your answers

the survey you just did · anonymous · n = 38

last class
understood now · average, 1 not at all … 5 very well
allocation mechanisms4.0
perfect competition4.4
barriers to entry4.3
today
known already · average, 1 nothing … 5 know it well
demand3.9
inverse demand2.6
law of demand2.8
supply3.8
inverse supply2.5
equilibrium3.2
shortage and surplus3.1
the interface
how useful · average, 1 not useful … 5 very useful
the chat4.6
the pulse4.4
reading material for each topic (up to 10 pages)?
% of you
yes63% (24)
maybe21% (8)
no11% (4)
problem set 1: have you looked at it?
% of you · Wednesday next to today
Wednesday · n = 38today · n = 38
not yet21% (8)26% (10)
looked at it21% (8)13% (5)
started21% (8)24% (9)
done37% (14)37% (14)
problem set 1: how hard was each question?
average, 1 easy … 5 very hard · and how many of you tried it · Wednesday next to today
Wednesday · n = 38today · n = 38
Q1 musical ticket (a)2.1 · 211.9 · 21
Q2 musical ticket (b)2.3 · 202.5 · 17
Q3 Mr Schmidt (a)3.2 · 193.2 · 15
Q4 Mr Schmidt (b)3.6 · 173.7 · 15
Q5 course average2.0 · 172.0 · 15
Q6 Tom's boats (a)2.3 · 161.6 · 14
Q7 Tom's boats (b)2.5 · 142.0 · 14
Q8 the car from your parents2.8 · 132.6 · 12

today

1 demand
2 inverse demand
3 supply
4 synthetic market
5 equilibrium
6 shortage and surplus

demand for a single person

your streaming subscriptions · each costs €10 a month

subscription1st2nd3rd4th
worth to you (a month)€16€12€8€4
how many do you keep?
how the class answered · n = 41 · first answer per person
17% (3)
2 ✓ right88% (36)
30% (0)
45% (2)
answer
2
Keep one more as long as it is worth at least €10: 1st €16 ✓ · 2nd €12 ✓ · 3rd €8 ✗

demand for a single person

€161€122€83€44subscriptionsP = 20 − 4Q

your reservation prices for ChatGPT Plus

reservation price: the most you would pay a month · 41 of you (lecture 1), sorted from the highest

rank1234567891011121314
€10060505040403530302928252525
rank1516171819202122232425262728
€2020202020201513121010101010
rank29303132333435363738394041
€101098855555100
€50at €50: 4 of you buyeach bar = one of you

how many of you buy?

the rule · at a price, everyone whose reservation price is at least the price buys
€50at €50: 4 of you buyeach bar = one of you
€0€50€60
at €25 a month, how many of you buy?
how the class answered · n = 37 · first answer per person
93% (1)
14 ✓ right62% (23)
2011% (4)
2524% (9)
answer
14
Ranks 1 to 14 would pay €25 or more; the 15th only €20.
€25at €25: 14 of you buyeach bar = one of you

your demand curve

€0€20€40€60010203040at €20: 20 of you buyhow many of you buy
€0€20€60

demand

Demand: what quantity do consumers want to buy at a given price?
formula · Q = 40 − P
at €20, 20 of you buy
PQ€4040demand€2020→
the line is drawn through your reservation prices (the faded bars)

inverse demand

Inverse demand: what price are consumers willing to pay for a given quantity?
formula · P = 40 − Q
for 20 subscriptions, €20
PQ€4040demand20€20↑
another way to read it: the 20th buyer, the marginal buyer, is willing to pay €20: the marginal willingness to pay

law of demand

price ↑ → quantity ↓
demand form
price ↑ → quantity demanded ↓ (all else equal)
inverse form
the marginal willingness to pay decreases as the quantity increases: to sell one more, the price must fall to the next buyer in line, whose reservation price is lower (the marginal buyer)
the picture
a downward-sloping curve

demand function

QD = f(P) for an AI subscription
price per month, thousand subscribers

demand
Q = 120 − 4P
120
if it were free, 120,000 would subscribe
− 4P
each €1 more a month: 4,000 fewer subscribers
at €20
120 − 4·20 = 40 → 40,000 subscribers

inverse demand function

demand · Q = 120 − 4P (Q in thousands of subscribers)
what is the inverse demand? then draw it
how the class answered · n = 38 · first answer per person
P = 30 − Q/4 ✓ right66% (25)
P = 120 − Q/416% (6)
P = 30 − 4Q11% (4)
P = Q/4 − 308% (3)
answer
P = 30 − Q/4
4P = 120 − Q, then divide every term by 4.
30: at €30 or more, nobody subscribes · − Q/4: to win 4,000 more subscribers, the price must fall by €1
Q = 120 − 4P
4P = 120 − Q
P = 30 − Q/4
PQ€30120demand40€20↑
6 more of this kind, with answers
market · a music-streaming plan in Austria (P in € per month, Q in thousand subscribers)
demand · Q = 100 − 5P
what is the inverse demand? then draw it
how the class answered · n = 27 · first answer per person
P = 20 − Q/5 ✓ right100% (27)
P = Q/5 − 200% (0)
P = 20 − Q0% (0)
P = 100 − 5Q0% (0)
answer
P = 20 − Q/5
5P = 100 − Q, then divide EVERY term by 5.
20: at €20 or more nobody buys · check: Q = 0 gives P = 20 in both forms
To draw it: price axis at €20, quantity axis at 100; join the two points.
Q = 100 − 5P
5P = 100 − Q
P = 20 − Q/5
market · an AI tutor app (P in € per month, Q in thousand users)
demand · Q = 60 − 3P
where does the demand curve meet the price axis and the quantity axis?
how the class answered · n = 23 · first answer per person
price axis at €20, quantity axis at 60,000 ✓ right83% (19)
price axis at €3, quantity axis at 60,0004% (1)
price axis at €180, quantity axis at 60,0000% (0)
price axis at €60, quantity axis at 20,0009% (2)
no idea4% (1)
answer
price axis at €20, quantity axis at 60,000
Price axis: nobody buys, Q = 0 → 0 = 60 − 3P → P = 20.
Quantity axis: free, P = 0 → Q = 60 (60,000). Join the two points with a straight line.
Q = 0:  3P = 60  →  P = 20
P = 0:  Q = 60
market · printing at a print shop in Vienna (P in cents per page, Q in pages per day)
demand · Q = 160 − 4P
160 · if printing were free, customers would print 160 pages a day
− 4P · each extra cent per page: 4 fewer pages a day
what is the inverse demand? then draw it
how the class answered · n = 13 · first answer per person
P = 160 − 4Q15% (2)
P = Q/4 − 400% (0)
P = 40 − Q8% (1)
P = 40 − Q/4 ✓ right77% (10)
answer
P = 40 − Q/4
Move the P-term to one side, then divide EVERY term by 4.
Check: Q = 0 gives P = 40 in both forms.
Read it: 40 = at 40 cents a page or more, nobody prints · − Q/4 = to sell 4 more pages a day, the price must fall by 1 cent.
To draw it: price axis at 40 cents, quantity axis at 160 pages; join the two points.
Q = 160 − 4P
4P = 160 − Q
P = 40 − Q/4
market · monthly memberships at Vienna gyms (P in € per month, Q in members)
demand · Q = 1600 − 40P
what is the inverse demand? then draw it
how the class answered · n = 6 · first answer per person
P = Q/40 − 4017% (1)
P = 40 − Q0% (0)
P = 40 − Q/40 ✓ right83% (5)
P = 1600 − 40Q0% (0)
answer
P = 40 − Q/40
40P = 1600 − Q, then divide EVERY term by 40.
40: at €40 or more nobody buys · check: Q = 0 gives P = 40 in both forms
To draw it: price axis at €40, quantity axis at 1600; join the two points.
Q = 1600 − 40P
40P = 1600 − Q
P = 40 − Q/40
market · day passes at Vienna gyms (P in € per pass, Q in passes per day)
demand · Q = 240 − 6P
where does the demand curve meet the price axis and the quantity axis?
how the class answered · n = 5 · first answer per person
price axis at €40, quantity axis at 240 ✓ right80% (4)
price axis at €1440, quantity axis at 24020% (1)
price axis at €6, quantity axis at 2400% (0)
price axis at €240, quantity axis at 400% (0)
answer
price axis at €40, quantity axis at 240
Price axis: nobody buys, Q = 0 → 0 = 240 − 6P → P = 40.
Quantity axis: free, P = 0 → Q = 240. Join the two points with a straight line.
Q = 0:  6P = 240  →  P = 40
P = 0:  Q = 240
market · an AI writing-assistant app (P in € per month, Q in thousand users)
demand · Q = 150 − 2.5P
what is the inverse demand? then draw it
how the class answered · n = 5 · first answer per person
P = 150 − 2.5Q20% (1)
P = 0.4Q − 6040% (2)
P = 60 − 0.4Q ✓ right40% (2)
P = 60 − Q0% (0)
answer
P = 60 − 0.4Q
2.5P = 150 − Q, then divide EVERY term by 2.5.
60: at €60 or more nobody buys · check: Q = 0 gives P = 60 in both forms
To draw it: price axis at €60, quantity axis at 150; join the two points.
Q = 150 − 2.5P
2.5P = 150 − Q
P = 60 − 0.4Q

draw the demand curve

demand · Q = a − bP
price axis · Q = 0 → P = a/b (the choke price)
quantity axis · P = 0 → Q = a

supply

price ↑ → quantity ↑
any firm
producing more needs overtime, extra staff, new machines: each extra unit costs more (rising marginal cost)
miners
when the price is low, it barely pays the electricity: fewer miners mine

supply: ten miners

each miner has one machine · what differs is the electricity bill

miner (cheapest first)12345678910
cost of mining (€)200250300350400450500550600650
the price is €460: how many miners switch on their machines and mine?
how the class answered · n = 36 · first answer per person
411% (4)
53% (1)
6 ✓ right83% (30)
70% (0)
no idea3% (1)
answer
6
A miner mines when the price covers her cost: miners 1–6 (up to €450) · miner 7 needs €500.
12345678910€460miners who mine

supply

Supply: what quantity do sellers want to sell at a given price?
formula · Q = P/50 − 3 (for P ≥ €150)
at €450, 6 miners mine
PQ€150supply€4506→
Q = the number of miners who mine

inverse supply

Inverse supply: what price do sellers need to offer a given quantity?
supply · Q = P/50 − 3
what is the inverse supply? then draw it
how the class answered · n = 37 · first answer per person
P = 150 + 50Q ✓ right70% (26)
P = 50Q − 1508% (3)
P = 3 + Q/505% (2)
P = 150Q + 5014% (5)
no idea3% (1)
answer
P = 150 + 50Q
Bring −3 across (it becomes +3), then multiply every term by 50. Keep the sign.
150: at €150 or less, nobody mines (on the line) · 50: each further miner needs a €50 higher price
Q = P/50 − 3
Q + 3 = P/50
P = 50Q + 150
PQ€150supply6€450↑
6 more of this kind, with answers
market · freelance video editors on a jobs platform (P in € per hour, Q in hundred hours per week)
supply · Q = 4P − 100
what is the inverse supply? then draw it
how the class answered · n = 7 · first answer per person
P = 25 + Q29% (2)
P = Q/4 − 2543% (3)
P = 4Q − 1000% (0)
P = 25 + Q/4 ✓ right29% (2)
answer
P = 25 + Q/4
Bring −100 across (it becomes +100), then divide every term by 4. Keep the sign.
25: below €25 nobody offers anything · check: P = 25 gives Q = 0
To draw it: start at €25 on the price axis; one more point: at €30, Q = 20.
Q = 4P − 100
4P = Q + 100
P = 25 + Q/4
market · seats of an AI note-taking tool sold by resellers (P in € per seat per month, Q in thousand seats)
supply · Q = 5P − 40
where does the supply curve start on the price axis, and how many are offered at €12?
how the class answered · n = 5 · first answer per person
starts at €40; at €12, 20,000 are offered0% (0)
starts at €8; at €12, 60,000 are offered20% (1)
starts at €0; at €12, 20,000 are offered20% (1)
starts at €8; at €12, 20,000 are offered ✓ right60% (3)
answer
starts at €8; at €12, 20,000 are offered
Start: Q = 0 → 5P = 40 → P = 8. Below €8 nobody offers anything.
One more point: at €12, Q = 5·12 − 40 = 20 (20,000). Join the two points.
Q = 0:  P = 40/5 = 8
P = 12:  Q = 60 − 40 = 20
market · dog-walking by students (P in € per walk, Q in walks per week)
supply · Q = 5P − 15
what is the inverse supply? then draw it
how the class answered · n = 6 · first answer per person
P = 3 + Q/5 ✓ right83% (5)
P = 3 + Q17% (1)
P = Q/5 − 30% (0)
P = 5Q − 150% (0)
answer
P = 3 + Q/5
Bring 15 to the other side (it becomes +15), then divide every term by 5.
Check: at P = 3 nobody offers a walk (Q = 0).
To draw it: start at €3 on the price axis; one more point: at €5, Q = 10.
Q = 5P − 15
5P = Q + 15
P = 3 + Q/5
market · cloud storage plans (P in € per month, Q in thousand plans)
supply · Q = 10P − 30
what is the inverse supply? then draw it
class results: only 4 answered in class
answer
P = 3 + Q/10
Bring −30 across (it becomes +30), then divide every term by 10. Keep the sign.
3: below €3 nobody offers anything · check: P = 3 gives Q = 0
To draw it: start at €3 on the price axis; one more point: at €5, Q = 20.
Q = 10P − 30
10P = Q + 30
P = 3 + Q/10
market · satellite launch slots on reusable rockets (P in € million per slot, Q slots per year)
supply · Q = 2P − 30
where does the supply curve start on the price axis, and how many are offered at €35?
class results: only 4 answered in class
answer
starts at €15; at €35, 40 are offered
Start: Q = 0 → 2P = 30 → P = 15. Below €15 nobody offers anything.
One more point: at €35, Q = 2·35 − 30 = 40. Join the two points.
Q = 0:  P = 30/2 = 15
P = 35:  Q = 70 − 30 = 40
market · sponsored posts by Instagram creators (P in € per post, Q in posts per week)
supply · Q = 0.5P − 60
what is the inverse supply? then draw it
class results: only 4 answered in class
answer
P = 120 + 2Q
Bring −60 across (it becomes +60), then divide every term by 0.5. Keep the sign.
120: below €120 nobody offers anything · check: P = 120 gives Q = 0
To draw it: start at €120 on the price axis; one more point: at €200, Q = 40.
Q = 0.5P − 60
0.5P = Q + 60
P = 120 + 2Q

inverse supply

Inverse supply: what price do sellers need to offer a given quantity?
formula · P = 150 + 50Q
for 6 miners to mine, the price must be €450
PQ€150supply6€450↑
another way to read it: the 6th miner, the marginal miner, needs €450 to switch on

draw the supply curve

supply · Q = dP − c
price axis · Q = 0 → P = c/d: below this price nobody sells

a synthetic market

10 buyers and 10 sellers · each buyer wants one unit · each seller has one unit

12345678910
buyers: the most they'd pay€100€90€80€70€60€50€40€30€20€10
sellers: the least they'd accept€10€20€30€40€50€60€70€80€90€100
the question
who should trade with whom, and at what price?
gain of a trade
buyer's value − seller's cost

matching rule 1: same with same

rule: each buyer trades with the seller whose cost equals her value, at that price

buyer€100€90€80…€20€10
seller€100€90€80…€20€10
gain000…00
how much value do these 10 trades create?
how the class answered · n = 35 · first answer per person
€0 ✓ right71% (25)
€2503% (1)
€55014% (5)
€1,0006% (2)
no idea6% (2)
answer
€0
Each buyer pays exactly what the good is worth to her; each seller gets exactly his cost.
Ten trades, and nobody is better off: more trades is not the same as more value.

matching rule 2: best with best

rule: the buyer who values it most trades with the cheapest seller, the second with the second, …

buyer€100€90€80€70€60€50€40€30€20€10
seller€10€20€30€40€50€60€70€80€90€100
gain9070503010−10−30−50−70−90
how many of these pairs should trade?
how the class answered · n = 35 · first answer per person
33% (1)
5 ✓ right86% (30)
63% (1)
106% (2)
no idea3% (1)
answer
5 pairs · total gain €250
The first 5 pairs gain; from the 6th pair on, every gain is negative (−10, −30, −50, −70, −90): those trades would destroy value.
No other matching creates more than €250.
90 + 70 + 50 + 30 + 10 = €250

price rule 1: all gain to the sellers

the 5 best pairs trade · rule: every pair trades at the buyer's value

pair100–1090–2080–3070–4060–50
price€100€90€80€70€60
seller gains9070503010
buyer gains00000
what is the total value created by these trades?
how the class answered · n = 35 · first answer per person
€017% (6)
€250 ✓ right74% (26)
€4009% (3)
€5000% (0)
answer
€250
Sellers gain 90 + 70 + 50 + 30 + 10 = €250; buyers gain 0. Together: €250, the same as with any other prices.
All of it goes to the sellers (average price €80).

price rule 2: all gain to the buyers

rule: every pair trades at the seller's cost

pair100–1090–2080–3070–4060–50
price€10€20€30€40€50
seller gains00000
buyer gains9070503010
what is the total value created by these trades?
how the class answered · n = 35 · first answer per person
€014% (5)
€1500% (0)
€250 ✓ right80% (28)
€4003% (1)
no idea3% (1)
answer
€250
Buyers gain 90 + 70 + 50 + 30 + 10 = €250; sellers gain 0. Together: €250 again.
Now all of it goes to the buyers (average price €30). The price only decides who gets the €250.

price rule 3: any price in between

rule: each pair agrees on some price between the seller's cost and the buyer's value

one pair
buyer €100 and seller €10 agree on €40
how much do the two gain together?
how the class answered · n = 36 · first answer per person
€306% (2)
€608% (3)
€90 ✓ right83% (30)
€1003% (1)
answer
€90
Buyer: 100 − 40 = 60 · seller: 40 − 10 = 30 · together 90 = 100 − 10.
The price only splits the gain; the total is the same at €40, at €100 or at €10.

the price splits the gain

the total gain is the same at any price
rule 1 (buyer's value)
sellers get all €250, buyers 0
rule 2 (seller's cost)
buyers get all €250, sellers 0
any price in between
the €250 is shared
so
the price does not change how much value trade creates; it decides how much goes to the sellers and how much to the buyers

one price for everyone: who buys, who sells?

at each price: buyers with value ≥ price want to buy, sellers with cost ≤ price want to sell · the most eager buyers and the cheapest sellers trade · at exactly €50 the €50 buyer, and at exactly €60 the €60 seller, doesn't care whether to trade: counted in here (6); if they stay out, 5 buy and 5 sell

price€30€40€50€55€58€60€70€80
buyers who want to buy (value ≥ price)87655543
sellers who want to sell (cost ≤ price)34555678
trades (the smaller number)34555543
total value created (€)210240250250250250240210

one price for everyone

in a market, everyone pays the same price

buyers (blue) from the highest value down · sellers (orange) from the lowest cost up · at a price in the shaded band, the 5 buyers above it trade with the 5 sellers below it
PQ€50€60€0€10012345678910buyerssellers5 buyers and 5 sellers trade (solid steps)
at which prices do buyers = sellers?
how the class answered · n = 33 · first answer per person
only at €5515% (5)
any price from €50 to €60 ✓ right85% (28)
any price from €30 to €800% (0)
at no price0% (0)
answer
any price from €50 to €60
Anywhere in that band: 5 buyers (€100 … €60) and 5 sellers (€10 … €50); exactly the 5 best pairs trade, €250 is created.
At exactly €50 the €50 buyer is indifferent, at exactly €60 the €60 seller. Below €50 more buyers than sellers; above €60 more sellers than buyers.

a trading floor

20 AI traders, the same synthetic market

the traders · 10 AI buyers (values €10 … €100) and 10 AI sellers (costs €10 … €100); each knows only its own number
the floor · they post bids and asks in one chat, 4 rounds
what happened · round 1: prices from €40 to €70 · round 4: €60–65 · nobody knew the curves, yet the prices narrowed
open full screen ↗

equilibrium

Equilibrium: demand and supply give the same quantity and the same price.
where demand and supply cross, both give the same Q and the same P
demand P = 750 − 50Q · supply P = 150 + 50Q
PQ€750€150demandsupply

find the equilibrium

curve 1 · P = 750 − 50Q
curve 2 · P = 150 + 50Q
what are Q* and P*?
how the class answered · n = 29 · first answer per person
Q* = 6, P* = €450 ✓ right69% (20)
Q* = 9, P* = €3003% (1)
Q* = 6, P* = €30010% (3)
Q* = 450, P* = €614% (4)
no idea3% (1)
answer
Q* = 6, P* = €450
1 · which is which? curve 1 falls: demand · curve 2 rises: supply
2 · in equilibrium both give the same Q and the same P, so the two right-hand sides are equal
3 · solve for Q*, then put Q* into both curves: the same P*
750 − 50Q = 150 + 50Q
600 = 100Q  →  Q* = 6
P* = 750 − 300 = 150 + 300 = €450
4 more of this kind, with answers
market · streaming subscriptions in Austria (P in € per month, Q in thousand subscribers)
demand · Qd = 300 − 10P
supply · Qs = 5P − 60
what are Q* and P*?
how the class answered · n = 7 · first answer per person
Q* = 120, P* = €360% (0)
Q* = 24, P* = €6029% (2)
Q* = 60, P* = €24 ✓ right43% (3)
Q* = 140, P* = €1629% (2)
answer
Q* = 60, P* = €24
In equilibrium Qd = Qs: set the two right-hand sides equal.
Collect P on one side and divide by the SUM of the slopes (10 + 5); then Q* from either curve.
300 − 10P = 5P − 60
360 = 15P  →  P* = 24
Q* = 300 − 10·24 = 60  (check: 5·24 − 60 = 60)
market · launches on reusable rockets (P in € million per launch, Q launches per year)
demand · P = 100 − 3Q
supply · P = 16 + Q
what are Q* and P*?
class results: only 4 answered in class
answer
Q* = 21, P* = €37 million
Both are already P(Q): set them equal and solve for Q*.
Put Q* into BOTH curves: they must give the same P*.
100 − 3Q = 16 + Q
84 = 4Q  →  Q* = 21
P* = 16 + 21 = 37  (check: 100 − 3·21 = 37)
market · monthly memberships at Vienna climbing gyms (P in € per month, Q in hundred members)
demand · Qd = 192 − 4P
inverse supply · P = 12 + Q/2
what are Q* and P*?
class results: only 2 answered in class
answer
Q* = 48, P* = €36
Same form first: P = 12 + Q/2 → Qs = 2P − 24.
Then Qd = Qs, solve for P*, and Q* from either curve.
Qs = 2P − 24
192 − 4P = 2P − 24
216 = 6P  →  P* = 36,  Q* = 48
market · personal-trainer sessions at Vienna gyms (P in € per session, Q in hundred sessions per week)
demand · Qd = 70 − 2P
inverse supply · P = 15 + Q/2
what is the equilibrium price?
class results: only 2 answered in class
answer
€25
Same form first: P = 15 + Q/2 → Qs = 2P − 30. Then Qd = Qs.
Qs = 2P − 30
70 − 2P = 2P − 30
100 = 4P  →  P* = 25,  Q* = 20

shortage

Shortage (excess demand): at a price below P*, buyers want more than sellers offer.
the story · the government caps the price at €350 (below P* = €450)
buyers want · 750 − 50Q = 350 → Qd = 8
miners mine · 150 + 50Q = 350 → Qs = 4
shortage · Qd − Qs = 8 − 4 = 4 · only 4 are sold (the short side)
demand P = 750 − 50Q · supply P = 150 + 50Q

surplus

Surplus (excess supply): at a price above P*, sellers offer more than buyers want.
the story · the government sets a minimum price of €550 (above P* = €450)
buyers want · 750 − 50Q = 550 → Qd = 4
miners mine · 150 + 50Q = 550 → Qs = 8
surplus · Qs − Qd = 8 − 4 = 4 · only 4 are sold (the short side)
demand P = 750 − 50Q · supply P = 150 + 50Q

summary

demand / inverse demand
price → quantity wanted · quantity → price buyers would pay
supply / inverse supply
price → quantity offered · quantity → price sellers need
the price splits the gain
the total gain is the same at any price; the price decides who gets it
equilibrium
the same Q and P for demand and supply · the marginal buyer and seller pin down the price
shortage / surplus
price below / above P* · the short side is sold