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Summary Intermediate Microeconomics Hal R. Varian,
complete
Microeconomics (Wageningen University & Research)
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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 ο Market equilibrium P*
Monopoly ο 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 ο 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: π1 π₯1 + π2 π₯2 ≤ m.
X2 = can be used as composite good (everything else the consumer buys)
Budget set = All bundles ≤ π. (area left of the line)
π
Budget line slope = − 1
π2
Budged line = Set of bundles that cost exactly m (the line):
π1 π₯1 + π2 π₯2 = m
The budged line can be rewritten as:
π
π
π₯2 = π − π1 π₯1 ο if x1 = 0 everything of m is used for x2.
2
2
Two formulas given:
Budget line before change: π1 π₯1 + π2 π₯2 = m
Change of consumption: π1 (π₯1 + βπ₯1 ) + π2 (π₯2 + βπ₯2 ) = π gives:
βπ₯2
π1
π1 βπ₯1 + π2 βπ₯2 = 0 →
=−
βπ₯1
π2
Slope measures opportunity cost. (of consuming good 1)
Income increase ο Budget Line shifts outwards parallel
Price increase ο Budget line becomes steeper
Numaire price ο Relative price to which we are measuring the other price and income: e.g:
p1
π
π1 π₯1 + π2 π₯2 = m → π₯1 + π₯2 =
→ π»πππ π2 ππ πππ’ππ π‘π 1
π2
π2
π2
π1
π₯1 + π₯2 = 1 π»πππ π ππ πππ’ππ π‘π 1
π
π
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Quantity tax: π1 + π‘ Value tax (%): (1 + π)π1
Quantity Subsidy: π1 − π Ad valorem Subsidy (%):(1 + π)π1
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 ο Complete list of goods and services (π₯1 , π₯2 )
(π₯1 , π₯2 ) > (π¦1 , π¦2 ): π΅π’ππππ π₯1 , π₯2 ππ π π‘ππππ‘ππ¦ πππππππππ ππ£ππ π¦1 , π¦2
(π₯1 , π₯2 ) ~ (π¦1 , π¦2 ): πΌππππππππππ‘. (π₯1 , π₯2 ) ≥ (π¦1 , π¦2 ): π€πππππ¦ πππππππππ
Assumptions about consumer preference:
Complete: Any two bundles can be compared: (π₯1 , π₯2 ) ≥ (π¦1 , π¦2 )
Reflexive: Any bundle is at least as good as itself: (π₯1 , π₯2 ) ≥ (π₯1 , π₯2 )
Transitive: If (π₯1 , π₯2 ) ≥ (π¦1 , π¦2 ) and (π¦1 , π¦2 ) ≥ (π§1 , π§2 ) then (π₯1 , π₯2 ) ≥ (π§1 , π§2 )
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: (xΜ 1 , xΜ 2 )
Well-behaved indifference curves features:
- Monotonicity: More is better; negative slope
- Averages preferred to extremes
- Convex
Weighted average:
πΌπ (π₯1 , π₯2 )~(π¦1 . π¦2 ) π‘βππ
(π‘1 + (1 − π‘)π¦1 , π‘π₯2 + (1 − π‘)π¦2 ) ≥ (x1 , π₯2 )
βπ₯
Marginal Rate of Substitution (MRS): slope of indifference curve; βπ₯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:
βπ (π(π’2 ) − π(π’1 ))
=
βπ’
π’2 − π’1
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Perfect substitutes: π’(π₯1 , π₯2 ) = ππ₯1 + ππ₯2 or the monotonic transformation (e.g. square root) ο
(π₯1 , π₯2 ) = π₯12 + 2π₯1 π₯2 + π₯22
π
a & b represent the ‘value’ of goods 1 and 2 to the consumer: The slope is − π
Perfect complements: π’(π₯1 , π₯2 ) = min{ππ₯1 , ππ₯2 }
a & b are the proportions in which the good is consumed
Quasilinear Preferences: π’(π₯1 , π₯2 ) = π = π£(π₯1 ) + π₯2
So the good can be non-linear in good x1 ο e.g.: π’(π₯1 , π₯2 ) = √π₯1 + π₯2
Cobb-douglas Preferences: π’(π₯1 , π₯2 ) = π₯1π π₯2π
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:
π
π
1
π
π
π+π
π+π
π’(π₯1 , π₯2 ) = (π₯1 π₯2 ) π‘π π‘βπ πππ€ππ ππ
→ = π₯1 π₯2 πππ€ ππππππ π
π+π
π
=
π‘βπππππππ: π£(π₯1 , π₯2 ) = π₯1π π₯21−π
π+π
βπ
Marginal Utility (of good 1): ππ1 = βπ₯ =
1
π’(π₯1 +βπ₯1 ,π₯2 )−π’(π₯1 ,π₯2 )
βπ₯1
ο good 2 is kept fixed.
So for the full change of utility if good x1 changes: βπ = ππ1 βπ₯1.
If: ππ1 βπ₯1 + ππ2 βπ₯2 = βπ = 0 ο so a change in x1 and x2 changes consumption along the
indifference curve then: ππ
π =
to keep the same level of utility
βπ₯2
βπ₯1
ππ
= − ππ1 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: (π₯1∗ π₯2∗ )
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: π₯1 (π1 , π2 , π) & π₯2 (π1 , π2 , π)
π€βππ π1 < π2 βΆ π/π1
For perfect substitutes π₯1 = {π€βππ π1 = π2 βΆ πππ¦ ππ’ππππ πππ‘π€πππ 0 πππ π/π1
π€βππ π1 > π2 βΆ 0
π
For perfect complements π1 π₯1 + π2 π₯2 = π → π₯1 = π₯2 = π₯ = π +π
1
2
π
π
π₯1 =
∗
π+π π1
Cobb-douglas preferences: π’(π₯1 , π₯2 ) = π₯1π π₯2π → π€ππ‘β ππππππ ππ’πππ‘ππππ : {
π
π
π₯2 =
∗
π+π π2
π1 π₯1
π1
π
π
π
π‘βπ ππππ’ππ‘ π ππππ ππ π₯1 =
→ π π’ππ π‘ππ‘π’π‘πππ π₯1 πππ£ππ : ∗
∗ =
π
π π + π π1 π + π
π
πππ π₯2 π‘βππ ππ
π+π
An optimum quantity tax applied: (π1 + π‘)π₯1∗ + π2 π₯2∗ = π → πππ£πππ’π πππ ππ ππ ππ¦ π‘ππ₯: π
∗ = π‘π₯1∗
This leads to a change of the slope pf the budget line −
π
π1 +π‘
π2
Income tax: π1 π₯1∗ + π2 π₯2∗ = π − π‘π₯1∗ ο slope stays − π1 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.
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The utility maximization problem: (workbook 5.2,5.4) p91 book
1: max π’(π₯1 , π₯2 )
such that π1 π₯1 + π2 π₯2 = π
π π1
π₯2 (π₯1 ) = − π₯1
π2 π2
Now substitute for the unconstrained maximization problem:
π
π1
max π’(π₯1 , − ( ) π₯1 )
π2
π2
To solve the unconstrained maximization (since we used π₯2 (π₯1 ) to ensure π₯2 will always satisfy the
budget constraint we have to differentiate with respect to π₯1 :
ππ’(π₯1 , π₯2 (π₯1 )) ππ’(π₯1 , π₯2 (π₯1 )) ππ₯2
+
∗
=0
ππ₯1
ππ₯2
ππ₯1
First part tells us how x1 increases the utility
ππ’
The second part tells us: 1) the rate of increase of utility as x2 increases: ππ₯
2
2) the rate of increase of x2 as x1 increases in order to continue to satisfy the
Differentiate π₯2 (π₯1 ) =
ππ₯
budged equation ππ₯2
π
π2
−
π1
π₯
π2 1
Substituting this formula gives:
ππ₯
π
to calculate 2)’s derivative ππ₯2 = − π1
ππ’(π₯∗1 ,π₯∗2 )
)
ππ₯1
∗
π₯
ππ’(π₯1∗ ,ππ₯2 )
2
(
1
π
= π1
1
2
2
this will give us two equations with two unknowns as π1 π₯1 + π2 π₯2 = π
2: Lagrange multiplier
Step 1: Lagrangian function:
πΏ = π’(π₯1 , π₯2 ) − π(π1 π₯2 + π2 π₯2 − π)
Step 2: The optimal choice has to satisfy the three first-order conditions:
ππΏ
ππ’(π₯1∗ , π₯2∗ )
=
− ππ1 = 0
ππ₯1
ππ₯1
ππΏ
ππ’(π₯1∗ , π₯2∗ )
=
− ππ2 = 0
ππ₯2
ππ₯2
ππΏ
= π1 π₯2∗ + π2 π₯2∗ − π = 0
ππ
Example for both ways on p93
Chapter 6
π₯ = π₯1 (π1 , π2 , π)
Consumer demand functions: { 1
π₯2 = π₯2 (π1 , π2 , π)
βπ₯
Normal good: βπ1 > 0 ο 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
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For perfect substitutes this means that if p1<p2 the income offer curve will be on the horizontal axis
of x1 (as only x1 is being consumed). The Engel curve will be upwards with slope π1 ππ π = π1 π₯1
For perfect complements the income offer line is diagonal and the Engel curve sloped as π1 + π2
With a Cobb-Douglas function π’(π₯1 , π₯2 ) = π₯1π π₯2π−1 → {
The Engel curve will be slopes as
π1
π
π₯1 = ππ/π1
π₯2 =
(1−π)π
ο both linear so diagonal line
π2
Luxury good = demand for a good goes up by a greater proportion than income
Necessary good = demand for a good goes up by a lesser proportion than income
For Quasilinear Preferences all indifference curves are vertically shifted, therefore the income offer
curve is a vertical line. The Engel curve will be vertical as well eventually because an increase in
income doesn’t matter for good x1 to be changed. (the Engel curve starts from the 0, then shifts
diagonal and after some point it will go vertical when x1 is satisfied).
Ordinary good: demand increases when price increases
Giffen good: demand decreases when the price decreases
Price offer curve: p2 and m are fixed, whilst p1 can change. The curve you can
draw represents the bundles that would be demanded at different prices for
good 1. If you look at the optimal level of consumption of good 1 (again with p2
and m fixed) you will get the corresponding
Perfect substitutes:
ππππ π‘ π1 > π2 π‘βππ π1 = π2 π‘βππ π1 < π2 ο
Perfect complements:
π
π₯1 = π +π so if m and p2 are fixed: diagonal line
1
2
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:
π’(0, π) = π’(1, π − π1 )
π’(π₯1 , π₯2 ) = π£(π₯1 ) + π₯2
|π£(0) + π = π£(1) + π − π1 → π1 = π£(1) ππ π£(0) = 0
{
π£(0) = 0
If r2 is the price where the consumer is indifferent between consuming 1 or 2 units of good 1:
π’(1, π − π2 ) = π’(2, π − 2π1 )
π’(π₯1 , π₯2 ) = π£(π₯1 ) + π₯2
|π£(1) + π − π2 = π£(2) + π − 2π2 → π2 = π£(2) − π£(1)
{
π£(0) = 0
The same can be done with r3,r4,…,r∞
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How to determine a substitute (perfect or imperfect):
good up the demand for good 1 will go up.
βπ₯1
βπ2
> 0 this means that if the price of good 2
βπ₯
How to determine a complement (perfect or imperfect): βπ1 < 0 this means that if the price of good
2 goes up the demand for good 1 will go down
2
Inverse demand function: demand function viewing price as a function of quantity (inverse because
negative sloped).
e.g.: Cobb Douglas: {
π₯1 =
π1 =
ππ
π1
ππ
π₯1
→ ππππ’πππ ππππππ ππ’πππ‘πππ
→ πΌππ£πππ π ππππππ ππ’πππ‘πππ
π
The absolute value of the MRS equals the price ratio: |ππ
π| = π1
2
At the optimal level of demand for good 1 we must have: π1 = π2 |ππ
π|
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 π£(π₯1 ) + π₯2 π€ππ‘β π1 π₯2 + π2 π₯2 = π
π π1 π₯1
max π£(π₯1 ) + −
π2
π2
π1
π£ ′ (π₯1∗ ) =
π2
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)
π1 (π₯1 ) = π£ ′ (π₯1 )π2
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 ο 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
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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 (π₯1 , π₯2 ) ππ ππππππππππ ππ‘ (π1 . π2 , π) πππ (π1′ , π2 , π′ )
π′ = π1′ π₯1 + π2 π₯2
π = π1 π₯1 + π2 π₯2
Subtracting the second equation from the first gives:
π′ − π = π₯1 [π1′ − π1 ]
π1′ − π1 = βπ1
π′ − π = βπ
βπ = π₯1 βπ1
*note: (π₯1 , π₯2 ) 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:
βπ₯1π = π₯1 (π1′ , π′ ) − π₯1 (π1 , π) ο p140 example with numbers
(slutsky) Income effect: the second shift, keeping the prices constant and changing m’ to m:
βπ₯1π = π₯1 (π1′ , π) − π₯1 (π1′ , π′ )
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 π1 > π1′ (P’ = new price) then π₯1 (π1′ , π′ ) ≥ π₯1 (π1 , π), π π π‘βππ‘ βπ₯1π ≥ 0
Total change in demand: only holding income constant
βπ₯1 = π₯1 (π1′ , π) − π₯1 (π, π)
Or:
The Slutsky identity: Total change in demand equals the substitution effect plus the income effect
βπ₯1 = βπ₯1π + βπ₯1π
π₯1 (π1′ , π) − π₯1 (π, π) = [π₯1 (π1′ , π′ ) − π₯1 (π1 , π)] + [π₯1 (π1′ , π) − π₯1 (π1′ , π′ )]
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
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The slutsky equation expressed in rates of change:
βπ₯1π ππ πππππππ ππ π‘βπ πππππ‘ππ£π ππ π‘βπ ππππππ ππππππ‘:
βπ₯1π = π₯1 (π1′ , π′ ) − π₯1 (π1′ , π) = −βπ₯1π
The slutsky equation becomes:
βπ₯1 = βπ₯1π − βπ₯1π
divide by βπ1
βπ₯1 βπ₯1π βπ₯1π
=
−
βπ1 βπ1 βπ1
We know that βπ = π₯1 βπ1
π π βΆ βπ1 =
Substituting in the last term gives:
βπ₯1
βπ1
=
βπ₯1π
βπ1
−
βπ₯1π
βπ
βπ
π₯1
π₯1 ο the slutsky equation expressed in rates of change
Each term can be interpret as followed:
βπ₯1 π₯1 (π1′ , π) − π₯1 (π, π)
=
βπ1
βπ1
βπ₯1π π₯1 (π1′ , π′ ) − π₯1 (π1 , π)
=
βπ1
βπ1
π₯1 (π1′ , π′ ) − π₯1 (π1′ , π)
βπ₯1π
π₯1 =
π₯1
π′ − π
βπ
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)
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If a tax is imposed and rebated (e.g. tax reduction elsewhere):
π′ = π + π‘ → π ππππ π’πππ πππ πππππ ππ¦ πβππππππ π₯ π‘π π₯ ′
The Revenue raised by the tax will be: π
= π‘π₯ ′ = (π′ − π)π₯ ′
Note: the revenue raised by the tax depends on x’ and not on x.
πππ ππ’ππππ‘ ππππ π‘πππππ‘: ππ₯ + π¦ = π
πππ€ ππ’ππππ‘ ππππ π‘πππππ‘: (π + π‘)π₯ ′ + π¦ ′ = π + π‘π₯ ′ → ππ₯ ′ + π¦ ′ = π
πβπ’π (π₯ ′ , π¦ ′ ) π€ππ πππ π ππππππ‘ππππ π’ππππ π‘βπ ππππππππ ππ’ππππ‘ ππππ π‘πππππ‘ πππ ππππππ‘ππ ππ πππ£ππ’π ππ (π₯, π¦)
Conclusion: (π₯, π¦) is preferred over (π₯ ′ , π¦ ′ ) 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:
ο¨ 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: π£1 , π£2 πππ π‘βπ π£πππ’ππ , π1 , π2 πππ π‘βπ ππππ
ππππ ππ π‘βπ ππππππππππ‘π¦ ππ βππ£πππ π‘βπ βππβππ π‘ πππ
The expected payoff for bidder 1 is: ππππ(π1 ≥ π2 )[π£1 − π2 ]
If π1 < π2 : ππππππ 1 πππ‘π π π π’ππππ’π ππ 0
If π£1 > π2 : ππππππ 1 π€πππ π ππ‘ π1 = π£1 to have the highest probability of winning
If π£1 < π2 : ππππππ 1 π€πππ π ππ‘ π1 = π£1 to have the lowest probability of winning
An optimal strategy for bidder 1 is to make his bid equal to his value.
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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
ππππππ‘ ππ π‘βπ πππ£πππ‘ππ ππ ππ π πππ‘ π : (π£π − ππ +1 )π₯π
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
π£ = π£πππ’π π = πππ π = πππ πππ£π πππππ
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.
πΌπ π > π2 π¦ππ’ πππ‘ π πππ¦πππ ππ (π£ − π2 )π₯1
πΌπ π ≤ π2 π¦ππ’ πππ‘ π πππ¦πππ ππ (π£ − π)π₯2
Therefore the expected payoff will be:
ππππ(π > π2 )(π£ − π2 )π₯1 + [1 − ππππ(π > π2 )]π£ − π(π₯2 )
(π£ − π)π₯2 + ππππ(π > π2 )[π£(π₯1 − π₯2 ) + ππ₯2 − π2 π₯1 ]
You want ππππ(π > π2 ) 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: ππ₯1 = π£(π₯1 − π₯2 ) + ππ₯2
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:
(π£3 − π)π₯3 ≥ (π£3 − π2 )π₯2
π£3 (π₯2 − π₯3 ) ≤ π2 π₯2 ππ₯3
So bound on the cost of clicks in position 2:
π2 π₯2 ≤ ππ₯3 + π£3 (π₯2 − π₯3 )
Bidder in position 2:
π1 π₯1 ≤ π2 π₯2 + π£2 (π₯1 − π₯2 )
Substituting gives:
π1 π₯1 ≤ ππ₯3 + π£3 (π₯2 − π₯3 ) + π£2 (π₯1 − π₯2 )
Total revenue: π1 π₯1 + π2 π₯2 + π3 π₯3
Lower bound total revenue (adding the two inequality’s and the revenue for slot 3:
π
πΏ ≤ π£2 (π₯1 − π₯2 ) + 2π£3 (π₯2 − π₯3 ) + 3ππ₯3
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When there are 4 bidders for 3 slots:
π
πΏ ≤ π£2 (π₯1 − π₯2 ) + 2π£3 (π₯2 − π₯3 ) + 3π£4 π₯3
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:
πππ π‘
ππππππ
πππ π‘
∗
=
ππππππ ππππππ π ππππ ππππππ π ππππ
e.g.: should well-known brands buy advertisement?
π£ = π£πππ’π ππ π πππππ π₯π = ππ’ππππ ππ ππ ππππππ π₯ππ = πππππππ ππππππ π€βππ ππ ππ ππππ πππ‘
π₯ππ = πππππππ ππππππ π€βππ π‘βπ ππ ππ πππ‘ ππππ πππ‘ π(π₯π ) = πππ π‘ ππ π₯π ππ ππππππ
If a website advertises the profit is: π£π₯π + π£π₯ππ − π(π₯π )
If a website does not advertise: π£π₯ππ
A website owner find is profitable to advertise when:
π£π₯π + π£π₯ππ − π(π₯π ) > π£π₯ππ
π(π₯π )
π£>
π₯π − (π₯ππ − π₯ππ )
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)
ππ π‘ππππ‘ππ π£πππ’π ππ ππππππ π = π£ + ππ
Where ππ is the error term associated with I’s estimate and π£ is the real value.
What bid should the bidder place?
Winners curse: π‘βπ ππππ ππ π€ππ‘β ππππ₯ will get the good, however if ππ ππππ₯ > 0 this person is
paying more than v (the true value). The optimal strategy here is to bet below your estimated value.
Deferred acceptance algorithm: p346 a way to make two-way matching possible
Economic mechanisms: They define a game or market that will yield some desired outcome. (e.g.
auctions and two-sided matching model)
Economic mechanism is the opposite of game theory: with game theory we are given the description
of the rules of the game and we want to determine what the outcome will be.
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Chapter 19: technology
Factors of production: Inputs to production (raw materials, capital, labour etc.)
Technological constraints: Only a certain combinations of inputs are feasible ways to produce
a given amount of output.
The maximum possible output when using input is described in the production function.
If there is a two way input π(π₯1 , π₯2 ) we use isoquants (otherwise 3d) ο the set of all possible
combinations of inputs 1 and 2 that are just sufficient to produce a given amount of output. (similar
as indifference curves)
Examples of production functions:
Fixed proportions:: π(π₯1 , π₯2 ) = min{π₯1 , π₯2 )
e.g.: Production holes: we need 1 man and 1 shovel.
Perfect substitutes: π(π₯1 , π₯2 ) = π₯1 + π₯2
e.g.: Production of homework: we need blue pencils or black pencils
Cobb-Douglas: π(π₯1 , π₯2 ) = π΄π₯1π π₯2π
The parameter A measures the scale of production; how much output we would get if we used only
one unit of each input. The a and b measure how the amount of output responds to changes in the
inputs.
Assumptions about technology:
Monotonic (if you increase the amount of at least one of the inputs, it should be possible to produce
at least as much output a you were producing originally: free disposal)
Convex (This means that if you have two ways to produce y units of output (π₯1 , π₯2 ); (π§1 , π§2 ) then
their weighted average will produce at least y units of output)
Marginal product of factor 1 (π₯1 , π₯2 ): use a little bit more of 1 & keep 2 fixed at x2:
βπ
π(ππ + βππ , ππ ) − π(ππ , ππ )
=
βππ
βππ
Factor 2 can be done the same. General denotation is: ππ1 = (π₯1 , π₯2 ) & ππ2 (π₯1 , π₯2 )
Diminishing MP: MP is normally positive but in a decreasing rate (if you add 1 more cows to a farm
and keep the land fixed the MP is bigger when you add 100)
Technical rate of substitution ππ
π(π₯1 , π₯2 ): giving up a little of 1 and adding more of 2 to get the
same output of y:
βπ¦ = ππ1 (π₯1 , π₯2 )βπ₯1 + ππ2 (π₯1 , π₯2 )βπ₯2 = 0
Solving gives:
βπ₯2
ππ1 (π₯1 , π₯2 )
ππ
π(π₯1 , π₯2 ) =
=−
βπ₯1
ππ2 (π₯1 , π₯2 )
Diminishing TRS: if you increase factor 1 and adjust factor 2 so as to stay on the same isoquant, the
TRS declines. So how the slope of the isoquant changes. With RTS it’s about the ratio of the marginal
products.
Short run: some factor fixed (e.g. land) Long run: all factors varied.
Constant returns to scale:
π‘π(π₯1 , π₯2 ) = π(π‘π₯1 , π‘π₯2 ) π€βπππ π‘ ππ π‘βπ π ππππ ππ ππππ’π‘ πππππππ πππππ‘
Increasing returns to scale: π(π‘π₯1 , π‘π₯2 ) > π‘π(π₯1 , π₯2 ) πππ πππ π‘ > 1
Decreasing returns to scale: π(π‘π₯1 , π‘π₯2 ) < π‘π(π₯1 , π₯2 )πππ πππ π‘ > 1
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Chapter 20:
Π = Revenues-Costs. Note: costs also incl. opportunity costs.
A firm produces n outputs (π¦π ), uses m inputs (π₯π ). Prices goods are (ππ ) Prices inputs: (π€π )
π
π
π=1
π=1
π = ∑ ππ π¦π − ∑ π€π π₯π
Fixed & Variable factors ο in the long run all variable.
Quasi-fixed factors: If production is 0 a company doesn’t need to pay if it’s > 0 it has to pay a fixed
amount (e.g. lighting).
Short-Run Profit Maximization:
max ππ(π₯1 , π₯Μ 2 ) − π€1 π₯1 − π€2 π₯Μ 2
π₯1
Where p is the price of the output, f is the production function and W is the price of the output.
π₯1∗ is the profit-maximizing choice of factor 1, then the output price times the marginal product of
factor 1 should equal the price of factor 1:
πππ1 (π₯1∗ , π₯Μ 2 ) = π€1
ο The value of the marginal product of a factor should
equal it’s price
Isoprofit lines: (y denotes the output of the firm)
ππππππ‘π : π = ππ¦ − π€1 π₯1 − π€2 π₯Μ 2
Transform so y is a function of x1:
π€1
π π€2
π₯Μ 2 +
π₯
π¦= +
π
π 1
π
This describes the isoprofit line: All combinations of the input goods and the output good that give a
π€
constant level of profit π. If π varies we get parallel straight lines with a slope of 1 each having a
π
π
vertical intercept of: +
π€2 π₯Μ 2
.
π
π
Since the slope of the profuction function is the
π€
marginal product, and the slope of the isoprofit is π1 .
We can write the maxpf function as:
π€1
ππ1 =
π
Comparative statics:
e.g.: how does the optimal choice of factor 1 vary as we increase its factor price π€1 .
ο The isoprofit line will be steeper, so the tangency will occur more to the left and decreases π₯1∗
Decreasing the output price (p) will cause the tangency to occur more to the left and decrease π₯1∗ .
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Profit Maximization in the Long Run: (both goods can vary)
max ππ(π₯1 , π₯2 ) − π€1 π₯1 − π€2 π₯2
π₯1 ,π₯2
Now do the same but with both factors:
πππ1 (π₯1∗ , π₯2∗ ) = π€1
πππ2 (π₯1∗ , π₯2∗ ) = π€2
If a firm has made their optimal choices of factors 1 and 2, the value of the marginal product of each
factor should equal it’s price.
‘
The two conditions above give us two unknowns: If we know how the marginal products behave as a
function of π₯1 & π₯2 we will be able to solve for the optimal choice of each factor as a function of the
prices ο Factor demand curves
Inverse factor demand curve: Measures what the factor prices must be for some given quantity of
inputs to be demanded. Downward sloping by the assumption of diminishing marginal product.
πππ1 (π₯1 , π₯2∗ ) = π€1
Example:
Firm has chosen:
Profits are:
π¦ ∗ = π(π₯1∗ , π₯2∗ )
π ∗ = ππ¦ ∗ − π€1 π₯1∗ − π€2 π₯2∗
20.11: revealed profitability
Example:
Suppose we observe two choices that a firm makes at two different sets of prices.
At time t, it faces prices: (ππ‘ , π€1π‘ , π€2π‘ ) πππ πβπππππ (π¦ π‘ , π₯1π‘ , π₯2π‘ )
At time s, it faces prices: (π π , π€1π , π€2π ) πππ πβπππππ (π¦ π , π₯1π , π₯2π )
If the production function didn’t change during t and s; the firm is a profit maximizer, therefore:
ππ‘ π¦ π‘ − π€1π‘ π₯1π‘ − π€2π‘ π₯2π‘ ≥ ππ‘ π¦ π − π€1π‘ π₯1π − π€2π‘ π₯2π
π π π¦ π − π€1π π₯1π − π€2π π₯2π ≥ π π π¦ π‘ − π€1π π₯1π‘ − π€2π π₯2π‘
If one of these properties is violated the firm is not maximizing profits in at least one of the periods.
Also known as: Weak Axiom of Profit Maximization (WAPM):
Adding the two previous equations you get:
(ππ‘ − π π )π¦ π‘ − (π€1π‘ − π€1π )π₯1π‘ − (π€2π‘ − π€2π )π₯2π‘ ≥ (ππ‘ − π π )π¦ π − (π€1π‘ − π€1π )π₯1π − (π€2π‘ − π€2π )π₯2π
(ππ‘ − π π )(π¦ π‘ − π¦ π ) − (π€1π‘ − π€1π )(π₯1π‘ − π₯1π ) − (π€2π‘ − π€2π )(π₯2π‘ − π₯2π ) ≥ 0
βπβπ¦ − βπ€1 βπ₯1 − βπ€2 βπ₯2 ≥ 0
If βπ€1 = βπ€2 = 0 → βπβπ¦ ≥ 0
βπ = βπ₯2 = 0 → −βπ€1 βπ₯1 ≥ 0 = βπ€1 βπ₯1 ≤ 0
In order to estimate the Technology level you can use the isoprofit lines for all the periods.
Example: (ππ‘ , π€1π‘ , π¦π‘ , π₯1π‘ ) πππ (π π , π€1π , π¦ π . π₯1π )
ππ‘ = ππ‘ π¦ − π€1π‘ π₯1 πππ ππ = π π π¦ − π€1π π₯1
ο¨ P377.
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Profit maximization example:
Which has first-order conditions:
max π π(π₯1 , π₯2 ) − π€1 π₯1 − π€2 π₯2
π₯1 ,π₯2
π
ππ(π₯1∗ , π₯2∗ )
− π€1 = 0
ππ₯1
ππ(π₯1∗ , π₯2∗ )
− π€2 = 0
ππ₯2
When a Cobb-Douglas function is given: π(π₯1 , π₯2 ) = π₯1π π₯2π
The two first-order conditions become:
πππ₯1π−1 π₯2π − π€1 = 0
πππ₯1π π₯2π−1 − π€2 = 0
Multiply the first equation by x1 and the second equation by x2 and use x1x2 = y:
πππ¦
πππ¦ = π€1 π₯1 → π₯1∗ =
π€1
πππ¦
πππ¦ = π€2 π₯2 → π₯2∗ =
π€2
π
πππ¦ π πππ¦ π
ππ
ππ π
Solve for optimal choice of output:( π€ ) ( π€ ) = π¦ → (π€ )π (π€ ) π¦ π+π = π¦
1
1
π
1−π−π
ππ
π¦=( )
π€1
(
1
π
1−π−π
ππ
)
π€2
2
Chapter 21
Cost minimization: given prices w, we want to figure out the cheapest way to produce a given level
of output ,y.
min π€1 π₯2 + π€2 π₯2 π π’πβ π‘βππ‘ π(π₯1 , π₯2 ) = π¦
π₯1 ,π₯2
The solution depends on π(π€1 , π€2 , π¦)ο Cost function
π‘βπ ππ πππ’πππ‘π πππ£π π’π π‘ππβππππππππ ππππ π‘πππππ‘π : πππ ππππππππ‘ππππ ππ π₯1 πππ π₯2 π‘βππ‘ πππ πππππ’ππ π¦.
Plotting all the combinations of inputs for some level of cost, C:
π€1 π₯2 + π€2 π₯2 = πΆ
π€1
πΆ
−
π₯
π₯2 =
π€2 π€2 1
If C can vary you get a lot of isocost lines. (higher isocost lines = higher costs)
Isoquant line with the lowest possible isocost line is optimal solution.
If the isoquant is a smooth curve, the cost-mimimizing point will be characterized by a tangency
condition: the slope of the isoquant must be equal to the slope of the isocost curve at the optimal
point. So TRS = factor price ratio:
ππ1 (π₯1∗ , π₯2∗ )
π€1
∗ ∗
−
∗ ∗ ) = ππ
π(π₯1 , π₯2 ) = −
(π₯
ππ2 1 , π₯2
π€2
Note: this has to be with a tangency condition; does not solve with a corner solution, discrete with
kinks etc.
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If we are at cost minimum the change cannot be lower costs so: π€1 βπ₯1 + π€2 βπ₯2 ≥ 0
The same goes for a negative change: −π€1 βπ₯1 − π€2 βπ₯2 ≥ 0 ο π€1 βπ₯1 + π€2 βπ₯2 = 0
Solving for:
βπ₯2
π€1
ππ1 (π₯1∗ , π₯2∗ )
=−
=−
βπ₯1
ππ2 (π₯1∗ , π₯2∗ )
π€2
Which is the condition for cost minimization derived above by a geometric argument.
Conditional factor demand functions & Derived factor demands: π₯1 (π€1 , π€2 , π¦)& π₯2 (π€1 , π€2 , π¦)
These are the choices of inputs that yield minimal costs for the firm; they depend on the input prices
and the level of output. They measure the relationship between the prices and the output and the
optimal factor choice of the firm.
With factor demand functions: give the cost-minimizing choices for a given output, y.
With profit-maximizing factor demands (ch20) they give the profit-maximizing choices for a given
price of output, p.
Examples:
Complements: π(π₯1 , π₯2 ) = min{π₯1 , π₯2 }
π(π€1 , π€2 , π¦) = π€1 π¦ + π€2 π¦ = (π€1 + π€2 )π¦ as you you need y of good 1 and 2.
Substitutes: π(π₯1 , π₯2 ) = π₯1 + π₯2
π(π€1 , π€2 , π¦) = min{π€1 π¦, π€2 π¦}
Cobb-douglas: π(π₯1 , π₯2 ) = π₯1π π₯2π ο Appendix p392 for detailed calculation
π(π€1 , π€2 , π¦) =
π
π
1
π+π π+π π+π
πΏπ€1 π€2 π¦
Weak Axiom of Cost minimization (WACM)
π€1π‘ π₯1π‘ + π€2π‘ π₯2π‘ ≥ π€1π‘ π₯1π + π€2π‘ π₯2π
π€1π π₯1π + π€2π π₯2π ≥ π€1π π₯1π‘ + π€2π π₯2π‘
Adding those two equation together (after rewriting the second one) gives:
(π€1π‘ − π€1π )π₯1π‘ + (π€2π‘ − π€2π ) ≤ (π€1π‘ − π€1π )π₯1π + (π€2π‘ − π€2π )π₯2π
Rearranging and using delta notations:
βπ€1 βπ₯1 + βπ€2 βπ₯2 ≤ 0
Returns to scale van be expressed in terms of the behaviour of the Average cost function: the cost
per unit to produce y units of output:
π(π€1 , π€2 , π¦)
π΄πΆ(π¦) =
π¦
If Technology exhibits constant returns to scale, we the cost function has the form: π(π€1 , π€2 , π¦) =
π(π€1 , π€2 , 1)π¦ ο (if you want the output to be x2 just make the inputs x2.)
The average cost function will be (with constant returns to scale)
π(π€1 , π€2 , 1)π¦
= π(π€1 , π€2 , 1)
π΄πΆ(π€1 , π€2 , π¦) =
π¦
This means that the cost per output is the same on average no matter what output (y) is being used
with constant returns to scale.
With increasing returns to scale the costs on average will be less, with decreasing returns to scale the
ACF will be higher.
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Short run cost function:
ππ (π¦, xΜ ) = min π€1 π₯1 + π€2 xΜ 2 π π’πβ π‘βππ‘ π(π₯1 , xΜ 2 ) = π¦
π₯1
Look at the smallest x1 so π(π₯1 , xΜ 2 ) = π¦
If there are many factors of production that are variable in the short run the cost-minimization
problem will involve more calculation:
In general it will depend on factor prices and the levels of the fixed factors:
π₯1 = π₯1π (π€1 , π€2 , xΜ 2 , π¦)
π₯2 = xΜ 2
Short-run function: ππ (π¦, xΜ ) = min π€1 π₯1π (π€1 , π€2 , xΜ 2 , π¦) + π€2 xΜ 2
π₯1
Long run cost funtion: π(π¦) = min π€1 π₯1 + π€2 π₯2 π π’πβ π‘βππ‘ π(π₯1 , π₯2 ) = π¦
π₯1 ,π₯2
Both factors free to vary: π₯1 = π₯1 (π€1 , π€2 , π¦) πππ π₯2 = π₯2 (π€1 , π€2 , π¦)
So it can also be written as: π€1 π₯1 (π€1 , π€2 , π¦) + π€2 π₯2 (π€1 , π€2 , π¦)
Sunk costs: costs that are not recoverable
Cost minimization example: constrained-minimization problem ο
min π€1 π₯2 + π€2 π₯2 π π’πβ π‘βππ‘ π(π₯1 , π₯2 ) = π¦
π₯1 ,π₯2
To solve, use the lagrangian:
πΏ = π€1 π₯2 + π€2 π₯2 − π(π(π₯1 , π₯2 ) − π¦)
If differentiating in respect to π₯1 , π₯2 πππ π we get the three conditions:
ππ(π₯1 , π₯2 )
π€1 − π
=0
ππ₯1
ππ(π₯1 , π₯2 )
=0
ππ₯2
π(π₯1 , π₯2 ) − π¦ = 0
Last condition is the constraint; the first two can be rearranged and divide the first by the second to
get:
π€1 ππ(π₯1 , π₯2 )/(ππ₯1 )
=
π€2 ππ(π₯1 , π₯2 )/(ππ₯2 )
Thus the this is the same as: the technical rate of substitution must equal the factor price
π€2 − π
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Cost minimization example: cobb-douglas production function:
π(π₯1 , π₯2 ) = π₯1π π₯2π
The problem is: min π€1 π₯1 + π€2 π₯2 π π’πβ π‘βππ‘ π₯1π π₯2π = π¦
π₯1 ,π₯2
Because we have specific functional form here we can use either the Lagrangian method or the
substitution method.
Substitution method:
First solve the constraint for x2 as a function of x1:
1
π₯2 = (π¦π₯1−π )π
Then substitute this into the objective function to get the unconstrained minimization problem:
1
min π€1 π₯1 + π€2 (π¦π₯1−π )π
π₯1
Now you can differentiate in respect to x1 and set resulting derivative equal to zero. The result can be
solved to get x1 as a function of w1, w2 and y, to get the conditional factor demand for x1.
Langrangian method: p394.
Chapter 22
In this chapter π(π€1 , π€2 , π¦) = π(π¦) ππ ππππππ πππ πππ₯ππ.
Total costs are: π(π¦) = ππ£ (π¦) + πΉ (variable + fixed)
Average cost function: π΄πΆ(π¦) =
Marginal cost curve:
In terms of marginal costs:
π(π¦)
π¦
=
ππ£ (π¦)
ππΆ(π¦) =
ππΆ(π¦) =
π¦
π
+ π¦ = π΄ππΆ(π¦) + π΄πΉπΆ(π¦)
βπ(π¦) (π(π¦ + βπ¦) − π(π¦))
=
βπ¦
βπ¦
βππ£ (π¦) ππ£ (π¦ + βπ¦) − ππ£ (π¦)
=
βπ¦
βπ¦
The same as the first equation as F does not change at βπ¦
- AC first falls due to the declining average fixed
costs but the it raises due to the increasing
variable costs.
- Mc&AVC are the same at the first unit of
output
- Marginal cost curve passes through the
minimum point of both the average variable
cost and the average cost curves.
- Area under MC until y = total variable costs
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Cost curves for online auctions:
Relationship between the number of clicks (x) and the cost of those clicks c(x);
max π£π₯ − π(π₯)
π₯
Optimal solution: value equal to marginal costs.
Long-Run Costs:
e.g. plant size: in short run: ππ (π¦, π) ππ’π‘ ππ π‘βπ ππππ ππ’π π = π(π¦) → ππ π πππ πβππππ
so we get: π(π¦) = ππ (π¦, π(π¦))
In the optimal point the long-term cost to produce output y needs to be at least equal or lower in
the long term run. (As k* can change in the long run):
π(π¦) ≤ ππ (π¦, π ∗ )
They touch at π¦ ∗ : π(π¦ ∗ ) = ππ (π¦ ∗ , π ∗ )
Because at π¦ ∗ , π ∗ is the optimal choice. So at π¦ ∗ , the
long-run costs and the short-run costs are the same.
The long-run average cost curve must be tangent to
the short run average cost curve: SAC=LAC at point π¦ ∗ .
If you pick π¦1 , π¦2 … π¦π and accompanying plant sizes:
π(π¦1 ) = π1 ππ‘π
You get a graph like on the left:
With discrete levels op plant sizes you get a graph like
fig. 22.9 (p409).
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Chapter 12
Contingent consumption plan: being a
specification of what will be consumed in
each different state of nature (= each
different outcome of the random process).
Contingent: depending on something not
yet certain.
πΆπ = ππππ π’πππ‘πππ ππ πππ ππ’π‘ππππ & πΆπ = ππππ π’πππ‘πππ ππ π‘βπ ππππ ππ’π‘ππππ
The consumption you lose in the good state, divided by the extra consumption you gain in the bad
state is:
βπΆπ
πΎπΎ
πΎ
=−
=−
βπΆπ
πΎ − πΎπΎ
1−πΎ
Reinsurance market: selling ‘risks’ to other parties (in the insurance market)
Utility functions and probabilities:
Utility also relies on the probability a person thinks an event will happen (will it rain or not).
The utility function will be:
π’(π1 , π2 , π1 , π2 ) → π2 = 1 − π1 if mutually exclusive: π are probabilities.
Perfect substitutes: π’(π1 , π2 , π’1 , π’2 ) = π1 π1 + π2 π2 ο expected value = average level of
consumption that you would get.
Cobb-douglas function: π’(π1 , π2 , π, 1 − π) = π1π π21−π
Monotonic transformation on this function (still representing the same preferences):
ln π’(π1 , π2 , π1 , π2 ) = π1 πππ1 + π2 πππ2
Another (convenient form) that the utility function might take:
π’(π1 , π2 , π1 , π2 ) = π1 π£(π1 ) + π2 π£(π2 )
This means that utility can be written as a weighted sum of some function of consumption in each
state π£(π1 ), π£(π2 ). And the weights are given by the possibilities π1 , π2 .
With perfect substitutes (above) π£(π) = π.
This formula measures the expected utility: expected utility function = von Neumann-Morgenstern
utility function.
Positive affine transformation: If a function π£(π’) can be written in the form: π£(π’) = ππ’ + π where
π > 0.
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Independence assumption: The choices a person plans to make in one state of nature should be
independent from the choices that they plan to make in other states of nature.
Thus if you have π1 , π2 πππ π3 , π1 , π2 πππ π3 under the independence assumption you must get:
π(π1 , π2 , π3 ) = π1 π’(π1 ) + π2 π’(π2 ) + π3 π’(π3 )
The MRS is independent between two goods is independent of how much is there of the third good:
βπ(π1 , π2 , π3 ) βπ(π1 , π2 , π3 )
/
ππ
π12 =
βπ2
βπ1
π1 βπ’(π1 ) π2 βπ’(π2 )
ππ
π12 =
/
βπ1
βπ2
So MRS only depends on the amount of good 1&2 not of the third good.
Risk averse: try to avoid the gamble (if expected utility is lower than expected wealth) & concave
utility function
If a person loves to risk: the utility is higher than expected value: convex utility function
Risk neutral: utility line Is equal to the expected value.
Example; starting wealth=35k$. state 1 is the situation with no loss, state 2 loss situation (-10k$)
π1 = $35,000 − πΎπΎ
π2 = $35000 − $10000 + πΎ − πΎπΎ
πβπ’(π2 )
πΎ
βπ2
ππ
π = −
=−
(1 − π)βπ’(π1 )
1−πΎ
βπ1
Profit insurance company is: π = πΎπΎ − ππΎ − (1 − π) ∗ 0 = πΎπΎ − ππΎ
π
If the profit is 0: πΎ = π therefore: − 1−π
Therefore the optimal amount of insurance must satisfy:
βπ’(π1 ) βπ’(π2 )
=
βπ2
βπ1
This says: The marginal utility of an extra dollar of income if the loss occurs should be equal to the
marginal utility of an extra dollar of income if the loss does not occur.
Not: appendix ch12.
Chapter 14
Reservation prices are defined to be the difference in utility:
π1 = π£(1) − π£(0)
π2 = π£(1) − π£(2)
So if you want to calculate v(3): π1 + π2 + π3 = π£(3) − π£(0): = gross consumer surplus.
(only utility associated of good 1). Total utility is then: π£(π) + π − ππ
Where m is the income and pn the expenditure of the other good.
Net consumer’s surplus: π(π) − ππ
It measures the utility minus the reduction in the expenditure on consumption of the other good.
(p254 graphical display) ο area under demand curve (of discrete good) displays the utility.: later it is
shown how to calculate this area.
Other interpretation of the surplus: π1 − π (the value he places is r1 but he only has to pay p)
So the total consumers surplus would be:
πΆπ = π1 − π + π2 − π + β― + ππ − π = π1 + β― + ππ − ππ
This gives us: πΆπ = π£(π) − ππ because the sum of reservation prices is the utility
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Consumers’ surplus: A sum of surpluses
(consumer’s surplus is only one consumer)
The example with quasilinear discrete goods
the reservation prices are independent of the
amount of money the consumer has to spend
on other goods. (in general the reservation
prices for good 1 will depend on how much
good 2 is being consumed)
Utility changes without using consumers surplus:
Compensating variation (CV): The change in income necessary to restore the consumer to point
(π₯1∗ , π₯2∗ ): So how much money the consumer has to get extra to get compensated for the price
change
Equivalent variation (EV): How much money needs to be taken away from the consumer to leave
him as well off as he would be after the price change. Thus you can say it’s a maximum a consumer is
willing to pay to avoid the price change.
When using quasilinear preferences the CV and EV are equal; as the indifference curves are parallel.
Example:
1
1
π’(π₯1 , π₯2 ) = π₯12 π₯22 with π(1,1) and π = 100 if price of good 1 increases from 1 to 2, calculate EV and
CV:
π
π
π₯1 =
, π₯2 =
2π1
2π2
∗
∗
′
Thus the demand changes from (π₯1 , π₯2 ) = (50,50) to (π₯1 , π₯2′ ) = (25,50)
If the prices were 2,1 with income π, we can substitute into the demand function. This function
needs to be set equal to the utility bundle for prices 1,1 ο (50/50) so we can solve EV.
1
1
1
1
π 2 π 2
π’(π₯1 , π₯2 ) = ( ) ( ) = 502 502
2
4
π = 100√2 ≈ 141
So the consumer needs about 141 – 100 = 41$ additional money to be as well off as he was before
the price change: EV
To calculate CV we need to ask how much money would be necessary at the prices (1,1) to make the
consumer as well off as he would be consuming the bundle (25,50):
1
1
1
1
π 2 π 2
( ) ( ) = 252 502
2
2
π = 50√2 ≈ 70
The consumer would be willing to pay 100-70 = 30$ to avoid the price change: CV
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With Quasilinear preferences:
π£(π₯1 ) + π₯2 depends on only p1 so π₯1 (π1 ) suppose the price changes from π∗ π‘π π′ . The consumer
chooses π₯ ∗ = π₯1 (π1∗ ) with utility π£(π₯1∗ ) + π − π1∗ π₯1∗
π₯ ′ = π₯1 (π1′ ) with utility π£(π₯1′ ) + π − π1′ π₯1′ Let C be the compensating variation, set both equal:
π£(π₯1′ ) + π + πΆ − π1′ π₯1′ = π£(π₯1∗ ) + π − π1∗ π₯1∗
Solving for C: πΆ = π£(π₯1∗ ) − π£(π₯1′ ) + π1′ π₯1′ − π1∗ π₯1∗
Now let E be the equivalent variation (money you can take away before price change):
π£(π₯1∗ ) + π − πΈ − π1∗ π₯1∗ = π£(π₯1∗ ) + π − π1∗ π₯1∗
πΈ = π£(π₯1∗ ) − π£(π₯1′ ) + π1′ π₯1′ − π1∗ π₯1∗
E = C.
Producers surplus: area above supply curve: willing to sell for ππ but the producer gets π∗ for it.
The difference between the minimum account the producers is willing to sell ππ and the amount she
actually gets for it π∗ is called the net producer’s surplus. You will get a figure like on p263, where
the rectangular shape represents the extra gain from the higher price and the rectangular shape
depicts the extra gain from selling additional products.
Ration coupon: effective price -price ceiling
Appendix ch14:
max π£(π₯) + π¦ π π’πβ π‘βππ‘ ππ₯ + π¦ = π
π₯,π¦
max π£(π₯) + π − ππ₯
π₯
π£ ′ (π₯) = π
πΉπππ π‘ πππππ ππππππ‘πππ
Thus inverse demand π(π₯)function: π(π₯) = π£ ′ (π₯)
For the discrete-good framework the price at which the consumer is just willing to consume x units is
equal to the marginal utility; here the inverse demand curve measures the derivative of utility, we
can simply integrate under the inverse demand function to find the utility function:
π₯
π₯
π£(π₯) = π£(π₯) − π£(π) = ∫ π£ ′ (π‘)ππ‘ = ∫ π(π‘)ππ‘
0
0
This is the utility associated with the consumption of the x-good; the area under the demand curve.
If the demand function is linear: π₯(π) = π − ππ so change in surplus is:
π
π
π‘2
π 2 − π2
∫ (π − ππ‘)ππ‘ = ππ‘ − π ∗ ] = π(π − π) − π ∗
2 π
2
π
If a demand function is: π₯(π) = π΄π∈ π€βπππ ∈ < 0 πππ π΄ πππ ππ‘ππ£π ππππ π‘πππ‘
π
π
π∈+1 − π∈+1
π‘ ∈+1
∈
∫ π΄π‘ ππ‘ = π΄ ∗
] =π΄∗
πππ ∈≠ −1
∈ +1
∈ +1 π
π
If ∈= −1 the demand function is x(p)=a/p
The change in consumer surplus for the cobb-douglas demand is:
π
ππ
π
ππ‘ = ππ ln π‘]π = ππ(ln π − ln π)
∫
π‘
π
The Consumer surplus always lies between CV and EV
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Chapter 15
Inverse demand function: price as a function of quantity P(X) ο measures what market price is if x
goods are demanded. The price of a good measures the MRS between it and all other goods. If all
consumers are facing the same prices for goods; all consumers will have the same MRS at π ∗ . Thus
P(X) measures the MRS.
Intensive margin & extensive margin: You will always keep consuming some of good x even though
the price changed; same goes when deciding to enter a new market.
βπ
The slope of the demand function βπ responds on what units you use; (using centimetres or meters
as q)
An unit free measurement: Elasticity of demand: π ο percental change
βπ
π βπ
π
= ∗
π=
βπ π βπ
π
Elastic demand: elasticity greater than 1
Inelastic demand: elasticity lower than 1
Unit elastic demand: exactly -1
Revenue = R = pq
If the price and q changes the revenue will be
π
′ = (π + βπ)(π + βπ) = ππ + πβπ + πβπ + βπβπ
π
′ − π
= βπ
= πβπ + πβπ + βπβπ
For small change values p and q the last term will be very small so it can be neglected:
βπ
= πβπ + πβπ
To express this for the rate of change of revenue per change in price, we divide this expression by βπ
βπ
βπ
ο βπ = π + π ∗
βπ
General formula for a demand with a constant elasticity of π is:
π = π΄ππ
A is positive constant, π is elasticity so negative.
You can transform this and take logarithms:
ln π = ln π΄ + π ln π
Here the logarithm of q depends in a linear way on the logarithm of p.
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The change in revenue is was: βπ
= πβπ + πβπ
Marginal Revenue MR:
βπ
βπ
ππ
=
=π+π
βπ
βπ
Rearranging:
βπ
πβπ
= π [1 +
]
βπ
πβπ
1
The last term is the reciprocal of elasticity: π =
Thus Marginal Revenue:
1
π
π
βπ
βπ
πβπ
= πβπ
1
βπ
= π(π) [1 +
]
|π(π)|
βπ
Where elasticity is a positive number. (if inelastic the revenue decreases (π < 1) if output is
decreased, if elastic the revenue is increased. )
Special case of the linear demand curve (inverse)
π(π) = π − ππ
Here the slope of the inverse demand curve is constant:
βπ
= −π
βπ
Thus MR becomes:
βπ(π)
βπ
= π(π) +
π
βπ
βπ
= π(π) − ππ = π − ππ − ππ = π − 2ππ
Income elasticity of demand:
% πβππππ ππ ππ’πππ‘ππ‘π¦
% πβππππ ππ ππππππ
Normal(between 0-1)/inferior(negative income elasticity)/luxury goods (income elasticity greater
than 1)
ππππππ ππππ π‘ππππ‘π¦ ππ ππππππ =
Expenditure share:
π π =
ππ π₯π
π
The weighted average of the income elastics is 1 where the weights are the expenditure shares
βπ₯1
βπ₯2
π₯
π₯1
π 1
+ π 2 2 = 1
βπ
βπ
π
π
This means that luxury goods with an elasticity of more than one needs to be counterbalanced by
goods with an elasticity below 1 so the average elasticity equals 1.
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Chapter 16
Market supply curve: All supply curves from individuals are added. The prices are given so it is an
competitive market.
Normal: individual demand curves are normally viewed as giving the optimal quantities demanded as
a function of the price. (same with supply)
Inversed curves: Price is measured with a given amount of supply.
Normal equilibrium:
π·(π∗ ) = π(π∗ )
Inversed demand + supply equilibrium
ππ (π∗ ) = ππ(π∗ )
Solving for p gives the inverse demand. (with q still in the function).
Value tax ο consumer end up paying for it; absolute price increase
Perfectly elastic: supply curve horizontal
Perfectly inelastic: supply curve vertical
Deadweight of tax:
Loss consumers’ surplus is the top area (tax + deadw.),
the loss in producers’ surplus at the bottom area
(Revenue + deadw.).
The deadweight area is the loss of consumers surplus +
loss of producers surplus – tax revenue = deadweight
loss/excess burden.
The producers/consumers are willing to pay the whole
area to ‘avoid’ the tax whilst the government only gets
the tax revenue from it; the excess burden is found by
subtracting those two areas.
The excess burden is the loss of value to the consumers and producers due to the reduction in the
sales of the good.
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Example 1: Market for loans ο equilibrium interest rate:
π·(π ∗ ) = π(π ∗ )
If a tax is applied on the interest the earned from lending money (everyone face same tax bracket):
π·((1 − π‘)π ′ ) = π((1 − π‘)π ′ )
We saw that π ∗ solves the first equation so π ∗ = (1 − π‘)π ′ must solve the second:
π ∗ = (1 − π‘)π ′
π∗
π′ =
1−π‘
1
So first the interest rate will be higher by 1−π‘, the after-tax will be π ∗ . p308 graphically
-
-
1
First the Supply curve will tilt up by a factor of 1−π‘
1
Then interest payments are tax deductible so this will tilt the demand curve up by (1−π‘)
1
Result: a net raise of the interest rate by 1−π‘
Example 2: Inverse demand and supply functions ο borrowers and lenders equilibrium:
ππ΅ (π∗ ) = ππ (π∗ )
Introduction of a tax (Buyers and seller can have different tax brackets π‘π πππ π‘π ):
The after-tax rate facing borrowers will be (1 − π‘π΅ )π with interest rate r gives the quantity they
choose to borrow:
(1 − π‘π )π = ππ (π)
ππ (π)
π=
1 − π‘π
Same with lenders:
ππ (π)
π=
1 − π‘π
So:
ππ (π)
ππ (π)
π=
=
1 − π‘π 1 − π‘π
If they are in the same tax brackets: π‘π = π‘π → π′ = π ∗
If in different tax brackets:
1 − π‘π
ππ (π′ ) =
π (π′ )
1 − π‘π π
1−π‘
Borrowers will face a higher price than lenders if: 1−π‘π > 1
π
This means that π‘π > π‘π so that the tax of lenders I greater than the tax of the borrowers, this is a net
tax on borrowing. If π‘π > π‘π it’s a net subsidy.
Pareto efficient: There is no way to make any person better without hurting anybody else.
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Chapter 28 (skip 28.5-28.8)
Oligopoly: A few competitors that have effect on price (between pure competition and monopoly)
Duopoly: Only two firms
Price leader (sets price; has information before the other firm) vs price follower
Similarly, one firm may get to choose its quantity first: quantity leader vs quantity follower.
Simultaneous game: no information about the other firm; so it’s a guess: each simultaneously
choose prices or quantities.
The two firms can also make price agreements or quantity agreements that maximize their profits:
colluding ο cooperative game
Quantity leadership:
Stackelberg model
The total output π = π¦1 + π¦2
Follower’s problem: Follower wants to max his output so:
max π(π¦1 + π¦2 )π¦2 − π2 (π¦2 )
MR should equal MC:
π¦2
βπ
π¦ = ππΆ2
βπ¦2 2
This means that if the follower increases its output, it increases its revenue by selling more output at
the market price. However, this increase in its output will decrease the price.( as βπ ↑)
ππ
2 = π(π¦1 + π¦2 ) +
The profit –maximizing choice of the follower will depend on the choice made by the leader:
Reaction function: π¦2 = π2 (π¦1 )
In the case of linear demand (inverse function) the reaction function is: (costs 0)
π(π¦1 + π¦2 ) = π − π(π¦1 + π¦2 )
Then the profit function for firm 2 is:
π2 (π¦1 , π¦2 ) = [π − π(π¦1 + π¦2 )]π¦2
Or
π2 (π¦1 , π¦2 ) = ππ¦2 − ππ¦1 π¦2 − ππ¦22
From this equation we can derive isocost lines: for a level π2
MR:
ππ
2 (π¦1 , π¦2 ) = π − ππ¦1 − 2ππ¦2
Setting equal to MC (0 here):
0 = π − ππ¦1 − 2ππ¦2
The reaction curve will be:
π¦2 =
π − ππ¦1
2π
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Leader’s problem:
Profit maximization problem for the leader becomes:
π π’πβ π‘βππ‘ π¦2 = π2 (π¦1 )
max π(π¦1 + π¦2 )π¦1 − π1 (π¦1 )
Substituting gives:
π¦1
max π(π¦1 + π2 (π¦1 ))π¦1 − π1 (π¦1 )
π¦1
When the leader contemplates changing its output it has to recognize the influence it exerts on the
follower. Demand function follower was:
(π − ππ¦1 )
π2 (π¦1 ) = π¦2 =
2π
Leaders profits are (MC = 0):
π1 (π¦1 , π¦2 ) = π(π¦1 + π¦2 )π¦1 = ππ¦1 + ππ¦12 − ππ¦1 π¦2
Reaction function: π¦2 = π2 (π¦1 )
π1 (π¦1 , π¦2 ) = ππ¦1 + ππ¦12 − ππ¦1 π2 (π¦1 )
π − ππ¦1
= ππ¦1 + ππ¦12 − ππ¦1
2π
Simplifying gives:
π
π
π(π¦1 , π¦2 ) = π¦1 − π¦12
2
2
The MR is:
π
ππ
= − ππ¦1
2
π
∗
ππ =
ππ
The follower’s output is substitute π¦1∗ πππ‘π π‘βπ πππππ‘πππ ππ’πππ‘πππ
ππ¦1∗
π¦2∗ = π −
2π
π
∗
ππ =
ππ
The total industry output is:
3π
π¦1∗ + π¦2∗ =
4π
Stackelberg equilibrium: Firm 1 chooses the point on firms 2’s reaction curve that touches firm 1’s
lowest possible isoprofit line, thus yielding the highest possible profits for firm 1 (p522 fig. 28.2)
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Price leadership:
Instead of setting quantity the leader also may set the price.
The follower wants to max his profits (p is fixed; set by the leader)
max ππ¦2 − π2 (π¦2 )
π¦2
The follower will supply: π(π).
The amount of output the leader will sell will be:
Residual demand curve: π
(π) = π·(π) − π(π)
The leader has a constant marginal cost of production c. The profits then will be:
π1 (π) = (π − π)[π·(π) − π(π)] = (π − π)π
(π)
In order to maximize profits the leader want to choose a price and output combination where MR
(for the residual demand curve; the curve that measures how much output it will be able to sell at
each given price) equals MC.
Example:
Cost functions: π2 (π¦2 ) =
π¦22
2
π1 (π¦1 ) = ππ¦1
Price equal to MC: ππΆ(π¦2 ) = π¦2 ο π¦2 = π
Followers supply curve: π¦2 = π(π) = π
π·(π) = π − ππ
The demand curve facing the leader (Residual) is:
π
(π) = π·(π) − π(π) = π − ππ − π = π − (π + 1)π
π
1
π=
−
→ πππ£πππ π ππππππ ππππππ π‘βπ ππππππ
π+1 π+1
MR has the same intercept and is twice as steep so:
2
π
−
π¦
ππ
1 =
π+1 π+1 1
MR = c
π
2
−
π¦ = π = ππΆ1
π+1 π+1 1
Solving for the leader’s profit maximizing output:
π − π(π + 1)
π¦1∗ =
2
Simultaneous Quantity setting:
Cournot model:
Firm 1 thinks the total output will be: π = π¦1 + π¦2π (expected)
This output will yield a market price: π(π) = π(π¦1 + π¦2π )
Profit max:
max π(π¦1 + π¦2π )π¦1 − π(π¦1 )
Expected output (=reaction function):
Firms 2 reactions curve:
π¦1
π¦1 = π1 (π¦2π )
π¦2 = π2 (π¦1π )
The cournot equilibrium is optimal; neither of the two firms want to change their output once they
find out the other’s choice, because they won’t get a higher profit from it.
Thus:
π¦1∗ = π1 (π¦2∗ ) π¦2∗ = π2 (π¦1∗ )
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Chapter 29.1
Game theory: analysis of strategic interaction ο payoff matrix; dominant strategy.
Chapter 27 (skip 27.3,27.4,27.11)
Suppose a firm has a monopoly for its output. Production function:
π¦ = π(π₯)
The revenue depends on its production of output:
π
(π¦) = π(π¦)π¦
How does an increase in the amount of the input affect the revenue of the firm?
βπ₯ π€πππ πβππππ → π π βπ¦
Marginal product:
βπ¦ π(π₯ + βπ₯) − π(π₯)
πππ₯ =
=
βπ₯
βπ₯
This increase in output will cause the revenue to change
Marginal revenue:
βπ
π
(π¦ + βπ¦) − π
(π¦)
ππ
π¦ =
=
βπ¦
βπ¦
The effect on revenue due to the marginal increase in the input is called the marginal revenue
product.
βπ
βπ
βπ¦
ππ
ππ₯ =
=
= πππ₯ ∗ πππ¦
βπ₯ βπ¦ βπ₯
We can use our standard expression for marginal revenue to write this as:
1
1
βπ
π¦] πππ = π(π¦) [1 + ] πππ₯ = π(π¦) [1 − ] πππ₯
ππ
ππ = [π(π¦) +
|π|
π
βπ¦
The elasticity of the demand curve facing an individual firm in a competitive market is infinite;
consequently the marginal revenue for a competitive firm is just equal to the price. So the marginal
revenue product of an input for a firm in a competitive market is just the value of the marginal
product of that input, pMPx.
With a monopoly the MRP is always less than the value of the MP:
1
ππ
ππ₯ = π [1 − ] πππ₯ ≤ ππππ₯
|π|
Only if demand is perfectly elastic it’s equal to each other; otherwise it’s less.
This means that at any level of employment of the factor, the marginal value of an additional unit is
less for a monopolist than for a competitive firm. ο This is because an increase in the output will
decrease the price for a monopolist
Hence a monopolist is using less input than a competitive firm.
How much should the employ of one factor?
Competitive market:
πππ(π₯π ) = π€
Monopolist:
ππ
π(π₯π ) = π€
Since ππ
π(π₯) < πππ(π₯) → π‘βπ πππππ‘ π€βπππ ππ
π(π₯π ) = π€ will always be to the left of the point
where πππ(π₯π ) = π€
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Monopsony: market with a single buyer. Monopsonist is a price maker
The profit maximization problem facing the Monopsonist is:
max ππ(π₯) − π€(π₯)π₯
π₯
The condition for profit maximization : MR from hiring an extra unit of the factor is equal the
marginal cost of that unit.
Marginal revenue: ππππ₯
Marginal costs:
Total change in costs from hiring βπ₯ more of the factor will be:
βπ = π€βπ₯ + π₯βπ€
w is changing because of the increase in demand of the factor; x.
Change in costs per unit change in βπ₯ is:
βπ
βπ€
= ππΆπ₯ = π€ +
π₯
βπ₯
βπ₯
We can write the marginal cost of hiring additional units of the factor as:
π₯ βπ€
ππΆπ₯ = π€ [1 +
]
π€ βπ₯
1
ππΆπ₯ = π€ [1 + ]
π
π is the supply elasticity of the factor. Since supply curves typically slope upward, π is a positive
number.
Inverse linear supply curve: (S)
π€(π₯) = π + ππ₯
So total costs:
πΆ(π₯) = π€(π₯)π₯ = ππ₯ + ππ₯ 2
Thus the marginal cost of an additional unit of the input
equals: (MFC)
ππΆπ₯ (π₯) = π + 2ππ₯
So a Monopsonist operates at a Pareto inefficient point.
(just as in a monopoly). But now the inefficiency lies in the
factor market rather than in the output market.
Figure 1: MC=MRP
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Chapter 31
Behavioural economics
Framing effects: e.g.: a book might sell more copies at 29,95€ but fewer at 29€
Anchoring effect: People’s choices can be influenced by completely spurious information
Bracketing: Difficult to predict what they will choose in different circumstances
Too much choice: Makes it more difficult to choose
Asset integration hyphothesis: individuals care about the total amount of wealth that they ended up
with in various outcomes. In general: people accept more higher risk and dismiss too many smaller
risks.
Excess risk eversion: over-insuring certain risks.
Sunk cost fallacy: Once you have bought something, the amount you paid is no longer recoverable.
(In reality however people do care about this)
Exponential discounting: people discount the future at a constant fraction.
π’(π) ππ π‘βπ π’π‘πππ‘π¦ π‘ππππ¦ π‘βππ π π‘ π’(π)ππ π‘βπ ππππ π’πππ‘πππ ππ π‘βπ ππ’π‘π’ππ π€βπππ π < 1
1
Hyperbolic discounting: the discount factor takes not the form π π‘ but
This is time consistent.
1+ππ‘
Example of exponential discounting and it’s time consistency: e.g. a person with a 3-period planning
horizon:
π’(π1 ) + ππ’(π2 ) + π 2 π’(π3 )
πππ(π2 )
ππ
π12 =
ππ(π1 )
π 2 ππ(π3 ) πππ(π3 )
ππ
π23 =
=
πππ(π2 )
ππ(π2 )
This shows that with exponential discounting the willing to substitute consumption in period 2&3 is
the same whether viewed from the perspective of period 1&2. With hyperbolic discounting discounts
the long-term future more heavily than he discounts the short-term future.
Self-control: hard to refrain yourself from doing something (not). ο commitment devices can help
with this.
Overconfidence ο People can risk more due to this as they are too confident about it.
Behavioural game theory: examines how actual people interact.
- Ultimatum game: 2 players, proposer and the responder. E.g.: One is given 10$, he needs to
propose a share to the other player (e.g. 1$) and the responder can then either accept the
proposal: they get the money, or refuse it: they both walk away with nothing.
People seem not to make rational choices with this game.
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Strategy method: responders can ask for a minimum first (without the proposer knowing this). ο
the offers seem to be higher this way.
Fairness norms: people value fairness even not directly in their interest to do so
Punishment games: a 3rd party can take away profits from the proposer (at some cost of himself)
Conclusion behavioural economics: basic theory of economics choice is incomplete.
Chapter 33
Production
Robinson Crusoe economy: one consumer, one firm, and two goods.
π = πΆ − π€πΏ
πΆ = π + π€πΏ
= isoprofit line. Maximization is normally the
tangency point: the slope of the production
function (MPL) equals w. (πΏ∗ )
Maximization problem v2: Optimal point will
occur where the indifference curve is tangent
to the budget line.
In the Robinson example: decreasing returns
to scale (see graph). With constant returns to
scale the budget line = production function
(straight line).
Increasing returns to scale: non-convexity ο
production set is not a convex set: Pareto efficient allocation cannot be achieved by a competitive
market.
A competitive equilibrium is Pareto efficient (first theorem of welfare economics)
All firms act as competitive profit maximizes, then a competitive equilibrium will be Pareto efficient.
Caveats:
- It has nothing to do with distribution; profit max. only ensures efficiency
- This result only makes sense when a competitive equilibrium actually exists
- Choices of one firm don’t affect production possibilities of other firms (production
externalities) same goes for consumers (consumption externalities)
If another good comes into play ο production possibilities set: all the goods that can be produced
by devoting a different amount of time to each production of this good. The boundary of the
production possibilities set is the possibilities frontier. The Marginal rate of transformation is the
slope of this frontier.
When another worker is added you get a join production set which can have a ‘kink’ in it (due to the
different slopes of the individual sets). If there are a lot of ways to produce output there is more of a
rounded structure.
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Pareto set: describes the set of Pareto efficient bundles given the amounts of goods 1 and 2 available
within the production possibilities set.
Pareto efficiency: MRS of each consumer should be equal to MRT, because if it’s not there will be a
Pareto improvement. P642 fig 33.9 shows a Pareto efficient allocation with the Edgeworth box.
A competitive market with profit-maximizing firms and utility-maximizing consumers must result in a
Pareto efficient allocation:
Example book:
Two outputs: C and F. Two kinds of labour: Lc and Lf.
Prices: Pc and Pf. plus both wages: wc and wf.
The profit-maximization problem is:
max ππ πΆ + ππ πΉ − π€π πΏπ − π€π πΏπ
πΆ,πΉ,πΏπ ,πΏπ
The firm finds it optimal in equilibrium to hire πΏ∗π πππ πΏ∗π
Labour costs of production: πΏ∗ = π€πΆ πΏ∗πΆ + π€π πΏ∗π
The profits of the firm are now:
π = ππ πΆ + ππ πΉ − πΏ∗
π + πΏ∗ ππ πΉ
πΆ=
−
ππ
ππ
ππ
This last equation describes the isoprofit lines of the firm. Sloped as − π and vert. intercept:
π
π+πΏ∗
ππ
For profit maximization the isoprofit line must be tangent to the production possibilities set (the
MRS) thus: ο figure 33.10 p645
ππ
ππ
π = −
ππΆ
If you look to the consumers perspective; their optimal bundle: MRS = common price ratio
This price ratio equals the MRT thus: MRS=MRT
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