Economics 312MacroeconomicsNoah WilliamsProblem Set 7Due in class on May 4.1. Williamson, Chapter 17, Problem 2.2. Williamson, Chapter 15, Problem 4.3. Consider a search model of the labor market in which workers have linear utilityU(c) = c, all jobs are the same and pay a wage of w units of consumption, un-employed workers get b < w in unemployment benefits, the separation rate is 50%(s = 1/2), the job finding rate is 50% (p = 1/2), and agents discount the futurewith factor β = 1/2.(a) The equilibrium unemployment rate occurs when flows into and out of unem-ployment are equal. Find the equilibrium unemployment rate.(b) Find the values (i.e. the total expected discounted utility) of an employedworker and an unemployed worker.(c) Instead of the scenario outlined above, suppose that the job finding rate wasp = 1, but that 1/2 of jobs paid a wage w1< b while 1/2 of jobs paid a wagew2> b. How would this change your results?4. Consider a simple IS-LM model given by the following equations:IS : Y = a0− a1rLM :MP= b0+ b1Y − b2rwhere all aiand bicoefficients are positive. Suppose the price level is fixed in thecurrent period to P = 1, and that there is a constant full employment level ofoutput given by¯Y . The monetary authority has the option to either control thereal interest rate r or the money supply M.(a) Find the values of r and M (call them r∗and M∗) that lead to the full em-ployment level of output. Does it matter which policy instrument is chosen?(b) Now suppose that there are shocks u to the IS equation and shocks to moneydemand v so that the equations become:IS : Y = a0− a1r + uLM :MP= b0+ b1Y − b2r + v1Policymakers must set policy before the shocks are realized, and the goal ofpolicy is to minimize a loss function that depends on fluctuations of outputfrom the full employment level:min(Y −¯Y )2i. Find the values of the loss (Y −¯Y )2which results (after the shocks arerealized) when the policymakers setr = r∗and when they set M = M∗.ii. Consider the special case when a1= 1, b1= 3, b2= 1. When is it optimalfor policymakers to choose to set the interest rate instead of the
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