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foundations of microeconomics

Questions and Answers of Foundations Of Microeconomics

where yt is output, xt is some (vector of) deterministic exogenous variable(s), and Et is a white noise stochastic error term with mean zero and variance aE2 .
Asymptotic variance Rational expectations models often use the asymptotic variance of output as a welfare measure. Intuitively, the asymptotic variance measures the degree of fluctuations over time
There are a number of other reasons why PIP fails-see Buiter (1980) for an interesting discussion. For example, private agents may not have rational expectations, or there may be nominal price
Chapter 3: Rational Expectations and Economic Policy conclusion is very surprising indeed: there is an optimal contract length of n* > 0, which Chadha estimates to be around 3.73 quarters for the US
when the oldest contracts In the oldest contract.her's (1977) analysis to the _ is of Calvo (1982) that odel, he is able to analyse d and two-period contracts the asymptotic variance in n is given in
In a recent paper, Chadha (1989) has extended Fischer's (1977) analysis to the multi-period overlapping contract setting using the insights of Calvo (1982) that are discussed in detail below in
So, to the extent that fluctuations in output are a good proxy for loss of economic welfare, the policy maker could attempt to minimize the asymptotic variance of output by choosing its reaction
The Foundation of Modern Macroeconomics output. Using standard (but tedious) techniques, the asymptotic variance of the output path described by (3.49) can be written as:a y2 ac2[41 1 —P uu +
(3.43)REH solution for the price 3Et_2 (vt - ut)• (3.44)obtain the expression for yt :— tit) 3Et-2(Vt Ut)] Vt _2 (Vt — Ut) vt , (3.45)+ 2pu] Et-i + 3 [An + PV] + p6ut-2• (3.49)n*Figure 3.7.
(3.42)mean? In words, it repmation)to expect in period nown in period t - 1, so it is the constant itself). By Ibtained:1 ti-t -2 y" t .-4-2.4-ti1
The appropriate measure is the asymptotic variance of yt, designated by QY (see the Intermezzo below).Intuitively, the asymptotic variance measures the severity of the fluctuations in+ Et _iEt_2Pr-)]
Clearly, output can be affected by monetary policy. But should it be affected, and if so, how? Clearly, (3.49) implies that output fluctuates stochastically, so some measure of the degree of
But Fischer's blow to the new classicals was made even more devastating by the following.
This is the crucial counter-example to the PIP. Equation (3.49) contains the policy parameters An and ,u21 , so that output can be affected by monetary policy even under rational expectations. As
Equation (3.48) is intuitive. Agents can perfectly forecast the money supply one period ahead (i.e. Et_imt = mt) but not two periods ahead. That is because in period t - 1 an innovation in the demand
where we have used the fact that (3.33) implies Et_2Ur-1 = puut_2 and Et_2 1/t-1 =--PVVt-2. Using (3.46) and (3.47) we find:-Et_2mt = [ut-i - Auut-2] + [vt-i Pvvt-2]r: = + (3.48)
Chapter 3: Rational Expectations and Economic Policy rst, (3.25) and (3.38) can be The monetary surprise (mt - Et_2mt ) must now be calculated. Using (3.29), we find that:(3.39) nit = Anut, +
(3.40)t -1 information of both 75
This is the crucial coil]parameters p H and under rational eXpet that "...between the 1 ci oration of that cor -information about ret two-period contracts. I on "stale" informai,,,i, But Fischer's
If we now substitute (3.41) and (3.43) into (3.39), the REH solution for the price level is obtained:5pt 4Et-imt + Et_2mt + 1(vt - ) + -61Et_i(vt - ut) + 3Et_2(vt - ut ). (3.44)This can be
(3.43)
Obviously, Et_iEt_ipt = Et_ipt, but what does Et_1Et_2pt mean? In words, it represents what agents expect (using period t — 1 information) to expect in period t — 2 about the price level in
(3.42)
Similarly, by taking expectations conditional upon period t —1 information of both sides of (3.39), we obtain:Et—lPt = i [Et-1 mt + Et-1 vt — Et-1 Ut (Et—lEt—lPt Et—lEt-2Pt)]
Expectations. In words this law says that you do not know ahead of time how you are going to change your mind. Only genuinely new information makes you change your expectation. Hence, (3.40) can be
We already know that Et_2Et_2Pt = Et-2Pt, but what does Et_2Et_ipt mean? In words, it represents what agents expect (using period t — 2 information) to expect in period t — 1 about the price
(3.40)
(3.39)By taking expectations conditional upon period t — 2 information of both sides of(3.39), we obtain:Et-2Pt = 2 [Et-2Mt Et-2Vt Et-2Ut (Et--2Et--iPt + Et--zEt--2Pt)]
The model can be solved by repeated substitution. First, (3.25) and (3.38) can be solved for pt:Pt = z [Mt Vt Ut (Et—lPt Et-2Pt)]
The rest of the model consists of the aggregate demand curve (3.25) and the money supply rule (3.29).I e the contract is specified ce the expected price level expectation is pet , then Mal price
Notice the difference in the information set used for the two contracts. The economy is perfectly competitive, so that there is only one output price, and aggregate supply is equal to:Yt = i [Pt —
Now consider the case where nominal contracts are decided on for two periods. We continue to assume that nominal wages are set such that the expected real wage is consistent with full employment.
The coefficients of the policy rule (i.e. Ali and /12i) do not influence the path of output, so that PIP holds. In other words, anticipated monetary policy is unable to cause deviations of output
By using these forecasts in equation (3.32), and substituting the price surprise into(3.28), the REH solution for output is obtained:Yt = i brit — + ut. (3,35)
What does the surprise term (3.32) look like? First, (3.29) implies that agents know the money supply in period t once they have lagged information (there is no stochastic element in the policy
put. so that PIP uiciaduons of mail 72 becomes:1.? expectational error:I£t_ lit)] (3.32)►n. i.e.:1, (3.33)Preferred to as innovations):(3.30)(3.31)Chapter 3: Rational Expectations and Economic
Now assume that the shock terms display autocorrelation, i.e.:ut punt-i + Et, IPul < 1, vt = pvvt-i + rlr, IPvl < 1, (3.33)where Et and rit are uncorrelated white noise terms (often referred to as
(3.32)
(3.31)Deducting (3.31) from (3.30) yields the expression for the expectational error:Pt — Et- iPt = 2 [(mt — Et_imt) + (vt — Et-ivt) — (ut — Et_iut)]
(3.30)By taking conditional expectations of both sides, (3.30) becomes:Et_iPt = 2 [Et_imt + Et_ivt — Et-tut + Et_ipt]
Not surprisingly, in view of the similarities with our earlier model, Fischer's oneperiod contract model implies that the PIP is valid. The REH solution is constructed as follows. First, solving
Hence, the policy maker is assumed to react to past shocks in aggregate demand and supply (below we shall see that it is in fact sufficient to react to shocks only lagged once and lagged twice, so
We assume that the policy rule adopted by the policy maker has the following form:mt = E -FE (3.29)
where we can simplify notation further by normalizing y = 0. The supply of output depends on the actual real wage:yt = [pt wt(t - 1)] + ut, (3.27)so that (3.26) and (3.27) imply a Lucas-type supply
The Foundation of Modern Macroeconomics labour, and estimate the market clearing real wage. Since the contract is specified before the price in period t is known, the workers use the expected price
1 Notice the difference i ctly compt •ki equal to:= [Pt — vvtlt —where the first term in:s on one-yt r unx&ers on two-year o_ obi .n the yr = [Pr — Et- iPrj r._nce, this sui e •y don set.
This is illustrated in Figure 3.6. Workers know the supply and demand schedules for Figure 3.6. Wage setting with single-period contracts 71 Overlapping%consider the case 1 CC 1 ue to assur, .,,Lilt
We assume that workers aim (and settle) for a nominal wage contract for which they expect full employment in the next period, when the wage contract is in operation.
which can be seen as a special case of (3.15) with po = /32 = 0 and /31 = 1. The supply side of the economy consists of workers signing one-period or two-period nominal wage contracts, after which
Lucas argued that a strong Keynesian flavour)tsed, namely the evaluation s so-called Lucas critique can the economy has operated rlw and understand it, and lutput follows the stochastic'1.21), it is
3.3.1 One-period nominal wage contracts Fischer's (1977) model is very simple. The AD curve is monetarist in nature:ranks of the professional that macroeconomists are fective, then why should g
3.3 Should We Take the PIP Seriously?Shortly after the publication of Sargent and Wallace's (1976) seemingly devastating blow to advocates of (Keynesian) countercyclical policy, it was argued that
1 -As embodying the REH are Chapter 3: Rational Expectations and Economic Policy
4. find a well-fitting model.that a monetary expansion. Indeed, many use simuladations.Lucas pointed out, imulations because its coef-REH. Indeed, suppose that cal viewpoint, reflected in a the model
ai +131(3.23)(3.24)
For example, an increase in 1/121 would increase the actual value of 10 1 1.Of course, Lucas is right in principle. Provided one compares only stochastic steady states, the effects mentioned by him
suppose that the government would switch to a strong countercyclical viewpoint, reflected in a more negative value for the parameter 11,2. Predictions with the model based on the existing estimates
An innocent but popular interpretation might suggest that a monetary expansion would yield an expansion of employment and output. Indeed, many use simulations of econometric models to give policy
An econometrician running regressions like (3.22) would find a well-fitting model.
By solving (3.16) for et and substituting the result into (3.21), it is clear that output can be written as follows:Yt = 4)o + Yr-1 + 4)2 m t + 03mt- +where ao(ai + Si) — 1t2a1S1aiSi Oo al + Pi , =
On top of this came the second strike of the new classicals against the then predominantly Keynesian army of policy-oriented macroeconomists. Lucas argued that the then popular large macroeconometric
The Foundation of Modern Macroeconomics Of course, the PIP caused an enormous stir in the ranks of the professional economists. Indeed, it seemed to have supplied proof that macroeconomists are
•e qualitative nature of our icy whatsoever.rns out that the solution. First, we equate aggregate level:- ut (3.17)Et-iPt = ai +/32Et-iEt-i [Pt+i - Pt] + Et-i (vt - ut)al + Pi 69
1. Investment depends on al interest rate, a higher terest, and a higher rate equation (3.16) is the pol-1 nests several special cases:;ince there is no real growth- ill) a Keynesian like Tobin= 0
In this system, there is no sense in which the authority has the option to conduct countercyclical policy. To exploit the Phillips curve, it must somehow trick the public. By virtue of the assumption
Equation (3.21) has an implication that proved very disturbing to many economists in the early 1970s. It says that monetary policy is completely ineffective at influencing output (and hence
where the parallel with equation (3.11) should be obvious. Equation (3.21) represents the stochastic steady-state solution for output. Given the model and the REH, output fluctuates according to
Only unanticipated shocks to AD and AS, and unanticipated changes in the money supply can cause agents to be surprised. Indeed, (3.16) implies that mt mt = et, so that (3.20) and (3.14) imply the
But the conditional expectation of a conditional expectation is just the conditional expectation itself, i.e. we only need to write Et_i once on the right-hand side of(3.18). The shock terms vt and
Chapter 3: Rational Expectations and Economic Policy Second, we take expectations of pt , conditional on the information set Qt-i Po - ao + PiEt-i nit +(3.18)
How do we solve the model given in (3.14)-(3.16)? It turns out that the solution method explained above can be used in this model also. First, we equate aggregate supply (3.14) and demand (3.15) and
. ., but that does not affect the qualitative nature of our conclusions regarding the effectiveness of monetary policy whatsoever.
, Yt-2, Yt-3,
•
Mt-2, Mt-3,
• •1
(i) Friedman would advocate a constant money supply (since there is no real growth in the model) and would set p,i = ,u2 = 0, so that mt = (ii) a Keynesian like Tobin would believe in a
Finally, equation (3.16) is the policy rule followed by the government. This specification nests several special cases:
The real balance term, mt - pt, reflects the influence of the LM curve, i.e. the Keynes effect, and the expected inflation rate, Er-i (Pr+i - pt), represents a Tobin effect. Investment depends on the
Equation (3.14) is the expectations based short-run aggregate supply curve(e.g. (2.2)). If agents underestimate the price level, they supply too much labour and output expands. Note that the
supply, and the price level, all measured in logarithms. The random terms are given by ut , vt, and et , and are assumed to be independent from themselves in time, and from each other, i.e. Evt = 0,
where Yt log Yt , mt log Mt, and pt log Pt are, respectively, output, the money
Their basic idea can be illustrated with a simple loglinear model, that is based on Sargent and Wallace (1975).yt = ao + ai(pt — Et_ipt) + ut, al > 0, Yr = fio + 8i (mr - pt) + 132Et_i(pt+i — pt)
The Foundation of Modern Macroeconomics applied it to macroeconomic issues. They took most of their motivation from Friedman's (1968) presidential address to the American Economic Association, and
_ employ and expec •authority cannot expk(3.14)(3.15)(3.16)
.ich simplified t , Et-ipt =1 F (3.L' 1 obtained:Pt - Et- =t unanticii _J •si.pply can cause agLi.:so t' at (3.20) and (3 Yr = cro + U the parali,, scents the stochastic ste c :. Actuates
!S L..), L. : c sbu...xs in period 1. _A,
The idea behind rational expectations remained unused for a decade, before new classicals like Robert Lucas, Thomas Sargent, Neil Wallace, and Robert Barro 67 aSe and, we take expec LI --1Pr = fio
So there are good reasons to believe that the use of the REH cannot be justified as an outcome of an informational cost-benefit analysis. Yet, many economists today accept the REH as the standard
Other authors investigate the question whether agents can learn to converge to rational expectations—see, for example, Friedman (1979), DeCanio (1979), and Pesaran (1987). The conclusion of this
One of the reasons is that (i) information is costly to get, and (ii) is at least partially a public good. Agents that possess information can, by their actions in the market place, unwittingly
First, if the conditional expectation of the price level based on the model (Et_ iPt ) were considerably better at forecasting Pt than the subjective expectation of suppliers (PO, there would be an
In the previous section we have postulated the REH in the form of a statement like(3.5). Muth (1961) offers an intuitive defence for the equality of conditional and subjective expectations.
Chapter 3: Rational Expectations and Economic Policy expected and actual price levels have been drawn for the same stochastic Ut terms as before. Not surprisingly, there is a clear pattern in the way
Obviously, from (3.10) it is clear that under the REH, Pt = P = 1. This is the dashed line in Figure 3.4. The actual price level under the REH is given by (3.11), and is drawn as a solid line
The Foundation of Modern Macroeconomics Ut of the model. The parameters of demand and supply were set at ao = 3, al = 1, bo = 1, and b1 = 1, which implies that the deterministic equilibrium price is
The issue can be illustrated with the aid of Figures 3.4 and 3.5, which show the paths of the price level and the expectational errors that are made under, respectively, the REH and the AEH. The
Equation (3.13) shows that the equilibrium price Pt under the AEH displays a clearly recognizable pattern, because Pt depends on its own lagged value Pt- i and the error term displays autocorrelation.
What would have been the case under the AEH? Obviously, under AEH, the expectational errors do display a predictable pattern. Recall (from (1.14)) that the AEH says that the expected price level can
supply shock, for example due to an agricultural disaster, the price level rises.

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