Term 2 Week 10: Oligopoly

Question 1

What are differences in strategy in different market structures (2)

Answer 1

-In a monopoly or perfectly competitive market, there are no strategic interactions
-When we have a handful of dominant firms, then we must consider the impact of strategic interaction

Question 2

What are 2 common ways to measure market power in an industry (2,2)

Answer 2

-The Herfindahl-Hirschman Index (HHI) measures the sum of market shares squared
-For the HHI, 1800 or above are seen as very concentrated

-Concentration ratio’s measure the % of total sales coming from the top n firms
-Typically, oligopolies are where the CR4 > 60%

Question 3

What does the Cournot model explain (2)

Answer 3

-The Cournot model explores how firms would compete in a market where prices are determined by total output
-How do firms compete in an oligopolistic industry

Question 4

What are the assumptions made in the Cournot model (4)

Answer 4

-Products are homogeneous
-Firms pick their output
-Prices are identical and determined by the output produced
-Firms can’t commit to producing a certain output (by moving first for example)

Question 5

How can you set up a basic set up of the Cournot model (4)

Answer 5

-A basic set up has 2 firms selecting output levels, where firm I picks q_i ≥ 0, and firm j picks q_j ≥ 0
-The market price for the good depends on total output Q = q_i + q_j
-There is only a marginal cost of c, with no fixed costs
-Profit is therefore π_i = P(Q)q_i - cq_i

Question 6

What are the three ingredients in the Cournot model (3)

Answer 6

-2 players: firms i and j
-Possible strategies q_i ≥ 0, q_j ≥ 0
-Payoff is profit:π_i = P(q_i + q_j)q_i - cq_i, π_j = P(q_i + q_j)q_j - cq_j

Question 7

How can we plug the linear demand curve into the profit function in the Cournot Model (2)

Answer 7

Remember π_i = Pq_i - cq_i
-π_i = (A-b(q_i + q_j))q_i - cq_i
-The other firm’s output lowers the price we get, and hence our optimal q_i

Question 8

How can we draw firm i’s profits as a function of q_j in the cournot model (3,2)

Answer 8

-Draw a graph with output (q_i) on the x axis, profit (π_i) on the y axis
-Draw many concave curves, all starting from (0,0) and all within each other
-The highest curve will be q_j = 0, and as firm j’s quantity rises, you shift inwards in curves

-The maximum profit for firm i is where their gradient = 0, and as you can see, their best output level depends on how much firm q produces
-As q_j rises, q_i falls, and as does maximum profit

Question 9

How can we find the profit maximising quantity for firm i in the cournot model (5,1)

Answer 9

-max q_iP(q_i + q_j) - cq_i
-Subbing in the linear inverse demand P(q_i + q_j) = (A-b(q_i + q_j)) gives us π_i = q_i(A-b(q_i + q_j)) - cq_i
-Rearranging gives us π_i = -bq_i² + q_i(A - bq_j - c)
-∂π_i/∂q_i = -2bq_i + (A - bq_j - c) = 0

-BR_i(q_j) = 1/2((A-c)/b - q_j) (make q_i the subject)

Question 10

What is firm i’s best response profit function in the cournot model, and what does this tell us (1, 2, 4)

Answer 10

-BR_i(q_j) = 1/2((A-c)/b - q_j)

-This function is decreasing in q_j
-There is no dominant strategy in the game, as our best response is dependent on what the other player picks

-The more our competitor produces, the less we should
-The higher our cost (c), the lower we should produce
-If inverse demand shifts out (A) increases, we produce more (if we have a larger market)
-If the inverse demand curve is steeper (b increases) we produce less

Question 11

How can we draw the best response curves for firms i and j in the Cournot model (1,2,2)

Answer 11

-Put q_i on the x axis, and q_j on the y axis

-q^m = A-C/2b is the monopoly output level, how much a firm would produce if the other firm produced nothing (for firm i, this is the x intercept)
-q^ZP = A-C/b is where the optimal output level is 0, when we’re indifferent between producing nothing and leaving the market (for firm i, this is the y intercept)

-The intersection is the outcome where both firms play their best responses
-This intersection q_i^c is the Nash equilibrium

Question 12

What are the two best response quantity equations in the Cournot model (2)

Answer 12

-q_i = (1/2)[(A-c/b)-q_j]
-q_j = (1/2)[(A-c/b)-q_i]

Question 13

How can we find the equation for the intersection of the best response quantities in the cournot model (5)

Answer 13

-q_i = (1/2)[(A-c/b)-q_j], -q_j = (1/2)[(A-c/b)-q_i]
-q_i = (1/2)[(A-c/b)-(1/2)((A-c)/b - q_i))]
-q_i = 1/4((A-c)/b + q_i)
-4q_i = ((A-c)/b + q_i)
-q_i = q_j = (A-c)/3b

Question 14

How can we draw the cournot outputs, and the profit maximising/minimising level (1,3)

Answer 14

-Draw the diagram with q_i on the x axis, and q_j on the y axis

-The industry’s optimal profit maximising level occurs on the line between q^M for both firms, and the price determines how it is split
-If the industry is producing on the line between q^ZP for both firms, industry profit is 0, below it profit > 0, above it profit < 0
-The nash equilibrium is not maximum profit in the industry, as they are competing

Question 15

How can we work out profit levels for firms in the cournot model (4)

Answer 15

-q_i = q_j = (A-c)/3b
-Q = q_i + q_j = 2(A-c)/3b
-π_i = (p-c)q_i = (A-b(2(A-c)/3b)-c)(A-c)/3b
-π_i = (A-c)²/9b

Question 16

What are the efficiency properties in the cournot model (4)

Answer 16

-Total surplus is maximised where marginal social benefit = marginal social cost
-P = MC
-A - bQ = c
-Q = (A-c)/b = q^ZP

Question 17

How can we graphically represent monopolies not being socially optimal (2,2,3)

Answer 17

-Draw a monopoly diagram, with Q on the x axis, P on the y axis
-Draw a horizontal MC curve, a downwards linear sloping D = AR curve, and a downwards linear sloping MR curve twice as steep as an AR curve

-Socially optimum output would be Q^ZP, where D = MC at (A-c)/b
-However, Q^M is where MR = MC at (1/2)(A-c)/b

-CS is the area to the left of Q^M, above the price and below the AR curve
-PS is the area to the left of Q^m, below the price and above MC
-DWL is all the area above MC and below AR which is not part of CS and PS

Question 18

How can we graphically represent DWL in the cournot case (2,3,3,2)

Answer 18

-Draw a monopoly diagram, with Q on the x axis, P on the y axis
-Draw a horizontal MC curve and a downwards linear sloping D = AR curve

-Socially optimum output would be Q^ZP, where D = MC at (A-c)/b
-However, Q^C is at (2/3)(A-c)/b

-CS is the area to the left of Q^C, above the price and below the AR curve
-PS is the area to the left of Q^C, below the price and above MC
-DWL is all the area above MC and below AR which is not part of CS and PS

-In the cournot case, there is less DWL than in a monopoly, due to increased competition
-There is also increased CS and decreased PS

Question 19

How can we introduce isoprofit curves into the cournot model (3,5)

Answer 19

-The isoprofit curves ask the question which combinations of q_i and q_j give the same profit
-π_i = (A-b(q_i + q_j)-c)q_i = k
-K is a number, and varying k gives a different isoprofit curve

-Draw a diagram with q_i on the x axis, and q_j on the y axis
-Draw 2 best response curves, BR_j(q_i) being downwards sloping with a y intercept, and BR_i(q_j) having a x intercept, and sloping upwards to the left (both linear)
-Then draw our isoprofit curves, concave on the x axis
-The isoprofit we will be on is the one which intersects the NE (where the 2 BR curves intercept)
-The lower down on the isoprofits, the better for i

Question 20

How can we draw each firms’ isoprofit curves in the cournot model, and then the pareto improvement area (2,3)

Answer 20

-Draw a graph with q_i on the x axis, and q_j on the y axis
-Draw each firms isoprofit curve, concave from the x axis for firm i, (this one = blue curve) concave from the y axis for firm j (this one = green curve)

-The area below the blue curve is better for i, and the area behind the green curve is better for j
-The area inbetween the 2 curves (the 2 curves intersect twice) is the pareto improvement area (for the firms)
-Within this area the Q^M curve (connecting both firms’ Q^M) should go through this point

Question 21

How in the cournot model can we diagramatically represent collusion (2,3)

Answer 21

-Draw a graph with q_i on the x axis, and q_j on the y axis
-Draw each firms isoprofit curve, concave from the x axis for firm i, (this one = blue curve) concave from the y axis for firm j (this one = green curve)

-This case of isoprofits in the cournot model is similar to the prisoners dilemma in terms that the NE is not pareto efficient, so by collusion, they could reach the outcome
-This is done by pushing the isoprofits together so that they are tangential on the Q^M line
-However, this collusion will need to be forced as it is not stable

Question 22

How can we drop the assumption of entry barriers in the Cournot model (2,2,2,3,3)

Answer 22

-If we drop the assumption of entry barriers, new firms can join the market
-If we allow entry, there is now N total producers

-Call the total quantity produced Q = q_i + Σq_j
-The price is the same as before: p = A - bQ

-The profit function is π_i = (A - bQ - c)q_i
-(∂π_i/∂q_i) = -bq_i + (A - bQ - c) = 0

-When firms are identical, the result of Cournot models are always symmetric: q_i^C = q_j^C = q
-Now let Q = nq, if we assume all firms are identical
-(∂π_i/∂q_i) = -bq + (A - bNq - c) = 0

-q = (A-c)/b(N+1)
-p = A - (N(A-c))/(N+1)
-π_i = (A-c)²/(b(N+1)²)

Question 23

What happens when N -> ∞ in the Cournot model (3, 3)

Answer 23

-q = (A-c)/b(N+1)
-p = A - (N(A-c))/(N+1)
-π_i = (A-c)²/(b(N+1)²)

As n -> ∞
-p -> c
-π_i -> 0
-This shows how entry barriers are a source of profit

Question 24

How can we adapt the profit functions in the cournot model to different marginal costs, fixed costs of production and imperfect substitutes (1,2,1,1)

Answer 24

Normally:
-π_i = (A-bQ)q_i - c_iq_i

Different marginal costs:
-π_i = (A-bQ)q_i - c_iq_i
-π_j = (A-bQ)q_j - c_jq_j

Fixed cost of production:
-π_i = (A-bQ)q_i - c_iq_i - F

Products can be imperfect substitutes:
π_i = (A-bq_i - γbq_j)q_i - cq_i

Question 25

Why does Bertand criticise the Cournot model (1)

Answer 25

-Bertrand argues quantity competition is not realistic, as firms usually pick prices

Question 26

What are the assumptions of the Bertrand model (5)

Answer 26

-Products are homogeneous -Firms pick prices -Buyers know all prices and buy from the cheapest firm -Firms compete just once and pick prices without observing the other's decision -No fixed cost and constant marginal cost

Question 27

What is the demand function in the Bertrand model (1,3,1)

Answer 27

-If one firm has lower price p_i < p_j, then market price is p = p_i q_i(p_i, p_j) = -Q(p) if p_i < p_j -Q(p)/2 if p_i = p_j -0 if p_i > p_j -This is where Q(p) couold be Q(p) = A - bp

Question 28

How can we draw the demand function for a firm i in the bertrand model (1,3)

Answer 28

-Draw p_i on the x axis, q_i on the y axis -For p_i < p_j, draw a normal downward sloping linear demand curve -For p_ij, demand suddenly drops to a half as they share the market -For p_i > p_j, there is no demand curve

Question 29

How do we usually find the Nash Equilibria, and why can't we do this in the Bertrand model (3,3)

Answer 29

-Put all the payoffs in a matrix and find the best response -Differentiate the profit function -Iterate on the best responses Reasons for not working: -Matrix = not possible, as price being continuous means there is unlimited information -Differentiate = not possible, function is not continuous -Iterate = may work, but not sure if will go to equilibrium or move in circles

Question 30

How can we find the nash equilibrium in the Bertrand model (1,4)

Answer 30

-Split the model into a number of cases and see if players are playing best responses These responses are: -Where either p_i or p_j < c -Where either c < p_i < p_j or cj < p_i -Where either c = p_i < p_j, c = p_j < p_i or c < p_i = p_j -Where c = p_i = p_j

Question 31

How can we diagramatically analyse where either p_i or p_j < c is a NE in the bertrand game (3,1,2)

Answer 31

-Draw a diagram with p_i on the x axis, p_j on the y axis -Now draw a 45 degree upward sloping linear line from the origin, where p_i = p_j -Then draw a point where p_i = c, p_j = c, and draw those dashed lines from the origin -For the case where either p_i or p_j < c, fill in the areas to the left of p_i = c, and below p_j = c -This is not a best response, as pricing below cost means they are making a loss -Either they both share the market and both lose, or one firm takes the whole market and would want to raise prices (incentive to change = not a NE)

Question 32

How can we diagramatically analyse where either c < p_i < p_j or cj < p_i is a NE in the bertrand game (3,2,2)

Answer 32

-Draw a diagram with p_i on the x axis, p_j on the y axis -Now draw a 45 degree upward sloping linear line from the origin, where p_i = p_j -Then draw a point where p_i = c, p_j = c, and draw those dashed lines from the origin -For p_i < p_j, fill in the triangle to the left of the p_i = p_j line, above both p_i = c and p_j = c -For p_i > p_j, fill in the triangle to the right of the p_i = p_j line, above both p_i = c and p_j = c -This is not a nash equilibrium -For the left triangle, player j has an incentive to cut prices to recieve demand

Question 33

How can we diagramatically analyse where either c = p_i < p_j, c = p_j < p_i or c < p_i = p_j is a NE in the bertrand game (3,3,3)

Answer 33

-Draw a diagram with p_i on the x axis, p_j on the y axis -Now draw a 45 degree upward sloping linear line from the origin, where p_i = p_j -Then draw a point where p_i = c, p_j = c, and draw those dashed lines from the origin -For c = p_i < p_j, colour in the c = p_i line above p_j = c -For c = p_j < p_i, colour in the c = p_j line above p_i = c -For c < p_i = p_j, colour in the p_j = p_i line above the intersection point -This is not a Nash equilibrium -For the firm pricing above cost, they have nothing to change -However, for the firm pricing at cost, they would want to bump up prices a little bit

Question 34

How can we diagramatically analyse where either c = p_i = p_j is a NE in the bertrand model (3,1,3)

Answer 34

-Draw a diagram with p_i on the x axis, p_j on the y axis -Now draw a 45 degree upward sloping linear line from the origin, where p_i = p_j -Then draw a point where p_i = c, p_j = c, and draw those dashed lines from the origin -c = p_i = p_j is the intersection point between all 3 dashed lines -This is the NE, as neither firm can improve -Lowering price means the firms make a loss -Increasing prices mean they lose all market demand

Question 35

What are the differences in profit in the cournot and bertrand model, and what is the bertrand paradox (4)

Answer 35

-Two firms competing on price (bertrand model) make zero profit, but two firms competing on quantity (cournot model) make positive profit -This is because firms produce at marginal cost in the Bertrand model, but not in the Cournot model -This is known as the Bertrand paradox -This is a surplus maximising outcome, as we get social efficiency without needing free entry or a large N value

Question 36

Why does the switch to price setting make a difference (1,3)

Answer 36

-Firms in the Bertrand model have much stronger incentives to compete, and the power to do so easily -If we marginally undercut, we take the whole market in the bertrand model -If we marginally increase q in the Cournot model, the benefits are smaller -In the Bertrand model, the firms control the variables demanders care about (price)

Question 37

How can we remove the assumption that products are identical (2,3)

Answer 37

-In Bertrand, the products are identical, and thus the lowest price firm takes the whole market -A new assumption which can be made is that products are imperfect substitutes -q_i(p_i, p_j) = A - bp_i + (b/2)p_j -q_j(p_j, p_i) = A - bp_j + (b/2)p_i -A firms demand is more responsive to their price than their competitors, but demand is no longer infinitely elastic

Question 38

What is the profit function in the bertrand model once products are imperfect substitutes (2,3,2)

Answer 38

-π_i = p_iq_i(p_i, p_j) - cq_i(p_i, p_j) -π_i = (p_i - c)(A - bp_i + (b/2)p_j) -Since the payoff function is differentiable, we can now find the BR this way -∂π_i/∂p_i = A - bp_i + (b/2)p_j - bp_i + bc = 0 -A + (b/2)p_j + bc = 2bp_i The best response functions then become: -p_i = (1/2)(A/b + (1/2)(p_j + c) -p_j = (1/2)(A/b + (1/2)(p_i + c)

Question 39

How can we draw the best response diagrams in the Bertrand model with imperfect substitutes (1,2,3,2)

Answer 39

-Put p_i on the x axis, and p_j on the y axis -Draw the BR_i(p_j) curve, upward linear sloping from the x axis -Draw the BR_j(p_i) curve, upward linear sloping from the y axis -Start from point (c,c) (below the x and y intercept) -Overtime, through drawing this point moving from one diagram to the other, the NE will be tended to -NE = where BR_i(p_j) = BR_j(p_i) at p_i* = p_j* -Actions are strategic complements, as both BR curves slope upwards -The higher one firm prices, the higher the other wants to price

Question 40

How do we find the NE in the bertrand model with imperfect substitutes mathematically (1,1,2,1)

Answer 40

-To find the NE mathematically, we solve the 2 best response functions simultaneously -As this is a symmetric equilibrium, we assume the 2 prices are the same -p_i = (1/2)(A/b + (1/2)(p_j + c) -p_j = (1/2)(A/b + (1/2)(p_i + c) -p_i* = p_j* = (2/3)(A/b + c)

Question 41

What are the differences in profit levels in Cournot and Bertrand modelling (2,1)

Answer 41

-In Cournot, profitability is higher in markets with more concentration of market power -In Bertrand, profitability is low even with only 2 firms -This thus makes implications for welfare and competition policy

Question 42

What are the different types of oligopoly model (2,1)

Answer 42

Oligopoly models are typically either: -Quantity setting (Cournot) -Price setting (Bertrand) -The differences in the predictions of the baseline models will narrow if we allow for product differentiation