Buyer Optimization

Budget Set and Budget Constraint

Understanding how buyers allocate their income among goods is central to microeconomics. The budget set includes all affordable combinations, while the budget constraint represents combinations that exhaust the entire budget.

• Budget Set: All allocations where .

• Budget Constraint: Allocations where .

• Inside the Line: Affordable, leaves income unspent.

• On the Line: Affordable, spends entire income.

• Outside the Line: Unaffordable.

• Allocation: A specific combination of goods chosen by the buyer.

Intercepts and Slope:

• Horizontal intercept: (maximum units of good 1).

• Vertical intercept: (maximum units of good 2).

• Slope: (opportunity cost of good 1 in terms of good 2).

Budget Line Movements:

• Income changes: Parallel shift.

• Price changes: Rotation about one axis.

Utility, Marginal Benefit, and Optimization

Buyers seek to maximize their utility, or satisfaction, from consumption. Marginal benefit (MB) is the extra benefit from consuming one more unit.

• Diminishing Marginal Benefit: Each additional unit provides less extra benefit.

• Optimization Rule: At the optimum, .

• If , buy more of good 1.

Demand, Quantity Demanded, and Consumer Surplus

The demand curve shows how quantity demanded varies with price. Consumer surplus measures the benefit buyers receive beyond the price paid.

• Demand Curve: Relationship between price and quantity demanded.

• Quantity Demanded: Specific amount at a given price.

• Consumer Surplus: per unit.

• Triangle Consumer Surplus: for linear demand.

Elasticity

Elasticity measures how responsive quantity demanded or supplied is to changes in price or other variables.

• Price Elasticity of Demand:

• Midpoint Formula:

• Classification:

  • 0: Perfectly inelastic

  • Between 0 and 1: Inelastic

  • 1: Unit elastic

  • Greater than 1: Elastic

  • Infinite: Perfectly elastic

• Cross Price Elasticity:

• Income Elasticity:

Seller Optimization

Perfect Competition and Firm Objectives

In perfect competition, many firms sell identical products and are price takers. Firms maximize economic profit, which includes opportunity costs.

• Total Revenue:

• Economic Profit:

• Zero Economic Profit: Revenue covers all costs, including opportunity cost.

Production and Time Periods

• Short Run: At least one input is fixed.

• Long Run: All inputs can be adjusted.

• Physical Capital (K): Equipment, buildings.

• Labor (L): Workers.

• Land (T): Natural resources.

• Marginal Product:

• Diminishing Marginal Returns: Marginal product falls as more of a variable input is added.

Cost Definitions and Calculations

• Cost Curve Relationships:

  • MC < ATC: ATC is falling

  • MC > ATC: ATC is rising

  • MC = ATC: ATC at minimum

  • MC < AVC: AVC is falling

  • MC > AVC: AVC is rising

  • MC = AVC: AVC at minimum (shutdown point)

Output, Profit, and Shutdown Decisions

• Competitive Firm:

• Optimal Output: Choose where

• Profit:

• Shutdown Rule: Operate if ; otherwise, shut down.

• Exit: Leave the market in the long run if persistent .

Firm Supply and Elasticity of Supply

• Short-run Supply Curve: Rising part of MC at and above minimum AVC.

• Price Elasticity of Supply:

• Classification:

  • 0: Perfectly inelastic

  • Between 0 and 1: Inelastic

  • 1: Unit elastic

  • Greater than 1: Elastic

  • Infinite: Perfectly elastic

Long Run Scale, Entry, and Exit

Perfect Competition and Market Efficiency

Reservation Values and Surplus

• Buyer Reservation Value: Maximum willingness to pay (WTP).

• Seller Reservation Value: Minimum willingness to accept (WTA).

• Consumer Surplus:

• Producer Surplus:

• Social Surplus:

• Efficient Trade: Occurs when WTP WTA.

Efficiency Results

1. Goods go to buyers with highest WTP; sellers with lowest WTA supply them.

2. Production is allocated so marginal cost is equal across firms.

3. Free entry and exit move resources to their highest-valued uses.

Invisible Hand and Economic Systems

• Invisible Hand: Market prices coordinate decentralized decisions, maximizing social surplus under strict competition.

• Market Economy: Prices direct resources.

• Command Economy: Central planner allocates resources.

Efficiency, Equity, and Pareto Efficiency

• Efficiency: Maximizing social surplus.

• Equity: Fairness in distribution.

• Pareto Efficiency: No one can be made better off without making someone else worse off.

• Deadweight Loss: Lost social surplus from omitted beneficial trades.

Price Controls

Examples and Applications

• Uber Surge Pricing: Higher prices after events ration rides to buyers with higher WTP and attract more drivers.

• Parking Meters: Prices allocate scarce spaces to those with higher WTP.

Decision Trees and Formula Reference

Competitive Firm Problem

1. Find where (under perfect competition, ).

2. Find and at that .

3. Compare with for profit/loss.

4. Compare with for shutdown decision.

5. Long run: persistent loss leads to exit.

Elasticity Problem

1. Identify responding quantity (numerator).

2. Identify changing variable (denominator).

3. Compute percentage changes (use midpoint formula).

4. Divide quantity percentage change by relevant percentage change.

5. Interpret and classify absolute value.

Surplus Problem

1. Identify each buyer's WTP and seller's WTA.

2. Complete trades while WTP WTA.

3. Calculate consumer surplus: .

4. Calculate producer surplus: .

5. Add and or directly sum for social surplus.

Final Formula List

• (Budget set)

• (Budget constraint)

• Budget slope:

• (Buyer optimum)

• (Consumer surplus per unit)

• (Price elasticity of demand)

• Cross elasticity:

• Income elasticity:

• (Marginal product)

• (Total cost)

• ; ; (Average costs)

• (Marginal cost)

• (Total revenue)

• (Economic profit)

• (Competitive firm optimum)

• Operate if (Short-run production decision)

• Social surplus: (Market efficiency)

• DWL: Lost social surplus (Cost of omitted beneficial trades)

Test Day Checklist

• Translate every symbol before calculating.

• Identify whether the question concerns a level, average, marginal value, or percentage change.

• Determine if the time period is short run or long run.

• Use MR vs MC for quantity, P vs ATC for profit, and P vs AVC for shutdown.

• On a market table, equilibrium is where QD = QS.

• For controlled price, actual trades equal the smaller of QD and QS.

• Check if a question changes the good's own price or shifts the entire curve.

• State what the answer means in words before selecting a choice.