1 π
π +
=
Π (2.4) First, the firms do not merge, resulting in the competition between one public and two private firms. We denote this no merger case as N,
Second, the firms this merger case is denoted as M.
2.3 A pre-merger equilibrium
To develop the analysis, a two-stage game is stated. First examine the second stage of the game in case N. The public firm chooses
q to maximize (2.2), Solving these
0 maximization problems simultaneously, we obtain the Nash equilibrium in the second stage:α α
− +
= −
g q
Na
10 4
) 4 (
0 > 0
α
−
= +
=
g
q ag q
N N10 4
2
2
1 > 0
2 2
0 (4 10 )
) 2 )(
4 (
α α π α
− +
+
= −
g g
N
a
> 0
2 2 2 2
1 (4 10 )
8 π α
π = = + −
g g
N
a
N > 0
2 2 2
) 10 4 ( 2
] ) 1 ( 4 [
α α
− +
−
= +
g g
CS
Na
> 02 2
) 10 4 ( 2
) 3 20 11 )(
8 4 (
α
α α
− +
− +
−
= +
g g g
SWN a > 0
As a result of we assume that private firm are more cost efficient than the public firm, the profit of each private firm is larger than the profit of the public firm.
Following José Méndez Naya (2011) discusses that the comprising on public firm and two private firms in a mixed oligopoly model. We assume that the decision by firms
８
to merger in market that have three types: the merger between public firm and a private firm without foreign penetration, that the merger between public firm and a private firm with foreign penetration, and that the merger involving no public.
Proposition 2.1 Under private firm is more cost efficient than the public firm, the public firm’s profit always lower than each private firm’s profit.
Next, we analyze both the public and private firm is incentives to merge in the first stage of the game.
2.4 Case 1: a post-merger involving the Public Firm and the Private Firm 1
This section moves on to the situation in which the public firm and a private firm without foreign penetration. Therefore, the profit function of firm is given by
2 1 , 0 1 , 0 2 1 , 0 1
, 0 1
, 0 1
,
0 2
)] 1 (
[ )
(
q a q q q q
C
NF =
Pq
− = − + −π ………..(2.5)
2 2 2 2 1 , 0 2
2
2NF =
Pq
−C
(q
)=[a
−(q
+q
)]q
−q
π
……….(2.6)This is assumed to be given by the sum of consumer’s surplus and producer’s surplus, and can be represented as follow:
] 1 , 0
2 [
1 , 0 1
,
0NF =
CS
+π
NF +απ
NFα
∈SW
……….(2.7)The aggregate domestic social welfare, which is assumed to be given by the sum of the consumer surplus, the producer surplus, and that can be represented as follow：
NF NF
NF
SW
V
0,1 =θπ
0,1 +(1−θ
) 0,1 ………..(2.8)９
The merged firm chooses
q
0NF,1 to maximize (2.8), the private firm choosesq
2NF to maximize (2.6). Solving these maximization problems simultaneously, we obtain the Nash equilibrium:θ θ α
θ θ α
5 ) 1 ( 8
] ) 1 ( 4 [
1 ,
0
− − −
−
−
=
a−
qNF > 0
θ θ α
θ
5 ) 1 ( 8
) 1 (
2
− − −
=
a−
qNF > 0
2 2
1 ,
0 2[8 (1 ) 5 ]
] 5 ) 1 ( 2 ][
) 1 ( 4 [
θ θ α
θ θ α θ θ π α
−
−
−
−
− +
−
−
=
a
−NF > 0
2 2 2
2 [8 (1 ) 5 ]
) 1 ( 2
θ θ α π θ
−
−
−
=
a
−NF > 0
2 2 2
1 ,
0 2[8 (1 ) 5 ]
] 2 ) 1 ( 5 [
θ θ α
θ θ α
−
−
−
−
−
=
a
−CS
NF > 0Substituting the equation (2.5), and (2.6) into the equation (2.7)
2 2
1 ,
0 2[8 (1 ) 5 ]
) 1 ](
9 ) 1 ( 4 33 [
θ θ α
θ θ θ α
−
−
−
−
−
−
=
a
−SW
NF > 0Where the superscript NF stands for the post-merger involving the public firm and the private firm without foreign penetration, and that the subscript 0,1 stands for the post-merger between the public firm and the private firm 1, and that the subscript 2 stands for the private firm 2.
] 0 5 ) 1 ( 8 [
) 1 11 ( 3
3 2
1 ,
0 =
−
−
−
+
= −
∂
∂
θ θ α
θ θ
SW
NFa
We obtain the optimal degree of privatization as：
= 0
θ∗NFThe manager of a fully nationalized firm (θ∗NF=0) maximizes social welfare.
We see that how the equity affects the degree of privatization and the proportion of domestic stockholding, the following expression.
１０
=0
∂
∂ ∗ α θ NF
For 0<θ∗NF <1 and
0 <
α< 1
. The proportion of domestic stockholding unchanged as a higher degree of privatization for public firm.Proposition 2.2 Under a merger between the public firm and the private firm 1, along
with the proportion of domestic stockholding unchanged, when the public firm increase the degree of privatization.
2.5 Case 2: a post-merger involving the Public Firm and a Private Firm 2
This section moves on to the situation in which the public firm and a private firm with foreign penetration. Therefore, the profit function of firm is given by
2 2 , 0 2 , 0 1 2 , 0 2
, 0 2
, 0 2 ,
0 2
)] 1 (
[ )
(
q a q q q q
C
F =
Pq
− = − + −π ……… (2.9)
2 1 1 1 2 , 0 1
1
1F =
Pq
−C
(q
)=[a
−(q
+q
)]q
−q
π
………..(2.10)This is assumed to be given by the sum of consumer’s surplus and producer’s surplus, and can be represented as follow:
] 1 , 0
2 [
, 0 1
2 , 0 2 ,
0F =
CS
F +π
F +απ
Fα
∈SW
………..(2.11)The aggregate domestic social welfare, which is assumed to be given by the sum of the consumer surplus, the producer surplus, and that can be represented as follow：
F F
F
SW
V
0,2 =θπ
0,2 +(1−θ
) 0,2………..(2.12) The merged firm choosesq
0F,2to maximize (2.12), the private firm choosesq
1F to maximize (2.10). Solving these maximization problems simultaneously, we obtain the１１
Nash equilibrium:
4
Substituting the equation (2.9), and (2.10) into the equation (2.11)
2
Where the superscript F stands for the post-merger involving the public Firm and the private Firm with foreign penetration. The subscript 0,2 stand for the merger between the public firm and the private firm 2, and the subscript 1 stand for the private firm 1.
The quantity of all firm may be positive, that may also be negative. We find that the quantity of all firm be positive if
2
0≤α <1 and 1 15
4 <θ < .
We then obtain the optimal degree of privatizations as：
2
To see how the properties command that affects the degree of privatization and the proportion of domestic stockholding, we have the following expression.
)]2
１２
For
0 <
θ∗F< 1
and0 <
α< 1
. An increasing proportion of domestic stockholding when a high degree of privatization for public firm.Proposition 2.3 Under a merger between the public firm and the private firm 2, along of
an increasing proportion of domestic stockholding, the public firm should increase the degree of privatization.
2.6 Case 3: a Post-Merger Involving No Public Firm
This case moves on to the situation in which the public firm and a private firm with foreign penetration. Therefore, the profit function of firm is given by
] ))
( [(
2 1,2 0 1,2 12,2
2
1+ =
a
−q
+q q
−q
=
Π
π π
………...……..(2.13)2 0 0 0 2 , 1 0
0
0 =
Pq
−C
(q
)=[a
−(q
+q
)]q
−gq
π
………(2.14)This is assumed to be given by the sum of consumer’s surplus and producer’s surplus, and can be represented as follow:
] 1 , 0
0
+ Π ∈ [
+
=
CS π α αSWNP ……….(2.15)
The aggregate domestic social welfare, which is assumed to be given by the sum of the consumer surplus, the producer surplus, and that can be represented as follow：
NP NP
NP
SW
V
1,2 =θπ
0 +(1−θ
) 1,2 ……….(2.16)The merged firm chooses
q
1NP,2 to maximize (2.13), the public firm chooses q0NP to maximize (2.16). Solving these maximization problems simultaneously, we obtain the Nash equilibrium:１３
Substituting the equation (2.13), and (2.14) into the equation (2.15)
2
Where the superscript NP stands for the post-merger involving the both private firms. The subscript 1,2 stand for the merger between the both private firms, and the subscript 0 stand for the public firm.
We then obtain the optimal degree of privatizations as：
]
To see how the properties command that affects the degree of privatization and the proportion of domestic stockholding, we have the following expression.
2 a higher degree of privatization for public firm.
１４
Proposition 2.4 Under a merger between the both private firms, along with the
proportion of domestic stockholding, the governments should increase the degree of privatization.
2.7Comparative Static’s Analysis
In the pre-merger, we first check the case that the degree of domestic stockholding.
2 0
) 4 10 (
10
−
−
= −
∂
∂
g ag q
Nα
α < 0
2 2
1
) 4 10 (
2
−
= −
∂
=∂
∂
∂
g ag q
q
N Nα α
α > 0
3 2
0
) 10 4 (
)]
1 ( 10 ) 4 ( 3 [ 2
α α α
α π
− +
− +
= −
∂
∂
g g g
N
a
> 0
3 2 2 2
1
) 10 4 (
16 α α
π α π
−
= +
∂
= ∂
∂
∂
g g
N
a
N
> 0
3 2
) 10 4 (
] ) 1 [(
6
α α
α + −
− +
= −
∂
∂
g g g a CS
N< 0
3 2
2
) 10 4 ( 2
) 3 20 11 ( 4
α α
α + −
−
= +
∂
∂
g g g
SW
Na
> 0
For a given of parameter θ, when parameter α is sufficiently high, the output of the public firm and the consumer surplus decrease as α increase, while profit of all the firms, the output of all the private firms, and social welfare increase as α increase.
Although the productivity improving merger enhances social welfare within the bounds of high α.
In the post-merger between the public firm and the private firm 1, we first check the case that the degree of domestic stockholding sets up the merger.
１５
For a given of parameter θ, the output of the merged firm, and consumer surplus decreases as α increase when the value of α is sufficiently high, that the output and profit of the private firm 2, the profit of merged firm, and social welfare increases as α increase. Although the productivity improving merger enhances social welfare within the bounds of high α.
In the post-merger between the public firm and the private firm 2, we first check the case that the degree of domestic stockholding sets up the merger.
2
１６
For a given of parameter θ, when parameter α is sufficiently high, the output of the merged firm and the consumer surplus decrease as α increase, while the output and profit of the private firm 1, the profit of the merged firm, and social welfare increase as α increase. Although the productivity improving merger enhances social welfare within the bounds of high α.
In the post-merger between no public firms, we first check the case that the degree of foreign penetration sets up the merger.
2
For a given of parameter θ, the output of the public firm, and consumer surplus decreases as α increase when the value of α is sufficiently high, that the output of the merged firm , the profit of the all firms and social welfare increases as α increase.
Although the productivity improving merger enhances social welfare within the bounds of high α.
2.8 Comparison Effects of Mergers
１７
2.8.1 Effect on total quantity
Case 1：Comparison the total quantity of the post-merger between the public firm and the private firm 1 and the pre-merger
]
Case 2：Comparison the total quantity of the post-merger between the public firm and the private firm 2 and the pre-merger
]
Case 3：Comparison the total quantity of the post-merger between the both private firms and the pre-merger
]
In the three cases, the total quantity and the consumer surplus are positive
correlation. There have three merger approach, the total quantity of the all post-merger is large than the total quantity of the pre-merger.
2.8.2 Effect on Profit
Case 1：Comparison effects of the profit of the merger between the public firm and the private firm 1 and the profit of the public firm before merger.
2
１８
Therefore,
π
0NF,1 >π for any 0Nα ∈ [ 0 , 1 )
andθ ∈ [ 0 , 1 )
The public firm and the private firm 1 decide to merge, the profit of post-merger is large than the profit of public firm. The profit increases, that which enhance the public firm incentive to merge.
Case 2：Comparison effects of the profit of the merger between the public firm and the private firm 2 and the profit of the public firm before merger.
2
The public firm and the private firm 2 decide to merge, the profit after the merger is large than the profit before the merger. The profit increases, that which enhance the public firm incentive to merge.
Case 3：Comparison effects of the profit of the merger between the public firm and the private firm 1 and the profit of the merger between the public firm and the private firm 2.
１９
The profit of the public firm and the private 1 decide to merge, that is large than the profit of the public firm and the private 2 decide to merge since the proportion of foreign penetration 1-α were taken away. We analyze the public firm and the private firm 1 incentive to merge.
Case 4：Comparison effects of the profit of the public firm after the both private firm decide to merge, that and the profit of the public firm before merger.
2 2
2 2
2 2
3 2
2
0
0 ( 10 4) [4 2 (1 ) 3 ]
))]
)) 5 18 ( 4 51 ( ) 8 19 )(
2 ( 4 ) 2 ( 12 )(
4 (
)) )) 2 9 ( 24 ( ) 2 )(
4 ( )(
) 1 ( 2 4 ( 5 ) ) 1 ( 2 4 (
25 ) ) 1 ( 2 4 ( ) 4 ( ( 2 ( 2 ) ) 1 ( 2 4 ( ) 4 10 ( [
θ θ α α
θ α α
θ α α
α α
θ α α α
α θ
θ α θ
θ α
θ θ θ α α
θ θ α α
θ
π
π − − − − +
−
− +
−
−
−
−
− +
−
− +
−
−
−
−
−
−
−
−
−
−
+
−
−
−
−
−
−
−
−
−
−
−
−
=
− g
g
g
g a
g a
a
N
NP <0
NP
π < 0 π for any 0N
α ∈ [ 0 , 1 )
andθ ∈ [ 0 , 1 )
The profit of both private firm decide to merge, that the profit of the public firm after the merger is large the profit of the public firm before the merger. Next, we analyze the public firm and another private firm incentive to merger in the first stage of the game.
Proposition 2.5 By way of merge that public firm improves cost efficiency, and the public firm and another private firm decide to merge that is profitable.
2.8.3 Welfare effects of a merger
Case 1: Comparison effects of the post-merger Involving Public Firm and the Private Firm 1
２０
We examine that the public firm decide to merge with the private firm 1. Since the public firm maximizing social welfare, it does not has an incentive to merge if
SW
0NF,1 >SWN. Let θ and 1∗ θ denote the values of parameter θ such that 1∗∗２１
Case 2: Comparison effects of the post-merger involving public firm and the private firm 2
We determine that the private firm wants to merge with the private firm 2 in the mixed oligopoly, let θ and 2∗ θ denote the values of parameter θ such that 2∗∗
SW
0F,2=SWN.２２
Case 3: Comparison effects of the post-merger involving no public firm
2
Due to the merger involving the both private firms, the social welfare of merger is deteriorating, and that the profit of public firm is negative. Most of the market share taken away by the merged firm, the public firm is the commitment for the social welfare.
Since the aims of the public firm maximizing social welfare, it has an incentive to merger with another private firm.
２３
Proposition 2.8 Due to the profit of the public firm is negative; the social welfare of the
merger between the both private firms may be deteriorating. SW
1,NP2 <SWNif and only
∗
∗
∗ < < 3
3 θ θ
θ
Case 4: Comparison welfare effects of the merger between the public firm and the private firm 1, and that the merger between the public firm and the private firm 2
2
The social welfare of the merger between the public firm and the private firm without foreign penetration is large than the social welfare of the merger between the public firm and the private firm with foreign penetration. Due to the foreign penetration 1-α taken away, the public firm choose to merger with the private firm 1, it may be profitable, and that is best strategy.
Proposition 2.9 The merger between the public firm and the private firm without foreign penetration, and is the best strategy of the public firm.
2.9 Concluding Remark
In the chapter, we see that the effects of the post-merger equilibrium. Under the
２４
equilibriums of the post-merger, the public firm decides to merge with another the private firm, the profit after the merger is large than the profit before the merger. The public firm will have incentives to merge, if the profit and social welfare increases after the merger. An increase in the degree of privatization and the proportion of domestic stockholding, the social welfare of post-merger is large than the social welfare of pre-merger, due to the market price decrease, the market quantity increase, and the consumer surplus increase. When the both private firms decide to merge, the profit of the public firm is negative. In the light, the social welfare after merger is lower than the social welfare before merger. There exists a traditional merger paradox in the mixed oligopoly model since the market price increase, the market quantity increase.
２５
CHAPTER THREE: Mixed Oligopoly, Horizontal Merger and Foreign Penetration In a Endogenous Timing Framework
3.1 Introduction
3.1.2 Pre-merger1) There have three firms decide their quantity simultaneously.
2) The public firm as a Stackelberg leader. The public firm decides their quantity in stage 1, two firms decide their quantity in stage 2.
3) The private firm 1 as Stackelberg leader. The private firm 1 decides their quantity in sage 1, the public firm and the private firm 2 decides their quantity in stage2.
4) The private firm 2 as Stackelberg leader. The private firm 2 decides their quantity in sage 1, the public firm and the private firm 1 decides their quantity in stage2.
This literature of endogenous timing in mixed oligopoly, this result from we see Naya (2009), can state that the most existing literature on mixed oligopoly, they
consider that their firms make their decisions simultaneously, such a situation is not the equilibrium. In literature of Naya (2009), for example of simultaneous and sequential equilibrium, Pal (1998), there are two subgame perfect Nash equilibrium, which the private firm decides their quantity simultaneously in stage 1, and the public firm decides their quantity in other stage.
3.1.2 Post-merger
In this chapter, there are three situations in Endogenous Timing of Stackelberg after the merge. In this case, three different alternative scenarios are considered: merger involving public firm and the private firm 1, merger involving public firm and the private firm 2, and merger involving no public firm.
２６
In this context, seven alternatives scenarios should be stated to determine
endogenous timing:1) This simultaneous movement, there have two post-merger firms decide their quantity in the same stage.
2) Merged between public firm and the private firm 1 as Stackelberg leader. In this case, the private firm’s reaction functions (firm 1) are derived in stage 2, and taking into account the said reaction functions, the merged firm optimizes in stage 1.
3) Merged between public firm and the private firm 1 as Stackelberg follower.
4) Merged between public firm and the private firm 2 as Stackelberg leader. In this case, a private firm’s reaction functions (firm 2) are derived in stage 2, and taking into account the said reaction functions, the merged firm optimizes in stage 1.
5) Merged between public firm and a private firm 2 as Stackelberg follower.
6) Merged between both private firms as Stackelberg leader. Both private firms decide their quantity in stage 1, and the public firm makes its decision in stage 2.
7) Merged between both private firms as Stackelberg follower.
Table 1: Endogenous Timing of merged firm and another firm
Leader (L) Follower (F)
Leader (L) Cournot
Case 2 Stackelberg
Follower (F)
Case 1 Stackelberg
Cournot
q
Mqi
２７
Table 1 summarizes three cases situation with simultaneous play (LL and FF), sequential play with merged firm as leadership (LF) and sequential play with other firm as leadership (FL).
In the chapter, there are two variables that play a very important role in determining the endogenous timing of the game in the stated model: the degree of foreign penetration and the degree of privatization of the mixed firm. There are six subgame perfect Nash equilibrium of the game under endogenous timing of post-merger are: the merged firm as a Stackelberg leader, and another firm as Stackelberg follower.
From the analysis of the subgame perfect Nash equilibrium in the seven scenarios, the follow result is obtained:
3.2 Merger Involving Public Firm and a Private Firm 1 of the Perfect Nash Equilibrium
In this case, we assume that the public firm and the private firm 1 without foreign penetration decide to merger in the existing market structure, that it is a duopoly composed of a mixed firm after the merger. We have two alternative scenarios: the merged firm as a Stackelberg leader and another firm as Stackelberg leader, denoted by the superscripts NF, in equilibrium, the output level of the firms, the consumer, the profits of the firms and the social welfare are, respectively:
3.2.1 The merged firm as a Stackelberg leader
θ θ α
θ α θ
9 ) 1 ( 4 31
)]
1 ( 4 ) 5 ( 3 [
1 ,
0
− − +
−
−
=
a−
qLNF
θ θ α
θ θ α
9 ) 1 ( 4 31
] 3 ) 1 )(
( 4 [
2
− − +
+
−
−
=
a+
aqFNF >0
２８
3.2.2 The merged firm as a Stackelberg follower
)
3.3 Merger Involving Public Firm and a Private Firm 2 of the Perfect Nash Equilibrium
In this case, we assume that the public firm and the private firm 2 with foreign penetration decide to merger in the existing market structure, that it is a duopoly composed of a mixed firm after the merger. We have two alternative scenarios: the merged firm as a Stackelberg leader and another firm as Stackelberg leader, denoted by the superscripts, F, in equilibrium, the output level of the firms, the consumer, the
２９
profits of the firms and the social welfare are, respectively:
3.3.1 The merged firm as a Stackelberg leader
13
3.3.2 The merged firm as a Stackelberg follower
]
3.4 A Merger Involving No Public Firm
In this case, we assume that the both private firm decide to merger in the existing market structure, that it is a duopoly composed of a mixed firm in the post-merger. We
３０
have two alternative scenarios: the merged firm as a Stackelberg leader and public firm as Stackelberg leader, denoted by the superscripts, NP, in equilibrium, the output level of the firms, the consumer, the profits of the firms and the social welfare are,
respectively:
Case1: The merged firm as a Stackelberg leader
)
Case2: The merged firm as a Stackelberg follower
θ
３１
3.5 Comparative Static’s Analysis
Results of the basic setting, we discuss that the foreign penetration of exogenous
parameters in three situations.
In the post-merger between the public firm and the private firm 2, we first check the case that the foreign penetration sets up the merger.
Case1：The merged firm as Stackelberg Leader
2 Under the merger between the public firm and a private firm with foreign penetration, the result from comparative static’s analysis, an increase in domestic stockholding of merged firm decreases the quantity of merged firm, the consumer surplus, and the social welfare, but it may increase the quantity of private firm 1, and the profit of all firms if the merged firm as Stackelberg leader.
Case2：The private firm 2 as Stackelberg Leader
３２ The result from comparative static’s analysis, along with an increasing proportion of domestic stockholding of merged firm, the decreases the quantity and profit of merged firm, the consumer surplus, and the social welfare, but it may increase the quantity and profit of the private firm 1 if the private firm 2 as Stackelberg leader.
3.6 The analysis of profit and welfare in Stackelberg equilibrium
Case 1: The perfect Nash equilibrium of merger involving the public firm and the private firm 1
Table 2: Endogenous Timing equilibrium of merger between the public firm and the private firm 1
３３
Merged firm as Leader (L) Merged firm as Follower (F)
L
SW0NF,1 = a2[332[−84−αα((11−−θθ))−−95θθ](]21−θ) 2 22 2 2
2 8(2 ) [4 (1 ) 3 ]
))]
3 20 ( 24 )(
1 ( 2
) ) 11 99 ( 200 ( ) 1 ( 2 132 )[
1 (
θ θ α θ
θ θ θ α
θ θ θ
θ θ
−
−
−
−
−
−
−
−
−
−
−
− +
−
=
a a
SWLNF
F
22
1 ,
0 2[31 4 (1 ) 9 ]
)]
13 ) 1 ( 4 15 ( 21 ) 1 (
) 4 31 ( 124 ][
4 3 ) 1 )(
[(
θ θ α
θ θ α α θ θ
α θ
θ α
+
−
−
+
−
− + +
−
− +
−
−
−
−
−
=
a a
a
SWLNF 2
2 1 ,
0 2[8 (1 ) 5 ]
) 1 ](
9 ) 1 ( 4 33 [
θ θ α
θ θ θ α
−
−
−
−
−
−
=a − SWNF
We substitute
a = 1 , θ = 0 . 6 , g = 1 . 5
draw graphics to illustrate.Figure 1：The social welfare of endogenous timing for merger between the public firm and the private
firm 1
Under the existence of endogenous time, the social welfare equilibrium
SW
LNF2 is large than the social welfare equilibriumSW
LNF0,1,SW
LNF2 >SW
LNF0,1and α∈ ( 0 , 1 ]
. The３４
social welfare of all is positive. In this light, the social welfare equilibrium of the private firm 2 as leader that is Subgame Perfect Nash Equilibrium. The merged firm as leader is dominated strategy. The merged firm decides to merge with the private firm 1, which should take the strategy as follower.
Proposition 3.1: under merger involving public firm and a private firm without foreign
penetration, SW
LNF2is optimal subgame perfect Nash equilibrium.
Case 2: The perfect Nash equilibrium of merger involving the public firm and the private firm 2
Table 3: Endogenous Timing equilibrium of merger between the public firm and the private firm 2
Merged firm as Leader (L) Merged firm as Follower (F)
L
Figure 2：The social welfare of endogenous timing for merger between the public firm and the private
firm 2
３５
The social welfare equilibrium
SW
0F,2 is large than the social welfareThe social welfare equilibrium