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Unformatted text preview: Econ 202, Winter 2010 Problem set 3: Answers. 1. See answers to question 3 of the midterm. Using the basic shortrun model of the text, what is the e¤ect of a permanent increase in government spending on the output gap and in‡ ation? (a) Assume the central bank does not change its policy rule and that the economy starts out with a zero output gap and in‡ ation equal to . Explain how the economy eventually returns to a zero output gap but with in‡ ation above . (b) If the the central bank whats to prevent a …scal expansion from having a permanent e¤ect on in‡ ation, how does ti need to adjust its policy rule? 2. The basic short run model of the text book takes the form ~ Yt = a
t b (Rt r) = t1 ~ + v Yt + o plus a policy rule that speci…es the behavior of Rt . Suppose, however, s that the central bank’ policy rate Rt is related to the interest rate that s appears in the IS relationship Rt according to
s Rt = Rt + p, where p is a mean zero shock to the risk premium. Policy is described by
s Rt = r + m ( t ). (1) ~ (a) Derive the aggregate demand curve linking Yt and t . How is it a¤ected by a positive realization of p? (i.e., is it shifted to the left or right?) To derive the ADI curve, we need to eliminate the interest rate from the IS curve to obtain a relationship between the output gap and in‡ation: ~ Yt = a b (Rt bm (
t ~ r) = Yt = a
t s b (Rt + p r) =a =a b (r + m ( )+p bp, r) ) so a positive realization of p is just like a negative a – i.e., it shifts the ADI curve to the left (or down). (b) Assume a = o = 0 but suppose p takes on a positive value (assume it remains permanently at this positive value). Describe the adjustment of the economy to such a shock. (Assume the economy starts out at ~ Y = 0 and = .) See answers to question 3 of the midterm which deals with a positive a – a positive p is the same in reverse, i.e., a negative a. 1 (c) Instead of (1), assume the central bank follows the policy rule given by s Rt = r + m ( t ) p. (2) What happens now if p takes on a positive value? How does your answer di¤er from that of part (b)? Why does it di¤er? With this policy rule, the ADI curve becomes ~ Yt = = = a a a b (Rt bm (
t ~ r) = Yt = a
t s b (Rt + p r) b (r + m ( ) p+p r) ), which is independent of p – i.e., p shock no longer a¤ ect aggregate demand or in‡ation. (d) Suppose that p takes on a value so big that with the policy rule (2), s Rt hits the lower bound of zero. How does this a¤ect your answer to part (c)? What are the dangers posed by the zero lower bound? Will discuss in class. 3. What is meant by the time inconsistency of optimal policy? Give an example other than the one leading to the in‡ ation bias of discretion. Explain how an economy could end up with an in‡ ation rate that was higher than desired if monetary policy is conducted with discretion. We discussed this in class. A timeinconsistent policy is one that it is not optimal to actually carry out when the time comes to do so. For the last part, if policy is set under a regime of discretion, then promises to keep in‡ation will not be credible. Private agents will expect higher in‡ation because they recognize that the policy maker has an incentive to try to stimulate the economy. Once expectations of in‡ation rise, the central bank can only keep output from falling by delivering high in‡ation. The economy becomes caught in a high in‡ation equilibrium. 4. Suppose the economy is described by the following IS and Phillips curves: ~ Y= = b (R
e r) ~ + vY The central bank wants to minimize ‡ uctuations in around and the output gap around k > 0. We can represent these objectives by assuming the central bank minimize an objective function given by 1 ( 2 )+
2 ~ Y 2 k subject to the constraints imposed by the IS and the Phillips curves. The central bank acts with discretion, meaning it sets R taking expectations e as given. 2 ~ (a) Use the IS curve to substitute Y out of the Phillips curve and use your result to substitute out of the central bank’ objective funcs tion. The Phillips curve becomes = e v b (R r) and then the objective function becomes 1 2
e v b (R r) 2 + ~ Y 2 k ~ (b) Use the IS curve to substitute Y out of the objective function. (Your objective function should now depend on R r, , and e .) i 1h e 2 2 v b (R r) + b (R r ) k 2 (c) Find the R that minimizes the objective function you obtained in part (b). What is the equilibrium output gap? What is the equilibirum in‡ ation rate under rational expectations? The …rst order condition is v b e v b (R r) b b (R r) k = 0 Solving for R R r yields r= vb ( e ) bk v( e ) k = 2 + b2 2 + )b (v b) (v The output gap is ~ Y= and in‡ation is = =
e b (R r) = v( e ) + k v2 v b (R
e r) = + e v2 ( e (v 2 ) vk +) v2 + v2 + v k v2 + Under rational expectations, = e (since there are no shocks or other disturbances that would lead to any prediction errors), so = )
e v2 + = + + v2 + v k = v2 + v2 + + v2 + v k v2 + k > v The output gap is then equal to v( ~ Y=
e ) k v2 + = v( ) v2 + k = v k v v2 + k =0 3 (d) Does equal ? If not, is greater than or less than ; explain why. In‡ation exceeds . This is do to the bias under discretion. ~ At = and Y = 0, the marginal bene…ts of a bit more expansion exceed the marginal costs of the resulting in‡ation. So the public does not believe in‡ation will be . Expected in‡ation rises, and then to avoid a recession (a negative output gap), the best the central bank can do is deliver in‡ation that exceeds . 5. See ps3_q5_output.txt on class web site. Answer this question using the data on the output gap, the natural rate of unemployment, CPI in‡ ation, and the actual rate of unemployment in ps3_202_w10_data.xls (the data run from 1984Q1  2007Q4 and are available on the class web site). (a) Using ordinary least squares, estimate the basic Phillips curve
t =a t1 ~ + v Yt + ot , where ot is the disturbance terms. Test Ho : a = 1. ~ (b) Estimate Okun’ Law: Yt = g (ut un ). s (c) Using your estimates in (a) and (b), how much would the unemployment rate need to rise to reduce in‡ ation by 1%? 4 ...
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 Winter '08
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