# eco331h4 - Northwestern University Fall 2009 Marciano...

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Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 HOMEWORK 4 DUE MONDAY, NOVEMBER 16, 2009, IN CLASS. PLUS ONE PRACTICE PROBLEM 1 Consider an economy with one risk-free asset, with return r f = 1 . 01, and two risky returns ˜ r 1 , ˜ r 2 . The latter are jointly normally distributed, with means μ 1 = 1 . 02 and μ 2 = 1 . 05, standard deviations σ 1 = 0 . 02 and σ 2 = 0 . 05, and covariance σ 12 = - 0 . 0003. (a) Why is it OK to use the mean-variance approach in this setting? (b) Construct the eﬃcient frontier. You need to ﬁnd an expression for the minimum amount of σ required to achieve any target level μ of expected return; the expression will be of the form “constant plus μ times another constant,” and you must specify what these constants are. If you wish, you can also plot the eﬃcient frontier (using Excel, Matlab etc.: remember σ goes on the horizontal axis); however, you do need to ﬁnd an explicit expression for the eﬃcient frontier. 2 Consider the following variant of the monopoly example we discussed in class. Demand is given by Q ( p,s ) = e - sp , the marginal cost is zero, and the possible values of s (i.e. the states) are S = [1 , 9]. Beliefs over S are uniform, as in the lecture notes. The information partition is I = { [1 , 5] , [5 , 9] } ; in other words, the ﬁrm learns whether s [1 , 5] or s [5 , 9]. Assume that the ﬁrm is risk-neutral, as in the lecture notes. Calculate the optimal prices the ﬁrm should set without any additional information, and conditional upon receiving each of the signals [1 , 5] and [5 , 9]. Finally, compute the value of the information partition I . 3 A mine is oﬀered for sale at 2 million dollars. If the mine is rich in ore, it is worth 3 million dollars; otherwise, it is only worth 1 million dollars. A test balloon can be lowered into a well; if the balloon encounters a solid object, it breaks; otherwise the balloon is retrieved, and can be

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eco331h4 - Northwestern University Fall 2009 Marciano...

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