ECO 220 Exam 1

Commodity and market basket

A commondity is a good or service defined in space and time. A market basket is a bundle of commodities.

Completeness

Any two bundles can be compared.

A>B ( A preferred)

A~B (Indifferent)

A>B (Weakly preferred)

Transitivity

Preferences are internally consistent

If A>B and B>C, then A>C

Monotonicity or nonsatiation

More is preferred to less, holding other goods fixed.

Identify bundleds that must be preferred.

Convexity

Averages are preferred to extremes.

Recognize bowed-in indifference curves and diminishing MRS.

Continuity

SMall changes in bundles do not cause abrupt preference reveresals

Use continuous curves and optimization methods.

Utility Function

A numerical representation of preferences

Rank bundles; remember utility is ordinal (no actually meaning, jsut ranks usefulness), not cardinal

Well behaved preferences

Satisfy all our standard class assumptions

• Rationality, monotonicity, convexity, and continuity

Indifference curves

• Every point on one indifference curve provides the same utility

• An indifference curve with a larger number is more preferred

• Indifference curves cannot cross if preferences are complete and transitive

• An indifference curve passes through every bundle in the goods space.

• For standard convex preferences, the curve becomes flatter as q1 increases

Marginal Rate of Substitution (MRS)

Key formula: MRS12 = MRS(q2 for q1) = -change in q2/change in q1, while utility is held constant.

Special preference types

Perfect Substitutes: Straight Lines

• constant MRS; the consumer often chooses a corner (zero of one good) unless MRS exactly equals the opportunity cost.

Perfect complements: L-shaped

• Goods are consumed in a fixed proportion; optimum is usally at the kink

HOmothetic: MRS depends only on the ratio q2/q1

• proportional income changes ofte nexpand choices along a ray

Quasilinear: parallel shifts in one direction, MRS only depends on level of q1 or q2.

• MRS depends only on the amount of one good; that good may have little or no income effect over a region.

Choice Set

Describes ability to purchase.

Preferences

Descrives willingness to trade.

Budget LIne

P1Q1+P2Q2 = m

Budget Set

P1Q1+P2Q2 < m

Slope-intercept form (for graphing)

Slope: q2 = m/p2 - (p1/p2) x q1

Vertical Intercept: m/p2

Horizontal intercept: m/p1

Opportunity Cost

Opportunity cost of one more unit of q1, measured in q2, is p1/p2. the budget line slope is -p1/p2

Changes in economic circumstances: Income rise, price fixed

Graphical effect: parallel outward shift.

What happens to OC: Unchanged, relative prives are unchanged.

What happens to buying power: buying power increases; more affordable options.

Changes in Economic Circumstances: Income falls, prices fixed

Graphical effect: Parallel inward shift

What happens to OC: unchanged; relative prices are unchanged

What happens to buying power: Buying power falls; fewer affordable options

Changes in Economic Circumstances: p1 falls

Graphical effect: q1-intercept moves outward; line pivots around q2-intercept

What happens to OC: q1 becomes cheapter relative to q1, p1/p2 falls.

What happens to buying power: Buying power increases; more affordable options

Changes in Economic circumstances: p2 rises

Graphical effect: q2-intercept moves inward; line pivots are q1-intercept

What happens to OC: q2 becomes more expensive relative to q1; p1/p2 rises

What happens to buying power: buying power falls; fewer affordable options

Kinked and multi-good choice sets

Coupons, taxes, subsidies, rationing, endowments, and nonlinear pricing can create kinks or segments with different slopes.

With N>2 goods, the budget constrain becomes: p1q1+p2q2+…+pnqn < m

A composite or numeraire good converts spending on all other goods into one variable: p1q1 + q < m. Its price is normalized to 1

Kinked and multi-good choice sets Example

Income is $500, flying monkeys cost $50, and raincoats cost $25

• Budge line: 50qm + 25qr = 500

• Intercepts: 10 monkeys or 20 raincoats

• Slope of budget constraint (raincoats on the vertical axis): -50/25 = -2

• Opportunity cost: for one more monkey MUST GIVE UP 2 raincoats

• Opportunity cost: for one more raincoat MUST GIVE UP ½ monkey

• Relative prices (defined by the market) are defining the buyer’s ABILITY to trade between goods

Optimal Bundle

Both affordable and at least as preffered as every other affordable bundle

Optimal Choice: Interior Solution

Key Formula: MRS1,2 = p1/p2 = OC12 (MRS = OC)

p1q1+p2q2 = m (exhaust income)

At tangency, willing and able at the margin. Always check for corners, kinks, and nonconvex preferences

Corner and Kink solutions: perfect substitutes

Compare MRS12 with OC12 = p1/p2. If MRS12> OC12 everywhere, only buy good 1 (corner). If MRS12<OC12 everywhere, only buy good 2 (corner). If equal, every bundle on the budget line may be optimal.

Corner and kind soltutions: Perfect complements

Use the fixed-proportion condition (kink for preferences) and the budget equation. The optimum is at the kink.

Corner and kink solutions: well-behaved preferences

Look for a tangency; then verify the bundle is feasible and preferrred to endpoints.

Corner and kink solutions: essential goods

Indifference curves do not intersect the axis for the other good, helping rule out some corner solutions

Uniqueness

A linear budget and strictly convex preferences generally produce one optimal bundle. Weak convextiy indifferent between averages and extremes; perfect subtitutes) can allow multiple optima; nonconvexity (extremes preffered to averages) can make endpoint comparasions necessary

Optimization Checklist

1. write the budget equation and calculate intercepts and slope

2. sketch the relevant indifference curves or identify the preference type

3. generate interior and/or cornder candidates

4. verify affordability

5. compare utility if more than one candidate remains

6. state the economic meaning of the result

Income and substitution effects

A change in circumstances changes behavior (the chosen bundle), not necessarily tastes (preferences). A price change usually changes both relative prices and purchasing power.

Income Effect

The change in consumption caused by a change in income, wealth, or purchasing power while opportunity cost is held fixed.

• Good type: Normal Good

  • When income rises: quanitty demanded rises

  • Income elasticity sign: positive

• Good Type: Inferior

  • When income rises: quantity demanded falls

  • Income elasticity sign: negative

• Good Type: Quasilinear good over a region

  • When income rises: quantity may remain unchanged

  • Income elsasticity sign: zero or near zero for the quasilinear good

Budget Logic

With two goods and nonsatiated preferences, both goods cannot be inferior everywhere: when income rises, the consumer must use the added purchasing power somewhere.

Hicksian substitution effect

Isolates the response to a change in relative prices while holding utility at its original level.

• Draw original budget line and original optimal choice (point A)

• Draw new budget line after the price change (note the new slope/opportunity cost)

• Draw the hypothetical compensated budget constraint (slope equal to new OC) tanget to the original indifference curve and identify hypothetica lchoice (point b)

• Movement from the original optimum to the compensated optimum is the substitution efect (a to b)

Direction of Hicksian substitution effect

With two goods, the direction of the Hicksian substitution effect is unambigous. The hicksian substitution effect always moves consumption toward the good tthat becomes relatively cheaper. Its size depends on substitutability: flatter adjustment for complements (less substitutables), larger adjustment for close substitutes (more subtitutable).

Total Effect of Substitution Effect

1. Movement from the original optimum to the compensated optimum is the substitution effect (A→B)

2. Movement from the compensated optimum to the final optimum is the income effect (B→C)

3. the total effect is the movement from the original optimum to the final optimum (A→C)

Price and income changes and types of goods

• Classification: Effect description

  • Income Response: income increase raises buying power

  • Own-price response: A price increase makes good relatively more expensive and lowers uying power

• Classification: Normal Good

  • Income response: positive income effect, consumption rises

  • Own-price response: consumption falls: income and substitution effects work together

• Classification: Regular inferior

  • Income response: negative income effect; consumption falls

  • Own-price response: consumption falls: the (negative) substitution effect is stronger than the opposing (positive) income effect

• Classification: Giffen

  • Income response: negative income effect; consumption falls

  • Own-price response: consumption rises: the (positive) income effect is stronger than the opposing (negative) substitution effect.

Individual Demand: Quantity Demanded and demand function

Quantity Demanded: The optimal quantity of a good at a particular set of prices, income, and time.

Demand Function: derived from constrained choice.

Key Formula: qd1 = q1(pq,p2,…,pn,m)

Three Comparative Relationships

• Own-price Demand

  • Horizontal or input variable: the good’s own price p1

  • Question answered: how does optimal q1 change as p1 changes

• Cross-price Demand

  • Horizontal or input variable: Another goods price

  • Question answered: how does optimal q1 change as p2 changes

• Income demand or engel relationship

  • Horizontal or input variable: income m

  • Question answered: how does optimal q1 change as m changes?

Movement vs Shift

• A change in a good’s own price causes movement along its own-price demand curve

• A change in income or another good’s price changes the entire own-price demand relationship, producing a shift

• On a cross--price graph, p2 changes along the curve; other determinants shift it

Reading a demand equation

Example: qd1 = 100 - 0.5p1 - 0.75p2 + 0.25m

• Own price: a $1 rise in pq reduces qd1 by 0.5 units, holding p2 and m fixed

• Cross price: a $1 rise in p2 reduces qd1 by 0.75 units. In this equation goods 1 and 2 are gross complements

• Income: a $1 rise in m raises qd1 by 0.25 units. Good 1 is normal over the relevant range

Sign Interpretation

A positive cross-price response suggests gross substitutes; a negative response suggets gross complements. These are behavioral classifications and can reflect both substitution and income effects

Demand curves to recognize

• Downward-sloping ordinary demand

• Vertical demand: perfectly inelastic over the displayed range

• Horizontal demand: perfectly elastice at one price

• Upward-sloping ordinary demand in the exceptional giffen case

Types of Goods

Lecture applies to private goods

Horizontal summation for private goods

At each common price Qd(p) = SUMi * qdi(p). Add quantities not prices

Example: at a price of $3, setlla demands 2 units, akib demands 6, and kody demands 10. Market quantity demanded is 2 + 6 + 10 = 18

Price elasticity of demand

𝐸𝑑 = %Δ𝑄𝑑 / %Δp = Δ𝑄𝑑 / Δp × 𝑝 / 𝑄𝑑

Elasticity Table

• Elasticity: 0

  • Classification: Perfectly inelastic

  • quantity response relative to price response: no quantity response

• Elasticity: Between -1 and 0

  • Classification: Relatively inelastic

  • Quantity response: Smaller percentage response

• Elasticity: -1

  • Classification: Unit elastic

  • Quantity response: equal percentage response

• Elasticity: less than -1

  • classification: relatively elastic

  • Quantity response: larger percentage response

• Elasticy: Approaches -infinity

  • Classification: perfectly elastic

  • Quantity response: extremely large response at the market price

Elasticity is not slope

Slope depends on units. Elasticity is unitless because it compares percentage changes. On a linear demand curve, slope is constant while elasticity generally varies along the curve

What effects elasticity

• Availability and closeness of substitutes: more or closer substitutes generally make dmeand more elastic

• Time horizon: demand is often more elastic when consumers have more time to adjust

Elasticity Equations

Own-Price Elasticity: 𝐸𝑑 = Δ𝑞1 / Δp × 𝑝 / 𝑞1

Cross-price elasticity: 𝐸𝑐 = Δ𝑞1 / Δp2 × 𝑝2 / 𝑞1

Income Elasticity: 𝐸𝑚 = Δ𝑞1 / Δm × 𝑚 / 𝑞1

Inverse Demand

An ordinary demand function gives quanitty as function of price. An inverse demand function solves for price as a function of quantity. It measures marginal willingness to pay or marginal benefit for the last unit.

Key formula: Demand: q = q(p)

Inverse Demand: p = p(q) = MB(q)

Consumer Surplus

CS = total willingness to pay - expendigture = area under demand - pQ