A brief introduction to economics Part II Tyler Moore Computer - - PDF document

A brief introduction to economics

Part II Tyler Moore

Computer Science & Engineering Department, SMU, Dallas, TX

September 6, 2012

Expected utility Budget constraints Markets

Outline

1

Expected utility Definitions Attitudes toward risk

2

Budget constraints Definition Changing budgets Making optimal choice Consumer demand

3

Markets From individual to aggregate Equilibrium

2 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Why isn’t utility theory enough?

Only rarely do actions people take directly determine outcomes Instead there is uncertainty about which outcome will come to pass More realistic model: agent selects action a from set of all possible actions A, and then outcomes O are associated with probability distribution

4 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Lotteries

Definition (Lottery) A lottery is a mapping from all outcomes (o1, o2, . . . , on) ∈ O to probabilities corresponding to each

• utcome (p1, p2, . . . , pn), where n

1 pi = 1. A lottery l1 is

represented as l1 = o1 : p1, o2 : p2, . . . , on : pn.

5 / 31

Notes Notes Notes Notes

Expected utility Budget constraints Markets Definitions Attitudes toward risk

Where does randomness come from?

Indeterminism in nature Lack of knowledge Incompleteness in the model Uncertainty concerns which outcome will occur

⇒ Known unknowns, NOT unknown unknowns

6 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Expected utility

Definition (Expected utility (discrete)) The expected utility of an action a ∈ A is defined by adding up the utility for all outcomes weighed by their probability of occurrence: E[U(a)] =

• ∈O

U(o) · P(o|a) (1) Agents make a rational decision by maximizing expected utility: a∗ = arg max

a∈A E[U(a)]

(2)

7 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Example: process control system security

Source: http://www.cl.cam.ac.uk/~fms27/papers/2011-Leverett-industrial.pdf 8 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Example: process control system security

Actions available: A = {disconnect, connect} Outcomes available: O = {attack, no attack} Probability of successful attack is 0.01 (P(attack|connect) = 0.01) If systems are disconnected, then P(attack|disconnect) = 0

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Notes Notes Notes Notes

Expected utility Budget constraints Markets Definitions Attitudes toward risk

Example: process control system security

attack no attack Action U P(attack|action) U P(no attack|action) E[U(action)] disconnect 100 0.01 5 0.99 5.95 connect

• 100

0.01 10 0.99 8.90

⇒ risk-neutral IT security manager chooses to connect since E[U(connect)] > E[U(disconnect)].

10 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Let’s make a deal

Option 1: Take $10 Option 2: Get $20 with a 50% chance, $0 otherwise Which would you choose? E[U] = 0.5 ∗ $20 + 0.5 ∗ $0 = $10 Prefer option 1: you’re risk-averse Prefer option 2: you’re risk-seeking Are you indifferent? If so-you’re risk-neutral

11 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Let’s make a deal (round 2)

Option 1: Take $10 Option 2: Get $150 with a 10% chance, $0 otherwise Which would you choose? E[U] = 0.1 ∗ $200 + 0.5 ∗ $0 = $15 Prefer option 1: you’re risk-averse Prefer option 2: you’re risk-neutral or seeking

12 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Let’s make a deal (round 3)

Option 1: Take $10 Option 2: Get $50 with a 10% chance, $0 otherwise Which would you choose? E[U] = 0.1 ∗ $50 + 0.5 ∗ $0 = $5 Prefer option 1: you’re risk-averse or risk-neutral Prefer option 2: you’ve got a gambling problem

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Notes Notes Notes Notes

Expected utility Budget constraints Markets Definitions Attitudes toward risk

Risk attitudes depend on the behavior of the utility function

• utcomes (o)

U(o) risk-neutral risk-averse risk-seeking

14 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Risk-averse prefer utility of expected value over lottery

Source: Varian, Intermediate Microeconomics, p. 225 15 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Risk-seekers prefer lottery over utility of expected value

Source: Varian, Intermediate Microeconomics, p. 226 16 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

From attitudes to utility

Suppose that outcomes are numeric O ∈ R When might that happen? Then we can define risk-attitudes by how the utility function behaves Definition (Risk neutrality) An agent is risk-neutral when U(o) is a linear function on o.

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Notes Notes Notes Notes

Expected utility Budget constraints Markets Definitions Attitudes toward risk

From attitudes to utility

Definition (Risk aversion) An agent is risk-averse when U(o) is a concave function (i.e., U′′(x) < 0 for a twice-differentiable function). Definition (Risk seeking) An agent is risk-seeking when U(o) is a convex function (i.e., U′′(x) > 0 for a twice-differentiable function).

18 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Example: antivirus software

Suppose you have $10,000 in wealth. You have the option to buy antivirus software for $75. Outcomes available: O ={hacked (decreases wealth by $2,000), not hacked (no change in wealth)} Without AV software, probability of being hacked is 0.05 (P(hacked|no antivirus) = 0.05) With AV software, probability of being hacked is 0 (P(hacked|antivirus) = 0) Exercise: compute the expected utility if you are risk-neutral (so that U(o) = o). Would you buy AV software?

19 / 31 Expected utility Budget constraints Markets Definitions Attitudes toward risk

Example: antivirus software

What if you are risk-averse (so that U(o) =

• (o))?

Risk-averse hack no hack Action U P(hack|action) U P(no hack|action) E[U(action)] buy AV √9, 925 √9, 925 1 99.6 don’t buy √8, 000 0.05 √10, 000 0.95 99.4

Exercise (on your own): How much would you pay for antivirus software if you were risk-neutral and the probability of getting hacked is 0.1 if you don’t have AV installed?

20 / 31 Expected utility Budget constraints Markets Definition Changing budgets Making optimal choice Consumer demand

Budget constraints – preferences meet limits

The utility model we have presented so far is still incomplete Rational actors must allocate a finite budget m Every outcome oi having an associated price pi Definition (Budget constraint) Agents may select outcomes subject to the budget constraint where

n

• i=1

pi ∗ oi <= m

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Notes Notes Notes Notes

Expected utility Budget constraints Markets Definition Changing budgets Making optimal choice Consumer demand

Budget constraints

We can simplify modeling by supposing there are only two

• utcomes o1 and outcome2 in O

Really, we’re only interested in o1, so outcome2 can be viewed as “everything else”. Our simpler budget constraint is o1 ∗ p1 + o2 ∗ p2 ≤ m

23 / 31 Expected utility Budget constraints Markets Definition Changing budgets Making optimal choice Consumer demand

Budget constraints

• 1
• 2

b u d g e t l i n e m p1 m p2 budget set slope = − p1

p2

• pportunity cost of o1

Diagrams adapted from Varian’s Intermediate Microeconomics 24 / 31 Expected utility Budget constraints Markets Definition Changing budgets Making optimal choice Consumer demand

Budget constraints

• 1
• 2

increased budget line

• l

d b u d g e t l i n e slope = − p1

p2

m′ p1 m′ p2

Diagrams adapted from Varian’s Intermediate Microeconomics 24 / 31 Expected utility Budget constraints Markets Definition Changing budgets Making optimal choice Consumer demand

Budget constraints

• 1
• 2

new budget line

• l

d b u d g e t l i n e slope = − p1

p′

2

Diagrams adapted from Varian’s Intermediate Microeconomics 24 / 31

Notes Notes Notes Notes

Expected utility Budget constraints Markets Definition Changing budgets Making optimal choice Consumer demand

Making an optimal choice under budget constraint

• 1
• 2

Indifference curves

b u d g e t l i n e

m p1 m p2

p2∗ p1∗

Diagrams adapted from Varian’s Intermediate Microeconomics 25 / 31 Expected utility Budget constraints Markets Definition Changing budgets Making optimal choice Consumer demand

Making an optimal choice under budget constraint

We just saw one example of an optimal choice for prices p1 and pri2 given a budget m What happens when prices change? The optimal choice does too We can systematically vary prices and obtain the optimal demand Definition (Demand function) A demand function for outcomes o1 and o2 and budget m returns the optimal choice of outcomes demanded do1(p1, p2, m) for given prices and budget.

26 / 31 Expected utility Budget constraints Markets From individual to aggregate Equilibrium

What about other agents?

We can measure overall market demand by adding up all individual market demand functions Definition (Market demand function) A market demand function for outcome

• 1 for n agents is given by

Do1(p1, p2, m1, m2, . . . , mn) =

n

• i=1

do1(p1, p2, mi) (3)

28 / 31 Expected utility Budget constraints Markets From individual to aggregate Equilibrium

What about supply?

Thus far we have focused on consumer preferences But production also matters (the supply side) Suppliers of goods are individually willing to produce goods at different prices We can construct a market supply function S(p1) similar to the market demand function D(p1)

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Notes Notes Notes Notes

Expected utility Budget constraints Markets From individual to aggregate Equilibrium

Market equilibrium

Definition (Market equilibrium) Market equilibrium is achieved at the price p∗ where D(p∗) = S(p∗). The market price is in equilibrium because agents are individually optimizing their demand functions d based on the prices they observe

30 / 31 Expected utility Budget constraints Markets From individual to aggregate Equilibrium

Market equilibrium

quantity price

demand curve constant supply

p∗ q∗

Diagrams adapted from Varian’s Intermediate Microeconomics 31 / 31 Expected utility Budget constraints Markets From individual to aggregate Equilibrium

Market equilibrium

quantity price

demand curve constant price

p∗ q∗

Diagrams adapted from Varian’s Intermediate Microeconomics 31 / 31 Expected utility Budget constraints Markets From individual to aggregate Equilibrium

Market equilibrium

quantity price

demand curve supply curve

pd = ps q∗

Diagrams adapted from Varian’s Intermediate Microeconomics 31 / 31

Notes Notes Notes Notes

Expected utility Budget constraints Markets From individual to aggregate Equilibrium

Market equilibrium

quantity price

demand curve supply curve

pd = ps q∗ pd Willing to buy ps Willing to sell

Diagrams adapted from Varian’s Intermediate Microeconomics 31 / 31

Notes Notes Notes Notes