# 微观经济学｜寡头垄断

Posted by Derek on March 16, 2019

# 2. 古诺模型

$$\mathrm{Br}_i(x_{-i})=\begin{cases}\frac{100-x_{-i}}{4},\ \ \ \ &x_i\leq100 \\ 0, &x_{-i}>100\end{cases}.$$但这并不意味着，其中一个企业生产超过100个单位的商品，能让另一个企业放弃生产。

# 3. 斯塔克伯格模型

$$\mathrm{Br}_2(x_1)=\begin{cases}\frac{100-x_1}{4},\ \ \ \ &x_1 \leq 100-2\sqrt{2F_2}\\ 0, &x_1>100-2\sqrt{2F_2}\end{cases}.$$

# 4. 一个更加丧心病狂关于固定成本的例子

We know \begin{aligned}\pi_1&=(480-200+\frac{5}{12}x_{-i}-x_{-1})(200-\frac{5}{12}x_{-1})-\frac{(200-\frac{5}{12}x_{-1})^2}{5}-F \\ &=(40-\sqrt{30}-\frac{\sqrt{30}}{12}x_{-1})^2-F.\end{aligned} Let $\pi_1\geq0,$ we have $x_{-1} \leq 480-\frac{2\sqrt{30}}{5}\sqrt{F}.$

Besides, \begin{aligned}\pi_i&=(480-120+\frac{1}{4}x_{-i}-x_{-i})(120-\frac{1}{4}x_{-i})-(120-\frac{1}{4}x_{-i})^2-F \\ &=2(120-\frac{1}{4}x_{-i})^2-F.\end{aligned} Let $\pi_i\geq0,$ we have $x_{-i}\leq480-2\sqrt{2F}.$

Thus, $$\mathrm{Br}_1(x_{-1})=\begin{cases}200-\frac{5}{12}x_{-1},\ \ \ \ &x_{-1}\leq480-\frac{2\sqrt{30F}}{5} \\ 0, &x_{-1}\geq480-\frac{2\sqrt{30F}}{5}\end{cases},$$ and $$\mathrm{Br}_i(x_{-i})=\begin{cases}120-\frac{1}{4}x_{-i},\ \ \ \ &x_{-i}\leq480-2\sqrt{2F} \\ 0, &x_{-i}\geq480-2\sqrt{2F}\end{cases}.$$ (i) If all three firms produce, then we have $$\begin{cases}134.4\leq480-\frac{2\sqrt{30F}}{5} \\ 211.2\leq480-2\sqrt{2F}\end{cases} \Rightarrow F\leq9031.68,$$ i.e., when $F\leq9031.68, (144, 67.2, 67.2)$ is the Nash Equilibrium.

(ii) If only two frims produce, then we have $x_1=\frac{7200}{43}, x_i=\frac{3360}{43},$ then we have $$\begin{cases}\frac{3360}{43}\leq480-\frac{2\sqrt{30F}}{5} \\ \frac{7200}{43}\leq480-2\sqrt{2F} \\ \frac{10560}{43}\geq480-2\sqrt{2F}\end{cases} \Rightarrow 27476.04 \leq F \leq 33644.13.$$ (iii) If only one firm produces, we have $x_1=200,$ then we have $$\begin{cases}0\leq480-\frac{2\sqrt{30F}}{5} \\ 200\geq480-2\sqrt{2F}\end{cases} \Rightarrow 39200 \leq F \leq 48000.$$ (iv) If no firms produce, then we have $F\geq48000.$