Monopoly

Definitions §

• Monopoly is when a single firm has all the Market Power. Monopolies normally have:
  • High barriers to entry
  • Production in elastic portion of demand
  • Single firm with market power
  • (usually) economies of scale
• A Natural Monopoly is when it’s hard for firms to achieve break-even without a monopoly market strucutre

Market Demand for Monopolies §

MR > 0, i.e. where ε > 1

• Monopolies face the whole market demand curve.
• Marginal Revenue (MR) curve will slope down with the same intercept and twice the gradient of the Demand curve.
• Visual Profit Maximization when $MC=MR$ (See below for math)
  1. Firm produces $x^M$ (=monopoly quantity)
  2. Market price settles at $p^M$ (=monopoly price)
  3. Firm makes profit of $c+d$
  4. Deadweight loss is $e$
     1. DWL happens because monopolist increase price more than optimum

Profit Maximization for Monopolies §

Profit maximization for monopolies:

$$ma{x}_{x}p(x)\cdot x-C(w,r,x)$$

• $p(x)$ is the inverse of the Ordinary Demand of goods
• $C(x)$ is the Cost Function that is derived w.r.t $x$
  • ! Not simply $wl+rk$!
• To solve, use Unconstrained Maximization
  1. First Order Condition: $\frac{\partial \pi }{\partial x}=0$
     1. This is the same condition as $MR=MC$ condition.
     2. ! In third-degree price discrimination, $MR=MC$ can only be used if $MC$ is constant.
  2. When elasticity is constant ⇒ Use the Elasticity condition: $p(x)=\frac{MC}{1+\frac{1}{{ϵ}_d}}$
     1. Elastic: $p>MC$ thus excess profit
     2. Unit elastic: $p=MC$ thus zero profit
     3. Inelastic: $p<MC$ thus negative profit (monopolist doesn’t produce)

Price Discrimination for Monopolies §

First Degree Price Discrimmination §

Everybody pays exactly how much they’re willing to pay.

• Demand curve is same as marginal revenue curve
• Pareto efficient because there is no deadweight loss; everybody pays exactly as they want to pay, and no extra profit is lost
  • → but the benefit goes to the monopolist only

Two-part Tariff §

• Consumers are made to pay amount A to enter the market
• Then charge consumers $p=MC$
• Same as Equivalent Variation

Process of enforcing the tariff

1. Consumers at utility $\overline{u_1}$ and only buys good $x_2$
2. Firm enters the market and starts selling good $x_1$ at price $p=MC$. The consumer gains utility $\overline{u_2}$
3. Firm realizes they could get more profit. They charge amount $A$ to enter the market. Thus the consumer loses amount of $\text{difference}$.
4. Consumers are back to $u_1$, but they’re still buying $x_1$ because they’re just as happy as when they only bought $x_2$.

You can get market ticket price by either

• using Equivalent Variation on the right graph
• using integration on the left graph

Third Degree Price Discrimination §

• Idea: Charge differently based on consumer characteristics (=based on Price Elasticity of Demand)
• ⇒ Split consumers into two groups $A,B$ with elasticity ${ϵ}_D^A,{ϵ}_D^B$ and…
  • If $MC$ is constant → use the elasticity condition
  • If $MC$ is not constant → use normal profit maximization: $ma{x}_{x_A,x_B}\pi =\stackrel{\text{Total Revenue}}{\overbrace{p^A(x^A)\cdot x^A+p^B(x^B)\cdot x^B}}-C(x^A+x^B)$
• You will eventually get $M{R}^A=MC$ and $M{R}^B=MC$