Oligopolistic Competition

Cournot (2 firms)

• There are $n=2$ firms, outputs $q_1$ and $q_2$
  • zero MC and FC, market demand $p=a-bQ$, where $Q=q_1+q_2$
  • firm 1: ${\pi }_1=[a-b(q_1+q_2)]q_1=a{q}_1-b{q}_1^2-b{q}_1{q}_2$
  • 1’s FOC: $a-2b{q}_1-b{q}_2=0$
  • 2’s FOC: $a-2b{q}_2-b{q}_1=0$
  • Cournot equilibrium: $q^{\ast }=q^{\ast }=\frac{a}{3b}\Rightarrow Q_{n=2}^C=\frac{2a}{3b}$
  • First best: $SMC=SMR\Rightarrow 0=a-bQ\Rightarrow Q^{FB}=\frac{a}{b}$
  • Monopoly: ${\pi }_M=(a-bQ)Q=aQ-b{Q}^2\Rightarrow Q^M=\frac{a}{2b}$
  • $Q^M<Q_{n=2}^C<Q^{FB}$

Cournot (n films)

• There are $n$ firms, outputs $q_1,q_2,\cdots ,q_n$
  • symmetric equilibrium: $q_1=q_2=\cdots =q_n=q^e$
  • firm i:$${\pi }_i=[a-b(q_i+(n-1)q^e)]q_i$$$$\Rightarrow {\pi }_i=a{q}_i-b{q}_i^2-(n-1)b{q}^e{q}_i$$
  • FOC: $a-2b{q}_i-(n-1)b{q}^e=0$
  • in equilibrium, it is optimal for $i$ to choose $q_i=q^e$
  • FOC $\Rightarrow q^e=\frac{a}{(n+1)b}\Rightarrow Q_n^C=\frac{na}{(n+1)b}$, clearly, $Q_n^C$ increases in $n$
  • when $n\to \infty$, $Q_n^C\to Q^{FB}=\frac{a}{b}$

Stackelberg

• Two firms, 1 is the leader, 2 is the follower
  • 1 chooses $q_1$ first, 2 chooses $q_2$ after observing $q_1$
  • BI: 2’s view, ${\pi }_2=[a-b(q_1+q_2)]q_2=a{q}_2-b{q}_1{q}_2-b{q}_2^2$
  • 2’s FOC: $a-b{q}_1-2b{q}_2=0\Rightarrow$ 2’s BR: $q_2^{BR}(q_1)=\frac{a}{2b}-\frac{q_1}{2}$
  • BI: 1’s view:$${\pi }_1=[a-b(q_1+q_2^{BR}(q_1))]q_1$$$$\Rightarrow {\pi }_1=[a-b(q_1+\frac{a}{2b}-\frac{q_1}{2})]q_1$$$$\Rightarrow {\pi }_1=[a-b(\frac{a}{2b}+\frac{q_1}{2})]q_1$$$$\Rightarrow {\pi }_1=\frac{a}{2}{q}_1-\frac{b}{2}{q}_1^2$$
  • 1’s FOC: $\frac{a}{2}-b{q}_1=0\Rightarrow q_1^{St}=\frac{a}{2b},q_2^{St}=\frac{a}{4b}\Rightarrow Q^{St}=\frac{3a}{4b}$
  • ${\pi }_1^{St}>{\pi }_1^C$ and ${\pi }_2^{St}<{\pi }_2^C$ (Revealed Profitability)

Stackelberg_cont

• In equilibrium $q_1^{St}=\frac{a}{2b},q_2^{St}=\frac{a}{4b},p=a-b\frac{3a}{4b}=\frac{a}{4}$
• Profits:
  • ${\pi }_1^{St}=\frac{a}{4}\left(\frac{a}{2b}\right)=\frac{a^2}{8b}$
  • ${\pi }_2^{St}=\frac{a}{4}\left(\frac{a}{4b}\right)=\frac{a^2}{16b}$
  • π i C = ( a − b 2 a 3 b ) a 3 b = a 2 9 b , ∀ i ∈ { 1 , 2 } \pi_i^C = \left( a - b \frac{2a}{3b} \right) \frac{a}{3b} = \frac{a^2}{9b}, \forall i \in \{1, 2\} πiC​=(a−b3b2a​)3ba​=9ba2​,∀i∈{1,2}
  • We confirm that:$${\pi }_1^{St}>{\pi }_1^C,{\pi }_1^{St}+{\pi }_2^{St}<2{\pi }_i^C,{\pi }_2^{St}<{\pi }_2^C$$

Bertrand

• Two firms, price competition, $FC=0,MC=c\ge 0$
  • $p=a-bQ$, from 1’s perspective:
    • $q_1=0$ if $p_1>p_2$
    • $q_1=\frac{1}{2b}(a-p)$ if $p_1=p_2$
    • $q_1=\frac{1}{b}(a-p)$ if $p_1<p_2$
  • A unique equilibrium: $p_1^B=p_2^B=c$
  • Logic:
    1. (1) Rule out min ⁡ { p 1 , p 2 } < c \min\{p_1, p_2\} < c min{p1​,p2​}<c
    2. (2) Rule out $p_1\ne p_2$ ($p_i>p_j>c$ and $p_i>p_j=c$)
    3. (3) Rule out $p_1=p_2>c$
    • Since every firm has an incentive to lower its price.

Make the invisible hands visible

1 Market Economy

• Two types of firms, low and high, denoted by F1 and F2
  • 10 homogeneous firms of each type ⇒ price takers
  • F1: $c_1(q_1)=\frac{q_1^2}{2}\Rightarrow m{c}_1(q_1)=q_1$
  • F2: $c_2(q_2)=\frac{q_2^2}{4}\Rightarrow m{c}_2(q_2)=\frac{q_2}{2}$
• For any market price $p$, using $MC=MR$
  • F1: $p=q_1\Rightarrow q_1=p$
  • F2: $p=\frac{q_2}{2}\Rightarrow q_2=2p$
  • Market supply: $Q=10p+20p=30p\Rightarrow p=\frac{Q}{30}$
  • Market equilibrium:
    $$q_1^m=\frac{1}{30}Q\&q_2^m=\frac{1}{15}Q$$$$\Rightarrow \frac{q_1^m}{q_2^m}=\frac{1}{2}$$

2 Central-Planned Economy (CPE)

• Central-Planned Economy (CPE)
  • Complete information.
  • Government (G) maximizes social welfare (SW).
  • On the supply side: Given $\bar{Q}=Q$, choose $(q_1,q_2)$ optimally
    • $\bar{Q}=10(q_1+q_2)\Rightarrow q_2=\frac{1}{10}\bar{Q}-q_1$
    • $\Rightarrow$ For G: choosing $(q_1,q_2)$ is equivalent to choosing $q_1$
      $$\underset{q_1}{\min }TC=10\left(\frac{1}{2}{q}_1^2+\frac{1}{4}{q}_2^2\right)=10\left[\frac{1}{2}{q}_1^2+\frac{1}{4}{\left(\frac{1}{10}\bar{Q}-q_1\right)}^2\right]$$$$\Rightarrow \frac{dTC}{d{q}_1}=10\left[q_1-\frac{1}{2}\left(\frac{1}{10}\bar{Q}-q_1\right)\right]=0$$$$\Rightarrow q_1^{\ast }=\frac{1}{30}\bar{Q}\&q_2^{\ast }=\frac{1}{10}\bar{Q}-q_1=\frac{1}{15}\bar{Q}$$
• Under 1 & 2, CPE = Market E, $\bar{Q}$ is produced efficiently.

3 Minimum Price

• Start with a market equilibrium
  • Demand: $p=3-\frac{Q}{15}$; Supply: $p=\frac{Q}{30}$
  • Market equilibrium: $p=1$, $Q=30$ $(q_1=1$, $q_2=2)$
• G sets a minimum price $\underline{p}=2$, demand: $Q^d=15$

  • Efficient way to produce $Q^s=Q^d=15$ is:
    $$q_1^{\ast }=\frac{1}{30}\bar{Q}=\frac{1}{2}\&q_2^{\ast }=\frac{1}{15}\bar{Q}=1$$

  • At $p=2$, two types’ ideal outputs are
    $$q_1^{idl}=2\&q_2^{idl}=4$$

  • Thus, each firm has an incentive to overproduce.

4 Inefficiency of Minimum Price

• Efficient way to produce 15 is $(q_1^{\ast }=\frac{1}{2},q_2^{\ast }=1)$
  • $TC(q_1^{\ast }=\frac{1}{2},q_2^{\ast }=1)=10[\frac{1}{2}(\frac{1}{2}{)}^2+\frac{1}{4}(1{)}^2]=\frac{30}{8}$
• Any other way is less efficient—i.e., more costly
  • $TC(q_1^{\ast }=\frac{3}{4},q_2^{\ast }=\frac{3}{4})=10[\frac{1}{2}(\frac{3}{4}{)}^2+\frac{1}{4}(\frac{3}{4}{)}^2]=\frac{270}{64}>\frac{30}{8}$
  • $TC(q_1^{\ast }=0,q_2^{\ast }=\frac{3}{2})=10[\frac{1}{2}(0{)}^2+\frac{1}{4}(\frac{3}{2}{)}^2]=\frac{45}{8}>\frac{30}{8}$
  • $TC(q_1^{\ast }=1,q_2^{\ast }=\frac{1}{2})=10[\frac{1}{2}(1{)}^2+\frac{1}{4}(\frac{1}{2}{)}^2]=\frac{45}{8}>\frac{30}{8}$
  • $TC(q_1^{\ast }=\frac{3}{2},q_2^{\ast }=0)=10[\frac{1}{2}(\frac{3}{2}{)}^2+\frac{1}{4}(0{)}^2]=\frac{90}{8}>\frac{30}{8}$
• $TC\uparrow \Rightarrow PS\downarrow \Rightarrow DWL\uparrow$

An Example on Public Good

PG under private contribution

• Two agents 1 and 2, agent $i$’s income $w_i>0$, consumption on private good $x_i$, public good contribution $g_i$; both prices are normalized to unity.
  • $u_1(x_1,g_1)=x_1+\sqrt{G}=w_1-g_1+\sqrt{g_1+g_2}$
  • $u_2(x_2,g_2)=x_2+\sqrt{G}=w_2-g_2+\sqrt{g_1+g_2}$
  • Let the equilibrium contribution be $(g_1^{\ast },g_2^{\ast })$.
  • Given $g_2=g_2^{\ast }$, FOC for agent 1: $\frac{1}{2\sqrt{g_1^{\ast }+g_2^{\ast }}}=1$
  • Given $g_1=g_1^{\ast }$, FOC for agent 2: $\frac{1}{2\sqrt{g_1^{\ast }+g_2^{\ast }}}=1$
  • Agent $i$ choosing $g_i=g_i^{\ast }$ is optimal in equilibrium, from above FOCs.
    $$G^{\ast }=g_1^{\ast }+g_2^{\ast }=\frac{1}{4}$$

Efficient PG and a Comparison

• Social Planner’s Choice
  • $u_{(1+2)}=w_1+w_2-g_1-g_2+2\sqrt{g_1+g_2}=w_1+w_2-G+2\sqrt{G}$
  • FOC implies that
    $$1=\frac{1}{\sqrt{G}}⟹G^{FB}=1$$
• A Comparison
  • $G^{FB}=1>G^{\ast }=\frac{1}{4}$
  • Under private contribution, we have multiple equilibria:
  • e.g., (1) $(g_1^{\ast }=0,g_2^{\ast }=\frac{1}{4})$; (2) $(g_1^{\ast }=\frac{1}{8},g_2^{\ast }=\frac{1}{8})$
  • In (1), agent 1 free rides on agent 2; in (2), no obvious free-riding
  • However, PG in (2) is also underprovided

More Analysis on Externalities

Coase theorem

The owner is B

• The case in which the owner is B (with quasi-linear preference)
  $$u_A=m_A+f\left(s_A\right)=y_A-p{s}_A+f\left(s_A\right)$$$$\frac{d{u}_A}{d{s}_A}=0\Rightarrow f^{\prime }\left(s_A\right)-p=0(1)$$$$u_B=m_B+g\left(1-s_B\right)=y_B+p{s}_B+g\left(1-s_B\right)$$$$\frac{d{u}_B}{d{s}_B}=0\Rightarrow g^{\prime }\left(1-s_B\right)-p=0(2)$$
• $\exists$ a unique $p$ and $s^{\ast }=s_A=s_B$ that satisfy (1) and (2)
  • (a) $f^{\prime }(s)$ decreases in $s$, and $g^{\prime }(1-s)$ increases in $s$
  • (b) $f^{\prime }(s)>g^{\prime }(1-s)$ if $s=0$ and $f^{\prime }(s)<g^{\prime }(1-s)$ if $s=1$
  • $\Rightarrow$ unique cross point of $f^{\prime }(s)$ and $g^{\prime }(1-s)$, for $s\in [0,1]$

The owner is A

• The case in which A is the owner (with quasi-linear preference)
  $$u_A=m_A+f(s_A)=y_A+\tilde{p}(1-s_A)+f(s_A)$$$$\frac{d{u}_A}{d{s}_A}=0\Rightarrow f^{\prime }(s_A)-\tilde{p}=0(3)$$$$u_B=m_B+g(1-s_B)=y_B-\tilde{p}(1-s_B)+g(1-s_B)$$$$\frac{d{u}_B}{d{s}_B}=0\Rightarrow g^{\prime }(1-s_B)-\tilde{p}=0(4)$$
  • $\exists$ a unique $\tilde{p}$ and ${\tilde{s}}^{\ast }=s_A=s_B$ that satisfy (3) and (4).
    • for the same logic in the previous case
  • using (1)-(4), we obtain that $\tilde{p}=p$ and ${\tilde{s}}^{\ast }=s^{\ast }$

A comparison of the two cases

• The case in which B is the owner
  • $u_A^{OB}=y_A-p{s}^{\ast }+f(s^{\ast })$
  • $u_B^{OB}=y_B+p{s}^{\ast }+g(1-s^{\ast })$
• The case in which A is the owner
  • $u_A^{OA}=y_A-p{s}^{\ast }+f(s^{\ast })+p$
  • $u_B^{OA}=y_B+p{s}^{\ast }+g(1-s^{\ast })-p$
• Obviously, we obtain that
  $$u_A^{OB}<u_A^{OA}\&u_B^{OB}>u_B^{OA}$$

Example with a river

No one owns the river (NOR)

• No one owns the river (NOR)
  • steel factory: ${\pi }_s^{NOR}=p_{s}s-c_s(s,x)$
    • Where $\frac{\partial c_s}{\partial x}\le 0$ if $x\le \bar{x}$
    • Its cost decreases with the level of pollution for $x\le \bar{x}$
    • FOC:
      $$p_s=\frac{\partial c_s}{\partial s}(5)$$$$-\frac{\partial c_s}{\partial x}=0(6)$$
    • (6) implies that $x=\bar{x}$
  • fish company: ${\pi }_f^{NOR}=p_{f}f-c_f(f,x)$
    • Where $\frac{\partial c_f}{\partial x}>0$ for any $x$
    • Its cost increases with the level of pollution
    • FOC:
      $$p_f=\frac{\partial c_f}{\partial f}(7)$$

Fish company owns the river (FOR)

• Fish company owns the river (FOR)
  • $q$ is the unit price of pollution.
  • steel: ${\pi }_s^{FOR}=p_{s}s-qx-c_s(s,x)$
  • fish: ${\pi }_f^{FOR}=p_{f}f+qx-c_f(f,x)$
  • FOC:
    $$p_s=\frac{\partial c_s}{\partial s}(8)$$$$q=-\frac{\partial c_s}{\partial x}(9)$$$$p_f=\frac{\partial c_f}{\partial f}(10)$$$$q=\frac{\partial c_f}{\partial x}(11)$$

Steel factory owns the river (SOR)

• Steel factory owns the river (SOR)
  • $\tilde{q}$ is the unit price of pollution.
  • steel: ${\pi }_s^{SOR}=p_{s}s+\tilde{q}(\bar{x}-x)-c_s(s,x)$
  • fish: ${\pi }_f^{SOR}=p_{f}f-\tilde{q}(\bar{x}-x)-c_f(f,x)$
  • FOC:
    $$p_s=\frac{\partial c_s}{\partial s}(12)$$$$\tilde{q}=-\frac{\partial c_s}{\partial x}(13)$$$$p_f=\frac{\partial c_f}{\partial f}(14)$$$$\tilde{q}=\frac{\partial c_f}{\partial x}(15)$$

Comparison between FOR and SOR

• The unit price of pollution $q$, the pollution level $x$, the production levels of the two firms $f$ and $s$
  • $(q,x,f,s)$ that solves (8)-(11) must equal $(\tilde{q},x,f,s)$ that solves (12)-(15).
• However, firm’s profits in the two cases are different.
  • let $(q,x,f,s)$ be the solution in either case
  • ${\pi }_s^{FOR}=p_{s}s-qx-c_s(s,x)$
  • ${\pi }_s^{SOR}=p_{s}s-qx-c_s(s,x)+q\bar{x}$
  • ${\pi }_f^{FOR}=p_{f}f+qx-c_f(f,x)$
  • ${\pi }_f^{SOR}=p_{f}f+qx-c_f(f,x)-q\bar{x}$
  • it is clear to see that
    $${\pi }_s^{FOR}<{\pi }_s^{SOR}\&{\pi }_f^{FOR}>{\pi }_f^{SOR}$$

When two firms merge (MER)

• When the two firms merge
  • ${\pi }_{s+f}^{MER}(s,f,x)=p_{s}s+p_{f}f-c_s(s,x)-c_f(f,x)$
  • FOC:
    $$p_s=\frac{\partial c_s}{\partial s}(16)$$$$p_f=\frac{\partial c_f}{\partial f}(17)$$$$-\frac{\partial c_s}{\partial x}=\frac{\partial c_f}{\partial x}(18)$$
  • $(x,f,s)$ that solves (16)-(18) equals $(x,f,s)$ that solves (8)-(11)
    • which also equals $(x,f,s)$ that solves (12)-(15)
• NOR and MER Comparison
  • NOR: $-\frac{\partial c_s}{\partial x}=0\Rightarrow x=\bar{x}$
  • MER: $-\frac{\partial c_s}{\partial x}=\frac{\partial c_f}{\partial x}>0\Rightarrow x=x^{\ast }<\bar{x}$
  • Revealed Profitability: ${\pi }_{s+f}^{MER}>{\pi }_s^{NOR}+{\pi }_f^{NOR}$

More on tragedy of the commons

Tragedy of the commons

• When government chooses $c$ (total number of the cows)
  • total profit $\pi (c)=f(c)-p_{c}c$, where $f^{\prime }(c)>0$, $f^{\prime \prime }(c)<0$
  • First Order Condition (FOC): $f^{\prime }(c)-p_c=0$ $\Rightarrow$ at optimum, $c=c^{\ast }$, where $$f^{\prime }(c^{\ast })-p_c=0$$
  • When each individual chooses freely
    • given $c$ cows in the meadow, each player $i$ owns exactly one cow, ∀ i ∈ { 1 , 2 , . . . , c } \forall i \in \{1, 2, ..., c\} ∀i∈{1,2,...,c}
    • for a new player $j$, j ∉ { 1 , 2 , . . . , c } j \notin \{1, 2, ..., c\} j∈/{1,2,...,c}, ${\pi }_j=\frac{f(c+1)}{c+1}-p_c>0$
    • a new player will not choose to enter when $c=\hat{c}$, where $$\frac{f(\hat{c})}{\hat{c}}-p_c=0$$
    • from the graph, it is clear to see that $c^{\ast }<\hat{c}$

Further on tragedy of the commons

• Given $c$, π i = f ( c ) c − p c > 0 , ∀ i ∈ { 1 , 2 , … , c } \pi_{i} = \frac{f(c)}{c} - p_{c} > 0, \forall i \in \{1, 2, \ldots, c\} πi​=cf(c)​−pc​>0,∀i∈{1,2,…,c}
• Consider $j$ 's choice of adding one more cow
  • for player $i$, $\Delta {\pi }_i=\frac{f(c+1)}{c+1}-\frac{f(c)}{c}$
  • when player $j$ cares about every $i$
    • social marginal cost (SMC): $c\Delta {\pi }_i=\frac{c}{c+1}f(c+1)-f(c)$
    • social marginal revenue (SMR): ${\pi }_j=\frac{f(c+1)}{c+1}-p_c$
    • social marginal profit (SMP): $$SMP=SMR+SMC=f(c+1)-f(c)-p_c$$
  • when the G (government) considers adding one more cow:
    $$\begin{gathered} \pi (c+1)-\pi (c)=[f(c+1)-p_c(c+1)]-[f(c)-p_{c}c] \\ =f(c+1)-f(c)-p_c=SMP \end{gathered}$$
  • decision of player $j$ (who cares about every $i$) is equivalent to the decision of G (government)