**iBusiness** Vol.2 No.2(2010), Article ID:1952,7 pages DOI:10.4236/ib.2010.22014

Inventories and Mixed Duopoly with State-Owned and Labor-Managed Firms

^{ }^{ }^{ }^{}

Institute for Basic Economic Science, Osaka, Japan.

Email: ohnishi@e.people.or.jp

Received February 25^{th}, 2010; revised April 2^{nd}, 2010; accepted May 8^{th}, 2010.

**Keywords:** Inventory investment, State-owned firm, Labor-managed firm

ABSTRACT

This paper considers a two-period mixed market model in which a state-owned firm and a labor-managed firm are allowed to hold inventories as a strategic device. The paper then shows that the equilibrium in the second period occurs at the Stackelberg point where the state-owned firm is the leader.

1. Introduction

The analysis of mixed market models that incorporate state-owned public firms has been performed by many researchers, such as [1-16].^{1} However, these studies consider mixed market models in which state-owned firms compete with profit-maximizing capitalist firms, and do not include labor-managed firms.

Mixed market models that incorporate labor-managed firms have also been studied by many researchers, such as [22-34].^{2} However, these studies consider mixed market models in which labor-managed firms compete against profit-maximizing capitalist firms, and do not include state-owned firms.

Some studies examine mixed market models with state-owned and labor-managed firms. For example, Delbono and Rossini [40] explore the creation of 1) a duopoly formed by a labor-managed firm and a state-owned firm in a Cournot-Nash setting, and 2) a horizontal merger between the same agents. In addition, Ohnishi [41] investigates the behaviors of a state-owned firm and a labormanaged firm in a two-stage mixed market model with capacity investment as a strategic instrument. There are few studies that examine mixed market models with state-owned and labor-managed firms.

Therefore, we consider a two-period mixed market model in which a state-owned firm and a labor-managed firm can hold inventories as a strategic device.^{3} In the first period, each firm simultaneously and independently chooses how much it sells in the current market and the level of inventory it holds for the second-period market. We analyze the equilibrium of the mixed duopoly model, and show that the equilibrium in the second period occurs at the Stackelberg point where the state-owned firm is the leader.

The remainder of this paper is organized as follows. In Section 2, we describe the model. Section 3 gives supplementary explanations of the model. Section 4 analyzes the equilibrium of the model. Section 5 concludes the paper. All proofs are given in the appendix.

2. The Model

Let us consider a mixed market with one state-owned welfare-maximizing firm (firm 1) and one labor-managed profit-per-worker-maximizing firm (firm 2), producing perfectly substitutable goods. There is no possibility of entry or exit. In the remainder of this paper, subscripts 1 and 2 refer to firms 1 and 2, respectively, and superscripts 1 and 2 refer to periods 1 and 2, respectively. In addition, when and are used to refer to firms in an expression, they should be understood to refer to 1 and 2 with. The price of each period is determined by, where is the aggregate sales of each period. We assume that and. The game runs as follows. In the first period, each firm simultaneously and independently chooses its first-period production and its first-period sales. Firm’s inventory becomes. At the end of the first period, firm knows and. In the second period, each firm simultaneously and independently chooses its second-period production. At the end of the second period, each firm sells and holds no inventory. For notational simplicity, we consider the game without discounting.

Since, social welfare is

(1)

where denotes firm’s constant marginal cost. The demand and cost conditions that firms face remain unchanged over time. We assume that firm 1 is less efficient than firm 2, i.e..^{4} We define

(2)

Furthermore, since, firm 2’s profit per worker is

(3)

where is firm 2’s fixed cost, and is the amount of labor in firm 2. We assume that is the function of with and. This assumption means that the marginal quantity of labor used is increasing. We define

(4)

We analyze the subgame perfect Nash equilibrium of the mixed market model.

3. Supplementary Explanation

In this section, we give supplementary explanations of the model described in the previous section. First, we derive firm 1’s reaction functions from (2). In the first period, since there is no inventory available, firm 1’s reaction function is defined by

(5)

In the second period, firm 1’s reaction function without inventory is defined by

(6)

and thus its best response is shown as follows:

(7)

Firm 1 maximizes social welfare with respect to, given. When the inventory is zero, the first-order condition for firm 1 is

(8)

Furthermore, we have

(9)

In the first period, the slope of the reaction function of firm 1 is –1. In the second period, the slope of the best response of firm 1 is –1 for, and it is zero for. This means that firm 1 treats as strategic substitutes.^{5}

Second, we derive firm 2’s reaction functions from (4). In the first period, since there is no inventory available, firm 2’s reaction function is defined by

(10)

In the second period, firm 2’s reaction function without inventory is defined by

(11)

and thus its best response is shown as follows:

(12)

Firm 2 maximizes its profit per worker with respect to, given. The equilibrium must satisfy the following conditions: When the inventory is zero, the first-order condition for firm 2 is

(13)

and the second-order condition is

(14)

Furthermore, we have

(15)

Since, , so that is positive; that is, is upward sloping. This means that firm 2 treats as strategic complements.

Third, we consider Stackelberg games. If firm 1 is the Stackelberg leader, then firm 1 selects, and firm 2 selects after observing. Firm 1 maximizes with respect to. On the other hand, if firm 2 is the Stackelberg leader, then firm 2 selects, and firm 1 selects after observing. Firm 2 maximizes with respect to. We present the following lemma:

Lemma 1. Each firm’s Stakelberg leader sales exceed its Cournot sales without inventory.

Lemma 1 means that each firm prefers sales higher than its Cournot sales without inventory.

4. Equilibrium

In this section, we analyze the equilibrium outcomes of the mixed market model. The equilibrium in the first period is stated by the following proposition:

Proposition 1. In the first-period of the mixed market model, the equilibrium coincides with the Cournot Nash solution without inventory.

The intuition behind Proposition 1 is as follows. There is no inventory available in the first period, and further does not affect and. Since and decrease by deviating from the Cournot Nash solution, each firm has no incentive to do so, and therefore the equilibrium is at.

We now consider the equilibrium of the second period. It is thought that the equilibrium of the second period is decided by the level of. We discuss the following three cases:

1) The case in which only firm 1 can hold inventory 2) The case in which only firm 2 can hold inventory 3) The case in which each firm can hold inventory We discuss these cases in order.

1) The case in which only firm 1 can hold inventory First, consider Figure 1, where denotes firm’s second-period reaction curve without inventory. is downward sloping, whereas is upward sloping. Suppose that firm 1 holds in the second period. By holding inventory, firm 1’s best response becomes (7). Firm 1’s inventory investment thus creates a kink in its reaction curve at the level of. That is, firm 1’s reaction curve becomes the kinked bold lines as drawn in the figure. The equilibrium is decided in a Cournot fashion, i.e., the intersection of firm 1’s and firm 2’s reaction curves gives us the equilibrium of the game. Figure 1 shows that the intersection of new reaction curves is not affected by the kink. Hence, the equilibrium occurs at.

Next, consider Figure 2. Suppose that firm 1 holds. From (7), firm 1’s reaction curve becomes the kinked bold lines. The intersection of firm 1’s and firm 2’s reaction curves gives us the equilibrium of the game.

Figure 1. The equilibrium occurs at N

Figure 2. The equilibrium occurs at B

From Figure 2, we see that the inventory level of changes the equilibrium of the game. The intersection of the new reaction curves is the equilibrium sales in the second period. That is, if firm 1 holds, then the equilibrium occurs at.

We can now state the following proposition:

Proposition 2. Suppose that only firm 1 can hold inventory. Then the equilibrium coincides with the Stackelberg solution where firm 1 is the leader.

2) The case in which only firm 2 can hold inventory First, consider Figure 3. Suppose that firm 2 holds. From (12), firm 2’s reaction curve becomes the kinked bold broken lines. In this figure, the reaction curves of both firms do not cross each other. That is, if firm 2 maintains the inventory level of, then there is no solution.

Next, consider Figure 4. Suppose that firm 2 holds. From (12), firm 2’s reaction curve becomes the kinked bold broken lines. The reaction curves of both firms cross twice as in Figure 4. We can see easily that and are stable solutions. That is, there are two stable solutions. However, we see that firm 2’s profit per worker is higher at than at.

We can now state the following proposition:

Proposition 3. Suppose that only firm 2 can hold inventory. Then the equilibrium coincides with the Cournot Nash solution without inventory.

3) The case in which each firm can hold inventory First, consider Figure 5. Suppose that firms 1 and 2 hold and, respectively. In this figure, the reaction curves of both firms do not cross each other. That is, if firms 1 and 2 maintain the inventory levels of and, respectively, then there is no solution.

Figure 3. The reaction curves do not cross each other

Next, consider Figure 6. Suppose that each firm holds. Firm 1’s reaction curve becomes the kinked bold

Figure 4. The reaction curves cross twice

Figure 5. The new reaction curves do not cross each other

Figure 6. The new reaction curves cross twice

lines, and firm 2’s reaction curve becomes the kinked bold broken lines. The new reaction curves of both firms cross twice. We see easily that and are stable solutions. That is, there are two stable solutions. However, we see that firm 2’s profit per worker is higher at than at these points.

The main result of this study is described by the following proposition:

Proposition 4. In the second period of the mixed market model, the equilibrium coincides with the Stackelberg solution where firm 1 is the leader. At equilibrium, firm 2’s profit per worker is lower than in the Cournot mixed duopoly game without inventory.

5. Conclusions

We have considered a two-period mixed market model in which a state-owned firm and a labor-managed firm are allowed to hold inventories as a strategic device. We have then shown that the equilibrium in the second period occurs at the Stackelberg point where the stateowned firm is the leader and at equilibrium the labormanaged firm’s profit per worker is lower than in the Cournot mixed duopoly game without inventory. As a result, we see that the introduction of inventory investment into the analysis of mixed market competition with state-owned and labor-managed firms is profitable for the state-owned firm while it is not profitable for the labor-managed firm.

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Appendix

Proof of Lemma 1

First, we prove that firm 1’s Stakelberg leader sales are higher than its Cournot sales without inventory. Firm 1 maximizes with respect to. Therefore, firm 1’s Stackelberg leader sales satisfy the firstorder condition:

, (16)

where is positive from (8) and, and is also positive from (10), (11) and (15). To satisfy (16), must be negative.

Second, we prove that firm 2’s Stakelberg leader sales are higher than its Cournot sales without inventory. Firm 2 maximizes with respect to. Therefore, firm 2’s Stackelberg leader sales satisfy the first-order condition:

, (17)

where is negative from, and is also negative from (5), (6) and (9). To satisfy (17), must be negative. Thus, the lemma follows. Q. E. D.

Proof of Proposition 1

Since does not affect and, has no strategic value. Thus, the result follows easily from (5) and (10). Q. E. D.

Proof of Proposition 2

The equilibrium is decided in a Cournot fashion, i.e., the intersection of firm 1’s and firm 2’s reaction functions gives us the equilibrium of the game. In, social welfare is the highest at firm 1’s Stakelberg leader point. Lemma 1 states that firm 1’s Stakelberg leader sales exceed its Cournot sales without inventory. is downward sloping, whereas is upward sloping. From (7), we see that the equilibrium in the second period is decided by the value of. can take values of zero and above. Our equilibrium concept is the subgame perfect equilibrium and all information in the model is common knowledge. In the first period, firm 1 chooses associated with its second-period Stackelberg leader solution. Thus, Proposition 2 follows. Q. E. D.

Proof of Proposition 3

Lemma 1 states that firm 2’s Stakelberg leader sales exceed its Cournot sales without inventory. is downward sloping, whereas is upward sloping. From (12), if, then it is impossible for firm 2 to change in equilibrium. That is, if, then inventory does not function as a strategic device. Let be assumed to be continuous and concave in. The further a point on gets from firm 2’s Stackelberg leader point, the more firm 2’s profit per worker decreases. Thus, Proposition 3 follows Q. E. D.

Proof of Proposition 4

First, consider the possibility that firm 2 holds inventory as a strategic device. Lemma 1 states that firm 2’s Stakelberg leader sales exceed its Cournot sales without inventory. is downward sloping, whereas is upward sloping. From (12), firm 2 cannot choose its Stackelberg leader point. Let be assumed to be continuous and concave in. The further a point on gets from firm 2’s Stackelberg leader point, the more firm 2’s profit per worker decreases. Hence, firm 2 does not choose. If and, then there is no solution. If and, then the solution becomes. However, at solution, firm 2’s profit per worker is lower than in the Cournot mixed duopoly game without inventory. Hence, firm 2 does not hold inventory as a strategic device whether firm 1 holds inventory or not.

Next, consider the possibility that firm 1 holds inventory as a strategic device. Our equilibrium concept is the subgame perfect equilibrium and all information in the model is common knowledge. Hence, firm 1 knows that firm 2 does not hold inventory as a strategic device whether firm 1 holds inventory or not. Lemma 1 states that firm 1’s Stakelberg leader sales exceed its Cournot sales without inventory. From (7), we see that the equilibrium in the second period is decided by the value of. can take values of zero and above. In the first period, firm 1 chooses associated with its second-period Stackelberg leader solution. Thus, the equilibrium coincides with the Stackelberg solution where firm 1 is the leader.

NOTES

^{1}See [17-21] for excellent surveys.

^{2}The pioneering work on a theoretical model of a labor-managed firm is conducted by Ward [35]. See also [36-39] for excellent surveys.

^{3}For private market models with inventories as a strategic device, see Rotemberg and Saloner [42] and Matsumura [43].

^{4}This assumption is justified in Nett [2,21] and Gunderson [44], and is often used in literature studying mixed markets. See, for instance, George and La Manna [5], Mujumdar and Pal [7], Pal [8], Nishimori and Ogawa [11], Matsumura [13], Ohnishi [14], and Fernández-Ruiz [16]. If firm 1 is equally or more efficient than firm 2, then firm 1 chooses and such that price equals marginal cost. Therefore, firm 2 has no incentive to operate in the market, and firm 1 supplies the entire market, resulting in a social-welfare-maximizing public monopoly. This assumption is made to eliminate such a trivial solution.

^{5}The concepts of strategic substitutes and complements are due to Bulow, Geanakoplos, and Klemperer [45].