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Choice under Uncertainty

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Introduction Many choices made by consumers take place under conditions of uncertainty Therefore involves an element of risk We examine how the theory of consumer choice can be used to describe such behaviour

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Description of Risky Alternatives The consumer is concerned with the probability distribution of getting different consumption bundles. A probability distribution consists of a list of different outcomes or consumption and the probability associated with each outcome E.g. decision on how much vehicle insurance to buy or how much to invest on the stock market, is a decision on a pattern of probability distribution across different amounts of consumption. Each of these decisions involves a choice among risky alternatives.

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Example: Lottery Suppose a consumer has K100 & he is contemplating buying a lotto ticket with a particular number. If that number is drawn in the lotto, the holder will be paid K200. This ticket costs K5. The two outcomes that are of interest are the event that the ticket is drawn and the event that it is not.

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The endowment amount the consumer would have if they did not buy the lotto ticket is: – K100 if the number of the ticket is drawn – K100 if it is not drawn. If the consumer did buy the ticket for K5 then the amount is: – K295 if the ticket is a winner – K95 if the ticket is not a winner The purchase of the lotto ticket changes the original endowment of probabilities of wealth in the two different circumstances.

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Example: insurance Suppose an individual has assets worth K35,000 but there is a possibility that K10,000 worth of assets may be lost, e.g. car stolen. Suppose probability of this happening is P = 0.01. Then the probability distribution the person is facing is: – 1% probability of having K25,000 of assets – 99% probability of having K35,000.

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Insurance offers a way to change this probability distribution. Suppose an insurance contract will pay the person K100 if the loss occurs in exchange for a K1 premium. If the person decides to buy K10,000 worth of insurance, it will cost him K100. In this case he will have a 1% chance of having: K34,900 = (K35,000 assets - K10,000 loss + K10,000 insurance payment - K100 insurance premium) if loss occurs. and 99% chance of having: K34,900 = (K35,000 assets -K100 insurance premium) if loss does not occur. The consumer ends up with the same wealth no matter what happens. He is fully insured against loss.

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In general

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What amount of insurance will person choose? This depends on his preferences. If conservative, he will purchase a lot of insurance or take risks and not purchase any at all. Therefore, people have different preferences over probability distributions just like they have over the consumption of ordinary goods.

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Contingent Consumption The different outcome of a random event can be thought of as being different states of nature. In the insurance example, there are 2 states of nature: the loss occurs or it does not. A contingent consumption plan is a specification of what will be consumed in each different state of nature. Contingent means depending on something note yet certain. Therefore a contingent consumption plan depends on the outcome of some event.

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People have preferences over different plans of contingent consumption just like they have over actual consumption. We can use the theory of choice developed so far to analyze the choices people make over consumption in different circumstances Using the example of insurance, we can describe the purchase of insurance in terms of indifference curves. Two states of nature: event that loss occurs and event that it does not. Contingent consumption plan: values of how much money you would have in each circumstance

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K25,000 K35,000 Endowment Choice K35,000 - γK K25,000 + K - γKCbCb CgCg

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The endowment of contingent consumption is K25,000 in the bad state and K35,000 in the good state. Insurance offers a way to move away from this endowment point. By purchasing K Zmw of insurance, γK Zmw of consumption is given up in the good state in exchange of K – γK Zmw of consumption in the bad state.

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Utility Functions and Probabilities How person values consumption in one state of nature compared to another will depend on the probability that the state in question will actually occur. In other words, the rate at which a consumer is willing to substitute consumption if loss occurs for consumption if it does not has something to do with how likely he thinks the loss will occur. Thus the utility function depends on the probabilities as well as the consumption levels.

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Expected Utility

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If you subject an expected utility function to a positive affine transformation, it not only represents the same preferences, but it will also still has the expected utility property. Thus we say that an expected utility function is unique up to an affine transformation. This means that you can apply an affine transformation to it and get another expected utility function that represents the same preferences. But any other kind of transformation will destroy the expected utility property.

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Risk Aversion

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Fig 1 Risk averse Wealth Utility K15 K5

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Fig 2 Risk loving Wealth Utility K15 K5

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In Fig 1 the consumer prefers K10 with certainty to the gamble (lottery) itself But there will be some amount of wealth we could offer with certainty that would make the consumer indifferent between accepting that wealth and facing the gamble. We call this wealth the certainty equivalent of the gamble.

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Certainty Equivalent & Risk Premium Risk averse Wealth Utility K15 K5 CE P

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Insurance Example 1

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Insurance Example

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Measuring risk aversion

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Risk aversion and wealth Does risk aversion increases or decreases with wealth? Intuitively one might think that the willingness to pay to avoid a gamble decline as wealth increases bcos diminishing marginal utility would make potential loses less serious for high wealth individuals The intuitive answer is not correct bcos diminishing marginal utility makes gains from winning gambles less attractive

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Decreasing absolute risk aversion (DARA) is generally a plausible restriction to impose: the higher the level of wealth, the less averse to taking small gambles (risks) Under constant absolute risk aversion (CARA), there is no greater willingness to pay to avoid a gamble at higher levels of wealth Under increasing absolute risk aversion, the greater wealth, the more one is willing to pay to avoid a gamble, i.e. the more averse one becomes to accepting the same small gamble__a rather perverse behaviour.

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Examples

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Relative Risk Aversion

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Chapter 12 Uncertainty Consider two lotteries L 1 : 500,000 (1) L 1 ’: 2,500,000 (0.1), 500,000 (0.89), 0 (0.01) Which one would you choose? Another two.

Chapter 12 Uncertainty Consider two lotteries L 1 : 500,000 (1) L 1 ’: 2,500,000 (0.1), 500,000 (0.89), 0 (0.01) Which one would you choose? Another two.

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