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This workbook is designed to explain the basic principles of game theory.
Important terms in the outline below are in red text.
An important idea is that of interdependence.
This means that one firm's decision affects another firm's decision and vice versa.
The degree of interdependence is a function of the number of firms in the industry
Number
of Firms Name
Degree of
Interdependenc
e
1 Monopoly None
2 Duopoly Very Hgh
Several Oligopoly High
ManyPerfect
CompetitionNone
This file will explore the Perfect Competition, Monopoly, and Duopoly cases.
The product is electricity, which comes in units of kilowatt hours, kwh. A kilowatt is one thousand watts. A watt is a unit of energy.
Electric power companies charge a price per kwh.
In Nov of 2008, the US average retail price was 8.90 cents per kwh.
Source: http://www.eia.doe.gov/fuelelectric.html
The Parameters sheet contains the market demand curve and the cost function of each firm.
The PerfectCompetition sheet shows the outcome if the industry is perfectly competitive.
Electricity is not a perfectly competitive industry, but perfect competition is our benchmark.
The Monopoly sheet finds the profit-maximizing output and price that would be selected by a monopolist.
We offer two ways of finding the optimal solution: graph and Excel's Solver.
The ResidualDemand sheet introduces the Duopoly game. It presents the idea that one firm optimizes based on a conjecture about what the other firm will do.
Once again, there are two ways to find the optimal solution: graph and Excel's Solver.
Given a conjecture, there is a best response and the resulting relationship between conjecture and best response is called a Best Response, or Reaction, Function.
The Duopoly sheet presents the two firms playing the game. There is a Best Response Function for each firm and the intersection pinpoints the Nash equilibrium.
The Summary sheet compares the results from the Perfect Competition, Monopoly, and Duopoly market structures.
Begin your study of Game Theory by going to the Parameters sheet.
The product is electricity, which comes in units of kilowatt hours, kwh. A kilowatt is one thousand watts. A watt is a unit of energy.
The ResidualDemand sheet introduces the Duopoly game. It presents the idea that one firm optimizes based on a conjecture about what the other firm will do.
Given a conjecture, there is a best response and the resulting relationship between conjecture and best response is called a Best Response, or Reaction, Function.
The Duopoly sheet presents the two firms playing the game. There is a Best Response Function for each firm and the intersection pinpoints the Nash equilibrium.
The Summary sheet compares the results from the Perfect Competition, Monopoly, and Duopoly market structures.
DEMAND
Market Demand is given by Q = a - bP Inverse Demand is P = d0 - d1Q
where d0=a/b and d1=1/b
a 20000 d_0 20b 1000 d_1 0.001
Price Quantity
0 20000
2 18000
4 16000
6 14000
8 12000
10 10000
12 8000
14 6000
16 4000
18 2000
20 0
SUPPLY
Each firm's cost function is TC = c3Q3 + c2Q
2 + c1Q
1 + c0
c_3 0
c_2 0 0
c_1 5
c_0 0
Industry cost = Sindividual cost
Quantity Total Cost Marginal Cost
0 0 5
2000 10000 5
4000 20000 5
6000 30000 5
8000 40000 5
10000 50000 5
12000 60000 5
14000 70000 5
16000 80000 5
18000 90000 5
20000 100000 5
0
5
10
15
20
25
0
Price
(cents/kwh)
0
20000
40000
60000
80000
100000
120000
0
cents
0
5
10
15
20
25
0
Price
(cents/kwh)
A linear cost function is not especially realistic, but it makes the problem easier to understand and does not change the basic conclusions of the analysis.
With each firm's MC constant, the number of firms does not affect the industry MC. Whether the industry is perfectly competitive, monpolistic, or anything in between, industrywide MC
0
5
10
15
20
25
0 5000 10000 15000 20000 25000
(cents/kwh)
Quantity (kwh)
Market Demand
0 5000 10000 15000 20000 25000
Quantity (kwh)
Total Cost
0 5000 10000 15000 20000 25000
(cents/kwh)
Quantity (kwh)
Marginal Cost
This sheet assumes that there are many firms producing electricity.
Together, they have a Market Supply curve, which is the sum of the individual firm supply curves.
The market will settle down to an equilibrium price and quantity combination where Market Supply and Market Demand intersect.
The graph makes clear that the equilibrium price will be 5 cents per kwh and equilibrium quantity will be 15,000 kwh.
The Monopoly sheet examines what kind of outcome we would get if one firm had a monopoly in the market for electricity.
Price
Quantity
Demanded
Quantity
Supplied
0 20000 5
2 18000 5
4 16000 5
6 14000 5
8 12000 5
10 10000 5
12 8000 5
14 6000 5
16 4000 5
18 2000 5
20 0 5
Demand
Supply
0
5
10
15
20
25
0 5000 10000 15000 20000 25000
Price (cents/kwh)
Quantity (kwh)
Market for Electricity
The market will settle down to an equilibrium price and quantity combination where Market Supply and Market Demand intersect.
The graph makes clear that the equilibrium price will be 5 cents per kwh and equilibrium quantity will be 15,000 kwh.
The Monopoly sheet examines what kind of outcome we would get if one firm had a monopoly in the market for electricity.
This sheet assumes that only one firm produces electricity.
This sheet offers two ways to find the optimal output-price combination.
1) Use the Choose Q control to find the optimal output and price.
2) Use Solver to find the monopolist's profit-maximizing solution
Parameters
a 20000
b 1000
c_3 0
c_2 0
c_1 5
c_0 0
Objective Function
profits 0 cents
Choice Variables
Q 15000 kwh
P 5 cents per kwh
Price
Quantity
Demanded
Quantity
Supplied MR Choose Q
0 20000 5 -20 15000 0
2 18000 5 -16 15000 5
4 16000 5 -12 0 5
6 14000 5 -8
8 12000 5 -4
10 10000 5 0
12 8000 5 4
14 6000 5 8
16 4000 5 12
Demand
MC
Marginal Revenue
-5
0
5
10
15
20
25
0 5000 10000 15000 20000
Price (cents/kwh)
Quantity (kwh)
Market for Electricity with a Monopolist
Choose Q
18 2000 5 16
20 0 5 20
Demand
MC
20000 25000
This sheet assumes that two firms produce electricity and models Firm 1's optimal decision-making.
Firm 1 has to guess Firm 2's output decision before maximizing.
We introduce a new parameter, Conjectured Q2. A conjecture is a guess.
Given a value of Conjectured Q2, Firm 1 subtracts the amount of output produced by its rival from the Market Demand curve.
Those units of output are presumably sold and, thus, decrease the remaining demand for the product.
Firm 1 will choose the quantity that maximizes profit using the Residual Marginal Revenue curve, then read the price from the Demand curve.
This sheet offers two ways to find the optimal output-price combination.
1) Use the Choose Q control to find the optimal output and price.
2) Use Solver to find the monopolist's profit-maximizing solution.
Parameters
Conjectured Q2 0
a 20000
b 1000
c_3 0
c_2 0
c_1 5
c_0 0
Objective Function
profits 56250 cents
Choice Variables
Q 7500 kwh
P 12.5 cents per kwh
By varying the Conjectured 2, we generate Firm 1's Reaction, or Best Response, Function.
Conjectured
Q2
Firm 1's
Optimal
Output
Market
Price
0
2500
5000
7500
10000
12500
15000
Marginal Revenue
-5
0
5
10
15
20
25
0 5000 10000 15000
Price (cents/kwh)
Quantity (kwh)
Market for Electricity with Two Firms
Choose Q
Price
Quantity
Demanded
Quantity
Supplied MR Residual Qd Residual MR Choose Q
0 20000 5 -20 20000 -20 7500 0
2 18000 5 -16 18000 -16 7500 12.50$
4 16000 5 -12 16000 -12 0 12.50$
6 14000 5 -8 14000 -8
8 12000 5 -4 12000 -4
10 10000 5 0 10000 0
12 8000 5 4 8000 4
14 6000 5 8 6000 8
16 4000 5 12 4000 12
18 2000 5 16 2000 16
20 0 5 20 0 20
Given a value of Conjectured Q2, Firm 1 subtracts the amount of output produced by its rival from the Market Demand curve.
Those units of output are presumably sold and, thus, decrease the remaining demand for the product.
Firm 1 will choose the quantity that maximizes profit using the Residual Marginal Revenue curve, then read the price from the Demand curve.
By varying the Conjectured 2, we generate Firm 1's Reaction, or Best Response, Function.
Demand
MC
15000 20000 25000
Market for Electricity with Two Firms
This sheet assumes that two firms produce electricity and you can have one firm maximize profits based on a conjecture about the output of the other firm.
Instead of using both the Residual Demand graph and Excel's Solver, we'll let the buttons do the heavy lifting of finding the optimal solution given the other firm's chosen level of output.
The buttons on this sheet use analytical solutions to crank out the optimal solution without you having to bother run Solver itself.
Thanks to Frank Howland for eliminating the need for Solver.
Firm 1's Maximization Problem Firm 2's Maximization Problem
Parameters Parameters
a 20000 a 20000
b 1000 b 1000
c_3 0 c_3 0
c_2 0 c_2 0
c_1 5 c_1 5
c_0 0 c_0 0
Conjectured Q2 0.00 Conjectured Q1 0.00
Objective Function Objective Function
profits 0 cents profits 0 cents
Choice Variables Choice Variables
q1 0.00 kwh q2 0.00 kwh
P 20 cents per kwh P 20 cents per kwh
This sheet assumes that two firms produce electricity and you can have one firm maximize profits based on a conjecture about the output of the other firm.
Instead of using both the Residual Demand graph and Excel's Solver, we'll let the buttons do the heavy lifting of finding the optimal solution given the other firm's chosen level of output.
The buttons on this sheet use analytical solutions to crank out the optimal solution without you having to bother run Solver itself.
0
0
1
1
1
1
1
1
q2
Firm 1's Inverse Reaction Function
0
0
0
1
1
1
1
1
1
q2
Firm 2's Reaction Function
0
0
0
0
0
1
1
1
1
1
1
0 0 0 1 1 1
q2
q1
Both Reaction Functions
Firm 2
Firm 1
0
0
0
0
0 0 0 1 1 1
q1
0
0
0
0
0 0 0 1 1 1
q1
Parameters
Demand: Q=a-bP Inverse Demand: P=d0-d1Q Total Cost = c3Q3 + c2Q2 + c1Q1 + c0
a 20000 d_0 20 c_3 0
b 1000 d_1 0.001 c_2 0
c_1 5
c_0 0
Firm Industry Price
Monopoly 7,500 7,500 12.5
Duopoly at Nash
Equilibrium5,000 10,000 10
Perfect
Competition
15,000/n
w/ n
large
15,000 5
Output
