A model of price competition on a network : a case study of fuel excises in the European Union

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A model of price

competition on a network

A case study of fuel excises in the European Union

University of Amsterdam

Master thesis master Economics – Behavioural Economics and Game Theory Max ‘t Hart – 10384383

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A model of price competition on a network

A case study of fuel excises in the European Union

Max ‘t Hart - 10384383 University of Amsterdam

Master thesis master Economics – Behavioural Economics and Game Theory Supervisor: dr. A. Ule

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by Max ‘t Hart

A model of price

competition on a network

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Table of contents MASTERTHESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM

4

Introduction

The model

2.1 Basic model notation 2.2 The game

Applying the model to a variety of configurations

3.1 A single isolated country

3.2 Two countries with identical domestic demand 3.3 Two countries with different domestic demand 3.4 Three countries with identical domestic demand

3.5 A complete network of n countries with identical domestic demand 3.6 A star network with centre j and n periphery countries i

3.7 Overall characteristics of the model

Applying the model to excises in the European Union

4.1 The game in the European Union 4.2 Calibrating the data

4.3 Computing the relative price levels

Conclusion

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9 10

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12 13 14 15 16 17 19

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21 27 28

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM Table of contents 5

References

Appendix

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM

Introduction 6

Introduction

2014 saw a political debate in the Netherlands on the ideal rate of excise on fuel. The Dutch Ministry of Finance has increased fuel excises, which was followed by a drop in fuel sales volume. The Dutch government suspected that the loss of excise revenue due to the drop in sales volume would be offset by an increase in the revenue per litre, and from current numbers it seems that this is the case. The drop in sales volume is most visible in border areas, where Dutch fuel stations have seen their revenue (which was already provided for a major part by foreign customers) disappear. Fuel stations in the centre of the country also noticed a drop in diesel sales. From casual conversations with major trucking companies it seems that the fuel purchasing behaviour of those companies is a major driver of the drop in sales of fuel stations in the centre of the country: trucks have a range of up to 3.000 kilometres which causes a sensitivity to fuel prices in multiple countries (State Secretary of Finance, 2014).

It appears that European countries may, and do, engage in strategic tax (or excise) competition in the case of excises on fuel1_{. This strategic tax competition is essentially a}

price competition and is characterized by a trade-off between excise revenue per good sold and taxable base: a decrease of the excise decreases excise revenue per good, but this may be offset by the gain in taxable base. This observation inspired me to develop a model of this price competition as a strategic game on a network.

In this game the governments of countries are the players, who aim to maximize fuel excise revenue by effectively setting the price for fuel products. Countries are on a network and may therefore be linked to one another. Inhabitants of a country are able to purchase fuel in their own country and in countries with which their home country is “linked”. In this thesis, a link exists between country i and country j if there exists a relationship between the price in country i and the demand in country j, and vice versa. This definition is intentionally quite broad: a relationship between price of country i and demand of country j may exist if countries share common borders (are direct neighbours), if they are trade partners, or

1_{Note that an excise is not really a tax: the amount of excise due is, unlike a tax, not dependent on the price of}

the underlying good, but on the amount of good sold. For example, Value Added Tax (VAT) on fuel is a percentage of the base price per litre, while a fuel excise is a fixed amount per litre independent of base price. This

difference is not relevant for this thesis as both tax and excise affect price.

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM Introduction 7

maybe if there is a high degree of tourism between countries. For now, I simply describe the strength of the link between country i and country j with the variable xij, where xij = 0

implies no link between countries. Note that a larger xij implies a stronger link; that is, the

relationship between the price in country i and the demand in country j (and vice versa) will be stronger.

If a country is not linked to another country (e.g. it is an island state), inhabitants may only purchase fuel from a single supplier. The government of that country will then behave as a monopolist. However, if a country is linked to other countries, inhabitants of those countries are not limited to buying fuel in their home country. I assume that the fuel of foreign countries is an imperfect substitute for fuel from the home country: inhabitants will have to deviate their route and / or pay a cost to purchase fuel in another country. The governments of countries are competitors for these fuel consumers, and compete by setting the price for fuel. As such the excise revenue of a country depends on its own price and the prices set by the other countries. These interrelations in turn depend on the links between countries: a stronger link implies a stronger influence of the price of the connected country. I assume that countries differ from each other with respect to the size of their ‘domestic demand’: the maximum market demand of inhabitants of each country (I will use GDP as a proxy). I thus model the tax competition between countries as a Bertrand competition game with imperfect substitutes on a network of heterogeneous countries.

Network game theory can help explain and optimize the pricing behaviour of European countries engaged in a strategic tax competition. In this thesis I investigate the question if strategic fuel tax competition in the European Union can be modelled using network game theory. I am interested in the dependence of relative price levels on demand and links, and I show how the network structure influences this kind of strategic tax competition. These insights from network game theory expand the traditional microeconomic price competition theory.

This thesis draws from and expands on two strands of literature: on games on networks, and on strategic tax competition. Galeotti et al. (2010) describe network game theory and competition on networks in general. Authors show how several factors, such as substitutability and knowledge over players’ actions, influence payoffs. Blume et al. (2009)

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8 Introduction

research a network game where an intermediary between buyer and seller has the power to set prices. They show that the price-setter can only set prices above marginal cost if he has some essential position on the network of buyers and sellers. Evers et al. (2004) form a theoretical model of strategic tax competition in fuel excises in Europe, where goods are imperfect substitutes. They model a two-country case and find that the smaller country will have the lowest tax rate. In addition, the authors statistically test empirical data on excises and find that an increase in the excise of a neighbouring country correlates with an increase in the domestic excise, implying that countries do engage in strategic tax competition. My thesis will expand on these articles by applying network game theory on the case of a strategic tax / price competition between multiple countries. I aim to provide insights on price competition between multiple players on a fixed network.

I develop the basics of my model in section 2. In section 3 I apply my model to configurations of simple, fictitious country sets. Here I show how the network structure and differences between countries can influence relative prices. In section 4 I demonstrate the use of my model by applying it to the empirical case of fuel excises in the European Union, to test whether my model as it stands as some predictive power in this case. Individual countries are free to set fuel excises, and are thus able to strategically set the fuel prices in their country. I focus on the case of diesel prices. Diesel is used by trucks, which have far greater range than consumer cars which should resulting in greater interdependence of prices. The existence of common borders and data on trade flows between countries serve as indicators for the strength of the link between countries. I use available data on diesel prices in the European Union to show that my model has some predictive power on relative price levels. Section 5 concludes.

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM The model 9

2.1 Basic model notation

The basic notation of this model draws from Goyal (2007) and Galeotti et al. (2010). The model describes a set of countries. Countries differ in size and the links they share with other countries. There is a finite set of countries N = {1, 2, …, n} embedded in an exogenous and fixed network. Links from country i to countries j are described by xij, where xij > 0

implies that a relationship between the price of country i and the demand of country j (and vice versa) exists. There are several possibilities to determine the existence of the relationship between countries in the empirical case of fuel excises in the European Union, for example:

- Relationships only exist if countries share a common border. The variable xij could

be a dummy in this case: 1 if countries share a common border, zero otherwise; - Trade flows determine the presence of a relationship between countries. The size

of the trade flow could determine the strength of the relationship: more consumers (trucks) are able to purchase gasoline in foreign countries if trade flows are higher, increasing substitutability. The variable xij may then be continuous.

- A combination of these two possibilities: countries are only linked if they share a common border but the strength of the link is determined by the trade flows. I simply use the variable xij in the calculations of section 3. Note that xii = 0. I assume

throughout this thesis that xij = xji. This simplifies my calculations. Section 4 expands on

different possibilities for determining the relationship between countries in the empirical case of fuel excises in the European Union, and estimates different values for the variable xij.

It also shows why the assumption of xij = xji may be reasonable for these different estimates

of links.

I assume that countries have perfect information about the network, and the actions and specifics of each country. I define a specific set of countries on a specific network as a configuration.

2 The model

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10 The model

2.2 The game

The countries are producers in a (non-cooperative) Bertrand competition game played on a network, meaning that countries try to increase revenue by setting price. I assume that there are no limits to the supply of goods. Goods are imperfect substitutes. The link between countries determines the substitutability of goods: a stronger link (a higher value for xij)

implies that goods are better substitutes, increasing the sensitivity to prices of connected countries. Countries face the demand function:

Qi defines the “local demand”, the demand faced by country i. Vi defines the “domestic

demand”. This describes the maximum market demand of the citizens of i and is exogenous. Parameter b describes the sensitivity of local demand Qi to price in country i, Pi. I assume

this relation to be linear. Note that I assume that b is equal and exogenous for all countries, which enables me to normalize b to 1 for simplicity. The strength of the link between countries i and j is described by xij. The sum signifies the importance of

the network in this game: a price difference between j and i (Pj – Pi) will influence local

demand of i if and only if xij > 0. Local demand of country i will decrease if countries i and j

are connected and the price in the foreign country j is smaller than the domestic price. Note that this does not affect domestic demand Vi but does affect the local demand Qi. Vice versa,

local demand of country i will increase if countries i and j are connected and the price in the foreign country j is larger than the domestic price. Note that xij determines the strength

of the link and therefore the influence of the neighbouring country’s price on the domestic price. A higher x implies a greater substitutability of goods.

Local demand could realistically not go below zero or towards infinity. A more realistic demand function may look something like this:

In addition x should be limited to, e.g., x<b. I am focussed on relative price levels throughout this thesis; I will use the simpler demand equation (1) in further sections.

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM The model 11

The demand function could have had different specifications, such as a non-linear relation between demand and price. Also, one could aim to normalize the influence of foreign prices, for example:

I have chosen my specification of the demand function as it keeps the calculations relatively simple while keeping the element of domestic demand and influence of foreign prices. Choosing a different demand function may yield different results.

I assume that countries maximize the following payoff function:

Cost c represents the base price of diesel and is equal and exogenous for all countries2_.

For the following analyses it will be normalized to zero. It is of course bold to assume that countries act as excise revenue maximizers. However, I assume this throughout the thesis for ease of modelling.

Specifying the model like this enables me to capture important differences between countries. The variable Vi expresses the difference between the demand for fuel of

consumers of different countries (approximated by GDP), while the variable xij expresses

the links and thus the structure of the network. Section 4 expands on what data I use for each of theses variables.

Throughout this thesis I will calculate the equilibrium conditions of this model: I show how in equilibrium price depend on domestic demand and links. Detailed calculations for some of the derivations can be found in the appendix.

2_{The empirical data of the European Union shows that base price differs somewhat across member states. I take}

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM Applying the model

to a variety of configurations

3

12

Applying the model to a variety of configurations

In this section I discuss the effect of different configurations on relative prices. I study in particular the effect of domestic demand and link strength. Countries are represented as rectangles for illustration. The relative size of domestic demand is represented by the relative size of rectangles. For illustrative purposes I assume that a link is only present when two rectangles share a common border.

The configurations form the building blocks of modelling more complex situations, such as the entire European Union. They provide insight in how networks play a role in competition between several countries.

3.1 A single isolated country

This situation represents an isolated country, such as an island. It serves as a benchmark for the case where all links fall to zero. This country optimizes its payoff function as a monopolist. I calculate the optimal price by taking the derivative of the payoff function with respect to price, and setting this derivative to zero.

In this case country i charges the payoff-maximizing price of Pi = ½. Network game theory

provides no additional insights to this configuration, as no neighbours are present. It does provide some insight to the European Union: some island states such as Malta have (almost) no link to other member states. As such it can behave as a monopolist when setting the price for fuel.

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM Applying the model 13 to a variety of configurations

3.2 Two countries with identical domestic demand

This is the simplest configuration where interaction is possible. Two countries share a border but are isolated from the rest of the world. The size of the boxes represents the domestic demand of each country, Vi and Vj. Domestic demand is equal in both countries,

meaning I can generalize to V. Countries compete for each other’s domestic demand by undercutting each other’s price. If country i chargers a lower price than country j, customers of country j will purchase diesel in country i. The strength of the link between countries is determined by xij. A higher xij means that more customers of country j travel to country i

to purchase diesel at a given price difference. Since I assume that xij equals xij I can simply

write xij = xij = x. Countries face the following demand function:

Countries are identical which means they will charge the same price as well. Solving for the equilibrium condition by setting the derivative of the payoff function to zero yields the following:

Prices in this case are lower than in the case of a monopolist for all x > 0. Countries aren’t linked if x = 0 and will charge the monopoly price I calculated in the previous paragraph. A stronger link (a higher x) implies a fiercer price competition and lower prices. Equation (8) shows that a larger V implies a higher price.

In the equilibrium of a standard Bertrand competition with heterogeneous goods, prices decrease as substitutability of the good increases. Prices reach the competitive outcome (which is the same as the Bertrand competition outcome with homogenous goods) as goods become prefect substitutes. Payoffs equal monopoly payoffs if substitutability is zero, and decrease as substitutability increases. The model analysis presented in this paragraph

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14 Applying the model to a variety of configurations

follows the same intuition. Link strength determines substitutability: prices are decreasing in x. Prices converge to the competitive outcome of marginal costs (which equal zero) as x approaches infinity. This follows the idea of the excise competition in the European Union: high link strength implies that more customers are able to (easily) purchase diesel in the connected country, enabling stronger price competition and driving prices down.

The case of two isolated countries with identical domestic demand does not seem present in the European Union. European countries typically differ in the size of their domestic demand and usually have links to more than one country.

3.3 Two countries with different domestic demand

In this configuration, countries i and j differ in the size of their domestic demand. The country with the highest domestic demand, country j, will forego a relative large amount of revenue if it lowers its price, but can gain relatively little since the domestic demand of country i is relatively small. In other words, I expect that it is relatively more profitable for the smallest country to undercut prices. Countries are unequal so generalizing to V and P is not possible. Since only two countries are present I can write xij = x. Setting the derivative

of the payoff function to zero yields the following best-response functions:

I use these best-response functions to calculate the following equilibrium conditions:

These conditions imply that in equilibrium, Pj is larger than Pi if and only if Vj is larger than

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will have relatively more to gain by undercutting its opponent and will therefore charge a lower price.

A configuration like this seems to occur in some form in Europe. Cyprus is effectively split into two parts due to the occupation of Turkey. The Occupied Areas of the Republic of Cyprus (“Northern Cyprus”) have a GDP approximately four times smaller than the remaining part of Cyprus, and some trade occurs between both parts. No excise is due in Northern Cyprus, creating the possibility of different prices3_{. The price of diesel in Northern Cyprus}

currently is approximately 17% lower than in the rest of Cyprus, following the example in this paragraph.

3.4 Three countries with identical domestic demand

This configuration of three countries is similar to that of two countries with identical domestic demand, and again I can generalize to V. However, the particular configuration causes country j to be in the centre while countries i and k are in the periphery. The demand function of the centre country differs from the demand function of a periphery country:

The profit maximizations by countries yield the following best responses, which can be computed by setting the derivative of each profit to zero:

3_{I confirmed this with the Government of Cyprus. Prices of petrol in Northern Cyprus at www.alpetkibris.com,}

retrieved 28 July 2014. However, since Northern Cyprus is not a part of the European Union I will assume Cyprus is isolated for the estimations in section four.

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For simplicity I assume that xij = xjk = x. The configuration above shows that country i

and k don’t share a common border. Therefore, xik = 0. Using the best-response functions

and substitution I find the following equilibrium conditions (detailed calculations in the appendix):

The equations show that the price in centre country j will be lower for all x > 0. At x = 0 all prices are equal to the monopoly price calculated previously. Similar to the calculation in the previous case, the result shows that the country that can attract relatively more foreign customers, centre country j, has the lowest price.

A configuration like this doesn’t seem present in the European Union. However, its intuition can be useful when investigating a set of countries as if isolated from the rest of the Union.

3.5 A complete network of n countries with identical domestic demand

In a complete network, all countries are connected. This particular configuration assumes that n countries have an identical domestic demand V (n = 3 in the figure). Unlike the previous configuration, each country is linked to all other countries. The following demand function applies to all countries:

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Setting the derivative of the payoff function to zero gives me the following best-response function (detailed calculations in the appendix):

Countries are identical, so I can generalize to x and P. I now find the following equilibrium condition (detailed calculations in the appendix):

The condition is similar to that in the case of two identical countries (section 3.2). However, price is also decreasing in n. In this case n has an effect similar to x: it drives down prices and payoffs. This follows the intuition of the case of the European Union: an extra neighbouring country enables more customers to purchase diesel in other countries increasing price competition.

At first sight, complete networks of more than two identical countries don’t seem to exist in the European Union. This however depends on the definition of a link. A link, and therefore the value of x, may be estimated by a common border, but also by trade flows between countries. Section four elaborates on this in greater extent. Taking this in mind, complete networks of more than two countries may exist. The country set of Lithuania, Latvia and Estonia (when viewed in isolation) could be a complete network as discussed in this paragraph if trade flows between each of these countries are approximately similar.

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A star network is formed in the shape of a star. The n periphery countries are identical and can therefore all be identified as i. The periphery countries i are only linked to the centre-country j. The figure shows a star network where n = 4 and Vj > Vi. In this configuration each

periphery country is identical in its domestic demand and its link strength x to the centre country j. That means that all periphery countries charge the same price. The following demand function applies:

These give the following best-response functions and equilibrium conditions:

This resolves to Pj > Pi if Vj > nVi (detailed calculations in the appendix).

From the equilibrium condition (19) we can see that the price of the centre j may be larger or smaller than prices in the periphery, dependent on the relative sizes of the domestic demand of each country. This follows intuitively from previous paragraphs: countries that have relatively more foreign customers to attract have the lowest prices. Country j can attract a relatively large amount of customers by undercutting prices if Vj is relatively small.

If Vj is relatively large, j can maximize payoff by charging a higher price than its neighbours.

Specifically, the centre country j charges a higher price than the periphery countries only if it has a higher domestic demand than all periphery countries combined.

Similar to the case of two countries with different domestic demand (section 3.3) country

j earns the highest payoff when it can charge Pj > Pi due to its significantly larger domestic

demand.

A star network where the periphery countries are only linked to the centre country does not seem present in the European Union. An approximate star network may be present when

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looking at certain countries in isolation and when considering trade flows as a proxy for link strength. For example, Germany and its surrounding countries form a star network: trade flows between each of the periphery countries may be dwarfed by the trade flows between the periphery and the centre, meaning that the periphery countries are (practically) not linked to each other but only to Germany. As such this set of countries can behave as a star network, suggesting that the price of diesel in Germany can only be larger than prices in the periphery countries if Germany’s domestic demand is significantly large.

3.7 Overall characteristics of the model

The previous analyses provide the following general ideas of the Bertrand competition with imperfect substitutability on a network.

- Prices and payoffs are decreasing in x and n for all configurations.

- Usually the country with smaller domestic demand has relatively the most to gain from ‘stealing’ domestic demand from neighbours, ceteris paribus. In the case of two isolated countries like on the island of Cyprus it is the country with the smaller domestic demand that charges the lower price.

- However, cases like the star network show that relative size of domestic demand alone does not determine relative price levels. The larger, centre country will charge the higher price only if its domestic demand is larger than the combined domestic demand of neighbouring countries.

Using my model I have shown that network analysis can shed more light on price competitions where the network may influence the equilibrium prices.

Modelling more complex configurations, such as the European Union, requires matrices that cover all countries and their links. The basis for computing the relative price levels is a complete network of all actors, which has the following best-response function for the countries in the network:

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All networks can be described by filling in the correct values for x and V in this best-response function (examples can be found in the appendix). However, I require some rewriting to find the matrix of equilibrium conditions:

In matrix notation, the case of three countries yields the following condition:

I’ll call the matrix on the left hand side matrix V, and the matrices on the right hand side matrix X and matrix P. Hence condition (22) can be rewritten to V = X * P. Values for xij

and Vi determine the configuration of the network. For example a star network with large

centre country i and three identical periphery countries j, k, and l has Vi > Vj = Vk = Vl, xij =

xik = xil > 0 and xjk = xjl = xkl = 0.

The matrices V and X only contain the variables x and V. The data for x and V can be plugged into these matrices. Then I can calculate matrix P by using the inverse of matrix X as follows: P = X-1_*V.

The next section describes the data I can use for modelling the European Union and uses the equation above to calculate relative fuel prices in the European Union.

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM Applying the model to 21 excises in the European Union

The model described in sections 2 and 3 applies network game theory to a Bertrand price competition between producers of imperfectly substitutable goods. I’ve specified the model so it can be applied to excises in the European Union. In this section I formulate the excise debate as a game by defining the variables P, V, and x. Next, I compare calculations using my model and the data to actual prices as posted by the European Commission.

4.1 The game in the European Union

Excises have been harmonized in the European Union to a limited extent. Member states are free to set excise levels, aside from some minimum level. European countries can therefore engage in strategic tax competition: lowering excises so that the excise revenue per litre fuel sold decreases but is compensated by an increase of the local demand. Empirical studies have shown that European countries engage in such a competition with different taxes (Evers et al., 2004).

In my model, the 28 Member States in the European Union are in a Bertrand price competition. I make the assumption that countries outside the European Union are not on the network. This is mainly because I have no uniform data on non-European Union States. The goods the countries produce are fuels. Each country differs in the size of its domestic demand and its links to other countries. The size of each country’s domestic demand combined with the links each country has to other countries determines the configuration of the network.

The good I am interested in is diesel. About 36% of all diesel bought in the European Union is bought by commercial trucks (Planbureau voor de Leefomgeving, 2010). In contrast with consumer cars, trucks have a long range (up to 3.000 kilometres per tank) and, thanks to the free trade arrangements within the European Union, travel extensively between countries (European Commission, 2012). Excises account for about 35% of diesel prices, on average, and diesels costs account for around 30% of total transport costs. This explains the sensitivity of transporting companies to excises (State Secretary of Finance, 2014) and in turn the effect their demand can have on the price-setting behaviour of countries. The following figure displays the average prices of diesel (excluding VAT as it is tax deductible for transporting companies and I therefore assume that it does not influence behaviour) over 2013 in the European Union (data source: European Commission, 2014).

4 Applying the model to excises in the European Union

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22 Applying the model to excises in the European Union

Figure 1 - Average diesel prices in the European Union

I approximate domestic demand of countries V by their GDP (Eurostat, 2014). This follows research suggesting that demand for fuel is increasing with income (Planbureau voor de Leefomgeving, 2010). I do not account for different price levels: the mobile consumers consist mainly of international trucking companies who are not influenced by national price levels (as they only purchase fuel).

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The relationship between country pairs, link strength x, can be approximated in several ways. The most straightforward possibility represents link strength as a dummy variable: x equals one if countries share a common border, or zero otherwise.

Link strength can also be approximated continuously. I believe that truck traffic accounts for a major part of the price competition between Member States. Eurostat provides data on trade flows both by weight and value between Member States. Weight seems a better proxy for the number of trucks. The link strength between two countries can then be approximated by total trade flows by weight between these countries: more weight indicates more trucks that drive between countries. That means that more consumers are able to choose where to buy fuel, increasing the influence the foreign price has on local demand. Here, I only take trade with neighbours with a common border into account.

A third way of computing link strength takes into account the 3.000 kilometre range of trucks. This means that if a truck must drive from France to Portugal, it is in principle free to choose to purchase diesel in France, Spain or Portugal. An increase in trade between France and Portugal therefore increases the number of trucks who can choose to buy fuel in France, Spain or Portugal. Trade from France to Portugal therefore also influences the link strength between France and Spain. Similarly, an increase in trade from Portugal to any country for which its trucks have to drive through France increases the number of trucks that can choose to buy fuel in Portugal, France or Spain. So the link strength x between France and Spain is influenced by all trade routes that go through this country pair. Note that this means that link strength between France and Spain is captured in one variable, that is: xSpain.France = xFrance.Spain. Also, country pairs who do not share a common border, such as

France and Portugal, can still be linked. I will call the inclusion of all trade routes through a country pair the “drive-through effect”.

Concluding, I model the European Union in three ways:

1. Countries set the price of diesel, GDP represents domestic demand, a dummy for shared common border represents link strength;

2. Countries set the price of diesel, GDP represents domestic demand, trade flows with countries sharing a common border represent link strength;

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3. Countries set the price of diesel, GDP represents domestic demand, trade flows with any country and taking into account the drive-through effect represent link strength.

The European Union can be represented in the same way as the configurations of section three. Figure 2 displays the European Union modelled according to the first method, where the size of each rectangle is relative to the GDP of the country it represents:

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The black areas represent Switzerland and non-EU Balkan states. Clearly visible is the effect of using GDP instead of actual country size: it is now visible that the GDP in Poland is much smaller than that of Germany even though the land area doesn’t differ that much. Parts of the entire network shown in figure 1 resemble configurations I discussed in section three: Germany seems the large centre of a star network; Luxembourg seems a small centre of a star network. Czech Republic, Slovakia and Austria, when viewed in isolation, appear as a complete network. It is also apparent that countries vary greatly in the number of countries they share a common border with: Germany has eight direct neighbours, Portugal one, and Malta zero4_.

Figure 3 depicts links between the subset of France, Germany, Belgium, the Netherlands and Luxembourg according to the second method: trade flows to countries with a common border.

Figure 3 - The network of Europe, limited country set, second method

4_{I have assumed in my calculations that the United Kingdom and Ireland are not connected. This is due to the}

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The blue lines in this figure represent trade flows (by weight) between countries sharing common borders. Thicker lines represent higher trade-flows and thus stronger links. Difference in link strength is the major difference with the previous picture. Trade flows are approximately to scale, scaled to trade between Luxembourg and Belgium, the smallest trade in the configuration. It is obvious from this figure that xNetherlands.Germany does not equal

xLuxembourg.Germany, a difference with the previous (dummy) method.

Figure 4 depicts links between the same subset according to the third method: trade flows to countries taking into account the drive-through effect.

Figure 4 - The network of Europe, limited country set, third method

The green lines are the added flows compared to the last figure. Lines are now scaled to trade flows from Belgium to France, the smallest in this configuration. There are two notable differences: countries that don’t share a common border, such as the Netherlands and France, can be connected in this model. Second, countries that trade little themselves,

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such as Luxembourg, can still have relatively strong links due to the drive-through effect. The figure shows that the difference between xNetherlands.Germany and xLuxembourg.Germany is much

smaller than in the previous method.

These figures illustrate how different methods create different networks. Each method differs in the value assigned to x, while the last model is the only model where x takes on a positive value for almost any country pair (notable exceptions: country pairs who are more than 3.000 kilometres driving distance apart, such as Portugal and Lithuania, or those where trade occurs mainly by sea, such as the United Kingdom).

I calculate the relative price levels in Europe using the different values for x each method gives. Earlier research statistically estimated the effect of tax rates in neighbouring countries on one’s own tax rate. Researchers found no statistical difference between including only tax rates of countries with a common border, or countries within a 1.000 kilometre range (Evers et al., 2014). This may suggests that the drive-through effect does not occur in the European Union.

4.2 Calibrating the data

Countries face the following familiar payoff and (local) demand functions:

I have data on GDP, common borders and trade flows in kilos per year to estimate the variables V and x. However, I cannot directly use this data, as I don’t know what order of magnitude the data should be in (e.g. GDP in millions, billions, or some calibrated value). This means I have to calibrate the data first. To do this, I take several countries (Portugal, the Netherlands, Italy), data for V and x for each method, and actual price levels for all countries in 2013 (European Commission, 2014). I selected these countries due to the differences in their links and GDP. I use the matrix equation (22) described in section 3.7, displayed in matrices as V = X * P. I plug in the values for V, P and x. Countries are only linked to direct neighbours in methods one and two. That means that Portugal is only linked to Spain, the Netherlands is linked to Belgium and Germany, and Italy is linked to France, Austria and Slovenia. The appendix displays equation (22) tailored to Portugal, the

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Netherlands and Italy.

I can now fill in the data for V, x and P to determine the different matrices V, X and P. The matrix equation V = X * P does not hold, partly due to differences in magnitude. I can now find some “calibration matrix” β with to multiply V such that β * V = X * P. The matrix β is build up from the different calibration numbers for each country: βP, βNL and βIT.

For the third method, the calibration follows the same principle. However, the matrix equation V = X * P becomes longer as the number of links increases. In the third method countries are linked to (almost) all other countries, meaning that the equation becomes too long to be displayed here. The principle is however the same.

I use the calculated values for β to calibrate my data. The next section computes prices for the entire European Union following the different methods described above.

4.3 Computing the relative price levels

Using the available dataset, calibrated as described in section 4.2, I compute relative price levels according to my model for each method. The dataset can be found in the appendix and consists of the values for GDP and actual prices, and three matrices with the data for x for all three methods. The following table presents the ranking of prices, from low to high, for the actual data and each method:

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Table 1 - Price levels ranked from low to high in 2013 according to actual data (European Commission, 2014) and own calculations

The black line separates the 28 countries in a low price group and a high price group5_{. If}

a method has correctly placed a country in a group (as compared to the actual data in the column most left) that country is coloured green. If countries were randomly assigned to the low or high price group I would expect 14 out of 28 countries in the correct group. Method 2 places 18 out of 28 countries in the correct group, methods 1 and 3 places 20 out of 28 correctly.

5_{Due to the specification of my model, estimates for fuel prices of isolated countries (such as the UK and Malta)}

are far higher than those for non-isolated countries. Testing if my estimates are correctly above or below the average of all estimated fuel prices thus proves not useful. Therefore I have chosen to divide countries in a low- and high price group.

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Conclusion 30

Conclusion

I created a model where countries engage in a Bertrand price competition with imperfect substitutes on a network. Countries differ in the size of their domestic demand and their position on the network. I showed that both the strength of the links between countries and the number of links of a country determine the substitutability of the goods the countries produce.

Specifically I showed the following:

- Prices and payoffs are decreasing in x and n for all configurations.

- Usually the country with smaller domestic demand has relatively the most to gain from ‘stealing’ domestic demand from neighbours (ceteris paribus). In the case of two isolated countries like on Cyprus it is the smaller country that charges the lower price.

- However, cases like the star network show that relative size of domestic demand alone does not determine relative price levels. The larger, centre country will charge the higher price only if its domestic demand is larger than the total domestic demand of neighbouring countries.

I have thus shown with my model that networks can influence the equilibrium prices of a price competition.

I have applied my model to the empirical case of fuel excises in the European Union. I have used several approaches of modelling the this case as a game in my model, specifically modelling the links as follows:

1. Countries set the price of diesel, GDP represents domestic demand, a dummy for shared common border represents link strength;

2. Countries set the price of diesel, GDP represents domestic demand, trade flows with countries sharing a common border represent link strength;

3. Countries set the price of diesel, GDP represents domestic demand, trade flows with any country and taking into account the drive-through effect represent link strength.

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM Conclusion 31

Using each of these methods, I have shown that my relatively simple model, even under the basic assumptions that I made, is better at sorting the member states into a low or high price group than random assignment. To better capture the empirical case of fuel excises in the European Union in my network price competition model, some expansions to my model must be made. Examples are investigating the specification of the demand equation (e.g. linear or non-linear), relaxing the assumption of revenue-maximizing behaviour (e.g. include variables on consumer welfare in the government “profit” function), or revisiting the nature of the links (in a way that it may include behaviour of non-trucker consumers). Concluding, network analysis has added value in price competition analysis and strategic tax competition analysis. Network structure should be taken into account when investigating such competitions.

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References 32

References

Blume, L.E., Easley, D., Kleinberg, J., Tardos, E. (2009). ‘Trading networks with price- setting agents’, Games and Economic Behavior, 67, pp. 36–50

European Commission (2012). EU Transport in figures, Statistical Pocketbook 2012. Luxembourg: Publications Office of the European Union.

European Commission (2014). Oil Bulletin. Retrieved 1 May 2014 from http://ec.europa.eu/energy/observatory/oil/bulletin_en.htm Eurostat (2014). EU Trade Since 1988 By CN8. Retrieved 5 July 2014 from

http://epp.eurostat.ec.europa.eu/portal/page/portal/international_trade/data/ database

Eurostat (2014). Gross domestic product at market prices. Retrieved 5 July 2014 from http://epp.eurostat.ec.europa.eu/tgm/table.do?tab=table&init=1&language=en& pcode=tec00001&plugin=1

Evers, M., De Mooij, R.A., Vollebergh, H.R.J. (2004). ‘Tax competition under minimum rates: the case of European diesel excises’, CESifo working papers, No. 1221

Galeotti, A., Goyal, S., Jackson, M.O., Vega-Redondo, F., Yariv, L. (2010). ‘Network Games’, Review of Economic Studies 77(1), pp. 218-244

Goyal, S. (2007). Links: an Introduction to the Economics of Networks. Princeton: Princeton University Press.

Planbureau voor de Leefomgeving (2010). ‘Effecten van prijsbeleid in verkeer en vervoer, een kennisoverzicht’, Publication number 500076011, retrieved 5 July 2014 from vervoer

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM References 33

State Secretary of Finance, letter of 28 May 2014, ‘Evaluatie accijnsverhoging op diesel en LPG’. Retrieved 5 June 2014 from

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Appendix 34

Appendix

Page 14:

On page 14 I display equation (10):

To calculate when the price of country i equals the price of country j I take the following steps:

This implies that Pj is larger than Pi if and only if Vj is larger than Vi.

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MASTER THESIS MAX ‘T HART - UNIVERSITY OF AMSTERDAM Appendix 35

Page 15:

To get from equation (12) to equation (13) I take the following steps:

First I assume xij = xjk = x. Then I use substitution:

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36 Appendix

Page 16:

To get from equation (14) to the best-response function of equation (15)

I assume b = 1, take the derivative of the payoff function and set it to zero:

This results in equation (15):

Generalizing to P and x gives:

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Page 18:

I simplify equation (24) to the condition Pj > Pi if Vj > nVi as follows:

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38 Appendix

Page 19:

All networks can be described by filling in the appropriate values for x and V into equation (20):

For example the configuration of section 3.4 is such that n = 3, xij = xjk = x > 0, xik = 0, and Vi

= Vj = Vk = V. This means the equations take the following form:

The example star network of section 3.5 has 4 periphery countries i that are linked to centre country j. Each periphery country is identical in its domestic demand, its link strength x to the centre country j, and the fact that it is not linked to any other periphery country. Therefore the periphery countries follow the same best-response function. Hence:

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Page 27:

In section 4.2 I calibrate the game by plugging the values for V, P, and x into equation (22). To equalize both sides I need to multiply the LHS by some “calibration number” βi. Hence

the below conditions form the matrix condition β * V = X * P. The conditions show that Portugal is only linked to Spain, the Netherlands to Belgium and Germany, and Italy to France, Austria and Slovenia. Obviously a larger number of links complicates the formulas, which is why I have not included the formulas for calibration according to the third method here.

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40 Appendix

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Values for xij for method 1 (dummy common border), method 2 (trade with direct neighbour) and method 3 (trade including drive-through effect)