### 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

ii

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

1

## 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 (x_{1}, . . . , x_{n}), 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 Ω_{i} 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

2bq^{2}−1

2γe^{2}. (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 z_{j}, using a linear cost technology c_{j}(z_{j}) = cz_{j}. 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
k_{i}(q_{i}) = (w + d)q_{i}, assuming that one unit of output q_{i} requires one unit of input z_{j}, 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(z_{1}+ ... + z_{m}) − c)z_{j}. (2.5)
and the downstream firm i solves:

maxqi

(p(q_{1}+ ... + q_{n}) − w − d)q_{i}. (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) + mπ_{j}^{u}+ nπ_{i}^{d}. (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 mπ_{j}^{u} and nπ_{i}^{d} respectively.

Proposition 1:

The total welfare W is maximised at the production level:

q^{∗}= a − c − d

b + γδ^{2} . (2.8)

The associated consumer price would be:

p^{∗} = a − ba − c − d

b + γδ^{2} . (2.9)

Proof :

Working out equation (2.7) further:

W = (B − pq + aq − 1

2bq^{2}−1

2γδ^{2}q^{2}) + (pq − (w + d)q + (w − c)q) =

= B + aq −1

2bq^{2}−1

2γδ^{2}q^{2}− (c + d)q.

(2.10)

Working out The first-order conditions (F.O.C.) for a maximum is:

∂W

∂q = a − bq − γδ^{2}q − c − d = 0. (2.11)
The optimal total quantity q* and p* follow as:

q^{∗}= a − c − d

b + γδ^{2} . (2.12)

p^{∗} = a − ba − c − d

b + γδ^{2} . (2.13)

Checking The second-order conditions for a maximum (S.O.C.) gives:

∂^{2}W

∂q^{2} = −(b + γδ^{2}) < 0, (2.14)

meaning q^{∗} is indeed the maximum. Q.E.D.

When there is no externality, i.e. γδ^{2} = 0, the production and price are:

q^{0∗}= a − c − d

b . (2.15)

p^{0∗}= c + d. (2.16)

CHAPTER 2. THE MODEL 11

Figure 2.3: W as a function of γδ^{2} (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 γδ^{2} 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:

π^{d}_{i} = q_{i}(a − bq − w − d) = q_{i}(a − bq_{i}− 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:

∂π^{d}_{i}

∂q_{i} = 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.:

∂^{2}π_{i}^{d}

∂q_{i}^{2} = −2b < 0, (2.26)

meaning q_{i} is indeed the maximum of the profit function π_{i}^{d}. 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:

z_{j} = 1
m + 1

n n + 1

a − c − d

b ⇔ (2.27)

Proof:

Upstream firm j takes on w(q) and solves:

π^{u}_{j} = (a − d − n + 1

n bq)zj− cz_{j}. (2.28)

Given q =Pn

i=1qi =Pm

j=1zj = z, equation (2.28) is re-written as:

π_{j}^{u} = (a − d −n + 1

n bz)z_{j} − cz_{j} = (a − d − z_{j}b(1 + 1

n) − z−jb(1 + 1

n))z_{j}− cz_{j}. (2.29)
where z_{j} is the individual production decision of firm j.

The F.O.C. are worked out as:

∂π_{j}^{u}

∂zj

= a − d − 2z_{j}b − z−jb − 2z_{j}b

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:

∂^{2}π^{u}_{j}

∂z_{j}^{2} = −2b(1 + 1

n) < 0, (2.32)

meaning zj is indeed the maximum of the profit function π^{u}_{j}. Q.E.D.

Equilibria in price and quantity

Given the production technology q =Pn

i=1q_{i} =Pm

j=1z_{j} = 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 + γδ^{2})

n

n + 1 m m + 1

a − c − d b

2

=

= B +

b − 1

2(b + γδ^{2}) 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 − ^{1}_{2}(b + γδ^{2})_{n+1}^{n} _{m+1}^{m}

∗

∗ _{n+1}^{n} _{m+1}^{m} ^{a−c−d}_{b} 2

is non-negative. To ensure production indeed brings good the the market,
the γδ^{2} < 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 + γδ^{2}) 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 + γδ^{2}) 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 + γδ^{2}. (2.38)

For δγ^{2} = 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, γδ^{2} = 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 - γδ^{2} = 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

mγδ^{2}− b. (2.40)

Figure 2.6: Optimal n as a function of γδ^{2} (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 γδ^{2}
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 + γδ^{2}) m
m + 1

m

m + 1

a − c − d b

2

−

b − 1

2(b + γδ^{2}) 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 γδ^{2} ≤ 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+1}^{n} is
replaced by _{m+1}^{m} , 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 + γδ^{2}) a − c − d
b

2

−

b − 1

2(b + γδ^{2}) 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 γδ^{2} ≤ 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:

W_{ban}> W (n, 1) or W (1, m) ⇔ (2.47)

0 > (a − c − d) m m + 1

a − c − d 2b −1

2(b + γδ^{2})

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
z_{j}, i.e. a quota on z_{j}. The quota value would then have to fall between (0;_{m+1}^{m} _{n+1}^{n} ^{a−c−d}_{b} ) 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

2bl^{2}−1

2γδ^{2}l^{2}− (c + d)l. (2.49)
where the optimal value of l* would be equal to the optimal production q^{∗} = ^{a−c−d}_{b+γδ}2 = l^{∗}. q^{∗}= l^{∗}
holds when:

a − c − d

b + γδ^{2} < m
m + 1

n n + 1

a − c − d

b . (2.50)

which is the case when:

b + γδ^{2} > 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 γδ^{2}< 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 + γδ^{2} > ^{m+1}_{m} ^{n+1}_{n} 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−d}_{b+γδ}2 − _{m+1}^{m} ^{a−c−d}_{2b} (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−d}_{b+γδ}2 − _{m+1}^{m} ^{a−c−d}_{2b} 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−d}_{b+γδ}2 −_{n+1}^{n} ^{a−c−d}_{2b} . 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−d}_{b+γδ}2 − ^{a−c−d}_{4b} . A simulation gives
4l > 0.015

where in all cases the values of γ, δ, and b must meet the conditions γδ^{2} < 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+1}^{m} _{n+1}^{n} ^{a−c−d−ρ}_{b} , 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 + γδ^{2} ⇔ (2.52)

ρ^{∗}= a − c − d − bn + 1
n

m + 1 m

a − c − d

b + γδ^{2} . (2.53)

Figure 2.8: Tax as a function of γδ^{2} (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. γδ^{2}= 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+1}^{m} _{n+1}^{n} ^{a−c−d}_{b} < ^{a−c−d}_{b+γδ}2 which accounts for 5% of the cases according to the simulation in
the quota subsection.

2. _{m+1}^{m} _{n+1}^{n} ^{a−c−d}_{b} ≥ ^{a−c−d}_{b+γδ}_{2}, 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 + γδ^{2} − m
m + 1

n n + 1

1 b

(2.54)

or

2(a − c − d)

1 −m + 1 m

n + 1 n

b
b + γδ^{2}

< 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 γδ^{2} < 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 + γδ^{2}

(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}_{∂q}^{m}

i . There
is Cournot competition in the market when ^{∂q}_{∂q}^{m}

i = 0 for each company i in a certain layer
(_{∂q}^{∂q}

i = 1 when looking at the total production q), collusion when ^{∂q}_{∂q}^{m}

i = 1 (_{∂q}^{∂q}

i = n) and perfect
competition when ^{∂q}_{∂q}^{m}

i = −_{n−1}^{1} (_{∂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 + γδ^{2})

b(m + 1 + (m − 1)θu)(n − 1)− n − 1 (2.58) Proof :

The F.O.C. for a downstream firm:

∂π_{i}^{d}

∂q_{i} = a − 2bqi− w − d − bq_{−i}− bq_{i}(n − 1)∂qm

∂q_{i} ⇔ (2.59)

q_{i}= a − w − d
b(n + 1 + (n − 1)^{∂q}_{∂q}^{m}

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}_{∂q}^{m}

i = θd is the conjectural variation for downstream firms.

For the upstream firms:

∂π^{u}_{j}

∂z_{j} = a − d − c − 2z_{j}b(n + 1 + (n − 1)θ_{d})

n −

−z−j

b(n + 1 + (n − 1)θ_{d})

n − z_{j}(m − 1)∂z_{k}

∂zj

b(n + 1 + (n − 1)θ_{d})

n ⇔

(2.62)

zj = n(a − c − d)

b(n + 1 + (n − 1)θ_{d})(1 + m + (m − 1)θ_{u}) (2.63)
where ^{∂z}_{∂z}^{k}

j = θu is the conjectural variation for upstream firms.

CHAPTER 2. THE MODEL 23 The total production then becomes:

z^{cv} = q^{cv}= mn(a − c − d)

b(n + 1 + (n − 1)θ_{d})(1 + m + (m − 1)θ_{u}) (2.64)
It could happen that for certain values of θ_{d} and θu, the production from Cournot compe-
tition is optimal, namely it is the case when:

q^{cv} = q^{∗} = m

m + 1 + (m − 1)θu

n

n + 1 + (n − 1)θd

a − c − d

b = a − c − d

b + γδ^{2} ⇔ (2.65)
Solving for θ_{d}:

θ_{d}= mn(b + γδ^{2})

b(m + 1 + (m − 1)θ_{u})(n − 1)− n − 1 (2.66)

Figure 2.10: θ_{d} as a function of θ_{u} (a=5, b=20, c=3, d=1, n=10, m=11): blue - γδ^{2}=15; green
- γδ^{2}=5

Then the effects of externality would be corrected. Q.E.D.

Figure 2.10 shows that conjectural variation θ_{d}is negatively correlated with θu on the either
side of the vertical asymptote. Equally, the higher the externality γδ^{2}, 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:

π^{vc}_{i} = pqi− (c + d)q_{i} = (a − bqi− bq−i− c − d)q_{i} (2.67)
From where the F.O.C.:

∂π_{i}^{vc}

∂qi

= a − 2bqi− bq−i− c − d = 0 ⇔ (2.68)

q_{i}= a − c − d

b(n + 1) (2.69)

which is indeed the maximum, given the second-order condition:

∂^{2}π^{vc}_{i}

∂q_{i}^{2} = −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:

π_{i}^{vc}= (a − c − d)^{2}
b(n + 1)

1 −n(a − c − d) n + 1

(2.71) And the profits of unmerged firms are:

π^{ud}_{i} = (a − c − d)^{2}
b

n
(n + 1)^{2}

n

n + 1 + 1 − 2 n^{2}

(n + 1)^{2} − n
(n + 1)^{2}

, (2.72)

which is derived from π^{u}_{i} + π^{d}_{i} 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:

π_{i}^{vc}> π^{ud}_{i} (2.73)

or

1 −n(a − c − d) n + 1 > n

n + 1

1 + n

n + 1 − n

(n + 1)^{2} − 2 n^{2}
(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 + γδ^{2}) 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 γδ^{2}< 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 γδ^{2} < 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 γδ^{2} < 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: γδ^{2} < 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: γδ^{2} < 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