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Berkeley ECON 202A - Lecture Notes

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1Lecture IV Economics 202A Fall 2007At the end of this morning’s class I promised you that I would go over Sargent’smodel of the Lucas Critique.That will be the topic of this class. Before I begin the Sargent model, I must go over an important technical detail.Sargent has terms in his equations, such as pt - pt-1where lower case pt is the log of the price level at tand lower case pt-1 is the log of the price level at t-1.In fact, such a term is almost equal to the percentage change in the price level. Itis almost equal to the rate of inflation.Let me show you why.Let upper case Pt be the Price Level.pt - pt-1 = ln Pt - ln Pt-1= ln Pt/ Pt-1.Let me make an assertion.My assertion is that ln Pt /Pt-1 is approximately equal to or,which is the percentage change in the price level.How do I know that2Consider any number close to one.I will show you that ln x – x - 1.I know by Taylor series expansion that:ln 1 = 0, POINT AND ALSOSo ln x – (x - 1).Using this approximationIf you do not follow what I have said here now I want you to be able to accept myinterpretation of Sargent’s formulas, and then you can later come back and verifythat in fact this formula follows.ERASE BBI am now going to present to you Sargent’s 3 equations. I will use his notation,which will make it easier for you to read it.3David Romer explains in detail how they arise out of microfoundations.I am going to just write them down and explain why these correspond to standardmacroeconomics from your intermediate course.Equation (1) is an aggregate supply equation.(1) yt = kt + ( ( pt - tp*t-1) + utwhere yt is real incomekt is potential GDPpt is the actual price level at time ttp*t-1 is the expected price level at time t,with the expectations made at time t-1, andut is an uncorrelated random variable.All of these variables are in logarithms, so I should have been more careful andsaid yt is the log of real incomekt is the log of potential GDPpt is the log of the actual price level at time t, andtp*t-1 is the log of the expected price level at time t,with the expectations made at time t-1. Equation (2) is an IS curve.Again the variables are in logs with the exception of the nominal rate of interest rt.(2) yt = kt + c ( rt - (t+1p*t - pt )) + d zt + etwhere rt is the nominal rate of interest,zt is a vector of exogenous variables,including government spending and tax rates, and4et is a random error term.Equation (3) is an LM curve(3) m t = pt + yt + brt + 0t,where m t is the log of the money supply,pt, yt, and rt are as beforeand 0t is a random error term.While this notation is slightly hard to read, it turns out that this is exactly thestandard Keynesian model with the standard Phillips Curve describing laborsupply.Let me now review the equations in reverse order.Equation (3) is the demand for money. The level of this demand will beproportional to prices and real income. And it will depend negatively on thenominal rate of interest.Equation (2) is an IS Curve.kt is potential GDP.Demand depends on the real rate of interest and other variables.rt is the nominal rate of interest.And using our previous reasoning you can check that t+1p*t - pt is the expected rate of change of prices.So rt - (t+1p*t - pt ) is the nominal rate of interest minus the expected rate ofprice change. That then is the expected real rate of interest.If we accept that rt - (t+1p*t - pt ) is the expected real rate of interest, we cansee that except for possible quibbles over functional form, this equation is an ISCurve for a closed economy.The standard IS condition, to recall, is that Sales = Production in a closed economy.5This condition is thereforeC + I + G = YC depends on real income and the real rate of interest and taxes.I depends upon real income and the real rate of interest.FILL IN:C (Y, re r, T) + I (Y, re r) + G = YWe can linearize and find as an approximation:Y = k + c re r + d zwhere z includes taxes and government spending.So equation (2) is just new compact notation for an IS curve.Now let’s return to equation (1).Equation (1) turns out to be just a standard Phillips Curve in a new notation.kt is GDP potential.yt - kt is then the deviation between current income and GDP potential, andsince it is in log form it is really the percentage deviation from GDP potential.According to Okun’s law the unemployment rate is a multiple of thepercentage deviation from GDP potential.So we could rewriteyt - kt as b( U* - Ut) where U* - Ut is the deviation of the unemployment ratefrom the natural rate.So at this point we can see thatb(U* - Ut) = ( ( pt - tp*t-1) + ut,merely by substituting for yt - kt its equivalent in unemployment rates rather thanin percentage deviation from GDP potential.6Let me now take one minor algebraic step and I will show you that this equationis equivalent to the Standard Accelerationist Phillips Curve.Let me make the merely algebraic operation of adding and subtracting (pt-1 to getb(U* - Ut) = ( ( pt - pt-1 ) - (( tp*t-1 - pt-1 ) + ut.Now pt - pt-1 is the difference in log prices and is therefore approximately the rateof inflation, which I will denote Bt.And, tp*t-1 - pt-1 is approximately the expected rate of inflation, which I will denoteBet.We then get b(U* - Ut) = (( Bt - Bet) + ut.We can solve for Bt and obtain:This is the standard accelerationist Phillips Curve that you must have beentaught in your intermediate macroeconomics class.Sargent gives a rather different interpretation of this equation. I will go over thatlater.So far I have shown you that all three equations here correspond to what youwere taught in the equivalent of Economics 100B, or Economics 101B.To that I am going to add the assumption of rational expectations.The assumption of rational expectations is the following:tp*t-1 = E(pt*2t-1).<Add this as (4) below other three equations.>That is: the expectations people make at t-1 about the price level at t is:7the expected value of the price level which will actually occur given theinformation available at t-1. The symbol 2t-1 here represents the informationavailable at t-1.Two implications follow from Sargent’s model + Rational expectations.(1) FIRST, a systematic monetary policy will have no expected effect onequilibrium income in this model.(2) SECOND, deviations in income from GDP potential will not be seriallycorrelated.And, yet more strongly, these deviations will have no correlation with anypreviously observable economic variable.I now want to put this exact model


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