Monopoly Notes

Monopoly

Introduction

  • This lecture covers monopoly profit maximization.
  • Topics include market power, market failure due to monopoly pricing, and causes of monopoly.
  • Optional government regulation will also be discussed.

Monopoly Profit Maximization

  • Monopoly Definition: The only supplier of a good with no close substitutes.
  • Price Maker: A monopoly can set its price, unlike competitive firms which are price takers.
  • Profit Maximization: All firms, including competitive firms and monopolies, maximize profits by setting marginal revenue (MR) equal to marginal cost (MC).

Marginal Revenue and Price

  • Firm's Revenue: R = p \cdot q where p is price and q is quantity.
  • Marginal Revenue (MR): The change in revenue from selling one more unit.
  • If the firm sells exactly one more unit: \Delta q = 1, then MR = \Delta R

Marginal Revenue and Price (Comparison to Competitive Firm)

  • Monopoly vs. Competitive Firm: A monopoly faces a downward-sloping demand curve, unlike a competitive firm.
  • Marginal Revenue Curve: The monopoly’s marginal revenue curve lies below the demand curve at every positive quantity.

Average and Marginal Revenue

  • Competitive Firm: Initial Revenue R1 = A, Revenue with One More Unit R2 = A + B, Marginal Revenue R2 - R1 = B = P_1
  • Monopoly: Initial Revenue R1 = A + C, Revenue with One More Unit R2 = A + B, Marginal Revenue R2 - R1 = B - C = p_2 - C

Deriving the Marginal Revenue Curve

  • Marginal Revenue Formula: MR = p + Q \frac{\Delta p}{\Delta Q}

Elasticity of Demand and Total, Average, and Marginal Revenue

  • Perfectly Elastic Demand: \epsilon \to -\infty
  • Elastic Demand: \epsilon < -1
  • Unit Elastic Demand: \epsilon = -1
  • Inelastic Demand: -1 < \epsilon < 0
  • Perfectly Inelastic Demand: \epsilon = 0

Solved Problem: Deriving the Marginal Revenue Curve

  • Inverse Demand Function: p = 24 - Q
  • Marginal Revenue Function: MR = 24 - 2Q
  • Slope Comparison: The marginal revenue curve is twice as steeply sloped as the inverse demand curve.

Mathematical Explanation for Solved Problem

  • p = 24 - Q
  • \Delta p = - \Delta Q
  • MR = p + Q \frac{\Delta p}{\Delta Q} = 24 - Q + Q(-1) = 24 - 2Q
  • The slope of the demand curve is \frac{\Delta p}{\Delta Q} = -1
  • The slope of the marginal revenue curve is \frac{\Delta MR}{\Delta Q} = -2

Marginal Revenue and Price Elasticity of Demand

  • Formula: MR = p(1 + \frac{1}{\epsilon}) where \epsilon is the price elasticity of demand.
  • Elasticity and Marginal Revenue: Marginal revenue is closer to price as demand becomes more elastic.

Special Cases of Marginal Revenue and Price Elasticity of Demand

  • Perfectly Elastic Demand (Q = 0): \epsilon \to -\infty, MR = p
  • Unitary Elasticity: \epsilon = -1, MR = 0
  • Inelastic Demand: - \infty < \epsilon < -1, MR < 0

Quantity, Price, Marginal Revenue, and Elasticity for the Linear Inverse Demand Curve p = 24 - Q

  • Table showing the relationship between quantity, price, marginal revenue, and elasticity of demand:
    • Q = 0, p = 24, MR = 24, \epsilon = -\infty
    • Q = 6, p = 18, MR = 12, \epsilon = -3
    • Q = 12, p = 12, MR = 0, \epsilon = -1
    • Q = 24, p = 0, MR = -24, \epsilon = 0

Choosing Price or Quantity

  • Profit Maximization: Any firm maximizes profit by setting MR = MC.
  • Monopoly's Choice: Unlike a competitive firm, a monopoly can choose either price or quantity to maximize profit.
  • Demand Curve Constraint: The monopoly is constrained by the market demand curve, facing a trade-off between higher prices and lower quantity, or lower prices and higher quantity.

Maximizing Profit

  • Monopoly profit is maximized in the elastic portion of the demand curve.
  • A monopoly never operates in the inelastic portion of its demand curve.

Shutdown Decision

  • Short Run: A monopoly shuts down to avoid losses if its price is below its average variable cost (AVC) at its profit-maximizing quantity.
  • Long Run: The monopoly shuts down if the price is less than its average cost (AC).

Mathematical Approach

  • Cost Function: C(Q) = Q^2 + 12 where Q^2 is the variable cost and $12 is the fixed cost.
  • Marginal Cost Function: MC = 2Q

Average Variable Cost

  • AVC Formula: AVC = \frac{Q^2}{Q} = Q
  • It is a straight line through the origin with a slope of 1.

Profit-Maximizing Output (Mathematical Approach)

  • Set MR = MC: 24 - 2Q = 2Q
  • Solve for Q: Q = 6
  • Substitute Q = 6 into the inverse demand function: p = 24 - 6 = $18

Verification of Shutdown Decision

  • At Q = 6:
    • AVC = $6 < $18, so the firm does not shut down.
    • AC = \frac{12}{6} + 6 = $8 < $18, so the firm makes a profit.

Solved Problem: Apple iPad Pricing

  • Marginal Cost: Apple's constant marginal cost of producing an iPad was approximately $220.
  • Average Cost: Average cost was approximately AC = \frac{220}{Q} + 2000 .
  • Inverse Demand Function: p = 770 - 11Q, where Q is measured in millions of iPads.
  • The problem asks to find Apple’s marginal revenue function, profit-maximizing price and quantity, and profit.

Solved Problem: Answer (Apple iPad)

  • Marginal Revenue: The marginal revenue curve is MR = 770 - 22Q
  • Profit Maximization: Set MR = MC: 770 - 22Q = 220. Solving for Q, Q = 25 million iPads.
  • Profit-Maximizing Price: p = 770 - 11(25) = $495
  • Profit: \pi = (p - AC) \cdot Q = (495 - 220) \cdot 25 = $4875 million.

Market Power

  • Definition: The ability of a firm to charge a price above marginal cost and earn a positive profit.
  • Formula: MR = p(1 + \frac{1}{\epsilon}) = MC
  • Rearranging Terms: \frac{p - MC}{p} = - \frac{1}{\epsilon}
  • The ratio of price to marginal cost depends only on the elasticity of demand at the profit-maximizing quantity.

Elasticity of Demand, Price, and Marginal Cost Table

  • Table illustrates the relationship between elasticity of demand, the ratio of price to marginal cost, and the Lerner Index.
  • \frac{p}{MC} = \frac{1}{1 + \frac{1}{\epsilon}}
  • \text{Lerner Index} = \frac{p - MC}{p} = - \frac{1}{\epsilon}

Lerner Index

  • Definition: The ratio of the difference between price and marginal cost to the price.
  • Formula: \frac{p - MC}{p}
  • In terms of elasticity of demand: - \frac{1}{\epsilon}
  • Range: 0 to 1 for a profit-maximizing firm.

Solved Problem: Microsoft Surface Pro 4

  • Price: $735
  • Marginal Cost: $470
  • The problem asks to calculate the Lerner Index and the elasticity of demand.

Solved Problem: Answer (Microsoft Surface Pro 4)

  • Lerner Index: \frac{735 - 470}{735} = 0.361
  • Elasticity: 0.361 = - \frac{1}{\epsilon} \implies \epsilon = -2.77

Sources of Market Power

  • The demand curve a firm faces becomes more elastic as:
    1. Better substitutes are introduced.
    2. More firms enter the market selling the same product.
    3. Firms providing the same service locate closer to this firm.

Market Failure Due to Monopoly Pricing

  • Welfare: Welfare is lower under monopoly than under competition.
  • Competition: Maximizes welfare because price equals marginal cost.
  • Monopoly: By setting its price above its marginal cost, a monopoly causes consumers to buy less than the competitive level of the good, resulting in a deadweight loss.

Deadweight Loss of Monopoly

  • Monopoly results in a deadweight loss due to reduced output and higher prices compared to a competitive market.

Causes of Monopoly

  • A firm has a cost advantage over other firms.
  • A government created the monopoly.

Cost Advantages

  • Reasons:
    • The firm uses a superior technology or has a better way of organizing production.
    • The firm controls an essential facility: a scarce resource that a rival needs to use to survive.

Natural Monopoly

  • Definition: A situation in which one firm can produce the total output of the market at a lower cost than several firms could.
  • Government Involvement: Governments frequently grant monopoly rights to public utilities, believing they are natural monopolies.

Condition for Natural Monopoly

  • C(Q) < C(q1) + C(q2) + … + C(q_n)
  • Where Q is the total output and q_i is the output of firm i.

Barriers to Entry

  • Governments create many monopolies by:
    • Owning and managing monopolies.
    • Preventing competing firms from entering a market.

Patents

  • Definition: An exclusive right granted to the inventor to sell a new and useful product, process, substance, or design for a fixed period of time.
  • Government Grant: Governments grant patent monopolies, despite the potential for deadweight loss, to incentivize innovation.

Optimal Price Regulation

  • In some markets, the government can eliminate the deadweight loss of monopoly by requiring that a monopoly charge no more than the competitive price.

Change Under Optimal Regulation

  • Consumer Surplus (CS): Increases
  • Producer Surplus (PS): Decreases
  • Welfare (W): Increases (Deadweight Loss is eliminated)

Problems in Regulating Monopolies

  • Limited Information: Governments may set the price at the wrong level due to limited information about demand and marginal cost curves.
  • Regulatory Capture: Regulation may be inefficient when regulators are influenced by the firms they regulate.
  • Inability to Subsidize: Regulators generally cannot subsidize the monopoly, they may be unable to set the price as low as they want because the firm may shut down.

Nonoptimal Price Regulation

  • If the regulated price is not optimal, a deadweight loss results.
  • If the price is set below the firm’s minimum average cost, the firm will shut down.
  • The deadweight loss equals the sum of the consumer plus producer surplus under optimal regulation.

Solved Problem: Nonoptimal Price Regulation

  • Suppose the government sets a price, p2, that is below the socially optimal level, p1, but above the monopoly’s minimum average cost.
  • The problem asks how the price, quantity sold, quantity demanded, and welfare compare to those under optimal regulation.

Answer to Solved Problem: Nonoptimal Price Regulation

  • Price: Set at p2 which is lower than p1.
  • Quantity Sold: Lower than under optimal regulation, constrained by the demand curve at p_2.
  • Quantity Demanded: There may be excess demand if more is demanded than is supplied at p_2.
  • Welfare: Lower than under optimal regulation due to deadweight loss.

Application: Natural Gas Regulation

  • Example of government regulation in the natural gas market.
  • Highlights potential deadweight loss implications.