Economics > SOLUTIONS MANUAL > University of California, Berkeley Problem Set-4-Summer2020-Solutions Econ 100B: Economic Analysis { (All)
University of California, Berkeley Problem Set-4-Summer2020-Solutions Econ 100B: Economic Analysis { Macroeconomics
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Econ 100B: Economic Analysis { Macroeconomics Problem Set #4 { Solutions Due Date: July 24, 2020 General Instructions: • Please upload a PDF of your problem set to Gradescope by 11:59 pm. • L... ate homework will not be accepted. • Please put your name & student ID at the upper right corner of the front page. 1. Write out the steps in the derivation of the optimal rate rule given on slides 21{23 of lecture 12. Include a derivation of the the first equation on slide 22 by minimizing the central bank loss function. In your derivation include comments that explain what you are doing. Solution: We begin with the Phillips curve πt = πt−1 + γ yt − yP (1) the IS curves Yt = Y − ζY rt−1 (2) and Y P = Y − ζY r∗ (3) and the central-bank loss function L = yt − yP 2 + β πt − πT 2 (4) The minimization of the loss function with respect to yt and πt can be reduced to a minimization with respect to only yt by substituting the Phillips curve into the loss function as follows L = yt − yP 2 + β πt−1 + γ yt − yP − πT 2 (5) Taking the derivative with respect to yt dL dyt = 2 yt − yP + 2ββ πt−1 + γ yt − yP − πT γ (6) or dL dyt = 2 yt − yP + 2βγ πt − πT (7) and setting it to zero to find the minimum yields yt − yP = −βγ πt − πT (8) 1With the IS curves we can express the output gap in terms of the rate gap by Yt − Y P Y P = − ζY YP (rt−1 − r∗) (9) or yt − yP = −ζy (rt−1 − r∗) (10) where we have used Yt − Y P Y P ≈ ln YYPt = ln (Yt) − ln Y P = yt − yP (11) and where ζy = ζY =YP. Returning to Eq. (8), the relationship between the output gap and the inflation gap when the central-bank loss function is at a minimum, let’s expand the inflation term using the Phillips curve yt − yP = −βγ πt−1 + γ yt − yP − πT (12) which, because the dynamic IS curve expresses yt in terms of rt−1, will set us up to have a rate function in terms of inflation at the same time. Dividing through by −γβ and collecting terms in the output gap −1 βγ yt − yP = πt−1 + γ yt − yP − πT (13) and − γ + βγ 1 yt − yP = πt−1 − πT : (14) Substituting the gap form of the IS equation gives us an equation relating the rate gap to the output gap we get ζy γ + βγ 1 (rt−1 − r∗) = πt−1 − πT (15) from which our rate equation emerges as rt−1 = r∗ + 1 ζy γ + βγ 1 πt−1 − πT (16) which holds for all times t − 1 and can be written in r(t) form as [Show More]
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