Cournot Competition Under Asymmetric Information

Transcription

1 Cournot Competition Under Asymmetric Information Imagine two businesses are trying to decide how much of a given good to produce. In the Cournot competition model we find how many goods the businesses will choose to produce if they both have the same utility functions but are unsure of what the other will make. In the Stackelberg Model we found how many goods the businesses will choose to produce if they both have the same utility functions but instead firm 1 moves first and then firm moves. What if we again have two businesses making these decisions at the same time, but now one business is unsure about the other s utility function? This is the dilemma Cournot s Competition Model under Asymmetric Information sheds light on. Imagine that two businesses are trying to decide how much of a homogeneous good to produce. As in the Cournot model we must first specify their utility functions. To start with we define q 1 and q as the quantities firm 1 and firm will choose to produce. Further, we define Q as the sum of q 1 and q. Thus, Q = q 1 + q (1) Next we must specify a price function, in this case it will be equal to demand (α) minus the quantities the firms choose to produce (Q) minus the costs of producing each item (c). Thus we can specify the price function (P(Q)) as... P (Q) = α Q c () And if we substitute equation 1 in for Q in the price function we are left with... P (Q) = α (q 1 + q ) c () P (Q) = α q 1 q c (4) Finally we must specify a cost function for the two firms. In this case we will assume that firm 1 s cost function is known and that each good costs c to produce. Thus firm 1 s cost function is... C 1 (q 1 ) = cq 1 (5) Firm s cost functions in this game will be different. Instead of having a cost function which is certain firm will have one of two possible cost functions. In one cost function firm will pay c H to produce each good and in one cost function firm will pay c to produce each good. C H (q ) = c H q (6) C L (q ) = c L q (7) If firm pays c H for each good they are paying the high cost while if firm pays c L for each good they are paying the low cost. Thus we have the following inequality... c H > c L (8) Although it is uncertain how often firm will experience each cost function it is possible to assign a probability to each possible outcome. We will say that firm pays the high cost with probability θ and pays the low cost with probability 1 θ. Thus we can specify firm s cost function... C H (q ) = c H q with probability θ (9) C L (q ) = c L q with probability (1-θ) (10) 1

2 Now that we have information on each firms cost and price functions we can now specify their utility functions. First, we will start with firm 1 s utility function (this should look familiar) U 1 (q 1 ) = q 1 (P (Q) C 1 (q 1 )) (11) If we insert the price function and cost function into the previous equation we are left with the following as firm 1 s utility function. U 1 (q 1 ) = q 1 (α Q c) (1) U 1 (q 1 ) = q 1 (α (q 1 + q ) c) (1) U 1 (q 1 ) = q 1 (α q 1 q c) (14) Since firm two does not have a set cost function (but rather two possible cost functions) we must write out two separate utility functions for firm. In the first case they pay the high cost and in the second case they pay the low cost. These two utility functions can be written out as the following. U (q c H ) = q (P (Q) C H (q )) (15) U (q c L ) = q (P (Q) C L (q )) (16) If we insert the price and cost functions into these two equations we are left with. U (q c H ) = q (α q 1 q c H ) (17) U (q c L ) = q (α q 1 q c L ) (18) Now that we know the varying utility functions we can begin to find an equilibrium for this game. Let s start with firm 1. They will look at firm and say...how much will those guys produce? They do not know which cost function firm possesses so they don t know how much they will produce exactly. But they can use the probability distribution to maximize their utility function. In other words, we must rewrite firm 1 s utility function so that it accounts for the two possible situations firm may find itself in. To do this we write their utility function as the following where q H and represent the amount of quantity firm will choose to produce given high costs and low costs. q L U 1 (q 1 ) = θ(q 1 (α q 1 q H c)) + (1 θ)(q 1 (α q 1 q L c)) (19) Now that we know firm 1 s utility function we must optimize it by taking the first order condition and setting it equal to zero. The first thing we should is some algebra to make the utility function comprehensible. U 1 (q 1 ) = θ(q 1 (α q 1 q H c)) + (1 θ)(q 1 (α q 1 q L c)) (0) U 1 (q 1 ) = q 1 θα q 1θ q 1 q H θ q 1 cθ + q 1 α q 1 q 1 q L q 1 c q 1 αθ q 1θ + q 1 q L θ + q 1 cθ (1) Now we take the first derivative of firm 1 s utility function with respect to q 1. U 1 (q 1 ) = q 1 θα q1θ q 1 q H θ q 1 cθ + q 1 α q1 q 1 q L q 1 c q 1 αθ q1θ + q 1 q L θ + q 1 cθ () U 1 (q 1 ) = θα q 1 θ q H θ cθ + α q 1 q L c αθ q 1 θ + q L θ + cθ q 1 ()

3 We must do some algebra again. At this point we will just let the q 1 θ s cancel each other out so that we can reformulate this first order condition in an easier way to handle later on. U 1 (q 1 ) q 1 = θα q H θ cθ + α q 1 q L c αθ + q L θ + cθ (4) q 1 = θα q H θ cθ + α q L c αθ + q L θ + cθ (5) q 1 = θα - qh θ - cθ + α - ql - c - αθ + q Lθ + cθ q1 = θ(α - qh - c) + (1 - θ)(α - ql - c) Now that we know what firm 1 will do in equilibrium we must figure out what firm will produce in any equilibrium (e.g. they have high costs or low costs). The first step is to figure out how much firm will produce given a high cost structure. To do this we must take the first order condition of their utility function when costs are high. First let s do some algebra. Now we need to take the first order condition. (6) (7) U (q c H ) = q (α q 1 q c H ) (8) U (q c H ) = q α q q 1 q q c H (9) U (q c H ) = q α q q 1 q q c H (0) U (q ) = α q 1 q c H q (1) Now we set the first order condition equal to zero and solve for q. U (q ) = α q 1 q c H q () 0 = α q 1 q c H () q = α q 1 c H (4) q = α - q 1 - c H Since this is the quantity firm will choose to produce given the high cost structure we will denote this critical value q H. Thus q H = α - q 1 - c H Now we must find out what firm will do when they are in a low cost structure. To do this we will follow the same process as we did when firm was under a high cost structure. To do this we must take the first order condition of their utility function when costs are low. First let s do some algebra. (5) (6) U (q c L ) = q (α q 1 q c L ) (7) U (q c L ) = q α q q 1 q q c L (8)

4 Now we need to take the first order condition. U (q c L ) = q α q q 1 q q c L (9) U (q ) = α q 1 q c L q (40) Now we set the first order condition equal to zero and solve for q. U (q ) = α q 1 q c L q (41) 0 = α q 1 q c L (4) q = α q 1 c L (4) q = α - q 1 - c L Since this is the quantity firm will choose to produce given the low cost structure we will denote this critical value q L. Thus q L = α - q 1 - c L Now we should return to firm 1 s level of production in equilibrium. Now that we know what firm will produce we can figure out what firm 1 will actually produce. To do this we will take firm 1 s q 1 and substitute in the values we now know for qh and ql. q1 = θ(α - qh - c) + (1 - θ)(α - ql - c) q1 = θ(α - α - q 1 - c H - c) + (1 - θ)(α - α - q 1 - c L - c) We can now do some algebra to simplify this equation. q 1 = θ(α - α - q 1 - c H - c) + (1 - θ)(α - α - q 1 - c L - c) q 1 = θ(α α - q 1 - c H c) + (1 θ)(α α - q 1 - c L (44) (45) (46) (47) (48) c) (49) q 1 = θ(α 1 α + 1 q c H c) + (1 θ)(α 1 α + 1 q c L c) (50) q 1 = θ( 1 α + 1 q c H c) + (1 θ)( 1 α + 1 q c L c) (51) q1 = θ 1 α + θ 1 q 1 + θ 1 C H cθ + 1 α + 1 q c L c 1 θα 1 q 1 θ c 1 L θ + cθ (5) q1 = θ 1 c H + 1 α + 1 q c 1 L c c L θ (5) q 1 = θ 1 c H + 1 α + 1 c 1 L c c L θ (54) q1 = θ 6 c H + 6 α + 6 c L c c L 6 θ (55) 4 q 1 = θc H + α + c L - c - θc L q 1 = θc H + α - c + (1-θ)c L (56) (57)

5 Now that we know what firm 1 will do we can figure out what firm will do when they are both under the high cost structure and the low cost structure. We ll start with the high cost structure. We take the quantity we know firm will produce under this incentive structure and place the value that we now know firm 1 will produce. q H q H = α - q 1 - c H = α - θc H + α - c + (1-θ)c L - c H Now we do some algebra to make this equation make some sense. (58) (59) q H = α - θc H + α - c + (1-θ)c L - c H (60) q H = α 1 (α c + c Hθ + c L c L θ) c H (61) q H = α 1 α c + 1 c Hθ + 1 c L 1 c Lθ c H (6) q H = α c + 1 c Hθ + 1 c L 1 c Lθ c H (6) q H = 6 α 6 c c Hθ c L 1 6 c Lθ 1 c H (64) q H = 6 α 4 6 c H 6 c c Hθ c L 1 6 c Lθ c H (65) q H = 6 α 4 6 c H 6 c c 1 H 6 c Hθ 1 6 c L c Lθ (66) q H = α - c H + c 1 - θ 6 (c H c L ) (67) So we now know what firm will do given a high cost environment. We should also find out what they will do given a low cost environment. To do this we take the quantity we know they will produce and substitute in the quantity we know firm 1 will produce. q L = α - q 1 - c L q L = α - θc H + α - c + (1-θ)c L - c L (68) (69) 5

6 Now we do some algebra to make this equation make some sense. q L = α - θc H + α - c + (1-θ)c L - c L q L = α θc H + α - c + (1-θ)c L (70) c L (71) q L = α 1 α + c 1 c Hθ 1 c L + 1 c Lθ c L (7) q L = α + c 1 c Hθ 4 c L + 1 c Lθ (7) q L = α 4 c L + c 1 c Hθ + 1 c Lθ (74) q L = 6 α 4 6 c L + 6 c 1 6 c Hθ c Lθ (75) q L = 1 α c L + 1 c 1 6 c Hθ c Lθ (76) q L = α - c L + c θ 6 (c H c L ) (77) We have now found the nash equilibrium of the Cournot Competition Model with Asymmetric Information! 6