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Long-Run Effects of Tax Policies in a Mixed Market. Joint work with Susumu Cato. Long-Run Effects of Tax Policies in a Mixed Market.

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Long-Run Effects of Tax Policies in a Mixed Market

Joint work with Susumu Cato

Long-Run Effects of Tax Policies in a Mixed Market

(1)Mixed Oligopoly at Free Entry Markets (2005, Journal of Economics) (2) What Role Should Public Enterprises Play in Free-Entry Markets? (2010, Journal of Economics)

(3) Long-Run Effect of Foreign Penetration on the Optimal Degree of Privatization (forthcoming in Journal of Institutional and Theoretical Economics)

(4) Mixed Duopoly, Privatization, and Subsidization with Excess Burden of Taxation (forthcoming, Canadian Journal of Economics)

(5) Market Structure and Privatization Policy under International Competition (forthcoming in Japanese Economic Review)

Plan of the Presentation

(1) Two lines of related literature on mixed oligopoly

(a) Free entry mixed market

(b) Privatization Neutrality Theorem

(3) Non-Neutral Results

(4) Model formulation

(5) Results, intuition, and implications

Related Literature on Mixed Oligopoly (1)

(1) Free Entry Market

(1-a) Monopolistic Competition

Anderson et al (1997),Matsumura et al (2009)

(1-b) Cournot Competition

Matsumura and Kanda (2005), Brandao and Castro (2007), Fujiwara (2007).

(1-c) Stackelberg

Ino and Matsumura (2010)←Ino and Matsumura (forthcoming in IER)

Matsumura and Kanda (2005)

Long-run analysis on mixed oligopoly

(1) Cournot competition, simultaneous-move, no product differentiation)

(2) No restrictions on the cost differences between public and private firms.

(3) The objective function of the public firm is the weight sum of social welfare and its own profits.

U0= (1-θ)W + θπ0

(4) General demand and general cost (increasing marginal costs).

(5) Free entry of private firms.

Time Line

(1) The government chooses whether to build a public firm (firm 0). The set-up cost F0 is sunk if it chooses to build it.

(2) The government chooses θ.

(3) After observing θ, each private firm chooses whether to enter the market.

(4) After observing the number of entering firm n, firms face Cournot competition.

The equilibrium of the subgame given α

  • Given θ, the following four variables are endogenously derived:

  • n(the number of firms),q1(the output of each private firm),q0(the output of the public firm),Q(total output)

  • from

  • the first order condition of the public firm

  • the first order condition of each private firm

  • zero profit condition of each private firm

  • Q=nq1+q0

The equilibrium of the subgame given θ

Comparative Statistics for these four variables


(a) q1 and Q are independent of θ.

(b)q0 is (increasing in, decreasing in, independent of ) θ.

(c)n is (increasing in, decreasing in, independent of ) θ.

Free entry equilibrium


private firm's residual demand

private firm's AC



private firm's output

a decrease in θ


private firm's residual demand

private firm's AC



private firm's output

long run ~ reduction of the number of private firms

Optimal θ

θ=0 is optimal. ←marginal cost pricing restrict wasteful entries (excess entry theorem).


Suppose that θ>0. A decrease in θ

(1) does not affect Q.→Consumer surplus remains unchanged.

(2) increases q0.→ Production cost increases by (firm 0’s marginal cost)・Δq0.

(3) Decreases n.→Production cost decreases by (firm 1’s average cost) ・q1・Δn.

q1・Δn =Δq0 (because Q remains unchanged).

Firm 1’s average cost =price >firm 1’s marginal cost

⇒An increase in α reduces production cost without changing the production level. (Welfare-improving production substitution)

First stage choice

・The government sets up firm 0 if and only if π0>0.

~The public firm which has deficit should be abolished in the ling-run.


・If the public firm is as efficient as private firms, it obtains (strictly positive, strictly negative, zero) profits.

Free entry equilibrium


private firm's residual demand





private firm's output

public firm's output


Effect of Privatization

Related Literature on Mixed Oligopoly (2)

(2) Privatization Neutrality Theorem (PNT)

(2-a) Privatization does not affect welfare under simple optimal subsidy policy, unit production subsidy.

White (1996),Tomaru (2006), Kato and Tomaru (2007), Hashimzade (2007)

(2-b) Public Leadership, private leadership, mixed Cournot, and private oligopoly yield the same welfare under optimal subsidy policy above.

Poyago-Theotoky (2001), Tomaru and Saito (2010)

Privatization Neutrality Theorem

Privatization Neutrality Theorem: Privatization does not matter under optimal subsidy policy.

It implies that if the optimal subsidy policy is adopted, discussing mixed oligopoly or privatization policy does not make sense.

Most of the results in mixed oligopoly literature have quite limited implications and importance if this theorem is really robust.

Destructive Result, Disaster for researchers in this field.

Intuition behind PNT

Suppose that all firms are symmetric. Consider the private oligopoly.

The first best is achieved when P=ci' (price =marginal cost) ~ all firms choose the same output level

It is achieved by the production subsidy s*.

Intuition behind PNT

Suppose that one firm is nationalized. Suppose that

all of remaining firms do not change their outputs.

The nationalized firm, which is welfare-maximizer, never changes its output .

All remaining private firms obviously have no incentive to change their outputs.

→s* yields the first best outcome in the mixed oligopoly.

Condition for PNT

When I explain the intuition behind PNT, I do not use any of the conditions

(1) profit-maximizing private firms

(2) homogeneous product market,

(3) single public firm

and so on.

All we use is the conditions that the first best is achieved at the symmetric equilibrium, that the first best is achieved by controlling outputs only, and that the pubic firm is welfare maximizer.

Robustness of PNT

Privatization Neutrality Theorem is far from robust:

(1) PNT obviously does not hold when there is cost difference between public and private firms.

(2) PNT does not hold unless all firms are domestic.

~ Matsumura and Tomaru (forthcoming in JER)

(3) PNT does not hold at free entry markets~This paper

(4) If there is an excess burden of taxation, PNT does not hold. ~Matsumura and Tomaru (forthcoming in CJE)

(5) PNT does not hold if firms control two or more independent variables.


  • The same demand and cost functions as in Matsumura and Kanda ~increasing marginal cost

  • Introducing unit subsidy s(lump-sum subsidy T).

  • The government chooses s or/and T to maximize welfare.

  • Each private firm chooses whether or not to enter the market.

  • Firms face Cournot competition.

Results ~ Unit subsidy

  • Lemma 1.

  • ∂qM1 /∂s>(=)0 if P’’ < (=) 0;

  • qM1 (s) = qP1(s) and QM(s) = QP(s) for all s;

  • nM(s) < nP(s) and dnM/ds > dnP/ds > 0 for all s.

  • Proposition 1. sM ≤ sP with the equality being satisfied if and only if P’’=0.

  • Proposition 2. W P(sP)<WM(sM).

  • Non-neutrality results

Results ~ Entry-License Tax

Lemma 2. (i) ∂qM0 /∂T>0;(ii) ∂qM1 /∂T>0; (iii)dnM/dT<0.

Lemma 3. The optimal entry-license tax TM is positive.

Lemma 4. (i) ∂qP1 /∂T>0;(ii) dnP/dT <0.

Lemma 5. The optimal entry-license tax TP is positive.

Lemma 6. For all T, (i)qM1(T)=qP1(T) and QM(T)=QP (T);

(ii) CSM(T) = CSP (T); (iii) nM(T) < nP(T) and dnM/dT<dnP/dT;

(iv)WM(T)≥WP(T) if and only if Π0(T) + {nM(T)+1}T≥nP(T)T.

Proposition 3. TM < TP .

Results ~ Two-Part Tax-Subsidy Policy

Proposition 4. In both mixed and private oligopolies, the first-best outcome is attained by the same tax-subsidy combination and the government budget is balanced.


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