An upstream monopolist (the producer) sells q units of intermediate goods to a ...
1) Suppose that the upstream and the downstream firms are integrated in a ...
3. VERTICAL RESTRAINTS, COLLUSION, VERTICAL DIFFERENTIATION
Exercise 3.1 An upstream monopolist (the producer) sells q units of intermediate goods to a downstream monopolist (the retailer). The producer’s costs are given by C(q)=q2/2, while the retailer’s are null. Consumers’ demand equals q=1-p where q denotes total quantity, and p is the price of the final good. 1) Suppose that the upstream and the downstream firms are integrated in a unique company: compute the aggregate profit, consumers’ surplus and social welfare. 2) Give a graphical representation of the results found at 1). 3) Consider, now, the separation case in which the two firms are independent. The producer charges the retailer a linear tariff T(q)=pwq. Compute the overall profit, consumers’ surplus and social welfare. Discuss. 4) Describe briefly which vertical restraints may mitigate the double marginalization problem, and eventually replicate the vertical integration equilibrium. Exercise 3.2* Consider a situation in which there is an upstream firm U (the manufacturer), with marginal costs equal to c, and two downstream firms D1 and D2, which have to decide on the level of effort ei they want to provide to sell U’s product. The downstream firms then compete in prices with marginal costs equal to the wholesale price w they pay to U. Assume that efforts increase the perceived quality of the good u according to the following specification: u = v + e, where the perceived quality (above the basic level v) depends on the sum of individual efforts, i.e. e = e1+ e2. This specification implies that there is a complete spill over of efforts between the two firms, namely firm i’s effort increases the perceived quality of the good both if it is sold by i, and if it is sold by j, with i=1, 2 and j=1, 2. Total costs of the downstream firms are given by C(q, ei) = wq + ei2. Consumers’ demand is q = (v+e) – p. Notice that is it an increasing function of the aggregate effort. The timing of the game is as follows: given the market structure defined above, U decides over the contract that specifies the compensation from Di to U; then D1 and D2 simultaneously choose their levels of effort; then D1 and D2 simultaneously set their prices. 1) Assume that U selects a linear contract Ti = wqi. Compute the equilibrium prices in the downstream market and the effort levels of D1 and D2. What is the wholesale price w chosen by U? 2) Compute the optimal price p and the effort levels in case of vertical integration among U, D1 and D2. 3) Go back to the original scenario with three independent firms. Assume that each retailer receives from U an exclusive territory; for simplicity you can further assume that each retailer Di can sell to half of the population, thus facing a downward demand qi=[(v+e1+e2)-pi]/2 (Notice that the spill over effect is still at play in the two sub-markets). Moreover, assume that U employs non-linear contracts of the type Ti=wqi+Fi. The timing is as follows: at time 1 U chooses the contract; at time 2 the retailers decide over the prices and efforts. Compute the equilibrium prices and efforts and compare with 1) and 2). Discuss. Exercise 3.3* A monopolist (P) produces final goods bearing marginal (and average) costs equal to c0 i=1..n. Assuming that the above game is infinitely repeated, and that firms make use of trigger strategies with Nash punishment, compute the value of the discount factor that prevents deviation. Exercise 3.8 In market Alpha, two firms compete à la Cournot, facing a demand function of the type p=1-q where p is the price and q denotes total quantity; in market Beta, two firms compete à la Bertrand, facing a demand function of the type p=1-q where p is the price and q denotes total quantity; both firms entail zero production costs. Assuming that the constituent game is infinitely repeated in each market, and that firms make use of trigger strategies with Nash punishment, find the discount factors δalfa and δbeta that prevent deviation in market Alpha and Beta. Exercise 3.9 Consider n firms competing in prices with marginal (and average) costs constant and equal to c. The demand function is given by q=D(p) and the discount factor is δ. Assume that the constituent game is infinitely repeated and that firms make use of trigger strategies with Nash punishment. 1) Compute the present value of firm i’s profit in case of collusion. 2) Compute the present value of firm i’s profit in case of deviation. 3) What is the threshold for the discount factor in order to prevent deviation? Exercise 3.10* Consider a duopoly with homogeneous product and marginal (average) costs constant and equal to c; assume an infinite horizon and call δ the discount factor. In every period firms simultaneously choose their price and they observe the competitor’s price with one period lag. Market demand is decreasing and it is given by q=D(p). If firms set the same price, they share total demand with weight λ for firm 1 and 1-λ for firm 2 with λ belonging to the interval (1/2, 1) (To solve the exercise you do not need to explain why firms have different shares of total demand, even if they set the same price). If one firm chooses a price lower than the other one, all consumers buy from the cheapest seller. The duopolists make use of trigger strategies with Nash punishment. 1) Derive the condition according to which firm1 and firm 2 do not find it profitable to deviate from the collusive price p^. 2) Derive the condition according to which there is one equilibrium in the repeated game that implements the collusive price p^. Show that such the higher λ the more stringent this condition. 3) Show that previous results hold for each collusive price p^ belonging to the interval (c, pM), where pM denotes the monopoly price. Exercise 3.11 Consider a market in which three firms compete in prices, with zero production costs. The demand function is p=1-q, where p is the price and q denotes total quantity. 1) Assume that the above game is infinitely repeated and that firms implement trigger strategies with Nash punishment. Do you think that a discount factor equal to 13/21 prevents defection? Why? 2) If firms compete in quantity, what is your answer to 1)?
3
Exercise 3.12* Two firms work on two independent markets. In the first market exchanges occur every period (t=0, 1, 2) while in the second one, they occur every two periods (t=0, 2, 4…). Firms tend to collude on both markets, making use of trigger strategies with Nash punishment. 1) Assume Bertrand competition and independent trigger strategies on each market; if collusion takes place, show that a discount factor equal to 0.6 guarantees collusion on the first market, but not on the second one. 2) Consider the case in which the trigger strategies employed by the two firms make punishment occur on both markets even if deviation takes place only in one of them. Show that a discount factor equal to 0.6 guarantees collusion on both markets. Exercise 3.13 Consider a market in which two firms compete in prices, and they offer a vertical differentiated product (i.e. by quality) xi (i=1, 2). In particular assume x1=2 and x2=4. The utility of consumer t∈[0, 1] if he/she buys variety xi at price pi is given by ut= txi – pi . 1) Offer a graphical representation of the above setting. 2) Derive the equilibrium demands for the two varieties and the optimal prices, given zero production costs for both firms. Exercise 3.14 Consider a market in which two firms compete in prices, and they offer a vertical differentiated product (i.e. by quality) xi (i=1, 2). In particular assume x1=1 and x2=2 and call ut= txi – pi the utilità of consumer t ∈[0, 1] if he/she buys variety xi at price pi. 1) Offer a graphical representation of the above setting. 2) Assuming zero production costs, derive the equilibrium prices and demands. 3) Assume instead that firm’s 2 costs equal TC2=q2. Compute the new equilibrium prices and demands. 4) Compute the equilibrium profits at 2) and 3). Discuss. Exercise 3.15* Consider a market in which two firms offer a vertical differentiated product xi (i=1, 2); assume x2>x1 and zero production costs; ut= txi – pi denotes the utility of consumer with income t ∈[0, 1] and it depends on the bought variety xi and the price pi. 1) Offer a graphical representation of the above setting. 2) Assume that firms compete à la Bertrand. Derive the equilibrium demands and prices. 3) Assume that firms compete à la Cournot. Derive the equilibrium demands and prices. 4) Compare firms’ profits in 1) and 2). Discuss. Exercise 3.16 Consider a market in which n firms compete à la Cournot, with zero marginal (and average) costs. Market demandi s p = a –q, where p is the price and q denotes total quantity. Suppose that the game is infinitely repeated and that firms make use of trigger strategies with Nash punishment. Found the value for discount factor that prevents deviation.
4
3. VERTICAL RESTRAINTS, COLLUSION, VERTICAL DIFFERENTIATION – solutions
Exercise 3.1 1) The integrated firm solves the following maximization problem: 1− p ⎞ ⎛ max π = ( p − AC )q = ⎜ p − ⎟(1 − p ) 2 ⎠ p ⎝
This yields: pI =
2 3
qI =
1 3
πI =
1 6
CS I = WI =
1 18
2 9
2) The graphical representation is as follows: p
MC
1
AC
E
2/3
π
1/3
1
q
Firm’s profit corresponds to the dotted rectangle; consumers’ surplus is the above triangle, while social welfare is the trapezium sum of the two areas. 3) If firms are independent, we need to solve two distinct maximization problems, starting with the retailer, and moving backwards. The retailer max π = ( p − AC )q = ( p − p w )(1 − p ) p
5
From FOC we get: p* =
1 + pw 2
q* =
1 − pw 2
1 − pw ⎞ 1 − pw ⎛ The producer max π = ( p w − AC )q = ⎜ p w − ⎟ 4 ⎠ 2 pw ⎝ From FOC we get: 3 5
p wNI =
Therefore: q NI =
1 5
p NI =
4 5
The aggregate profit, consumers’ surplus and social welfare are given by:
π NI =
7 1 < 50 6
CS NI = W NI =
1 1 < 50 18 4 2 < 25 9
Notice that they are all lower than before (i.e. integration case). 4) The franchising fee and the fixed price are a possible solution. Exercise 3.2* 1) First of all, focus carefully on the choice variables of the different players at the different stages of the game: Stage 1: the upstream monopolist U chooses w Stage 2: the downstream retailers D1 and D2 choose e1 and e2 Stage 3:the downstream retailers D1 and D2 choose p1 and p2 The game is solved by backward induction.
Stage 3: due to Bertrand competition, the retailers choose a price equal to their marginal cost, so p1 = p2 = w. 6
Stage 2: the retailer D1 max π 1 = p1 q1 − C (q, e1 ) = ( w − w) e1
(v + e1 + e 2 − w) − e12 2
This yields e1NI = 0 the retailer D2 max π 2 = p 2 q 2 − C ( q, e 2 ) = ( w − w) e2
(v + e1 + e 2 − w) − e 22 2
NI This yields e2 = 0
Stage 1: the monopolist U max π = ( w − c)q = ( w − c)(v − w) w
v+c 2 v+c NI = p2 = 2
This yields w NI = therefore p1NI
2) In this case, firm U chooses p and e directly, without intermediation. It thus solves the following maximization problem: max π = ( p − c)q − C (q, e1 ) − C (q, e 2 ) = ( p − c)(v + e1 + e 2 − p ) − e12 − e 22 p , e1 , e2
First order conditions, with respect to the three choice variables, are as follows: FOCp: v + e1 + e2 − p − p + c = 0 This yields: p* =
v + c + e1 + e 2 2
FOCe1: p − c − 2e1 = 0 This yields: e1* =
p−c 2
FOCe2: p − c − 2e 2 = 0 This yields: e2* =
p−c 2
In the end, we find: pI = v e1I = e2I =
v−c 2
3) The game consists now of two stages: Stage 1: the upstream monopolist U chooses w and F Stage 2: the downstream retailers D1 and D2 choose pi and ei. The game is solved by backward induction: 7
v + e1 + e 2 − p1 − e12 − F1 2 v + e1 + e2 + w From FOCp1 we get p1 = 2 p1 − w From FOCe1 we get e1 = 4
Stage 2: the retailer D1 max π 1 = p1 q1 − C (q, e1 ) − F1 = ( p1 − w) p1 , e1
v + e1 + e2 − p1 − e22 − F2 2 v + e1 + e2 + w From FOCp2 we get p 2 = 2 p −w From FOCe2 we get e2 = 2 4
The retailer D2 max π 2 = p 2 q 2 − C (q, e 2 ) − F2 = ( p 2 − w) p 2 , e2
Therefore: p1* = p 2* =
e1* = e2* =
2v + w 3
v−w 6 ⎛ ⎝
Stage 1: the monopolist U max π = ( w − c)q + F1 + F2 = ( w − c)⎜ v + w, F
From FOCw we get w E =
v − w 2v + w ⎞ − ⎟ + F1 + F2 3 3 ⎠
v+c 2
Substituing wE in the expressions for prices and efforts, we find the following results: p1E = p 2E =
e1E = e 2E =
5v + c 6
v−c 12
Now we are ready to compare the efforts found at 1), 2) and 3). Through exclusive territories the v−c = eiE > eiNI = 0 ), but this is not enough to restore the first 12 v−c v−c = eiE < eiI = best solution of vertical integration ( ). At 3), the effort levels are sub-optimal, 12 2
situation improves compared to 1) (
because the retailers know that the higher their ei the higher the market demand, but each of them faces only one half of it. Exercise 3.3* 1) This is equivalent to the vertical integration case, which yields the efficient outcome. Therefore:
max π = ( p − AC )q = ( p − c)(1 − p ) p
From FOCp and the demand function, we get: 8
pI =
1+ c 2
qI =
1− c 2
πI =
(1 − c) 2 4
2) If the retailers compete à la Bertrand on the downstream market, they choose a price equal to their marginal costs. As a result: p1B = p 2B = p B = w qB=1-w Given w, to choose the optimal contract, the upstream monopolist solves the following maximization problem: max π = ( p − AC )q = ( w − c)(1 − w) + 2T w,T
This yields: wB =
1+ c = pB 2
TB =0 With this contract, the monopolist is able to replicate the efficient solution, indeed: qB = qI =
1− c 2
pB = pI =
1+ c 2
πB =π I =
(1 − c) 2 4
3) If the retailers compete à la Cournot on the downstream market, they maximize their profit, with respect to the quantity. In particular: Firm D1: max π 1 = ( p − AC1 )q1 − T = (1 − q1 − q 2 − w)q1 − T q1
From FOC q1 we derive firm 1’s best reply function: 9
q1* (q 2 ) : q1 =
1 − q2 − w 2
By simmetry: q 2* (q1 ) : q 2 =
1 − q1 − w 2
Solving the system of best replies, we find: q1C = q 2C =
1− w 3
To replicate the efficient solution of 1), the monopolist needs to choose w such that qC=qI: 1− c 2 = (1 − w) 2 3 This yields: wC =
1 3 + c 4 4
Notice that the equilibrium value for w is greater than c: w =
1 3 + c > c is equivalent to c π 2** . Exercise 3.15* 1) The graphical representation is as follows:
tx2-p2
ut
tx1-p1
0 -p1
t01
t12
t
1
-p2 2) To find demands, we first identify the consumer indifferent between buying nothing and buying variety x1 (called t01) and the consumer indifferent between buying variety x1 and x2 (called t12): t 01 : 0 = t 01 x1 − p1
Therefore: t 01 =
p1 x1
t12 : t12 x1 − p1 = t12 x 2 − p 2
Therefore: t12 =
p1 − p 2 x1 − x 2
Call D1 and D2 demands for varieties 1 and 2:
25
D2 = 1 − t12 = 1 −
D1 = t12 − t 01 =
p1 − p 2 x1 − x 2
p1 x 2 − p 2 x1 x1 ( x1 − x 2 )
Now we maximize firms’ profit with respect to the prices: max π 1 = ( p1 − AC1 ) D1 = p1 p1
p1 x 2 − p 2 x1 x1 ( x1 − x 2 )
From FOC p1 we derive firm 1’s best reply function: p1* ( p 2 ) : p1 =
p 2 x1 2x2
⎛ p − p2 max π 2 = ( p 2 − AC 2 ) D2 = p 2 ⎜⎜1 − 1 p2 x1 − x 2 ⎝
⎞ ⎟⎟ ⎠
From FOC p2 we derive firm 2’s best reply function: p 2* ( p1 ) : p 2 =
p1 + x 2 − x1 2
Solving the system of best replies: p1* =
( x 2 − x1 ) x1 4 x 2 − x1
p 2* =
2 x 2 ( x 2 − x1 ) 4 x 2 − x1
Substituting back into the expressions for D1 and D2 found above, we compute the equilibrium demands: D1* =
x2 4 x 2 − x1
D2* =
2 x2 4 x 2 − x1
3) If firms compete à la Cournot, quantity is their choice variable, so we need to write down p1 and p2 as a function of q1 and q2 before maximizing firms’ profits (Notice that q1=D1, q2=D2). From D1 we get: p1 =
p 2 x1 q1 x1 ( x 2 − x1 ) − x2 x2
26
From D2 we get: p 2 = x 2 − x1 + p1 − q 2 ( x 2 − x1 )
By means of straightforward substitution: p1 = x1 (1 − q 2 − q1 ) p 2 = x 2 (1 − q 2 ) − x1 q1
Now we maximize firms’ profits with respect to the quantities: max π 1 = ( p1 − AC1 )q1 = x1 (1 − q 2 − q1 )q1 q1
From FOC q1 we derive firm 1’s best reply function: q1* (q 2 ) : q1 =
1 − q2 2
max π 2 = ( p 2 − AC 2 )q 2 = [x 2 (1 − q 2 ) − x1 q1 ]q 2 q2
From FOC q2 we derive firm 2’s best reply function: q 2* ( q1 ) : q 2 =
x 2 − x1 q1 2 x2
Solving the system of best replies: q1* =
x2 4 x 2 − x1
q 2* =
2 x 2 − x1 4 x 2 − x1
Substituting back into the expressions for p1 and p2 found above, we compute the optimal prices: x1 x 2 4 x 2 − x1 x ( 2 x 2 − x1 ) p 2* = 2 4 x 2 − x1 p1* =
4) Firms’ profits under vertical differentiation, with Bertrand and Cournot competition, are as follows: π 1* =
x1 x 2 ( x 2 − x1 ) (4 x 2 − x1 ) 2
27
π 2* =
4 x 22 ( x 2 − x1 ) ( 4 x 2 − x1 ) 2
π 1** =
x 22 x1 (4 x 2 − x1 ) 2
π 2** =
x 2 (2 x 2 − x1 ) 2 (4 x 2 − x1 ) 2
Notice that π 1* < π 1** and π 2* < π 2** . To interpret these results think that quantity competition, being milder than price competition, guarantees higher profits. Exercise 3.16 The critical value for the discount factor, in order to prevent deviation, results from the following formula:
*
δ =
π iD − π iC π iD − π iP
d where π iD , π iC e π iP denote respectively firmi’s deviation, collusion and punishment profits. To compute δ * , we first derive π iD , π iC e π iP . Collusion This is equivalent to the monopoly equilibrium. The monopolist solves the following maximization problem, with respect to the quantity: max π = ( p − AC )q = (a − q )q q
From the first order conditions and the demand function, we compute:
a 2 a pC = 2 a2 π iC = 4n qC =
Punishment This is equivalent to a Cournot game with n firms. Firm i solves the following maximization problem: n ⎛ ⎞ ⎜ max π i = ( p − AC i )qi = ⎜ a − ∑ q j ⎟⎟qi qi j =1 ⎝ ⎠
28
From FOC with respect to qi we get:
qi = a − q Given that the n firms are symmetric, on the cost side, by symmetry:
nqi=q Therefore:
qiP =
a (1 + n)
Substituting into the demand function, we find:
pP =
a 1+ n
π iP =
a2 (1 + n) 2
Deviation We compute firm i’s profit when it deviates but all the others stick on their collusive behaviour. Put another way, we need to derive firm i’s best reply to n-1 collusive quantities. Recall from above that:
qi* (q) : qi = a − q n
Given q = ∑ q j , the previous expression can be re-written as: 1
⎛ ⎞ qi = a − ⎜⎜ qi + ∑ q j ⎟⎟ i≠ j ⎝ ⎠
By means of straightforward substitution, we get:
qiD =
an + a 4n
pD =
an + a 4n
π iD =
a 2 (n + 1) 2 16n 2
In the end: 29
δ* =
π iD − π iC (n + 1) 2 = π iD − π iP (n + 1) 2 − 4n
30
