Equilibrium in Production Chains with Multiple Upstream Partners
EEquilibrium in Production Chains with MultipleUpstream Partners (cid:73)
Meng Yu a,b , Junnan Zhang c, ∗ a Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academyof Sciences, Beijing 100190, China b School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing100049, China c Research School of Economics, Australian National University, Australia
Abstract
In this paper, we extend and improve the production chain model introduced byKikuchi et al. (2018). Utilizing the theory of monotone concave operators, weprove the existence, uniqueness, and global stability of equilibrium price, henceimproving their results on production networks with multiple upstream partners.We propose an algorithm for computing the equilibrium price function that ismore than ten times faster than successive evaluations of the operator. Themodel is then generalized to a stochastic setting that offers richer implicationsfor the distribution of firms in a production network.
Keywords:
Production Network, Firm Boundaries, Monotone ConcaveOperator Theory, Equilibrium Uniqueness, Computational Techniques
JEL:
C62, C63, L11, L14
1. Introduction
Over the past several centuries, firms have self-organized into ever morecomplex production networks, spanning both state and international boundaries,and constructing and delivering a vast range of manufactured goods and services.The structures of these networks help determine the efficiency (Levine, 2012;Ciccone, 2002) and resilience (Carvalho, 2007; Jones, 2011; Bigio and LaO, 2016;Acemoglu et al., 2012, 2015a) of the entire economy, and also provide new insightsinto the directions of trade and financial policies (Baldwin and Venables, 2013;Acemoglu et al., 2015b).We consider a production chain model introduced by Kikuchi et al. (2018)that examines the formation of such structures. They connect the literatureon firm networks and network structure to the underlying theory of the firm. (cid:73)
The authors thank John Stachurski for his helpful comments and valuable suggestions.We also thank Simon Grant, Ronald Stauber, Ruitian Lang, and seminar participants at theAustralian National University for their helpful comments and discussions. We are also gratefulfor the inputs from the editor and two anonymous referees. This research is supported by anAustralian Government Research Training Program (RTP) Scholarship. ∗ Corresponding author
Email addresses: [email protected] (Meng Yu), [email protected] (Junnan Zhang )
Preprint submitted to Journal of Mathematical Economics February 16, 2019 a r X i v : . [ ec on . T H ] A ug n particular, the model in Kikuchi et al. (2018) formalizes the foundationalideas on the theory of the firm presented in Coase (1937), embedding them inan equilibrium model with a continuum of price taking firms, and providingmathematical representations of the determinants of firm boundaries suggestedby Coase (1937).A single firm at the end of the production chain sells a final product toconsumers. The firm can choose to produce the whole product by itself orsubcontract a portion of it to possible multiple upstream partners, who thenmake similar choices until all the remaining production is completed. The mainreason for firms to produce more in-house is to save the transaction costs ofbuying intermediate products from the market. In fact, Coase (1937) regardsthis as the primary force that brings firms into existence. An opposing force thatlimits the size of a firm is the costs of organizing production within the firm . Aprice function governs the choices firms make and is determined endogenously inequilibrium when every firm in the production chain makes zero profit.Considering that all firms are ex ante identical, a notable feature of thismodel is its ability to generate a production network with multiple layers offirms different in their sizes and numbers of upstream partners. The source ofthe heterogeneity lies solely in the transaction costs and firms’ different stagesin the production chain. This feature provides insights into the formation ofpotentially more complex structures in a production network. Kikuchi et al.(2018) prove the existence, uniqueness, and global stability of the equilibriumprice function restricting every firm to have only one upstream partner. In thiscase, the resulting production network consists of a single chain.There are however, several significant weaknesses with the analysis in Kikuchiet al. (2018). First, while they provide comprehensive results on uniqueness ofequilibrium prices and convergence of successive approximations in the singleupstream partner case, they fail to provide analogous results for the more inter-esting multiple upstream partner case, presumably due to technical difficulties.Second, their model cannot accurately reflect the data on observed productionnetworks because their networks are always symmetric, with sub-networks ateach layer being exact copies of one another. Real production networks do notexhibit this symmetry . Third, they provide no effective algorithm for computingthe equilibrium price function in the multiple upstream partner case.This paper resolves all of the shortcomings listed above. As our first contri-bution, we extend their existence, uniqueness, and global stability results to themultiple partner case. To avoid the technical difficulties faced in their paper, weemploy a different approach utilizing the theory of monotone concave operators,which enables us to give a unified proof for both cases.Theoretically, the concave operator theory ensures the global stability of the One justification also mentioned in Kikuchi et al. (2018) is that firms usually experiencediminishing return to management: when a firm gets bigger it also bears increasing coordinationcosts. See also Coase (1937), Lucas (1978), and Becker and Murphy (1992). Mathematically, the equilibrium price function is determined as the fixed point of a Bellmanlike operator (see Section 3). Globally stability means that the fixed point can be computedby successive evaluations of the operator on any function in a certain function space. For instance, for a mobile phone manufacturer, most subcontractors who supply complicatedcomponents like display or CPUs have multiple upstream partners of their own, while thosewho supply raw materials usually don’t (Dedrick et al., 2011; Kraemer et al., 2011). f such that f ( x ) > x and f ( x ) < x with x < x . Then it must be true that f has aunique fixed point on [ x , x ], and by the concavity of f , the fixed point canbe computed by successive evaluations of f on any x ∈ [ x , x ]. No contractionproperty is needed here while we still get all the results from the ContractionMapping Theorem. A full-fledged theorem owing to Du (1989) for arbitraryBanach spaces is stated in Theorem 3.1. For similar treatments in the mathliterature, also see Krasnoselskii et al. (1972), Krasnosel’skii and Zabre˘iko (1984),Guo and Lakshmikantham (1988), Guo et al. (2004), and Zhang (2013).Apart from Theorem 3.1, there are other similar techniques that utilizeconcavity to show uniqueness of the fixed point. Krasnosel’skii (1964) shows We thank an anonymous referee for referring us to some of the works mentioned here. The existence of fixed points can be tackled in various ways. For the operator T , existencehas already been proved in Kikuchi et al. (2018) who use the classical Knaester–Tarskifixed point theorem. Alternatively, it can be proved, as an anonymous referee suggests, bydemonstrating that T is completely continuous (see, e.g., Krasnosel’skii, 1964, Chapter 4).The Schauder typed fixed point theorems also apply; see, for example, Section 7.1 in Cheney(2013). u -concave). For applications ofthis technique in the economic literature, see, for example, Lacker and Schreft(1991) and Becker and Rinc´on-Zapatero (2017). Following Krasnosel’skii (1964),Coleman (1991) proves uniqueness under slightly different concavity and mono-tonicity conditions (pseudo-concave and x -monotone). See also Datta et al.(2002b), Datta et al. (2002a), and Morand and Reffett (2003) for other economicapplications along this line.Marinacci and Montrucchio (2010, 2017) link concavity to contraction in theThompson metric (Thompson, 1963), which allows one to apply the ContractionMapping Theorem to operators that are not contractive under the supnorm.In a similar vein, Marinacci and Montrucchio (2017) establish existence anduniqueness results for monotone operators under a range of weaker concavityconditions using Tarski-type fixed point theorems and the Thompson metric.Among all of these results, the theorem by Du (1989) turns out to be the mostsuitable for our work.The monotone concave operator theory has seen some recent success in theeconomic literature. Lacker and Schreft (1991) study an economy with cash andtrade credit as means of payment and show that the equilibrium interest rateis a unique fixed point of a monotone concave operator. Coleman (1991, 2000)studies the equilibrium in a production economy with income tax and proves theexistence and uniqueness of consumption function by constructing a monotoneconcave map. Following this approach, Datta et al. (2002b) prove the existenceand uniqueness of equilibrium in a large class of dynamic economies with capitaland elastic labor supply. Similar work in the same vein includes Morand andReffett (2003) and Datta et al. (2002a). Rinc´on-Zapatero and Rodr´ıguez-Palmero(2003) exploit the monotonicity and convexity properties of the Bellman operatorand give conditions for existence and uniqueness of fixed points in the case ofunbounded returns. Balbus et al. (2013) study the existence and uniquenessof pure strategy Markovian equilibrium using theories concerning decreasingand “mixed concave” operators. More recently, this theory has been appliedextensively to models with recursive utilities since Marinacci and Montrucchio(2010); other contributions along this line include Balbus (2016), Boroviˇckaand Stachurski (2017, 2018), Becker and Rinc´on-Zapatero (2017), Marinacci andMontrucchio (2017), Pavoni et al. (2018), Bloise and Vailakis (2018), and Renand Stachurski (2018).Our work connects to this literature in that the operator which determinesthe equilibrium price is shown to be increasing and concave but does not satisfyany contraction property. To prove existence and uniqueness, Kikuchi et al.(2018) use an ad hoc and convoluted method for the case when every firm canonly have one upstream partner but fail to generalize it to the multiple partnercase. Using the monotone concave operator theory, we are able to extend theirresults and give a much simpler proof.Section 2 describes the model in detail. Section 3 introduces the monotoneconcave operator theory and gives existence and uniqueness results. The algo-rithm is described in Section 4. Section 5 generalizes the model, allowing for Among these works, Balbus (2016), Boroviˇcka and Stachurski (2017), and Ren andStachurski (2018) use versions of fixed point theorems similar to Theorem 3.1 in this paper.
2. The Model
We study the production chain model with multiple partners in Kikuchi et al.(2018). The chain consists of a single firm at the end of the chain which sells asingle final good to consumers and firms at different stages of the production,each of which sells an intermediate good to a downstream firm by producing thegood in-house or subcontracting a portion of the production process to possiblymultiple upstream firms. We index the stage of production by s ∈ X = [0 , p : X → R + and a cost function c : X → R + . Subcontracting incurs a transaction costthat is proportionate to the price with coefficient δ > g : N → R + that is a function of thenumber of upstream partners. The cost g can be seen as the costs of maintainingpartnerships such as legal expenses and communication costs.We adopt the same assumptions as in Kikuchi et al. (2018). For the costfunction c , we assume that c (0) = 0 and it is differentiable, strictly increasing,and strictly convex. In other words, each firm experiences diminishing returnto management as mentioned in the introduction. This assumption is neededhere because otherwise no firm would want to subcontract its production. Wealso assume c (cid:48) (0) >
0. For the additive transaction cost function g , we assumethat it is strictly increasing, g (1) = 0, and g ( k ) goes to infinity as the number ofupstream partners k goes to infinity. To summarize, we have the following twoassumptions. Assumption 2.1.
The cost function c is differentiable, strictly increasing, andstrictly convex. It also satisfies c (0) = 0 and c (cid:48) (0) > Assumption 2.2.
The additive transaction cost function g is strictly increasing, g (1) = 0, and g ( k ) → ∞ as k → ∞ .Therefore, a firm at stage s solves the following problem:min t ≤ sk ∈ N { c ( s − t ) + g ( k ) + δkp ( t/k ) } . (1)In (1), the firm chooses to produce s − t in-house with cost c ( s − t ) and subcontract t to k upstream partners. Since each subcontractor is in charge of t/k part ofthe product, this results in a proportionate transaction cost δkp ( t/k ) and anadditive transaction cost g ( k ). Then the firm sells the product to its downstreamfirm at price p ( s ).
3. Equilibrium
Following Kikuchi et al. (2018), we consider the equilibrium in a competitivemarket with free entry and free exit. The price adjusts so that in the long run Here we follow Kikuchi et al. (2018). This transaction cost can be the cost of gatheringinformation, drafting contract, bargaining, or even tax, all of which tend to increase with thevolume of the transaction. p ( s ) = min t ≤ sk ∈ N { c ( s − t ) + g ( k ) + δkp ( t/k ) } . (2)Let R ( X ) be the space of real functions and C ( X ) the space of continuousfunctions on X . Then we can define an operator T : C ( X ) → R ( X ) by T p ( s ) := min t ≤ sk ∈ N { c ( s − t ) + g ( k ) + δkp ( t/k ) } . (3)The equilibrium price function is thus determined as the fixed point of theoperator T . Before proceeding to our main result, we first introduce a theorem due to Du(1989), which studies the fixed point properties of monotone concave operatorson a partially ordered Banach space.Let E be a real Banach space on which a partial ordering is defined bya cone P ⊂ E , in the sense that x ≤ y if and only if y − x ∈ P . If x ≤ y but x (cid:54) = y , we write x < y . An operator A : E → E is called an increasing operator if for all x, y ∈ E , x ≤ y implies that Ax ≤ Ay . It is called a concave operator if for any x, y ∈ E with x ≤ y and any t ∈ [0 , A ( tx + (1 − t ) y ) ≥ tAx + (1 − t ) Ay . For any u , v ∈ E with u < v , wecan define an order interval by [ u , v ] := { x ∈ E : u ≤ x ≤ v } . We have thefollowing theorem (see, e.g., Guo et al., 2004, Theorem 3.1.6 or Zhang, 2013,Theorem 2.1.2). Theorem 3.1 (Du, 1989) . Suppose P is a normal cone , u , v ∈ E , and u < v . Moreover, A : [ u , v ] → E is an increasing operator. Let h = v − u .If A is an concave operator, Au ≥ u + (cid:15)h for some (cid:15) ∈ [0 , , and Av ≤ v ,then A has a unique fixed point x ∗ in [ u , v ] . Furthermore, for any x ∈ [ u , v ] , A n x → x ∗ as n → ∞ . This theorem gives a sufficient condition for the existence, uniqueness, andglobal stability of the fixed point of an operator without assuming it to be acontraction mapping. It is particularly useful in cases where we study a monotoneconcave operator but the contraction property is hard or impossible to establish.This is the case in our model. The operator T is not a contraction because thetransaction cost coefficient δ is greater than 1, but as will be shown below, T isactually an increasing concave operator.Based on Theorem 3.1, we have the following theorem. A cone P ⊂ E is said to be normal if there exists δ > (cid:107) x + y (cid:107) ≥ δ for all x, y ∈ P and (cid:107) x (cid:107) = (cid:107) y (cid:107) = 1. To be more rigorous, T is not a contraction under the supnorm, but it might be acontraction in some other complete metric. In fact, Bessaga (1959) proves a partial converse ofthe Contraction Mapping Theorem, which ensures that under certain conditions there exists acomplete metric in which T is a contraction. Also see Leader (1982); for the construction ofsuch metrics, see Janos (1967) and Williamson and Janos (1987). For an application of thistheorem in the economic literature, see Balbus et al. (2013). We wish to thank an anonymousreferee for referring us to this literature. heorem 3.2. Let u ( s ) = c (cid:48) (0) s , v ( s ) = c ( s ) , and [ u , v ] be the order intervalon C ( X ) with the usual partial order. If Assumption 2.1 and 2.2 hold, then T hasa unique fixed point p ∗ in [ u , v ] . Furthermore, T n p → p ∗ for any p ∈ [ u , v ] . This theorem ensures that there exists a unique price function in equilibriumand it can be computed by successive evaluation of the operator T on anyfunction located in that order interval . Furthermore, as is clear in the proof(see Appendix A), the existence of the minimizers t ∗ ( s ) and k ∗ ( s ) can also beproved, although they might not be single valued for some s . In the case where each firm can only have one upstream partner, the equi-librium price function is strictly increasing and strictly convex (Kikuchi et al.,2018). In this model, however, complications arise since firms at different stagesmight choose to have different numbers of upstream partners. In fact, theequilibrium price is usually piece-wise convex due to this fact. An example of the equilibrium price function is plotted in Figure 1 where c ( s ) = e s − g ( k ) = β ( k −
1) with β = 50, and δ = 10. As is shown in the plot, the pricefunction as a whole is not convex, but it is piece-wise convex with each piececorresponding to a choice of k . Monotonicity of p ∗ remains true. Proposition 3.3.
The equilibrium price function p ∗ : X → R + is strictlyincreasing. As for comparative statics, we have some basic results also present in Kikuchiet al. (2018) about the effect of changing transaction costs on the equilibriumprice function. If either transaction cost ( δ or g ) increases, the equilibrium pricefunction also increases. Proposition 3.4. If δ a ≤ δ b , then p ∗ a ≤ p ∗ b . Similarly, if g a ≤ g b , p ∗ a ≤ p ∗ b . In Figure 2, we plot how the equilibrium price function changes when trans-action cost increases. The baseline model setting is the same as Figure 1. Wecan see that if δ or β increases, the equilibrium price function also increases.
4. Computation
To compute an approximation to the equilibrium price function given δ , c ,and g , one possibility is to take a function in [ u , v ] and iterate with T . However,in practice we can only approximate the iterates, and, since T is not a contractionmapping the rate of convergence can be unsatisfactory for some model settings.On the other hand, as we now show, there is a fast, non-iterative alternativethat is guaranteed to converge.Let G = { , h, h, ..., } for fixed h . Given G , we define our approximation p to p ∗ via the recursive procedure in Algorithm 1. In the fourth line, the For the choice of the order interval we also follow Kikuchi et al. (2018). The parameterization here is merely chosen to highlight the shape of the price functionand is not economically realistic. The price is computed using a faster algorithm introduced inSection 4 with m = 5000 grid points instead of successive evaluation of T . . . . . . . p r i ce f un c t i o n p ∗ ( s ) Figure 1: An example of equilibrium price function. . . . . . . p r i ce f un c t i o n p ∗ ( s ) δ = 10 δ = 15 0 . . . . . . β = 50 β = 100 Figure 2: Equilibrium price function when c ( s ) = e s − g ( k ) = β ( k − evaluation of p ( s ) is by setting p ( s ) = min t ≤ s − hk ∈ N { c ( s − t ) + g ( k ) + δkp ( t/k ) } . (4)In line five, the linear interpolation is piecewise linear interpolation of grid points0 , h, h, . . . , s and values p (0) , p ( h ) , p (2 h ) , . . . , p ( s ).The procedure can be implemented because the minimization step on theright-hand side of (4), which is used to compute p ( s ), only evaluates p on [0 , s − h ],and the values of p on this set are determined by previous iterations of the loop.Once the value p ( s ) has been computed, the following line extends p from [0 , s − h ]to the new interval [0 , s ]. The process repeats. Once the algorithm completes,the resulting function p is defined on all of [0 ,
1] and satisfies p (0) = 0 and (4)for all s ∈ G with s > { G n } , and the corresponding functions Algorithm 1
Construction of p from G = { , h, h, ..., } p (0) ← s ← h while s ≤ do evaluate p ( s ) via equation (4)define p on [0 , s ] by linear interpolation of p (0) , p ( h ) , p (2 h ) , . . . , p ( s ) s ← s + h end while C o m pu t a t i o n t i m e ( s ) δ = 1 . δ = 1 .
01 Method 1Method 2
Figure 3: Computation time comparison for the two methods. { p n } defined by Algorithm 1. Let G n = { , h n , h n , . . . , } with h n = 2 − n . Inthis setting we have the following result, the proof of which is given in AppendixB. Theorem 4.1.
If Assumption 2.1 and 2.2 hold, then { p n } converges to p ∗ uniformly. The main advantage of this algorithm is that, for any chosen number ofgrid points, the number of minimization operations required is fixed, and wecan improve the accuracy of this algorithm by increasing the number of gridpoints. For the iteration method, however, the rate of convergence is differentfor different model settings and to achieve the same accuracy it usually requireslonger computation time.In Figure 3, we plot the computation time of successive iterations of T with p = c (method 1) and Algorithm 1 (method 2) for ten different modelsettings when the number of grid points is set to be m = 1000. The first andlast five models are the same except δ = 1 . δ = 1 .
01 forthe latter. In each model, we also compute an accurate price function usingAlgorithm 1 with a very large number of grid points ( m = 50000) and compareit with results from both methods when m = 1000. We find that the error frommethod 2 is comparable or smaller than that from method 1 in each model. Thealgorithm achieves more accurate results at a much faster speed. As we can seein Figure 3, method 2 completes the computation in around 3 seconds in eachmodel while the computation time of method 1 ranges from 7 seconds to morethan 2 minutes. The speed difference is especially drastic when δ is close to 1,since it takes T more iterations to converge with smaller δ but the number ofoperations for the algorithm is fixed. In model 1 with δ = 1 .
01, the algorithm is40 times faster than successive iterations of T ! The computations were conducted on a XPS 13 9360 laptop with i7-7500U CPU. Theprogram only utilizes a single core. The cost function c and additive transaction cost function g for the five models are:(1) c ( s ) = e s − g ( k ) = k −
1; (2) c ( s ) = e s − g ( k ) = 0 . k − c ( s ) = e s − g ( k ) = 0 . k − c ( s ) = s + s , g ( k ) = 0 . k − c ( s ) = e s + s − g ( k ) = 0 . k − . Stochastic Choices So far we have discussed the case in which each firm can choose the optimalnumber of upstream partners according to (1). In reality, however, firms usuallyface uncertainty when choosing their partners. The result is that some firmsmight choose fewer or more partners than what is optimal. For instance, afirm might not be able to choose a certain number of upstream partners due toregulation or failure to arrive at agreements with potential partners. Conversely,the upstream partners of a firm might experience supply shocks and fail to meetproduction requirements, causing it to sign more partners than what is optimaland bear more transaction costs. In this section, we model this scenario andincorporate uncertainty into each firm’s optimization problem.We assume that each firm chooses an amount of “search effort” λ and theresulting number of upstream partners follows a Poisson distribution withparameter λ that starts from k = 1. In other words, the probability of having k partners is f ( k ; λ ) = λ k − e − λ ( k − λ >
0. We also assume that when λ = 0, Prob( n = 1) = 1, that is, eachfirm can always choose to have only one upstream partner with certainty. Forexample, if a firm chooses to exert effort λ = 2 .
5, the probabilities of it endingup with 1, 2, 3, 4, 5 partners are, respectively: 0.08, 0.2, 0.26, 0.21, 0.13. Onecharacteristic of the Poisson distribution is that both its mean and varianceincrease with λ , which makes it suitable for our model since the more partners afirm aims for, the more uncertainty there will be in the contracting process.Hence, a firm at stage s solves the following problem:min t ≤ sλ ≥ (cid:8) c ( s − t ) + E λk [ g ( k ) + δkp ( t/k )] (cid:9) (5)where E λk stands for taking expectation of k under the Poisson distribution withparameter λ . Specifically, E λk [ g ( k ) + δkp ( t/k )] = ∞ (cid:88) k =1 [ g ( k ) + δkp ( t/k )] f ( k ; λ ) . Similar to Section 3, we can define another operator ˜ T : C ( X ) → R ( X ) by˜ T p ( s ) := min t ≤ sλ ≥ (cid:8) c ( s − t ) + E λk [ g ( k ) + δkp ( t/k )] (cid:9) . (6)As will be shown in Appendix C, all of the above results still apply in thestochastic case and we summarize them in the following theorem. Theorem 5.1.
Let u ( s ) = c (cid:48) (0) s , v ( s ) = c ( s ) . If Assumption 2.1 and 2.2 hold,then the operator ˜ T has a unique fixed point ˜ p ∗ in [ u , v ] and ˜ T n p → ˜ p ∗ for any p ∈ [ u , v ] . Furthermore, ˜ p n from Algorithm 1 converges to ˜ p ∗ uniformly. Note that in the usual sense, if a random variable X follows the Poisson distribution, X takes values in nonnegative integers. Here we shift the probability function so that k startsfrom 1. (a) β = 0 . δ = 1 . θ = 1 . (b) β = 0 . δ = 1 . θ = 1 . (c) β = 0 . δ = 1 . θ = 1 . (d) β = 0 . δ = 1 . θ = 1 . Figure 4: Production networks with stochastic choices of upstream partners
By Theorem 5.1, there exists a unique equilibrium price function ˜ p ∗ and wecan compute it either by successive evaluation of ˜ T or by Algorithm 1. Thealgorithm is particularly useful here since it now takes much longer time tocomplete one minimization operation with firms choosing continuous values of λ instead of discreet values of k .Similarly, there exist minimizers t ∗ and λ ∗ so that firm at any stage s hasan optimal choice t ∗ ( s ) and λ ∗ ( s ). With the optimal choice functions, we cancompute an equilibrium firm allocation recursively as in Kikuchi et al. (2018).Specifically, we start at the most downstream firm at s = 1 and compute itsoptimal choices t ∗ and λ ∗ . Next, we pick a realization of k according to thePoisson distribution with parameter λ ∗ and repeat the process for each of itsupstream firm at s (cid:48) = t ∗ /k . The whole process ends when all the most upstreamfirms choose to carry out the remaining production process by themselves. Notethat due to the stochastic nature of this model, each simulation will give adifferent firm allocation.In Figure 4, we plot some production networks for different model parame-terizations using the above approach. Each node represents a firm and the oneat the center is the firm at s = 1. The size of each node is proportionate to the11ize of the corresponding firm. The cost function is set to be c ( s ) = s θ andthe additive transaction cost is g ( k ) = β ( k − . . Compared with productionnetworks in Kikuchi et al. (2018), the graphs here are no longer symmetric sinceeven firms on the same layer can have different realized numbers of upstreampartners and thus different firm sizes. The prediction that downstream firms arelarger and tend to have more subcontractors, on the other hand, is also valid inour networks.Comparing (a) and (b), an increase in transaction cost makes firms in(b) outsource less and produce more in-house, resulting in fewer layers in theproduction network. Similarly, comparing (c) with (a), a decrease in additivetransaction costs encourages firms at each level to find more subcontractors. Theresults are more but smaller firms at each level and fewer layers in the network.Comparing (d) with (a), the difference is a decrease in curvature of the costfunction c , which makes outsourcing less appealing. The firms in (d) tend toproduce more in-house, resulting in a production network of fewer layers.
6. Conclusion
In this paper, we extend the production chain model of Kikuchi et al. (2018)to more realistic settings, in which each firm can have multiple upstream partnersand face uncertainty in the contracting process. We prove the uniqueness ofequilibrium price for these extensions and propose a fast algorithm for computingthe price function that is guaranteed to converge.The key to proving uniqueness of equilibrium price in this model is the theoryof monotone concave operators, which gives sufficient conditions for existence,uniqueness, and convergence. This theory has been proven useful in findingequilibria in a range of economic models as mentioned in the introduction andcan potentially be applied to more problems where contraction property is hardto establish.Our model also has some predictions regarding the shape of productionnetworks and the size distribution of firms. In our extended model with uncer-tainty, we generate a series of production networks (Figure 4) under differentmodel settings. A notable observation from this exercise is that increasing theproportionate transaction cost δ or decreasing the additive transaction cost g will reduce the number of layers in a network. In the former case, the cost ofmarket transactions increases; this encourages vertical integration and henceleads to larger firms along each chain. In the latter case, the cost of maintainingmultiple partners decreases; this discourages lateral integration and leads tomore firms in each layer. This prediction can potentially be tested with suitablechoice of proxies for δ and g .Another observation is that different model settings lead to different sizedistributions of firms. For example, smaller δ seems to lead to more extremedifferences in firm sizes as shown in the comparison between (a) and (b) inFigure 4. The underlying mechanism is unclear in our model, which provides apossible channel for future research. Here the firm size is calculated using its value added c ( s − t ∗ )+ g ( k ) where k is a realizationof the Poisson distribution with parameter λ ∗ .
12 notable feature of our model is that firms are ex-ante identical but ex-post heterogeneous in equilibrium in terms of sizes, positions in a network, andnumber of subcontractors. However, the cost function c and transaction costs δ and g are assumed to be fixed throughout this paper. Introducing heterogeneityinto these costs might offer richer implications for firm distribution and industrypolicies. We also leave this possibility for future research. Appendix A. Proofs from Section 3
Let U = N × [0 ,
1] equipped with the Euclidean metric in R and X beequipped with the Euclidean metric in R . To simplify notation, we can write T as T p ( s ) = min ( k,t ) ∈ Θ( s ) f p ( s, k, t )where Θ : X → U is a correspondence defined by Θ( s ) = N × [0 , s ], and f p ( s, k, t ) = c ( s − t ) + g ( k ) + δkp ( t/k ). Lemma A.1.
T p ∈ C ([0 , for all p ∈ C ([0 , .Proof. We use Berge’s theorem to prove continuity. By Assumption 2.2, we canrestrict Θ to be Θ( s ) = { , , . . . , ¯ k } × [0 , s ] for some large ¯ k ∈ N . Then Θ iscompact-valued.To see Θ is upper hemicontinuous, note Θ( s ) is closed for all s ∈ X . Sincethe graph of Θ is also closed, by the Closed Graph Theorem (see, e.g., Aliprantisand Border, 2006, p. 565), Θ is upper hemicontinuous on X .To check for lower hemicontinuity, fix s ∈ X . Let V be any open setintersecting Θ( s ) = { , , . . . , ¯ k } × [0 , s ]. Then it is easy to see that we can finda small (cid:15) > s (cid:48) ) ∩ V (cid:54) = ∅ for all s (cid:48) ∈ [ s − (cid:15), s + (cid:15) ]. Hence Θ is lowerhemicontinuous on X .Because p ∈ C ([0 , f p is jointly continuous in its three arguments. ByBerge’s theorem, T p is continuous on X .Note that by Berge’s theorem, the minimizers t ∗ and k ∗ exist and are upperhemicontinuous. Lemma A.2.
T is increasing and concave.Proof.
It is apparent that T is increasing. To see T is concave, let p, q ∈ C ([0 , α ∈ (0 , αT p ( s ) + (1 − α ) T q ( s ) = min ( k,t ) ∈ Θ( s ) αf p ( s, k, t ) + min ( k,t ) ∈ Θ( s ) (1 − α ) f q ( s, k, t ) ≤ min ( k,t ) ∈ Θ( s ) { αf p ( s, k, t ) + (1 − α ) f q ( s, k, t ) } = min ( k,t ) ∈ Θ( s ) { c ( s − t ) + g ( k ) + δk [ αp ( t/k ) + (1 − α ) q ( t/k )] } = min ( k,t ) ∈ Θ( s ) f αp +(1 − α ) q ( s, k, t )= T [ αp + (1 − α ) q ] ( s )which completes the proof. Lemma A.3.
T u ≥ u + (cid:15) ( v − u ) for some (cid:15) ∈ (0 , . roof. Define ¯ s := max { ≤ s ≤ c (cid:48) ( s ) ≤ δc (cid:48) (0) } . Then we have T u ( s ) = min ( k,t ) ∈ Θ( s ) f u ( s, k, t )= min ( k,t ) ∈ Θ( s ) { c ( s − t ) + g ( k ) + δc (cid:48) (0) t } = min t ≤ s { c ( s − t ) + δc (cid:48) (0) t } = (cid:40) c (¯ s ) + δc (cid:48) (0)( s − ¯ s ) , if s ≥ ¯ sc ( s ) , if s < ¯ s Since
T u ( s ) > u ( s ) for all s except at 0, we can find (cid:15) ∈ (0 ,
1) such that
T u ≥ u + (cid:15) ( v − u ). Lemma A.4.
T v ≤ v .Proof. Choose k = 1 and t = 0. We have T v ( s ) ≤ c ( s −
0) + g (1) + δc (0) = c ( s ) = v ( s ). Proof of Theorem 3.2.
Since P = { f ∈ C ( X ) : f ( x ) ≥ x ∈ X } is a nor-mal cone, the theorem follows from the previous lemmas and Theorem 3.1. Proof of Proposition 3.3.
We first show that T maps a strictly increasing func-tion to a strictly increasing function. Suppose p ∈ [ u , v ] and is strictly increas-ing. Pick any s , s ∈ [0 ,
1] with s < s . Let t ∗ and k ∗ be the minimizers of T .To simplify notation, let t ∈ t ∗ ( s ), t ∈ t ∗ ( s ), k ∈ k ∗ ( s ), and k ∈ k ∗ ( s ).If t ≤ s , then we have T p ( s ) = c ( s − t ) + g ( k ) + δk p ( t /k ) > c ( s − t ) + g ( k ) + δk p ( t /k ) ≥ T p ( s ) . If s < t ≤ s , then t + s − s ≤ s . Since p is strictly increasing, we have T p ( s ) = c ( s − ( t + s − s )) + g ( k ) + δk p ( t /k ) > c ( s − ( t + s − s )) + g ( k ) + δk p (( t + s − s ) /k ) ≥ T p ( s ) . Since c ∈ [ u , v ], by Theorem 3.2, T n c → p ∗ as n → ∞ . Furthermore,since c is strictly increasing, it follows from the above result that p ∗ is strictlyincreasing. Proof of Proposition 3.4. If δ a ≤ δ b , then T a p ≤ T b p for any p ∈ [ u , v ]. Since T is increasing by Lemma A.2, we have T na p ≤ T nb p for any p ∈ [ u , v ] andany n ∈ N . Then by Theorem 3.2, p ∗ a ≤ p ∗ b . The same arguments applies if g a ≤ g b . Appendix B. Proof of Theorem 4.1Lemma B.1.
The function p n is increasing for every n . roof. As p n is piecewise linear, we shall prove it by induction. Since p n (0) = 0and p n ( h n ) = c ( h n ), p n is increasing on [0 , h n ]. Suppose it is increasing on [0 , s ]for some s = h n , h n , . . . , − h n , then we have p n ( s + h n ) = min t ≤ s, k ∈ N { c ( s + h n − t ) + g ( k ) + δkp n ( t/k ) } = c ( s + h n − t ∗ ) + g ( k ∗ ) + δk ∗ p n ( t ∗ /k ∗ )where t ∗ and k ∗ are the minimizers. If t ∗ ≤ s − h n , it follows from the monotonicityof c that p n ( s + h n ) ≥ c ( s − t ∗ ) + g ( k ∗ ) + δk ∗ p n ( t ∗ /k ∗ ) ≥ min t ≤ s − h n , k ∈ N { c ( s − t ) + g ( k ) + δkp n ( t/k ) } = p n ( s ) . If t ∗ ∈ ( s − h n , s ], then s + h n − t ∗ ≥ h n . Because p n is increasing on [0 , s ], wehave p n ( s + h n ) ≥ c [ s − ( s − h n )] + g ( k ∗ ) + δk ∗ p n [( s − h n ) /k ∗ ] ≥ min t ≤ s − h n , k ∈ N { c ( s − t ) + g ( k ) + δkp n ( t/k ) } = p n ( s ) , which completes the proof. Lemma B.2.
The sequence { p n } ∞ n =1 is uniformly bounded and equicontinuous.Proof. To see { p n } is uniformly bounded, note that for each n , p n ( s + h n ) = min t ≤ s, k ∈ N { c ( s + h n − t ) + g ( k ) + δkp n ( t/k ) }≤ c ( s + h n ) + g (1) + δp n (0)= c ( s + h n ) ≤ c (1)for all s = 0 , h n , . . . , − h n .Due to Lemma B.1, to see { p n } is equicontinuous, it suffices to show thatthere exists K > p n ( s + h n ) − p n ( s ) ≤ Kh n for all n ∈ N and all s = 0 , h n , h n , . . . , − h n . Fix such n and s . If s = 0, p n ( h n ) − p n (0) = c ( h n ) ≤ c (cid:48) (1) h n . If s ≥ h n , denote the minimizers in the definition of p n ( s ) by t ∗ and k ∗ ,i.e., p n ( s ) = min t ≤ s − h n , k ∈ N { c ( s − t ) + g ( k ) + δkp n ( t/k ) } = c ( s − t ∗ ) + g ( k ∗ ) + δk ∗ p n ( t ∗ /k ∗ ) . Since t ∗ ≤ s , it follows that p n ( s + h n ) = min t ≤ s, k ∈ N { c ( s + h n − t ) + g ( k ) + δkp n ( t/k ) }≤ c ( s + h n − t ∗ ) + g ( k ∗ ) + δk ∗ p n ( t ∗ /k ∗ ) . Hence, p n ( s + h n ) − p n ( s ) ≤ c ( s + h n − t ∗ ) + c ( s − t ∗ ) ≤ c (cid:48) (1) h n , which completes the proof. 15 emma B.3. There exists a uniformly convergent subsequence of { p n } . Fur-thermore, every uniformly convergent subsequence of { p n } converges to a fixedpoint of T .Proof. Lemma B.2 and the Arzel`a-Ascoli theorem imply that p n has a uniformlyconvergent subsequence. To simplify notation, let { p n } be such a subsequenceand converge uniformly to ¯ p . Because p n are continuous, ¯ p is continuous. ByBerge’s theorem, T ¯ p ( s ) = min t ≤ s, k ∈ N { c ( s − t ) + g ( k ) + δk ¯ p ( t/k ) } is also continuous. To see ¯ p is a fixed point of T , it is sufficient to show that ¯ p and T ¯ p agree on the dyadic rationals ∪ n G n , i.e.,lim n →∞ min t ≤ s − h n k ∈ N { c ( s − t ) + g ( k ) + δkp n ( t/k ) } = min t ≤ s, k ∈ N { c ( s − t ) + g ( k ) + δk ¯ p ( t/k ) } for every s ∈ ∪ n G n .Fix (cid:15) >
0. Since p n → ¯ p uniformly, there exists N ∈ N such that n > N implies that p n ( x ) > ¯ p ( x ) − (cid:15)/ ( δ ¯ k )for all x ∈ [0 ,
1] where ¯ k is the upper bound on the possible values of k . It followsthat for n > N we havemin t ≤ s − h n k ∈ N { c ( s − t ) + g ( k ) + δkp n ( t/k ) } > min t ≤ s − h n k ∈ N { c ( s − t ) + g ( k ) + δk ¯ p ( t/k ) } − (cid:15) ≥ min t ≤ sk ∈ N { c ( s − t ) + g ( k ) + δk ¯ p ( t/k ) } − (cid:15). Therefore,lim n →∞ min t ≤ s − h n k ∈ N { c ( s − t ) + g ( k ) + δkp n ( t/k ) } ≥ min t ≤ s, k ∈ N { c ( s − t ) + g ( k ) + δk ¯ p ( t/k ) } . For the other direction, there exists N ∈ N such that n > N implies that p n ( x ) < ¯ p ( x ) + (cid:15)/ (2 δ ¯ k )for all x ∈ [0 , n > N we havemin t ≤ s − h n k ∈ N { c ( s − t ) + g ( k ) + δkp n ( t/k ) } < min t ≤ s − h n k ∈ N { c ( s − t ) + g ( k ) + δk ¯ p ( t/k ) } + (cid:15)/ . Since c, g, ¯ p are continuous and h n →
0, we can choose N such that n > N implies thatmin t ≤ s − h n k ∈ N { c ( s − t ) + g ( k ) + δk ¯ p ( t/k ) } < min t ≤ sk ∈ N { c ( s − t ) + g ( k ) + δk ¯ p ( t/k ) } + (cid:15)/ . Hence, for n > max { N , N } we havemin t ≤ s − h n k ∈ N { c ( s − t ) + g ( k ) + δkp n ( t/k ) } < min t ≤ sk ∈ N { c ( s − t ) + g ( k ) + δk ¯ p ( t/k ) } + (cid:15). n →∞ min t ≤ s − h n k ∈ N { c ( s − t ) + g ( k ) + δkp n ( t/k ) } ≤ min t ≤ s, k ∈ N { c ( s − t ) + g ( k ) + δk ¯ p ( t/k ) } . Therefore, ¯ p = T ¯ p . Lemma B.4.
Every uniformly convergent subsequence of { p n } converges to p ∗ .Proof. Let { p n } be the subsequence that converges uniformly to ¯ p . By Theorem3.2, to see ¯ p = p ∗ , it suffices to show that ¯ p is continuous and c (cid:48) (0) x ≤ ¯ p ( x ) ≤ c ( x )for all x ∈ [0 , p n is continuousand p n → ¯ p uniformly. To show the second one, we again prove this holds on ∪ n G n , and it is sufficient to show that c (cid:48) (0) s ≤ p n ( s ) ≤ c ( s ) for all s ∈ G n andall n ∈ N . It is apparent that p n ( s ) ≤ c ( s ) (choose t = 0 and k = 1). We show p n ( s ) ≥ c (cid:48) (0) s by induction. Suppose p n ( x ) ≥ c (cid:48) (0) x for all x ≤ s . Then we have p n ( s + h n ) = min t ≤ s, k ∈ N { c ( s + h n − t ) + g ( k ) + δkp n ( t/k ) }≥ min t ≤ s, k ∈ N { c (cid:48) (0)( s + h n − t ) + g ( k ) + δc (cid:48) (0) t } = min t ≤ s { c (cid:48) (0)( s + h n − t + δt ) } = c (cid:48) (0)( s + h n ) . Since p n (0) = 0 ≥ c (cid:48) (0) ·
0, it follows that p n ( s ) ≥ c (cid:48) (0) s . This concludes theproof. Appendix C. Proof of Theorem 5.1
Similar to Appendix A, we can write the operator ˜ T in (6) as˜ T p ( s ) = min ( λ,t ) ∈ ˜Θ( s ) (cid:8) c ( s − t ) + E λk [ g ( k ) + δkp ( t/k )] (cid:9) where ˜Θ( s ) = [0 , ∞ ) × [0 , s ]. Upon close inspection, all of the above lemmasstill hold for ˜ T if we can restrict ˜Θ( s ) to be a compact set. To be more specific,Lemma A.2 and B.1 can be proved in the exact same way; Lemma A.3, A.4, B.2,and B.4 hold since each firm can choose k = 1 with probability 1; Lemma B.3and A.1 need the compactness of ˜Θ( s ). To avoid redundancy, we omit the proofsand shall only show that there exists an upper bound on the choice set of λ .Let ν be the median of the Poisson distribution and denote the ceiling of ν (i.e., the least integer greater than or equal to ν ) by ¯ ν . Then we have ∞ (cid:88) k =¯ ν f ( k ; λ ) ≥ g ( k ) E λk g ( k ) = ∞ (cid:88) k =1 g ( k ) f ( k ; λ ) ≥ ∞ (cid:88) k =¯ ν g ( k ) f ( k ; λ ) ≥ g (¯ ν ) ∞ (cid:88) k =¯ ν f ( k ; λ ) ≥ g (¯ ν )where the second inequality follows from Assumption 2.2. Choi (1994) givesbounds for the median of the Poisson distribution: λ − ln 2 ≤ ν − < λ + 13 . So we have E λk g ( k ) ≥ g (¯ ν ) ≥ g ( ν ) ≥ g ( λ − ln 2 + 1) . Therefore, we can find ¯ λ such that E λk g ( k ) ≥ c (1) for all λ ≥ ¯ λ and hence Θ( s )is essentially [0 , ¯ λ ] × [0 , s ] which is a compact set. References
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