Econ 4170 – where a determines the effect of the drug and ǫ is random noise

Econ 4170Fall 2015Prof. Leonard MirmanFinal Exam1. Consider a consumer with income I. Suppose the consumer gets utilityfrom a good x and health y. But health depends on the drug z asfollows:y = az + ǫ˜where a determines the effect of the drug and ǫ is random noise. The˜drug effect a takes two values a with prior ρ and a with prior 1 − ρ,¯where a > a, i.e. the maximal and minimal effect of the drug on¯health. The random noise ǫ ∼ f (ǫ) and E(ǫ) = 0, where f (·) satisfiesthe monotonic ratio property. Let u(x, y) be the utility function for xand y. The expected utility function is thenEu(x, az + ǫ)˜So the choice for the consumer is between good x and the drug z.Suppose the utility function has the form u(x, y) = x1/2 + y and theprice of x is px = 1, the price of z is pz = 1, i.e. the budget constraintis z + x = I.(a) Set up the one-period expected utility maximization problem interms of z for this utility function. Then solve for z, rememberboth a and ǫ are random and write a = ρ¯ + (1 − ρ)a.ˆa1(b) After solving for the optimal z, find the utility derived from optimal consumption as a function of ρ. Is it increasing or decreasingin ρ? Is it convex in ρ? Explain.(c) Set up the two period maximization problem so that the consumertakes account of the decision of the first period amount of drug onthe expected return in the second period. In this expression writeout the entire two period expected utility with the second periodutility as a function of the posterior ρ.ˆ(d) Determine the effect of the information in this case on the amountof drug z consumed in the first period. Explain why you get thisresult.(e) Identify all the effects of the information in the second period.2. Consider a risk averse monopolist with utility function u(π), u′ (·) > 0,u′′ (·) < 0. The firm has no cost. The firm does not know the parameterof the demand curve, in other words, there are two possible demandcurves(there is no learning):p=a − bq¯a − bqwith probability ρwith probability 1-ρ(a) Set up the problem for this firm to maximize expected utility(b) Find the first order condition. Are the second order conditionssatisfied?(c) Draw a picture showing the solution to the first order conditionsusing the two parts of the first order conditions in separate curves.Which crosses the q axis first?2(d) Now suppose the monopolist becomes more risk averse. Writedown the maximization problem for the more risk averse firm andthe associated first order conditions. How does this change in riskaversion affect the first order conditions? What effect does it haveon the optimal solution? Explain.3. Consider a monopolist selling a product of unknown quality. Thereare two types of consumers, one group knows the quality of the goodbeing sold and the other group learns the quality of the good from theprice. In equilibrium the firm maximizes profit given the belief of theconsumers are consistent with the choice of the monopolist and theconsumers learn the true quality of the good.The demand function for the good isλI (θ − p) + λL (χ(p) − p) = ywhere λI is the number of informed consumers who know that thequality of the good is θ, and λL is the number of learning consumers whohave the updating function χ(p), which must be correct in equilibrium.(a) If both types of consumers know θ (there is no need to update),calculate the optimal quantity produced by the monopolist. Whatis the relationship between the price and the quality in this fullinformation case? Also, what is the relationship between outputof the firm and the quality θ?(b) If now there are λL learning consumers and λI informed consumers, show that the relationships for question (a) cannot bean equilibrium. Finally show that there is a fully revealed equiand the price is p = (λI +2λL )θ . What2(λI +λL )are the equilibrium conjecture of the uninformed consumers?librium if output is y =λI θ23

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