Econ808H2ans.02

# Econ808H2ans.02 - The Ohio State University Department of...

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The Ohio State University Department of Economics Econ 808–Problem Set #2 Questions and Answers Spring 2002 Levin and Peck 1. Consider the following economy with one physical commodity per state of nature and three consumers, each of whom seek to maximize expected utility. For i = 1,2, consumer i is risk averse, with utility of certain consumption given by u i ( x i ) = log( x i ) . For i = 1,2, consumer i is endowed with 1 unit of consumption when she does not have an accident, 0 units of consumption when she has an accident. Consumer 1 is a “low risk” consumer, with a probability of an accident equal to 1/3. Consumer 2 is a “high risk” consumer, with a probability of an accident equal to 1 2 . Consumer 1 having an accident and consumer 2 having an accident are independent events. Consumer 3 is risk neutral, with utility of certain consumption given by u 3 ( x 3 )= x 3 , and has an endowment of 2 units of consumption in all states of nature. For parts (i) and (ii), assume that consumer 3 knows that consumer 1 is low risk and that consumer 2 is high risk, so information is symmetric. (i) De…ne a competitive equilibrium for the economy with complete state- contingent commodity markets. Specify how many states of nature there are and the probability of each state. (ii) Calculate the competitive equilibrium price vector and allocation for the economy with complete state-contingent commodity markets. For parts (iii) and (iv), suppose that consumers 1 and 2 are in a Rothschild- Stiglitz world. That is, instead of consumer 3, there are many risk-neutral …rms who cannot observe which consumer is low risk and which consumer is high risk. Firms compete by o¤ering contracts, specifying consumption a policyholder receives when she has an accident and consumption she receives when she does not have an accident. (iii) Find the pooling contract providing full insurance (consumption is inde- pendent of whether the policyholder has an accident) and yielding zero expected pro…ts when both consumers accept the contract. (iv) Show that the pooling contract of part (iii), call it ® , is not an equilib- rium. That is, …nd another contract, ¯ , that would be chosen by the low risk consumer, would not be chosen by the high risk consumer, and yields expected pro…ts for the …rm o¤ering the contract. Be as explicit as you can. Answer: (i) There are 4 states of nature, corresponding to the four possible outcomes: consumer 1 does not have an accident and consumer 2 does not have an accident, and so on. Call the four states gg, gb, bg, and bb, where the …rst letter (g for good, b for bad) describes whether consumer 1 had an accident and the second 1

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letter describes whether consumer 2 had an accident. The probabilities of the four states are: (1/3, 1/3, 1/6, 1/6).
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Econ808H2ans.02 - The Ohio State University Department of...

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