Market Power

Definition (Market Power). The ability to alter profitably prices away from competitive levels.

Monopoly

Prices Choice

Assumptions Monopolist: A firm is the only producer of a good. The demand for the good is $x(p)$, which is continuous and strictly decreasing at all $p$ such that $x(p)>0$. There exists a price $\overline{p}<\mathrm{\infty }$ such that $x(p)=0$ for all $p>\overline{p}$. The monopolist knows the demand function for its products and can produce output level $q$ at a cost of $c(q)$.

$$\underset{p}{max}px(p)-c(x(p))$$

Quantity Choice

An equivalent formulation in terms of quantity choices can be derived by thinking instead of the monopolist as deciding on the level of output that it desires to sell, $q\ge 0$, letting the price at which it can sell this output be given by the inverse demand function $p(\cdot )=x^{-1}(\cdot )$.

$$\underset{q\ge 0}{max}p(q)q-c(q)$$

Assumptions $p(\cdot )$ and $c(\cdot )$ are continuous and twice differentiable at all $q\ge 0$. $p(0)>c^{\prime }(0)$. There exists a unique output level $q^o\in (0,\mathrm{\infty })$ such that $p(q^o)=c^{\prime }(q^o)$.

First-Order Condition

$$p^{\prime }(q^m)q^m+p(q^m)\le c^{\prime }(q^m),\text{with equality if }{q}^m>0$$

Under our assumptions:

$$p^{\prime }(q^m)q^m+p(q^m)=c^{\prime }(q^m)$$

For the typical case in which $p^{\prime }(q)<0$ at all $q\ge 0$, it implies that we must have $p(q^m)>c^{\prime }(q^m)$, and so $q^m<q^o$.

Definition (Deadweight Loss of monopoly). The welfare loss from this quantity distortion of monopoly.

• It can be measured using the change in Marshallian aggregate surplus

$${\int }_{q^m}^{q^o}[p(s)-c^{\prime }(s)]ds>0.$$

The behavioral distortions arising under monopoly are not limited to pricing decisions.

Proposition (Monopolistic Price and Marginal Cost). $\mathrm{\forall }q$, if $c_2^{\prime }(q)>c_1^{\prime }(q)$, then $p_2^m\ge p_1^m$.

Price discrimination

Definition (价格歧视). 一个垄断厂商以不同的价格销售相同单位的产品的做法称为价格歧视。价格歧视是一种重要的垄断定价行为，是垄断企业通过差别价格来获取超额利润的一种策略.

1. 一级价格歧视：又称“完全价格歧视”，对每一单位产品都有不同的价格，此时，垄断者知道每一个消费者对任何数量的产品所要支付的最大货币量，并以此决定其价格，并正好等于产品的需求价格。
2. 二级价格歧视：此时，垄断厂商了解消费者的需求曲线，并将需求曲线分为不同段，根据不同购买量，确定不同价格，垄断者获得一部分，而不是全部的消费者剩余。
3. 三级价格歧视：此时，垄断厂商拥有关于顾客愿意支付的价格的信息，可以对不同市场的不同消费者实行不同的价格，在实行高价格的市场上获得超额利润。
   • 三级价格歧视是最普遍的价格歧视形式，三级价格歧视中销售者将购买者分组，对每一组制定不同的价格，这种行为也称为“市场分割”。

一级价格歧视

Assumptions 垄断者 厂商了解消费者的全部信息，知道每位消费者的最高支付意愿

$$\pi ={\int }_0^{q}f(t)dt-c(q)$$$$\frac{d\pi }{dq}=f(q)-c^{\prime }(q)=0\Rightarrow p(q)=mc(q)$$$$CS=0$$

三级价格歧视

Assumptions 同质商品不同市场需求弹性不同 不同组群消费者可被识别且相互分离

$$\pi =R_1(q_1)+R_2(q_2)-C(q_1+q_2)$$$$R_1^{\prime }(q_1)=R_2^{\prime }(q_2)=C^{\prime }(q_1+q_2)$$$$R^{\prime }=p+p^{\prime }q=p(1-\frac{1}{\epsilon })$$$$p_1(1-\frac{1}{{\epsilon }_1})=p_2(1-\frac{1}{{\epsilon }_2})$$

三级价格歧视要求厂商在需求价格弹性小的市场上制定较高的产品价格，在需求价格弹性大的市场上制定较低的产品价格。

二级价格歧视

Assumptions 消费者存在不同类型需求 $\mathrm{\forall }q>0$, $u_1(q)>u_2(q)$, $u_1^{\prime }(q)>u_2^{\prime }(q)$. 厂商不知道消费者的类型

$$\begin{array}{rl}\underset{p_1,q_1,p_2,q_2}{max}& p_1{q}_1+p_2{q}_2-c(q_1)-c(q_2)\\ & \text{s.t.}\text{(1)}{u}_1(q_1)-p_1{q}_1\ge 0\\ & \text{(2)}{u}_2(q_2)-p_2{q}_2\ge 0\\ & \text{(3)}{u}_1(q_1)-p_1{q}_1\ge u_1(q_2)-p_2{q}_2\\ & \text{(4)}{u}_2(q_2)-p_2{q}_2\ge u_2(q_1)-p_1{q}_1\end{array}$$

可得

$$max\pi =u_1(q_1)-u_1(q_2)+u_2(q_2)-c(q_1)+u_2(q_2)-c(q_2)$$

FOC

$$\frac{d\pi }{d{q}_1}=0\Rightarrow u_1^{\prime }(q_1^{\ast })=M{C}_1=c^{\prime }(q_1^{\ast })$$$$\frac{d\pi }{d{q}_2}=0\Rightarrow u_2^{\prime }(q_2^{\ast })=M{C}_2+[u_1^{\prime }(q_2^{\ast })-u_2^{\prime }(q_2^{\ast })]>M{C}_2$$

Static Models of Oligopoly

Definition (Oligopoly). A case in which more than one, but still not many, firms compete in a market.

Assumptions There is only one period of competitive interaction and firms take their actions simultaneously.

The Bertrand Model of Price Competition

Assumptions The two firms have constant returns to scale technologies with the same cost, $c>0$, per unit produced. The two firms simultaneously name their prices $p_1$ and $p_2$.

Sales for firm $j$ are then given by

$$x_j(p_j,p_k)=\{\begin{array}{ll}x(p_j)& \text{if }{p}_j<p_k\\ \frac{1}{2}x(p_j)& \text{if }{p}_j=p_k\\ 0& \text{if }{p}_j>p_k.\end{array}$$

Proposition (Bertrand Model Equilibrium). There is a unique Nash equilibrium $(p_1^{\ast },p_2^{\ast })$ in the Bertrand duopoly model. In this equilibrium, both firms set their prices equal to cost: $p_1^{\ast }=p_2^{\ast }=c$.

Competition between the two firms makes each firm face an infinitely elastic demand curve at the price charged by its rival.

• The idea can be extended to any number of firms greater than two.

The Cournot Model

Assumptions The two firms have constant returns to scale technologies with the same cost, $c>0$, per unit produced. The two firms simultaneously decide how much to produce, $q_1$ and $q_2$. Given these quantity choices, price adjusts to the level that clears the market, $p(q_1+q_2)$, where $p(\cdot )=x^{-1}(\cdot )$ is the inverse demand function.

Proposition (Cournot Model Equilibrium). In any Nash equilibrium of the Cournot duopoly model with cost $c>0$ per unit for the two firms and an inverse demand function $p(\cdot )$ satisfying $p^{\prime }(q)<0$ for all $q\ge 0$ and $p(0)>c$, the market price is greater than $c$ (the competitive price) and smaller than the monopoly price.

Capacity Constriants and Decreasing Returns to Scale

Assumptions Suppose that firms operate under conditions of eventual decreasing returns to scale, at least in the short run when capital is fixed. We make a minimal adjustment to the rules of the Bertrand model by taking price announcements to be a commitment to supply demand only up to capacity. We also assume that capacities are commonly known among the firms.

In this case, the Bertrand outcome $p_1^{\ast }=p_2^{\ast }=c$ is no longer an equilibrium.

Whenever the capacity level $\overline{q}$ satisfies $\overline{q}<x(c)$, each firm can assure itself of a strictly positive level of sales at a strictly positive profit margin by setting its price below $p(\overline{q})$ but above $c$.

Product Differentiation

Often, consumers perceive differences among the products of different firms. When product differentiation exists, each firm will possess some market power as a result of the uniqueness of its product.

Assumptions There are $J>1$ firms. Each firm produces at a constant marginal cost of $c>0$. The demand for firm $j$'s product is given by the continuous function $x_j(p_j,p_{-j})$, where $p_{-j}$ is a vector of prices of firm $j$'s rivals.

In a setting of simultaneous price choices, each firm $j$ takes its rivals' price choices ${\overline{p}}_{-j}$ as given and chooses $p_j$ to solve

$$\underset{p_j}{max}(p_j-c)x_j(p_j,{\overline{p}}_{-j}).$$

As long as $x_j(c,{\overline{p}}_{-j})>0$, firm $j$'s best response necessarily involves a price in excess of its costs ($p_j>c$).