T f f and f¯ T f¯
Proof. We first show that T f f . For q ≥ q0 this is immediate, as T f (q) ≤ 1 = f (q). For q < q0 ,
we have
−λ̃
T f (q) = e
R∞
0
1−f
1
( zq ) α
dH(z)
−λ̃
=e
R∞
α [1−ρ]dH(z)
q/q0
1−α
≤ e−λ̃[1−ρ](1−H (q0
)) = ρ = f (q)
∞
¯
We now proceed to f¯. First, for q < q2 , we have T f¯(q) = e−λ̃ 0 (1−f )dH(z) ≥ e−λ̃(1−ρ) = ρ = f (q).
Next, as an intermediate step, we will show that for q ≥ q2 :
#
"
−β
Z q/qα
2
β
β
q
α
α
> λ̃ exp
ln z α dH (z) − ln (q/q2 ) α H (q/q2 )
q2
0
R
α
We start with q21−α > λ̃ β exp
q21−α
R∞
ln (z) dH (z) which can be written as
#
"Z
Z ∞
q/q2α
α
> λ̃ β exp
ln (z) dH(z)
ln (z) dH (z) +
0
q/q2α
0
Using the fact that q ≥ q2 and that
"
Z
α
1−α
β
q
> λ̃ exp
R∞
x
ln(z)dH(z) ≥ ln(x) [1 − H(x)] gives
q/q2α
#
ln (z) dH (z) + ln (q/q2α ) [1 − H (q/q2α )]
0
Raising both sides to
β
α
β
and dividing (q/q2α ) α gives the desired result.
52
Next, beginning with 1 − T f¯(q) ≤ − ln T f¯(q)
Z ∞ 1 − f¯ (q/z) α1
¯
1 − T f (q)
≤ λ̃
dH (z)
1−ρ
1−ρ
0
1
Z ∞
¯
α
1 − f (q/z)
dH (z)
ln
≤ λ̃ exp
1−ρ
0
Z
1
q/q2α
ln
= λ̃ exp
0
"Z
= λ̃ exp
q/q2α
(q/z) α
q2
ln z
β
α
!−β
dH (z)
dH (z) − ln
β
(q/q2α ) α
#
H
(q/q2α )
0
q −β
q2
1 − f¯ (q)
1−ρ
<
=
This then gives, for q ≥ q2 , T f¯(q) ≥ f¯(q).
Lemma 2 There exist least and greatest fixed points of the operator T in F, given by limN →∞ T N f¯
and limN →∞ T N f respectively.
Proof. Proof: The operator T is order preserving, and the F is a complete lattice. By the Tarski
fixed point theorem, the set of fixed pints of T in F is also a complete lattice, and hence has a least
and a greatest fixed point given by lim T N f¯ and lim T N f respectively.
C.2
Existence of a Limit
We first show that the planer’s problem is well defined. Given the set of techniques at a point in
time, Φ, let Cj∞ be the set of distinct infinite chains available to produce good j. For each chain
c ∈ Cj∞ and each n ≥ 0, let zn (c) is the productivity of the nth technique in chain c. In other
words, for Q
a chain c of techniques φ0 φ1 φ2 ... with φ0 furthest downstream, zn (c) = z(φn ). Finally
αn
let q(c) ≡ ∞
n=0 zn (c) .
We want to define qj to be the efficiency provided by the most cost effective supply chain
available to produce j, or more formally that qj = supc∈Cj∞ q(c). To do this, we first argue that
nQ
o
N
αn might
for each c, q(c) is well defined. The concern is that for some c the sequence
z
(c)
n=0 n
not converge. The following
proposition
shows
that
it
does.
QN −1
n
Define qN (c) ≡ n=0
zn (c)α for N ≥ 1 with q0 (c) ≡ 1.
Lemma 3 Assume z0 > 0. Then for each c ∈ Cj∞ , limN →∞ qN (c) exists.
Proof. For each n, we can decompose log zn (c) into log zn+ (c) − log zn− (c), P
where log zn+ (c) =
−1 n
−
+
max{log zn (c), 0} and log zn (c) = max{−logzn (c), 0}. We then have log qN (c) = N
n=0 α log zn (c)−
53
PN −1
PN −1 n
αn log zn− (c).
α log zn+ (c) is a monotone sequence so it converges to a (possibly inPN −1 n n=0 −
0)
α log zn (c) is a monotone sequence bounded by log(1/z
finite) limit.
so it converges to a
n=0
1−α
h
i
log(1/z0 )
limit in the range 0, 1−α . Therefore qN (c) converges to a (possibly infinite) limit.
Next, we define the random variable XN,j = maxc∈Cj∞ qN (c). Roughly, the remainder of this
subsection shows that XN,j converges qj . Since qj = supc∈Cj∞ limN →∞ qN (c), we are essentially
proving that the limit can be passed through the sup.
A useful property of the variable XN,j is that its CDF is T N f 1 . It will also be useful to define
the random variables ȲN,j and YN,j . These variables will be constructed so that their CDFs are
T N f¯, and T N f respectively.
To do this, we expand the probability space as follows: For each realization of Φ, let q̃(c) be a
¯−ρ
∞ .
random variable with CDF f1−ρ
drawn independently for each c ∈ CN,j
n=0
N
N
∞ qN (c)q α . We also define
∞ qN (c)q̃(c)α
For N ≥ 1, let ȲN,j ≡ maxc∈CN,j
and YN,j ≡ maxc∈CN,j
0
Ȳ0,j and Y0,j to have CDFs f¯ and f respectively.
To improve readability, the subscript j will be suppressed when not necessary.
Lemma 4 {XN }N ∈N , ȲN
N ∈N
, and {YN }N ∈N are uniformly integrable in Lε−1 .
Proof. First, Recall that Ȳ0 is defined to have the cumulative distribution f¯. Since the T is
order preserving, the relations T N f 1 T N f¯ and T N f¯ T N −1 f¯ imply that T N f 1 f¯. As
a consequence, Ȳ0 first-order stochastically dominates each XN and ȲN , and, by the identical
argument, YN . Therefore E Ȳ0
E ȲN
ε−1
ε−1
=
q2ε−1
1− ε−1
β
< ∞ serves as a uniform bound on each E |XN |ε−1 ,
, and E |YN |ε−1 .
Lemma 5 There exists a random variable X sp such that XN converges to X sp almost surely and
in Lε−1 .
Proof. Let MN ≡
X
QN N
n=0 µn
where µn ≡
λ̃(1−ρ)H 0 (w)e−λ̃(1−ρ)[1−H(w)]
dw.
1−e−λ̃(1−ρ)
integrable in Lε−1 .
R∞
z0
wα
n
We first show that
MN is a submartingale that is uniformly
To do this, we define a set DN as follows: Let c∗N = arg maxc∈CN∞ qN (c) so that XN = qN (c∗N ).
∞ be the set of chains in C ∞ for which the first N − 1 links are c∗
Let DN ⊆ CN
N
N −1 . In other words,
∗
all chains in DN are of the form cN −1 φ for some φ.
∞ , it must be that X ≥ D .
Define the random variable DN = maxc∈DN qN (c). Since DN ⊆ CN
N
N
We now show that E [DN |CN −1 ] ≥ µN −1 XN :
k
e−λ̃[1−ρ] [λ̃(1−ρ)]
The number of chains in DN is at least one. The probability that |DN | = k is
[1−e−λ̃[1−ρ] ]k!
for k ≥ 1. To see this, note that for any node, the number of techniques is poisson with mean λ̃.
Each of those has probability 1 − ρ of being viable (having a chain that continues infinitely), and
we are conditioning on at least one viable technique.
Each of those techniques has a productivity drawn from H. For any φ such that c∗N −1 φ ∈ DN ,
we have that
α−N
∞
αn
Pr qN (c∗N φ) < x|CN
=
Pr
z(φ)
<
x/X
=
H
(x/X
)
N −1
N −1
−1
54
Therefore the probability that all k of the chains in DN give qN (c∗N φ) < x given XN −1 is
k
α−N
∞
Pr DN < x|CN
−1 , |DN | = k = H (x/XN −1 )
∞ , is
With this, the CDF of DN , given CN
1
∞
Pr DN < x|CN
−1
=
∞
X
Pr (DN < x|XN −1 , |DN | = k) Pr (|DN | = k)
k=1
h
ik
−λ̃[1−ρ] λ̃ (1 − ρ)
e
−N k
h
i
=
H (x/XN −1 )α
−λ̃[1−ρ] k!
1
−
e
k=1
∞
X
=
e
−N
λ̃[1−ρ]H (x/XN −1 )α
eλ̃[1−ρ]
−1
−1
We can now compute the conditional expectation of DN (using the change of variables w =
−N
(x/XN −1 )α ):
∞
E DN |CN
=
−1
Z
∞
xα−N
0
∞
Z
= XN −1
1
XN −1
x
α−N −1
XN −1
N
wα λ̃ (1 − ρ) H 0 (w)
0
e
−N
λ̃ (1 − ρ) H 0 (x/XN −1 )α
eλ̃[1−ρ]H(w)
eλ̃[1−ρ] − 1
−N
λ̃[1−ρ]H (x/XN −1 )α
eλ̃[1−ρ] − 1
dw
= µN XN −1
Putting this together, we have
1
∞
E MN |CN
−1 = QN
n=0 µn
1
∞
E XN |CN
−1 ≥ QN
n=0 µn
1
∞
E DN |CN
−1 = QN
n=0 µn
µN XN −1 = MN −1
We next need to show that {MN } is uniformly integrable, i.e., that supN
QE [MN ] < ∞. We
know that supN E [XN ] < ∞. So it suffices to show a uniform lower bound on N
n=0 µn . Since each
1
Q
Q
Q
n
n
n
∞
N
1−α
α
α
µn ≥ z0α and z0 < 1, we have that N
.
n=0 z0 = z0
n=0 µn ≥
n=0 z0 ≥
We have therefore established that {MN }N ∈N is a uniformly integrable (in L1 ) submartingale,
so by the martingale convergence theorem, there exists an M such that MN converges to M almost
surely. By the continuous mapping theorem, there exists an X sp such that XN converges to X sp
ε−1
almost surely. Since each XN
is dominated by the integrable random variable Ȳ0ε−1 , by dominated
convergence we have that XN converges to X sp in Lε−1 .
Proposition 5 With probability one, X sp = supc∈C ∞ q(c)
Proof. We first show that X sp ≥ supc∈C ∞ q(c) with probability one. Consider any realization of
techniques, Φ. For any ν > 0, there exists a c∗ ∈ C ∞ such that q(c∗ ) > supc∈C ∞ q(c) − ν. There
also exists an N1 such that N > N1 implies qN (c∗ ) > q(c∗ ) − ν. Lastly, with probability one there
55
dx
exists an N2 such that N > N2 implies XN < X sp + ν. We then have for N > max N1 , N2 that
X sp > XN − ν = max
q (c) − ν ≥ qN (c∗ ) − ν > q(c∗ ) − 2ν > sup q(c) − 3ν,
∞ N
c∈CN
w.p.1
c∈C ∞
This is true for any ν > 0, so X sp ≥ supc∈C ∞ q(c). We next show the opposite inequality. For any
N , we have
αN
1−α
sup q(c) ≥ sup qN (c)z0
c∈C ∞
c∈C ∞
αN
1−α
Since this is true for any N and limN →∞ z0
with probability one.
C.3
αN
1−α
= XN z0
= 1, we can take the limit to get supc∈C ∞ q(c) ≥ X sp
Characterization of the Limit
We will show below that log ȲN − log YN converges to 0 in probability. Since XN ∈ [YN , ȲN ], it
must be that both ȲN and YN converge to X sp in probability. Convergence in probability implies
convergence in distribution, which gives to implications. First, T N f¯ and T N f converge to the
same limiting function. Since these are the least and greatest fixed points of T in F, this limiting
function, f sp , is the unique fixed point of T in F. Second, since T N f¯ T N f 1 T N f , f sp is the
CDF of X sp .
!
−N
Lemma 6 For any η > 1, limN →∞ 1 − ϕ(N )
f¯ η α
q0 −ρ
1−ρ
=0
d (N )
Proof. We first show that for x ∈ [0, 1], dx
ϕ (x) ≤ λ̃N . To see this, note that ϕ is convex and
0
0
ϕ (1) = λ̃, so that ϕ (x) ≤ λ̃ for x ≤ 1. In addition, if x ∈ [0, 1] then ϕ(x) ∈ (0, 1], which implies
ϕ(N ) (x) ∈ (0, 1] for each N . We then have
N
Y
d (N )
ϕ (x) =
ϕ0 ϕ(n−1) (x) ≤ λ̃N
dx
n=1
With this, for any x, we can bound ϕ(N ) (x) by
Z 1
ϕ(N )0 (w)dw
ϕ(N ) (x) = ϕ(N ) (1) −
x
Z 1
≥ 1 − λ̃N
dw
x
≥ 1 − λ̃N [1 − x]
56
"
To complete the proof, we show that limN →∞
(
λ̃N
1−
#
−N
f¯ η α
q0 −ρ
1−ρ
= 0. Recall that
f¯(q)−ρ
1−rho
=
0
q < q2
−β
. We have for q ≥ q2
1 − qq2
q ≥ q2
−N
f¯ η α q0 − ρ
= lim λ̃N q2β η −βα−N = 0
lim λ̃N 1 −
N →∞
N →∞
1−ρ
Next we use this to show that log ȲN − log YN converges to zero in probability.
Lemma 7 For any η > 1, limN →∞ Pr ȲN /YN > η = 0.
N
Proof. For any chain c ∈ CN , define the following two objects: q̄N (c) ≡ qN (c)q̃(c)α (recall that
∞ q̃(c) was a random variable with CDF f¯−ρ ) and q (c) ≡ q (c)q αN (q is the same
for each c ∈ CN
0
N
N
0
1−ρ
f −ρ
∞ q̄N (c) and
as a random variable with CDF 1−ρ ). With these definitions, we have ȲN,j ≡ maxc∈CN,j
∞ qN (c).
YN,j ≡ maxc∈CN,j
∞ (if any exist) that
Conditional on the set of techniques Φ, we have that for each chain c in CN
α N
q̄N (c)/qN (c) = q̃(c)
. We therefore have:
q0
Pr q̄N (c)/qN (c) < η|Φ = Pr
q̃(c)
q0
αN
−N
f¯ η α q0 − ρ
!
< η|Φ
=
1−ρ
If there are bN such chains of length N , the probability that every one of them of them gives
!bN
−N
q̄N (c)/qN (c) < η is
f¯ η α
q0 −ρ
1−ρ
Pr ȲN /YN
so that
bN
−N
f¯ η α q0 − ρ
< η|Φ ≥
1−ρ
Recall from Appendix B that for any x, E xbN = ϕ(N ) (x) where ϕ(N ) is the N -fold composition
of ϕ and the expectations are taken over realizations of Φ. We can use this to show
Pr ȲN /YN < η
= E Pr ȲN /YN < η|Φ
bN
−N
−N
α
¯
¯ η α q0 − ρ
f
η
q
−
ρ
f
0
(N )
≥ E
=ϕ
1−ρ
1−ρ
57
Put differently, limN →∞ Pr ȲN /YN > η ≤ limN →∞ 1 −
ϕ(N )
!
−N
f¯ η α
q0 −ρ
1−ρ
. We complete the
proof by applying the previous lemma.
We now come to the main result.
Proposition 6 There is a unique fixed point of T on F, F sp . F sp is CDF of X sp . Aggregate
1
R∞
productivity is Q = 0 q ε−1 dF sp (q) ε−1 with probability one.
p
p
Proof. The combination of log ȲN − log YN → 0, ȲN ≥ XN ≥ YN , and XN → X sp implies that
p
p
ȲN → X sp and YN → X sp .
We first show that there is a unique fixed point, which is also the CDF of X sp . The CDFs
of ȲN and YN are T N f¯ and T N f respectively. The least and greatest fixed points of T in F are
limN →∞ T N f¯ and limN →∞ T N f respectively. Convergence in probability implies convergence in
distribution, so the least and greatest fixed point are the same, and that the fixed point is the CDF
sp
of X sp . Call
this fixed point F .
Since ȲN and {YN } are uniformly integrable in Lε−1 , we have by Vitali’s convergence theorem
that Ȳ N → X sp in Lε−1 and Y N → X sp in Lε−1 .
Putting all of these pieces together, we have that the CDF of qj is F sp . We next show that
1
R∞
aggregate productivity is the Q = 0 q ε−1 df sp (q) ε−1 . For this we simply apply the law of large
numbers for a continuum economy of Uhlig (1996). To do this, we must verify that the efficiencies
are pairwise uncorrelated. This is trivial: consider two firms, j and i. Since the number of firms in
any of j’s supply chains is countable, the probability that i and j have overlapping supply chains is
zero. The theorem in Uhlig (1996) also requires that the variable in question has a finite variance,
and if it does, the then the L2 integral exists. Here we are interested in the Lε−1 norm, so we require
1
R∞
the X sp is Lε−1 integrable. Therefore we have that Q = 0 q ε−1 df sp (q) ε−1 with probability one.
D
D.1
Size Distribution
Number of Customers
Proposition 7 Let pn be the mass of firms with n customers. Then
Z
pn =
0
∞
w−α
Γ(1−α)
n
−
e
w−α
Γ(1−α)
n!
The counter cumulative distribution has a tail index of 1/α.
58
e−w dw
Proof.
∞
Z
pn =
αζ
λ αζ n − λ
e θq
θq
n!
0
∞
Z
=
dF (q)
−α n − λθα (θq−ζ )−α
−ζ
θ
θq
e
λθα
θ
n!
0
∞
Z
=
w−α
Γ(1−α)
n
e
−α
w
− Γ(1−α)
e−w dw
n!
0
−ζ
ζθq −ζ−1 e−θq dq
The second line uses the fact that θ = Γ(1 − α)λθα . We can rewrite this as
∞
Z
pn =
0
−1/α
1
un e−n e−[Γ(1−α)u]
u− α −1 du
1
n!
Γ(1 − α) α α
−1/α
Since
e−[Γ(1−α)u]
1
Γ(1−α) α α
is slowly varying as u → ∞, applying Theorem 2.1 of Willmot (1990) gives
pn ∼
e−[Γ(1−α)n]
−1/α
1
α
Γ(1 − α) α
1
n− α −1 ,
n→∞
Therefore the counter-cumulative distribution, Pr (# customers > n), has a tail index of 1/α.
D.2
The Distribution of Employment
Let χ(s) be the characteristic function associated with the cross sectional distribution of employment. This is the central object of interest, but to get at it, we take several intermediate steps. If
L0 (q) is the (deterministic) quantity of labor used producing final output for a firm with efficiency
q, let χ0 (s|q) be the characteristic function for labor used for final demand. We have
Z ∞
Z ∞
0
isl
0
isL0 (q)
eisx δ(x)dx
χ (s|q) =
e δ l − L (q) dl = e
−∞
−∞
isL0 (q)
= e
where δ is the Dirac delta function.
We showed above that the quantity of labor used by a single customer can be treated as an IID
random variable. Recall also the convenient fact that if firm j uses Lj units of labor, j’s supplier
will use αLj units of labor to make the inputs for j. If Lj is an IID random variable, αLj is as
well. With this in mind, let χ1 (s) be the characteristic function associated with the labor required
to make the inputs for a single customer.
If the density of labor is Pr(Lj = l), the density of labor
used for a supplier is α1 Pr Lj = α1 , so that we have
59
Z ∞
1
l
l
l
−isl
−i(αs) αl
Pr Lj =
χ (s) =
e
dl =
e
d
Pr Lj =
α
α
α
−∞ α
−∞
= χ(αs)
Z
1
∞
Let χint (s|q) be the characteristic function associated with the labor used for all intermediates
among firms with efficiency q. Using the fact that the characteristic function of the sum of independent random variables is the product of the characteristic functions of each of the random
variables, we can write χint (s|q) as
χ
int
(s|q) =
∞
X
1
n
χ (s) Pr (n customers|q) =
n=0
= e
∞
X
1
χ (s)
n
αζ
λ αζ n −λ q θ
e
θq
n!
n=0
λ αζ
1
= e− θ q [1−χ (s)]
−λ
q αζ [1−χ(αs)]
θ
where the second equality uses the fact that among firms with efficiency q the distribution of
customers is Poisson with parameter λθ q αζ .
We can put these together to derive an expression for χ(s|q), the characteristic function associated with the distribution of employment among firms with efficiency q:
χ (s|q) = χ0 (s|q) χint (s|q) = eisL
0 (q)
λ αζ
[1−χ(αs)]
e− θ q
Lastly we can integrate across firms, which delivers a single recursive equation that defines χ(s):
Z ∞
Z ∞
λ αζ
0
χ (s) =
χ (s|q) dF (q) =
eisL (q) e− θ q [1−χ(αs)] dF (q)
0
0
We now plug in the functional forms
L0 (q)
− ε−1
ζ
=
(θq−ζ )
Γ 1− ε−1
ζ
−ζ
(1 − α) L and dF (q) = θζq −ζ−1 e−θq dq
to give
Z
χ (s) =
ε−1
−ζ −
(θq ) ζ
is
(1−α)L
∞
Γ(1− ε−1 )
e
ζ
e
−µ
θ α θq −ζ
(
θ
)
−α
[1−χ(αs)]
−ζ
θζq −ζ−1 e−θq dq
0
and using a change of variables along with the fact that θ = Γ(1 − α)λθα gives
Z ∞
− ε−1
−α
ζ
t
t
(1 − α) L −
[1 − χ(αs)] − t dt
χ(s) =
exp is
Γ 1 − ε−1
Γ(1 − α)
0
ζ
E
(16)
Pairwise Stable Equilibria
We are careful here to use notation that allows for the possibility that a firm uses a production
chain with a cycle: at least some of the input from j is used in as an intermediate good in the
supply chain used by j. In this case, if a firm decides to lower its price of final output, it sells more
60
final output but also raises the demand for its good as an intermediate
input.
B∗
In particular, given an arrangement, for each φ ∈ Φj , let xb(φ) yj0 be the quantity of good
j sold to firm b(φ) given yj0 , holding constant the final output of other firms. xb(φ) yj0 is an
nondecreasing and weakly concave function of yj0 . The profit for firm j can be written as:
X
p(φ)xb(φ) yj0 + τ (φ) − wL − p (φ∗ ) x − τ (φ∗ )
πj = max p0j yj0 +
p0j ,yj0 ,L,x
φ∈ΦB∗
j
subject to
yj0 +
X
xb(φ) yj0
φ∈ΦB∗
j
≤
1
z (φ∗ ) xα L1−α
− α)1−α
αα (1
yj0 ≤ D p0j
The first order conditions with respect to inputs deliver cost minimization, so that firm j’s marginal
cost is cj ≡ z(φ1 ∗ ) p (φ∗ )α w1−α . The price of final output p0j then satisfies:
X
p0j = arg max (p − cj ) D(p) +
p
(p(φ) − cj ) xb(φ) (D(p))
φ∈ΦB∗
j
Note that if good j is in the supply chain to produce good j (there is a cycle), firm j internalizes
how changing final output would affect there sales of j as an intermediate.
Proposition 8 In any pairwise stable equilibrium, for each technique φ∗ that is used, p(φ) = cs(φ) .
Proof. Assume there is a φ∗ = {j, i, z} such that x (φ∗ ) > 0 and p (φ∗ ) 6= ci . Consider the deviation
p̂ (φ∗ ) = ci
τ̂ (φ∗ ) = τ (φ∗ ) + xj (p (φ∗ ) − ci ) + K
h
i
P
where K ∈ 0, p̂0j − ĉj ŷj0 − p0j − ĉj yj0 + φ∈ΦB∗ (p(φ) − ĉj ) xb(φ) ŷj0 − xb(φ) yj0
. Given
j
i’s cost ci , the cost for j of the new technique is ĉj =
the deviation is
1
α 1−α .
z(φ∗ ) ci w
The change in profit for i from
π̂i − πi = τ̂ (φ∗ ) − τ (φ∗ ) + (p̂ (φ∗ ) − ci ) x̂j − (p (φ∗ ) − ci ) xj = K > 0
The change in profit for j from the deviation is
π̂j − πj
= − [τ̂ (φ∗ ) − τ (φ∗ )] + p̂0j ŷj0 − p0j yj0 +
X
φ∈ΦB∗
j
−
nh
i
o
wL̂j + ci x̂j − [wLj + p (φ∗ ) xj ]
61
p(φ) xb(φ) ŷj0 − xb(φ) yj0
substituting in the expressions for τ̂ , p̂, and ĉ, we have
X
(p(φ) − ĉj ) xb(φ) ŷj0 − xb(φ) yj0 − K
π̂j − πj = p̂0j − ĉj ŷj0 − p0j − ĉj yj0 +
φ∈ΦB∗
j
+ (1 − α) +
ci
α
p (φ∗ )
p (φ∗ )
ci
α
− 1 ĉj yj0 +
X
xb(φ) yj0
φ∈ΦB∗
j
We know that [(1 − α) + xα] x−α ≥ 1 for α ∈ [0, 1] (since xt is a convex function of t we
−α
1−α ≥ x(1−α)(−α)+(1−α)α = 1). Therefore
can
use Jensen’s inequality
to get (1 − α)x + αx
(1 − α) +
ci
p(φ∗ ) α
p(φ∗ )
ci
α
≥ 1, so that
X
π̂j − πj ≥ p̂0j − ĉj ŷj0 − p0j − ĉj yj0 +
(p(φ) − ĉj ) xb(φ) ŷj0 − xb(φ) yj0 − K > 0
φ∈ΦB∗
j
Therefore the deviation is mutually beneficial.
Proposition 9 In any pairwise stable equilibrium, for each technique φ∗ that is used, τ (φ∗ ) ≥ 0.
Proof. If τ (φ∗ ) < 0, for some φ∗ = {j, i, z} then consider the unilateral deviation  (φ∗ ) = 0 (the
supplier i refuses the contract).
After the deviation, every firm’s input price remains the same except j. j’s price will change
and will use a technique from an alternative supplier (the next one on his list if there is one)
and hence an alternative supply chain. The quantity produced by i will change: it will no longer
produce inputs for j, and if the i is in j’s alternative supply chain, i may produce more for that
chain. However, since price equals marginal cost, this change in quantity produced has no effect
on i’s profit. We therefore have
π̂i = πi − τ (φ∗ ) > πi
which confirms the unilateral deviation.
n
o
α
Proposition 10 In any pairwise stable equilibrium, qj = maxφ∈ΦS z(φ)qs(φ)
.
j
Proof. Assume that φ∗j = φ1 ≡ j, i1 , z 1 and let cj = z 1 qiα1 . Toward a contradiction, assume
there exists a φ2 = j, i2 , z 2 ∈ ΦSj and that cj > ĉj ≡ z 2 qiα2 .
Consider the pairwise deviation: Ôj φ2 = 1, Ôj φ1 > 1, Âi2 φ2 = 1, p̂ φ2 = ci2 , and
maxp (p−ĉj )D(p)−maxp (p−cj )D(p)
τ̂ φ2 =
> 0.
2
In the deviation, j 0 s marginal cost falls, but every other firm’s marginal cost is unchanged.
Therefore j lowers its price of final output, but no other firm does, so the quantity of final output
of j rises.
Firm j finds the deviation profitable, π̂j > πj , because marginal cost has fallen: (i) j now makes
a positive profit on each unit of output sold as intermediate output (if j is in a cycle, then profit
from intermediates
would rise even more); (ii) The change in profit from final sales is greater than
the fee τ̂ φ2 ; and (iii) j no longer as to pay τ φ1 (which is nonnegative).
62
Firm i2 also finds the deviation profitable, π̂i2 > πi2 , because it now collects τ̂ φ2 . Since price
equals marginal cost, changes in quantity produced as intermediates have no impact on its profit.
This confirms the unilateral deviation.
We now show that if there if each firm has access to a technology to produce its good using only
labor, yj = qL with common productivity q, then in every pairwise stable equilibrium aggregate
productivity Q is the same as in the planner’s problem.
If qjpw is the efficiency of firm j in a pairwise stable equilibrium, let F pw = Pr qjpw < q be the
cross sectional CDF of efficiencies in such an equilibrium. We can compare this to the distribution
of efficiencies that solve the planner’s problem, F sp .
Proposition 11 In any pairwise stable equilibrium, F pw = F sp and Qpw = Qsp .
Proof. In both the planner’s problem and in any pairwise stable equilibrium, each firm chooses
the technique that delivers the most cost effective combination:
(
)
n
o
α
qj = max q, max z(φ)qs(φ)
φ∈ΦB
j
For each firm j, we are interested in the efficiency in the social planner’s problem, qjsp , and efficiency
in a pairwise equilibrium. These satisfy the following three equations:
sp
αN
,
w.p.1
qj = max sup q(c), sup max qN (c)q
c∈C ∞
qjpw ≥
qjpw
≤
N ∈N c∈CN
sup max qN (c)q α
N
N ∈N c∈CN
qjsp ,
w.p.1
These can be interpreted as follows: First, the chain with the maximum feasible efficiency available
to the planner is either an infinite chain or a finite chain that ends with the furthest firm upstream
using the outside option. Second, in any pairwise stable equilibrium, the efficiency of a firm must
be at least that of any finite chain that ends with a firm using its outside option q. If otherwise, at
least one of the firms in that chain is must be using a suboptimal technique. Third, efficiency in a
pairwise stable equilibrium must be feasible. The first and third equations are satisfied only with
probability one because the planner may deviate from the maximum feasible efficiency for a set of
firms with measure zero.
The remainder of the proof follows broadly along the same lines as Appendix C with some
minor modifications. As a result I will only provide a sketch, pointing out the differences. We can
construct an operator in the same manner as before:
0,
q<q
1
R∞
q α
T f (q) ≡
e−λ̃ 1− 0 f ( z ) dH(z) , q ≥ q
63
We can also define three functions.
0, q < q
1, q ≥ q
0, q < 1
1, q ≥ 1
f (q) ≡
1
f (q) ≡
(
f¯(q) ≡
1−
0,
−β
q
q2
q < q2
, q ≥ q2
With the additional restriction that q2 > q.
Let F be the subset of F̄ defined by {f |f¯ f f }. In the same manner as before, we can
show that T f¯ f¯ and T f f . This means that there is a least and greatest fixed point of T in
F, given by limN →∞ T N f¯ and limN →∞ T N f respectively.
Q −1
αN̂
Define YN ≡ maxN̂ ≤N maxc∈CN̂ N̂
variables XN ≡
n=0 q(c)q n . With this we can define o
n
o
QN −1
QN −1
N
α
max YN , maxc∈CN n=0 q(c) and ȲN ≡ max YN , maxc∈CN n=0 q(c)q̃(c)
, where q̃(c) is
a random variable drawn independently for each c in CN for each realization of Φ. The CDFs of
XN , ȲN , and YN are T N f 1 , T N f¯, and T N f respectively.
Let X sp ≡ limN→ ∞ XN and X pw ≡ limN→ ∞ YN In the same manner as before we can show that
these limits exists and that with probability 1 we have both X sp = q sp and X pw ≤ q pw ≤ X sp .
Lastly, we can show that F sp is the unique fixed point of T on F, and is the CDF of both X sp and
X pw . Applying the law of large numbers gives Qsp = Qpw .
F
Multiple Inputs
Fraction of tasks that deliver efficiency no greater than v is K(v):
−λ̃
R ∞h
K(v) = e
Since q =
QN
0
1−F
( vz )
1/α
i
dH(z)
1
N
n=1 vn , we the cross sectional distribution of efficiency F (q) is
!
Z ∞ Z ∞
qN
F (q) =
...
K QN
dK (v2 ) dK (v3 ) ...dK (vN )
v
0
0
n
n=2
We now impose the parametric assumptions H(z) = 1 − (z/z0 )−ζ and λ̃ = λz0−ζ . This gives
K(v) = e
−λ
R ∞h
= e−λv
z0
−ζ
1−F
( vz )
R (v/z0 )1/α
0
Taking a limit as z0 → 0 gives
K(v) = e−θv
where θ = λ
R∞
0
[1 − F (r)] αζrαζ−1 dr
64
1/α
i
ζz −ζ−1 dz
[1−F (r)]αζrαζ−1 dr
−ζ
Next, we solve for θ. To do this we can rewrite F (q) as (using the substitution wn = θvn−ζ )
Z ∞ Z ∞
QN
−ζ
−ζ
−ζ
N
−ζ−1 −θvN
e−θ(q / n=2 vn ) θζv2−ζ−1 e−θv2 dv2 ...θζvN
e
dvN
...
F (q) =
Z0 ∞ Z0 ∞
QN
−ζ N
e−(θq ) / n=2 wn e−w2 dw2 ...e−wN dwN
...
=
0
0
so that the density is
Z
N
1
F (q) = N ζ θq −ζ
q
0
∞
∞
Z
...
0
−ζ
e−(θq )
N
/
QN
n=2
wn −w2 dw2
e
0
w2
...e−wN
dwN
wN
We can take the expression for θ, integrate by parts, and substitute in for F 0 to get
Z ∞
[1 − F (r)] αζrαζ−1 dr
θ = λ
Z0 ∞
= λ
F 0 (r) rαζ dr
0
Z ∞
N Z ∞ Z ∞
QN
−ζ N
dw2 −wN dwN αζ
1
−ζ
e−(θr ) / n=2 wn e−w2
...
= λ
N ζ θr
...e
r dr
r
w2
wN
0
0
0
N
Making the substitution u =
∞Z ∞
Z
θ = λθα
(θr−ζ )
QN
,
n=2 wn
∞
Z
...
0
0
this becomes
0
dw2 −wN dwN 1− α
e−u e−w2
u N
...e
w2
wN
N
Y
n=2
!1− Nα
wn
du
u
α N
= λθα Γ 1 −
N
Aggregate productivity Q can be computed as follows:
Z ∞
Qε−1 =
q ε−1 dF (q)
Z0 ∞
N Z ∞ Z ∞
QN
−ζ N
dw2 −wN dwN
ε−1 1
−ζ
q
...
e−(θq ) / n=2 wn e−w2
=
N ζ θq
...e
dq
q
w2
wN
0
0
0
N
Making the substitution u =
ε−1
Q
= θ
ε−1
ζ
Z
0
∞
(θq−ζ )
QN
n=2 wn
N
Y
and this becomes
!1− ε−1
/N
ζ
wn
1− ε−1
/N
ζ
Z
u
65
Z
...
0
n=2
N
ε−1
(ε
−
1)
/ζ
= θ ζ Γ 1−
N
∞
0
∞
e−u e−w2
dw2 −wN dwN du
...e
w2
wN u
so that aggregate productivity is
(ε − 1) /ζ
Q=Γ 1−
N
F.1
N
ε−1
1
1
α N (1−α)ζ
λ (1−α)ζ Γ 1 −
N
Number of Customers
Given efficiency qi , efficiency that a single technique
efficiency less
that uses that gooddelivers
v
v
1
0
than by a technique is probability that a v is H qα , with density qα H qα . The probability
i
i
i
that the potential buyer has no other techniques better than v is simply K(v) (more properly, the
K(v)−e−λ̃
R ∞ v 1/α
dH(z)
(1−e−λ̃ ) 0 F ( z )
which converges to K(v) as z0 → 0). So the probability that
R∞
a technique is the potential buyer’s best is 0 q1α H 0 qvα K(v)dv. Since the number of potential
probability is
i
i
buyer’s follows a Poisson distribution with mean N λ̃, the number of actual buyers is Poisson with
mean:
−ζ−1
Z ∞
Z ∞
1 0 v
1
v
−ζ
N λ̃
H
K
(v)
dv
→
N
λ
ζ
e−θv dv
α
α
α
α
q
q
q
q
0
0
i
i
i
i
λ αζ
= N qi
θ
66
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Sumit Agarwal and Faye H. Wang
WP-09-08
Pay for Percentile
Gadi Barlevy and Derek Neal
WP-09-09
The Life and Times of Nicolas Dutot
François R. Velde
WP-09-10
Regulating Two-Sided Markets: An Empirical Investigation
Santiago Carbó Valverde, Sujit Chakravorti, and Francisco Rodriguez Fernandez
WP-09-11
2
Working Paper Series (continued)
The Case of the Undying Debt
François R. Velde
Paying for Performance: The Education Impacts of a Community College Scholarship
Program for Low-income Adults
Lisa Barrow, Lashawn Richburg-Hayes, Cecilia Elena Rouse, and Thomas Brock
Establishments Dynamics, Vacancies and Unemployment: A Neoclassical Synthesis
Marcelo Veracierto
WP-09-12
WP-09-13
WP-09-14
The Price of Gasoline and the Demand for Fuel Economy:
Evidence from Monthly New Vehicles Sales Data
Thomas Klier and Joshua Linn
WP-09-15
Estimation of a Transformation Model with Truncation,
Interval Observation and Time-Varying Covariates
Bo E. Honoré and Luojia Hu
WP-09-16
Self-Enforcing Trade Agreements: Evidence from Antidumping Policy
Chad P. Bown and Meredith A. Crowley
WP-09-17
Too much right can make a wrong: Setting the stage for the financial crisis
Richard J. Rosen
WP-09-18
Can Structural Small Open Economy Models Account
for the Influence of Foreign Disturbances?
Alejandro Justiniano and Bruce Preston
WP-09-19
Liquidity Constraints of the Middle Class
Jeffrey R. Campbell and Zvi Hercowitz
WP-09-20
Monetary Policy and Uncertainty in an Empirical Small Open Economy Model
Alejandro Justiniano and Bruce Preston
WP-09-21
Firm boundaries and buyer-supplier match in market transaction:
IT system procurement of U.S. credit unions
Yukako Ono and Junichi Suzuki
Health and the Savings of Insured Versus Uninsured, Working-Age Households in the U.S.
Maude Toussaint-Comeau and Jonathan Hartley
WP-09-22
WP-09-23
The Economics of “Radiator Springs:” Industry Dynamics, Sunk Costs, and
Spatial Demand Shifts
Jeffrey R. Campbell and Thomas N. Hubbard
WP-09-24
On the Relationship between Mobility, Population Growth, and
Capital Spending in the United States
Marco Bassetto and Leslie McGranahan
WP-09-25
The Impact of Rosenwald Schools on Black Achievement
Daniel Aaronson and Bhashkar Mazumder
WP-09-26
3
Working Paper Series (continued)
Comment on “Letting Different Views about Business Cycles Compete”
Jonas D.M. Fisher
WP-10-01
Macroeconomic Implications of Agglomeration
Morris A. Davis, Jonas D.M. Fisher and Toni M. Whited
WP-10-02
Accounting for non-annuitization
Svetlana Pashchenko
WP-10-03
Robustness and Macroeconomic Policy
Gadi Barlevy
WP-10-04
Benefits of Relationship Banking: Evidence from Consumer Credit Markets
Sumit Agarwal, Souphala Chomsisengphet, Chunlin Liu, and Nicholas S. Souleles
WP-10-05
The Effect of Sales Tax Holidays on Household Consumption Patterns
Nathan Marwell and Leslie McGranahan
WP-10-06
Gathering Insights on the Forest from the Trees: A New Metric for Financial Conditions
Scott Brave and R. Andrew Butters
WP-10-07
Identification of Models of the Labor Market
Eric French and Christopher Taber
WP-10-08
Public Pensions and Labor Supply Over the Life Cycle
Eric French and John Jones
WP-10-09
Explaining Asset Pricing Puzzles Associated with the 1987 Market Crash
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
WP-10-10
Prenatal Sex Selection and Girls’ Well‐Being: Evidence from India
Luojia Hu and Analía Schlosser
WP-10-11
Mortgage Choices and Housing Speculation
Gadi Barlevy and Jonas D.M. Fisher
WP-10-12
Did Adhering to the Gold Standard Reduce the Cost of Capital?
Ron Alquist and Benjamin Chabot
WP-10-13
Introduction to the Macroeconomic Dynamics:
Special issues on money, credit, and liquidity
Ed Nosal, Christopher Waller, and Randall Wright
WP-10-14
Summer Workshop on Money, Banking, Payments and Finance: An Overview
Ed Nosal and Randall Wright
WP-10-15
Cognitive Abilities and Household Financial Decision Making
Sumit Agarwal and Bhashkar Mazumder
WP-10-16
Complex Mortgages
Gene Amromin, Jennifer Huang, Clemens Sialm, and Edward Zhong
WP-10-17
4
Working Paper Series (continued)
The Role of Housing in Labor Reallocation
Morris Davis, Jonas Fisher, and Marcelo Veracierto
WP-10-18
Why Do Banks Reward their Customers to Use their Credit Cards?
Sumit Agarwal, Sujit Chakravorti, and Anna Lunn
WP-10-19
The impact of the originate-to-distribute model on banks
before and during the financial crisis
Richard J. Rosen
WP-10-20
Simple Markov-Perfect Industry Dynamics
Jaap H. Abbring, Jeffrey R. Campbell, and Nan Yang
WP-10-21
Commodity Money with Frequent Search
Ezra Oberfield and Nicholas Trachter
WP-10-22
Corporate Average Fuel Economy Standards and the Market for New Vehicles
Thomas Klier and Joshua Linn
WP-11-01
The Role of Securitization in Mortgage Renegotiation
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-11-02
Market-Based Loss Mitigation Practices for Troubled Mortgages
Following the Financial Crisis
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-11-03
Federal Reserve Policies and Financial Market Conditions During the Crisis
Scott A. Brave and Hesna Genay
WP-11-04
The Financial Labor Supply Accelerator
Jeffrey R. Campbell and Zvi Hercowitz
WP-11-05
Survival and long-run dynamics with heterogeneous beliefs under recursive preferences
Jaroslav Borovička
WP-11-06
A Leverage-based Model of Speculative Bubbles (Revised)
Gadi Barlevy
WP-11-07
Estimation of Panel Data Regression Models with Two-Sided Censoring or Truncation
Sule Alan, Bo E. Honoré, Luojia Hu, and Søren Leth–Petersen
WP-11-08
Fertility Transitions Along the Extensive and Intensive Margins
Daniel Aaronson, Fabian Lange, and Bhashkar Mazumder
WP-11-09
Black-White Differences in Intergenerational Economic Mobility in the US
Bhashkar Mazumder
WP-11-10
Can Standard Preferences Explain the Prices of Out-of-the-Money S&P 500 Put Options?
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
WP-11-11
5
Working Paper Series (continued)
Business Networks, Production Chains, and Productivity:
A Theory of Input-Output Architecture
Ezra Oberfield
WP-11-12
6
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