Exercise 1 Suppose that W" & %, δ & $, and π

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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.

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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.

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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¤erence 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

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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

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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.

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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

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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 ^r^m ^r^f

m measures the price of risk.

The equilibrium allocation takes place at the point where the

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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

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Figure 13.2 from Varian

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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 (or

‘pure exchange economy’) 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.