Microeconomics

Scarcity, Choice, and Opportunity Cost

The Fundamental Economic Problem

Scarcity is the central problem of economics: human wants are infinite, but the resources available to satisfy those wants are finite. Because resources are scarce, individuals, firms, and governments must make choices about how to allocate them. Every choice involves a trade-off: choosing one alternative means forgoing another.

The four factors of production are:

• Land: all natural resources (land, minerals, water, forests)
• Labour: the physical and mental effort contributed by workers
• Capital: manufactured goods used to produce other goods and services (machinery, tools, factories). Capital is distinct from financial capital (money), which is not itself a factor of production
• Entrepreneurship: the ability to organise the other factors of production and take risks in pursuit of profit

Opportunity Cost

The opportunity cost of a decision is the value of the next best alternative foregone. It is not the sum of all alternatives, but only the single most valuable one that was rejected.

Opportunity cost applies at every level:

• Individual: a student choosing to attend university forgoes the full-time salary they could have earned
• Firm: a factory producing cars cannot simultaneously use the same factory floor to produce buses
• Government: spending on healthcare means less spending available for education or defence

Economic vs. Accounting Profit

• Accounting profit $= \text{Total Revenue} - \text{Explicit Costs}$
• Economic profit $= \text{Total Revenue} - \text{Explicit Costs} - \text{Implicit Costs}$

Implicit costs include the opportunity cost of the owner's time and capital. A firm may earn a positive accounting profit but a negative economic profit if it could earn more by deploying its resources elsewhere.

Production Possibilities Frontier (PPF)

Definition and Interpretation

The PPF is a curve showing the maximum possible combinations of two goods or services an economy can produce when all resources are fully and efficiently employed, given the current state of technology.

Assume an economy produces two goods, capital goods ($K$) and consumer goods ($C$). The PPF curves outward (concave to the origin) because resources are not perfectly adaptable: as more of one good is produced, increasingly larger sacrifices of the other good are required. This reflects the law of increasing opportunity cost.

Points on, inside, and beyond the PPF

• On the PPF: the economy is productively efficient -- all resources are fully employed and allocated to their best use
• Inside the PPF: the economy is productively inefficient -- resources are unemployed or underemployed (e.g., during a recession)
• Beyond the PPF: currently unattainable given the existing resources and technology

Shifts of the PPF

The PPF can shift outward (economic growth) due to:

• Increases in the quantity of factors of production (more labour, capital, land)
• Improvements in the quality of factors of production (better education, technological progress)
• Institutional improvements (property rights, reduced corruption)

The PPF can shift inward due to:

• Natural disasters, war, or disease that destroy resources
• Resource depletion (exhaustion of non-renewable resources)

An asymmetric shift (where one axis shifts more than the other) occurs when a technological improvement or resource increase is specific to one industry.

PPF and Opportunity Cost

The slope of the PPF at any point represents the marginal opportunity cost of producing one more unit of the good on the x-axis, measured in terms of the good on the y-axis:

$\text{Slope of PPF} = -\frac{\Delta C}{\Delta K}$

A linear PPF implies constant opportunity costs (resources are equally suited to producing both goods). A concave PPF implies increasing opportunity costs (resources are not perfectly transferable between uses).

Supply and Demand

The Law of Demand

The law of demand states that, ceteris paribus, as the price of a good rises, the quantity demanded falls. The relationship is captured by a downward-sloping demand curve. A movement along the curve reflects a change in price (change in quantity demanded). A shift of the curve reflects a change in a non-price determinant of demand.

Non-price determinants of demand:

• Income (normal vs. inferior goods)
• Price of related goods (substitutes and complements)
• Tastes and preferences
• Population and demographics
• Expectations of future prices or income

A substitute has a positive cross-price elasticity of demand: if the price of coffee rises, demand for tea shifts rightward. A complement has a negative cross-price elasticity: if the price of petrol rises, demand for cars shifts leftward.

The Law of Supply

The law of supply states that, ceteris paribus, as price rises, quantity supplied increases. The supply curve slopes upward because higher prices increase the profitability of production, incentivising firms to expand output.

Non-price determinants of supply:

• Costs of production (wages, raw materials, energy)
• Technology
• Indirect taxes and subsidies
• Number of firms in the market
• Expectations of future prices
• Supply shocks (natural disasters, political instability)

Market Equilibrium

Equilibrium occurs where quantity demanded equals quantity supplied. The equilibrium price $P^*$ and quantity $Q^*$ are found at the intersection of the demand and supply curves.

If the market price is above $P^*$, a surplus (excess supply) exists, placing downward pressure on price. If the market price is below $P^*$, a shortage (excess demand) exists, placing upward pressure on price.

The rationing function of price allocates scarce resources to those willing and able to pay. The signalling function communicates information about scarcity and consumer preferences to producers.

Consumer and Producer Surplus

Consumer surplus is the difference between the maximum price a consumer is willing to pay and the actual market price. Graphically, it is the area below the demand curve and above the equilibrium price.

$\mathrm{CS} = \int_{0}^{Q^*} D(Q) \, dQ - P^* \cdot Q^*$

For a linear demand curve $P = a - bQ$:

$\mathrm{CS} = \frac{1}{2}(a - P^*) \cdot Q^*$

Producer surplus is the difference between the actual market price and the minimum price a producer is willing to accept. Graphically, it is the area above the supply curve and below the equilibrium price.

$\mathrm{PS} = P^* \cdot Q^* - \int_{0}^{Q^*} S(Q) \, dQ$

For a linear supply curve $P = c + dQ$:

$\mathrm{PS} = \frac{1}{2}(P^* - c) \cdot Q^*$

At equilibrium, total welfare (consumer surplus + producer surplus) is maximised. Any deviation from equilibrium creates a deadweight loss (DWL):

benefit and marginal cost at the new quantity{'\}'})$$ ### Functions of Price Prices serve several critical functions in a market economy: 1. **Rationing**: prices allocate scarce goods to those who value them most (measured by willingness and ability to pay) 2. **Signalling**: prices convey information about scarcity, consumer preferences, and production costs 3. **Incentive**: high prices incentivise producers to increase supply and consumers to reduce demand; low prices do the reverse ## Elasticity ### Price Elasticity of Demand (PED) PED measures the responsiveness of quantity demanded to a change in price: $$\mathrm{PED} = \frac{\% \Delta Q_d}{\% \Delta P}$$ Since the demand curve slopes downward, PED is negative, but it is conventionally expressed as an absolute value. For precise calculations, the **midpoint (arc) formula** avoids the ambiguity of which price and quantity to use as the base: $$\mathrm{PED} = \frac{\frac{Q_2 - Q_1}{(Q_1 + Q_2)/2}}{\frac{P_2 - P_1}{(P_1 + P_2)/2}}$$ | PED Value | Classification | Characteristics | | ----------------------------- | ------------------- | ----------------------------- | | $\|\mathrm{PED}\| > 1$ | Elastic | $\% \Delta Q_d > \% \Delta P$ | | $\|\mathrm{PED}\| = 1$ | Unit elastic | $\% \Delta Q_d = \% \Delta P$ | | $\|\mathrm{PED}\| < 1$ | Inelastic | $\% \Delta Q_d < \% \Delta P$ | | $\|\mathrm{PED}\| = 0$ | Perfectly inelastic | Vertical demand curve | | $\|\mathrm{PED}\| \to \infty$ | Perfectly elastic | Horizontal demand curve | **Determinants of PED:** - **Availability of substitutes**: more substitutes = more elastic. A good with many close substitutes (e.g., Coca-Cola vs. Pepsi) has highly elastic demand - **Proportion of income**: goods that take a larger share of income (e.g., cars, holidays) tend to be more elastic than inexpensive goods (e.g., salt, matches) - **Necessity vs. luxury**: necessities (food, medicine) tend to be inelastic; luxuries (designer clothing) tend to be elastic - **Time horizon**: demand becomes more elastic over time as consumers adjust behaviour and find alternatives. Short-run PED for petrol is low; long-run PED is higher as consumers switch to electric vehicles or public transport - **Addictiveness**: addictive goods (cigarettes, alcohol) tend to be inelastic in the short run, though demand may become more elastic in the long run as addiction is broken **Relationship with total revenue:** $$\mathrm{TR} = P \times Q$$ - If demand is elastic ($\|\mathrm{PED}\| > 1$), a price decrease increases total revenue - If demand is inelastic ($\|\mathrm{PED}\| < 1$), a price decrease decreases total revenue - If demand is unit elastic ($\|\mathrm{PED}\| = 1$), total revenue is unchanged (TR is maximised where PED $= -1$) ### Income Elasticity of Demand (YED) $$\mathrm{YED} = \frac{\% \Delta Q_d}{\% \Delta Y}$$ | YED Value | Classification | Example | | --------------- | ---------------------------- | ---------------------------------- | | YED $> 0$ | Normal good | Organic food, restaurant meals | | $0 <$ YED $< 1$ | Necessity (income inelastic) | Bread, basic clothing | | YED $> 1$ | Luxury (income elastic) | Designer goods, holidays | | YED $< 0$ | Inferior good | Instant noodles, second-hand goods | YED is important for understanding how demand changes as an economy grows. In a booming economy, demand for luxuries rises disproportionately, while demand for inferior goods falls. ### Price Elasticity of Supply (PES) $$\mathrm{PES} = \frac{\% \Delta Q_s}{\% \Delta P}$$ **Determinants of PES:** - **Spare production capacity**: firms with unused capacity can respond more quickly to price increases - **Mobility of factors of production**: labour and capital that can be easily reallocated increase PES - **Ability to store stocks**: goods that are non-perishable and inexpensive to store tend to have higher PES - **Time period**: supply is more elastic in the long run than in the short run. In the _momentary run_, supply is perfectly inelastic (output cannot change). In the _short run_, supply is somewhat elastic. In the _long run_, all factors are variable, and supply is highly elastic ### Cross-Price Elasticity of Demand (XED) $$\mathrm{XED} = \frac{\% \Delta Q_{d,A}}{\% \Delta P_B}$$ - XED $> 0$: goods are substitutes (the higher the positive value, the closer the substitute) - XED $< 0$: goods are complements (the more negative, the stronger the complement relationship) - XED $= 0$: goods are unrelated XED is used by firms to assess competitive threats. A high positive XED between two products indicates they are close competitors. ## Market Failure Market failure occurs when the free market fails to allocate resources efficiently, resulting in a loss of social welfare. The condition for allocative efficiency is: $$\mathrm{MSB} = \mathrm{MSC}$$ When MSB differs from MSC, the market produces either too much or too little of the good relative to the socially optimal quantity. ### Externalities An externality is a spill-over effect of production or consumption on third parties who are not involved in the transaction. **Negative production externalities** (e.g., factory pollution): - Marginal Social Cost (MSC) $>$ Marginal Private Cost (MPC) - MSC $=$ MPC $+$ Marginal External Cost (MEC) - Overproduction occurs relative to the socially optimal level - Deadweight loss arises from units produced where MSC $>$ Marginal Social Benefit (MSB) The welfare analysis: $$\text{DWL} = \frac{1}{2} \times \text{MEC} \times (Q_{\text{private}} - Q_{\text{social}})$$ **Positive production externalities** (e.g., research and development): - Marginal Social Benefit (MSB) $>$ Marginal Private Benefit (MPB) - MSB $=$ MPB $+$ Marginal External Benefit (MEB) - Underproduction occurs relative to the socially optimal level **Negative consumption externalities** (e.g., smoking in public, driving a car that emits pollution): - MSC $>$ MSB at the private equilibrium (or equivalently, MPB $>$ MSB) - Overconsumption relative to the social optimum **Positive consumption externalities** (e.g., vaccination, education): - MSB $>$ MPB at the private equilibrium - Underconsumption relative to the social optimum ### Diagrammatic Analysis of Externalities For a negative production externality, the MSC curve lies above the MPC curve. The free market equilibrium is at the intersection of MPC and MPB (demand). The socially optimal equilibrium is at the intersection of MSC and MSB. The difference in quantity between the two represents the overproduction, and the triangle between MSC, MSB, and the two quantities is the deadweight loss. For a positive consumption externality, the MSB curve lies above the MPB (demand) curve. The free market underproduces relative to the social optimum. The deadweight loss is the triangle between MPB, MSB, and the two quantities. ### Public Goods Public goods are non-excludable and non-rivalrous. Because of these properties, they suffer from the **free-rider problem**: individuals have no incentive to pay for the good since they cannot be excluded from consumption. This leads to market under-provision or zero provision. **Characteristics:** - **Non-excludable**: it is not possible (or prohibitively costly) to prevent someone from consuming the good - **Non-rivalrous**: one person's consumption does not reduce the amount available to others **Classification of goods:** | | Excludable | Non-excludable | | ------------------- | ----------------- | ------------------ | | **Rivalrous** | Private goods | Common resources | | **Non-rivalrous** | Club goods | Public goods | Pure public goods (e.g., national defence, street lighting, lighthouses) contrast with **private goods** (excludable and rivalrous), **quasi-public goods** (partially excludable or congestible, such as toll roads), and **common resources** (non-excludable but rivalrous, such as fish stocks, leading to the tragedy of the commons). ### Tragedy of the Commons Common resources are non-excludable but rivalrous. Since no individual has property rights, each user has an incentive to consume the resource before others do, leading to **overexploitation and depletion**. Examples include overfishing, deforestation of unowned land, and groundwater depletion. Solutions include: - Privatization (establishing property rights) - Government regulation (quotas, licences, seasonal bans) - Community management with social norms and local enforcement ### Merit and Demerit Goods **Merit goods** are under-consumed in the free market because individuals underestimate their private benefits (information failure). Examples include education and healthcare. They generate positive externalities. **Demerit goods** are over-consumed because individuals underestimate their private costs. Examples include tobacco, alcohol, and illicit drugs. They generate negative externalities. ### Information Failure Information failure occurs when consumers or producers lack accurate or complete information, leading to inefficient market outcomes. **Types of information failure:** - **Asymmetric information**: one party to a transaction has more information than the other. In the market for used cars, the seller knows the vehicle's condition better than the buyer (**adverse selection**). In insurance markets, individuals who purchase insurance may take greater risks (**moral hazard**) - **Imperfect information**: consumers lack full knowledge of product quality, long-term health effects, or environmental impacts - **Information overload**: when the volume of information available is so large that consumers cannot process it effectively ### Factor Immobility **Occupational immobility**: workers lack the skills to move between industries (e.g., a coal miner cannot easily become a software engineer). This contributes to structural unemployment. **Geographical immobility**: workers cannot or will not relocate to areas with job vacancies due to housing costs, family ties, language barriers, or lack of information about opportunities. ## Government Intervention ### Price Controls **Price ceilings** (maximum prices) are set below the equilibrium price, typically to make essential goods more affordable (e.g., rent controls). Consequences include: - Shortages (excess demand): at the ceiling price, quantity demanded exceeds quantity supplied - Black markets where goods are sold illegally above the ceiling - Reduced quality as producers cut costs to maintain profitability - Misallocation of resources: goods may not reach those who value them most (rationing mechanisms such as queues or favouritism replace the price mechanism) - Reduced producer surplus and potential long-run supply reductions **Price floors** (minimum prices) are set above the equilibrium price, typically to protect producers' incomes (e.g., minimum wages, agricultural price supports). Consequences include: - Surpluses (excess supply): at the floor price, quantity supplied exceeds quantity demanded - Government purchases or destruction of surplus (in agriculture) - Inefficient overproduction - Higher costs passed to consumers - Potential for government budget burdens ### Indirect Taxes An indirect tax is levied on goods and services. A **specific tax** is a fixed amount per unit sold; an **ad valorem tax** is a percentage of the price. Taxes shift the supply curve upward (or leftward) by the amount of the tax. The **tax incidence** (burden distribution between consumers and producers) depends on PED and PES: $$\frac{\text{Consumer burden}}{\text{Producer burden}} = \frac{\text{PES}}{|\text{PED}|}$$ - The more inelastic side of the market bears a larger share of the tax burden - If demand is perfectly inelastic, consumers bear the entire burden - If supply is perfectly elastic, consumers bear the entire burden A tax creates a deadweight loss because some mutually beneficial transactions no longer occur: $$\mathrm{DWL} = \frac{1}{2} \times t \times (Q_0 - Q_t)$$ Where $t$ is the tax per unit, $Q_0$ is the pre-tax quantity, and $Q_t$ is the post-tax quantity. Tax revenue is: $$\mathrm{Tax revenue} = t \times Q_t$$ ### Subsidies A subsidy is a payment by the government to producers (or consumers) per unit of output. It shifts the supply curve downward (or rightward), lowering the market price and increasing quantity. Effects of subsidies include: - Lower prices for consumers - Higher producer revenues - Correction of positive externalities (if set at the socially optimal level) - Government expenditure (opportunity cost) - Potential overproduction and inefficiency - Deadweight loss if the subsidy causes production beyond the socially optimal quantity The total cost to the government is: $$\text{Subsidy cost} = \text{subsidy per unit} \times Q_{\text{new}}$$ ### Regulation Governments can directly regulate markets by setting standards, banning certain activities, or requiring permits. Examples include: - Pollution emission limits (command-and-control regulation) - Minimum quality standards (food safety, building codes) - Bans on harmful substances (CFCs, certain pesticides) - Mandatory labelling requirements Regulation has the advantage of certainty (firms know exactly what is required) but may be inflexible and costly to enforce. ### Tradable Pollution Permits A market-based approach to correcting negative externalities. The government sets a total cap on emissions and issues permits that firms can trade among themselves. Firms with low abatement costs reduce emissions and sell their surplus permits; firms with high abatement costs buy permits instead of reducing emissions. Advantages: - Ensures the environmental target is met (total emissions are capped) - Allocates emission reductions efficiently (firms with the lowest abatement costs reduce first) - Provides incentives for innovation in pollution-reduction technology Disadvantages: - Initial allocation of permits (windfall profits if permits are given away rather than auctioned) - Price volatility of permits - Difficulty in setting the correct cap - Monitoring and enforcement costs ### Government Failure Government failure occurs when government intervention worsens the allocation of resources rather than improving it. Causes include: - **Regulatory capture**: regulators serve the interests of the industry they regulate rather than the public interest - **Information problems**: governments lacking sufficient information to set optimal policies - **Unintended consequences**: price ceilings causing shortages, subsidies causing overproduction - **Administrative costs**: the cost of implementing and enforcing policies may exceed the benefits - **Political pressures** leading to short-term thinking and populist measures - **Principal-agent problems**: government officials (agents) may not act in the interests of citizens (principals) ## Market Structures ### Classification Criteria Market structures are classified along several dimensions: - **Number of firms**: from many (perfect competition) to one (monopoly) - **Nature of the product**: homogeneous (identical) or differentiated - **Barriers to entry**: low to high - **Degree of market power**: the ability of firms to influence price - **Information availability**: perfect or imperfect ### Perfect Competition **Assumptions:** 1. Many buyers and sellers, none of whom can influence the market price (price takers) 2. Homogeneous (identical) products 3. Perfect information 4. Free entry and exit (no barriers) 5. Perfect factor mobility **Short-run equilibrium:** - The firm faces a perfectly elastic (horizontal) demand curve at the market price $P^*$ - Average Revenue (AR) $=$ Marginal Revenue (MR) $= P^*$ - The firm maximises profit where $\mathrm{MR} = \mathrm{MC}$ - In the short run, the firm may earn supernormal profit (if $P > \text{ATC}$), normal profit (if $P = \text{ATC}$), or make a loss (if $P < \text{ATC}$) - The firm continues to produce in the short run as long as $P \geq \text{AVC}$ (the shutdown point) **Long-run equilibrium:** - Free entry and exit drive economic profit to zero ($P = \text{ATC} = \mathrm{MC}$) - The firm produces at the minimum point of the ATC curve (productive efficiency) - $P = \mathrm{MC}$ (allocative efficiency) - Zero economic profit, but normal profit is earned $$\text{Short-run: } P \geq \text{AVC}, \text{ produce where MR} = \text{MC}$$ $$\text{Long-run: } P = \text{ATC}_{\min} = \text{MC}$$ ### Monopoly **Assumptions:** 1. Single seller (or one dominant firm) 2. No close substitutes 3. High barriers to entry (legal barriers such as patents, natural monopoly due to economies of scale, ownership of essential resources, predatory pricing) 4. Price maker **Revenue curves:** - The monopoly faces the market demand curve (downward-sloping) - $\mathrm{AR} = P$ (the demand curve) - MR lies below AR because to sell an additional unit, the monopolist must lower the price on all units sold - For a linear demand curve $P = a - bQ$: $\mathrm{MR} = a - 2bQ$ **Profit maximisation:** The monopolist produces where $\mathrm{MR} = \mathrm{MC}$ and charges the price found on the demand curve at that quantity. $$\text{Profit} = (P - \text{ATC}) \times Q$$ In the long run, the monopolist can earn supernormal profit because barriers to entry prevent new firms from entering the market. **Inefficiencies of monopoly:** - **Allocative inefficiency**: $P > \mathrm{MC}$, meaning the value consumers place on the last unit exceeds the cost of producing it. Too little is produced relative to the social optimum - **Productive inefficiency**: the monopolist does not produce at the minimum of ATC - **Deadweight loss**: the loss of consumer and producer surplus due to reduced output and higher prices compared to perfect competition - **X-inefficiency**: lack of competitive pressure may allow the monopolist to operate with higher costs than necessary **Arguments in favour of monopoly:** - **Economies of scale**: a single large firm may produce at lower average cost than multiple small firms (natural monopoly, e.g., water supply, electricity distribution) - **Research and development**: supernormal profits can fund R&D, leading to innovation and dynamic efficiency - **International competitiveness**: large firms may be better able to compete in global markets **Price discrimination** occurs when a monopolist charges different prices to different consumers for the same good or service, not justified by differences in cost. Conditions for price discrimination: 1. Market power (price maker) 2. Ability to separate markets (prevent resale) 3. Different price elasticities of demand in different markets **Degrees of price discrimination:** - **First-degree (perfect)**: charge each consumer their maximum willingness to pay. Captures all consumer surplus - **Second-degree**: charge different prices based on the quantity purchased (bulk discounts) - **Third-degree**: charge different prices to different consumer groups (e.g., student discounts, senior citizen discounts, peak/off-peak pricing) ### Monopolistic Competition **Assumptions:** 1. Many firms 2. Differentiated products (branding, quality, location, service) 3. Low barriers to entry and exit 4. Some degree of market power (firms are price makers to a limited extent) **Short-run equilibrium:** - Each firm faces a downward-sloping demand curve (due to product differentiation) - Profit maximisation at $\mathrm{MR} = \mathrm{MC}$ - Firms can earn supernormal profit, normal profit, or losses in the short run **Long-run equilibrium:** - Free entry and exit drives economic profit to zero ($P = \text{ATC}$) - Unlike perfect competition, the demand curve is downward-sloping, so the tangency with ATC occurs to the left of the minimum ATC - **Excess capacity**: the firm produces at a lower output than the output that minimises ATC - **Allocative inefficiency**: $P > \mathrm{MC}$ $$\text{Long-run: } P > \mathrm{MC} \text{ and } P = \text{ATC} > \text{ATC}_{\min}$$ **Non-price competition**: firms compete through advertising, branding, product differentiation, and customer service rather than solely through price. ### Oligopoly **Assumptions:** 1. A few large firms dominate the market 2. Interdependence: each firm's actions affect, and are affected by, the actions of rivals 3. High barriers to entry (economies of scale, brand loyalty, patents, strategic barriers) 4. Products may be homogeneous (e.g., oil) or differentiated (e.g., cars, smartphones) **Key features:** - **Kinked demand curve model**: if a firm raises its price, rivals do not follow (demand is elastic above the kink); if a firm lowers its price, rivals match the cut (demand is inelastic below the kink). This creates price rigidity and explains why prices in oligopolistic markets tend to be stable - **Collusion**: firms may coordinate to act as a monopoly, setting prices and output to maximise joint profit. Collusion may be **explicit** (cartel, e.g., OPEC) or **tacit** (price leadership, conscious parallelism) - **Game theory**: oligopoly is the market structure most naturally analysed using game theory (see the Game Theory and Behavioural Economics chapter) - **Non-price competition**: advertising, branding, product development, and loyalty programmes are more common than price competition - **Contestable market theory**: even if an industry has few firms, the threat of potential entry may discipline incumbents to keep prices close to competitive levels **Cartels:** A cartel is a formal agreement among competing firms to coordinate prices, output, or market shares. Cartels are inherently unstable because each member has an incentive to cheat by secretly lowering prices or exceeding its quota (prisoner's dilemma). Cartels are illegal in most countries under anti-trust/competition law. ## Theory of the Firm ### Costs of Production **Total cost (TC)** is the sum of total fixed cost (TFC) and total variable cost (TVC): $$\mathrm{TC} = \mathrm{TFC} + \mathrm{TVC}$$ - **Fixed costs**: costs that do not vary with output in the short run (rent, salaries of permanent staff, insurance) - **Variable costs**: costs that vary directly with output (raw materials, hourly wages, energy) **Average costs:** $$\mathrm{AFC} = \frac{\mathrm{TFC}}{Q} \qquad \mathrm{AVC} = \frac{\mathrm{TVC}}{Q} \qquad \mathrm{ATC} = \frac{\mathrm{TC}}{Q} = \mathrm{AFC} + \mathrm{AVC}$$ **Marginal cost (MC):** the additional cost of producing one more unit: $$\mathrm{MC} = \frac{\Delta \mathrm{TC}}{\Delta Q}$$ ### Short-Run Cost Curves - **MC initially falls** (due to increasing marginal returns from better utilisation of fixed factors) then **rises** (due to diminishing marginal returns as the fixed factor becomes a constraint) - **AVC follows MC**: when MC $<$ AVC, AVC is falling; when MC $>$ AVC, AVC is rising. MC intersects AVC at its minimum point - **ATC follows MC similarly**: MC intersects ATC at its minimum point - **AFC declines continuously** as output increases (spreading fixed costs over more units) - The gap between ATC and AVC equals AFC at every level of output ### The Law of Diminishing Marginal Returns In the short run, as more of a variable factor (e.g., labour) is added to a fixed factor (e.g., capital), the marginal product of the variable factor eventually declines. This is not caused by reduced quality of the variable factor but by the increasing ratio of variable to fixed factors. $$\mathrm{MP}_L = \frac{\Delta Q}{\Delta L} \qquad \text{Initially rises, then falls}$$ The relationship between production and cost: $$\mathrm{MC} = \frac{w}{\mathrm{MP}_L}$$ Where $w$ is the wage rate. When MP is rising, MC is falling; when MP is falling, MC is rising. ### Long-Run Costs In the long run, all factors of production are variable. The firm can choose its optimal scale of production. - **Long-run average cost (LRAC) curve**: the envelope of all short-run ATC curves. It shows the lowest average cost at each output level when the firm can adjust all inputs - **Economies of scale**: LRAC falls as output increases. Sources include: - **Technical economies**: specialisation, indivisibility of capital, the "container principle" - **Purchasing economies**: bulk buying discounts on raw materials - **Managerial economies**: specialisation of management functions - **Financial economies**: lower borrowing costs for larger firms - **Risk-bearing economies**: diversification of products and markets - **Constant returns to scale**: LRAC is flat; doubling all inputs exactly doubles output - **Diseconomies of scale**: LRAC rises as output increases. Causes include: - Communication and coordination problems in large organisations - Alienation and reduced motivation of workers - Bureaucratic inefficiency and red tape The **minimum efficient scale (MES)** is the lowest output at which LRAC is minimised. Firms that do not reach MES may be unable to compete with larger rivals. ### Revenue - **Total Revenue (TR)** $= P \times Q$ - **Average Revenue (AR)** $= \frac{\mathrm{TR}}{Q} = P$ (the demand curve) - **Marginal Revenue (MR)** $= \frac{\Delta \mathrm{TR}}{\Delta Q}$ Relationship between AR and MR: - Under perfect competition: AR $=$ MR $= P$ (horizontal demand curve) - Under monopoly and monopolistic competition: MR $<$ AR because the firm must lower price on all units to sell additional units ### Profit Maximisation A firm maximises profit by producing the quantity where: $$\mathrm{MR} = \mathrm{MC}$$ **Derivation:** profit $\pi = \mathrm{TR} - \mathrm{TC}$. Profit is maximised where the first derivative equals zero: $$\frac{d\pi}{dQ} = \frac{d\mathrm{TR}}{dQ} - \frac{d\mathrm{TC}}{dQ} = \mathrm{MR} - \mathrm{MC} = 0$$ **Second-order condition:** $\frac{d^2\pi}{dQ^2} < 0$, meaning MC must be rising at the point where MR $=$ MC (MC cuts MR from below). - If $\mathrm{MR} > \mathrm{MC}$: producing an additional unit adds more to revenue than to cost, so the firm should expand output - If $\mathrm{MR} < \mathrm{MC}$: producing an additional unit adds more to cost than to revenue, so the firm should reduce output **Shutdown condition:** in the short run, a firm should continue to produce as long as $P \geq$ AVC. If $P <$ AVC, the firm cannot cover its variable costs and should shut down immediately, losing only its fixed costs. If AVC $\leq P <$ ATC, the firm makes a loss but continues to produce because it covers its variable costs and contributes partially to fixed costs. **Normal profit**: the minimum profit necessary to keep a firm in the industry (the opportunity cost of the entrepreneur's time and capital). Normal profit is included in ATC as a cost. When economists say "zero economic profit," they mean normal profit is being earned. ## Common Pitfalls - Confusing a _movement along_ a demand curve (caused by a price change) with a _shift_ of the demand curve (caused by a change in a non-price determinant). Always state the cause clearly. - Forgetting that PED is expressed as an absolute value. A PED of $-0.5$ should be described as inelastic (since $|-0.5| < 1$), not as "negative elastic." - Confusing public goods with merit goods. Public goods are defined by their characteristics (non-excludable, non-rivalrous); merit goods are defined by their positive externalities and information failure. - Stating that subsidies "increase supply" without specifying that the supply curve _shifts rightward_. An increase in supply is a shift; an increase in quantity supplied is a movement along the curve. - Neglecting to discuss deadweight loss when analysing taxes and subsidies. Always identify the welfare loss triangle on the diagram. - Confusing negative externalities of _production_ with negative externalities of _consumption_. Pollution from a factory is a production externality; second-hand smoke from cigarettes is a consumption externality. - Confusing economic profit with accounting profit. Economic profit includes implicit costs (opportunity costs). - Stating that a monopolist charges "the highest possible price." A profit-maximising monopolist charges the price on the demand curve at the quantity where MR $=$ MC, not the highest price consumers would pay. - Confusing the shutdown condition with the exit condition. A firm shuts down in the short run if $P <$ AVC; a firm exits the industry in the long run if $P <$ ATC. - Drawing the MR curve incorrectly. For a linear demand curve, MR has the same vertical intercept but twice the slope (MR falls twice as fast as AR). - Confusing diminishing returns with diseconomies of scale. Diminishing returns occur in the short run (one factor is fixed); diseconomies of scale occur in the long run (all factors are variable). ## Practice Problems <details> <summary>Problem 1: PPF and Opportunity Cost</summary> An economy can produce the following combinations of capital goods ($K$) and consumer goods ($C$): | Point | K (units) | C (units) | | ----- | --------- | --------- | | A | 0 | 100 | | B | 10 | 95 | | C | 20 | 85 | | D | 30 | 70 | | E | 40 | 50 | | F | 50 | 0 | (a) Calculate the opportunity cost of increasing capital goods production from 20 to 30 units. (b) Does this PPF show constant or increasing opportunity costs? Explain. (c) If the economy is currently producing at point B, can it produce 30 units of $K$ and 70 units of $C$? Explain. (a) Moving from point C (20 $K$, 85 $C$) to point D (30 $K$, 70 $C$): $$\text{Opportunity cost} = \frac{\Delta C}{\Delta K} = \frac{85 - 70}{30 - 20} = \frac{15}{10} = 1.5 \; C \text{ per } K$$ Producing 10 additional units of capital goods requires sacrificing 15 units of consumer goods. (b) The opportunity costs are increasing: - From A to B: $\frac{100 - 95}{10 - 0} = 0.5 \; C$ per $K$ - From B to C: $\frac{95 - 85}{20 - 10} = 1.0 \; C$ per $K$ - From C to D: $\frac{85 - 70}{30 - 20} = 1.5 \; C$ per $K$ - From D to E: $\frac{70 - 50}{40 - 30} = 2.0 \; C$ per $K$ - From E to F: $\frac{50 - 0}{50 - 40} = 5.0 \; C$ per $K$ The opportunity cost of each additional 10 units of $K$ increases as more $K$ is produced, confirming increasing opportunity costs (a concave PPF). (c) Point D is on the PPF (30 $K$, 70 $C$), so yes, the economy can produce this combination if all resources are fully and efficiently employed. However, to move from point B to point D, the economy must reallocate resources from consumer goods to capital goods, which requires time and adjustment. </details> <details> <summary>Problem 2: PED, YED, and XED Calculations</summary> The demand for good $X$ is given by $Q_{d,X} = 200 - 2P_X + 0.5Y + 0.8P_Y$, where $P_X = 20$, $Y = 400$ (income), and $P_Y = 30$ (price of related good $Y$). (a) Calculate the quantity demanded of $X$. (b) Calculate PED at this point. (c) Calculate YED at this point. Is $X$ a normal or inferior good? (d) Calculate XED at this point. Are $X$ and $Y$ substitutes or complements? (a) $Q_{d,X} = 200 - 2(20) + 0.5(400) + 0.8(30) = 200 - 40 + 200 + 24 = 384$ (b) $\mathrm{PED} = \frac{\partial Q}{\partial P_X} \times \frac{P_X}{Q} = (-2) \times \frac{20}{384} = -0.104$ $\|\mathrm{PED}\| = 0.104 < 1$: demand is inelastic at this point. (c) $\mathrm{YED} = \frac{\partial Q}{\partial Y} \times \frac{Y}{Q} = 0.5 \times \frac{400}{384} = 0.521$ YED $> 0$: $X$ is a normal good. Since $0 <$ YED $< 1$, it is a necessity (income inelastic). (d) $\mathrm{XED} = \frac{\partial Q}{\partial P_Y} \times \frac{P_Y}{Q} = 0.8 \times \frac{30}{384} = 0.0625$ XED $> 0$: $X$ and $Y$ are substitutes. The positive but small value suggests they are weak substitutes. </details> <details> <summary>Problem 3: Negative Externality with Tax Correction</summary> The market for a chemical product has the following characteristics: - Demand: $P = 100 - Q$ (MPB $=$ MSB) - Private supply: $P = 20 + Q$ (MPC) - Marginal external cost: MEC $= 10$ per unit at all output levels (a) Find the market equilibrium (private equilibrium) price and quantity. (b) Find the socially optimal price and quantity. (c) Calculate the deadweight loss at the private equilibrium. (d) What specific tax would correct the market failure? (a) Market equilibrium: set demand $=$ supply: $$100 - Q = 20 + Q \implies 2Q = 80 \implies Q_{\text{private}} = 40$$ $$P_{\text{private}} = 100 - 40 = 60$$ (b) Social optimum: MSC $=$ MPC $+$ MEC $= (20 + Q) + 10 = 30 + Q$. Set MSB $=$ MSC: $$100 - Q = 30 + Q \implies 2Q = 70 \implies Q_{\text{social}} = 35$$ $$P_{\text{social}} = 100 - 35 = 65$$ (c) Deadweight loss: $$\mathrm{DWL} = \frac{1}{2} \times \mathrm{MEC} \times (Q_{\text{private}} - Q_{\text{social}}) = \frac{1}{2} \times 10 \times (40 - 35) = \frac{1}{2} \times 10 \times 5 = 25$$ (d) The specific tax should equal the marginal external cost: a tax of `USD 10` per unit. This shifts the supply curve from $P = 20 + Q$ to $P = 30 + Q$ (which is MSC), leading to the socially optimal quantity of 35 units. </details> <details> <summary>Problem 4: Monopoly Profit Maximisation and Welfare Analysis</summary> A monopoly faces the demand curve $P = 150 - 2Q$ and has a total cost function $\mathrm{TC} = 100 + 10Q + Q^2$. (a) Find the profit-maximising price and quantity. (b) Calculate the monopolist's economic profit. (c) What price and quantity would prevail under perfect competition? (d) Calculate the deadweight loss of monopoly. (a) $\mathrm{MR} = 150 - 4Q$ (twice the slope of demand). $\mathrm{MC} = \frac{d\mathrm{TC}}{dQ} = 10 + 2Q$. Set MR $=$ MC: $$150 - 4Q = 10 + 2Q \implies 6Q = 140 \implies Q_m = \frac{140}{6} = 23.33$$ $$P_m = 150 - 2(23.33) = 150 - 46.67 = 103.33$$ (b) $\mathrm{TR} = P_m \times Q_m = 103.33 \times 23.33 = 2411.1$ $\mathrm{TC} = 100 + 10(23.33) + (23.33)^2 = 100 + 233.3 + 544.3 = 877.6$ $\pi = 2411.1 - 877.6 = 1533.5$ (c) Under perfect competition, $P = \mathrm{MC}$: $$150 - 2Q = 10 + 2Q \implies 4Q = 140 \implies Q_c = 35$$ $$P_c = 150 - 2(35) = 80$$ (d) Deadweight loss: $$\mathrm{DWL} = \frac{1}{2} \times (P_m - P_c) \times (Q_c - Q_m) = \frac{1}{2} \times (103.33 - 80) \times (35 - 23.33)$$ $$\mathrm{DWL} = \frac{1}{2} \times 23.33 \times 11.67 = 136.1$$ </details> <details> <summary>Problem 5: Tax Incidence and Welfare</summary> The demand for a good is given by $Q_d = 120 - P$ and the supply is $Q_s = 2P - 40$. The government imposes a specific tax of `USD 15` per unit. (a) Find the pre-tax equilibrium price and quantity. (b) Find the post-tax equilibrium price paid by consumers and price received by producers. (c) Calculate the tax revenue and deadweight loss. (d) Determine the proportion of the tax borne by consumers and producers. (a) Pre-tax: $Q_d = Q_s$: $$120 - P = 2P - 40 \implies 3P = 160 \implies P_0 = 53.33$$ $$Q_0 = 120 - 53.33 = 66.67$$ (b) With a specific tax of $t = 15$, the supply curve shifts upward. The new supply (price received by producers) is $P_s = P_d - 15$. Setting $Q_d = Q_s$: $$120 - P_d = 2(P_d - 15) - 40 = 2P_d - 70$$ $$190 = 3P_d \implies P_d = 63.33$$ $$P_s = 63.33 - 15 = 48.33$$ $$Q_t = 120 - 63.33 = 56.67$$ (c) Tax revenue $= t \times Q_t = 15 \times 56.67 = 850.0$ Deadweight loss $= \frac{1}{2} \times t \times (Q_0 - Q_t) = \frac{1}{2} \times 15 \times (66.67 - 56.67) = \frac{1}{2} \times 15 \times 10 = 75$ (d) Consumer burden $= P_d - P_0 = 63.33 - 53.33 = 10$ Producer burden $= P_0 - P_s = 53.33 - 48.33 = 5$ Consumer share $= \frac{10}{15} = 66.7\%$. Producer share $= \frac{5}{15} = 33.3\%$. Consumers bear a larger share because demand is less elastic than supply at the equilibrium. </details> <details> <summary>Problem 6: Monopolistic Competition Long-Run Equilibrium</summary> A firm in monopolistic competition has the demand curve $P = 200 - Q$ and total cost function $\mathrm{TC} = 500 + 40Q + Q^2$. In the long run, entry and exit ensure zero economic profit. (a) Find the long-run equilibrium quantity and price. (b) Calculate the excess capacity. (c) Is the outcome allocatively efficient? Explain. (a) Zero economic profit: $\mathrm{TR} = \mathrm{TC}$. $\mathrm{TR} = P \times Q = (200 - Q)Q = 200Q - Q^2$ $\mathrm{TC} = 500 + 40Q + Q^2$ Set TR $=$ TC: $$200Q - Q^2 = 500 + 40Q + Q^2$$ $$2Q^2 - 160Q + 500 = 0$$ $$Q^2 - 80Q + 250 = 0$$ Using the quadratic formula: $$Q = \frac{80 \pm \sqrt{6400 - 1000}}{2} = \frac{80 \pm \sqrt{5400}}{2} = \frac{80 \pm 73.48}{2}$$ $Q = 76.74$ (the other root, $Q = 3.26$, gives a higher ATC). $P = 200 - 76.74 = 123.26$ (b) ATC is minimised where MC $=$ ATC. $\mathrm{MC} = 40 + 2Q$. $\mathrm{ATC} = 500/Q + 40 + Q$. $40 + 2Q = 500/Q + 40 + Q \implies Q = 500/Q \implies Q^2 = 500 \implies Q_{\min} = 22.36$ Excess capacity $= 76.74 - 22.36 = 54.38$ units. The firm operates well below its minimum efficient scale. (c) No. Allocative efficiency requires $P = \mathrm{MC}$. Here, $P = 123.26$ while $\mathrm{MC} = 40 + 2(76.74) = 193.48$. Since $P < \mathrm{MC}$, this suggests an issue. In practice, the firm should also satisfy the MR $=$ MC condition. $\mathrm{MR} = 200 - 2Q = 200 - 153.48 = 46.52$. Setting MR $=$ MC: $46.52 = 193.48$ is not satisfied. The correct approach recognises that in long-run equilibrium for monopolistic competition, the demand curve is tangent to ATC where MR $=$ MC, and $P > \mathrm{MC}$. The firm produces less than the socially optimal quantity. </details> <details> <summary>Problem 7: Subsidy Welfare Analysis</summary> The government provides a `USD 5` per unit subsidy on solar panels. Before the subsidy, the equilibrium price was `USD 100` and the equilibrium quantity was $10000$ units. After the subsidy, the quantity increases to $14000$ units. Calculate the total cost of the subsidy to the government and discuss potential government failures. Total subsidy cost $= \$5 \times 14000 = \$70000$. Potential government failures include: - The subsidy may lead to overproduction if the new quantity exceeds the socially optimal level, creating a deadweight loss. - Firms may become dependent on the subsidy and fail to innovate or reduce costs independently. - The subsidy is funded by taxpayers, representing an opportunity cost -- the funds could have been spent on other public services. - If solar panel production has its own negative externalities (e.g., manufacturing pollution), the subsidy could exacerbate those. - Administrative costs of implementing and monitoring the subsidy reduce its net benefit. </details> ## Consumer Choice Theory (HL Extension) ### Budget Constraints A consumer's budget constraint represents all combinations of two goods they can afford given their income and the prices of the goods. If a consumer has income $M$, the price of good $X$ is $P_X$, and the price of good $Y$ is $P_Y$: $$P_X \cdot X + P_Y \cdot Y = M$$ The budget line has: - **Slope** $= -P_X / P_Y$ (the opportunity cost of one more unit of $X$ in terms of $Y$) - **X-intercept** $= M / P_X$ (maximum quantity of $X$ affordable) - **Y-intercept** $= M / P_Y$ (maximum quantity of $Y$ affordable) The budget constraint shifts outward when income increases or when the price of both goods falls proportionally. A change in the price of one good rotates the budget line around the intercept of the other good. ### Indifference Curves An indifference curve shows all combinations of two goods that give the consumer the same level of utility (satisfaction). The consumer is indifferent between any two points on the same curve. **Properties of indifference curves:** 1. **Downward-sloping**: to maintain the same utility, consuming more of one good requires consuming less of the other 2. **Convex to the origin**: due to the diminishing marginal rate of substitution 3. **Cannot intersect**: if two curves intersected, a point would simultaneously represent two different utility levels, which is a logical contradiction 4. **Higher curves represent higher utility**: the further from the origin, the greater the combination of goods consumed and therefore the higher the utility ### Marginal Rate of Substitution (MRS) The MRS measures the rate at which a consumer is willing to trade one good for another while maintaining the same utility level: $$\mathrm{MRS}_{XY} = -\frac{\Delta Y}{\Delta X}$$ At any point on the indifference curve, the MRS equals the absolute value of the slope of the indifference curve. As the consumer moves down along the curve, the MRS diminishes: the more of $X$ and the less of $Y$ consumed, the less $Y$ the consumer is willing to give up for an additional unit of $X$. The MRS also equals the ratio of marginal utilities: $$\mathrm{MRS}_{XY} = \frac{MU_X}{MU_Y}$$ ### Consumer Equilibrium A rational consumer maximises utility subject to their budget constraint. The optimal consumption bundle occurs where the indifference curve is tangent to the budget line: $$\mathrm{MRS}_{XY} = \frac{P_X}{P_Y}$$ Or equivalently: $$\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}$$ This condition states that the marginal utility per dollar spent must be equal across all goods. If $\frac{MU_X}{P_X} > \frac{MU_Y}{P_Y}$, the consumer should buy more $X$ and less $Y$ until the ratio equalises. ### Income and Substitution Effects When the price of a good changes, the total effect on quantity demanded can be decomposed into: 1. **Substitution effect**: the change in consumption due to the change in relative prices, holding utility constant. The substitution effect is always negative (price and quantity move in opposite directions) 2. **Income effect**: the change in consumption due to the change in real purchasing power (real income). The direction depends on whether the good is normal or inferior **For a normal good:** - Both income and substitution effects work in the same direction - When price falls, both effects increase quantity demanded - The demand curve slopes downward **For an inferior good:** - The substitution effect increases quantity demanded when price falls - The income effect decreases quantity demanded (because the consumer is effectively richer and switches to superior goods) - If the income effect is smaller than the substitution effect, the demand curve still slopes downward (most inferior goods) - If the income effect outweighs the substitution effect, the demand curve slopes upward: this is a **Giffen good** (extremely rare; requires the good to be a staple that dominates the budget of very poor consumers) **Numerical example:** A consumer has income $M = 100$, $P_X = 5$, $P_Y = 2$. The consumer's utility function is $U = X^{0.5} \cdot Y^{0.5}$. Budget constraint: $5X + 2Y = 100$ $\mathrm{MRS} = \frac{MU_X}{MU_Y} = \frac{0.5 X^{-0.5} Y^{0.5}}{0.5 X^{0.5} Y^{-0.5}} = \frac{Y}{X}$ Setting $\mathrm{MRS} = P_X / P_Y$: $$\frac{Y}{X} = \frac{5}{2} \implies Y = 2.5X$$ Substituting into the budget constraint: $$5X + 2(2.5X) = 100 \implies 10X = 100 \implies X^* = 10, \quad Y^* = 25$$ ### Common Pitfalls in Consumer Choice - Drawing indifference curves that slope upward or that intersect. Indifference curves must be downward-sloping and non-intersecting. - Confusing the MRS with the budget line slope. At equilibrium they are equal, but they represent different concepts (subjective trade-off vs. market trade-off). - Assuming all inferior goods are Giffen goods. A Giffen good is a special case where the income effect dominates the substitution effect. Most inferior goods have a normally-shaped demand curve. ## Production Theory (HL Extension) ### Isoquants and Isocost Lines An **isoquant** shows all combinations of two inputs (typically capital $K$ and labour $L$) that produce the same level of output. Isoquants are analogous to indifference curves in consumer theory. **Properties of isoquants:** 1. Downward-sloping: to maintain the same output, using more of one input requires using less of the other 2. Convex to the origin: due to the diminishing marginal rate of technical substitution (MRTS) 3. Higher isoquants represent higher output levels The **marginal rate of technical substitution** (MRTS) measures the rate at which one input can be substituted for another while maintaining output: $$\mathrm{MRTS}_{LK} = -\frac{\Delta K}{\Delta L} = \frac{MP_L}{MP_K}$$ An **isocost line** shows all combinations of inputs that cost the same total amount. If the wage rate is $w$ and the rental rate of capital is $r$: $$wL + rK = C$$ The slope of the isocost line is $-w/r$. ### Cost Minimisation A firm minimises the cost of producing a given level of output where the isoquant is tangent to the isocost line: $$\mathrm{MRTS}_{LK} = \frac{w}{r}$$ Or equivalently: $$\frac{MP_L}{w} = \frac{MP_K}{r}$$ This condition states that the marginal product per dollar spent must be equal across all inputs. ### Returns to Scale Returns to scale describe how output changes when all inputs are increased proportionally: - **Constant returns to scale**: doubling all inputs exactly doubles output. $f(tK, tL) = t \cdot f(K, L)$ - **Increasing returns to scale**: doubling all inputs more than doubles output. $f(tK, tL) > t \cdot f(K, L)$ - **Decreasing returns to scale**: doubling all inputs less than doubles output. $f(tK, tL) < t \cdot f(K, L)$ Returns to scale determine the shape of the long-run average cost (LRAC) curve: increasing returns produce a downward-sloping LRAC, constant returns produce a flat LRAC, and decreasing returns produce an upward-sloping LRAC. ### Cobb-Douglas Production Function The Cobb-Douglas production function is widely used in economics: $$Q = A \cdot K^\alpha \cdot L^\beta$$ Where $A$ is total factor productivity, $\alpha$ is the output elasticity of capital, and $\beta$ is the output elasticity of labour. **Returns to scale with Cobb-Douglas:** - If $\alpha + \beta = 1$: constant returns to scale - If $\alpha + \beta > 1$: increasing returns to scale - If $\alpha + \beta < 1$: decreasing returns to scale **Marginal products:** $$MP_K = \frac{\partial Q}{\partial K} = \alpha \cdot A \cdot K^{\alpha - 1} \cdot L^\beta = \alpha \cdot \frac{Q}{K}$$ $$MP_L = \frac{\partial Q}{\partial L} = \beta \cdot A \cdot K^\alpha \cdot L^{\beta - 1} = \beta \cdot \frac{Q}{L}$$ **Worked example:** A firm has the production function $Q = 10K^{0.4}L^{0.6}$, $w = 20$, $r = 40$. Cost minimisation: $\frac{MP_L}{w} = \frac{MP_K}{r}$ $$\frac{0.6 \cdot 10K^{0.4}L^{-0.4}}{20} = \frac{0.4 \cdot 10K^{-0.6}L^{0.6}}{40}$$ $$\frac{0.6L^{-0.4}K^{0.4}}{20} = \frac{0.4K^{-0.6}L^{0.6}}{40}$$ $$\frac{0.6}{20} \cdot \frac{K^{0.4}}{L^{0.4}} = \frac{0.4}{40} \cdot \frac{L^{0.6}}{K^{0.6}}$$ $$0.03 \cdot \left(\frac{K}{L}\right)^{0.4} = 0.01 \cdot \left(\frac{L}{K}\right)^{0.6}$$ $$3 \cdot \left(\frac{K}{L}\right)^{0.4} = \left(\frac{L}{K}\right)^{0.6}$$ $$3 = \left(\frac{L}{K}\right)^{0.6} \cdot \left(\frac{L}{K}\right)^{0.4} = \frac{L}{K}$$ $$L = 3K$$ The optimal capital-labour ratio is $K/L = 1/3$. For every unit of capital, the firm should employ 3 units of labour. ## Oligopoly Models in Depth (HL Extension) ### Cournot Duopoly In the Cournot model (1838), each firm independently chooses its output quantity, taking the rival's output as given. Firms choose quantities simultaneously. **Setup:** Two firms (1 and 2) produce a homogeneous good with market demand $P = a - bQ$, where $Q = q_1 + q_2$. Each firm has constant marginal cost $c$. **Firm 1's profit:** $$\pi_1 = P \cdot q_1 - c \cdot q_1 = (a - bq_1 - bq_2 - c) \cdot q_1$$ **Best response function:** maximising $\pi_1$ with respect to $q_1$: $$\frac{\partial \pi_1}{\partial q_1} = a - 2bq_1 - bq_2 - c = 0$$ $$q_1 = \frac{a - c}{2b} - \frac{q_2}{2}$$ By symmetry, Firm 2's best response is: $$q_2 = \frac{a - c}{2b} - \frac{q_1}{2}$$ **Nash equilibrium (Cournot equilibrium):** substituting one best response into the other: $$q_1^* = q_2^* = \frac{a - c}{3b}$$ $$Q^* = \frac{2(a - c)}{3b}, \quad P^* = \frac{a + 2c}{3}$$ **Comparison with monopoly and perfect competition:** | Market Structure | Total Output | Price | | ------------------ | ----------------------- | ------------------------ | | Perfect competition | $Q_c = \frac{a-c}{b}$ | $P_c = c$ | | Cournot duopoly | $Q^{Co} = \frac{2(a-c)}{3b}$ | $P^{Co} = \frac{a+2c}{3}$ | | Monopoly | $Q_m = \frac{a-c}{2b}$ | $P_m = \frac{a+c}{2}$ | Cournot output is between monopoly and perfect competition. As the number of firms increases, Cournot output approaches the competitive level. ### Bertrand Duopoly In the Bertrand model (1883), each firm independently chooses its price, taking the rival's price as given. Firms set prices simultaneously. **Result with homogeneous goods:** the Nash equilibrium is both firms setting $P = MC = c$. If $P_1 > P_2 = c$, Firm 1 sells nothing. If $P_1 < P_2$, Firm 1 captures the entire market. Each firm has an incentive to undercut the other until price equals marginal cost. This is known as the **Bertrand paradox**: just two firms are sufficient to achieve the perfectly competitive outcome when products are homogeneous and firms compete on price. **With differentiated products:** the demand for each firm depends on both prices. If demand is: $$q_1 = a - bP_1 + dP_2$$ $$q_2 = a - bP_2 + dP_1$$ Where $d$ measures the degree of substitutability ($0 < d < b$). In this case, prices exceed marginal cost, and the equilibrium depends on the degree of differentiation. ### Stackelberg Duopoly In the Stackelberg model, one firm (the **leader**) moves first by choosing its quantity, and the other firm (the **follower**) observes this and then chooses its quantity. **Solution:** 1. The follower solves its Cournot best response given the leader's output $q_1$: $$q_2 = \frac{a - c}{2b} - \frac{q_1}{2}$$ 2. The leader anticipates this response and maximises its profit: $$\pi_1 = (a - bq_1 - bq_2(q_1) - c) \cdot q_1$$ $$\pi_1 = \left(a - bq_1 - b\left(\frac{a-c}{2b} - \frac{q_1}{2}\right) - c\right) q_1$$ $$\pi_1 = \left(\frac{a - c}{2} - \frac{bq_1}{2}\right) q_1$$ Maximising: $\frac{\partial \pi_1}{\partial q_1} = \frac{a - c}{2} - bq_1 = 0$ $$q_1^* = \frac{a - c}{2b}, \quad q_2^* = \frac{a - c}{4b}$$ **First-mover advantage:** the Stackelberg leader produces twice as much as the follower and earns higher profit. Total output is $Q^* = \frac{3(a-c)}{4b}$, which exceeds Cournot output but is still less than competitive output. This demonstrates the value of commitment and strategic leadership. ### Kinked Demand Curve: Formal Analysis The kinked demand curve model (Sweezy, 1939) explains price rigidity in oligopoly. The model assumes: - If a firm raises its price, rivals do **not** follow (demand is elastic above the kink) - If a firm lowers its price, rivals **match** the cut (demand is inelastic below the kink) This creates a discontinuity in the MR curve at the current price and quantity. The result is that small changes in marginal cost within the gap in the MR curve do not change the profit-maximising price, explaining price rigidity. **Limitations of the kinked demand curve model:** - It does not explain how the initial price is determined - It assumes asymmetric behaviour by rivals (matching price cuts but not price increases) - Empirical evidence for the model is mixed; price rigidity may be better explained by menu costs or implicit collusion ### Game Theory and Oligopoly Game theory provides a rigorous framework for analysing strategic interaction in oligopoly. **Payoff matrix example -- pricing game:** Two firms, A and B, can each set a High price or a Low price. Payoffs (profit in millions): | | B: High Price | B: Low Price | | ---------------- | ------------- | ------------ | | **A: High Price** | (10, 10) | (2, 12) | | **A: Low Price** | (12, 2) | (5, 5) | **Analysis:** - If B chooses High: A earns 10 (High) vs. 12 (Low). A chooses Low. - If B chooses Low: A earns 2 (High) vs. 5 (Low). A chooses Low. - Low is A's dominant strategy. By symmetry, Low is B's dominant strategy. - Nash equilibrium: (Low, Low) with payoffs (5, 5). - Both would be better off at (High, High) with (10, 10) -- a prisoner's dilemma. **Collusion and cartels as a prisoner's dilemma:** If firms successfully collude (agree to restrict output and raise prices), joint profits are maximised. However, each firm has an incentive to cheat by secretly lowering price or exceeding its quota. The payoff structure mirrors the prisoner's dilemma: | | B: Comply (cartel) | B: Cheat | | -------------------- | ------------------ | ----------- | | **A: Comply (cartel)**| (8, 8) | (3, 12) | | **A: Cheat** | (12, 3) | (5, 5) | The Nash equilibrium is (Cheat, Cheat), which is Pareto inferior to (Comply, Comply). This is why cartels are inherently unstable and require enforcement mechanisms (monitoring, penalties, or repeated interaction with the threat of punishment). **Sequential games in oligopoly:** In a sequential (Stackelberg) game, the first mover gains an advantage. The leader commits to a high output, and the follower responds with a lower output. This can be analysed using backward induction. ### Common Pitfalls in Oligopoly - Confusing Cournot and Bertrand models. Cournot assumes firms choose quantities; Bertrand assumes firms choose prices. The outcomes differ significantly. - Stating that the kinked demand curve explains why prices change. It actually explains why prices are stable (rigid). - Assuming that oligopoly always leads to higher prices than monopoly. In Bertrand competition with homogeneous goods, price equals marginal cost. - Forgetting that game theory predictions depend on the payoff structure. Different assumptions about demand, costs, and firm behaviour lead to different equilibria. ## Price Discrimination in Depth (HL Extension) ### First-Degree (Perfect) Price Discrimination The monopolist charges each consumer their exact maximum willingness to pay. The monopolist captures the entire consumer surplus. **Welfare effects:** - Consumer surplus falls to zero - Producer surplus increases by the full amount of the former consumer surplus - Deadweight loss is eliminated -- the monopolist produces the competitive quantity - Total output equals the perfectly competitive output (where $P = MC$) - Allocative efficiency is achieved, but all surplus goes to the producer $$\text{Profit} = \text{Total consumer surplus (under single price)} + \text{Producer surplus (under single price)}$$ **Requirements for first-degree price discrimination:** 1. Perfect knowledge of every consumer's willingness to pay 2. Ability to prevent resale 3. Market power In practice, perfect price discrimination is rare because firms cannot perfectly observe each consumer's willingness to pay. Approximations include personalised pricing (online, based on browsing history), negotiation (car dealerships, real estate), and auctions. ### Second-Degree Price Discrimination The monopolist offers different pricing options (quantity-based blocks, versioning, self-selection), and consumers self-select into the option that maximises their utility. **Block pricing:** the price per unit varies with the quantity purchased: $$\text{Total revenue} = P_1 Q_1 + P_2 (Q_2 - Q_1) + P_3 (Q_3 - Q_2) + \cdots$$ Where $P_1 > P_2 > P_3$ (declining block pricing). This captures some consumer surplus from consumers who buy in larger quantities. **Numerical example:** A cinema charges `USD 8` for the first ticket, `USD 6` for the second, and `USD 4` for each additional ticket (up to 5). A consumer who buys 4 tickets pays: $$\text{Total} = 8 + 6 + 4 + 4 = \text{USD 22}$$ Average price per ticket $= 22/4 = \text{USD 5.50$, which is less than the single-ticket price of USD 8.}$ ### Third-Degree Price Discrimination The monopolist separates consumers into groups with different price elasticities of demand and charges each group a different price. **Rule:** charge the higher price to the group with the less elastic demand. **Mathematical derivation:** For each market segment $i$, the monopolist maximises: $$\pi_i = P_i(Q_i) \cdot Q_i - C(Q_1 + Q_2 + \cdots + Q_n)$$ The first-order condition is: $$\mathrm{MR}_i = MC \quad \forall \; i$$ Since $\mathrm{MR}_i = P_i(1 + 1/\mathrm{PED}_i)$, and $\mathrm{MR}_i = MC$ for all segments: $$P_i(1 + 1/\mathrm{PED}_i) = P_j(1 + 1/\mathrm{PED}_j) = MC$$ Rearranging: $$\frac{P_i}{P_j} = \frac{1 + 1/\mathrm{PED}_j}{1 + 1/\mathrm{PED}_i}$$ If $|\mathrm{PED}_i| < |\mathrm{PED}_j|$ (market $i$ has less elastic demand), then $P_i > P_j$. **Worked example:** A monopolist serves two markets with demand curves: Market 1 (adults): $P_1 = 40 - Q_1$, $\mathrm{PED}_1 = -2$ at equilibrium Market 2 (students): $P_2 = 25 - Q_2$, $\mathrm{PED}_2 = -3$ at equilibrium Total cost: $\mathrm{TC} = 50 + 5(Q_1 + Q_2)$, so $MC = 5$. For each market: $MR = MC = 5$. Market 1: $\mathrm{MR}_1 = 40 - 2Q_1 = 5 \implies Q_1 = 17.5$, $P_1 = 40 - 17.5 = 22.5$ Market 2: $\mathrm{MR}_2 = 25 - 2Q_2 = 5 \implies Q_2 = 10$, $P_2 = 25 - 10 = 15$ Total profit $= P_1 Q_1 + P_2 Q_2 - \mathrm{TC} = 22.5(17.5) + 15(10) - 50 - 5(27.5)$ $= 393.75 + 150 - 50 - 137.5 = 356.25$ Verification: $\frac{P_1}{P_2} = \frac{22.5}{15} = 1.5$ $$\frac{1 + 1/\mathrm{PED}_2}{1 + 1/\mathrm{PED}_1} = \frac{1 + 1/(-3)}{1 + 1/(-2)} = \frac{1 - 1/3}{1 - 1/2} = \frac{2/3}{1/2} = \frac{4}{3} \approx 1.33$$ The slight discrepancy arises because PED values vary along the demand curve; the condition holds exactly only at the equilibrium quantities. **Conditions for price discrimination:** 1. Market power (price maker) 2. Ability to segment the market (prevent arbitrage/resale between segments) 3. Different price elasticities of demand in different segments **Welfare effects of third-degree price discrimination:** - Consumer surplus decreases in the less elastic market (higher price) and may increase in the more elastic market (lower price) - Total output may increase or decrease relative to single-price monopoly - If total output increases, welfare may increase (the DWL reduction from higher output may offset the DWL from misallocation between markets) - If total output decreases, welfare unambiguously decreases ### Common Pitfalls in Price Discrimination - Assuming that price discrimination always reduces consumer surplus. Consumers in the more elastic market may face lower prices than under uniform pricing. - Confusing price discrimination with price differentiation due to cost differences. Price discrimination requires the same product to be sold at different prices not justified by cost differences. - Stating that a monopolist charges "whatever they want" under price discrimination. Even with price discrimination, the monopolist must optimise within the constraints of each market's demand curve. ## Factor Markets (HL Extension) ### Factor Demand A firm demands factors of production (labour, capital, land) to produce output. The demand for a factor is a **derived demand** -- it depends on the demand for the product that the factor produces. ### Marginal Revenue Product (MRP) The marginal revenue product of labour is the additional revenue generated by employing one more unit of labour: $$\mathrm{MRP}_L = \mathrm{MR} \times \mathrm{MP}_L$$ Where $\mathrm{MP}_L$ is the marginal product of labour and $\mathrm{MR}$ is marginal revenue. Under perfect competition in the product market ($\mathrm{MR} = P$): $$\mathrm{MRP}_L = P \times \mathrm{MP}_L = \mathrm{VMP}_L$$ Where $\mathrm{VMP}_L$ is the value of the marginal product. Under monopoly ($\mathrm{MR} < P$): $$\mathrm{MRP}_L < \mathrm{VMP}_L$$ A profit-maximising firm hires labour up to the point where: $$\mathrm{MRP}_L = \mathrm{MRC}_L$$ Where $\mathrm{MRC}_L$ (marginal resource cost) is the additional cost of hiring one more unit of labour. ### Marginal Resource Cost (MRC) In a perfectly competitive labour market, the firm is a wage taker and $\mathrm{MRC}_L = w$ (the wage rate). The firm faces a horizontal supply curve for labour. $$\text{Labour demand rule (perfect competition): } \mathrm{MRP}_L = w$$ ### Monopsony A monopsony is a market with a single buyer of a factor of production (typically labour). The monopsonist faces an upward-sloping supply curve for labour and must pay a higher wage to attract additional workers. If the labour supply curve is $w = a + bL$, the total cost of labour is: $$\mathrm{TC}_L = w \cdot L = (a + bL) \cdot L = aL + bL^2$$ $$\mathrm{MRC}_L = \frac{d\mathrm{TC}_L}{dL} = a + 2bL$$ The MRC curve lies above the supply (average cost of labour) curve. The monopsonist hires where $\mathrm{MRP}_L = \mathrm{MRC}_L$, but pays the wage indicated by the supply curve at that quantity. **Comparison with competitive labour market:** - A monopsony hires **fewer** workers than a competitive market - A monopsony pays a **lower** wage than a competitive market - There is a deadweight loss from the underemployment of labour **Worked example:** A monopsonist faces labour supply $w = 20 + 2L$ and $\mathrm{MRP}_L = 100 - 2L$. $\mathrm{MRC}_L = 20 + 4L$ Setting $\mathrm{MRP}_L = \mathrm{MRC}_L$: $$100 - 2L = 20 + 4L \implies 80 = 6L \implies L^* = 13.33$$ Wage paid $= 20 + 2(13.33) = 46.67$ Under competitive conditions: $\mathrm{MRP}_L = w$, so $100 - 2L = 20 + 2L \implies 80 = 4L \implies L_c = 20$, $w_c = 20 + 2(20) = 60$. The monopsony hires 13.33 workers at a wage of 46.67, compared to 20 workers at a wage of 60 under competition. The deadweight loss arises from the 6.67 workers who would be employed competitively (between $L^* = 13.33$ and $L_c = 20$) but are not hired by the monopsonist. ### Transfer Earnings and Economic Rent **Transfer earnings** are the minimum payment required to keep a factor of production in its current use. If a factor earns less than its transfer earnings, it will move to its next best alternative. **Economic rent** is any payment above transfer earnings: $$\text{Economic rent} = \text{Total factor income} - \text{Transfer earnings}$$ The proportion of economic rent to total income depends on the elasticity of the factor supply: - **Perfectly inelastic supply** (vertical): all income is economic rent (e.g., land in the short run) - **Perfectly elastic supply** (horizontal): all income is transfer earnings (e.g., unskilled labour in a competitive market) - **Upward-sloping supply**: income is a mix of transfer earnings and economic rent **Factor mobility:** - **Occupationally mobile**: workers can move between different types of jobs (requires training, education) - **Geographically mobile**: workers can move between different locations - High factor mobility increases the elasticity of factor supply, reducing economic rent - Low factor mobility decreases the elasticity of factor supply, increasing economic rent ### Common Pitfalls in Factor Markets - Confusing MRP with VMP. MRP $= MR \times MP_L$; VMP $= P \times MP_L$. They are equal only under perfect competition in the product market. - Confusing monopsony with monopoly. Monopoly is a single seller; monopsony is a single buyer. - Stating that a minimum wage always reduces employment. In a monopsony, a minimum wage set at or below the competitive wage can increase both employment and wages. - Confusing transfer earnings with economic profit. Transfer earnings are a cost to the firm; they are part of the normal return to the factor. ## Government Intervention: Welfare Analysis (HL Extension) ### Deadweight Loss Calculations The deadweight loss (DWL) from any government intervention that creates a wedge between the marginal benefit and marginal cost of a good can be calculated using the formula: $$\mathrm{DWL} = \frac{1}{2} \times (P_{\text{distortion}}) \times (Q_{\text{loss}})$$ Where $P_{\text{distortion}}$ is the difference between the marginal benefit and marginal cost at the distorted quantity, and $Q_{\text{loss}}$ is the reduction in quantity from the efficient level. ### Comprehensive Welfare Analysis of a Tax For a specific tax $t$ on a good with linear demand $P = a - bQ$ and supply $P = c + dQ$: **Pre-tax equilibrium:** $$a - bQ = c + dQ \implies Q_0 = \frac{a - c}{b + d}, \quad P_0 = \frac{ad + bc}{b + d}$$ **Post-tax equilibrium:** supply shifts to $P = c + dQ + t$: $$a - bQ_t = c + dQ_t + t \implies Q_t = \frac{a - c - t}{b + d}$$ $$P_d = a - bQ_t = a - \frac{b(a - c - t)}{b + d} = \frac{a(b + d) - b(a - c - t)}{b + d} = \frac{ad + bc + bt}{b + d}$$ $$P_s = P_d - t = \frac{ad + bc - dt}{b + d}$$ **Welfare changes:** $$\Delta\mathrm{CS} = -\frac{1}{2}(P_d - P_0)(Q_0 + Q_t)$$ $$\Delta\mathrm{PS} = -\frac{1}{2}(P_0 - P_s)(Q_0 + Q_t)$$ $$\text{Tax revenue} = t \times Q_t$$ $$\mathrm{DWL} = \frac{1}{2} \times t \times (Q_0 - Q_t) = \frac{t^2}{2(b + d)}$$ ### Welfare Analysis of a Subsidy For a per-unit subsidy $s$, the supply curve shifts downward to $P = c + dQ - s$: $$a - bQ = c + dQ - s \implies Q_s = \frac{a - c + s}{b + d}$$ $$P_d = a - bQ_s = \frac{ad + bc - bs}{b + d}$$ $$P_s = P_d + s = \frac{ad + bc + ds}{b + d}$$ The subsidy cost is $s \times Q_s$. The DWL of the subsidy is: $$\mathrm{DWL} = \frac{1}{2} \times s \times (Q_s - Q_0) = \frac{s^2}{2(b + d)}$$ ### Welfare Analysis of a Price Ceiling A binding price ceiling $P_{\text{ceil}} < P_0$ creates: - Quantity demanded: $Q_d = (a - P_{\text{ceil}}) / b$ - Quantity supplied: $Q_s = (P_{\text{ceil}} - c) / d$ - Shortage: $Q_d - Q_s$ $$\Delta\mathrm{CS} = (P_0 - P_{\text{ceil}})Q_s - \frac{1}{2}(P_0 - P_{\text{ceil}})(Q_0 - Q_s) - \frac{1}{2}(P_0 - P_{\text{ceil}})(Q_d - Q_0)$$ The first term is the gain to consumers who still buy the good. The second and third terms are losses from reduced consumption. The net effect is ambiguous and depends on the parameters. $$\mathrm{DWL} = \frac{1}{2}(P_{\text{ceil}} - c)(Q_0 - Q_s) + \frac{1}{2}(a - P_{\text{ceil}})(Q_d - Q_0)$$ The first triangle is the loss from inefficiently low production. The second is the loss from foregone mutually beneficial transactions. ### Welfare Analysis of a Price Floor A binding price floor $P_{\text{floor}} > P_0$ creates: - Quantity demanded: $Q_d = (a - P_{\text{floor}}) / b$ - Quantity supplied: $Q_s = (P_{\text{floor}} - c) / d$ - Surplus: $Q_s - Q_d$ $$\mathrm{DWL} = \frac{1}{2}(P_{\text{floor}} - c)(Q_s - Q_0) + \frac{1}{2}(a - P_{\text{floor}})(Q_0 - Q_d)$$ If the government purchases the surplus at the floor price, the total cost is $P_{\text{floor}} \times (Q_s - Q_d)$, adding to the welfare loss. ### Common Pitfalls in Welfare Analysis - Confusing the change in consumer surplus with the change in total welfare. A tax reduces both CS and PS, and the net welfare loss is the DWL. - Forgetting that the DWL triangle depends on the change in quantity, not the change in price. - Calculating government revenue from a quota as positive. Unless licences are auctioned, the quota revenue may accrue to foreign producers or importers, not the government. - Drawing the DWL on the wrong side of the supply and demand curves. Always identify the pre- and post-intervention quantities and the area between the supply and demand curves. ## Additional Practice Problems <details> <summary>Problem 8: Consumer Choice and MRS</summary> A consumer has utility function $U = X \cdot Y$ and income $M = 200$. The price of $X$ is $P_X = 10$ and the price of $Y$ is $P_Y = 5$. (a) Find the optimal consumption bundle. (b) If the price of $X$ rises to $P_X = 20$, find the new optimal bundle. (c) Decompose the change in $X$ consumption into substitution and income effects. (a) $\mathrm{MRS}_{XY} = \frac{MU_X}{MU_Y} = \frac{Y}{X}$ Setting $\mathrm{MRS} = P_X / P_Y$: $$\frac{Y}{X} = \frac{10}{5} = 2 \implies Y = 2X$$ Budget constraint: $10X + 5Y = 200$ $10X + 5(2X) = 200 \implies 20X = 200 \implies X^* = 10, \quad Y^* = 20$ (b) New MRS condition: $\frac{Y}{X} = \frac{20}{5} = 4 \implies Y = 4X$ New budget constraint: $20X + 5Y = 200$ $20X + 5(4X) = 200 \implies 40X = 200 \implies X^{**} = 5, \quad Y^{**} = 20$ (c) To decompose: at the new prices, find the bundle that gives the original utility $U = 10 \times 20 = 200$. $XY = 200$ and $Y = 4X \implies 4X^2 = 200 \implies X^2 = 50 \implies X^C = \sqrt{50} \approx 7.07$ $Y^C = 4 \times 7.07 = 28.28$ **Substitution effect:** $X^C - X^* = 7.07 - 10 = -2.93$ (decrease in $X$ due to substitution toward $Y$) **Income effect:** $X^{**} - X^C = 5 - 7.07 = -2.07$ (decrease in $X$ due to reduced purchasing power) **Total effect:** $-2.93 + (-2.07) = -5$ (from 10 to 5) Both effects reduce consumption of $X$, confirming it is a normal good. </details> <details> <summary>Problem 9: Cournot Duopoly with Calculation</summary> Two firms compete in a Cournot duopoly. Market demand is $P = 100 - Q$ where $Q = q_1 + q_2$. Both firms have $MC = 10$. (a) Find the Cournot equilibrium output, price, and profit for each firm. (b) Compare with the monopoly outcome. (c) What would happen if the firms colluded? (a) Firm 1's profit: $\pi_1 = (100 - q_1 - q_2 - 10)q_1 = (90 - q_1 - q_2)q_1$ FOC: $90 - 2q_1 - q_2 = 0 \implies q_1 = 45 - 0.5q_2$ By symmetry: $q_2 = 45 - 0.5q_1$ Substituting: $q_1 = 45 - 0.5(45 - 0.5q_1) = 45 - 22.5 + 0.25q_1 = 22.5 + 0.25q_1$ $0.75q_1 = 22.5 \implies q_1^* = q_2^* = 30$ $Q^* = 60$, $P^* = 100 - 60 = 40$ $\pi_1 = \pi_2 = (40 - 10) \times 30 = 900$ (b) Monopoly: $\mathrm{MR} = 100 - 2Q = MC = 10 \implies Q_m = 45$, $P_m = 55$ $\pi_m = (55 - 10) \times 45 = 2025$ Under Cournot, total output is higher (60 vs. 45), price is lower (40 vs. 55), and total profit is lower (1800 vs. 2025). (c) If firms collude (act as a monopoly), each produces $Q_m / 2 = 22.5$. $\pi_{\text{collusion}} = (55 - 10) \times 22.5 = 1012.5$ per firm. Each firm earns more under collusion ($1012.5$) than under Cournot ($900$), but each has an incentive to cheat: if Firm 1 cheats and Firm 2 produces $22.5$: $q_1 = 45 - 0.5(22.5) = 33.75$, $Q = 56.25$, $P = 43.75$ $\pi_1^{\text{cheat}} = (43.75 - 10) \times 33.75 = 1139.06 > 1012.5$ This confirms the prisoner's dilemma structure of collusion. </details> <details> <summary>Problem 10: Third-Degree Price Discrimination</summary> A cinema serves two markets: adults and students. The demand curves are: Adults: $P_A = 30 - 0.5Q_A$ Students: $P_S = 20 - 0.5Q_S$ The cinema's marginal cost is constant at $MC = 4$ per ticket. There are no fixed costs. (a) Calculate the profit-maximising price and quantity for each market under third-degree price discrimination. (b) Calculate total profit with and without price discrimination (assuming a single price for all customers with combined demand). (a) Market A: $\mathrm{MR}_A = 30 - Q_A = 4 \implies Q_A = 26$, $P_A = 30 - 13 = 17$ Market S: $\mathrm{MR}_S = 20 - Q_S = 4 \implies Q_S = 16$, $P_S = 20 - 8 = 12$ $\pi_A = (17 - 4) \times 26 = 338$ $\pi_S = (12 - 4) \times 16 = 128$ Total profit $= 338 + 128 = 466$ (b) Combined demand: $P = 30 - 0.5Q$ for $Q \leq 20$ (when $P > 20$), and $P = 20 - 0.5(Q - 20) = 30 - 0.5Q$ for $Q > 20$. Actually, we need to sum horizontally. For $P \geq 20$: only adults buy. $Q_A = 60 - 2P$, $Q_S = 0$. For $P < 20$: both buy. $Q_A = 60 - 2P$, $Q_S = 40 - 2P$. Total $Q = 100 - 4P$, so $P = 25 - 0.25Q$. $\mathrm{MR} = 25 - 0.5Q = 4 \implies Q = 42$ $P = 25 - 0.25(42) = 14.5$ $Q_A = 60 - 2(14.5) = 31$, $Q_S = 40 - 2(14.5) = 11$ $\pi_{\text{single}} = (14.5 - 4) \times 42 = 441$ Price discrimination yields higher profit ($466 > 441$), confirming that the firm benefits from segmenting the market. Students pay less ($12$ vs. $14.5$), while adults pay more ($17$ vs. $14.5$). </details> <details> <summary>Problem 11: Monopsony and Minimum Wage</summary> A coal mine is the sole employer in a remote town. The labour supply is $w = 10 + L$ and the $\mathrm{MRP}_L = 50 - 2L$. (a) Find the profit-maximising wage and employment level. (b) Calculate the deadweight loss compared to a competitive labour market. (c) The government introduces a minimum wage of $w = 22$. Analyse the effect on employment and wages. (a) $\mathrm{MRC}_L = 10 + 2L$ Setting $\mathrm{MRP}_L = \mathrm{MRC}_L$: $50 - 2L = 10 + 2L \implies 40 = 4L \implies L^* = 10$ Wage $= 10 + 10 = 20$ (b) Competitive equilibrium: $\mathrm{MRP}_L = w$, so $50 - 2L = 10 + L \implies 40 = 3L \implies L_c = 13.33$, $w_c = 23.33$. The monopsony under-employs by $13.33 - 10 = 3.33$ workers. The DWL is: $$\mathrm{DWL} = \frac{1}{2} \times (\mathrm{MRP}_{L=10} - w_{L=10}) \times (L_c - L^*)$$ At $L = 10$: $\mathrm{MRP}_L = 50 - 20 = 30$, $w = 20$. $\mathrm{DWL} = \frac{1}{2} \times (30 - 20) \times (13.33 - 10) = \frac{1}{2} \times 10 \times 3.33 = 16.67$ (c) With minimum wage $w_{\min} = 22$: For $L \leq 12$ (where supply wage $= 22$), the $\mathrm{MRC}$ is constant at $22$. For $L > 12$, the firm must pay above $22$ to attract more workers. The firm now faces: $\mathrm{MRC} = 22$ for $L \leq 12$. $\mathrm{MRP}_L = 22 \implies 50 - 2L = 22 \implies 2L = 28 \implies L = 14$. But at $L = 14$, the supply wage would be $10 + 14 = 24 > 22$, so the firm must pay $24$ for the 14th worker. The $\mathrm{MRC}$ jumps at $L = 12$. For $L > 12$: $\mathrm{MRC}_L = 10 + 2L$ (back to the original supply curve). Setting $\mathrm{MRP}_L = \mathrm{MRC}_L$ at $22$: the firm hires $L = 14$ workers at a wage of $w = \max(22, 10 + 14) = 24$ for the 14th worker. But the minimum wage is only binding up to $L = 12$. More precisely, the firm faces $\mathrm{MRC} = 22$ for the first 12 workers. The $\mathrm{MRP}$ at $L = 12$ is $50 - 24 = 26 > 22$, so the firm wants to hire more. For $L > 12$, the $\mathrm{MRC}$ reverts to $10 + 2L$. Setting $50 - 2L = 10 + 2L \implies L = 10$, but this is at the original equilibrium. The firm hires where the horizontal portion of MRC (at 22) intersects MRP: $50 - 2L = 22 \implies L = 14$ Since $14 > 12$, the firm must check whether the 13th and 14th workers have $\mathrm{MRC} > 22$. At $L = 13$: $\mathrm{MRC} = 10 + 2(13) = 36 > \mathrm{MRP}_{L=13} = 50 - 26 = 24$. So the firm will not hire the 13th worker. The firm hires $L = 12$ workers at $w = 22$. Employment increased from 10 to 12 (compared to the monopsony without minimum wage), and the wage increased from 20 to 22. This demonstrates that in a monopsony, a minimum wage can increase both employment and wages -- the opposite of the competitive case. </details> <details> <summary>Problem 12: Comprehensive Welfare Analysis</summary> A market has demand $Q_d = 100 - P$ and supply $Q_s = P - 20$. (a) Find the equilibrium and calculate total welfare (CS + PS). (b) The government imposes a price floor at $P = 65$ and purchases the resulting surplus. Calculate the change in welfare and the cost to the government. (c) Alternatively, the government provides a subsidy of `USD 10` per unit. Compare the welfare effects with the price floor. (a) Equilibrium: $100 - P = P - 20 \implies 2P = 120 \implies P_0 = 60$, $Q_0 = 40$. $\mathrm{CS} = \frac{1}{2}(100 - 60)(40) = 800$ $\mathrm{PS} = \frac{1}{2}(60 - 20)(40) = 800$ Total welfare $= 1600$. (b) Price floor at $P_f = 65$: $Q_d = 100 - 65 = 35$, $Q_s = 65 - 20 = 45$. Surplus $= 45 - 35 = 10$ units. Government purchases 10 units at `USD 65` each. Cost to government $= 65 \times 10 = 650$. New CS $= \frac{1}{2}(100 - 65)(35) = 612.5$. $\Delta\mathrm{CS} = 612.5 - 800 = -187.5$ New PS $= \frac{1}{2}(65 - 20)(45) = 1012.5$. $\Delta\mathrm{PS} = 1012.5 - 800 = +212.5$ Government cost $= 650$. However, the government acquires goods worth $(100 - 65)/2 \times 10 + 35 \times 10$ ... more simply, the government's surplus from buying and destroying the goods is zero. $\mathrm{DWL} = \frac{1}{2}(65 - 60)(40 - 35) + \frac{1}{2}(65 - 60)(45 - 40) = \frac{1}{2}(5)(5) + \frac{1}{2}(5)(5) = 12.5 + 12.5 = 25$ Total welfare change $= -187.5 + 212.5 - 650 = -625$. The net welfare loss including government spending is `USD 625`. (c) Subsidy of $s = 10$: new supply is $Q_s = P - 20 + 10$, or $P = Q_s + 10$. The supply curve shifts down. In demand-supply form: $Q_d = 100 - P$, $P = Q_s + 10$, so $Q_s = P - 10$. $100 - P = P - 10 \implies 2P = 110 \implies P_d = 55$, $P_s = 45$. $Q_s = 45$. Subsidy cost $= 10 \times 45 = 450$. New CS $= \frac{1}{2}(100 - 55)(45) = 1012.5$. $\Delta\mathrm{CS} = +212.5$ New PS $= \frac{1}{2}(45 - 20)(45) = 562.5$. $\Delta\mathrm{PS} = 562.5 - 800 = -237.5$ $\mathrm{DWL} = \frac{1}{2} \times 10 \times (45 - 40) = 25$ Net welfare change (including government cost) $= 212.5 - 237.5 - 450 = -475$ Both policies create the same DWL of 25 (coincidentally, because the quantity distortions are the same). However, the price floor costs the government more (650 vs. 450) and creates a larger net welfare loss. The subsidy is less costly overall because consumers benefit from lower prices. </details> ## Prospect Theory and Behavioural Biases (HL Extension) ### Kahneman and Tversky's Prospect Theory **Prospect theory** (1979) challenges the standard expected utility model by demonstrating that individuals evaluate outcomes relative to a reference point, overweight small probabilities, and are loss-averse. **Key departures from expected utility theory:** 1. **Reference dependence:** utility is measured from gains and losses relative to a reference point, not from absolute wealth levels 2. **Loss aversion:** losses loom larger than equivalent gains. The pain of losing USD 100 is approximately twice the pleasure of gaining USD 100 3. **Diminishing sensitivity:** the marginal impact of gains or losses decreases with distance from the reference point 4. **Probability weighting:** individuals overweight small probabilities and underweight large probabilities **The value function:** $$v(x) = \begin{cases} x^\alpha & \text{if } x \geq 0 \text{ (gains)} \\ -\lambda(-x)^\beta & \text{if } x < 0 \text{ (losses)} \end{cases}$$ Where $\alpha, \beta < 1$ (diminishing sensitivity), $\lambda > 1$ (loss aversion; typically $\lambda \approx 2.25$). **The probability weighting function:** $$\pi(p) = \frac{p^\gamma}{(p^\gamma + (1-p)^\gamma)^{1/\gamma}}$$ Where $\gamma < 1$ for most individuals, causing the inverse-S-shaped weighting function. ### Implications for Economic Behaviour 1. **Endowment effect:** people value goods they own more than identical goods they do not own. This is explained by loss aversion: giving up a good is perceived as a loss, which is weighted more heavily than the equivalent gain from acquiring it 2. **Status quo bias:** the tendency to prefer the current state of affairs. Changing the status quo involves potential losses (what might go wrong) that are overweighted 3. **Framing effects:** the same outcome presented differently (as a gain or a loss) leads to different choices. A surgery with a "90% survival rate" is preferred to one with a "10% mortality rate" despite being identical 4. **Mental accounting:** individuals categorise money into different "accounts" (rent, food, entertainment) and treat money differently depending on the account, violating fungibility ### Numerical Example: Loss Aversion An individual with loss aversion parameter $\lambda = 2$ is offered a gamble: - 50% chance of winning USD 200 - 50% chance of losing USD 100 Expected value $= 0.5 \times 200 + 0.5 \times (-100) = +50$ Under expected utility theory, a risk-neutral individual would accept (positive expected value). Under prospect theory (assuming linear value function for simplicity): $V(\text{gamble}) = 0.5 \times v(200) + 0.5 \times v(-100) = 0.5 \times 200 + 0.5 \times (-2 \times 100)$ $= 100 - 100 = 0$ The individual is indifferent between the gamble and the status quo, despite the positive expected value. With $\lambda = 2.5$: $V = 0.5 \times 200 + 0.5 \times (-2.5 \times 100) = 100 - 125 = -25$ The individual rejects the gamble, illustrating loss aversion. ### Nudge Theory (Thaler and Sunstein) A **nudge** is any aspect of the choice architecture that alters people's behaviour in a predictable way without forbidding any options or significantly changing their economic incentives. **Examples of nudges in policy:** 1. **Default options:** organ donation (opt-out systems have 90%+ participation vs. 15% in opt-in systems), pension plan enrolment 2. **Social norms:** displaying energy consumption relative to neighbours reduces energy use by 2--6% 3. **Simplified information:** clearer nutrition labels, plain-language financial disclosures 4. **Commitment devices:** pre-commitment to savings goals, with penalties for withdrawal 5. **Framing:** presenting tax as a contribution to public goods rather than a burden **Evaluation of nudge theory:** - **Advantages:** low cost, preserves freedom of choice, empirically effective in many contexts - **Disadvantages:** may be manipulative, effects may be small and temporary, raises ethical questions about who decides what behaviour is "correct," may distract from structural reforms ## Anchoring and Adjustment Heuristic (HL Extension) ### Definition and Mechanism **Anchoring** is a cognitive bias whereby individuals rely too heavily on an initial piece of information (the "anchor") when making decisions, even when the anchor is irrelevant. **Mechanism:** individuals start from the anchor and adjust insufficiently to reach their final estimate. The adjustment process is cognitively effortful, so people tend to "stick" close to the anchor. ### Experimental Evidence **Kahneman and Tversky (1974):** participants were asked to estimate the percentage of African countries in the UN. Those who first saw the number 65 guessed 45% on average; those who saw 10 guessed 25%. A completely irrelevant number significantly influenced estimates. **Ariely, Loewenstein, and Prelec (2003):** students wrote down the last two digits of their social security number and then bid on items (wine, chocolate, books). Students with higher SSN digits bid 60--120% more than those with lower digits. ### Economic Applications 1. **Wage negotiations:** the first number mentioned in a salary negotiation serves as an anchor, significantly influencing the final outcome 2. **Pricing strategies:** "was USD 200, now USD 100" uses the original price as an anchor to make the discount appear larger 3. **Real estate:** listing prices anchor buyer expectations. Properties listed 10% above market value still sell close to that price 4. **Courtroom damages:** plaintiffs who request higher initial awards tend to receive higher settlements, even when judges and jurors are instructed to ignore the request ## Contestable Markets (HL Extension) ### Theory of Contestable Markets A **contestable market** is one with no barriers to entry or exit, even if there is only one firm (a natural monopoly). The threat of potential entry constrains the behaviour of the incumbent. **Key conditions:** 1. **No sunk costs:** firms can enter and exit without irrecoverable costs 2. **No barriers to entry:** no legal, technological, or strategic barriers 3. **Perfect information:** potential entrants know the incumbent's costs and demand 4. **Hit-and-run entry:** firms can enter, capture profits, and exit before the incumbent can respond ### The Contestable Market Outcome Even a monopoly in a perfectly contestable market will set $P = \text{AC}$ (average cost pricing) and produce at the efficient scale. If $P > \text{AC}$, potential entrants can profitably enter, drive the price down to AC, and earn zero economic profit. $$\text{Sustainable monopoly: } P = \text{AC}, \pi = 0$$ This is in contrast to an unregulated monopoly, which sets $P > \text{AC}$ and earns positive economic profit. ### Comparison: Contestable vs. Monopolistic Market | Feature | Unregulated monopoly | Contestable monopoly | |---|---|---| | Price | $P > \text{MC}$, $P > \text{AC}$ | $P = \text{AC} > \text{MC}$ | | Output | $Q < Q_{\text{efficient}}$ | Higher than unregulated monopoly | | Profit | $\pi > 0$ | $\pi = 0$ | | Efficiency | Allocatively and productively inefficient | Productively efficient ($P = \text{AC}$), allocatively inefficient ($P > \text{MC}$) | | X-inefficiency | Likely (no competitive pressure) | Unlikely (threat of entry disciplines the firm) | ### Limitations of Contestable Market Theory 1. **Sunk costs are pervasive:** most industries have significant sunk costs (brand development, specialised equipment, regulatory compliance) 2. **Strategic barriers:** incumbents can engage in limit pricing, predatory pricing, or excess capacity to deter entry 3. **Information asymmetry:** potential entrants rarely have perfect information about the incumbent's costs 4. **Speed of response:** in many industries, incumbents can respond quickly enough to prevent profitable hit-and-run entry ## Efficiency vs. Equity: Detailed Analysis (HL Extension) ### The Fundamental Trade-off **Efficiency** refers to maximising total surplus (the sum of consumer and producer surplus). **Equity** refers to the fairness of the distribution of surplus among members of society. The trade-off arises because policies that improve equity (e.g., redistribution through taxation) typically reduce efficiency (e.g., by creating deadweight loss from taxation). ### Types of Efficiency 1. **Allocative efficiency:** $P = \text{MC}$ (the value consumers place on the last unit equals the cost of producing it) 2. **Productive efficiency:** production at minimum average cost 3. **Dynamic efficiency:** the rate of innovation and technological progress over time 4. **Pareto efficiency:** no reallocation can make anyone better off without making someone worse off ### Types of Equity 1. **Horizontal equity:** equals should be treated equally (people in similar circumstances should pay similar taxes) 2. **Vertical equity:** unequals should be treated unequally (those with greater ability to pay should pay more) 3. **Intergenerational equity:** fairness between current and future generations 4. **Procedural equity:** fairness of the process by which outcomes are determined ### The Equity-Efficiency Trade-off in Practice **Progressive taxation:** A progressive income tax with a top marginal rate of 50% reduces inequality but creates a deadweight loss. The DWL depends on the elasticity of labour supply: $$\text{DWL} \approx \frac{1}{2} \times \epsilon_L \times t^2 \times w \times L$$ Where $\epsilon_L$ is the labour supply elasticity, $t$ is the tax rate, $w$ is the wage rate, and $L$ is labour supply. If $\epsilon_L = 0.5$ and $t = 0.5$: $\text{DWL} \approx \frac{1}{2} \times 0.5 \times 0.25 \times w \times L = 0.0625 \times w \times L$ The DWL is 6.25% of total labour income. Whether this trade-off is worthwhile depends on the social value placed on equality. **Rawlsian vs. utilitarian perspectives:** - **Rawls (maximin):** policy should maximise the welfare of the worst-off member of society. Redistribution is justified even at significant efficiency cost - **Utilitarian:** policy should maximise total welfare. Redistribution is justified only if the marginal utility of income for the poor exceeds the marginal efficiency loss from taxation - **Nozick (entitlement):** any redistribution beyond voluntary exchange is unjust. The distribution resulting from free exchange is inherently fair ## Government Failure (HL Extension) ### Definition **Government failure** occurs when government intervention in the market leads to a net reduction in economic welfare, i.e., the costs of intervention exceed the benefits. ### Types of Government Failure 1. **Information failure:** governments may lack the information needed to design effective policies. Central planners cannot aggregate dispersed knowledge as effectively as price signals (Hayek's knowledge problem) 2. **Regulatory capture:** regulated industries influence the regulators to act in the industry's interest rather than the public interest. Agencies may become "captured" by the firms they regulate 3. **Principal-agent problems:** elected officials (principals) may not be able to control bureaucrats (agents) who pursue their own objectives (budget maximisation, empire building) 4. **Short-termism:** democratic political cycles incentivise policies with immediate, visible benefits and deferred costs (e.g., debt accumulation, underinvestment in infrastructure) 5. **Rent-seeking:** resources are wasted on lobbying for government favours rather than productive activity. The total cost of rent-seeking can exceed the deadweight loss of the market failure it aims to correct 6. **Government bureaucracy and inflexibility:** government agencies may be slower to adapt than private firms, leading to inefficiency 7. **Unintended consequences:** policies may have secondary effects that offset or exceed the intended benefits ### Rent-Seeking: Detailed Analysis **Definition:** rent-seeking is the expenditure of resources to obtain a transfer of wealth without creating any new wealth. **Example:** Suppose the government introduces a tariff that transfers USD 100 million from consumers to producers. The DWL of the tariff is USD 20 million. Producers may spend up to USD 100 million on lobbying to secure the tariff. The total social cost is: $$\text{Total cost} = \text{DWL} + \text{Rent-seeking expenditure} = 20 + 100 = 120$$ The total social cost of the tariff exceeds the DWL by six times, because the rent-seeking expenditure is a pure waste of resources. **Tullock's paradox:** why is rent-seeking expenditure typically much smaller than the potential transfer? Possible explanations include: free-rider problems among beneficiaries, uncertainty about policy outcomes, and ethical constraints on bribery. ## Poverty and Inequality Measurement (HL Extension) ### Measuring Inequality **Lorenz curve:** plots the cumulative share of income (or wealth) received by the cumulative share of the population, ordered from poorest to richest. **Gini coefficient:** $$G = \frac{A}{A + B}$$ Where $A$ is the area between the line of perfect equality and the Lorenz curve, and $B$ is the area under the Lorenz curve. $G \in [0, 1]$, where 0 is perfect equality and 1 is perfect inequality. **Interpretation of Gini coefficients:** - $G < 0.25$: low inequality (e.g., Denmark 0.28, Norway 0.27) - $0.25 \leq G < 0.35$: moderate inequality (e.g., UK 0.35, France 0.32) - $0.35 \leq G < 0.50$: high inequality (e.g., USA 0.41, China 0.47) - $G \geq 0.50$: very high inequality (e.g., South Africa 0.63, Brazil 0.53) ### The Palma Ratio The Palma ratio is the ratio of the income share of the top 10% to the income share of the bottom 40%: $$\text{Palma} = \frac{S_{\text{top 10\%}}}{S_{\text{bottom 40\%}}}$$ The Palma ratio focuses on the "tails" of the distribution because the middle 50% of the population captures approximately 50% of income in most countries, so inequality is driven primarily by differences at the extremes. **Advantages over the Gini coefficient:** - More intuitive and easier to interpret - More sensitive to changes at the top and bottom of the distribution - Less sensitive to middle-income changes that are less policy-relevant ### Measuring Poverty 1. **Absolute poverty:** income below a fixed threshold (World Bank: USD 2.15/day at 2017 PPP) 2. **Relative poverty:** income below a percentage of median income (OECD: below 60% of median equivalised disposable income) 3. **Multidimensional Poverty Index (MPI):** measures poverty across health, education, and living standards (10 indicators including nutrition, school attendance, electricity, sanitation) 4. **Human Development Index (HDI):** composite of life expectancy, education (mean and expected years of schooling), and GNI per capita **Numerical example:** A country has the following income distribution (quintiles): | Quintile | Share of income | |---|---| | Bottom 20% | 5% | | Second 20% | 10% | | Middle 20% | 15% | | Fourth 20% | 20% | | Top 20% | 50% | Gini coefficient approximation: $$G = 1 - \sum_{i=1}^{5} (X_i - X_{i-1})(Y_i + Y_{i-1})$$ Where $X_i$ is the cumulative population share and $Y_i$ is the cumulative income share. | $i$ | $X_i$ | $Y_i$ | $X_i - X_{i-1}$ | $Y_i + Y_{i-1}$ | Product | |---|---|---|---|---|---| | 0 | 0 | 0 | -- | -- | -- | | 1 | 0.2 | 0.05 | 0.2 | 0.05 | 0.010 | | 2 | 0.4 | 0.15 | 0.2 | 0.20 | 0.040 | | 3 | 0.6 | 0.30 | 0.2 | 0.45 | 0.090 | | 4 | 0.8 | 0.50 | 0.2 | 0.80 | 0.160 | | 5 | 1.0 | 1.00 | 0.2 | 1.50 | 0.300 | $G = 1 - (0.010 + 0.040 + 0.090 + 0.160 + 0.300) = 1 - 0.600 = 0.400$ This indicates high inequality (comparable to the USA). Palma ratio $= 50\% / 15\% = 3.33$. ## Worked Examples: Microeconomics (HL Extension) <details> <summary>Problem 8: Prospect Theory Application</summary> An investor with loss aversion parameter $\lambda = 2$ and value function $v(x) = x^{0.88}$ for gains and $v(x) = -2|x|^{0.88}$ for losses is choosing between: Option A: 80% chance of gaining USD 4000, 20% chance of gaining nothing Option B: certain gain of USD 3000 (a) Which option does expected utility theory predict? (b) Which option does prospect theory predict (assuming linear probability weighting)? (c) How does the result change if the choices are framed as losses? (a) Expected utility: $E[A] = 0.8 \times 4000 + 0.2 \times 0 = 3200$. $E[B] = 3000$. Expected utility theory predicts A (higher expected value). (b) Prospect theory: $V(A) = 0.8 \times (4000)^{0.88} + 0.2 \times 0^{0.88}$ $(4000)^{0.88} = e^{0.88 \ln 4000} = e^{0.88 \times 8.294} = e^{7.299} = 1484$ $V(A) = 0.8 \times 1484 = 1187.2$ $V(B) = (3000)^{0.88} = e^{0.88 \times 8.006} = e^{7.045} = 1148$ $V(A) = 1187.2 > V(B) = 1148$, so prospect theory also predicts A. (c) Framed as losses: Option A: 80% chance of losing USD 4000, 20% chance of losing nothing Option B: certain loss of USD 3000 $V(A) = 0.8 \times (-2 \times 4000^{0.88}) + 0.2 \times 0 = 0.8 \times (-2968) = -2374.4$ $V(B) = -2 \times 3000^{0.88} = -2 \times 1148 = -2296$ $V(A) = -2374.4 < V(B) = -2296$, so prospect theory predicts B (the certain loss). This demonstrates the **reflection effect:** people are risk-averse for gains but risk-seeking for losses. Expected utility theory predicts A in both frames (same expected value). </details> <details> <summary>Problem 9: Contestable Market Analysis</summary> A natural monopoly has total cost function $\text{TC} = 200 + 10Q$. Market demand is $P = 50 - 2Q$. (a) Calculate the profit-maximising price, quantity, and profit for an unregulated monopoly. (b) If the market is perfectly contestable, what price and quantity will the firm set? (c) Calculate the deadweight loss in each case. (a) $\text{MR} = 50 - 4Q$. $\text{MC} = 10$. $\text{MR} = \text{MC}: 50 - 4Q = 10 \implies 4Q = 40 \implies Q = 10$ $P = 50 - 2(10) = 30$ $\pi = TR - TC = 300 - (200 + 100) = 0$ Interesting: profit is zero even for the unregulated monopoly in this case (the fixed cost exactly equals the monopoly profit). Let me verify: $TR = 30 \times 10 = 300$. $TC = 200 + 10(10) = 300$. $\pi = 0$. Yes. (b) In a contestable market, the firm sets $P = \text{AC}$ to deter entry: $\text{AC} = 200/Q + 10$ $P = \text{AC}: 50 - 2Q = 200/Q + 10 \implies 50Q - 2Q^2 = 200 + 10Q$ $-2Q^2 + 40Q - 200 = 0 \implies Q^2 - 20Q + 100 = 0 \implies (Q - 10)^2 = 0 \implies Q = 10$ $P = 30$. The contestable market outcome coincides with the monopoly outcome in this specific case because the monopoly was already earning zero profit. (c) Efficient quantity: $P = \text{MC}: 50 - 2Q = 10 \implies Q = 20$, $P = 10$. $\text{DWL} = \frac{1}{2}(30 - 10)(20 - 10) = \frac{1}{2}(20)(10) = 100$ The DWL is the same in both cases because the output is identical. Note: this is a special case where the monopoly earns zero profit. In general, contestable markets achieve higher output than unregulated monopolies because the threat of entry forces the incumbent to set lower prices. </details> <details> <summary>Problem 10: Government Failure --- Cost-Benefit Analysis of Regulation</summary> The government regulates a monopoly to set $P = \text{AC}$. The monopoly has $\text{TC} = 100 + 5Q$ and faces demand $P = 25 - Q$. (a) Compare the outcomes under: (i) unregulated monopoly, (ii) $P = \text{AC}$ regulation, (iii) $P = \text{MC}$ regulation. (b) The regulatory process costs USD 15 per period (administrative costs). Is regulation justified? (c) Identify potential sources of government failure in this regulation. (a) (i) Unregulated monopoly: $\text{MR} = 25 - 2Q$, $\text{MC} = 5$. $\text{MR} = \text{MC}: 25 - 2Q = 5 \implies Q = 10$, $P = 15$ $\pi = 150 - (100 + 50) = 0$. Again zero profit. (ii) $P = \text{AC}$: $25 - Q = 100/Q + 5 \implies 25Q - Q^2 = 100 + 5Q$ $Q^2 - 20Q + 100 = 0 \implies Q = 10$, $P = 15$. Same as monopoly (zero profit already). (iii) $P = \text{MC}$: $25 - Q = 5 \implies Q = 20$, $P = 5$. $\pi = 100 - (100 + 100) = -100$. The firm makes a loss and would exit without a subsidy. (b) DWL under unregulated monopoly: Efficient quantity: $Q^* = 20$. DWL $= \frac{1}{2}(15 - 5)(20 - 10) = 50$. The DWL savings from MC pricing $= 50$. Regulatory cost $= 15$. Net benefit $= 50 - 15 = 35$. Regulation is justified on efficiency grounds. However, the MC pricing option requires a subsidy of 100, which creates additional DWL from taxation. (c) Sources of government failure: 1. **Information asymmetry:** the regulator may not know the firm's true cost function. The firm has an incentive to overstate costs to justify higher prices (Averch-Johnson effect) 2. **Regulatory capture:** the firm may lobby the regulator to set prices above AC 3. **Averch-Johnson effect:** rate-of-return regulation incentivises the firm to over-invest in capital (gold-plating) to increase its rate base 4. **Dynamic inefficiency:** regulated prices reduce the incentive to innovate and reduce costs 5. **Time inconsistency:** as the regulatory cycle progresses, the regulator may be "captured" by the information asymmetry and set prices that are too high </details> <details> <summary>Problem 11: Income Inequality and Redistribution</summary> Country A has five income quintiles with the following shares: Bottom 20%: 3%, Second: 8%, Third: 14%, Fourth: 22%, Top 20%: 53%. (a) Calculate the Gini coefficient. (b) Calculate the Palma ratio. (c) The government introduces a progressive tax that transfers 5% of GDP from the top quintile to the bottom quintile. Calculate the new Gini coefficient. (a) Cumulative shares: | $i$ | $X_i$ | $Y_i$ | $X_i - X_{i-1}$ | $Y_i + Y_{i-1}$ | Product | |---|---|---|---|---|---| | 0 | 0 | 0 | -- | -- | -- | | 1 | 0.2 | 0.03 | 0.2 | 0.03 | 0.006 | | 2 | 0.4 | 0.11 | 0.2 | 0.14 | 0.028 | | 3 | 0.6 | 0.25 | 0.2 | 0.36 | 0.072 | | 4 | 0.8 | 0.47 | 0.2 | 0.72 | 0.144 | | 5 | 1.0 | 1.00 | 0.2 | 1.47 | 0.294 | $G = 1 - (0.006 + 0.028 + 0.072 + 0.144 + 0.294) = 1 - 0.544 = 0.456$ This indicates high inequality. (b) Palma ratio $= 53\% / (3\% + 8\%) = 53/11 = 4.82$. The top 10% earn nearly 5 times the combined income of the bottom 40%. (c) After redistribution: bottom quintile $= 3 + 5 = 8\%$, top quintile $= 53 - 5 = 48\%$. New cumulative shares: | $i$ | $X_i$ | $Y_i$ | Product | |---|---|---|---| | 0 | 0 | 0 | -- | | 1 | 0.2 | 0.08 | 0.016 | | 2 | 0.4 | 0.16 | 0.048 | | 3 | 0.6 | 0.30 | 0.092 | | 4 | 0.8 | 0.52 | 0.164 | | 5 | 1.0 | 1.00 | 0.304 | $G_{\text{new}} = 1 - (0.016 + 0.048 + 0.092 + 0.164 + 0.304) = 1 - 0.624 = 0.376$ The Gini coefficient falls from 0.456 to 0.376, a reduction of 0.080. This is a significant improvement, but the country still has moderate-to-high inequality. New Palma ratio $= 48\% / (8\% + 8\%) = 48/16 = 3.00$. </details> ## Common Pitfalls: Microeconomics (Comprehensive) - Confusing the Gini coefficient with the Palma ratio. The Gini captures overall inequality; the Palma focuses on the tails and may give a different picture - Assuming that contestable markets achieve allocative efficiency. Contestable markets achieve $P = \text{AC}$ (productive efficiency) but not $P = \text{MC}$ (allocative efficiency) unless MC is constant - Assuming that loss aversion is the same as risk aversion. Loss aversion is a separate bias from diminishing marginal utility; it applies even for small stakes where risk aversion should be negligible - Assuming that government intervention always improves outcomes. Government failure can make outcomes worse than the market failure it aims to correct - Confusing absolute and relative poverty measures. A country can have zero absolute poverty but high relative poverty - Ignoring the behavioural responses to redistribution. High marginal tax rates may reduce labour supply and investment, partially offsetting the redistribution - Assuming that nudges are always benign. Choice architecture can be manipulative and may not reflect genuine informed consent - Applying prospect theory's value function parameters universally. The loss aversion parameter $\lambda$ varies across individuals, cultures, and contexts ## Price Discrimination: Advanced Analysis (HL Extension) ### Conditions for Price Discrimination 1. **Market power:** the firm must face a downward-sloping demand curve 2. **Market segmentation:** the firm must be able to identify and separate consumer groups with different price elasticities of demand 3. **No arbitrage:** consumers must be unable to resell the good between market segments ### Three Degrees of Price Discrimination **First degree (perfect) price discrimination:** the firm charges each consumer their maximum willingness to pay. The firm captures the entire consumer surplus. $$\text{Profit} = \int_0^Q P(Q) dQ - \text{TC}(Q)$$ Output is at the socially efficient level ($P = \text{MC}$) because the firm captures the marginal benefit of each unit. There is no deadweight loss, but all surplus goes to the producer. **Second degree (self-selection):** the firm offers different quantity or quality options, and consumers self-select into the option designed for them. Examples: quantity discounts, versioning (software, airline tickets), two-part tariffs. **Third degree (market segmentation):** the firm charges different prices to different identifiable groups (e.g., student discounts, senior citizen discounts, geographic pricing). ### Third-Degree Price Discrimination: Formal Treatment A monopolist serves two markets with demand functions: Market 1: $P_1 = a_1 - b_1 Q_1$ Market 2: $P_2 = a_2 - b_2 Q_2$ Total cost: $\text{TC} = c(Q_1 + Q_2) + F$ The firm maximises: $\pi = P_1 Q_1 + P_2 Q_2 - c(Q_1 + Q_2) - F$ FOC for each market: $$\text{MR}_1 = \text{MC}: \quad a_1 - 2b_1 Q_1 = c \implies Q_1 = \frac{a_1 - c}{2b_1}$$ $$\text{MR}_2 = \text{MC}: \quad a_2 - 2b_2 Q_2 = c \implies Q_2 = \frac{a_2 - c}{2b_2}$$ The price in each market: $$P_1 = \frac{a_1 + c}{2}, \quad P_2 = \frac{a_2 + c}{2}$$ The higher-price market has the less elastic demand (more inelastic). **Inverse elasticity rule:** $$\frac{P_1 - \text{MC}}{P_1} = \frac{1}{|\epsilon_1|}, \quad \frac{P_2 - \text{MC}}{P_2} = \frac{1}{|\epsilon_2|}$$ The Lerner index (mark-up over MC as a proportion of price) equals the inverse of the price elasticity of demand. The market with the less elastic demand has the higher mark-up. ### Numerical Example A monopolist serves two markets: Market 1 (business travellers): $P_1 = 200 - Q_1$ Market 2 (leisure travellers): $P_2 = 120 - Q_2$ $\text{MC} = 20$, $F = 500$ **Without price discrimination (uniform price):** Total demand: $Q = Q_1 + Q_2 = (200 - P) + (120 - P) = 320 - 2P$ Inverse demand: $P = 160 - Q/2$ $\text{MR} = 160 - Q$ $\text{MR} = \text{MC}: 160 - Q = 20 \implies Q = 140$ $P = 160 - 70 = 90$ $\pi = 90 \times 140 - 20 \times 140 - 500 = 12\,600 - 2\,800 - 500 = 9\,300$ **With price discrimination:** Market 1: $Q_1 = (200 - 20)/2 = 90$, $P_1 = 110$. $\pi_1 = 110 \times 90 - 20 \times 90 = 8\,100$ Market 2: $Q_2 = (120 - 20)/2 = 50$, $P_2 = 70$. $\pi_2 = 70 \times 50 - 20 \times 50 = 2\,500$ $\pi = 8\,100 + 2\,500 - 500 = 10\,100$ Price discrimination increases profit by $10\,100 - 9\,300 = 800$. Total output: $90 + 50 = 140$ (same as without discrimination, because MC is constant). If MC were increasing, price discrimination would change total output. With increasing MC, the monopolist produces more total output under price discrimination than under uniform pricing, reducing deadweight loss. ## Two-Part Tariffs (HL Extension) ### Structure and Optimal Design A **two-part tariff** charges consumers a fixed fee $F$ plus a per-unit price $p$. Total charge for quantity $q$: $T(q) = F + pq$ ### Optimal Two-Part Tariff with Identical Consumers If all consumers have identical demand, the firm sets: - $p = \text{MC}$ (efficient per-unit price) - $F = \text{CS}$ at $p = \text{MC}$ (capture the entire consumer surplus) This achieves the first-best outcome: efficient quantity, no deadweight loss, and all surplus goes to the producer. ### Optimal Two-Part Tariff with Heterogeneous Consumers With different consumer types, the firm faces a trade-off: - Higher $F$ extracts more surplus from high-demand consumers but excludes low-demand consumers - Lower $p$ increases quantity consumed but reduces revenue per unit **Two consumer types:** Type H (high demand): $Q_H = 100 - P$, $n_H$ consumers Type L (low demand): $Q_L = 60 - P$, $n_L$ consumers $\text{MC} = 10$. **Strategy 1: serve both types.** Set $p$ and $F$ such that type L still buys. Type L's consumer surplus at price $p$: $\text{CS}_L = \frac{1}{2}(60 - p)(60 - p)/1 = \frac{(60-p)^2}{2}$. Wait, $\text{CS} = \frac{1}{2}(P_{\max} - p) \times Q(p) = \frac{1}{2}(60-p)(60-p) = \frac{(60-p)^2}{2}$. Set $F = \text{CS}_L$: $F = \frac{(60-p)^2}{2}$. Total profit: $\pi = (n_H + n_L)F + (p - 10)(n_H Q_H + n_L Q_L)$ $= (n_H + n_L)\frac{(60-p)^2}{2} + (p-10)[n_H(100-p) + n_L(60-p)]$ With $n_H = n_L = 1$: $\pi = (60-p)^2 + (p-10)(160 - 2p) = 3600 - 120p + p^2 + 160p - 2p^2 - 1600 + 20p$ $= 2000 + 60p - p^2$ $\frac{d\pi}{dp} = 60 - 2p = 0 \implies p = 30$ $F = (60-30)^2/2 = 450$ $Q_H = 70$, $Q_L = 30$. $\pi = 2(450) + (30-10)(70+30) = 900 + 2000 = 2900$. **Strategy 2: serve only type H.** Set $F = \text{CS}_H$ at $p = 10$. $F = \frac{1}{2}(100-10)(90) = 4050$. $\pi = 4050 + 0 = 4050$. With only type H: profit $= 4050 > 2900$. The firm prefers to exclude type L. This illustrates the **exclusion problem:** when consumer types are sufficiently different, the firm may prefer to exclude low-demand consumers to extract more surplus from high-demand consumers. ## Natural Monopoly Regulation: Extended Analysis (HL Extension) ### The Natural Monopoly Problem A natural monopoly arises when a single firm can supply the entire market at a lower cost than two or more firms. This occurs when there are large fixed costs and relatively low marginal costs: $$\text{AC}(Q) = \frac{F}{Q} + \text{MC} \text{ is decreasing for all } Q$$ Examples: electricity distribution, water supply, rail networks, telecommunications. ### Regulatory Options 1. **Marginal cost pricing ($P = \text{MC}$):** allocatively efficient but the firm makes a loss ($\text{AC} > \text{MC}$). Requires a subsidy, which creates deadweight loss from taxation and raises distributional questions 2. **Average cost pricing ($P = \text{AC}$):** the firm breaks even. Productively efficient but allocatively inefficient ($P > \text{MC}$). No subsidy required 3. **Price cap regulation ($P \leq P_{\text{cap}}$):** the regulator sets a maximum price, often using the formula $P_{\text{cap}} = P_{\text{RPI}} - X$, where RPI is the retail price index (inflation) and $X$ is an efficiency factor. This incentivises cost reduction (the firm keeps the difference between actual costs and the cap) 4. **Rate-of-return regulation:** the regulator allows the firm to earn a fair rate of return on its capital. This creates the Averch-Johnson effect: the firm over-invests in capital ("gold plating") to increase its rate base 5. **Yardstick competition:** the regulator compares the performance of the monopoly to similar firms in other regions or countries, using the comparison to set efficiency targets ### Numerical Example: Regulatory Comparison A natural monopoly has $\text{TC} = 500 + 10Q$ and faces demand $P = 100 - Q$. (a) Unregulated monopoly. (b) $P = \text{MC}$ regulation. (c) $P = \text{AC}$ regulation. (d) Price cap at $P = 30$. (a) $\text{MR} = 100 - 2Q$, $\text{MC} = 10$. $\text{MR} = \text{MC}: Q = 45$, $P = 55$. $\pi = 55 \times 45 - (500 + 450) = 2475 - 950 = 1525$. $\text{DWL} = \frac{1}{2}(55 - 10)(90 - 45) = \frac{1}{2}(45)(45) = 1012.5$. (b) $P = \text{MC} = 10$. $Q = 90$. $P = 10$. $\pi = 900 - (500 + 900) = -500$. The firm loses USD 500 and requires a subsidy. $\text{DWL} = 0$ (allocatively efficient). (c) $P = \text{AC}$. $\text{AC} = 500/Q + 10$. $100 - Q = 500/Q + 10$. $90Q - Q^2 = 500 \implies Q^2 - 90Q + 500 = 0 \implies Q = \frac{90 \pm \sqrt{8100-2000}}{2} = \frac{90 \pm 76.8}{2}$. $Q = 83.4$ or $Q = 6.6$. Taking the larger root: $Q = 83.4$, $P = 16.6$. $\pi = 0$ (by construction). $\text{DWL} = \frac{1}{2}(16.6 - 10)(90 - 83.4) = \frac{1}{2}(6.6)(6.6) = 21.8$. (d) Price cap at $P = 30$: $Q = 70$. $\pi = 30 \times 70 - (500 + 700) = 2100 - 1200 = 900$. The firm still earns positive profit but less than the unregulated case (900 vs. 1525). $\text{DWL} = \frac{1}{2}(30 - 10)(90 - 70) = \frac{1}{2}(20)(20) = 200$. | Regulation | $P$ | $Q$ | $\pi$ | DWL | |---|---|---|---|---| | None | 55 | 45 | 1525 | 1012.5 | | $P = \text{MC}$ | 10 | 90 | -500 | 0 | | $P = \text{AC}$ | 16.6 | 83.4 | 0 | 21.8 | | Price cap (30) | 30 | 70 | 900 | 200 | **Evaluation:** - $P = \text{MC}$ maximises allocative efficiency but requires subsidy (distributional issue) - $P = \text{AC}$ balances efficiency and financial sustainability - Price cap at 30 is less efficient but easier to implement and provides investment incentives ## Worked Examples: Microeconomics (Additional) <details> <summary>Problem 12: Third-Degree Price Discrimination with Increasing MC</summary> A cinema charges different prices for adults and students. Demand: Adults: $P_A = 20 - 0.05Q_A$ Students: $P_S = 12 - 0.05Q_S$ Total cost: $\text{TC} = 1000 + 2(Q_A + Q_S) + 0.01(Q_A + Q_S)^2$ (a) Calculate the profit-maximising prices and quantities with price discrimination. [6 marks] (b) Calculate the profit-maximising uniform price. [4 marks] (c) Calculate the difference in profit. [2 marks] (a) $\text{MC} = 2 + 0.02(Q_A + Q_S)$. This depends on total output, making the two markets interdependent. $\text{MR}_A = 20 - 0.10Q_A$. $\text{MR}_S = 12 - 0.10Q_S$. FOC: $\text{MR}_A = \text{MC}$ and $\text{MR}_S = \text{MC}$: $20 - 0.10Q_A = 2 + 0.02(Q_A + Q_S)$ ... (1) $12 - 0.10Q_S = 2 + 0.02(Q_A + Q_S)$ ... (2) From (1): $18 = 0.12Q_A + 0.02Q_S$ ... (1a) From (2): $10 = 0.02Q_A + 0.12Q_S$ ... (2a) Multiply (1a) by 6: $108 = 0.72Q_A + 0.12Q_S$ Multiply (2a) by 1: $10 = 0.02Q_A + 0.12Q_S$ Subtract: $98 = 0.70Q_A \implies Q_A = 140$. From (1a): $18 = 0.12(140) + 0.02Q_S = 16.8 + 0.02Q_S \implies Q_S = 60$. $P_A = 20 - 0.05(140) = 13$. $P_S = 12 - 0.05(60) = 9$. $\pi = 13 \times 140 + 9 \times 60 - [1000 + 2(200) + 0.01(200)^2]$ $= 1820 + 540 - [1000 + 400 + 400]$ $= 2360 - 1800 = 560$. (b) Without price discrimination, aggregate demand: For $P > 12$: only adults buy. $Q = (20-P)/0.05 = 400 - 20P$. For $P \leq 12$: both buy. $Q = (20-P)/0.05 + (12-P)/0.05 = (32 - 2P)/0.05 = 640 - 40P$. Since the uniform price is likely below 12 (both groups are served): $P = 16 - Q/40$ for $Q \geq 200$ (both markets). $\text{MR} = 16 - Q/20$. $\text{MC} = 2 + 0.02Q$. $\text{MR} = \text{MC}: 16 - Q/20 = 2 + 0.02Q \implies 14 = Q/20 + Q/50 = 5Q/100 + 2Q/100 = 7Q/100$ $Q = 1400/7 = 200$. $P = 16 - 200/40 = 11$. $\pi = 11 \times 200 - 1800 = 2200 - 1800 = 400$. (c) Price discrimination increases profit by $560 - 400 = 160$. Total output is the same (200 in both cases) because MC is constant at the margin when $Q_A + Q_S = 200$. The gain from discrimination comes purely from extracting more consumer surplus, not from producing more. </details> <details> <summary>Problem 13: Two-Part Tariff with Two Consumer Types</summary> A golf club serves two types of members: Type A (avid golfers, $n_A = 100$): demand for rounds $P = 50 - Q$ Type B (occasional golfers, $n_B = 200$): demand for rounds $P = 30 - Q$ $\text{MC} = 5$ per round. The club charges an annual membership fee $F$ and a per-round fee $p$. (a) Calculate the optimal two-part tariff if the club serves both types. [4 marks] (b) Calculate the optimal two-part tariff if the club serves only type A. [3 marks] (c) Which strategy should the club choose? [3 marks] (a) To serve both types, set $F = \text{CS}_B$ (the lower-demand type's consumer surplus): $\text{CS}_B = \frac{1}{2}(30 - p)(30 - p) = \frac{(30-p)^2}{2}$ Total profit: $\pi = (n_A + n_B)F + (p - 5)[n_A Q_A + n_B Q_B]$ $= 300 \times \frac{(30-p)^2}{2} + (p-5)[100(50-p) + 200(30-p)]$ $= 150(30-p)^2 + (p-5)(10000 - 300p)$ $= 150(900 - 60p + p^2) + 10000p - 50000 - 300p^2 + 1500p$ $= 135\,000 - 9000p + 150p^2 - 50000 + 11500p - 300p^2$ $= 85\,000 + 2500p - 150p^2$ $\frac{d\pi}{dp} = 2500 - 300p = 0 \implies p = 2500/300 = 8.33$ $F = (30 - 8.33)^2 / 2 = 21.67^2 / 2 = 469.4/2 = 234.7$ $\pi = 300(234.7) + 3.33[100(41.67) + 200(21.67)]$ $= 70\,410 + 3.33[4\,167 + 4\,334] = 70\,410 + 3.33(8\,501) = 70\,410 + 28\,308 = 98\,718$ (b) Serve only type A: $F = \text{CS}_A$ at $p = 5$: $\text{CS}_A = \frac{1}{2}(50-5)(45) = 1012.5$ $\pi = 100 \times 1012.5 + 0 = 101\,250$ (c) Strategy 2 (serve only type A) yields higher profit: $101\,250 > 98\,718$. The club should set a high membership fee ($F = 1012.5$) and a low per-round fee ($p = 5$), serving only avid golfers. This is the classic exclusion result: the firm excludes low-demand consumers to extract more surplus from high-demand consumers. **Note:** this assumes type B golfers will not join at $F = 1012.5$. Since their maximum willingness to pay is $\text{CS}_B$ at $p = 5$: $\frac{1}{2}(25)(25) = 312.5$, which is less than 1012.5, they will not join. The exclusion is self-selecting. </details> ## Oligopoly: Cournot and Bertrand Comparison (HL Extension) ### Cournot (Quantity) Competition Firms simultaneously choose quantities. Each firm treats the other's output as given. **Duopoly with linear demand:** $P = a - Q = a - q_1 - q_2$, $\text{MC} = c$ for both firms. Firm 1 maximises $\pi_1 = (a - q_1 - q_2)q_1 - cq_1$. FOC: $a - 2q_1 - q_2 - c = 0 \implies q_1 = \frac{a - c - q_2}{2}$ This is Firm 1's **best response function**. By symmetry: $q_1^* = q_2^* = \frac{a - c}{3}$. Total output: $Q^* = \frac{2(a-c)}{3}$. Price: $P^* = \frac{a + 2c}{3}$. Each firm's profit: $\pi_i = \frac{(a-c)^2}{9}$. ### Bertrand (Price) Competition Firms simultaneously choose prices. The firm with the lower price captures the entire market. With homogeneous goods and constant MC: If $p_1 < p_2$: Firm 1 serves all demand at $p_1$. If $p_1 = p_2$: firms split demand equally. **Nash equilibrium:** $p_1^* = p_2^* = c$ (marginal cost pricing). Each firm's profit: $\pi_i = 0$. ### Comparison: Cournot vs. Bertrand | Feature | Cournot | Bertrand | |---|---|---| | Strategic variable | Quantity | Price | | Equilibrium price | $P = (a + 2c)/3$ | $P = c$ | | Equilibrium profit | $(a-c)^2/9$ per firm | 0 | | Efficiency | Allocatively inefficient ($P > c$) | Allocatively efficient ($P = c$) | | Competitive pressure | Weaker (firms produce less than competitive output) | Stronger (drives price to MC) | ### Numerical Example Demand: $P = 100 - Q$. $\text{MC} = 10$ for both firms. **Cournot:** $q_1^* = q_2^* = (100 - 10)/3 = 30$. $Q = 60$. $P = 100 - 60 = 40$. $\pi_1 = \pi_2 = (40 - 10) \times 30 = 900$. **Bertrand:** $p_1^* = p_2^* = 10$. $Q = 90$. $\pi_1 = \pi_2 = 0$. **Monopoly:** $Q = 45$, $P = 55$, $\pi = 2025$. **Perfect competition:** $Q = 90$, $P = 10$, $\pi = 0$. The Cournot outcome lies between monopoly and perfect competition, while the Bertrand outcome coincides with perfect competition (for homogeneous goods with constant MC). ### When Does Bertrand Not Lead to MC Pricing? 1. **Capacity constraints:** if firms cannot produce enough to meet total demand at MC, prices exceed MC (Edgeworth model) 2. **Product differentiation:** with differentiated products, firms have some market power and prices exceed MC even in Bertrand equilibrium 3. **Repeated interaction:** if firms interact repeatedly, they can sustain collusive prices above MC (see repeated games) 4. **Dynamic competition:** firms may engage in price wars, limit pricing, or predatory pricing strategies that differ from the static Bertrand prediction ### Differentiated Bertrand If products are differentiated, each firm faces a downward-sloping demand: $Q_1 = a - bP_1 + dP_2$ and $Q_2 = a - bP_2 + dP_1$ (where $d < b$ captures differentiation). Best response functions: $P_1 = \frac{a + bc + dP_2}{2b}$ At equilibrium: $P_1^* = P_2^* = \frac{a + bc}{2b - d}$. As $d \to b$ (products become perfect substitutes), $P^* \to c$ (Bertrand result). As $d \to 0$ (products become independent), $P^* \to (a + bc)/(2b)$ (monopoly pricing for each). **Numerical example:** $Q_1 = 100 - 2P_1 + P_2$, $\text{MC} = 10$. $P_1 = (100 + 20 + P_2)/4 = 30 + P_2/4$. By symmetry: $P_1^* = P_2^* = 30 + P_1/4$. $3P_1/4 = 30 \implies P_1^* = 40$. $Q_1 = 100 - 80 + 40 = 60$. $\pi_1 = (40 - 10) \times 60 = 1800$. With differentiated products, prices are above MC and firms earn positive profits.