Second-best defence: Justification for mergers and cartels?
George Soroko
Student number: 11360313
Date of final version: August 15, 2021 Master’s programme: Econometrics
Specialisation: Complexity and Economic Behaviour Supervisor: Prof. dr. J. Tuinstra
Second reader: Prof. dr. M. P. Schinkel
Faculty of Economics and Business
i
Statement of originality
This document is written by George Soroko who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Contents
Abstract . . . 1
1 Introduction and literature review 2 2 The Model 7 2.1 Rationale . . . 7
2.2 Set-up . . . 8
2.2.1 Consumption side . . . 8
2.2.2 Production side . . . 8
2.2.3 Externality . . . 9
2.3 Analysis . . . 9
2.3.1 Welfare-maximising allocation . . . 9
2.3.2 Market equilibrium with Cournot competition . . . 11
2.3.3 Welfare with Cournot competition and collusion . . . 13
2.3.4 Cartels and other externality-mitigating policies . . . 17
2.3.5 Model extensions . . . 22
3 Results 26 3.1 Welfare-increasing cartels . . . 26
3.2 Cartelisation as the safest policy to mitigate an externality . . . 26
3.3 Conjectural variations can correct for an externality . . . 26
3.4 Vertical integration may and may not be welfare-enhancing . . . 27
4 Conclusion 28
Bibliography 28
A Programs 30
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Abstract
This research presents the defence for the second-best theory by studying the effects of mergers and cartels on the total economic welfare given the presence of an externality. The study also compares cartel-formation and mergers to other externality-mitigating policies, and shows that cartelisation is the most efficient.
Keywords: economics, microeconomics, welfare economics, mathematical economics, cartel, merger, monopoly, externality, first-best, second-best, conjectural variation, vertical inte- gration
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Introduction and literature review
Ever since the beginning of modern Economics with Adam Smith have most mainstream economists seen promotion of competition as a primary concern for policy-makers. Once left to the “invisible hand” power of the market competition, society is able to reach the highest
“common good”, or what Arthur Pigou would later call economic welfare. It is by unhindered competition that markets can reach their equilibria [Pareto-optimal allocation] – the situation when no consumer nor producer could be made better-off without making another worse-off – that state of highest possible “common good”. The way policy-makers have maintained competition is through restricting the power of monopolies and cartels. The company mergers have also traditionally been seen as a threat to free markets, as they inevitably entailed increased market power, which was what made them another target of competition policy.
The current paper introduces an exception to the rule that perfect competition is welfare- enhancing at all times, by giving an example of a market with an externality that produces higher ”common good” when cartels and monopolies are given free rein. It challenges traditional view that anti-trust policy always acts in the overall benefit of society, and gives reasons for its reviewal.
The assumption that perfect competition is always good has largely been unchallenged by mainstream (neo)classical economists, with critique mostly coming from the side of alternative schools of Economics, e.g. Marxist or Austrian. The first modern attack from inside the orthodox school came in 1956 with the paper “The General Theory of Second Best” by R.G.
Lipsey and K. Lancaster. What was their argument?
The Walrasian equilibrium is a well-known economic corollary: the whole economy (combin- ing all individual markets) is at equilibrium if and only if all optimum (first-best) conditions are fulfilled. Should one market fail to be at equilibrium, the highest possible welfare could still be reached if equilibria in all other markets are maintained, it was assumed. Lipsey and Lancaster disagreed, and proposed that in case first-best market conditions are not fulfillable, the new optimum situation could only be achieved by departing from all other Paretian conditions, i.e.
bringing all other markets away from equilibrium.
Lipsey and Lancaster found traces of the second-best theory elsewhere in other economic
2
CHAPTER 1. INTRODUCTION AND LITERATURE REVIEW 3 applications. For instance, in the field of customs unions and trade barriers. The default assumption that any reduction of tariffs would lead to a welfare improvement had already been challenged by the idea that the same reduction of tariffs could disrupt trade between the union area and the rest of the world. The consequence of which was the decrease, not the increase of total welfare. Another example is with mixed economies. In cases when one section of the economy was rigidly controlled by the state, and the other was not, there had been the debate as to whether it was better to increase control over the uncontrolled or decrease it over the controlled sections of the economy. It was found that there was no clear answer. Either tighter or looser control could lead to a rise in welfare, each depending on the circumstances. These and other examples let Lipsey and Lancaster argue that trying to apply the same welfare rules for a small part of the economy, as if they were applied everywhere, had the potential to move the economy away from and not towards the second-best optimum position. Further, it is not the necessary conditions of the welfare maximum that are of interest but the sufficient conditions for the welfare improvement. The problem with the second-best theory is that no such sufficient conditions can be found. In the general equilibrium model there are no conditions which are sufficient for an increase in welfare without also being sufficient for a welfare maximum.
Lipsey and Lancaster concluded by setting up their own second-best model and showing its validity, namely by finding the optimum of the function F (x1, . . . , xn), subject to a constraint on the variables φ(x1, . . . , xn) = 0. The solution is in the form Ωi(x1, . . . , xn) = 0, for i = 1, . . . , n − 1, which stood for the Paretian optimum. In case an extra constraint of the type Ωi 6= 0 is imposed for i = j, the optimum of F subject to both φ and Ωi 6= 0 would be such that none of the still available Paretian conditions Ωi= 0, i 6= j is satisfied.
Figure 1.1: Ng´s model
As the theory of second-best risked undermining welfare economics, 50 years after Lipsey
and Lancaster’s publication there came up a debate. On the one side was R. Lipsey himself and on the other Y.K. Ng, trying to salvage welfare economics. In his paper “Towards a Theory of Third-Best” published in 1977, Ng suggested that in a state of “Informational poverty” (the situation in which policy-makers do not have enough information to tell them in which direction they should move the variables that they control in the market) and in case neither first- nor second-best optima could be achieved sticking to first-best conditions is the best strategy (or what he called the third-best strategy). One should only attempt to move from the first-best conditions when there is enough information on the second-best constraint(s). Ng developed a model (Figure 1.1) where the objective function took the shape of a concave curve and was placed on a Cartesian plane with the degree of divergence from the first-best rule on the x-axis (in his case ad valorem tax) and the value of welfare on the y-axis. The function’s maximising x co-ordinate thus coincided with the 0-value on the x-axis. Next, he introduced the second- best constraint that shifted the function horizontally either left- or rightwards. In case of
“Informational poverty” (when the direction of shift is not known), Ng argued, the expected value of deviating from the 0-value on the x-axis is always less than doing nothing. Lipsey in his 2017 study “Generality versus context specificity: first, second, and third best in theory and policy” pointed out that it was not the first-best condition that Ng defended but rather the status quo wherever it was. In other words, the assumption that markets find themselves at 0 on the x-axis does not always hold. Next, Lipsey challenged another assumption of Ng about the concavity of the objective function. The perfectly concave function, he claimed, is only in place for a good that has no relation to any other goods in the market, i.e. in an isolated market for a single good. Instead, it is more realistic that the objective function levels out from both sides at some point, in case the effects of other goods on the good in question are being considered.
In this new setting, keeping the status quo or going back to 0 on x-axis has a negative effect on the eventual welfare. Lipsey concludes that attempting to reach the second-best optimum will have the higher expected welfare than either staying put or going to 0.
In defence of the third-best strategy, Ng in his 2017 paper “Theory of third best: how to interpret and apply” retorted by saying that the 0 value on x-axis still produces higher welfare than staying on the status quo wherever it may be. He based it on the analysis of the expected values of welfare when the direction of change (left- or rightwards) was not known.
The lesson of the second-best analysis is that one market failure can compensate another market failure, as first-best conditions no longer have to be maintained in all the markets in case one market fails. If this is so, could it be that the primary goal of policy-makers – maintaining competition – may in some cases be not the best strategy? P. Hammer in his 2000 paper “Antitrust beyond Competition: Market Failures, Total Welfare, and the Challenge of Intramarket Second-Best Tradeoffs” looked for cases when mergers and cartels may be welfare- enhancing, not depleting. He focussed on the intramarket second-best trade-offs, i.e. trade-offs involving multiple market failures in a single economic market, rather than on intermarket, where multiple market failures occur in distinct economic markets (Hammer, 2000). In other
CHAPTER 1. INTRODUCTION AND LITERATURE REVIEW 5 words, the focus is on looking at a single well-defined market in isolation, holding conditions elsewhere constant.
Hammer considered a few examples. First, he looked at the market for negative externali- ties. In a competitive market, price equals marginal costs. With the negative externality, price falls below the marginal costs (taking the costs for the externality into account), and the output is higher than optimal. Assuming two companies polluting the air merge, they get the power to raise prices. The price increase entails a fall in demand, which itself leads to less negative externality on the market. Hammer argues that in some cases the welfare society gets from cleaner air may exceed the costs that the merger produces. An example for that is cigarette smoking. The externality of smoking are the healthcare costs that now need to be covered by the National Health Service, and the costs from passive smoking. In the market where price equals marginal costs, there is overproduction and overconsumption of cigarettes compared to the market where the externality is internalised. Therefore, Hammer proposes that competi- tion policy ought to consider allowing cartel formation in cases where under the second-best conditions a higher net welfare is achieved.
But how can one model the harm or good that cartelisation creates? M. Han, M. P. Schinkel and J. Tuinstra in their paper “The Overcharge as a Measure for Antitrust Damages” built up a multilayer chain of production model (Figure 1.2), and let the firms in one of the layers form a cartel. Through modelling the upstream and downstream change in profits, prices, consumer surplus and the deadweight loss they concluded that the usual way of measuring damage from anticompetitive behaviour – direct purchaser overcharge – underestimates the total antitrust harm.
Figure 1.2: A longer vertical chain of production
This thesis draws on Han, Schinkel, and Tuinstra’s paper and builds its own multi(two)- layered production chain, with the externality released in the upstream (first) layer. Consumers
are replaced by a single representative consumer with one utility function. Next, the study looks at how collusion (merging of firms) in either (or both) of the layers affects the total market welfare.
The rest of the paper is organised as follows. Section 2 describes the model: Subsection 2.1 discusses the rationale behind the model, Subsection 2.2 sets it up, Subsection 2.3 analyses the set-up model and introduces a number of extensions for better generalisation. Section 3 sets out the results and Section 4 concludes. The study is finished by a Bibliography and an Appendix containing the listings of the code used throughout the paper.
Chapter 2
The Model
2.1 Rationale
As the title of this study suggests, the paper attempts to give validation to the theory of second-best, i.e. that in situations when each first-order condition is not fulfillable, it is better for the overall welfare to deviate from all first-order conditions, including the conditions that can be fulfilled (illustrated in Figure 2.1).
Figure 2.1: Second-best theory in brief
A clear application of this theory in Economics would be the presence of a market imper- fection in the form of an externality - a cost or benefit of a decision to one or multiple economic actors that did not make that decision. It is also often case that market competition is imperfect - in other words, there is a limited number of firms that do not take market prices as given, but instead attempt to maximise profits by basing their own prices or production quantities on the prices or production quantities set by their competitors in a given market (Bertand and Cournot modes of competition). In this situation is it better to leave the oligopolistic competi- tion intact, or would it be better for all market actors if companies merged or formed cartels?
Proving cartels or monopolies to be more welfare-enhancing in the presence of an externality than the status quo would give ample defence for the theory of the second-best, and it is this
7
defence that this paper attempts to present.
The chosen model consists of the two-layered production chain and the final consumer. The two layers are chosen over one for the sake of better model generalisation, and to be able to study the effects of a cartel in a layer different from a layer in which the externality is created. The number of layers is restricted to two to keep calculations and analysis succinct. The production of the externality is placed in the upstream layer.
2.2 Set-up
2.2.1 Consumption side
On the demand side of the market is the representative ”final” consumer whose utility function can be used to measure consumer welfare. The utility function is defined as:
U (x, q, e) = x + aq − 1
2bq2−1
2γe2. (2.1)
where q is the consumer product, x is a composite good representing all other commodities, and e is the level of pollution. a, b, γ are positive parameters. Further, the representative consumer is in possession of the budget B, which is assumed to be large enough to avoid the boundary solution with x = 0. The price for the composite good is fixed at 1, and price for the good q is equal to p.
Taken all together, the representative consumer attempts to maximise utility subject to the budget constraint, or in other words:
maxx,q U (x, q, e), s.t. x + pq ≤ B.
(2.2)
From the consumer problem above, the inverse demand function for good q can be derived through a Lagrangian method as:
p(q) = a − bq. (2.3)
Demand for the composite good follows from the budget equation and can then be expressed as:
x = B − (a − bq)q. (2.4)
2.2.2 Production side
The production process is assumed to be two-layered. There are m upstream firms producing the main input zj, using a linear cost technology cj(zj) = czj. These m firms sell inputs zj to n downstream firms at the price w. The downstream firms have the cost technology
CHAPTER 2. THE MODEL 9 ki(qi) = (w + d)qi, assuming that one unit of output qi requires one unit of input zj, or q = Pn
i=1qi=Pm
j=1zj. Further, firms in both layers are assumed to follow Cournot competition.
Put simply, the upstream firm j solves:
maxzj
(w(z1+ ... + zm) − c)zj. (2.5) and the downstream firm i solves:
maxqi
(p(q1+ ... + qn) − w − d)qi. (2.6) Figure 2.2 below sums up the set-up production-consumption chain.
Figure 2.2: Production chain
2.2.3 Externality
The production of input produced, z, leads to pollution, i.e. e = δPm
j=1zj, or in other words, each unit of inputs leads to the production of δ units of the externality (Belleflamme &
Peitz, 2015).
2.3 Analysis
2.3.1 Welfare-maximising allocation
In this subsection the allocation that leads to highest possible welfare is determined.
Let W be the total societal welfare. The welfare is seen as the sum of consumer and producer surpluses, which are the buyer’s utility and the total profits respectively. The welfare is then expressed as:
W = U (x, q, e) + mπju+ nπid. (2.7)
since all firms in each layer have identical production technologies, they are expected to turn identical profits. From there follows that the total of profits in the upstream and downstream layers is mπju and nπid respectively.
Proposition 1:
The total welfare W is maximised at the production level:
q∗= a − c − d
b + γδ2 . (2.8)
The associated consumer price would be:
p∗ = a − ba − c − d
b + γδ2 . (2.9)
Proof :
Working out equation (2.7) further:
W = (B − pq + aq − 1
2bq2−1
2γδ2q2) + (pq − (w + d)q + (w − c)q) =
= B + aq −1
2bq2−1
2γδ2q2− (c + d)q.
(2.10)
Working out The first-order conditions (F.O.C.) for a maximum is:
∂W
∂q = a − bq − γδ2q − c − d = 0. (2.11) The optimal total quantity q* and p* follow as:
q∗= a − c − d
b + γδ2 . (2.12)
p∗ = a − ba − c − d
b + γδ2 . (2.13)
Checking The second-order conditions for a maximum (S.O.C.) gives:
∂2W
∂q2 = −(b + γδ2) < 0, (2.14)
meaning q∗ is indeed the maximum. Q.E.D.
When there is no externality, i.e. γδ2 = 0, the production and price are:
q0∗= a − c − d
b . (2.15)
p0∗= c + d. (2.16)
CHAPTER 2. THE MODEL 11
Figure 2.3: W as a function of γδ2 (B=1, a=10, b=20, c=3, d=1)
From Figure 2.3 one could see the total welfare W at q* is a decreasing function of the externality γδ2 with a lower bound at W = 1.
2.3.2 Market equilibrium with Cournot competition
When a market follows Cournot competition, each company attempts to establish its best response to the actions of all its competitors (the Nash equilibrium), with individual quantities as strategic variables (Barr & Saraceno, 2005). Mathematically, it can be solved through backward induction beginning with solving for downstream firms´ optimal production q, finding the input price w, and then solving for optimal z in the upstream layer (Tuinstra, 2021).
Using the inverse demand function (2.3), downstream firm i’s quantity decision q is given in the following proposition:
Downstream firm Proposition 2:
The individual and total downstream productions respectively are:
qi= a − w − d
b(n + 1) ⇔ (2.17)
q(w) = n(a − w − d)
b(n + 1) , (2.18)
respectively, and the optimal input price is:
w(q) = a − d − n + 1
n bq. (2.19)
Proof :
The profits of a downstream firm i are:
πdi = qi(a − bq − w − d) = qi(a − bqi− bq−i− w − d). (2.20) where q−i is the aggregate quantity produced by all downstream firms except i.
Working out the F.O.C., and assuming symmetry (q1= ... = qn) yields:
∂πdi
∂qi = a − 2bqi− bq−i− w − d = 0 ⇒ (2.21)
qi(b + bn) = a − w − d. (2.22) from which the individual production qifollows. Multiplied by the number of downstream firms n, the total production is established:
qi= a − w − d
b(n + 1) ⇔ (2.23)
q(w) = n(a − w − d)
b(n + 1) . (2.24)
Expressing w in terms of the total production q gives the input price value w :
w(q) = a − d − n + 1
n bq. (2.25)
Checking the S.O.C.:
∂2πid
∂qi2 = −2b < 0, (2.26)
meaning qi is indeed the maximum of the profit function πid. Q.E.D.
Next, using the new-expressed input price value w in terms of q, the upstream firm’s j’s quantity decision z is given in the following proposition:
Upstream firm Proposition 3:
The profit-maximising individual upstream production is:
zj = 1 m + 1
n n + 1
a − c − d
b ⇔ (2.27)
Proof:
Upstream firm j takes on w(q) and solves:
πuj = (a − d − n + 1
n bq)zj− czj. (2.28)
Given q =Pn
i=1qi =Pm
j=1zj = z, equation (2.28) is re-written as:
πju = (a − d −n + 1
n bz)zj − czj = (a − d − zjb(1 + 1
n) − z−jb(1 + 1
n))zj− czj. (2.29) where zj is the individual production decision of firm j.
The F.O.C. are worked out as:
∂πju
∂zj
= a − d − 2zjb − z−jb − 2zjb
n −z−jb
n − c = 0, (2.30)
from which we find (assuming symmetry):
CHAPTER 2. THE MODEL 13
zj = 1 m + 1
n n + 1
a − c − d
b ⇔ (2.31)
Checking the S.O.C. for the maximum:
∂2πuj
∂zj2 = −2b(1 + 1
n) < 0, (2.32)
meaning zj is indeed the maximum of the profit function πuj. Q.E.D.
Equilibria in price and quantity
Given the production technology q =Pn
i=1qi =Pm
j=1zj = z, Propositions 1 and 2 show that the total quantity q = z is equal to:
q = z = m m + 1
n n + 1
a − c − d
b . (2.33)
from which the consumer price p follows as:
p(q) = (m + n + 1)a + nm(c + d)
(n + 1)(m + 1) . (2.34)
2.3.3 Welfare with Cournot competition and collusion
The total welfare with the quantities from a competitive Cournot equilibrium is:
W = B + (a − c − d) n n + 1
m m + 1
a − c − d
b −1
2(b + γδ2)
n
n + 1 m m + 1
a − c − d b
2
=
= B +
b − 1
2(b + γδ2) n n + 1
m m + 1
n
n + 1 m m + 1
a − c − d b
2
.
(2.35)
In cases when W < B, it is more efficient for the market to have no production at all. How- ever, production is always beneficial for the total welfare, if the term
b − 12(b + γδ2)n+1n m+1m
∗
∗ n+1n m+1m a−c−db 2
is non-negative. To ensure production indeed brings good the the market, the γδ2 < b condition is imposed for all further computations.
No externality: is collusion beneficial?
First, the general assumption that cartel formation leads to a decrease in welfare is confirmed for the model this study uses.
Proposition 4:
Greater competition always leads to greater total welfare in the absence of externality.
Proof :
Looking at the expression:
G(n, m) = (b −1
2(b + γδ2) m m + 1
n n + 1) n
n + 1 m
m + 1. (2.36)
the first-order derivatives ∂m∂G and ∂G∂n are identical to each other. Looking at the derivative with respect to m:
∂G
∂m =
b − (b + γδ2) n n + 1
m m + 1
n
n + 1 1
(m + 1)2. (2.37)
where the maximum is at:
n n + 1
m
m + 1 = b
b + γδ2. (2.38)
For δγ2 = 0:
n n + 1
m
m + 1 = 1. (2.39)
Figure 2.4: The F.O.C. of welfare as a function of n (m=9): red - with no externality; green - b=20, γδ2 = 5
meaning that more competition does indeed lead to more welfare (which could be seen on Figure 2.4 as a greater n leads the function closer to the value of 1), as stipulated by economic theory.
Q.E.D.
Figure 2.5 below plots welfare W as a function of n, showing that a higher competition (higher n) is always beneficial for the welfare when there is no externality, and lower competition (n ≈ 1.4 ≈ 1) is optimal when externality is in the model.
CHAPTER 2. THE MODEL 15
Figure 2.5: Welfare as a function of n (m=9, b=10): purple - no externality, green - γδ2 = 9 from which it follows that reducing competition can be welfare-enhancing in situations when there exists an externality. As mergers are one way of reducing competition, they can indeed be beneficial for the total welfare.
Next, this welfare-maximising number of downstream firms n is derived as:
n = (m + 1)b
mγδ2− b. (2.40)
Figure 2.6: Optimal n as a function of γδ2 (b=10): blue - m=3; black - m=9; blue - m=20 Figure 2.6 above shows that a lower number of firms in one opposite layer (m) leads to a greater optimal number of firms in another layer. Equally, a higher value of externality γδ2 leads to a reduction in optimal level of competition, as can be seen from the downward-sloping curves.
Does downstream collusion increase welfare?
When downstream collusion increases welfare, the following expression must hold:
W (1, m) − W (n, m) > 0 (2.41) where n = 1 corresponds to the situation where there is a single cartel in the downstream layer.
Expression 2.41 can be re-written as:
1 2
b − 1
4(b + γδ2) m m + 1
m
m + 1
a − c − d b
2
−
b − 1
2(b + γδ2) n n + 1
m m + 1
n
n + 1 m m + 1
a − c − d b
2
> 0.
(2.42)
which for certain values of b, n, m, γ, δ could indeed hold. Performing a simulation (the code is in Listing A.3 in Appendix A) 20,000 times on the values:
1. b, n, m ∈ [1, 20] and γ, δ ∈ [1, 5]: in 73% cases collusion is welfare-enhancing;
2. n, m ∈ [2, 5], b ∈ [1, 20], and γ, δ ∈ [1, 5]: 68.5%;
3. b, γ, δ ∈ [1, 20] and n, m ∈ [2, 20]: 98%.
where for each case the values of b, γ, and δ are conditioned by γδ2 ≤ b to avoid negative values of total welfare. The results show that downstream collusion is welfare-enhancing in more than a half of the situations (combinations of parameters).
Does upstream collusion increase welfare?
When upstream collusion increases welfare, the following expression must always hold:
W (n, 1) − W (n, m) > 0 (2.43)
which is identical to W(1, m) - W(n, m) situation (downstream collusion) except that n+1n is replaced by m+1m , meaning that simulating on the parameters would produce identical likeli- hoods, and so make upstream collusion welfare-enhancing in most situations (combinations of parameters).
Do collusions in upstream and downstream layers increase welfare?
When collusions in both layers increase welfare, the following expression must always hold:
W (1, 1) − W (n, m) > 0 (2.44)
or
1 4
b − 1
8(b + γδ2) a − c − d b
2
−
b − 1
2(b + γδ2) n n + 1
m m + 1
n
n + 1 m m + 1
a − c − d b
2
> 0.
(2.45)
CHAPTER 2. THE MODEL 17 which, likewise, could indeed hold for some values of the parameters. Performing another simulation (the code is in Listing A.4 in Appendix A) 20,000 times on the values:
1. b, n, m ∈ [1, 20] and γ, δ ∈ [1, 5]: in 99.7% cases is W(1, 1) welfare-enhancing;
2. n, m ∈ [2, 5] and b, γ, δ ∈ [1, 20]: 99.8%;
3. b, γ, δ ∈ [1, 20] and n, m ∈ [2, 20]: 99.96%.
where for each case the values of b, γ, and δ are conditioned by γδ2 ≤ b to avoid negative values of total welfare. The result shows that collusions in both layers are welfare-enhancing in almost all cases.
2.3.4 Cartels and other externality-mitigating policies
As mentioned in theory section, there are other policies that could potentially mitigate the externality’s effect on welfare, namely:
• Ban on the upstream production
• Quota on the upstream production
• Pigouvian tax/ subsidy on the upstream production
In this section each policy is analysed and compared to cartel formation.
I. Ban
In case of a ban, there is no more upstream production. No upstream firms producing inputs means no production in the downstream layer. The total welfare thus looks like:
W = x = B. (2.46)
where B is the consumer budget.
Ban is a better solution to the effects of externality than cartel formation in one of the layers iff:
Wban> W (n, 1) or W (1, m) ⇔ (2.47)
0 > (a − c − d) m m + 1
a − c − d 2b −1
2(b + γδ2)
m
m + 1
a − c − d 2b
2
. (2.48)
or, in other words, when the welfare from good associated with externality is negative.
II. Quota
The policy-makers can also choose to implement a maximum allowed production of the input zj, i.e. a quota on zj. The quota value would then have to fall between (0;m+1m n+1n a−c−db ) for it to be relevant.
If l is the value of the total quota (individual quota times m), the total welfare then becomes:
W = B + al − 1
2bl2−1
2γδ2l2− (c + d)l. (2.49) where the optimal value of l* would be equal to the optimal production q∗ = a−c−db+γδ2 = l∗. q∗= l∗ holds when:
a − c − d
b + γδ2 < m m + 1
n n + 1
a − c − d
b . (2.50)
which is the case when:
b + γδ2 > m + 1 m
n + 1
n b. (2.51)
otherwise quota l would have no effect on welfare (as it can only reduce the current production, and never increase), compared to the Cournot equilibrium, since W(l) is a concave parabola.
Figure 2.7 below shows why quota is only relevant if the competitive equilibrium is higher than the optimum.
Figure 2.7: Optimal quota
Since quotas are only capable to reduce production and never increase it, quotas could only be welfare-increasing when the current competitive equilibrium is greater (to the right side on the plot) of the optimal production.
CHAPTER 2. THE MODEL 19 To check how frequently the competitive equilibrium is greater than the optimum, a simu- lation (the code is in Listing A.5 in Appendix A) is performed 20,000 times on the parameter values:
1. General case: b ∈ [1, 20], γ, δ ∈ [1, 5], m, n ∈ [2, 20]: in 89.3% is quota workable.
2. Upstream collusion case: b ∈ [1, 20], γ, δ ∈ [1, 5], m ∈ [2, 20], and n = 1: 65.1%.
3. Downstream collusion case: b ∈ [1, 20], γ, δ ∈ [1, 5], n ∈ [2, 20], and m = 1: 65.1%.
4. Downstream and upstream collusions case: b ∈ [1, 20], γ, δ ∈ [1, 5], n = 1, and m = 1:
42.8%.
where in all cases the values of γ, δ, and b are conditioned by γδ2< b.
A significant condition on fixing l∗ is that policy-makers are aware of the consumer demand function, cost technologies and utility of consumers, i.e. have perfect information, which is often not the case. Too low quota could lead to a decrease in total welfare, i.e. government failure.
Assuming b + γδ2 > m+1m n+1n b, too low a quota leads to a decrease in total welfare, when (the code could be found in Listing A.6 in Appendix A):
1. Downstream collusion case: 4l > a−c−db+γδ2 − m+1m a−c−d2b (for values of l that lie outside the arch formed by the competitive equilibrium and its mirror image on the parabola in Figure 2.7). Taking the average from 20,000 simulations of a−c−db+γδ2 − m+1m a−c−d2b for the values of a, b, c, d ∈ [1, 20], γ, δ ∈ [1, 5], and m ∈ [2, 20] gives the average 4l > 0.017.
In other words, should policy-makers impose the quota 0.017 units greater or less than the optimum, the policy would lead to a welfare loss.
2. Upstream collusion case: 4l > a−c−db+γδ2 −n+1n a−c−d2b . Simulation akin to the one for down- stream leads to the same results of 4l > 0.017.
3. Upstream and downstream collusions case: 4l > a−c−db+γδ2 − a−c−d4b . A simulation gives 4l > 0.015
where in all cases the values of γ, δ, and b must meet the conditions γδ2 < b and a − c − d ≥ 0.
The likelihood of a government failure with the error margins estimated above is high indeed.
III. Pigouvian tax/ subsidy
The introduction of tax on the production of inputs changes the total production q into q = m+1m n+1n a−c−d−ρb , where ρ is the amount of tax.
The optimal value ρ∗ is found through:
m m + 1
n n + 1
a − c − d − ρ
b = a − c − d
b + γδ2 ⇔ (2.52)
ρ∗= a − c − d − bn + 1 n
m + 1 m
a − c − d
b + γδ2 . (2.53)
Figure 2.8: Tax as a function of γδ2 (a=5, b=20, c=3, d=1): red - n=10, m=11; blue - n=14, m=21
As can be seen from Figure 2.8, the tax size is positively correlated with the externality value. The decision to subsidise or tax is also dependent on the externality (e.g. γδ2= 4 is the break point between subsidy and tax for the red line case)
Similar to quotas, policy-makers must have perfect information about demand function, production technologies, and consumer utilities to avoid reducing the welfare, which is often not the case.
There are two cases:
1. m+1m n+1n a−c−db < a−c−db+γδ2 which accounts for 5% of the cases according to the simulation in the quota subsection.
2. m+1m n+1n a−c−db ≥ a−c−db+γδ2, which accounts for the rest 95% of the cases.
CHAPTER 2. THE MODEL 21
Figure 2.9: Values of q for which tax is optimal
For tax to lead to welfare-improvement, the updated (with tax added) total quantity q should lie between the current production and its mirror image on the parabola-shaped welfare function W (illustrated on Figure 2.9). When the current production is greater then the optimum, the following situation is considered:
I. m m + 1
n n + 1
a − c − d
b < m m + 1
n n + 1
a − c − d − ρ b <
< m m + 1
n n + 1
a − c − d
b + 2
a − c − d)( 1
b + γδ2 − m m + 1
n n + 1
1 b
(2.54)
or
2(a − c − d)
1 −m + 1 m
n + 1 n
b b + γδ2
< ρ < 0 (2.55) Simulating the left-hand side 20,000 times for values a, b, c, d ∈ [1, 20], m, n ∈ [2, 20], γ, δ ∈ [1, 5] with γδ2 < b and a − c − d > 0 (the code is in Listing A.7 in Appendix A) give the average value of -5.5. Meaning that if the tax ∈ [-5.5, 0), then welfare would go up.
When the current production is less than the optimum, the follwoing is solved:
II. m m + 1
n n + 1
a − c − d
b − 2(a − c − d) <
< m m + 1
n n + 1
a − c − d − ρ
b < m
m + 1 n n + 1
a − c − d b
(2.56)
or
0 < ρ < 2(a − c − d)
1 −m + 1 m
n + 1 n
b b + γδ2
(2.57)
where another simulation specifies (0, 5.5] as the average domain for the values of tax that increases welfare.
2.3.5 Model extensions
Conjectural variations
In certain situations, downstream and upstream firms may have expectations on how their competitors would react, should they vary their output - the so-called conjectural variations.
Mathematically, it is expressed as the rate of change of company m’s production in response to a change in the company i ’s production, located in the same production layer, or ∂q∂qm
i . There is Cournot competition in the market when ∂q∂qm
i = 0 for each company i in a certain layer (∂q∂q
i = 1 when looking at the total production q), collusion when ∂q∂qm
i = 1 (∂q∂q
i = n) and perfect competition when ∂q∂qm
i = −n−11 (∂q∂q
i = 0).
If these expectations exist, the first-order conditions (F.O.C.) for both downstream and upstream firms change.
Proposition 5:
The externality-correcting downstream conjectural variation is:
θd= mn(b + γδ2)
b(m + 1 + (m − 1)θu)(n − 1)− n − 1 (2.58) Proof :
The F.O.C. for a downstream firm:
∂πid
∂qi = a − 2bqi− w − d − bq−i− bqi(n − 1)∂qm
∂qi ⇔ (2.59)
qi= a − w − d b(n + 1 + (n − 1)∂q∂qm
i ) = a − w − d
b(n + 1 + (n − 1)θd) (2.60) and
w(q, θd) = a − d −bq(n + 1 + (n − 1)θd)
n (2.61)
where ∂q∂qm
i = θd is the conjectural variation for downstream firms.
For the upstream firms:
∂πuj
∂zj = a − d − c − 2zjb(n + 1 + (n − 1)θd)
n −
−z−j
b(n + 1 + (n − 1)θd)
n − zj(m − 1)∂zk
∂zj
b(n + 1 + (n − 1)θd)
n ⇔
(2.62)
zj = n(a − c − d)
b(n + 1 + (n − 1)θd)(1 + m + (m − 1)θu) (2.63) where ∂z∂zk
j = θu is the conjectural variation for upstream firms.
CHAPTER 2. THE MODEL 23 The total production then becomes:
zcv = qcv= mn(a − c − d)
b(n + 1 + (n − 1)θd)(1 + m + (m − 1)θu) (2.64) It could happen that for certain values of θd and θu, the production from Cournot compe- tition is optimal, namely it is the case when:
qcv = q∗ = m
m + 1 + (m − 1)θu
n
n + 1 + (n − 1)θd
a − c − d
b = a − c − d
b + γδ2 ⇔ (2.65) Solving for θd:
θd= mn(b + γδ2)
b(m + 1 + (m − 1)θu)(n − 1)− n − 1 (2.66)
Figure 2.10: θd as a function of θu (a=5, b=20, c=3, d=1, n=10, m=11): blue - γδ2=15; green - γδ2=5
Then the effects of externality would be corrected. Q.E.D.
Figure 2.10 shows that conjectural variation θdis negatively correlated with θu on the either side of the vertical asymptote. Equally, the higher the externality γδ2, the higher the absolute value of θd.
Vertical cartelisation
It could also happen that downstream and upstream firms merge with each other, forming pairwise cartels (as described in Figure 2.11) (Mas-Colell et al., 1995). These companies have an incentive to merge when their joint profits are greater than the sum of individual profits.
Figure 2.11: Production chain with vertical cartels The merger’s profit functions is:
πvci = pqi− (c + d)qi = (a − bqi− bq−i− c − d)qi (2.67) From where the F.O.C.:
∂πivc
∂qi
= a − 2bqi− bq−i− c − d = 0 ⇔ (2.68)
qi= a − c − d
b(n + 1) (2.69)
which is indeed the maximum, given the second-order condition:
∂2πvci
∂qi2 = −2b < 0 (2.70)
This quantity holds iff each company in one layer decides to merge with a company from another layer.
The profit in a vertically merged firm becomes:
πivc= (a − c − d)2 b(n + 1)
1 −n(a − c − d) n + 1
(2.71) And the profits of unmerged firms are:
πudi = (a − c − d)2 b
n (n + 1)2
n
n + 1 + 1 − 2 n2
(n + 1)2 − n (n + 1)2
, (2.72)
which is derived from πui + πdi with zj = (n+1)n 2a−c−d b = qi.
CHAPTER 2. THE MODEL 25 It is important to note that the number of merging firms in upstream and downstream layers are fixed at n, as if m > n in one of the layers, the m - n excess companies would not survive the competition and quit production.
And so, if:
πivc> πudi (2.73)
or
1 −n(a − c − d) n + 1 > n
n + 1
1 + n
n + 1 − n
(n + 1)2 − 2 n2 (n + 1)2
(2.74) then vertical mergers are beneficial and would take place.
Performing a simulation 20,000 times on the parameters a, c, d ∈ [1, 20], and n ∈ [2, 20]
(the code is in Listing A.8 in Appendix A), yields that vertical cartels are beneficial in 85% of the cases.
If all companies in both layers merge (and others leave production), the supply chain becomes one-layered.
The new welfare is then equal to:
W = B +
b −1
2(b + γδ2) n n + 1
n
n + 1
a − c − d b
2
(2.75) This collusion is welfare enhancing when
W (n) > W (n, m) (2.76)
Performing another simulation 20,000 times on the parameters a, b, c, d ∈ [1, 20], γ, δ ∈ [1, 5], and m, n ∈ [2, 20] with γδ2< b, a − c − d < 0, and m ≥ n (the code is in Listing A.9 in Appendix A), yields that vertical cartels are beneficial in 8% of the cases.
Next, it is checked if a single producer (monopoly) is preferable to competition between merged firms in one layer. It is the case when:
W (1) > W (n) (2.77)
A simulation performed 20000 times on the parameters a, b, c, d ∈ [1, 20], γ, δ ∈ [1, 5], and m, n ∈ [2, 20] with γδ2 < b, a − c − d < 0, and m ≥ n (the code is in Listing A.10 in Appendix A), yields that a single monopoly is preferable to a one-layered competition in 85.5%
of the cases.
Finally, a monopoly is compared to the Cournot competition in two layers (the default situation). Having a single producer on the market is beneficial when:
W (1) > W (n, m) (2.78)
Another simulation, performed 20000 times on the parameters a, b, c, d ∈ [1, 20], γ, δ ∈ [1, 5], and m, n ∈ [2, 20] with γδ2 < b, a − c − d < 0, and m ≥ n (the code is in Listing A.11 in Appendix A) shows that a monopoly is preferable in 88.14% of all cases.
Results
Based on the model analysis in Section 2.3, the following results are found.
3.1 Welfare-increasing cartels
Downstream, upstream, and cartels in two layers combined are shown to be beneficial for welfare in most of the cases after simulating for various combinations of the model parameters.
There is found no difference between allowing cartels in either upstream or downstream layers of production. That means that given two layers of production, locating externality formation upstream has no effect on where cartels can be welfare-enhancing.
3.2 Cartelisation as the safest policy to mitigate an externality
Allowing cartels is also found to be the safest policy to mitigate an externality, given a high likelihood of government failure in cases of ban, tax, and quota. Imposing a tax or quota requires a significant deal of knowledge on the part of the policy-makers in predicting the parameters of the model. The margin within which a tax or quota is welfare-enhancing is found to be relatively slim for the given model. A ban is found to be beneficial only when the welfare generated by consumption and production of the good in question is net negative.
Since allowing cartels does not require government involvement but rather is a laissez-faire solution, where markets balance themselves, it may be seen as a safer policy under the circum- stances described in the model. Furthermore, it is important to note that whether cartel is welfare-enhancing or not tightly depends upon model parameters.
3.3 Conjectural variations can correct for an externality
There is found a value of the downstream conjectural variation (expectation of the produc- tion change of another firm, given one’s own change of production), that fully corrects for the effects of externality. It means that given the possibility of cartel-formation, the welfare can
26
CHAPTER 3. RESULTS 27 be increased further should the downstream companies have expectations close to the found optimal value (2.66).
3.4 Vertical integration may and may not be welfare-enhancing
The effects of vertical integration are ambiguous. On the one hand, the vertical mergers are found to be overwhelmingly welfare-decreasing after simulating for different combinations of parameters. On the other hand, letting the vertically-merged firms merge further into a monopoly is found to have a positive effect on welfare in most simulated cases.
Conclusion
As it was found that allowing mergers and cartels can lead to a welfare-improvement given the presence of an externality, Lipsey, Lancaster, and Hammer’s claim that it is more beneficial to deviate from the first-best conditions, when each of them cannot be met, is validated by this thesis’s model. This conclusion presents the defence for the second-best theory and, therefore, questions the inviolability of the anti-trust policy.
Nevertheless, it is worth reiterating that the results of this study are limited to the set- up model. Further research on different models and extensions is necessary to establish more general applicability of the second-best theory.
First and foremost, the assumption on the production technology presented here (one-to-one technology) may be too unrealistic for the real-world production processes.
Another assumption used by this paper that is worth re-visiting in the future is the two- layered supply-chain. Likewise, the supply chain presented in this thesis may be too short and simple in comparison to most cases in reality.
Studies on different variations of the production technology and supply chains can, therefore, be valuable for a more comprehensive understanding of cartel’s effects on society, and, with it, the theory of second-best.
28
Bibliography
[1] Barr, J., Saraceno, F. (2005) Cournot competition, organization and learning. Journal of Economic Dynamics and Control, Volume 29, Issues 1–2, January 2005, Pages 277-295 [2] Belleflamme, P., Peitz, M. (2015) Industrial Organization: Markets and Strategies (second
edition). Cambridge University Press, ISBN 978-1-107-68789-9
[3] Hammer, P. J. (2000) Antitrust Beyond Competition: Market Failures, Total Welfare, and the Challenge of Intra-Market Second-Best Tradeoffs. SSRN Electronic Journal. Published.
https://doi.org/10.2139/ssrn.229958
[4] Han, M. A., Schinkel, M. P., & Tuinstra, J. (2009) The Overcharge as a Measure for Antitrust Damages. SSRN Electronic Journal. Published. https://doi.org/10.2139/ssrn.1387096 [5] Lipsey, R. G., & Lancaster, K. (1956) The General Theory of Second Best. The Review of
Economic Studies, 24(1), 11. https://doi.org/10.2307/2296233
[6] Lipsey, R. G. (2017) Generality Versus Context Specificity: First, Second and Third Best in Theory and Policy. Pacific Economic Review, 22(2), 167–177. https://doi.org/10.1111/1468- 0106.12220
[7] Mas-Colell, A., Whinston, M.D. and Green, J.R. (1995) Microeconomic Theory. Oxford University Press, ISBN 9780195102680
[8] Ng, Y. K. (2017) Theory of Third Best: How to Interpret and Apply. Pacific Economic Review, 22(2), 178–188. https://doi.org/10.1111/1468-0106.12221
[9] Smith, A. (1776) The Wealth of Nations. BookRix
[10] Tuinstra, J. (2021) Vertical Integration and Vertical Relations. Universiteit van Amsterdam
29
Programs
This appendix consists of the code sections used for simulations. All listed code sections are written in the R computer language.
# s e t s c o n d i t i o n s on gam , d e l , and b
gam d e l b c h e c k e r <− function (gam , del , b) { while ( gam ∗ d el ∗∗2 > b) {
gam = sample ( 1 : 5 , 1 ) d e l = sample ( 1 : 5 , 1 ) b = sample ( 1 : 2 0 , 1 ) }
}
Listing A.1: γδ2 < b condition checker
# s e t s c o n d i t i o n s on gam , d e l , a , b , c , d
a b c d gam d e l c h e c k e r <− function ( a , b , c , d , gam , d el ) { while ( gam ∗ d el ∗∗2 > b && a − c − d < 0) {
a = sample ( 1 : 2 0 , 1 ) b = sample ( 1 : 2 0 , 1 ) c = sample ( 1 : 2 0 , 1 ) d = sample ( 1 : 2 0 , 1 ) gam = sample ( 1 : 5 , 1 )
d e l = sample ( 1 : 5 , 1 ) }
}
Listing A.2: γδ2 < b and a − c − d < 0 conditions checker
# i s downstream c o l l u s i o n b e n e f i c i a l ?
down c o l count ben <− function (n , m, b , gam , d el ) { b ∗ (n/ (n + 1) − 1/ 2) + 1/ 2 ∗
30
APPENDIX A. PROGRAMS 31 ( b + gam ∗ d el ∗∗2) ∗ m/ (m + 1) ∗
( 1 / 4 − n∗∗2/ (n + 1)∗∗2) }
# c o u n t t h e number o f b e n e f i c i a l c o l l u s i o n s sum neg down = 0
f o r ( i i n 1 : 2 0 0 0 0 ) { n = sample ( 2 : 2 0 , 1 ) m = sample ( 2 : 2 0 , 1 ) b = sample ( 1 : 2 0 , 1 ) gam = sample ( 1 : 2 0 , 1 )
d e l = sample ( 1 : 2 0 , 1 )
gam d e l b c h e c k e r ( gam , d e l , b )
i f ( down c o l count ben ( n , m, b , gam , d e l ) < 0 ) { sum neg down = sum neg down + 1
} }
# f r a c t i o n o f b e n e f i c i a l downstream c o l l u s i o n s print (sum neg down/ 20000 ∗ 100)
Listing A.3: Downstream collusion checker
# i s downstream and u p s tr e a m c o l l u s i o n b e n e f i c i a l ? d u c o l count ben <− function (n , m, b , gam , d el ) {
( b − 1/ 2 ∗ (b + gam ∗ d el ∗∗2) ∗
n/ ( n + 1 ) ∗ m/ (m + 1 )) ∗ n/ (n + 1) ∗
m/ (m + 1 ) − 1/ 4 ∗ (b − 1/ 8 ∗ (b + gam ∗ d el ∗∗ 2 )) }
# c o u n t t h e number o f b e n e f i c i a l c o l l u s i o n s sum neg d u = 0
f o r ( i i n 1 : 2 0 0 0 0 ) { n = sample ( 2 : 2 0 , 1 ) m = sample ( 2 : 2 0 , 1 ) b = sample ( 1 : 2 0 , 1 ) gam = sample ( 1 : 2 0 , 1 )
d e l = sample ( 1 : 2 0 , 1 )
gam d e l b c h e c k e r ( gam , d e l , b )
i f ( d u c o l count ben ( n , m, b , gam , d e l ) ) { sum neg d u = sum neg d u + 1
} }
# f r a c t i o n o f b e n e f i c i a l u p s t re a m and downstream c o l l u s i o n s print (sum neg d u/ 20000 ∗ 100)
Listing A.4: Upstream and downstream collusions checker
# i s q u o t a b e n e f i c i a l ?
quota ben <− function (n , m, b , gam , d el ) {
b + gam∗ de l ∗∗2 − (m + 1)/ m ∗ (n + 1)/ n ∗ b }
# c o u n t t h e number o f s i t u a t i o n s w i t h b e n e f i c i a l q u o t a s quota ben sum = 0
f o r ( i i n 1 : 2 0 0 0 0 ) { n = 1
m = 1
b = sample ( 1 : 2 0 , 1 ) gam = sample ( 1 : 5 , 1 )
d e l = sample ( 1 : 5 , 1 )
gam d e l b c h e c k e r ( gam , d e l , b )
i f ( quota ben ( 1 , 1 , b , gam , d e l ) > 0 ) quota ben sum = quota ben sum + 1 }
# f r a c t i o n o f s i t u a t i o n s w i t h b e n e f i c i a l q u o t a s print ( quota ben sum/ 20000 ∗ 100)
Listing A.5: Beneficial quotas checker
# d e l t a q u o t a
av quota <− function ( a , b , c , d , gam , del , m, n) { ( a − c − d)/ (b + de l ∗ gamˆ2) − m/ (m + 1) ∗ n/ ( n + 1 ) ∗ ( a − c − d)/ b
}
# sum o f q u o t a s sum l = 0
f o r ( i i n 1 : 2 0 0 0 0 ) { a = sample ( 1 : 2 0 , 1 ) b = sample ( 1 : 2 0 , 1 )
APPENDIX A. PROGRAMS 33 c = sample ( 1 : 2 0 , 1 )
d = sample ( 1 : 2 0 , 1 ) n = 1
m = 1
gam = sample ( 1 : 5 , 1 ) d e l = sample ( 1 : 5 , 1 )
a b c d gam d e l c h e c k e r ( a , b , c , d , gam , d e l )
sum l = sum l + av quota ( a , b , c , d , gam , d e l , 1 , 1 ) }
# d e l t a a v e r a g e q u o t a print (sum l / 2 0 0 0 0 )
Listing A.6: Average quota counter
# c a l c u l a t e l e f t b o u n d a r y o f t a x
t a x bound l e f t <− function ( a , b , c , d , m, n , gam , de l ) { ( a − c − d) ∗ (1 − (m + 1)/ m ∗
( n + 1 ) / n ∗ b/ (b + gam ∗ d el ˆ2)) }
# sum o f t h e v a l u e s o f l e f t b o u n d a r i e s t a x check l e f t sum = 0
f o r ( i i n 1 : 2 0 0 0 0 ) { a = sample ( 1 : 2 0 , 1 ) b = sample ( 1 : 2 0 , 1 ) c = sample ( 1 : 2 0 , 1 ) d = sample ( 1 : 2 0 , 1 ) n = sample ( 2 : 2 0 , 1 ) m = sample ( 2 : 2 0 , 1 ) gam = sample ( 1 : 5 , 1 )
d e l = sample ( 1 : 5 , 1 )
a b c d gam d e l c h e c k e r ( a , b , c , d , gam , d e l ) t a x check l e f t sum = t a x check l e f t sum + t a x bound l e f t ( a , b , c , d , m, n , gam , d e l ) }
# a v e r a g e l e f t b o u n d a ry o f t a x print ( t a x check l e f t sum/ 2 0 0 0 0 )
Listing A.7: Average left boundary of tax
# p r o f i t o f a v e r t i c a l l y merged f i r m pr vc <− function ( a , c , d , n) {
1 − n ∗ ( a − c − d)/ (n + 1) }
# sum o f p r o f i t s o f an up s t r e a m and downstream f i r m s pr tdu <− function (n) {
n/ ( n + 1 ) ∗ (1 + n/ (n + 1) − n/ (n + 1)∗∗2 − 2 ∗ n∗∗2/ (n + 1)∗∗2) }
# c o u n t t h e t i m e s when v e r t i c a l c a r t e l s a r e b e n e f i c i a l num v e r c a r good = 0
f o r ( i i n 1 : 2 0 0 0 0 ) {
i f ( pr vc ( sample ( 1 : 2 0 , 1 ) , sample ( 1 : 2 0 , 1 ) , sample ( 1 : 2 0 , 1 ) , sample ( 2 : 2 0 , 1 ) ) >
pr tdu ( sample ( 2 : 2 0 , 1 ) ) )
num v e r c a r good = num v e r c a r good + 1 }
# f r a c t i o n o f b e n e f i c i a l v e r t i c a l c a r t e l s print (num v e r c a r good/ 20000 ∗ 100)
Listing A.8: Fraction of beneficial vertical cartels
# c h e c k i f a v e r t i c a l c a r t e l i s b e n e f i c i a l
v e r v s none <− function ( a , b , c , d , m, n , gam , de l ) { ( b − 0 .5 ∗ (b + gam ∗ d el ∗∗2) ∗
n/ ( n + 1 ) ) ∗ n/ (n + 1) ∗ ( a − c − d)∗∗2/ b∗∗2 −
( b − 0 .5 ∗ (b + gam ∗ d e l ∗∗2) ∗
n/ ( n + 1 ) ∗ m/ (m + 1 )) ∗ n/ (n + 1) ∗ m/ (m + 1 ) ∗ ( a − c − d)∗∗2/ b∗∗2 > 0 }
# c o u n t t h e number o f s i t u a t i o n s w i t h b e n e f i c i a l v e r t i c a l c a r t e l s sum v e r v s none = 0
f o r ( i i n 1 : 2 0 0 0 0 ) { a = sample ( 1 : 2 0 , 1 ) b = sample ( 1 : 2 0 , 1 ) c = sample ( 1 : 2 0 , 1 ) d = sample ( 1 : 2 0 , 1 )
APPENDIX A. PROGRAMS 35 n = sample ( 2 : 2 0 , 1 )
m = sample ( 2 : 2 0 , 1 ) while ( n > m) {
n = sample ( 2 : 2 0 , 1 ) m = sample ( 2 : 2 0 , 1 ) }
gam = sample ( 1 : 5 , 1 ) d e l = sample ( 1 : 5 , 1 )
a b c d gam d e l c h e c k e r ( a , b , c , d , gam , d e l ) i f ( v e r v s none ( a , b , c , d , m, n , gam , d e l ) ) {
sum v e r v s none = sum v e r v s none + 1 }
}
# f r a c t i o n o f b e n e f i c i a l v e r t i c a l c a r t e l s print (sum v e r v s none/ 20000 ∗ 100)
Listing A.9: Vertical cartels vs Cournot competition checker
# c h e c k i f a monopoly i s b e n e f i c i a l
mon v s none <− function ( a , b , c , d , m, n , gam , de l ) {
b − 0 . 5 ∗ (b + gam ∗ d el ∗∗2) ∗ 1/ 2 ∗ 1/ 2 ∗ ( a − c − d)∗∗2/ b∗∗2 − ( b − 0 . 5 ∗ (b + gam ∗ d el ∗∗2) ∗ n/ (n + 1) ∗ m/ (m + 1 )) ∗
n/ ( n + 1 ) ∗ m/ (m + 1) ∗ ( a − c − d)∗∗2/ b∗∗2 > 0 }
# c o u n t t h e number o f s i t u a t i o n s w i t h b e n e f i c i a l m o n o p o l i e s sum mon v s none = 0
f o r ( i i n 1 : 2 0 0 0 0 ) { a = sample ( 1 : 2 0 , 1 ) b = sample ( 1 : 2 0 , 1 ) c = sample ( 1 : 2 0 , 1 ) d = sample ( 1 : 2 0 , 1 ) n = sample ( 2 : 2 0 , 1 ) m = sample ( 2 : 2 0 , 1 )
while ( n > m) {
n = sample ( 2 : 2 0 , 1 ) m = sample ( 2 : 2 0 , 1 ) }
gam = sample ( 1 : 5 , 1 ) d e l = sample ( 1 : 5 , 1 )
a b c d gam d e l c h e c k e r ( a , b , c , d , gam , d e l ) i f (mon v s none ( a , b , c , d , m, n , gam , d e l ) )
sum mon v s none = sum mon v s none + 1 }
# f r a c t i o n o f b e n e f i c i a l m o n o p o l i e s print (sum mon v s none/ 20000 ∗ 100)
Listing A.10: Monopoly vs Cournot competition checker
# c h e c k i f a monopoly c a r t e l i s b e n e f i c i a l
mon v s none <− function ( a , b , c , d , m, n , gam , de l ) { b − 0 . 5 ∗ (b + gam ∗ d el ∗∗2) ∗ 1/ 2 ∗ 1/ 2 ∗
( a − c − d)∗∗2/ b∗∗2 − (b − 0 .5 ∗ (b + gam ∗ d el ∗∗2) ∗ n/ ( n + 1 ) ∗ m/ (m + 1 )) ∗ n/ (n + 1) ∗ m/ (m + 1) ∗ ( a − c − d)∗∗2/ b∗∗2 > 0
}
# c o u n t t h e number o f s i t u a t i o n s w i t h b e n e f i c i a l m o n o p o l i e s sum mon v s none = 0
f o r ( i i n 1 : 2 0 0 0 0 ) { a = sample ( 1 : 2 0 , 1 ) b = sample ( 1 : 2 0 , 1 ) c = sample ( 1 : 2 0 , 1 ) d = sample ( 1 : 2 0 , 1 ) n = sample ( 2 : 2 0 , 1 ) m = sample ( 2 : 2 0 , 1 )
while ( n > m) {
n = sample ( 2 : 2 0 , 1 ) m = sample ( 2 : 2 0 , 1 ) }
gam = sample ( 1 : 5 , 1 ) d e l = sample ( 1 : 5 , 1 )
a b c d gam d e l c h e c k e r ( a , b , c , d , gam , d e l ) i f (mon v s none ( a , b , c , d , m, n , gam , d e l ) )
sum mon v s none = sum mon v s none + 1 }
# f r a c t i o n o f b e n e f i c i a l m o n o p o l i e s
APPENDIX A. PROGRAMS 37 print (sum mon v s none/ 20000 ∗ 100)
Listing A.11: Monopoly vs vertical cartels checker
