Industrial Organization

Transcription

1 Industrial Organization Lecture Notes Sérgio O. Parreiras Fall, 2017

2 Outline Mathematical Toolbox Intermediate Microeconomic Theory Revision Perfect Competition Monopoly Oligopoly

3 Mathematical Toolbox How to use the toolbox: 1 Casually read it once so you can classify your understanding of the topics in three categories: mastered, familiar but not mastered, and never seen before. 2 Read it again but, skip the mastered topics. 3 In your second reading, make sure to have pen and paper at hand and, Mathematica open and running. 4 Work the learning-by-doing exercises using pen and paper and, verify using Mathematica if your answers are correct. 5 When you have problems with Mathematica as you will for sure refer to the crash tutorial and Mathematica s help documentation. As last resort, me your.nb file.

4 Mathematical Toolbox Matrices A matrix is just a convenient way of displaying information. A n by m matrix A is composed of n m entires. The entry A ij is displayed in the ith row and jth column. Example of a 3 by 3 matrix, A = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33.

5 Mathematical Toolbox Matrices A matrix is just a convenient way of displaying information. A n by m matrix A is composed of n m entires. The entry A ij is displayed in the ith row and jth column. Example of a 3 by 3 matrix, A = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33.

6 Mathematical Toolbox: Matrix Multiplication a 11 a a 1p a 21 a a 2p a n1 a n2... a np A : n rows p columns b 11 b b 1q b 21 b b 2q b p1 b p2... b pq B : p rows q columns c 11 c c 1q c 21 c c 2q c n1 c n2... c nq a 21 b 12 a 22 b 22 a 2p b p C = A B : n rows q columns

7 Matrix Multiplication Examples A 1 n = (p 1, p 2,..., p n ) and B n 1 = u(x 1 ) u(x 2 ). u(x n ) C 1 1 = A B = p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n )

8 Matrix Multiplication Examples A 1 n = (p 1, p 2,..., p n ) and B n 1 = u(x 1 ) u(x 2 ). u(x n ) C 1 1 = A B = p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n )

9 Matrix Multiplication Mathematica In Mathematica to enter the matrices, we type: A = ( ) A :={{1, 0, 3}, {5, 4, 7}} and B = B :={{1, 2}, {3, 4}, {7, 0}} shift+enter To multiply the matrices, type: A.B shift+enter. ( A B = ).

10 Mathematical Toolbox Partial Derivatives Often we wish to evaluate the marginal impact of ONE given variable on some function of several variables. M y f = f(x, y +, z) f(x, y, z) f(x, y, z) = lim y 0 We refer to M y as: 1 The marginal change f with respect to y. 2 The partial derivative of f wrt y. 3 The slope of f wrt y.

11 Partial Derivatives How to Compute M y f = f(x, y +, z) f(x, y, z) f(x, y, z) = lim y 0 To compute the partial derivative with respect a given variable y in the above example we use the exact same rules of derivation you learn in calculus with one variable. What about the other variables? We treat all the other variables that are not of interest (x and z in the example above) as constants. In Mathematica, we use the command D to compute partial derivatives. For example, we use D[f[x, y, z], y] to compute M y f. See the crash tutorial for additional examples.

12 Partial Derivatives How to Compute M y f = f(x, y +, z) f(x, y, z) f(x, y, z) = lim y 0 To compute the partial derivative with respect a given variable y in the above example we use the exact same rules of derivation you learn in calculus with one variable. What about the other variables? We treat all the other variables that are not of interest (x and z in the example above) as constants. In Mathematica, we use the command D to compute partial derivatives. For example, we use D[f[x, y, z], y] to compute M y f. See the crash tutorial for additional examples.

13 Partial Derivatives How to Compute M y f = f(x, y +, z) f(x, y, z) f(x, y, z) = lim y 0 To compute the partial derivative with respect a given variable y in the above example we use the exact same rules of derivation you learn in calculus with one variable. What about the other variables? We treat all the other variables that are not of interest (x and z in the example above) as constants. In Mathematica, we use the command D to compute partial derivatives. For example, we use D[f[x, y, z], y] to compute M y f. See the crash tutorial for additional examples.

14 Partial Derivatives How to Compute M y f = f(x, y +, z) f(x, y, z) f(x, y, z) = lim y 0 To compute the partial derivative with respect a given variable y in the above example we use the exact same rules of derivation you learn in calculus with one variable. What about the other variables? We treat all the other variables that are not of interest (x and z in the example above) as constants. In Mathematica, we use the command D to compute partial derivatives. For example, we use D[f[x, y, z], y] to compute M y f. See the crash tutorial for additional examples.

15 Partial Derivatives Examples

16 Partial Derivatives Learning-by-doing exercises Compute the marginal utilities MU X and MU Y for the following utility functions: 1 u(x, y) = 1 4 x y 2 u(x, y) = 1 2 x y 3 u(x, y) = 1 3 ln(x) ln(y)

17 Mathematical Tool Box, The Chain Rule: If h(x) = f(g(x)) then h (x) = f (g(x)) g (x)

18 Chain Rule Learning-by-doing exercises 1 For each of the composite functions below tell us, What are the corresponding f and g and compute h. a) h(x) = 2x b) h(x) = exp( ρ x) c) h(x) = (4 + x σ ) 1 σ 2 Use the Chain Rule to obtain the marginal utilities M X and M Y of the utility function, u(x, y) = 2 3 exp( x) 1 3 exp( y). 3 If k(x) = f(g(h(x))) is a composition of three functions, apply the chain rule twice to compute k (x). 4 Consider f(x, y) and g(x) compute the total derivative of f(x, g(x)) with respect to x using the Chain Rule and partial derivatives.

19 Mathematical Toolbox Taylor s Approximation Consider a function of one variable defined on the real line, f : R R. If f is differentiable, we write the first order Taylor approximation: f(x + h) f(x) f (x) h The approximation works well only if h is small. For a function of two variables and h = (h 1, h 2 ) we have a similar expression: f(x + h 1, y + h 2 ) f(x, y) x f(x, y) h 1 + y f(x, y) h 2

20 Mathematical Toolbox Taylor s Approximation Consider a function of one variable defined on the real line, f : R R. If f is differentiable, we write the first order Taylor approximation: f(x + h) f(x) f (x) h The approximation works well only if h is small. For a function of two variables and h = (h 1, h 2 ) we have a similar expression: f(x + h 1, y + h 2 ) f(x, y) x f(x, y) h 1 + y f(x, y) h 2

21 Mathematical Toolbox Taylor s Approximation Consider a function of one variable defined on the real line, f : R R. If f is differentiable, we write the first order Taylor approximation: f(x + h) f(x) f (x) h The approximation works well only if h is small. For a function of two variables and h = (h 1, h 2 ) we have a similar expression: f(x + h 1, y + h 2 ) f(x, y) x f(x, y) h 1 + y f(x, y) h 2

22 Mathematical Toolbox, Marginal Utility & Taylor s Approximation: U = U(x + x, y + y ) U(x, y) MU x x + MU y y

23 Taylor s Approximation Learning-by-doing exercises Using Taylor s approximation, for each of the utility functions below, compute the change in utility when the consumer moves from consuming the basket (100, 100) to consuming the basket (105, 99). 1 u(x, y) = 1 4 x y 2 u(x, y) = 1 2 x y 3 u(x, y) = 1 3 ln(x) ln(y) 4 u(x, y) = 2 3 exp( x) 1 3 exp( y) Hint: What is the value of X? What is the value of Y? What is the value of MU X and MU Y when the basket (100, 100) is consumed?

24 Mathematical Toolbox Implicit Functions Let f(x, y) be a real-valued function of two variables and let g(x) be a real-valued function of one-variable with the following property: If we set y = g(x), f remains constant as we change x. that is, f(x, g(x)) = c for all x, where c is a constant. We refer to the function g as an implicit function since g is implicitly defined by the equation f(x, g(x)) = c.

25 Mathematical Toolbox Implicit Functions Let f(x, y) be a real-valued function of two variables and let g(x) be a real-valued function of one-variable with the following property: If we set y = g(x), f remains constant as we change x. that is, f(x, g(x)) = c for all x, where c is a constant. We refer to the function g as an implicit function since g is implicitly defined by the equation f(x, g(x)) = c.

26 Mathematical Toolbox Implicit Functions Let f(x, y) be a real-valued function of two variables and let g(x) be a real-valued function of one-variable with the following property: If we set y = g(x), f remains constant as we change x. that is, f(x, g(x)) = c for all x, where c is a constant. We refer to the function g as an implicit function since g is implicitly defined by the equation f(x, g(x)) = c.

27 Implicit Functions Learning-by-doing exercises For the utility curves below, find the equation of the indifference curve that gives utility c: 1 u(x, y) = 1 4 x y 2 u(x, y) = 1 2 x y 3 u(x, y) = 1 3 ln(x) ln(y) 4 u(x, y) = 2 3 exp( x) 1 3 exp( y) Hint: set u(x, y) = c and solve for y, this is your g function...

28 Mathematical Toolbox Implicit Function Theorem If f(x, g(x)) = c for all x, where c is a constant. f(x, g(x)) Then, g (x) = x f(x, g(x)). y

29 Mathematical Tools The Implicit Function Theorem Proof: Taking the total derivative of f with respect to x, d dx f(x, g(x)) = x f(x, g(x)) x x + y f(x, g(x)) x g(x) = x f(x, g(x)) + y f(x, g(x)) g (x) Since, d f(x, g(x)) dx f(x, g(x)) = 0 g (x) = x. y f(x, g(x)) As f is constant along g(x), we also call g an iso-curve of f. Indifference curves and iso-cost curves are examples of iso-curves that you should be familiar.

30 The Implicit Function Theorem Learning-by-doing exercises For the utility curves below: a) find the marginal rate of substitution; b) in the MRS, replace y by the implicit function g you found in the previous learning-by-doing exercise and simplify the expression; c) compute g (x) for the implicit functions of the previous exercise; d) compare the results you found in items (b) and (c). 1 u(x, y) = 1 4 x y 2 u(x, y) = 1 2 x y 3 u(x, y) = 1 3 ln(x) ln(y) 4 u(x, y) = 2 3 exp( x) 1 3 exp( y)

31 Mathematical Toolbox Interior Solutions Maximizing a function of one variable defined on the real line, f : R R. Maximization Problem max f(x) x R (P) First order condition f (x) = 0 (FOC) Second order condition f (x) 0 (SOC) Any point x satisfying FOC and SOC is a candidate for an interior solution.

32 Mathematical Toolbox Interior and Corner Solutions Maximizing a function of one variable defined on an interval, f : [a, b] R. As before, Maximization Problem max f(x) b x a (P) First order condition f (x) = 0 (FOC) Second order condition f (x) 0 (SOC) Any point x satisfying FOC and SOC is a candidate for an interior solution and now, x = a is a candidate for a corner solution if f (a) 0. x = b is a candidate for a corner solution if f (b) 0.

33 Mathematical Toolbox Concavity and Convexity Consider any function f : R k R. Definition: f is concave if and only if, for all α [0, 1], and any two points x, y R k, we have f (α x + (1 α) y) α f(x) + (1 α) f(y). Another definition: We say that f is convex if f is concave.

34 Mathematical ToolBox Global Maxima Proposition. Assume f is concave and also assume that x satisfy the FOC then x is a solution to the maximization problem (i.e. x is a global maximum).

35 Consumer Choice ECON 410, Revision List of ingredients: U : R R utility function x = (x 1, x 2,..., x n) basket of goods p = (p 1, p 2,..., p n) price list I consumer s income Consumer s goal: max x subject to: x 0 p x I U(x)

36 Consumer Choice Learning by Doing Exercise Let ε > 0 be a fixed positive number. 1 How many dollars does the consumer save when he/she reduces consumption of good i by ε units? 2 How many units of good j can the consumer buy with the saved amount you found in (1)? 3 Use Taylor s approximation to estimate the change in utility as a function of MU i, MU j, ε, p i, and p j.

37 Consumer Choice Learning by Doing Exercise Let ε > 0 be a fixed positive number. 1 How many dollars does the consumer save when he/she reduces consumption of good i by ε units? 2 How many units of good j can the consumer buy with the saved amount you found in (1)? 3 Use Taylor s approximation to estimate the change in utility as a function of MU i, MU j, ε, p i, and p j.

38 Consumer Choice Learning by Doing Exercise Let ε > 0 be a fixed positive number. 1 How many dollars does the consumer save when he/she reduces consumption of good i by ε units? 2 How many units of good j can the consumer buy with the saved amount you found in (1)? 3 Use Taylor s approximation to estimate the change in utility as a function of MU i, MU j, ε, p i, and p j.

39 Consumer Choice ECON 410, Revision Assume the the optimal basket x has positive amounts of each good. Then it must be the case that: MU i p i for any two goods i and j, and = MU j p j p x = I.

40 410 Review Work in groups of two or three to answer the following. Assume two goods such that MU 1 = 3 and MU 2 = 10 for Anna. 1 If the utility of Anna s endowment is 2 but she gives up 1 unit of good 1 in exchange for 1/2 units of good 2, what is her utility after the trade? 2 If p 1 = 10 and p 2 = 20, should Anna buy more or less of good 1? and of good 2? 3 If p 1 = 10 and p 2 = 40, should Anna buy more or less of good 1? and of good 2? 4 If p 1 = 1 what is the maximum value of p 2 so that Anna would be willing to not reduce her consumption of good 2?

41 Consumer Choice Learning By Doing Exercises 1 Let x 1 be the (monthly) change in consumption of good 1 and x 2 the change in consumption of good 1. The changes for the other goods is zero. If prices and (monthly) income do not change and the consumer always spend the entirety of the income every month, show or argue that p 1 x 1 = p 2 x 2. 2 Using the Taylor s formula for estimating the change in utility, U = MU 1 x 1 + MU 2 x 2. Show that if MU 1 /p 1 > MU 2 /p 2 and x 1 > 0 with p 1 x 1 = p 2 x 2 then U > 0. Hint: First, try to show the result using numerical values for MU 1, MU 2, p 1, and p 2 and set x 1 = +1. Next, do the same but replace the numerical value for the economic variables.

42 Lecture 1 What IO is about? Trusts and cartels became pervasive in the US economy in the XIX century, watch this clip. But with the passage of the Sherman Antitrust Act (1890) and the Clayton Antitrust Act (1904), read Chapter 1 of section 15 of the U.S. code. However, it is not easy to answer if a dominant firm is exercising monopoly power and restricting competition. Listen to this podcast on Google. A major topic of Industrial Organization (IO) is market structure. That is, characterizing and identifying different market structures: competitive markets, monopoly, oligopoly,

43 Perfect Competition Definition Competitive Behavior A buyer or a seller is said to be competitive if he/she believes that the market price is given and that his/her actions do not influence the market price.

44 Perfect Competition Non-Increasing Returns to Scale Two firms: i = 1, 2 with costs TC i (q i ) = c i q i. Linear inverse demand: p = a b Q = a b (q 1 + q 2 ). Competitive Equilibrium The triplet (p, q 1, q 2 ) is a competitive equilibrium if: 1 give p, q i solves 2 p = a b (q 1 + q 2 ) max q i π i (q i ) = p q i TC i (q i )

45 Individual Supply Functions Constant Returns to Scale The supply function of firm i = 1, 2 is: + if p > c i, q i (p) = [0, + ] if p = c i, 0 if p < c i.

46 The Competitive Equilibrium If a > c 2 c 1, the unique competitive equilibrium price is p = c 1 and: 1 if c 1 < c 2 then q 2 = 0 and q 1 = (a c 1 )/b; 2 if c 1 = c 2 then Q = q 1 + q 2 = (a c 1 )/b and q 1, q 2 0.

47 Returns to Scale Definition Assume we increase all inputs by the same ratio, λ > 0. So that total costs increase by λ. If production increases by: 1 a ratio of λ 2 more than a ratio of λ 3 less than a ration of λ We say that the firm s technology has : 1 constant returns to scale: Q ATC(Q) = TC(Q)/Q stays constant 2 increasing returns to scale: Q ATC(Q) = TC(Q)/Q 3 decreasing returns to scale: Q ATC(Q) = TC(Q)/Q

48 Competitive Equilibrium Increasing Returns to Scale TC(q) = F + c q. Suppose that a > c, then if the firms technology exhibit increasing returns to scale (decreasing average cost), a competitive equilibrium does not exist.

49 Competitive Equilibrium Increasing Returns to Scale TC(q) = F + c q. Suppose that a > c, then if the firms technology exhibit increasing returns to scale (decreasing average cost), a competitive equilibrium does not exist.

50 Social Welfare Definition Given a market price p and N firms in the industry, we define the social welfare by: N W(p) = CS(p) + π i (p) i=1 That is, the social welfare is the consumer surplus (area under the demand curve) plus the firm s profits.

51 Social Welfare Recall (ECON 410) that the consumer surplus, CS(p), at a market price is p 0 measures how much consumers would be willing to pay for the quantity that they demand at this price minus the actual amount they pay. In turn, the willingness to pay to consume Q(p 0 ) units is measured by the area under the demand curve. p p = a b Q p 0 = a b Q 0 Q 0 Q

52 Social Welfare Recall (ECON 410) that the consumer surplus, CS(p), at a market price is p 0 measures how much consumers would be willing to pay for the quantity that they demand at this price minus the actual amount they pay. In turn, the willingness to pay to consume Q(p 0 ) units is measured by the area under the demand curve. p p = a b Q p 0 = a b Q 0 Q 0 Q

53 Social Welfare Recall (ECON 410) that the consumer surplus, CS(p), at a market price is p 0 measures how much consumers would be willing to pay for the quantity that they demand at this price minus the actual amount they pay. In turn, the willingness to pay to consume Q(p 0 ) units is measured by the area under the demand curve. p p = a b Q p 0 = a b Q 0 Q 0 Q

54 Social Welfare Recall (ECON 410) that the consumer surplus, CS(p), at a market price is p 0 measures how much consumers would be willing to pay for the quantity that they demand at this price minus the actual amount they pay. In turn, the willingness to pay to consume Q(p 0 ) units is measured by the area under the demand curve. p p = a b Q p 0 = a b Q 0 Q 0 Q

55 Social Welfare continued... The social welfare is: W(p) = CS(p) + N π i (p) That os, the social welfare is the consumer surplus (area under the demand curve minus payments to the firms) plus the firm s profits. Moreover, since the sum of the firms profits is p Q N i=1 TC i(q i ), where Q = N i=1, when we add the consumer surplus (which subtracts the payments to the firms) to the profits (which includes the payments to the firms), the term p Q (the payment to the firms/the firms revenue) is cancelled. Thus, we can also write i=1 W(p) = area under the demand curve at Q(p) N TC i (q i ). i=1 } {{ } total cost

56 Social Welfare continued... The social welfare is: W(p) = CS(p) + N π i (p) That os, the social welfare is the consumer surplus (area under the demand curve minus payments to the firms) plus the firm s profits. Moreover, since the sum of the firms profits is p Q N i=1 TC i(q i ), where Q = N i=1, when we add the consumer surplus (which subtracts the payments to the firms) to the profits (which includes the payments to the firms), the term p Q (the payment to the firms/the firms revenue) is cancelled. Thus, we can also write i=1 W(p) = area under the demand curve at Q(p) N TC i (q i ). i=1 } {{ } total cost

57 Social Welfare An Example Assume a linear inverse demand, p = a b Q and linear costs TC i (q i ) = c q i. Thus, the total costs are simply N i=1 TC i(q i ) = c N i=1 q i = c Q. Since fixed costs are zero (in this example), total cost is just the area under the marginal cost curve. The are under the demand curve is: (a p 0) Q 0 2 +p 0 Q 0 = (a + p 0 )Q 0 /2 = (2a bq 0 )Q 0 /2. Total costs are: c Q 0 p a p = a b Q p 0 = a b Q 0 c Q 0 Q

58 Social Welfare An Example Assume a linear inverse demand, p = a b Q and linear costs TC i (q i ) = c q i. Thus, the total costs are simply N i=1 TC i(q i ) = c N i=1 q i = c Q. Since fixed costs are zero (in this example), total cost is just the area under the marginal cost curve. The are under the demand curve is: (a p 0) Q 0 2 +p 0 Q 0 = (a + p 0 )Q 0 /2 = (2a bq 0 )Q 0 /2. Total costs are: c Q 0 p a p = a b Q p 0 = a b Q 0 c Q 0 Q

59 Social Welfare An Example Assume a linear inverse demand, p = a b Q and linear costs TC i (q i ) = c q i. Thus, the total costs are simply N i=1 TC i(q i ) = c N i=1 q i = c Q. Since fixed costs are zero (in this example), total cost is just the area under the marginal cost curve. The are under the demand curve is: (a p 0) Q 0 2 +p 0 Q 0 = (a + p 0 )Q 0 /2 = (2a bq 0 )Q 0 /2. Total costs are: c Q 0 p a p = a b Q p 0 = a b Q 0 c W(p 0 ) Q 0 Q

60 Social Welfare The competitive equilibrium maximizes the social welfare In the example: W =(2a bq)q/2 cq W =(a bq) c = p c = 0 p = c Q

61 Perfect Competition Discussion 1 In a competitive market, can we have few firms? or firms with different technologies? 2 In a competitive equilibrium, how the profits of the firms with the better technology (i.e., lower marginal and average costs) differ from the profits of the firms with the worse technology? 3 What are the long-run incentives that firms in a competitive market face? 4 Listen to the podcast How stuff gets cheap?

62 Monopoly Unlike competitive firms, the monopolist firm takes into account its actions affects the market price. Notice that: max π i(q) = p(q) Q TC(Q) Q Q π = p (Q) Q + p(q) MC(Q) = 0 }{{} MR MC(Q ) = p(q ) + p (Q ) (Q ) < p. The monopoly equilibrium price is higher than the marginal cost.

63 Monopoly Unlike competitive firms, the monopolist firm takes into account its actions affects the market price. Notice that: max π i(q) = p(q) Q TC(Q) Q Q π = p (Q) Q + p(q) MC(Q) = 0 }{{} MR MC(Q ) = p(q ) + p (Q ) (Q ) < p. The monopoly equilibrium price is higher than the marginal cost.

64 Monopoly Unlike competitive firms, the monopolist firm takes into account its actions affects the market price. Notice that: max π i(q) = p(q) Q TC(Q) Q Q π = p (Q) Q + p(q) MC(Q) = 0 }{{} MR MC(Q ) = p(q ) + p (Q ) (Q ) < p. The monopoly equilibrium price is higher than the marginal cost.

65 Monopoly with two markets max π i (Q) = p 1 (Q 1 ) Q 1 + p 2 (Q 2 ) Q 2 TC(Q 1 + Q 2 ) Q 1,Q 2 π = p Q 1(Q) Q 1 + p 1 (Q 1 ) MC(Q 1 + Q 2 ) = 0, 1 }{{} MR 1 π = p Q 2(Q) Q 2 + p 2 (Q 2 ) MC(Q 1 + Q 2 ) = 0 2 }{{} MR 2

66 Monopoly with two markets max π i (Q) = p 1 (Q 1 ) Q 1 + p 2 (Q 2 ) Q 2 TC(Q 1 + Q 2 ) Q 1,Q 2 π = p Q 1(Q) Q 1 + p 1 (Q 1 ) MC(Q 1 + Q 2 ) = 0, 1 }{{} MR 1 π = p Q 2(Q) Q 2 + p 2 (Q 2 ) MC(Q 1 + Q 2 ) = 0 2 }{{} MR 2

67 Monopoly Podcasts by Planet Money, NPR 1 What A 16th Century Guild Teaches Us About Competition 2 Mavericks, Monopolies And Beer 3 Why It s Illegal To Braid Hair Without A License

68 Cartel There are N firms in the industry which form a cartel whose aim is to maximize their joint profits. We assume a linear inverse demand, p = a b Q and quadratic costs, TC i (q i ) = F + c q i, in which F and c are positive constants (the fixed cost and marginal cost).

69 Cartel Equilibrium The cartel s profit is: π = (a b(q q N )) (q q N ) TC 1 (q 1 ) +... TC N (q N ) }{{}}{{}}{{} } p {{ Q } Sum of total costs Revenue To find the quantities that max. its profits we must solve the first-order conditions: π = bq + (a bq) MC i (q i ) = 0, i = 1,..., N. q i }{{} MR Since the marginal revenue is the same for all firms, in eq., their marginal cost, which equals the MR, must also be equal. But then, the individuals quantities must also be equal! So Q = N q where the value of q solves: bnq + (a bnq) c q = 0.

70 A crash review of Game Theory Consider N firms indexed by i = 1, 2,... N, and let π i denote the profit of firm i. Let s i denote a strategy of firm i (s i could be the quantity that firm i is producing, or the price it charges, or even the location choice...). Fix the strategy that the others are choosing s i = (s 1,..., s i 1, s i+1,..., s N ). We define the best response function of firm i to s i, and we write BR i (s i ) as the strategy or strategies that maximize the payoff of firm i when the others follow the strategy s i.

71 A crash review of Game Theory Notice that the best response is a function: its value depend on the choice of the others firms. Of course, when choices are simultaneous, the firm does not observe the strategy choices of the rival. So, you should think of the best response as the plan the firm intended to carry if the firm believes that the others will play a given strategy s i. We read the subscript i as not i. That is, s i is a list of strategies (one for each firm) of the firms that are distinct from firm i.

72 A crash review of Game Theory Nash equilibrium Definition An equilibrium (a Nash eq.) is a list of strategies (one for each firm) such that each firm is best responding to the choice of the others.

73 Cournot Oligopoly Competition in Quantities We consider an industry that produces an homogeneous good whose market demand is p = a b Q. There are N firm and each firm has a total cost function TC i (q i ) (notice that different firms might have different cost functions!).

74 Cournot Oligopoly Competition in Quantities We assume non-increasing returns to scale so that the marginal cost is increasing with quantity. In this case, to find the best-response function of firm i, we set the marginal profit equal to zero and solve for q i : π i =(a b (q q N )q i TC i (q i ) q i π i =a b (q q i 1 + 2q i + q i+1 MC i (q i ) = 0 BR i (q i ) solves the above equation

75 Finding a best response Consider the case of three firms with TC i (q i ) = c q i for i = 1, 2, 3 (in this case all firms have the same cost function). To find the best-response function of firm 1: π 1 =(a b (q 1 + q 2 + q 3 ) q 1 c q 1 π 1 =a b (2q 1 + q 2 + q 3 ) c = 0 q 1 q 1 = BR 1 (q 2, q 3 ) = a c bq 2 bq 3 2b solves the above equation

76 Finding a Nash eq. Continuing the example above, to find the Nash eq., we must solve the system of equations below, which is equivalent to every firm be best responding: q 1 = a c bq 2 bq 3, 2b q 2 = a c bq 1 bq 3, 2b q 3 = a c bq 1 bq 2 2b Exercise: Check that q 1 = q 2 = q 3 = a c 4b is a Nash eq.

77 Bertrand Oligopoly Competition in prices In the Cournot games, the best response was single valued but in general the best response is set-valued. We may have several (or even infinite) number of best responses to s i. It is even possible the best response to s i may fail to exist! This is the case, when firms compete in prices.

78 Bertrand Competition Duopoly There are no fixed costs and the marginal cost is constant, c. The market demand is P = a bq but firms compete in prices. The firm charging the lowest price captures the entire market. If both firms charge the same, they split the market. Profits are: ( ) a p1 (p 1 c) if p 1 < p 2, b ( π 1 (p 1, p 2 ) = a p1 ) b (p 1 c) 2 if p 1 = p 2 and 0 if p 1 > p 2. For player 2, π 2 (p 1, p 2 ) = π 1 (p 2, p 1 ).

79 Bertrand Competition Duopoly There are no fixed costs and the marginal cost is constant, c. The market demand is P = a bq but firms compete in prices. The firm charging the lowest price captures the entire market. If both firms charge the same, they split the market. Profits are: ( ) a p1 (p 1 c) if p 1 < p 2, b ( π 1 (p 1, p 2 ) = a p1 ) b (p 1 c) 2 if p 1 = p 2 and 0 if p 1 > p 2. For player 2, π 2 (p 1, p 2 ) = π 1 (p 2, p 1 ).

80 Bertrand Competition Duopoly There are no fixed costs and the marginal cost is constant, c. The market demand is P = a bq but firms compete in prices. The firm charging the lowest price captures the entire market. If both firms charge the same, they split the market. Profits are: ( ) a p1 (p 1 c) if p 1 < p 2, b ( π 1 (p 1, p 2 ) = a p1 ) b (p 1 c) 2 if p 1 = p 2 and 0 if p 1 > p 2. For player 2, π 2 (p 1, p 2 ) = π 1 (p 2, p 1 ).

81 Bertrand Oligopoly Equilibrium ( ) a p1 Suppose that p 2 > p 1 then π 1 = (p 1 c) and thus, b π 1 = a + c 2p 1 p 1 b > 0. That is, as long as firm 2 is charging a higher price and (a + c)/2 > p 1, firm 1 has incentives to raise its own price up to the point in which p 1 = (a + c)/2. But if p 1 > c, firm 2 has an incentive to charge a lower price and capture the entire market. Thus, in an eq. it must be that p 1 = p 2 c. Exercise: verify that p 1 = p 2 = c is a Nash equilibrium.

82 Oligopoly with differentiated products textbook, : pp We consider a demand system for two products given by: p 1 = α βq 1 γq 2, p 2 = α βq 2 γq 1, We define δ = γ 2 /β 2 and refer to it as a measure of product differentiation. The smaller δ is, the more differentiated are the products. The larger δ is, the more homogeneous are the products. Notice that since we assume γ < β, the measure δ lies in the interval (0, 1).

83 Oligopoly with differentiated products textbook, : pp Learning by doing exercise, assume that firms compete choosing quantities and face zero costs of production and: 1 Write the profits of the firms as a function of the quantities q 1, q 2 and the demand parameters α, β and γ. 2 Equate the marginal profit of each firm to zero and solve for the best-response functions. 3 Find the Nash equilibrium. 4 Show that profits decrease as γ > 0 increases. 5 Explain why product differentiation increases the equilibrium profits. Solution (Mathematica.nb file)

84 Oligopoly with Differentiated (by location) products textbook, , pp First, firms choose locations A and B on the interval [0, L]. Second, firms chose their prices, p A and p B. Consumers are uniformly distributed on the interval [0, L]. Assume that A B. The utility of a consumer who is located at point x is: p A τ A x if buys from firm A u i = p B τ B x if buys from firm B. Here, τ > 0 measures the consumer s transportation costs.

85 Oligopoly with Differentiated (by location) products textbook, , pp Assume A B and that there is a consumer located at A x (p A, p B ) B who is indifferent between buying from A or B. Then, p A τ(x A) = p B τ(b x ) x = (A + B)/2 + (p B p A )/(2τ) Learning by doing exercise, assume that firms compete choosing quantities and face zero costs of production and: 1 In the above case, explain why the demand for A is q A = x and the demand for B is q B = L x. 2 Write the profit of firm A. 3 Compute the best-response (price) function of firm A to p B. 4 Do the same for firm B and then, solve for the eq. prices. Solution (Mathematica.nb file)

86 Oligopoly with Differentiated (by location) products textbook, , pp Once we obtain the equilibrium prices p A and p B which should be functions of the locations. We analyze the choice of location. Plug the eq. prices you found previously and then compute A π A and show it is postive. That is, firm A wants to move closer to the location of firm B...

87 Mergers Measures of Industry Concentration Label firms so that q 1 q 2 q 3 etc... Definitions: s i = 100 q i Q is the market share of firm i. I 4 = s 1 + s 2 + s 3 + s 4 is the market share of the top four firms. I HH = N i=1 s2 i is the index of market concentration, note that: 0 < I HH

88 Mergers Horizontal Mergers Merger increases joint profits but: Firms outside the merger may profit from the merger. The merger reduces consumers surplus but the social surplus might increase (if the merger has strong cost saving efficiency properties).

89 Mergers Vertical Mergers, textbook, 8.2.2: pp

90 Research and Development A basic model: V value of innovation α probability of discovering a successful innovation. I research and development costs (R&D investment). With one firm, invest if and only the expected benefits are larger than the costs: αv I 0.

91 Research and Development With two firms, i=1,2, if both invest, profits are: V I only firm i is successful. π i = V/2 I both firms are successful. 0 firm i fails.

92 Research and Development With two firms, i=1,2: 1 If both invest, expected profits are: π 1 = π 2 = α 2 (V/2) + (1 α)αv I. 2 If firm 1 invests but firm 2 does not, expected profits are: π 1 = αv I and π 2 = 0.

93 Research and Development Depending on V, α and I there are several possible cases to consider: 1 If αv I < 0, no firms invest. 2 If α 2 (V/2) + (1 α)αv I > 0, both firms invest. 3 If α 2 (V/2) + (1 α)αv I < 0 < αv I, one firm invests while the other does not.

94 Research and Development Is investment efficient? The social surplus is the sum of profits (for simplicity we assume zero consumer surplus): if both firms invest, W = (1 (1 α) 2 )V 2I; and if only one firm invests, W = αv I. 1 If α 2 (V/2) + (1 α)αv I > 0 but (1 (1 α) 2 )V 2I < αv I there is over investment. 2 If α 2 (V/2) + (1 α)αv I < 0 < αv I but (1 (1 α) 2 )V 2I > αv I there is under investment.