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Oligopoly Theory 8. Irrelevance Results in Mixed Oligopoly

Oligopoly Theory 8. Irrelevance Results in Mixed Oligopoly. 今日の講義の目的 （１）税・補助金が公企業の行動に与える影響を理解する （２） Irrelevance Results が成立するメカニズムを理解する. 8-1 Tax Effect in Mixed Oligopoly 8-2 Irrelevance Results in Mixed Oligopoly 8-3 Robustness of Irrelevance Results

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Oligopoly Theory 8. Irrelevance Results in Mixed Oligopoly

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  1. Oligopoly Theory 8. Irrelevance Results in Mixed Oligopoly 今日の講義の目的 （１）税・補助金が公企業の行動に与える影響を理解する （２） Irrelevance Results が成立するメカニズムを理解する

  2. 8-1 Tax Effect in Mixed Oligopoly 8-2 Irrelevance Results in Mixed Oligopoly 8-3 Robustness of Irrelevance Results 8-4 Asymmetry of Order in Private Oligopoly and Irrelevance Results 8-5 Shadow Cost of Public Funding and Partial Privatization 8-6 Shadow Cost of Public Funding and Irrelevance Results 8-7 Irrelevance Results in Free Entry Markets Outline of the 8th Lecture

  3. Tax Effect in Mixed Oligopoly Mujumdar and Pal (1998) Introducing unit tax t into the Cournot-type model of De Fraja and Delbono (1998). Question: Consider the reaction function of a private firm. Given the outputs of other firms (one public and m-1 private firms), the optimal output of the private firm is (increasing in, decreasing in, independent of) t (tax rate).

  4. Tax Effect in Mixed Oligopoly : Public Firm Introducing unit tax t into the Cournot-type model of De Fraja and Delbono (1998). Question: Consider the reaction function of the public firm. Given the outputs of other firms (m private firms), the optimal output of the public firm is (increasing in, decreasing in, independent of) t (tax rate).

  5. Tax Effect in Mixed Oligopoly : Equilibrium Output of Each Private Firm Introducing unit tax t into the Cournot-type model of De Fraja and Delbono (1998). Question: Consider the equilibrium outputs. The equilibrium output of each private firm is (increasing in, decreasing in, independent of) t.

  6. Tax Effect in Mixed Oligopoly : Equilibrium Output of the Public Firm Introducing unit tax t into the Cournot-type model of De Fraja and Delbono (1998). Question: Consider the equilibrium outputs. The equilibrium output of the public firm is (increasing in, decreasing in, independent of) t.

  7. Tax Effect in Mixed Oligopoly : Welfare Implication Introducing unit tax t into the Cournot-type model of De Fraja and Delbono (1998). Question: Consider the equilibrium welfare (total social surplus). For t >0, the equilibrium welfare is (increasing in, decreasing in, independent of) t.

  8. Optimal Subsidy in Mixed Oligopoly White(1996) Introducing subsidy policy into the Cournot-type model of De Fraja and Delbono (1989). The government chooses unit subsidy s so as to maximizes resulting welfare Results: Privatization affects neither optimal subsidy rate nor resulting welfare →Privatization does not matter under optimal subsidy policy (Irrelevance Results)

  9. Optimal Subsidy in Mixed Oligopoly Introducing subsidy policy into the Cournot-type model of De Fraja and Delbono (1989). The government chooses unit subsidy s so as to maximizes resulting welfare. Consider the duopoly case. Suppose that s* yields the first best in private duopoly. Let y* denote the optimal output of each private firm in private duopoly at the first best outcome. (Henceforth we call it the base case). Question: Consider the reaction function of the public firm (firm 0) in mixed duopoly. Suppose that s=s*. R0(y*) (>,<,=) y*.

  10. Optimal Subsidy in Mixed Oligopoly Introducing subsidy policy into the Cournot-type model of De Fraja and Delbono (1989). The government chooses unit subsidy s so as to maximizes resulting welfare. Consider the duopoly case. Suppose that s* yields the first best in private duopoly. Let y* denote the optimal output of each private firm in private duopoly at the first best outcome. Answer: Consider the reaction function of the public firm in mixed duopoly. Suppose that s=s*. R0(y*) = y*.

  11. Optimal Subsidy in Mixed Oligopoly: Partial Privatization Tomaru (2006) Consider the base case. Suppose that the public firm is partially privatized and its objective is convex combination of welfare and its own Question: Consider the reaction function of the semi public firm (firm 0) in mixed duopoly. Suppose that s=s*. R0(y*) (>,<,=) y*. .

  12. Optimal Subsidy in Mixed Oligopoly: Non-Profit Maximizing Private Firm Kato and Tomaru (2007) Consider the base case. Suppose that the private firm's objective is convex combination of its own profit and its revenue , a la Fershtman and Judd (1987). Question: Consider the reaction function of the public firm (firm 1). Suppose that s=s* (the optimal subsidy rate in private duopoly). R0(y*) (>,<,=) y*.

  13. Optimal Subsidy in Mixed Oligopoly: Product Differentiation Hashimzade et al. (2007) Consider the base case. Suppose that the demand of firm 0 is p0=a-Y0-βY1 p1=a-Y1-βY2 where a and β are positive constants and β∈(0,1]. Question: Consider the reaction function of the private firm (firm 1). Suppose that s=s* (optimal subsidy rate in private oligopoly). R1(y*) (>,<,=) y*.

  14. Optimal Subsidy in Mixed Oligopoly: Public Leadership Poyago-Theotoky (2001) Consider the base case. Suppose that the public firm is Stackelberg Leader. Question: Suppose that s=s*. Consider the equilibrium output of the public firm (firm 0). y0E(>,<,=) y*.

  15. Optimal Subsidy in Mixed Oligopoly: Private Leadership Saito and Tomaru (2009) Consider the base case. Suppose that the public firm is Stackelberg Follower. Question: Suppose that s=s*. Consider the equilibrium output of the private firm (firm 0). y1E(>,<,=) y*.

  16. Optimal Subsidy in Mixed Oligopoly: Private Leadership Consider the base case. Suppose that the public firm is Stackelberg Follower. Answer : Suppose that s=s*. Consider the equilibrium output of the private firm (firm 0). y1E> y*, since firm 1 can reduce the rival's output by increasing its output. Irrelevance result on subsidy rate does not holds under private leadership. Question: Does irrelevance result on welfare hold under private leadership?

  17. Irrelevance Results The irrelevance result on subsidy rate does not hold under private leadership in mixed duopoly, but the irrelevance result on welfare is quite robust. Exception Fjell and Heywood (2004): Privatization is relevant under asymmetric order of moves among private firms. Asymmetry after privatization of the public firm yields the relevance result on welfare. ←Tinbergen Theorem

  18. Matsumura and Tomaru (unpublished) Introducing excess burden of taxation. One dollar of subsidy costs (1+λ) dollars (1)The first best outcome is not achieved. (2)The government has an incentive to economizes subsidy. (3)λaffects the behavior of the public firm. The output of public firm is increasing in λ. λ=0 →standard marginal cost pricing λ= ∞→profit maximizing Similar to partial privatization approach.

  19. The Model Players: government, firm 0 (public firm), firm 1 (private firm), Payoffs: welfare (government, public firm), Its own profits (private firm) (1) Government sets s. (2) Given s, firms faces Cournot competition.

  20. Notations s: unit subsidy rate λ: excess burden of taxation Ri: Firm i's reaction function at Cournot competition qi: Firm i's output, Q: Total output Ci(qi) =0.5k(qi)2 : Firm i's production cost P(Q)=a-Q: linear demand function πi: Firm i's profit, CS: Consumer surplus, W: social surplus, Superscript M: Equilibrium value in mixed duopoly Superscript P: Equilibrium value in private duopoly

  21. Welfare Before Privatization W=CS + profits of firms – total subsidy -λ (subsidy for the private firm). After Privatization W=CS + profits of firms – (1+ λ) total subsidy + λ( revenue from selling the stocks of the former public firm)

  22. (1) Either sM>sP or sM<sP is possible ~ Privatization affects optimal subsidy rate (Relevance result) When λ is large, sM>sP . The government has a stronger incentive to reduce s in private duopoly than in mixed duopoly since it must pay the subsidy for both firms. When λ is small (but positive) , sM<sP . In private oligopoly both firms' productions are too small when s=0. The government has a stronger incentive to raise s in private duopoly than in mixed duopoly since it stimulate production of both firms. Result on optimal subsidy

  23. (2) WM>WP for any λ>0 ~ Privatization affects welfare (Relevance result) The government has to pay subsidy for both firms in private duopoly. In mixed duopoly the public firm produces more than the private firm even when s is small ~ welfare improving since the government can economizes subsidy. Remark: Privatization can improve welfare if the privatization reduces firm 0's production cost. Nevertheless, privatization still affects welfare (Relevance Result still holds). Result on welfare

  24. Consider two Stackelberg models. One is Public Leadership (Firm 0 is the Stackelberg Leader) and the other is Private Leadership. Let superscript L denote the equilibrium value of the public leadership and let superscript F denote the equilibrium value of the public followership (private leadership). Result WF=WL=WM=WP if λ=0. WF>WL>WM>WP if λ>0 (Relevance Result). Extension 1:Stackelbergs

  25. Consider the observable delay game. There are two possible time periods for output choice . In the first stage, firm i simultaneously chooses whether it likes to be the leader (ti=L) or the follower (t=F). If two players' choices are consistent, i.e., one chooses to be the leader and the other does to be the follower, they get the equilibrium payoffs of a agreed timing Stackelberg. Otherwise, they receive the equilibrium payoffs in Cournot. After observing the timing the government chooses optimal tax rate so as to control the outputs of firms. Extension 2:Endogenous Timing

  26. Public Leadership constitutes an equilibrium regardless of λ, while Private leadership is not always. ~Desirable distribution of roles between public and private firms may not realized in observable delay game. →sharp contrast to Pal (1998) and Matsumura (2003b). Results in Endogenous Timing

  27. Introducing shadow cost of public funding (excess burden of taxation) changes the results in subsidized mixed oligopoly. Privatization matters under shadow cost of pubic funding. Summary

  28. Even without excess burden of taxation, privatization matters if we consider free entry (Cato and Matsumura ,unpublished). Introducing subsidy into Matsumura and Kanda (2005). One public firm compete against private firms. (1)The government chooses subsidy rate s. (2)Each private firm chooses whether or not to enter the market. (3) Firms face Cournot competition. Free Entry

  29. Subsidy affects both the number of entering private firms and the output of each private firm. Result 1 Optimal subsidy rate is 0 if the demand is linear, positive if it is concave, and negative if it is convex, in both mixed and private oligopoly. →Linear demand yields irrelevance result on subsidy rate but it crucially depends on the linearity of the demand. Result 2 Welfare is higher in Mixed Oligopoly than in Private Oligopoly (Relevance Result on Welfare). Irrelevance result again holds if we adopt two part tax-subsidy scheme. ←Tinbergen Theorem Free Entry

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