# ProblemSet-4-Summer2020-Solutions-v3.pdf - Econ 100B...

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Econ 100B: Economic Analysis – MacroeconomicsProblem Set #4 – SolutionsDue Date: July 24, 2020General Instructions:Please upload a PDF of your problem set to Gradescope by 11:59 pm.Late 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 oflecture 12. Include a derivation of the the first equation on slide 22 by minimizing thecentral bank loss function. In your derivation include comments that explain what youare doing. 1
With the IS curves we can express the output gap in terms of the rate gap byYt-YPYP=-ζYYP(rt-1-r*)(9)oryt-yP=-ζy(rt-1-r*)(10)where we have usedYt-YPYPlnYtYP= ln (Yt)-ln(YP)=yt-yP(11)and whereζy=ζY/YP. Returning to Eq. (8), the relationship between the outputgap 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 expressesytin terms ofrt-1, will set us up tohave a rate function in terms of inflation at the same time. Dividing through by-γβand collecting terms in the output gap-1βγ(yt-yP)=(πt-1+γ(yt-yP)-πT)(13)and-γ+1βγ(yt-yP)=(πt-1-πT).(14)Substituting the gap form of the IS equation gives us an equation relating the rategap to the output gap we getζyγ+1βγ(rt-1-r*) =(πt-1-πT)(15)from which our rate equation emerges asrt-1=r*+1ζyγ+1βγ(πt-1-πT)(16)which holds for all timest-1 and can be written inr(t) form asr(t) =r*+1ζyγ+1βγ(π(t)-πT).(17)2
2. The notion of contractionary and expansionary monetary policy can be expressed using the rate rule derived in question1by the expressionr(t) =r+1ζyγ+1γβπ(t)orr(t)-r(t-1) = Δr+1ζyγ+1γβ(π(t)-π(t-1))andΔr=r(t)-r(t-1)-1ζyγ+1γβ(π(t)-π(t-1))wherer(t)-r(t-1) is the rate change made at timetby the central bank in responseto the inflation changeπ(t)-π(t-1).The sign of Δrindicates whether the ratechange was contractionary or expansionary as follows:If Δr >0 then the rate change was greater than that indicated solely by the changein inflation and this is referred to as contractionary monetary policy because thisrate change would decrease aggregate outputYmore than the amount indicatedby the change in inflation.
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