# Stackelberg Leader Follower Models For Strategic Decision Making Engineering Essay

This paper reviews some Stackelberg Leader-Follower theoretical accounts used for strategic determination devising. The simple Stackelberg duopoly is looked at first, and a generalization of the Stackelberg duopoly job is given. By analyzing the theoretical accounts by Murphy et Al. ( 1983 ) and Smeers and Wolf ( 1997 ) , the paper reviews Stackelberg theoretical account from its classical signifier to the recent stochastic versions. The paper looks at the mathematical preparation of both a nonlinear mathematical scheduling theoretical account and a nonlinear stochastic scheduling theoretical account. Towards the terminal of this paper, a simple numeral illustration is given and practical applications of Stackelberg Leader-Follower theoretical accounts are discussed.

Chapter 1: Introduction

In economic sciences, an oligopoly is considered to be the most interesting and complex market construction ( amongst other constructions like monopolies and perfect competition ) . Most industries in the UK and world- from retailing to fast nutrient, nomadic phone webs to professional services- are oligopolistic. Given the current fiscal clime, it is imperative for houses to be certain that they make determinations accurately, maximizing non merely their net income, but besides their opportunities of staying competitory. Many mathematicians and economic experts have attempted to pattern the determination devising procedure and net income maximizing schemes of oligopolistic houses. For illustration, A. A. Cournot was one of the first mathematicians to pattern the behaviors of monopolies and duopolies in 1838. In Cournot ‘s theoretical account both houses choose their end product at the same time presuming that the other house does non change its end product ( Gibbons, 1992 ) . Subsequently, in 1934, H. V. Stackelberg proposed a different theoretical account where one of the duopoly houses makes its end product determination foremost and the other house observes this determination and sets its end product degree ( Stackelberg, 1934 ) .

The classical Stackelberg theoretical account has been extended to pattern a assortment of strategic determination devising. For illustration, Murphy et Al. ( 1983 ) model the end product determination doing procedure in an oligopoly. Later plants by Smeers and Wolf ( 1997 ) extend this theoretical account to include a stochastic component. More interestingly, in a theoretical account by He et Al. ( 2009 ) , the Stackelberg theory is used to pattern the interaction between a maker and a retail merchant when doing determinations about concerted advertisement policies and sweeping monetary values.

The aim of this paper is to reexamine the Stackelberg theoretical accounts from its authoritative signifier to the more recent stochastic versions. In chapter 2, the simple Stackelberg duopoly is reviewed and a generalization of the Stackelberg duopoly job is given. In chapter 3, more complicated and recent theoretical accounts are reviewed. The mathematical preparation of Murphy et Al. ‘s ( 1983 ) and Smeers and Wolf ‘s ( 1997 ) theoretical account is given. At the terminal of chapter 3, a numerical illustration is applied to Smeers and Wolf ‘s ( 1997 ) theoretical account. In chapter 4, practical applications of Stackelberg leader-follower theoretical accounts are discussed. Chapter 4 besides looks at the drawbacks of and possible extensions to Stackelberg theoretical accounts. Appendix 1 explains the Oligopoly market construction and economic sciences involved in net income maximization.

## Chapter 2: Classical Stackelberg Leader-Follower Model

## 2.1 Duopoly Behaviour

Stackelberg ( 1934 ) discussed monetary value formation under oligopoly by looking at the particular instance of a duopoly. He argued that houses in a duopoly can act either as dependant on or independent of the rival house ‘s behavior:

Mentioning to the two houses as house 1 and steadfast 2, severally, house 1 ‘s behavior can be generalised as follows:

Firm 1 positions the behavior of house 2 as being independent of steadfast 1 ‘s behavior. Firm 1 would see steadfast 2 ‘s supply as a given variable and adapts itself to this supply. Therefore, the behavior of house 1 is dependent on that of house 2 ( Stackelberg, 1934 ) .

Firm 1 can see the behavior of house 2 as being dependant on house 1 ‘s behavior. Therefore, house 2 ever accommodate itself to the former ‘s behavior ( steadfast 2 positions steadfast 1 ‘s behavior as a given state of affairs ) ( Stackelberg, 1934 ) .

However, harmonizing to Stackelberg ( 1934 ) , there is a difference in the houses ‘ existent places ; each of the houses could accommodate to either of these two places, doing monetary value formation progressive. Stackelberg ( 1934 ) describes three instances that arise from this state of affairs:

Bowler ( 1924 ) foremost described a state of affairs when both houses in the duopoly strive for market laterality. Harmonizing to Bowler ( 1924 ) , for this to go on the first house supplies the measure it would if it dominated the market with the 2nd house as a follower. This supply is referred to as the “ independent supply ” . By providing this end product level the first house tries to convert the 2nd house to see its behavior as a given variable. However, the 2nd house besides supplies the “ independent supply ” since it is besides endeavoring for market laterality. This duopoly is referred to as the Bowler duopoly with entire supply of the duopoly bing the amount of two “ independent supply ” . Harmonizing to Stackelberg ( 1934 ) , the monetary value formation under the Bowler duopoly is unstable because neither of the houses attempts to maximize net income “ under the given circumstance ” .

The 2nd instance described by Stackelberg ( 1934 ) is a state of affairs where both houses favour being dependant on the other house ‘s behavior. The first house would hold to fit ( in a net income maximising mode ) its end product degree to the each end product in the 2nd house ‘s executable set of end product. The 2nd house does the same and both houses are therefore followings. This is a Cournot duopoly, foremost described by A. A Cournot in 1838. Harmonizing to Stackelberg 1934, the monetary value formation here is unstable because neither of the houses attempts to accomplish the largest net income “ under the given circumstance ” .

The 3rd instance is a state of affairs where one house strives for independency and the other favor being dependent. In this instance both houses are better off making what the other house would wish. Both houses adapt their behavior to maximizing net income under the given circumstance. This state of affairs is referred to as the asymmetric duopoly or more normally as the Stackelberg duopoly. The monetary value formation is more stable in this instance because, harmonizing to Stackelberg ( 1934 ) , “ no 1 has an involvement in modifying the existent monetary value formation ” .

The Stackelberg theoretical account is based on the 3rd instance of a Stackelberg duopoly.

## 2.2 The Model

In the Stackelberg duopoly the leader ( Stackelberg house ) moves first and the follower moves 2nd. As opposed to other theoretical accounts like the Bertrand theoretical account and Cournot pattern where houses make determinations about monetary value or end product at the same time, houses in the Stackelberg duopoly make determinations consecutive.

The Stackelberg equilibrium is determined utilizing backwards initiation ( first find the follower house ‘s best response to an arbitrary end product degree by the Stackelberg house ) . Harmonizing to Gibbons ( 1992 ) , information is an of import component of the theoretical account. The information in inquiry is the Stackelberg house ‘s degree of end product ( or monetary value, Dastidar ( 2004 ) looks at Stackelberg equilibrium in monetary value ) . The follower house would cognize this end product one time the Stackelberg house moves foremost and, as significantly, the Stackelberg house knows that the follower house will cognize the end product degree and respond to it consequently.

Inspired by the work of Gibbons ( 1992 ) , Murphy et Al. ( 1983 ) and Dastidar ( 2004 ) , a general solution to the Stackelberg game ( duopoly ) is derived in the parts that follow.

## 2.2.1 Price map, cost maps, and net income maps

Suppose that two houses in a duopoly supply a homogenous merchandise.

Denote the demand map of this market as, where is the entire degree of end product supplied by the duopoly ( is the Stackelberg house ‘s end product degree and is the follower house ‘s end product degree ) . The monetary value map can be re-written as.

Denote the cost maps ( Appendix 1 ) as for the Stackelberg house, and for the follower house.

The net income map of the Stackelberg house is given by:

Similarly, the net income map of the follower house is given by:

## 2.2.2 Backward initiation to deduce the best response maps and Stackelberg equilibrium

Harmonizing to Gibbons ( 1992 ) , the best response for the follower will be one that maximises its net income given the end product determination of the Stackelberg house.

The follower ‘s net income maximization job can be written as:

This can be solved by distinguishing the nonsubjective map and comparing the derived function to nothing ( as seen in Appendix 1 ) .

Using concatenation regulation to distinguish equation [ 2 ] and puting the derived function to zero, the undermentioned consequence is obtained:

Note that this is a partial distinction of the net income map since the map depends on the demand map which depends on two variables. Equation [ 4 ] gives the follower ‘s best response map. For a given the best response measure satisfies equation [ 4 ] . As a consequence, the Stackelberg house ‘s net income maximization job becomes:

By distinguishing the nonsubjective map in equation [ 5 ] and comparing the derived function to zero, the following consequence which maximises the Stackelberg house ‘s net income is obtained:

By work outing equation [ 6 ] with the follower house ‘s best response net income maximising end product, is obtained by the Stackelberg house given the follower ‘s best response. Gibbons ( 1992 ) describes as the Stackelberg equilibrium ( or the Nash equilibrium of the Stackelberg game ) .

## 2.2.3 Example

Edward gibbons ( 1992 ) considers a simple duopoly selling homogenous merchandises. He assumes that both houses are indistinguishable and the fringy cost of production is changeless at. He besides assumes that the market faces a additive downward inclining demand curve. The net income map of the houses is given by:

where, with stand foring the Stackelberg house and stand foring the follower house.

Using backward initiation, the follower house ‘s best response map is calculated:

Solving equation [ 8 ] :

The Stackelberg house anticipates that its end product will be met by the follower ‘s response. Thus the Stackelberg house maximises net income by puting end product to:

Solving equation [ 10 ] :

Substituting this in equation [ 9 ] :

Equations [ 11 ] and [ 12 ] give the Stackelberg equilibrium. The entire end product in this Stackelberg duopoly is.

Note: Edward gibbons ( 1992 ) worked out the entire end product in a Cournot duopoly to be ( utilizing this illustration ) which is less than the end product in the Stackelberg duopoly ; the market monetary value is higher in the Cournot duopoly and lower in the Stackelberg duopoly. Each house in the Cournot duopoly produces ; the follower is worse off in the Stackelberg theoretical account than in the Cournot theoretical account because it would provide a lower measure at a lower market monetary value. Clear, there exists a first mover advantage in this instance.

In general, harmonizing to Dastidar ( 2004 ) , first advantage is possible if houses are indistinguishable and if the demand is concave and costs are bulging. Gal-or ( 1985 ) showed that first mover advantage exists if the houses are indistinguishable and have indistinguishable downward inclining best response maps.

## Chapter 3: Recent Stackelberg Leader-Follower Models

The classical Stackelberg theoretical account has been an inspiration for many economic experts and mathematicians. Murphy et Al. ( 1983 ) extend the Stackelberg theoretical account to an oligopoly. Later, Smeers and Wolf ( 1997 ) extended Murphy et Al. ‘s theoretical account to a stochastic version where demand is unknown when the Stackelberg house makes its determination. In a more recent study by DeMiguel and Xu ( 2009 ) the Stackelberg job is extended to an oligopoly with multi-leaders.

In this subdivision the theoretical accounts proposed by Murphy et Al. ( 1983 ) and Smeers and Wolf ( 1997 ) are reviewed.

## 3.1 A Nonlinear Mathematical Programming Version

The theoretical account proposed by Murphy et Al ( 1983 ) , is a nonlinear mathematical programming version of the Stackelberg theoretical account. In their theoretical account, they consider the supply side of an oligopoly that supplies homogenous merchandise. The theoretical account is designed to pattern end product determinations in a non-cooperative oligopoly.

There are followings in this market who are referred to as Cournot houses ( note that from now onwards the follower houses are referred to as Cournot house as opposed to merely follower houses ) and leader who is referred to as the Stackelberg house ( as earlier ) . The Stackelberg house considers the reaction of the Cournot houses in its end product determination and sets its end product degree in a net income maximising mode. The Cournot houses, on the other manus, observe the Stackelberg house ‘s determination and maximize their single net incomes by puting end product under the Cournot premise of zero divinatory fluctuations ( Carlton and Perloff, 2005, define divinatory fluctuation are outlooks made by houses in an oligopolistic market about reactions of the other house ) . It is assumed that all the houses have complete cognition about the other houses.

## 3.1.1 Notations and premises

For each Cournot house, allow stand for the end product degree. For the Stackelberg house, allow stand for the end product degree ( note that is used here alternatively of, as seen earlier, to separate the Stackelberg house from the Cournot houses ) .

is the entire cost map of degree of end product by Cournot houses and is the entire cost map of degree of end product by the Stackelberg house.

Let represents the reverse market demand curve ( that is, is the monetary value at which consumers are willing and able to buy units of end product ) .

In add-on to the Cournot premise and premise of complete cognition, Murphy et Al. ( 1983 ) make the undermentioned premise:

and are both convex and twice differentiable.

is a purely diminishing map and twice differentiable which satisfies the undermentioned inequality,

There exists a measure ( the maximal degree of end product any house is willing to provide ) such that,

For citing, these set of premise will be referred to as Assumption A.

Premise 2 implies that the industry ‘s fringy gross ( Appendix 1 ) decreases as industry supply additions. A cogent evidence of this statement can be found in the study by Murphy et Al. ( 1983 ) .

Premise 3 implies that at end product degrees the fringy cost is greater than the monetary value.

## 3.1.2 Stackelberg-Nash-Cournot ( SCN ) equilibrium

The Stackelberg-Nash-Cournot ( SCN ) equilibrium is derived at in a similar manner to the Stackelberg equilibrium seen in chapter 2.

Using backward initiation, Murphy et Al. ( 1983 ) foremost maximize the Cournot houses ‘ net income under the premise of zero divinatory fluctuation and for a given.

For each Cournot house let the set of end product degrees be such that, for a given and presuming are fixed, solves the following Cournot job:

Harmonizing to Murphy et Al. ( 1983 ) , the nonsubjective map in equation [ 15 ] is a purely bulging net income map over the closed, bulging and compact interval. This implies that a alone optimal exists.

The maps can be referred to as the joint reaction maps of the Cournot houses. Murphy et Al. ( 1983 ) specify the aggregative reaction curve as:

The Stackelberg job can be written as:

If solves, so the set of end product degrees is the SNC equilibrium with

To acquire this equilibrium, the end product degrees need to be determined. Murphy et Al. ( 1983 ) use the Equilibrating plan ( a household of mathematical plans designed to accommodate the supply-side and demand-side of a market to equilibrium ) to find:

Let the Lagrange multiplier associated with the maximization job [ 19 ] be. Murphy et Al. ‘s ( 1983 ) attack here is to find for which the optimal. The undermentioned consequence, obtained from Murphy et Al. ( 1983 ) , defines the optimum solution to job [ 19 ] :

Theorem 1: For a fixed, see Problem suppose that satisfy Assumption A. Denote by the alone optimum solution to and allow be the corresponding optimum Lagrange Multiplier associated with job [ 19 ] . ( In instance since alternate optimum multipliers associated with job [ 19 ] exist, allow be the minimal non-negative optimum Lagrange Multiplier. ) Then,

is a uninterrupted map of for.

is a uninterrupted, purely diminishing map of. Furthermore, there exist end product degrees and such that and.

A set of end product degrees optimal to Problem, where, fulfill the Cournot Problem [ 15 ] if and merely if, whence, for.

( This theorem is taken from Murphy et Al. ( 1983 ) with a few changes to the notation )

The cogent evidence of this consequence can be found in the study by Murphy et Al. ( 1983 ) .

This theorem provides an efficient manner of happening for each fixed. For illustration, one can simple carry on a univariate bisection hunt to happen the alone root of.

## 3.1.3 Properties of and

Murphy et Al. ( 1983 ) describes the sum Cournot reaction curve as follows:

is a uninterrupted, purely diminishing map of.

If the right manus derived function of with regard to is denoted as ( the rate of addition of with an addition in ) , so for each:

The cogent evidence to these two belongingss can be found in the study by Murphy et Al. ( 1983 ) .

Murphy et Al. ( 1983 ) province that if solves the Stackelberg job [ 17 ] , so the net income made by the Stackelberg house is greater than or equal to the net income it would hold made as a Cournot house. Suppose that is a Nash-Cournot equilibrium for the house oligopoly. is the end product the “ Stackelberg ” house would provide if it was a Cournot house. solves:

But since solves the Stackelberg job [ 17 ] , the following must keep:

In fact, is the lower edge of. The cogent evidence to this can be found in Murphy et Al. ( 1983 )

From premise 3 in Assumption A, it is clear that. Therefore, it is clear that is an upper edge. However, harmonizing to Murphy et Al. ( 1983 ) another upper edge exists. In a paper by Sherali et Al. ( 1980 ) on the Interaction between Oligopolistic houses and Competitive Fringe ( a monetary value taking house in an oligopoly that competes with dominant houses ) a different follower-follower theoretical account is discussed. In this theoretical account, the competitory periphery is content at equilibrium to hold adjusted its end product to the degree for which fringy cost peers monetary value. Murphy et Al. ( 1983 ) summarise this theoretical account as follows:

For fixed and say is a set of end product degrees such that for each house solves:

and

For the “ Stackelberg house ” , allow satisfy:

In add-on to Assumption A, if is purely bulging, so a alone solution exists and satisfies conditions [ 23 ] and [ 24 ] . The Equilibrating Program with a periphery becomes:

Theorem 1 holds for with and which implies that. In fact, if is purely bulging, is the upper edge of.

Jointly, is bounded as follows:

## 3.1.4 Being and singularity of the Stackelberg-Nash-Cournot equilibrium

Murphy et Al. ( 1983 ) turn out the being and singularity of the Stackelberg-Nash-Cournot ( SCN ) equilibrium. Their attack to the cogent evidence is summarised below:

## Being

For the SNC equilibrium to be, and for should fulfill Assumption A.

Since is bounded and is uninterrupted ( as is uninterrupted ) , the Stackelberg job [ 17 ] involves the maximization of a uninterrupted nonsubjective map over the compact set. This implies that an optimum solution exists. From Theorem 1 it is seen that a alone set of end product degrees, which at the same time solves the Cournot job [ 15 ] , exists. As a consequence the SNC equilibrium exists.

## Singularity

If is convex, so the equilibrium is alone.

Since is bulging, the nonsubjective map of the Stackelberg job [ 17 ] becomes purely concave on. This has been proven by Murphy et Al. ( 1983 ) and the cogent evidence can be found in their study. This implies the equilibrium is alone.

## 3.1.5 Algorithm to work out the Stackelberg job

Murphy et Al. ( 1983 ) provide an algorithm in their study to work out the Stackelberg job. This algorithm is summarised as follows:

To get down with the Stackelberg house needs the undermentioned information about the market and the Cournot houses:

Cost maps of the Cournot houses, fulfilling Premise A.

The upper edge as per Assumption A.

The reverse demand map for the industry, which besides satisfies Assumption A.

With this information, the Stackelberg house demand to find the lower edge and split the interval into grid points with, where and ( from [ 26 ] ) . A piecewise additive estimate of is made as follows:

Here,

is an estimate to and from equation [ 20 ] it follows that:

Note that at each grid point the estimate agrees with.

The Stackelberg job [ 17 ] , therefore, becomes:

can be re-written as:

Where and

Therefore job [ 30 ] becomes:

The nonsubjective map is purely concave and solvable.

Let be the nonsubjective map of the Stackelberg job [ 17 ] and the nonsubjective map of the piecewise Stackelberg job [ 32 ] , so:

Suppose is the optimal degree of end product. First, suppose that is an end point of the interval, so. Now suppose that, that is, . Then needs to be evaluated in order to find. Theorem 1 can be used here. Remember that is a uninterrupted, diminishing map of. To happen the point where ( portion three of Theorem 1 ) , the undermentioned method is suggested by Murphy et Al. ( 1983 ) :

## Figure: Method for finding

## Beginning: Smeers and Wolf ( 1997 ) ( changes made to the notation )

First determine utilizing the bounds. Next, determine utilizing the bounds. Then find utilizing the bounds.Next, determine utilizing the bounds and so on. If so measure utilizing the bounds.

Having evaluated for some grid points, the game can either be terminated with the best of these grid points as an optimum solution or the grid can be redefined at an appropriate part to better truth.

Murphy et Al. ( 1983 ) travel on to find the maximal mistake from the estimated optimum Stackelberg solution. This is summarised below:

Let be the derivative of with regard to, so:

Let be the fringy net income made by the Stackelberg house for providing units of end product,

Let be the existent optimum nonsubjective map value in the interval with the estimation being. Then the mistake of this estimation is defined as:

satisfies the followers:

This concludes the reappraisal of Murphy et Al. ‘s ( 1983 ) nonlinear mathematical programing theoretical account of the Stackelberg job in an oligopoly.

## 3.2 A Stochastic Version

Smeers and Wolf ( 1997 ) provide an extension to the nonlinear mathematical programming version of the Stackelberg theoretical account by Murphy et Al. ( 1983 ) discussed in subdivision 3.1. In the same manner as Murphy et Al. ‘s ( 1983 ) theoretical account, the Stackelberg game in this version is played in two phases. In the first phase, the Stackelberg house makes a determination about its end product degree. In the 2nd phase, the Cournot houses, holding observed the Stackelberg house ‘s determination, respond harmonizing to the Cournot premise of zero divinatory fluctuation. However, Smeers and Wolf ( 1997 ) add the component of uncertainness to this procedure. When the Stackelberg house makes its determination the market demand is unsure, but demand is known when the Cournot houses make their determination. This makes the Smeers and Wolf ‘s ( 1997 ) version of the Stackelberg theoretical account stochastic. Smeers and Wolf ( 1997 ) assume that this uncertainness can be modelled my demand scenarios.

## 3.2.1 Notations and Premise

For the costs maps, the same notations are used. is the entire cost map of degree of end product by Cournot houses and is the entire cost map of degree of end product by the Stackelberg house.

The demand map is changed somewhat to take into history the uncertainness. is a set of demand scenarios with matching chances of happening As such, is the monetary value at which clients are willing and able to buy units of end product in demand scenario. has a chance of happening.

The same Assumption set A discussed in subdivision 3.1.1 apply here with changes made to conditions [ 13 ] and [ 14 ] . Assumption set A can be re-written as:

and are both convex and twice differentiable, as earlier.

is a purely diminishing map and twice differentiable which satisfies the undermentioned inequality,

There exists a measure ( the maximal degree of end product any house is willing to provide in each demand scenario ) such that,

For citing, these set of premise will be referred to as Assumption B.

## 3.2.2 Stochastic Stackelberg-Nash-Cournot ( SSNC ) equilibrium

Smeers and Wolf ( 1997 ) use the same attack seen before to deduce the SSNC equilibrium.

The Cournot job [ 15 ] can be re-written as follows:

For each Cournot house and each demand scenario, allow the set of end product degrees be such that, for a given and presuming are fixed, solves the following Cournot job:

Note that is the end product degree of Cournot house when the demand scenario is.

For each, harmonizing to Murphy et Al. ( 1983 ) , the nonsubjective map in equation [ 40 ] is a purely bulging net income map over the closed, bulging and compact interval.

The maps can be referred to as the joint reaction maps of the Cournot houses for a demand scenario. The aggregative reaction curve becomes:

The Stackelberg job with demand uncertainness can be written as:

Note the Stackelberg job defined job [ 42 ] differs from that defined in [ 17 ] . This is because of the component of uncertainness. The Cournot job [ 40 ] is similar to the Cournot job [ 15 ] because the demand is known when the Cournot houses make their determination. In the Stackelberg job [ 42 ] note the component. This is the estimated average monetary value, that is, the Stackelberg house considers the reaction of the Cournot house under each demand scenario and works out the market monetary value in each scenario, and it so multiplies it by the chance of each scenario. The summing up of this represents the estimated average monetary value.

If solves the stochastic, so the set of end product degrees is the SSNC equilibrium for demand scenario.

To acquire this equilibrium, the end product degrees need to be determined. Smeers and Wolf ( 1997 ) use the same attack as Murphy et Al. ( 1983 ) in making so. The Equilibrating plan is the same as that in [ 19 ] , with alterations made to the Cournot end product and demand map: For each demand scenario,

Theorem 1 lays out a foundation on how to work out the Equilibrating plan in job [ 19 ] and can besides be used to work out [ 44 ] . Smeers and Wolf ( 1997 ) Summarise Theorem 1 as follows:

Theorem 2: For each fixed,

An optimum solution for the job satisfies the Cournot job [ 40 ] if and merely if the Lagrange multiplier, , associated with the Equilibrating plan [ 44 ] , is equal to zero.

This multiplier is a uninterrupted, purely diminishing map of. Furthermore, there exists and such that:

( This theorem is taken from Smeers and Wolf ( 1997 ) , with a few change to the notations )

The belongingss of are the same as those discussed in subdivision 3.1.3. The being and singularity of the SSNC equilibrium is shown in the same ways as the SNC equilibrium of Murphy et Al. ‘s ( 1983 ) theoretical account discussed in subdivision 3.1.4.

## 3.2.3 Algorithm to work out the Stackelberg job

The Stackelberg job here is solved in the same manner Murphy et Al. ( 1983 ) proposed ( discussed in subdivision 3.1.5 ) .

In their study, Smeers and Wolf ( 1997 ) do non stipulate the upper and lower edge of, therefore, it is assumed that is bounded by.The interval can be split into grid points with, where and. The piecewise additive estimate of in [ 27 ] can be re-written as follows:

Here,

has the same belongingss as [ 29 ] .

The Stackelberg job [ 42 ] , therefore, becomes:

Hereafter, the algorithm summarised in subdivision 3.1.5 can be used to work out this job.

## 3.3 Numeric Example

In Murphy et Al. ‘s ( 1983 ) describe a simple illustration of the Stackelberg theoretical account is given. They consider the instance of a additive demand curve and quadratic cost maps:

It is assumed that the Stackelberg house and Cournot houses are indistinguishable. The Cournot job [ 15 ] becomes as follow, with as the optimum solution:

Solving this job outputs:

Note the upper edge of is found by puting. The working to acquire equation [ 51 ] is shown in Appendix 2.

The aggregative reaction curve can be written as:

Using this information, this illustration is now extended to Smeers and Wolf ‘s ( 1997 ) theoretical account with numerical values.

Note that the maps listed in equations [ 49 ] , [ 50 ] , [ 51 ] and [ 52 ] satisfy Assumptions A & A ; B and other belongingss discussed in old subdivisions.

Suppose and. And suppose demand is unknown when the Stackelberg house makes its determination. The cost maps of the houses will be as follows:

## Figure: Different Demand Scenarios

The tabular arraies below describe the possible demand scenarios, chance of each scenario happening, the joint reaction curve and aggregative reaction curve for, and:

Scenario,

Demand,

Probability,

= Demand falls,

= Demand remains unchanged,

= Demand Increases,

Scenario,

Joint reaction curve,

Aggregate reaction curve,

Using this information, the Stackelberg job [ 42 ] can be solved. First, the estimated monetary value component can be calculated as follows:

Substituting this consequence back into the Stackelberg job [ 42 ] gives:

This job can easy be solved by distinguishing the nonsubjective map and happening the value of for which the derived function is equal to nothing. The working to obtain the undermentioned optimum solution is shown in Appendix 2.

Using this consequence, the undermentioned consequence is obtained for each demand scenario:

## Figure: Optimum Output, Price and Net income

1

260.870

98.02

652.96

147.04

2

260.870

134.39

798.42

201.58

3

260.870

170.75

943.87

256.13

Stackelberg house Net income,

Cournot house Net income,

Industry Net income,

1

21,243.87

12,010.81

69,387.12

2

35,573.12

22,574.95

125,872.92

3

49,802.37

36,444.87

195,581.87

The tabular arraies in figure 3 province the SSNC equilibriums for each scenario, and the net incomes made by each house in this oligopoly and the entire industry net income in each scenario. Note that since is purely bulging, the equilibrium obtained for each scenario is alone. Besides note that in all three scenarios, the Stackelberg Output and net income is greater than that of the Cournot houses, exemplifying the first mover advantage.

## Chapter 4: Discussion

In this subdivision, the practical applications, drawbacks and possible extensions to Stackelberg theoretical accounts are discussed.

## 4.1 Practical Applications of Stackelberg theoretical accounts

Stackelberg theoretical accounts are widely used by houses to help determination devising. Some illustrations include:

## Manufacturer-Retailer Supply Chain

He et Al. ( 2009 ) present a stochastic Stackelberg job to pattern the interaction between a maker and a retail merchant. The maker would denote its concerted advertisement policy ( per centum of retail merchant ‘s advertisement disbursals it will cover-participation rate ) and the sweeping monetary value. The retail merchant, in response, chooses its optimum advertisement and pricing policies. When the retail merchant ‘s advertisement and pricing is an of import constituent of the merchandise ‘s selling mix, it becomes critical for the maker to take into history the reaction of the retail merchant when puting the engagement rate and monetary value. The stochastic component of this theoretical account is the proportion of the market aware of the merchandise ( or served by the retail merchant ) , the engagement rate is a map of this proportion. Later work by Huang et Al. ( 2009 ) extends this theoretical account to include more retail merchants.

## Strategic determination devising in Marketing or Expansion

The Stackelberg theoretical account can besides be used to pattern strategic determinations in Marketing and enlargement. As seen for end product determinations, a Stackelberg house can be the first one to make up one’s mind on a promotional run or alter the bing run. The other oligopolistic houses will hold to alter their promotional schemes so as non to lose clients to competition.

Strategic determinations of enlargement can be modelled by stochastic Stackelberg theoretical accounts. For illustration, Tesco plc ( market leader in the UK retail industry ) may make up one’s mind to open more subdivisions in UK in the hereafter depending on the economic conditions. The rivals ( Sainsbury, Morrisons, and Asda ) will respond to Tesco ‘s determination and alter their enlargement programs. Tesco has to take into history the reaction of its rival in different future economic scenarios when make up one’s minding on its enlargement scheme.

## New entrants in a Monopolistic Market

The interactions between a new entrant to a market and an bing monopoly can be modelled utilizing the Stackelberg theoretical account. When barriers to entry a low and a new house enter the market, a monopoly needs to alter its end product to keep its laterality in the market. The monopoly ( now the incumbent house ) is likely to act as a Stackelberg leader ( Geroski and Ulph, 1988 )

In a different theoretical account, it is possible for the new entrant to act as a Stackelberg leader. Say, a foreign monopoly enters a domestic monopolistic market. Then the foreign house can seek to derive market laterality by acting like a Stackelberg leader.

## Merchandise Development

Stackelberg theoretical accounts can besides be used when developing a new merchandise. When developing a new merchandise, a house will hold to take into history that the rivals will follow suit and develop similar merchandises. For illustration, after the release of Xbox 360 in 2005, Sony and Nintendo followed suit and released PlayStation 3 and Wii, severally, in 2006.

## Future Production Level Decisions

The Stackelberg game can be used to pattern future production degrees when the hereafter is unsure. For illustration, Smeers and Wolf ( 1997 ) applied their theoretical account ( discussed in chapter 3, subdivision 3.2 ) to the European gas market. The chief oligopolistic houses ( at the clip when their study was written ) included CEI, Norway, Netherlands, Algeria and United Kingdom. Norway, in 1990, had to make up one’s mind the degree of development of a new field that would be effectual in 10 old ages. The other manufacturers were considered to hold adequate trim capacities to set their production to run into demand in 2000 ( Smeers and Wolf, 1997 ) . The demand for natural gas in 2000 was unknown in 1990, but known in 2000. Here, Norway ( the Stackelberg house ) makes a determination about its degree of development for the twelvemonth 2000 sing the reaction of the other states and under uncertainness about future demand.

## 4.2 Drawbacks of Stackelberg Models and Possible Extensions

Here the drawbacks of Stackelberg theoretical accounts discussed in this paper and Stackelberg theoretical accounts in general are discussed with suggestion of possible extensions where possible.

In Smeers and Wolf ( 1997 ) , future demand, at the point of doing the end product determination, is unsure. Smeers and Wolf ( 1997 ) fail to take into history how other factors would alter overtime. For illustration, in 10 old ages ‘ clip costs would alter ( unsteadily ) and the cost maps known in 1990 may non be valid in 2000. The theoretical account could perchance be extended by sing how costs could alter in the hereafter ( as done for demand, the theoretical account could take into history different cost scenarios ) .

Another drawback of Smeers and Wolf ‘s ( 1997 ) theoretical account is the preparation of the Stackelberg job [ 42 ] . Smeers and Wolf ( 1997 ) usage estimated monetary value in the Stackelberg job. Suppose because of utmost scenarios, the standard divergence of market monetary value is big. In this instance, the estimated monetary value could differ significantly from existent monetary value. The theoretical account could be extended to include a assurance interval around the estimated monetary value and optimum end product could be worked out around this interval.

Stackelberg theoretical accounts, in general, assume that homogenous merchandises face the same demand curve. However, because of stigmatization and trade name trueness, it is likely that some providers will confront Steeper demand curve ( if their trade names are more popular ) and some face a level demand curve ( less popular trade names ) . The Stackelberg house would necessitate to see single demand curves alternatively and, along with end product determinations, see how the other houses react to alterations in monetary value.

Stackelberg theoretical accounts assume complete information. In a competitory and non-cooperative oligopoly, this premise may non be practical. Firms will by and large non supply information about their cost maps, for illustration, to rivals. Geroski and Ulph ( 1988 ) expression at the Stackelberg job under symmetric and asymmetric information.

The Cournot premise may non be valid after one period. When the Cournot houses make their end product determination, it will be observed that Cournot premise was incorrect since the other Cournot house besides changed their degree of end product ( Carlton and Perloff 2005 ) .

## Chapter 5: Decision

In this paper, some Stackelberg leader-follower theoretical accounts have been reviewed. In chapter 2, it is seen that, in a Stackelberg duopoly ( or asymmetric duopoly ) , one house strives for independency and the other house paths for dependance. In acting like this, both houses are able to maximize their net incomes and the monetary value formation is more stable. Using backward initiation, the Stackelberg duopoly was generalised for any demand and cost map. In the illustration given in chapter 2, it is seen that the Stackelberg duopoly end product is higher than that of a Cournot duopoly, and the Stackelberg house is better off.

Chapter 3 looks at the mathematical preparation of the theoretical accounts by Murphy et Al. ( 1983 ) and Smeers and Wolf ( 1997 ) . The belongingss of the aggregative reaction curve ( by Cournot houses ) and the optimum Stackelberg end product are summarised. It is found that the optimum Stackelberg end product is bounded below by the optimum end product it would bring forth if it were a Cournot house, and bounded above by the optimum end product the house would bring forth if it were a competitory periphery with a purely convex cost map. The stochastic component of Smeers and Wolf ‘s ( 1997 ) theoretical account is besides reviewed. It is seen that the Stackelberg house considers the aggregative reaction curves of the Cournot houses in different future demand scenarios when puting its end product under future demand uncertainness. The numerical illustration shows that in all demand scenarios, the Stackelberg house will be better off.

Finally, in chapter 4 some practical applications of Stackelberg theoretical accounts are explained. It is seen that both nonlinear mathematical theoretical accounts and nonlinear stochastic theoretical accounts can be used in different determination devising procedures like advertisement, strategizing enlargement, and pricing. Some drawback of Stackelberg theoretical accounts are besides looked at. In peculiar, it is noted that the Cournot premise may non keep in a multi-period determination devising procedure and since it is hard to obtain complete cognition about the rivals, Stackelberg theoretical accounts may non be accurate in world.

## Appendix 1: Economic Background

To understand Stackelberg theoretical accounts, an apprehension of the market structures involved and a house ‘s net income maximization theory demands to be developed. In developing the classical Stackelberg theoretical account, Stackelberg ( 1934 ) looked at a market construction with lone two houses, a duopoly. The theoretical account has so been extended to include more houses and show how houses in an oligopolistic market behave.

## A1.1 Definitions and Characteristics of Oligopoly and Duopoly

A duopoly is a market construction with lone two houses in the market selling homogenous ( or similar ) merchandises ( Perloff, 2009 ) .

An oligopoly is a market construction where there are a little figure of houses ruling the industry. These houses are mutualist and rule the market due to the being of high barriers of entry ( for illustration, trade name trueness, high start-up costs, and unrecoverable costs ) . The houses may bring forth homogenous merchandises ( for illustration, metals, chemicals, sugar, and gasoline ) or differentiated merchandises ( for illustration, autos, soap pulverization, and electronic contraptions ) . A duopoly is a particular type of an oligopoly ( Perloff, 2009 ) .

Harmonizing to Perloff ( 2009 ) , other features of an oligopoly include:

High concentration ratios: Supply in the industry must be concentrated in the custodies of comparatively few houses

Mutuality: the actions of one big house will straight impact another big house.

Non-price competition: Often times, monetary value is non the most of import factor in the competitory procedure in an oligopoly. Firms engage in viing by make up one’s minding upon their selling mix and the stigmatization of their merchandise.

Price Rigidity: monetary values in an oligopolistic market tend to alter far less than in absolutely competitory markets

Collusion: oligopolistic houses frequently benefit form colluding and doing understandings amongst themselves to curtail competition.

## A1.2 Profit Maximisation

## A1.2.1 Definitions

Cost can be split into two parts: fixed costs and variable costs. The cost map represents the entire cost of bring forthing units of end product ( Chen, 2007 ) .

Fringy cost is the cost of bring forthing an extra unit of end product and can be thought of as the incline of the cost map ( Perloff, 2009 ) . That is,

Gross is the entire sum of money received by a house for the goods sold or services provided. The gross map represents the entire gross received from selling units of end product at monetary value per unit. The monetary value can frequently vary with the end product degree.

Fringy Revenue is the gross obtained from selling an extra unit of end product and can be thought of as the incline of the gross map ( Perloff, 2009 ) . That is,

## A1.2.2 Profit and Profit Maximisation

Net income is the difference between gross and costs. Therefore, the net income map is given by:

Net income can be maximised by puting the end product degree to such that the net income map is maximised. This can be done by taking the first derived function of with regard to and comparing it to zero:

Substituting equation [ 59 ] and [ 61 ] into equation [ 63 ] :

Comparing this to zero:

Therefore, net income is maximised if [ 65 ] is satisfied. Let be the net income maximizing degree of end product, so satisfies

This can be illustrated diagrammatically:

## Figure: Net income maximization

The graph above shows a house ‘s fringy cost and fringy gross. The net income maximizing degree of end product is where. At this point fringy net income is zero. At end product degrees less than, . Therefore, fringy net income is positive ; the house can go on to produce/supply more end product since it is doing net income for each extra unit. At end product degrees greater than, . Therefore, fringy net income is negative ; the house will do a loss for every extra end product it produces/supplies. Therefore, the maximal net income is attained merely when. The shaded part represents the maximal net income.

## A1.3 Non-cooperative Oligopoly

Earlier on it was mentioned that oligopolistic houses frequently benefit form colluding and doing understandings amongst themselves to curtail competition. They do this by moving like a monopoly jointly. However, Stackelberg theoretical accounts are based on the premise that oligopolistic houses make determinations independently but are cognizant of the other house ‘s actions ( Hallam, 2005 ) . These houses are so in a non-cooperative oligopoly.

The houses in a non-cooperative oligopoly face a kinked demand curve.

Harmonizing to Sweezy ( 1939 ) , houses “ match monetary value decreases but ignore monetary value additions ” . The curve is kinked at the current monetary value, with demand being significantly more elastic above this monetary value than below ( Carlton and Perloff 2005 ) . This occurs when two conditions hold:

If an oligopolistic house cuts its monetary value, its challengers will experience forced to follow suit and cut theirs, to forestall losing clients to the first house

If an oligopolistic house raised its monetary value, its challengers will non follow suit since, by maintaining their monetary values the same, they will derive clients from the first house.

## Appendix 2: Workss for Numeric Example

The Cournot job [ 50 ] is solved as follows:

Suppose,

Therefore,

the solution of [ 67 ] , is such that:

Note that if, so. However, in [ 67 ] is maximised capable to. Thus,

The Stackelberg job [ 56 ] is solved as follows:

Suppose,

Therefore,

, the solution to [ 71 ] , is such that