lecture 3 decisions under uncertainty

LECTURE 3

DECISIONS UNDER UNCERTAINTY

Consider this example. When you are driving in Bangkok, an accident may happen to your car with the probability 0.3. You have to incur a repair cost which is undoubtedly high. To avoid this loss, you can purchase an insurance in which you pay an insurance premium to an insurance company. When an accident does happen, an insurance company will cover the cost that happens.

Probability Distribution: 0.3 - accident, 0.7 - no accident States of nature: accident or no accident

Contingent consumption plan: an insurance company will pay for the cost if an accident happens.

REVIEW OF STATISTICAL CONCEPTS

For a random variable X that takes values of x₁; x₂; : : : ; x_n with probabilities ₁; ₂; : : : ; _n, the expected value of this random vari- able is

E (X) = ₁x₁ + ₂x₂ + : : : + _nx_n

=

Xn i=1

ix_i where

Xn i=1

i = 1

For continuous outcomes, with the probability density function f (x), the expected value is

E (X) =

Z ₁

xf (x) dx where

Z ₁

f (x) dx = 1

The variance of X is de…ned by

2X =

Xn i=1

i (x_i E (X))²

= E (X E (X))²

= E X² [E (X)]² For a continuous version

2

X =

Z ₁

1

(x E (X))² f (x) dx And the standard deviation is simply _X.

BACK TO OUR EXAMPLE

The driver has an endowment of m. If an accident occurs, he will face a repair cost of . Denote c_G as a consumption when

no accident happens (the good state of the world) and c_B as a consumption when accident happens (the bad state of the world).

Also, suppose that an accident will happen with a probability . Without any insurance, the contingent consumption plan is

c_G = m c_B = m

However, this driver can purchase $K of insurance and has to pay a premium K, i.e., the driver pays the insurance company K so that if an accident does happen, the company will pay him

$K. The contingent consumption plan becomes c_G = m K

From c_B, solving for K yields

K = c_B m + 1

Substituting into c_G,

c_G = m

1 [c_B m + ]

= m

1 1 c_B

The above expression is the budget line in which we can show this driver’s endowment (or what happens in the good and bad states without insurance) and another point on the budget line where this driver purchases an insurance with the premium :

The slope of this budget line is = (1 ). The numerator is as if the price of consumption in the good state while the denominator

the price of consumption in the bad state.

The point where the indi¤erence curve touches the budget line will determine how much insurance this driver wants to purchase.

From a standard indi¤erence curve, we can write dU = @U

@c_Bdc_B + (1 ) @U

@c_Gdc_G = 0 which gives us a familiar expression

dc_G

dc_B = @U=@c_B

@U=c_G 1

The RHS of the above expression is the marginal rate of substitu- tion (MRS). This driver will choose how much insurance he wants

to purchase where

@U=@c_B

@U=c_G 1 = 1

Figure 12.1 of Varian. In this example, the car is worth $35,000 but if the accident happens it will cost $10,000 to the driver. He has a choice of purchasing $K insurance with a premium K:

EXPECTED UTILITY

Given various choices of risky decision, we concentrate on the utility that consumers obtain from their decision over these uncer- tainty. We restrict our attention on consumers’income from these choices. For example, you can imagine we consider the utility a consumer obtains from buying a lottery. Suppose this consumer has an initial wealth of $200. He buys a lottery X which gives him $20 if he wins and nothing if he loses. The lottery costs him

$10 and the probability of winning is ¹₂. Let W be his …nal wealth, If he wins : W = $200 $10 + $20 = $210

If he loses : W = $200 $10 + 0 = $190 The expected utility of this consumer is given by

EU (W_win; W_loss) = 1

2U (W_win) + 1

2U (W_loss)

De…nition 1 The expected utility is the sum of the utilities as- sociated with all possible outcomes, weighted by the probability that each outcome will occur. (Pindyck and Rubinfeld, 2005: pp. 159- 160)

The expected utility function, also known as von Newmann- Morgenstern utility function, can be subjected to some kinds of

transformation. However, unlike the usual utility function we have studied, only a positive a¢ ne transformation that represents the same preferences. For an expected utility function U, a positive a¢ ne transformation f (U) takes a form of

f (U) = aU + b where a > 0 Also, please read section 12.4 of Varian !

PREFERENCES TOWARD RISK

Consider the following example. A consumer with an initial wealth of $10 buys a lottery X. If he wins, he will receive $5 but if he loses, it will cost him $5. He has a 50% chance of win-

ning.

If he wins: W_win = $10 + $5 = $15 If he loses : W_loss = $10 $5 = $5 The expected …nal wealth of this consumer is

E (W) = 1

2$15 + 1

2$5 = $10 while the expected utility is

EU (W) = 1

2U (W_win) + 1

2U (W_loss)

= 1

2U ($15) + 1

2U ($5) and the utility of the expected wealth is

U (E (W )) = U ($10)

Therefore, if

The consumer is risk averse if U (E (W )) > EU (W) The consumer is risk loving if U (E (W )) < EU (W) The consumer is risk neutral if U (E (W )) = EU (W) From our example, this consumer is risk averse if

U ($10) > 1

2U ($15) + 1

2U ($5) while he is a risk lover if

U ($10) < 1

2U ($15) + 1

2U ($5)

If U ($10) = ¹₂U ($15) + ¹₂U ($5) ; he is risk neutral.

Risk aversion : U ($10) > 1

2U ($15)+1

2U ($5) (Figure 12.2 of Varian)

Risk loving: U ($10) < 1

2U ($15)+1

2U ($5) (Figure 12.3 of Varian)

BACK TO THE INSURANCE EXAMPLE

Recall the budget line

c_G = m

1 1 c_B

Suppose that the insurance company o¤ers a fair insurance in which the expected value of the insurance is just equal to its cost

K = K or = Recall our optimum condition

@U=@c_B

@U=c_G 1 = 1 which becomes

@U

@c_B = @U

@c_G

FORMAL TREATMENT: INSURANCE

Let the driver has an initial wealth of W₀. An accident can hap- pen with the probability in which the driver will lose . He purchases the insurance that covers an amount of K at the pre- mium K where 2 (0; 1) is the insurance rate. He will choose K to maximise his expected utility. Assume that U⁰ ( ) > 0 and U⁰⁰ ( ) < 0:

maxK U (accident) + (1 ) U (no accident)

If the accident occurs, his …nal wealth is W₀ K + K:

If there is no accident, his …nal wealth is W₀ K: Therefore,

FOC implies

U⁰ (W₀ K + K) ( + 1)+(1 ) U⁰ (W₀ K) ( ) = 0 Rearranging to have

U⁰ (W₀ K + K) U⁰ (W₀ K) =

1

1 When the premium rate is actuarially fair, =

U⁰ (W₀ K + K) = U⁰ (W₀ K) W₀ K + K = W₀ K

K =

This driver will buy a full coverage under fair premium. But what will happen if > ?

Exercise 1 Suppose that W₀ = 5, = 2, and = ¹₂: Also the premium rate is actuarially fair and this consumer’s utility function is U = p

W. Will he buy a full coverage insurance?

A risk averse agent can pay the risk premium in order to avoid taking a risk

De…nition 2 The risk premium is the maximum amount of money that a risk-averse person will py to avoid taking a risk.

From …gure 5.4 (Pindyck and Rubinfeld), the risk premium is

$4000.

Income ($1,000) Utility

0 10 16

10 18

30 40

20

14

A

C

E

G

20

Risk Premium

F

Income ($1,000) Utility

0 10 16

10 18

30 40

20

14

A

C

E

G

10 18

30 40

20

14

A

C

E

G

20

Risk Premium Risk Premium

F F

Figure 5.4 of Pindyck and Rubinfeld: CF measures the risk premium.

REDUCING RISK

1. Diversi…cation: "Don’t put all your eggs in one basket."

2. Insurance: pay to aviod risk

3. Information: gather more information

De…nition 3 Value of complete information is the di¤er- ence between the expected value of a choice when there is complete information and the expected value when information is incomplete.

Mean-Variance Utility

We can express a consumer’s preferences of having a …nal wealth W_s in state s 2 S with a probability _s by a utility function that

probability distribution.

Recall that for a random variable X that takes value of x_s in state s 2 S = f1; : : : ; Sg with a probability _s;

Expected Value (or mean) E (X) =

XS s=1

sx_s = _x Variance

var (X) = E (X E (X))² =

XS s=1

s (x_{s s})² = ²_x

Standard Deviation

E (X E (X)) = vu

utX^S

s=1

s (x_{s s})² = _x

We can write the utility function of a representative consumer that faces a …nal wealth W_s in state s with a probability _s as U ( _W; _W) :

One crucial assumption of this approach is that people are risk averse, i.e., high risk (or high variance) is less preferable. This assumption results in an upward sloping indi¤erence curve in ( ; ) space.

Suppose that the consumer’s portfolio (P) consists of two assets.

2. A risky asset where its return (m_s) depends on the states of the nature (s 2 S) and their associated probability ( _s).

Let 2 [0; 1] be a fraction of wealth the consumer invest in the risky asset. The expected return of this portfolio, _P, will be

P =

XS s=1

s m_s + (1 ) r_f = r_m + (1 ) r_f

= r_f + r_m r_f

where r_m is the expected return of the risky asset. The variance of this portfolio is

var (P ) = var ( m_s) + var (1 ) r_f

2

P = ^{2 2}_m or = ^P

m

Note that in ( ; ) space if = 0 we have the point r_f; 0 while

= 1 would result in (r_m; _m). Connecting these two points will form the budget line describing the market tradeo¤ between risk and return. Substituting = ^P

m into _P to have the expression of this budget line

P = r_f + r_m r_f

m P

The slope of this line ^{rm rf}

m measures the price of risk.

The equilibrium allocation takes place at the point where the

indi¤erence curve touches the budget line. From U ( ; ), write dU = @U

@ d + @U

@ d = 0 d

d = @U=@

@U=@ = M RS _; At the optimal portfolio choice,

@U=@

@U=@ = r_m r_f

m

Figure 13.2 from Varian

READING

Varian Ch. 12, 13.1; Pindyck and Rubinfeld Ch. 5.1-5.4 NEXT LECTURE

We will put the consumption side and production side of the economy together. We begin with the economy with no production where we allow for consumers to trade their endowment with each other in order to achieve an e¢ cient allocation. Later on, we will introduce the production side into our simple model to study the economy’s production possibilities frontier and gains from trade.