垄断定义

  • 拥有定价权
  • 价格不受 Demand Quantity 影响

Profit Maximizing Rule

依然遵循利益最大化原则，对于垄断企业来说，只需要考虑在 Demand Curve 上找到 Marginal Revenue =0=0 的那个点即可。

Marginal Revenue

对于离散的 Quantity，有

MR(Q)=TR(Q)−TR(Q−1) MR(Q)=TR(Q)-TR(Q-1)

对于连续的 Quantity，有

P(Q)=a−bQ  ⟹  MR(Q)=a−2bQ P(Q)=a-bQ\implies MR(Q)=a-2bQ
证明

由于是连续性变量，考虑极小值 ΔQ\Delta Q 的 Marginal Revenue，则有

MR(Q)=lim⁡ΔQ→0(a−b(Q+ΔQ))(Q+ΔQ)−(a−bQ)QΔQ=a−2bQ \begin{aligned} MR(Q)&=\lim_{\Delta Q\to 0}\frac{\Big(a-b(Q+\Delta Q)\Big)(Q+\Delta Q)-(a-bQ)Q}{\Delta Q}\\ &=a-2bQ \end{aligned}

Select Price

选定 Quantity 之后，根据 Demand Curve 计算 Price。

Price Elasticity of Demand 与 Monopoly Markup

完全竞争的市场里，无法有溢价，即必须满足 P=MCP=MC。但垄断市场里可以有溢价：

P=ϵ1+ϵMCMarkup=P−MCMC=−11+ϵ P=\frac{\epsilon}{1+\epsilon}MC\\ Markup=\frac{P-MC}{MC}=\frac{-1}{1+\epsilon}

The More Inelastic the Demand Curve the More the Monopolist Raises Price Above Marginal Cost

Welfare Analysis

Producer Surplus

FC=0
FC=0

Monopoly Profit

FC>0
FC>0

政府干预

垄断企业相比其他企业，由于生产规模更大，其平均生产成本也更低，此时一定会将新入行的企业（其平均生产成本更高）挤出市场。

若政府设置价格上限……

如果设置在 MCMC，垄断企业会退出市场，因为 AC>MCAC>MC。所以如果要让垄断企业留在行业内，最低的上限必须设置为 ACAC.

the price that corresponding to the same quantity Q∗Q^\ast depends on the demand.

例题

Q1

A monopolist possesses the cost curves: TC=Q2+80Q+90000TC = Q^2 + 80Q + 90000, MC=2Q+80MC = 2Q + 80. The market demand is P=1100−QP = 1100 - Q.

(a) To maximize profit, the monopolist charges [Answer1].

(b) Continue above, the deadweight loss is [Answer2].

© If the government imposes a price ceiling at $700. The monopoly will earn a profit of [Answer3]. (Hint: Be careful with where MR(Q)>=MC(Q)MR(Q) >= MC(Q) in price ceiling case.)

(d) Suppose the government instead imposes a price ceiling to completely eliminate deadweight loss in the short run. The monopolist will earn a profit of [Answer4].

(e) Ignore price ceiling. Suppose the marginal cost of production is increased by $20 per unit sold. The monopolist will earn a profit of [Answer5]. (Hint: What are the new TC and MC curves?)

(f) Suppose instead only the fixed cost of production is increased by $20000. The monopolist will produce earn a profit of [Answer6]. (Hint: What are the new TC and MC curves?)

(g) Suppose instead the government provides a 20 per-unit subsidy to the monopoly. The monopolist will earn a profit of $[Answer7]. (Hint: What are the new TC and MC curves?)

(h) Suppose instead the government provides a 20 per-unit subsidy to consumers. The monopolist will earn a profit of $[Answer8].

(i) Suppose instead the government wants to completely eliminate deadweight loss. It can do so by providing per-unit subsidy of [Answer9][Answer9] to consumers.