microeconomic theory lecture

8/13/2019 Microeconomic theory lecture

1/23


2/23


3/23

Recall: “Uncertain Situations”

Expected Value of Uncertain SituationL g, g

×

=

Expected Utility of Uncertain SituationL g, g

×

=

Certainty Equivalence, , is such that

3

Uncertainty

$

$

..

$


4/23

Example: Agent with 200 Facing $600,$200;0.6,0.4

4

⇒ 200 0.6 600 0.4(200)

($600)

($200) 280

$280

0.6 $600 0.4 $200

($600)

87.99

Discount due to risk −


5/23

Recall: Attitudes Towards Risk for Decision Making Under Uncertainty

5

Suppose an agent faces the uncertain situation:

$, $, . . , $ ; , , . . ,

An agent has Utility of $ with > 0Risk Averse Agent Risk Neutral Agent Risk Loving Agent

Strictly concave () Linear () Strictly convex ()

> < <

Discount due to risk > 0

Discount due to risk 0 >

Discount due to risk < 0 Decides on the basis of

Expected utility

Decides on the basis of

Expected utility

or Expected Value

Decides on the basis of

Expected utility


6/23

Insurance Nomenclature

6

It was a good partyThe guestsThe party venue

∎ $Wealth before loss ∎ Probability of loss ∎ $Loss

$Insurance Policy

“Price per dollar of insurance” Insurance PremiumInsurance Policy Insurance Premium

Example

$0.2m insurance policy has a $10,000 insurance premium ⇒ $,$, $0.05

Insurance Premium =


7/23

Risk Averse Agents Willing to Pay Money to Avoid “Risk”

7

$

()

−L

=

Wealth

= −

− +( −)

() ()

( )

1

()

1

∎ $Wealth before loss ∎ Probability of loss ∎ $Loss

( )

()


8/23

Ahead: 4 Fundamental Questions in Insurance

8

❶ From the insurance company’s perspective:

What’s the minimum “price per dollar of insurance”?

❷ From policy holder’s perspective:

What is the optimal amount of insurance policy? ( ⋚ )

❸ From policy holder’s perspective:

What’s the and from being insured vs. being uninsured?

❹ What is the maximum insurance premium that an agent will pay for aninsurance policy?

Alternatively, what is the maximum insurance premium that insurance

companies can charge?

❶


9/23

❶ From the insurance company’s perspective: What’s the minimum “price per dollar ofinsurance”?

Scenario # 1: Suppose Insurance Industry is Regulated

Each insurance company required to have sufficient capital to cover expected payout.

For each insured person, we’d require:

1 0

Scenario # 2: Suppose Insurance Industry is Competitive

Each company’s expected profit will be zero:

Π (1 )(Insurance Premium)

(Insurance Premium – Payout) = 0

Π 1 0

9 is known as “actuarially fair insurance”


10/23

Assume actuarially fair insurance:

❷ From policy holder’s perspective: What is the optimal amount of insurance policy? ( ⋚ ) ?

10

Agent buysinsurance andpays insurancepremium

Nature

“Wealth” when there is no loss:

Insurance Premium “Wealth” when there is a loss:

Insurance Premium

Risk averse policy holders will choose optimal by solving “UMP”: max 1 ( )

When will FOC yield a “solution” to this “maximization” problem?

1

′ ′ 1 0

ll f


11/23

Assume actuarially fair insurance:

❷ From policy holder’s perspective: What is the optimal amount of insurance policy? ( ⋚ ) ?

11

(1 )′ ′ 1

′ ′ Provided agent is risk averse (why not risk neutral?):

() ()

A t i ll f i i


12/23

Assume actuarially fair insurance From policy holder’s perspective:What’s the and from being fully insured vs. being uninsured?

12

“Expected Wealth withoutInsurance”

1 ( )

Nature


“Wealth” when there is a loss:

“Certain Wealth withActuarially Fair Full

Insurance” Nature


“Wealth” when there is a loss:

When Agent is Uninsured

When Agent Purchases Actuarially Fair Full Insurance Policy

∎ $Pre-loss Wealth ∎ $Loss ∎ Probability of loss∎ $Insurance policy ∎ $Insurance Premium

A t i ll f i i


13/23

Assume actuarially fair insurance From policy holder’s perspective:What’s the and from being fully insured vs. being uninsured?

13

$

()

−L

=

Wealth

= −

− +( −)

−

() ()

( )

$

()

$ 1

( )

()

$

“Utility” of noinsurance

“Utility” of havingactuarially fair full

insurance


Actuarially fair full Insurance

Premium

Wh t i th i i i th t t ill f


14/23

What is the maximum insurance premium that an agent will pay for

an insurance policy?

14

$

()

−L

=

Wealth

= −

() ()

( )

()

( )

()

$

“Utility” of no

insurance

“Utility” of having fullinsurance with

insurance premium


Maximum Full Insurance Premium


15/23

“Insurance” Application: Credit Default Swaps

15

$

()

$ =

()

BondholdersCoupon Payments

Suppose there is no perceived risk of default


16/23

“Insurance” Application: Credit Default Swap Policies (CDS)

16

No risk of default foreseen


Risk of default foreseen. Suppose Bondholders buy Credit Default Swap

Bondholder Coupon Payments

Buy

CDSpolicy

Receives Payment in

case Samsung defaults


17/23

“Insurance” Application: Credit Default Swaps

17

$

()

$=

$ = −

Coupon $ − +

( Coupon )

(0)

()


Suppose there is a perceived risk of default

Max CDS premium under full “insurance”


18/23

“Risk Avoidance” Application: Forward Contracts in Commodities

18

$

()

L

=

ℎ Hg = −

Price − +

() ()

()

( Price )

“Utility” without Forward

Contract

The Seller will Sell the Commodity at any Forward Price ≥ Potential gains and losses to each party?

()

● “High” Commodity Price ● “Low” Commodity Price● Probability of low price


19/23

Review: “Conditional Probability”

19

Sample space of Probability Model

Suppose we are told that event has occurred

What is the probability that outcome is also in event ?


20/23

Decision Making Under Uncertainty on the Basis of Costly “Test” Information

Examples

∎ Mining ∎ Surgery ∎ Stock Initial Public Offerings (IPOs) ∎ Marketing ∎ Movies

Risk Neutral decision makers will decide on the basis of test information if:

[Decision with Test] [Decision without Test]

≥ Cost of the Test

20

Choose one option for making

decision under uncertainty

Decide on basis of (Costly)“Test” Information

(without knowing testresults)

Decision Tree with Testinformation

Decide without (Costly)“Test” Information

Decision Tree withoutTest information


21/23

Example: Drilling for Oil: With or Without Testing? Calculating Test and


22/23

Example: Drilling for Oil: With or Without Testing? Calculating Test andConditional Probabilities

∎

40

100 ∎

60

10

∎ 50100 ∎ 50

100

∎ and 3050 ∎ F and 2050

∎ and

10

50

∎ F and

40

50

22

Record of Past Test Results and Actual Outcomes

Cell entries = # of sites drilled # Actual Success (S) # Actual Failure (F) Total #

# Tested Positive () 30 20 50 # Tested Negative () 10 40 50

Total # 40 60 100

l ll f l h h ?


23/23

Example: Drilling for Oil: With or Without Testing?

23

Decision?

Decisionwithout testing

Drill with

$120

Drill

Drill $120 $120 $6 0 0

$200 Not Drill = $0

Decision WithTesting

Drill if Test +

Don’t drill ifTest –

$140

Test Result $280

Not Drill = $0

Drill $280 $280

$6 0 0

$200

Test Result $0

Not Drill = $0

Drill $40 $40 $6 0 0

$200

“Value” of Test Information = [Decision with Test]$

[Decision without Test]$

$20

Risk Neutral decision makers will decide on the basis of test information if cost of test≤$20

microeconomic theory lecture

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