_{1}

Utilizing the model with upstream mixed oligopoly in which there are one public and n private wholesaler s in the upstream market, and m private retailers in the downstream market, this paper examines the optimal privatization policy of the upstream public wholesaler. It shows that: Firstly, in an environment with mixed oligopoly in the wholesale market and many private firms in the retail market, the public wholesaler should be partially privatized in the short run. Besides, the more private firms are in wholesalers or retailers market, the higher degree of privatization the government should take. Secondly, the public wholesaler should be partially privatized in the long run; moreover, the more private retailers firms are in the market, the less degree of privatization the government should take. Thirdly, the difference of optimal degree of privatization between long run and short run is increasing in the market scale and decreasing in the entry cost. Hence, the optimal degree of privatization in long run is smaller than in short run, when the market scale is restricted and the entry cost is high.

Recently, a growing amount of literature on the privatization issue in mixed oligopoly has been seen. Regarding the policy of partial privatization^{1}, Matsumura [

Wang and Chen [

While Matsumura and Kanda [

The remainder of this paper is organized as follows. We present the basic model in Section 2. Major results are derived and explained in Sections 3 and 4. The final section is the concluding remarks.

We assume that in a mixed oligopoly structure, there are one partially state-owned wholesaler and n private wholesalers in the upstream, while there are m private retailers in the downstream. All the upstream wholesalers, no matter whether it is the public or the private ones, sell homogeneous intermediate goods. It requires one unit of intermediate goods for producing one unit of final goods. Both the upstream and the downstream markets are engaging in Cournot competition. The market demand is characterized by a linear function^{2}; namely,

The profit functions of the public wholesaler, the private wholesalers, and the private retailers are as follows respectively:

where

The objective function of the public firm is the weighted average of both its profit and social welfare. Thus, we express its objective function as

where

In the following, a three-stage game is used to explore how the government determines the optimal privatization policy. At Stage 1, by maximizing the social welfare, the government determines the optimal degree of privatization; at Stage 2, it is the Cournot competition where the public firm, given the privatization degree, determines the optimum wholesale amount and competes with other private wholesalers; at Stage 3, the private retailers are in Cournot competition as well. As usual, this game is solved by subgame perfection deduced through backward induction.

We start by considering Stage 3. Retailers are in Cournot competition to determine the optimal retail quantity (

We then get the equilibrium retail price

From Equation (5), we obtain the retailer’s profit, the wholesale price and the aggregate profit of the wholesale market respectively in the following expression:

Then, in Stage 2, we find the optimal wholesale output for the wholesalers. By substituting the preceding solutions into Equation (1), the profit functions of the public wholesaler and the private firm are rewritten as follows:

The first-order conditions of the above maximization problems are

The wholesale market is in homogeneous goods competition, we express

The equilibrium input price is

From comparative static analysis, we obtain that:

In the short run, the equilibrium input price of intermediate good will increase because the increase of the derive demand which is called the “pass through effect”; when the number of the firm in the wholesaler market increases, the equilibrium input price of intermediate goodwill decrease because the market is soften; the higher degree of the privatization will push up the equilibrium input price of intermediate good is because the decrease in the production of the public firm will decrease the supply of the intermediate goods.

Substituting

From Equations (8) and (9), we find that the output of the public wholesaler outnumbers that of the private wholesaler. Similar to the results obtained in other studies without concerning vertical market structure, our finding also indicates that the public firm will produce more for the consideration of social welfare.

Moreover, by using the foregoing equations, we can derive the impact of privatization degree

^{3}The second order condition

From the above equations, we see that as the degree of privatization increases, the output of the public wholesaler shrinks, while that of the private wholesaler expands and that of private retailer decreases. The aggregate market output goes down and that will boost up the market price. Hence, consumer surplus declines.

Finally, we go to the first stage, and solve the government’s social welfare maximization problem to get the optimal degree of privatization. From the first order condition for social welfare maximization with respect to ^{3}:

We obtain the optimal degree of privatization

When

Proposition 1: In an environment with mixed oligopoly in the wholesale market and many firms in the retail market, the public wholesaler should be partially privatized in the short run.

Besides, from Equation (14), we perceive how the number of private wholesalers or that of private retailers affects the optimal degree of privatization. It shows that:

The optimal degree of privatization for the public firm depends on the number of private wholesalers or that of private retailers. When the number of private firms increases in wholesalers and retailers market, the government should increase the optimal degree of privatization.

In the long run, the government may liberalize the market with free entry^{4} which leads to zero profit for all the private wholesalers. Substituting Equation (8) and Equation (9) into

From Equation (17), we derive the number of the private wholesalers under free entry:

From Equation (18), we find that the impact of the optimal privatization degree or private retailer number on the optimal number of private wholesaler in the free entry upstream market. We express them respectively in the following equations:

From Equation (19), we also perceive that with free entry in the upstream market, the optimal private wholesaler number (

Substituting Equation (18) into Equations (5), (8) and (9), we get the outputs of the public firm and the private firms and wholesale price in the following expression:

From comparative static analysis, we obtain that:

In the long run, the equilibrium input price of intermediate good will decrease rather than increase is due to the deterrence effect of entry cost that will make the number of the firm in the downstream market becomes smaller and consequently, decreases the derive demand. The long run effect of firm entry in downstream market on the equilibrium input price of intermediate good is different from the short run “pass through” effect.

Furthermore, we find that:

From the results of Equations (25) and (26), we find that, in the industry environment with free entry, the government’s optimal privatization policy will only affect the optimum wholesaler number in the market or the output of the public firm. Neither the output of any individual private firm nor the optimum wholesale price is affected^{5}.

Then, we go back to the first stage. In an industry environment with free entry, we solve the government’s social welfare maximization problem with respect to the optimal privatization degree. Substituting Equation (18) to Equation (2) gives us the long run social welfare level. From the first order condition, we get the long run optimal privatization degree as follows^{6}:

^{6}When

^{7} implies that the retail market is by no means monopoly.

^{8}Matsumura and Kanda [

From Equation (27), we perceive the impact of the private retailer number, the market scale and the entry cost on the long run optimal privatization degree as follows^{7}:

Lemma 1: The optimal degree of privatization is decreasing in the number of private retailers and the entry cost, while it is increasing in the market scale.

The reasons for Lemma 1 are that, firstly, when the competition of downstream market is intensive, the degree of privatization should decline in order to enhance the supply of intermediate goods. Secondly, when the entry cost is increasing and the competition of upstream market is moderate, the government should decline the optimal degree of privatization.

In addition, we find that the government’s optimum policy is always partially privatization no matter in the situation with entry barrier in the short run or with free entry in the long run. The consideration of wholesale/retail structure in this paper may explain why the above result differs from that in Matsumura and Kanda [^{8}.

Proposition 2: In an environment with mixed oligopoly in the wholesale market and many firms in the retail market, the public wholesaler should be partially privatized when there is free entry in the wholesale market in the long run.

As argued in Matsumura and Kanda [

Proposition 3. The difference of optimal degree of privatization between long-run and short-run is increasing in the market scale and decreasing in the entry cost. The optimal degree of privatization in long-run is smaller than in short-run, when the market scale is restricted or the entry cost is high.

The reasoning is provided in Lemma 1.

From the comparison of Equation (16) and (28), we see that the entry of firm in the downstream market will increase the optimal degree of privatization in the short run but it may adjust the optimal degree of privatization in the long run. The reasoning is that in short run, as the market competition in the downstream market became intensive and that will make the market overproduced which lead the government to use the privatization policy in the wholesaler market as the instrument to reduce the supply of the public firm and hence, mitigate the adverse effect of overproduction in the retail market; but in the long run, since the profit of the firms in the wholesaler market is driven to zero and the number of the firm is restricted by the industry scale, the government should reduce the scale of optimal degree of privatization and increase the production of the semi-public firm to cope with the intensive competition in the retailers market.

The major finding of this paper lies in the consideration of both the public wholesaler and the government’s privatization policy in a vertical structure. We analyze the economic effect of the optimal privatization policy in the short run and in the long run, and have the following conclusions: Firstly, in an environment with mixed oligopoly in the wholesale market and many private firms in the retail market, the public wholesaler should be partially privatized in the short run. Besides, the more private firms are in wholesalers or retailers market, the higher degree of privatization the government should take. Secondly, the public wholesaler should be partially privatized in the long run; moreover, the more private retailers firms are in the market, the less degree of privatization the government should take. Thirdly, the difference of optimal degree of privatization between long run and short run is increasing in the market scale and decreasing in the entry cost. Hence, the optimal degree of privatization in long run is smaller than in short run, when the market scale is restricted and the entry cost is high.

Hsu, C.-C. (2016) Partial Privatization in Upstream Mixed Oligopoly with Free Entry. Modern Economy, 7, 1444-1454. http://dx.doi.org/10.4236/me.2016.712132