Summary Intermediate Microeconomics Hal R. Varian, complete PDF

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Summary Intermediate MicroeconomicsHal R. Varian, ninth editionChapter 1Optimization principle: People try to choose the best patterns of consumption that they can affordThe equilibrium principle: Prices adjust until the amount that people demand of something is equalto the amount that is suppliedCo...

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Summary Intermediate Microeconomics Hal R. Varian, ninth edition Chapter 1 Optimization principle: People try to choose the best patterns of consumption that they can afford The equilibrium principle: Prices adjust until the amount that people demand of something is equal to the amount that is supplied Competetive market: Demand curve & Supply curve  Market equilibrium P* Monopoly  Normal monopolist: Picks prices with biggest revenue box (fig 1.7 p13) (Discriminating monopolist: different prices) Excess demand (Pmax )e.g. : rent control. Pareto improvement: A way in which someone gets better off without any other party worse. If an allocation calls for a Pareto improvement: Pareto inefficient. If the allocation cannot be improved: Pareto efficient. Chapter 2 Budget constraint  Consumption bundle (x1 , x2) = The set of goods a consumer can choose to consume from where p1, p2 are the prices. M = the money the consumer has to spend. The budget constraint is: 𝑃1 𝑥1 + 𝑃2 𝑥2 ≤ m. X2 = can be used as composite good (everything else the consumer buys) Budget set = All bundles ≤ 𝑚. (area left of the line) 𝑝 Budget line slope = − 𝑝1 2

Budged line = Set of bundles that cost exactly m (the line): 𝑃1 𝑥1 + 𝑃2 𝑥2 = m The budged line can be rewritten as: 𝑝1 𝑚 𝑥2 = 𝑝 − 𝑥1  if x1 = 0 everything of m is used for x2. 2

𝑝2

Two formulas given: Budget line before change: 𝑃1 𝑥1 + 𝑃2 𝑥2 = m Change of consumption: 𝑝1 (𝑥1 + ∆𝑥1 ) + 𝑃2 (𝑥2 + ∆𝑥2 ) = 𝑚 gives: ∆𝑥2 𝑝1 𝑝1 ∆𝑥1 + 𝑝2 ∆𝑥2 = 0 → =− 𝑝2 ∆𝑥1 Slope measures opportunity cost. (of consuming good 1) Income increase  Budget Line shifts outwards parallel Price increase  Budget line becomes steeper Numaire price  Relative price to which we are measuring the other price and income: e.g: p1 𝑚 → 𝐻𝑒𝑟𝑒 𝑝2 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 1 𝑥 + 𝑥2 = 𝑃1 𝑥1 + 𝑃2 𝑥2 = m → 𝑝2 𝑝2 1 𝑝1 𝑝2 𝑥1 + 𝑥2 = 1 𝐻𝑒𝑟𝑒 𝑚 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 1 𝑚 𝑚

Quantity tax: 𝑃1 + 𝑡 Value tax (%): (1 + 𝜏 )𝑝1 Quantity Subsidy: 𝑃1 − 𝑠 Ad valorem Subsidy (%):(1 + 𝜎)𝑝1 Lumpsum tax/subsidy: Budget line shifts in- or outwards. Rationing (see figure): Limit the amount of goods that can be consumed Tax, subsidy, rationing can be combined (e.g.: higher tax when a certain point is reached.

Chapter 3 Consumption bundle  Complete list of goods and services (𝑥1 , 𝑥2 ) (𝑥1 , 𝑥2 ) > (𝑦1 , 𝑦2 ): 𝐵𝑢𝑛𝑑𝑙𝑒 𝑥1 , 𝑥2 𝑖𝑠 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑝𝑟𝑒𝑓𝑒𝑟𝑟𝑒𝑑 𝑜𝑣𝑒𝑟 𝑦1 , 𝑦2 (𝑥1 , 𝑥2 ) ~ (𝑦1 , 𝑦2 ): 𝐼𝑛𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡. (𝑥1 , 𝑥2 ) ≥ (𝑦1 , 𝑦2 ): 𝑤𝑒𝑎𝑘𝑙𝑦 𝑝𝑟𝑒𝑓𝑒𝑟𝑟𝑒𝑑 Assumptions about consumer preference:

Complete: Any two bundles can be compared: (𝑥1 , 𝑥2 ) ≥ (𝑦1 , 𝑦2 ) Reflexive: Any bundle is at least as good as itself: (𝑥1 , 𝑥2 ) ≥ (𝑥1 , 𝑥2 ) Transitive: If (𝑥1 , 𝑥2 ) ≥ (𝑦1 , 𝑦2 ) and (𝑦1 , 𝑦2 ) ≥ (𝑧1 , 𝑧2 ) then (𝑥1 , 𝑥2 ) ≥ (𝑧1 , 𝑧2 ) Bad: commodity that the consumer doesn’t like: Indifferent curves with a negative slope Neutrals: if the consumer is indifferent: Indifferent curves vertical lines Satiation point: (x 1 , x 2 ) Well-behaved indifference curves features: - Monotonicity: More is better; negative slope - Averages preferred to extremes - Convex Weighted average: 𝐼𝑓 (𝑥1 , 𝑥2 )~(𝑦1 . 𝑦2 ) 𝑡ℎ𝑒𝑛 (𝑡1 + (1 − 𝑡)𝑦1 , 𝑡𝑥2 + (1 − 𝑡)𝑦2 ) ≥ (x1 , 𝑥2 )

∆𝑥

Marginal Rate of Substitution (MRS): slope of indifference curve; ∆𝑥 2

With perfect substitutes: -1 With ‘neutrals’: MRS is infinity

1

perfect complements: 0 or infinity

Chapter 4 Utility function: a way to assign a number to every possible consumption bundle such that morepreferred bundles get assigned larger numbers than less-preferred bundles. Cardinal utility: Ranking of utility’s and adding a significance to the difference between them Monotonic transformation: Transforming numbers in one way to another preserving the order: The rate of change in f(u) can be measured by looking at the change in f between two values of u, divided by the change in u: ∆𝑓 (𝑓(𝑢2 ) − 𝑓(𝑢1 )) = ∆𝑢 𝑢2 − 𝑢1

Perfect substitutes: 𝑢(𝑥1 , 𝑥2 ) = 𝑎𝑥1 + 𝑏𝑥2 or the monotonic transformation (e.g. square root)  (𝑥1 , 𝑥2 ) = 𝑥12 + 2𝑥1 𝑥2 + 𝑥22 𝑎 a & b represent the ‘value’ of goods 1 and 2 to the consumer: The slope is − 𝑏 Perfect complements: 𝑢(𝑥1 , 𝑥2 ) = min{𝑎𝑥1 , 𝑏𝑥2 } a & b are the proportions in which the good is consumed Quasilinear Preferences: 𝑢(𝑥1 , 𝑥2 ) = 𝑘 = 𝑣(𝑥1 ) + 𝑥2 So the good can be non-linear in good x1  e.g.: 𝑢(𝑥1 , 𝑥2 ) = √𝑥1 + 𝑥2 Cobb-douglas Preferences: 𝑢(𝑥1 , 𝑥2 ) = 𝑥1𝑐 𝑥2𝑑 C & d are positive numbers that describe the preferences of the consumer. If c + d are not equal to one you can monotonic transform it: 𝑐 𝑑 1 → = 𝑥1𝑐+𝑑 𝑥2𝑐+𝑑 𝑛𝑜𝑤 𝑑𝑒𝑓𝑖𝑛𝑒 𝑎 𝑢(𝑥1 , 𝑥2 ) = (𝑥1𝑐 𝑥2𝑑 ) 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 𝑐+𝑑 𝑐 = 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒: 𝑣(𝑥1 , 𝑥2 ) = 𝑥1𝑎 𝑥21−𝑎 𝑐+𝑑 Marginal Utility (of good 1): 𝑀𝑈1 =

∆𝑈 ∆𝑥1

=

𝑢(𝑥1 +∆𝑥1,𝑥2 )−𝑢(𝑥1 ,𝑥2 ) ∆𝑥1

 good 2 is kept fixed.

So for the full change of utility if good x1 changes: ∆𝑈 = 𝑀𝑈1 ∆𝑥1.

If: 𝑀𝑈1 ∆𝑥1 + 𝑀𝑈2 ∆𝑥2 = ∆𝑈 = 0  so a change in x1 and x2 changes consumption along the indifference curve then: 𝑀𝑅𝑆 = to keep the same level of utility

∆𝑥2 ∆𝑥1

𝑀𝑈

= − 𝑀𝑈1 if you consume more of good 1 your get less of good 2 2

Chapter 5 Optimal choice alias the highest budget line available is labelled as: (𝑥1∗ 𝑥2∗ ) In general: Where the budged line is tangent to the indifference curve: when (strictly) convex Also: Boundary optimum & more than once tangency (with curved indifference curves, here it is not necessary that the tangency condition leads to an optimum) Demand function: 𝑥1 (𝑝1 , 𝑝2 , 𝑚) & 𝑥2 (𝑝1 , 𝑝2 , 𝑚)

𝑤ℎ𝑒𝑛 𝑝1 < 𝑝2 ∶ 𝑚/𝑝1 For perfect substitutes 𝑥1 = {𝑤ℎ𝑒𝑛 𝑝1 = 𝑝2 ∶ 𝑎𝑛𝑦 𝑛𝑢𝑚𝑏𝑒𝑟 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 0 𝑎𝑛𝑑 𝑚/𝑝1 𝑤ℎ𝑒𝑛 𝑝1 > 𝑝2 ∶ 0 𝑚 For perfect complements 𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚 → 𝑥1 = 𝑥2 = 𝑥 = 𝑝1 +𝑝2

𝑐

𝑚

𝑥1 = ∗ 𝑐+𝑑 𝑝1 → 𝑤𝑖𝑡ℎ 𝑑𝑒𝑚𝑎𝑛𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠: { ) Cobb-douglas preferences: 𝑢(𝑥1 , 𝑥2 = 𝑚 𝑑 𝑥2 = 𝑐+𝑑 ∗ 𝑝 2 𝑝1 𝑥1 𝑐 𝑝1 𝑚 𝑐 𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑠𝑝𝑒𝑛𝑑 𝑜𝑛 𝑥1 = → 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑥1 𝑔𝑖𝑣𝑒𝑠: ∗ ∗ = 𝑚 𝑚 𝑐 + 𝑑 𝑝1 𝑐 + 𝑑 𝑑 𝑓𝑜𝑟 𝑥2 𝑡ℎ𝑖𝑠 𝑖𝑠 𝑐+𝑑 An optimum quantity tax applied: (𝑝1 + 𝑡)𝑥1∗ + 𝑝2 𝑥∗2 = 𝑚 → 𝑟𝑒𝑣𝑒𝑛𝑢𝑒 𝑟𝑎𝑠𝑖𝑠𝑒𝑑 𝑏𝑦 𝑡𝑎𝑥: 𝑅 ∗ = 𝑡𝑥1∗ 𝑥1𝑐 𝑥2𝑑

This leads to a change of the slope pf the budget line − 𝑝

𝑝1 +𝑡 𝑝2

Income tax: 𝑝1 𝑥1∗ + 𝑝2 𝑥2∗ = 𝑚 − 𝑡𝑥1∗  slope stays − 𝑝1 but it shifts back. 2

Conclusion: An income tax leads in general to a higher utility than a quantity tax. (this differs per person as not everyone consumes an equal amount of x1 and income(m) can be different.

The utility maximization problem: (workbook 5.2,5.4) p91 book 1: max 𝑢 (𝑥1 , 𝑥2 ) such that 𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚 𝑚 𝑝 𝑥2 (𝑥1 ) = − 1 𝑥1 𝑝2 𝑝2 Now substitute for the unconstrained maximization problem: 𝑝1 𝑚 max 𝑢(𝑥1 , − ( ) 𝑥1 ) 𝑝2 𝑝2 To solve the unconstrained maximization (since we used 𝑥2 (𝑥1 ) to ensure 𝑥2 will always satisfy the budget constraint we have to differentiate with respect to 𝑥1 : 𝜕𝑢(𝑥1 , 𝑥2 (𝑥1 )) 𝜕𝑢(𝑥1 , 𝑥2 (𝑥1 )) 𝑑𝑥2 ∗ + =0 𝜕𝑥2 𝜕𝑥1 𝑑𝑥1 First part tells us how x1 increases the utility The second part tells us: 1) the rate of increase of utility as x2 increases:

𝜕𝑢 𝜕𝑥2

2) the rate of increase of x2 as x1 increases in order to continue to satisfy the

Differentiate 𝑥2 (𝑥1 ) =

𝑑𝑥

budged equation 𝑑𝑥2

𝑚 𝑝2

−

𝑝1 𝑥 𝑝2 1

Substituting this formula gives:

1

to calculate 2)’s derivative

∗ ∗ 𝜕𝑢(𝑥1 ,𝑥2 ) ) ( 𝜕𝑥 1 ∗ 𝑥 𝜕𝑢(𝑥1∗ ,𝜕𝑥2 ) 2

=

𝑝1

𝑑𝑥2 𝑑𝑥1

𝑝

= − 𝑝1

2

𝑝2

this will give us two equations with two unknowns as 𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚 2: Lagrange multiplier Step 1: Lagrangian function:

𝐿 = 𝑢 (𝑥1 , 𝑥2 ) − 𝜆(𝑝1 𝑥2 + 𝑝2 𝑥2 − 𝑚)

Step 2: The optimal choice has to satisfy the three first-order conditions: 𝜕𝐿 𝜕𝑢(𝑥1∗ , 𝑥2∗ ) − 𝜆𝑝1 = 0 = 𝜕𝑥1 𝜕𝑥1 𝜕𝐿 𝜕𝑢(𝑥1∗ , 𝑥2∗ ) − 𝜆𝑝2 = 0 = 𝜕𝑥2 𝜕𝑥2 𝜕𝐿 = 𝑝1 𝑥2∗ + 𝑝2 𝑥2∗ − 𝑚 = 0 𝜕𝜆 Example for both ways on p93 Chapter 6

Consumer demand functions: { ∆𝑥1

𝑥1 = 𝑥1 (𝑝1 , 𝑝2 , 𝑚) 𝑥2 = 𝑥2 (𝑝1 , 𝑝2 , 𝑚)

> 0  if income goes up the demand for x1 increases Inferior good: If income goes up the demand for a good will decrease Income offer curve: Relation between both goods (if both normal this line is positive) Engel curve: if p1,p2 are held fixed and only m is changed: The Engel curve is the graph of the demand for one of the goods as a function of income Normal good:

∆𝑚

For perfect substitutes this means that if p1 𝑝2 𝑡ℎ𝑒𝑛 𝑝1 = 𝑝2 𝑡ℎ𝑒𝑛 𝑝1 < 𝑝2  Perfect complements: 𝑚 𝑥1 = so if m and p2 are fixed: diagonal line 𝑝1 +𝑝2

Discrete good: Reservation price: The price at which the consumer is just indifferent to consuming or not consuming the good e.g: If r1 is the price where the consumer is indifferent between consuming 0 or 1 units of good 1: 𝑢(0, 𝑚 ) = 𝑢 (1, 𝑚 − 𝑟1 ) 𝑢(𝑥1 , 𝑥2 ) = 𝑣(𝑥1 ) + 𝑥2 { |𝑣(0) + 𝑚 = 𝑣(1) + 𝑚 − 𝑟1 → 𝑟1 = 𝑣 (1) 𝑎𝑠 𝑣 (0) = 0 𝑣(0) = 0 If r2 is the price where the consumer is indifferent between consuming 1 or 2 units of good 1: 𝑢(1, 𝑚 − 𝑟2 ) = 𝑢(2, 𝑚 − 2𝑟1 ) 𝑢(𝑥1 , 𝑥2 ) = 𝑣(𝑥1 ) + 𝑥2 { |𝑣(1) + 𝑚 − 𝑟2 = 𝑣 (2) + 𝑚 − 2𝑟2 → 𝑟2 = 𝑣(2) − 𝑣(1) 𝑣(0) = 0 The same can be done with r3,r4,…,r∞

How to determine a substitute (perfect or imperfect): good up the demand for good 1 will go up.

∆𝑥1 ∆𝑝2

How to determine a complement (perfect or imperfect): 2 goes up the demand for good 1 will go down

> 0 this means that if the price of good 2

∆𝑥1

∆𝑝2

< 0 this means that if the price of good

Inverse demand function: demand function viewing price as a function of quantity (inverse because negative sloped). e.g.: Cobb Douglas: {

𝑥1 =

𝑝1 =

𝑎𝑚 𝑝1 𝑎𝑚 𝑥1

→ 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 𝑑𝑒𝑚𝑎𝑛𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 → 𝐼𝑛𝑣𝑒𝑟𝑠𝑒 𝑑𝑒𝑚𝑎𝑛𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛

The absolute value of the MRS equals the price ratio: |𝑀𝑅𝑆| =

𝑝1 𝑝2

At the optimal level of demand for good 1 we must have: 𝑝1 = 𝑝2 |𝑀𝑅𝑆| This tells us how much of good 2 the consumer would want to have to compensate him for a small reduction in the amount of good 1. Maximisation problem: p115-117 max 𝑣(𝑥1 ) + 𝑥2 𝑤𝑖𝑡ℎ 𝑝1 𝑥2 + 𝑝2 𝑥2 = 𝑚

𝑚 𝑝1 𝑥1 max 𝑣(𝑥1 ) + − 𝑝2 𝑝2

𝑣 ′(𝑥1∗ )

𝑝1 = 𝑝2

𝑝1 (𝑥1 ) = 𝑣 ′ (𝑥1 )𝑝2

Solving for x2 then substitute:

Differentiate gives u the first-order condition

The inverse demand curve is given by (derivative of the utility function times p2)

Chapter 8 Substitution effect: The change in demand due to the change in the rate of exchange between the two goods Income effect: The change in demand due to having more purchasing power To measure both of these effects  breaking the price movement into two steps - First let the relative prices change and adjust money income so as to hold purchasing power constant - Secondly we let the purchasing power adjust while holding the relative prices constant In this case p1 declines; two steps can be defined: 1) First it pivots and the purchasing power stays equal (Y-X is substitution effect) 2) Then it shifts out to the new demanded bundle (Y-Z is the income effect) If you apply both steps you can measure the substitution and the income effect

Pivoted formula How much we have to adjust money income (m) to keep the old bindle just affordable: m’= the amount of money income that will just make the original consumption bundle affordable (is the same as the pivoted line as (𝑥1 , 𝑥2 ) 𝑖𝑠 𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑙𝑒 𝑎𝑡 (𝑝1 . 𝑝2 , 𝑚) 𝑎𝑛𝑑 (𝑝′1, 𝑝2 , 𝑚 ′ ) 𝑚′ = 𝑝1′ 𝑥1 + 𝑝2 𝑥2 𝑚 = 𝑝1 𝑥1 + 𝑝2 𝑥2 Subtracting the second equation from the first gives: 𝑚′ − 𝑚 = 𝑥1 [𝑝1′ − 𝑝1 ] 𝑝1′ − 𝑝1 = ∆𝑝1 𝑚′ − 𝑚 = ∆𝑚 ∆𝑚 = 𝑥1 ∆𝑝1 *note: (𝑥1 , 𝑥2 ) is still affordable, but it doesn’t have to be optimal The movement from X to Y is the (slutsky) substitution effect (see picture), algebraic: ∆𝑥1𝑠 = 𝑥1 (𝑝1′ , 𝑚 ′) − 𝑥1 (𝑝1 , 𝑚)  p140 example with numbers (slutsky) Income effect: the second shift, keeping the prices constant and changing m’ to m: ∆𝑥1𝑛 = 𝑥1 (𝑝1′ , 𝑚 ) − 𝑥1 (𝑝1′ , 𝑚 ′ ) If price of a good goes down, then the change in the demand for the good due to the substitution effect must be nonnegative: If 𝑝1 > 𝑝1′ (P’ = new price) then 𝑥1 (𝑝1′ , 𝑚 ′ ) ≥ 𝑥1 (𝑝1 , 𝑚), 𝑠𝑜 𝑡ℎ𝑎𝑡 ∆𝑥1𝑠 ≥ 0 Total change in demand: only holding income constant ∆𝑥1 = 𝑥1 (𝑝1′ , 𝑚 ) − 𝑥1 (𝑝, 𝑚) Or: The Slutsky identity: Total change in demand equals the substitution effect plus the income effect ∆𝑥1 = ∆𝑥1𝑠 + ∆𝑥1𝑛 𝑥1 (𝑝1′, 𝑚 ) − 𝑥1 (𝑝, 𝑚) = [𝑥1 (𝑝1′ , 𝑚 ′) − 𝑥1 (𝑝1 , 𝑚 )] + [𝑥1 (𝑝′1, 𝑚 ) − 𝑥1 (𝑝1′ , 𝑚 ′ )] Normal good: income + substitution effect are negative: change in demand also Inferior good: substitution is negative, income is positive: change in demand may be both Giffen good: if the income negative effect is bigger than the positive substitution effect

The slutsky equation expressed in rates of change: ∆𝑥1𝑚 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠 𝑡ℎ𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑐𝑜𝑚𝑒 𝑒𝑓𝑓𝑒𝑐𝑡: ∆𝑥1𝑚 = 𝑥1 (𝑝1′ , 𝑚 ′ ) − 𝑥1 (𝑝1′ , 𝑚 ) = −∆𝑥1𝑛 The slutsky equation becomes: ∆𝑥1 = ∆𝑥1𝑠 − ∆𝑥1𝑚 divide by ∆𝑝1 𝑠 𝑚 ∆𝑥1 ∆𝑥1 ∆𝑥1 = − ∆𝑝1 ∆𝑝1 ∆𝑝1

We know that ∆𝑚 = 𝑥1 ∆𝑝1

𝑠𝑜 ∶ ∆𝑝1 =

Substituting in the last term gives: ∆𝑥1 ∆𝑝1

=

∆𝑥1𝑚 ∆𝑥1𝑠 𝑥1 − ∆𝑚 ∆𝑝1

∆𝑚

𝑥1

the slutsky equation expressed in rates of change

Each term can be interpret as followed: ∆𝑥1 𝑥1 (𝑝1′ , 𝑚 ) − 𝑥1 (𝑝, 𝑚) = ∆𝑝1 ∆𝑝1

∆𝑥1𝑠 𝑥1 (𝑝1′ , 𝑚 ′) − 𝑥1 (𝑝1 , 𝑚) = ∆𝑝1 ∆𝑝1

𝑥1 (𝑝1′, 𝑚 ′ ) − 𝑥1 (𝑝1′ , 𝑚) ∆𝑥1𝑚 𝑥1 𝑥1 = 𝑚′ − 𝑚 ∆𝑚

Law of demand: If the demand for a good increases when income increases, then the demand for that good must decrease when its price increases. Perfect substitutes & Perfect complements

The total effect with substitutes is only due the substitution effect, as there is a corner solution (there is no shift) The total effect with the perfect complements is due to the income effect as there will not be a new optimal point. Quasilinear: The total effect is due to the substitution effect. (a shift in income doesn’t cause a higher consumption of good x1 with quasilinear preferences)

If a tax is imposed and rebated (e.g. tax reduction elsewhere): 𝑝 ′ = 𝑝 + 𝑡 → 𝑎 𝑐𝑜𝑛𝑠𝑢𝑚𝑒𝑟 𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑠 𝑏𝑦 𝑐ℎ𝑎𝑛𝑔𝑖𝑛𝑔 𝑥 𝑡𝑜 𝑥 ′ The Revenue raised by the tax will be: 𝑅 = 𝑡𝑥 ′ = (𝑝 ′ − 𝑝)𝑥 ′ Note: the revenue raised by the tax depends on x’ and not on x. 𝑂𝑙𝑑 𝑏𝑢𝑑𝑔𝑒𝑡 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡: 𝑝𝑥 + 𝑦 = 𝑚 𝑁𝑒𝑤 𝑏𝑢𝑑𝑔𝑒𝑡 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡: (𝑝 + 𝑡)𝑥 ′ + 𝑦 ′ = 𝑚 + 𝑡𝑥 ′ → 𝑝𝑥′ + 𝑦 ′ = 𝑚 𝑇ℎ𝑢𝑠 (𝑥 ′, 𝑦 ′ ) 𝑤𝑎𝑠 𝑎𝑙𝑠𝑜 𝑎𝑓𝑓𝑜𝑟𝑡𝑎𝑏𝑙𝑒 𝑢𝑛𝑑𝑒𝑟 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑏𝑢𝑑𝑔𝑒𝑡 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 𝑎𝑛𝑑 𝑟𝑒𝑗𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑓𝑎𝑣𝑜𝑢𝑟 𝑜𝑓 (𝑥, 𝑦) Conclusion: (𝑥, 𝑦) is preferred over (𝑥 ′, 𝑦 ′ ) if a tax is rebated (niet in stof; 8.8/8.9) Hicks substitution effect: Instead of pivoting the original budget line is ‘rolled’ down. So the utility Is kept constant instead of the purchasing power. Hicksian demand curve (utility held constant) = compensated demand curve:  The consumer is ‘compensated’ for the price changes. The normal demand curve: consumer is worse off when there is a price raise. Chapter 18 Private-value auctions: Each participant has a different value for the good in mind Common-value auctions: The goods are worth the same to every bidder; their estimates may differ English auction: starting with a reserve price then bidder bid higher with a bid increment. Dutch auction: Starting high; then lower until someone wants to buy it. Sealed-bid auction: anonymously bidding; highest bidder wins (construction work) Philatelist auction/Vickrey auction: person who bids the highest gets the good for the second price that have been bid. How to pick the right auction? Two natural goals: - Pareto efficiency: (good has to end up if the person with the highest value) - Profit maximisation Example with 2 bidders in a Vicky auction: 𝑣1 , 𝑣2 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒𝑠, 𝑏1 , 𝑏2 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑏𝑖𝑑𝑠 𝑃𝑟𝑜𝑏 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 ℎ𝑎𝑣𝑖𝑛𝑔 𝑡ℎ𝑒 ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝑏𝑖𝑑

The expected payoff for bidder 1 is: 𝑃𝑟𝑜𝑏 (𝑏1 ≥ 𝑏2 )[𝑣1 − 𝑏2 ] If 𝑏1 < 𝑏2 : 𝑏𝑖𝑑𝑑𝑒𝑟 1 𝑔𝑒𝑡𝑠 𝑎 𝑠𝑢𝑟𝑝𝑙𝑢𝑠 𝑜𝑓 0 If 𝑣1 > 𝑏2 : 𝑏𝑖𝑑𝑑𝑒𝑟 1 𝑤𝑖𝑙𝑙 𝑠𝑒𝑡 𝑏1 = 𝑣1 to have the highest probability of winning If 𝑣1 < 𝑏2 : 𝑏𝑖𝑑𝑑𝑒𝑟 1 𝑤𝑖𝑙𝑙 𝑠𝑒𝑡 𝑏1 = 𝑣1 to have the lowest probability of winning An optimal strategy for bidder 1 is to make his bid equal to his value.

Other forms of Vicky auctions Goethe auction: auction (p335) Bidding agent: (telling an agent your max. bid, then he bids in increments) Escalation auction: the highest bidder wins the item, but the highest and the second-highest bidders both pay the amount they bid. Everyone pays auction: Same as escalation auction but everyone pays

Position auctions: for positions e.g. advertisement on google. Different value’s but the value of being ‘first’ in the line is valued more than being ‘second’. Everyone is placing a bid, and the highest bid is getting the first ‘slot’ of advertisement, the second highest bid the second slot. Generalized second price auction (GSP). By setting the payment of the advertiser in slot s to be the bid of the advertiser in slot s+1, each advertiser ends up paying the minimum bid necessary to retain its position 𝑃𝑟𝑜𝑓𝑖𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑑𝑣𝑒𝑟𝑡𝑖𝑠𝑒𝑟 𝑖𝑛 𝑠𝑙𝑜𝑡 𝑠: (𝑣𝑠 − 𝑏𝑠+1 )𝑥𝑠 The formula is just the value of the clicks minus the cost of the clicks (x1) that an advertiser receives (what he bids for it). Position auction with 2 slots and 2 bidders 𝑣 = 𝑣𝑎𝑙𝑢𝑒 𝑏 = 𝑏𝑖𝑑 𝑟 = 𝑟𝑒𝑠𝑒𝑟𝑣𝑒 𝑝𝑟𝑖𝑐𝑒 The high bidder gets x1 and pays the bid of the second highest bidder b2. The second highest bidder gets slot 2 and pays a reserve price r. 𝐼𝑓 𝑏 > 𝑏2 𝑦𝑜𝑢 𝑔𝑒𝑡 𝑎 𝑝𝑎𝑦𝑜𝑓𝑓 𝑜𝑓 (𝑣 − 𝑏2 )𝑥1 𝐼𝑓 𝑏 ≤ 𝑏2 𝑦𝑜𝑢 𝑔𝑒𝑡 𝑎 𝑝𝑎𝑦𝑜𝑓𝑓 𝑜𝑓 (𝑣 − 𝑟)𝑥2 Therefore the expected payoff will be: 𝑃𝑟𝑜𝑏 (𝑏 > 𝑏2 )(𝑣 − 𝑏2 )𝑥1 + [1 − 𝑃𝑟𝑜𝑏 (𝑏 > 𝑏2 )]𝑣 − 𝑟 (𝑥2 ) (𝑣 − 𝑟)𝑥2 + 𝑃𝑟𝑜𝑏(𝑏 > 𝑏2 )[𝑣(𝑥1 − 𝑥2 ) + 𝑟𝑥2 − 𝑏2 𝑥1 ]

You want 𝑃𝑟𝑜𝑏 (𝑏 > 𝑏2 ) to be as large as possible when the term in the brackets is positive, otherwise it needs to be as small as possible. Rearranging you get: 𝑏𝑥1 = 𝑣(𝑥1 − 𝑥2 ) + 𝑟𝑥2 In this auction you don’t bid your true value per click, you want to bid an amount that reflects your true value of the incremental clicks that you are getting

Position auction with more than two bidders 3 slots and 3 bidders: in equilibrium the bidder doesn’t want to move up to slot 2, therefore you get: (𝑣3 − 𝑟)𝑥3 ≥ (𝑣3 − 𝑝2 )𝑥2 𝑣3 (𝑥2 − 𝑥3 ) ≤ 𝑝2 𝑥2 𝑟𝑥3 So bound on the cost of clicks in position 2: 𝑝2 𝑥2 ≤ 𝑟𝑥3 + 𝑣3 (𝑥2 − 𝑥3 ) Bidder in position 2: 𝑝1 𝑥1 ≤ 𝑝2 𝑥2 + 𝑣2 (𝑥1 − 𝑥2 ) Substituting gives: 𝑝1 𝑥1 ≤ 𝑟𝑥3 + 𝑣3 (𝑥2 − 𝑥3 ) + 𝑣2 (𝑥1 − 𝑥2 ) Total revenue: 𝑝1 𝑥1 + 𝑝2 𝑥2 + 𝑝3 𝑥3 Lower bound total revenue (adding the two inequality’s and the revenue for slot 3: 𝑅𝐿 ≤ 𝑣2 (𝑥1 − 𝑥2 ) + 2𝑣3 (𝑥2 − 𝑥3 ) + 3𝑟𝑥3

When there are 4 bidders for 3 slots: 𝑅𝐿 ≤ 𝑣2 (𝑥1 − 𝑥2 ) + 2𝑣3 (𝑥2 − 𝑥3 ) + 3𝑣4 𝑥3 Notes: The bigger the gap the higher the revenue, the more competition the more the revenue, it’s about how many clicks you get. Quality Scores: The bids are multiplied by a quality score to get an auction ranking score: 𝑐𝑜𝑠𝑡 𝑐𝑙𝑖𝑐𝑘𝑠 𝑐𝑜𝑠𝑡 ∗ = 𝑐𝑙𝑖𝑐𝑘𝑠 𝑖𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛𝑠 𝑖𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛𝑠 e.g.: should well-known brands buy advertisement? 𝑣 = 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎 𝑐𝑙𝑖𝑐𝑘 𝑥𝑎 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑑 𝑐𝑙𝑖𝑐𝑘𝑠 𝑥𝑜𝑎 = 𝑜𝑟𝑔𝑎𝑛𝑖𝑐 𝑐𝑙𝑖𝑐𝑘𝑠 𝑤ℎ𝑒𝑛 𝑎𝑑 𝑖𝑠 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑥𝑜𝑛 = 𝑜𝑟𝑔𝑎𝑛𝑖𝑐 𝑐𝑙𝑖𝑐𝑘𝑠 𝑤ℎ𝑒𝑛 𝑡ℎ𝑒 𝑎𝑑 𝑖𝑠 𝑛𝑜𝑡 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑐 (𝑥𝑎 ) = 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑥𝑎 𝑎𝑑 𝑐𝑙𝑖𝑐𝑘𝑠 If a website advertises the profit is: 𝑣𝑥𝑎 + 𝑣𝑥𝑜𝑎 − 𝑐 (𝑥𝑎 ) If a website does not advertise: 𝑣𝑥𝑜𝑛 A website owner find is profitable to advertise when: 𝑣𝑥𝑎 + 𝑣𝑥𝑜𝑎 − 𝑐 (𝑥𝑎 ) > 𝑣𝑥𝑜𝑛 𝑐(𝑥𝑎 ) 𝑣> 𝑥𝑎 − (𝑥𝑜𝑛 − 𝑥𝑜𝑎 ) Second order statistic: The expected revenue will be the expected value of the second-largest valuation in a sample of size n. e.g. an interval like [0,1]: The higher the n the closer it will get to 1. Problem with English/Vickrey auctions: collusion and manipulation. Common-value auctions: (same value to all bidders, but the estimates may differ) 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏𝑖𝑑𝑑𝑒𝑟 𝑖 = 𝑣 + 𝜖𝑖 Where 𝜖𝑖 is the error term associated with I’s estimate and 𝑣 is the real value. What bid should the bidder place? Winners curse: 𝑡ℎ𝑒 𝑝𝑒𝑟𝑠𝑜𝑛 𝑤𝑖𝑡ℎ 𝜖𝑚𝑎𝑥 will get the good, however if 𝑖𝑓 𝜖𝑀𝑎𝑥 > 0 this person is paying more than v (the true value). The optimal strateg...