# Microeconomics- Risk Aversion Essay

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Q3 (a) According to the expected utility model, does a risk averter:

(i)     Always choose to be fully insured if offered an actuarially fair insurance premium?

Risk averse means willingness to pay money to avoid playing a risky game, even when the expected value of the game is in your favor - Microeconomics- Risk Aversion Essay introduction.  A risk-averse individual is one who prefers a guaranteed amount, x, to an uncertain prospect (e.g. a gamble) with the same expected value.  We thus observe that a risk-averse person has a diminishing marginal utility of income and prefers a certain income to a gamble with the same expected income. In relation to insurance, A risk averse person will pay more than the expected value of a game that lets him or her avoid a risk.  Suppose Harry faces a 1/100 chance of losing \$10,000.  Risk aversion means that you would pay more than \$100 (the expected or “actuarially fair” value) for an insurance policy that would reimburse you for that \$10,000 loss, if it happens.  Thus, a risk averse person is always prepared to pay an insurance premium in excess of the expected value of the loss. If actuarially-fair insurance is available, a risk-averse individual will insure fully against uncertainty. Lets us see Why? The insurance premium assures the individual of having the same income regardless of whether or not a loss occurs. Because the insurance is actuarially fair, this certain income is equal to the expected income if the individual takes the risky option of not purchasing insurance. This guarantee of the same income, whatever the outcome, generates more utility for a risk-averse person than the average utility of a high income when there was no loss and the utility of a low income with a loss (i.e., because of risk aversion, E[U(x)] £ (E[x]). Therefore, we can justify the statement that if the cost of insurance is equal to the expected loss, (i.e., if the insurance is actuarially fair), risk-averse individuals will fully insure against monetary loss. In an example, the insurer’s objective is to find the optimal amount of insurance coverage C given some premium per dollar of coverage, g . Thus the optimization problem is:

max U(w, c) = p su(wA + C – g C) + (1-p s)u(wN – g C).

Now, if insurance is “fair”, i.e. if g = p s, then this reduces to:

u¢ (wN – g C)/u¢ (wA + C – g C) = 1

so the marginal utility of a bad state is equal to that of a good state. This implies that (with state-independent utility), wA + C – g C = wN – g C, which implies, in turn, that C = wN – wA, i.e. the agent takes full coverage so that the entire income loss from an accident is recovered. The optimal coverage is shown in Figure 4 by the allocation c = (wN-a , wA + b ) where the highest indifference curve U(c) is tangent to the fair insurance line F on the 45° certainty line.

Figure illustrating Optimal Insurance;  source:  http://cepa.newschool.edu/het/essays/uncert/statepref.htm

The full coverage result depends crucially on the assumption of “fair insurance”, or g = p s.

Also in another example, we consider two persons A and B with the same initial wealth, both being risk averse but their preferences are not identical.  We assume that that each of them can loose 20% of wealth with probability p.  Thus, in this case, the price of unit insurance  is equal to the probability of accident: . As we know in case of fair insurance any risk averse agent insures completely. And since both are risk averse and losses are the same, we can conclude that both will purchase the same amount of insurance, equal to the value of losses, as it is shown in below figure.

Demand of the two risk averse individuals (with different preferences but the same initial endowment) for fair insurance; Source: http://www.icef.ru/study/materials/micro3/Micro.doc

(ii)Never pay an actuarially unfair insurance premium to avoid taking risk?

It is wrong to say that that an risk averter shall never pay an actuarially unfair insurance premium to avoid taking risk. When the insurance premium normally exceeds the expected value of the insured loss, it is referred as an actuarially unfair insurance premium. Thus in unfair insurance, we consider the situation when price of insurance exceeds the probability of accident (otherwise insurance company will have negative expected profit).  In this case, risk-averse individuals may choose only partial insurance.eg use of a “deductible” if there’s a fixed cost of any claim (to avoid paying premium to cover small losses, which the individual would prefer to bear, rather than insure at actuarially “unfair”rates). Thus, depending on the piece of insurance and degree of risk aversion agents will purchase partial insurance or will stay with their initial endowments.

In this example, we consider the case of unfair insurance premium. Suppose that the firm decides to make extraordinary profits, so (1-p s)a – p sb > 0. This implies that g /(1-g ) > p s/(1-p s) or, simply, g > p s, so that the premium per unit of coverage exceeds the probability of an accident. In the agent’s perspective, these is “unfair” insurance and it is captured by the unfair insurance line G the below figure with slope g ¢ /(1-g ¢ ). In this case, the consumer’s first order condition implies that: u¢ (wN – g C)/u¢ (wA + C – g C) = [p s/(1-p s)]·[(1-g )/g ] < 0 so the marginal utility of a good state is less than the marginal utility of a bad state. As we are assuming quasi-concave utility, this implies that the agent’s utility in a good state exceeds his utility in a bad state. This implies that the agent must be still making some loss in the case of an accident – i.e. he cannot be taking full coverage. The individual optimum under “unfair” insurance is depicted in Figure  at point c¢ , the tangency of the unfair insurance line G with the highest indifference curve source:  http://cepa.newschool.edu/het/essays/uncert/statepref.htm

U(c¢ ). We see we do not have full coverage as the allocation is off the 45° certainty line, so wA + b ¢ < wN – a ¢ , i.e. the loss is not fully covered. To quote another example, we consider two persons A and B with the same initial wealth, both being risk averse but their preferences are not identical.  We assume that that each of them can loose 20% of wealth with probability p.  Thus, in this case, the price of unit insurance  is equal to the probability of accident: . As we know in case of unfair Source: http://www.icef.ru/study/materials/micro3/Micro.doci         insurance, risk averse agents will purchase less than full insurance if insurance is unfair. But since agents have different preferences they will  not necessarily purchase the same amount of insurance. Agent, who is more risk averse will purchase more insurance, as it is shown in below figure.

Q3 (b) An individual prefers £ 300 with certainty to a gamble with an 80 percent chance of £400 and a 20 percent chance of £0. If the same individual is confronted with the following two gambles:

(i)a 25 percent chance of £300 and a 75 percent chance  of £ 0, and

(ii)a 20 percent chance  of £400 and an 80 percent chance of £0

Ans: The expected value for the individual when the individual prefers £ 300 with certainty

EV = (100/100) * 300 = £ 300                      ——- (1)

The expected value for the individual for a gamble with an 80 percent chance of £400 and a 20 percent chance of £0

EV = ((80/100) * 400) + ((20/100) * 0) = £ 320 —– (2)

It is clearly evident that the individual is risk averse since he prefers £ 300 with certainty to a bet inspite of the EV for the bet being higher. Risk premium in this case is £ 20.

However, a rational decision maker will always maximise expected utility and not expected profits. Hence in this case, we need to construct a utility function in order to calculate the expected utility for both the gambles.

Constructing utility function: Assuming that the individual is risk neutral instead of risk aversive in which case he is indifferent between the certainty of a gain and a gamble.

Let P be the probability to a gamble with an 80 percent chance of £ 400. And (1-P) be the probability to a gamble with a 20 percent chance of £ 0. Therefore,

300 = P * 400 + (1-P) * 0 => 300 = 400P =>  P = .75         —- (3)

Setting the utility attached to each possible outcome. Let U (£0) = 0 and U (£ 400) = 50

Since decision maker is indifferent, hence

U (£ 300) = P * U (£ 400) + (1-P) * U (£0)

U (£ 300) = .75 * 50 + .25 * 0

U (£ 300) = 37.50                              ——– (4)

(i)Expected utility (EU) for a gamble with 25 percent chance of £300 and a 75 percent chance of £ 0.

EU = .25 * U (£ 300) + .75 * U (£0)

EU = .25 * 37.50 + 0

EU = 9.375                                        —— (5)

(ii) Expected utility (EU) for a gamble with a 20 percent chance of £400 and an 80 percent chance of £0

EU = .20 * U (£ 400) + .80 * U (£0)

EU = .20 * 50 + 0

EU = 10                                             —— (6)

From (5) and (6), since EU (i) < EU (ii), hence the individual would choose gamble (ii).

Q3( c ) Suppose you know that an individual’s utility function is such that :u(£0)=0, u(£100)=0.5, u(£200)=0.75, u(£300)=0.875, u(£400)=1. Which one of the two gambles at (b) above will this individual choose and why?

Ans: Using the given utility function,

(i) Expected utility (EU) for a gamble with 25 percent chance of £300 and a 75 percent chance of £ 0

EU = .25 * U (£ 300) + .75 * U (£0)

EU = .25 * .875 + .75 * 0

EU = .21875                                      —— (1)

(ii) Expected utility(EU) for a gamble with a 20 percent chance of £400 and an 80 percent chance of £0

EU = .20 * U (£ 400) + .80 * U (£0)

EU = .20 * 1 + .80 * 0

EU = .20                                            —— (2)

From (1) and (2), since EU (i) > EU (ii), hence the individual would choose gamble (i).

Q2 (a) Outline the nature of the ‘Allais Paradox’. What is the significance of the paradox?

Ans. The Allais paradox, or the Allais problem designed by Maurice Allais. According to Wikipedia, the Encyclopedia, “It is a choice problem which shows that reasonable individual choices are inconsistent with the predictions of expected utility theory.” The problem consists of two choices presented to a participant, each between a pair of gambles:

Choice 1
Choice 2
Gamble 1A
Gamble 1B
Gamble 2A
Gamble 2B
Winnings
Chance
Winnings
Chance
Winnings
Chance
Winnings
Chance
\$1 million
100%
\$5 million
10%
\$1 million
11%
\$5 million
10%
\$1 million
89%
nothing
89%
nothing
90%
nothing
1%

Most people choose 1A and 2B. The point is that both gambles give the same outcome 89% of the time (\$1 million for Gamble 1, and zero for Gamble 2), so, in expected utility, these equal outcomes can have no effect on the desirability of the gamble. If the 89% ‘common consequence’ is disregarded, both gambles offer the same choice; a 10% chance of getting \$5 million and 1% chance of getting nothing as against an 11% chance of getting \$1 million.

Allais presented his paradox as a counterexample to the independence axiom of expected utility theory. Independence means that if an agent is indifferent between simple lotteries L1 and L2, the agent is also indifferent between L1 mixed with an arbitrary simple lottery L3 with probability p and L2 mixed with L3 with the same probability p.The Allais paradox is the most prominent example for behavioral inconsistencies related to the von Neumann Morgenstern axiomatic model of choice under uncertainty. The Allais paradox shows that the significant majority of real decision makers orders uncertain prospects in a way that is inconsistent with the postulate that choices are independent of irrelevant alternatives. Basically, it is this postulate that allows representing preferences over uncertain prospects as a linear functional of the utilities of the basic outcomes, viz. as the expectation of these utilities.

Consider the following choice situation (A) among two lotteries:

·  lottery L1 promises a sure win of \$30,

·  lottery L2 is a 80% chance to win \$45 (and zero in 20% of the cases).
Typically, L1 is strictly preferred to L2 (such observed behavior is called a revealed preference).

Now, consider another choice situation (B):

·  lottery K1 promises a 25% chance of winning \$30,

·  lottery K2 is a 20% chance to win \$45.
Here, the typical choice is K2 over K1 although situation B differs from situation A only in that in each lottery, three quarters of the original probability of winning a positive amount are cancelled. Assume the typical subject decides among lotteries in the following way. According to SFB 504 Glossary, “To each of the basic outcomes, a number is assigned that indicates its attractiveness; say u(0)=0, u(45)=1, and u(30)=v (0<v<1). The overall attractiveness of a lottery (compared to another lottery) derives as the sum of the outcomes’ elementary attractiveness’s, weighted by their respective probabilities. Among two lotteries, the preferred one is that which offers a higher expected level of overall attractiveness. Now in situation A, the revealed preference of L1 over L2 implies u(30) > 0.8 u(45), or v > 0.8; while the revealed preference of K2 over K1 in situation B shows that 1/4 v < 1/5, or v < 0.8.”

In relation to cognitive psychology, this inconsistency is explained as a certainty effect. In situation A, L2 differs from L1 by a winning probability that is 20% lower, just as lottery K2 differs from K1 in situation B (where 4/5 x 25 = 20). Empirically, it seems that cancelling a fixed proportion of winning probability has a higher cognitive impact in a lottery where winning was extremely likely than in a lottery where winning was “a rather unlikely event, anyway.”

Q2 (b) Draw a utility to wealth function that exhibits risk loving in relation to small gambles and risk aversion in relation to large gambles.

Differing attitudes to risk can be illustrated in the diagram below.

A risk loving behavior is the attitude exhibited by gamblers. In reality we can say that agents may be risk loving for ‘small’ gambles although risk averse for larger gambles. If we could measure the risk attitude of the decision maker, then we could replace monetary Source: http://www.econ.surrey.ac.uk/staff/rpierse/cop8.pdf

payoffs by utility and use a decision criterion that maximises expected utility.

Assigning utilities to monetary outcomes

Von-Neumann and Morgenstern showed how revealed preference could be used to assign utility values to different gambles. Consider the gamble represented by decision d1 in the numerical example below

Source: http://www.econ.surrey.ac.uk/staff/rpierse/cop8.pdf

This gamble is a 40% chance of winning 500 and a 60% chance of losing 250. The expected (average) outcome is a win of 50. Without loss of generality we can arbitrarily assign utility values of 1 and zero to the two payoffs 500 and -250 so that U (500)=1 U(-250)=0. Then the expected utility of the gamble p * U(500) + (1 – p) * U(-250) = p is given by the probability p. A risk neutral agent will be indifferent when the certain outcome is equal to the EMV of the gamble p * 500 + (1 – p) *(-250) e.g. for a certain outcome of Source50 a risk neutral agent will be indifferent for p=0.4. In the diagram, the indifference curve for a risk neutral agent lies on the straight line from -250, 0 to500, 1. A risk averse agent will have an indifference curve lying above this line since they require a probability of winning higher than the expected outcome to compensate them for bearing risk.

Source: http://www.econ.surrey.ac.uk/staff/rpierse/cop8.pdf

A risk loving agent will have an indifference curve lying below the straight line since they are prepared to take a gamble in which they will lose on average.

Q2 (c) Explain and illustrate why a risk averse individual will be willing to purchase an insurance contract at an actuarially fair premium

Ans.   A risk-averse person has a diminishing marginal utility of income and prefers a certain income to a gamble with the same expected income. The reason why a risk averse individual will be willing to purchase an insurance contract at an actuarially fair premium is because of the fact that the insurance premium assures the individual of having the same income regardless of whether or not a loss occurs. Since the insurance is actuarially fair, this certain income is equal to the expected income if the individual takes the risky option of not purchasing insurance. This guarantee of the same income, whatever the outcome, creates more utility for a risk-averse person than the average utility of a high income when there was no loss and the utility of a low income with a loss (i.e., because of risk aversion, E[U(x)] £  U(E[x]).

For instance, a risk-averse consumer has initial wealth w, a utility function. He owns a car of value L, and the probability of an accident which would total the car is p (assuming p as the current accident rate in the state where he lives) . Let’s say this price is r, for \$1 worth of insurance, so for \$x of insurance, he would be paying \$rx as a premium. For insurance to be actuarially fair, the insurance company should have zero expected profits. Thus, with probability p, the insurance company must pay \$x, while receiving \$rx in premiums. With probability (1-p), they pay nothing, and continue to receive \$rx in premiums. So their expected profit is:
p(rx – x) + (1-p)rx
If this equals zero, we have: px(r-1) + (1-p)rx = 0
Dividing throughout by x, we get: pr – p + r – pr = 0
i.e. p = r.
So for insurance to be actuarially fair, the premium rate must equal the probability of an accident. As a risk-averse consumer, you would want to choose a value of x so as to maximize expected utility, i.e.
According to Econ port, “Given actuarially fair insurance, where p = r, we can solve : max pu(w – px – L + x) + (1-p)u(w – px), since in case of an accident,  total wealth would be w, less the loss suffered due to the accident, less the premium paid, and adding the amount received from the insurance company. Differentiating with respect to x, and setting the result equal to zero, we get the first-order necessary condition as: (1-p)pu'(w – px – L + x) – p(1-p)u'(w – px) = 0,
which gives us: u'(w – px – L + x) = u'(w – px)
Risk-aversion implies u” < 0, so that equality of the marginal utilities of wealth implies equality of the wealth levels, i.e. w – px – L + x = w – px,
so we must have x = L.”
So, given actuarially fair insurance, you would choose to fully insure your car. Since you’re risk-averse, you’d aim to equalize your wealth across all circumstances – whether or not you have an accident.

Q2 (d) Suppose two individuals ( A & B) have the following utility of wealth functions:

A=u(w)= √w                          B=u(w)=w²

State for each whether the individual is risk loving or risk averse. Suppose wo= £100 and the individual is considering whether to bet £19 in the hope of   a return of £144 (£125 plus the stake of £19). Determine whether A and B undertake the bet if they judge the probability of winning p=0.1 and if p=0.5

Ans.  Since A‘s utility of wealth function is W0.5, A is risk averse, showing a diminishing marginal utility of wealth.  To show this, assume that she has \$10,000 and is offered a gamble of a \$1,000 gain with 50 percent probability and a \$1,000 loss with 50 percent probability. A’s utility of \$10,000 is 3.162, (u(I) = 100.5 = 3.162). A’s expected utility is EU = (0.5)(90.5 ) + (0.5)(110.5 ) = 3.158 < 3.162. A would thus avoid the gamble. On the other hand, B is risk loving because his w², showing an increasing marginal utility of wealth.

First situation when p=0.1

(1) A’s expected utility= (0.1* 1440.5) + (0.9* 00.5)

=1.2

A’s current utility =190.5=4.358

Since A’s expected utility is less than his current utility, he would not undertake this bet

(2) B’s expected utility = (0.1 * 144²) + (0.9 *0²)

=2073.6

B’s current utility = 19² =361

Since B’s expected utility is more than his current utility, so he would undertake this bet

Second situation when p=0.5

(1) A’s expected utility= (0.5* 1440.5) + (0.5* 00.5)

=6

A’s current utility =190.5=4.358

Since A’s expected utility is more than his current utility,  he would not undertake this bet

(2) B’s expected utility = (0.5 * 144²) + (0.5 *0²)

=10368

B’s current utility = 19² =361

Since B’s expected utility is more than his current utility, so he would undertake this bet

References

http://www.sfb504.uni-mannheim.de/glossary/allais.htm

http://cepa.newschool.edu/het/essays/uncert/statepref.htm

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