Modelo Novo Keynesiano de Mankiw-Weinzierl - Macroeconomia EC2B1

Lectures 7 8 New Keynesian Model MankiwWeinzierl version Macroeconomics EC2B1 Benjamin Moll These notes discuss one version of the New Keynesian model namely the one outlined by Mankiw and Weinzierl 2011 We will first cover this New Keynesian model because as you will see it ties in quite nicely with our microfounded approach to macroeconomics I will then give a history lesson on the old Keynesian model that some of you have seen before and which does not have any microfoundations and relies on a graphical analysis The reason for proceeding in this order is that I expect you will find the New Keynesian model easier to understand 1 Real Model see Lecture Notes 5 The MankiwWeinzierl version of the New Keynesian model builds on the general equilibrium model of investment and capital accumulation covered in lecture notes 5 see section 2 of the model writeup As a reminder the economy has the following primitives Preferences households have utility function UC1 βUC2 with UC C1 1 σ 1 1 1 σ 1 Technology firms have production function Yt AtKt t 1 2 and capital accumulates according to K2 I1 1 dK1 with d 1 ie K2 I1 Resource constraints feasibility goods in period 1 C1 I1 Y1 goods in period 2 C2 Y2 1 In the competitive equilibrium households maximize utility 1 subject to the budget constraint C₁ C₂ 1 r₁ W 2 and firms maximize profits W maxₖ₂ A₁K₁ K₂ A₂K₂ 1 r₁ 3 As we have shown in lecture 5 consumption satisfies an Euler equation C₁ 1β1r₁σ 1 r₁ 1 1β1r₁σ 1 r₁ W C₂ 1 r₁ 1 1β1r₁σ 1 r₁ W 4 and the competitive equilibrium allocation is C₁ 1βA₂σ A₂ 1 1βA₂σ A₂ A₁K₁ C₂ A₂ 1 1βA₂σ A₂ A₁K₁ I₁ 1 1 1βA₂σ A₁K₁ Y₁ A₁K₁ Y₂ A₂ 1 1βA₂σ A₁K₁ 5 This model is a baby twoperiod version of Real Business Cycle RBC model As we have discussed this model has the feature that the welfare theorems hold and so there is no room for macroeconomic stabilization policy monetary and fiscal policy The New Keynesian model adds nominal rigidities in the form of sticky prices to this model As you will see adding nominal rigidities changes the models behavior and policy implications quite dramatically 2 Introducing Money and Inflation 21 Price Level and Inflation So far we have ignored nominal factors and the whole model was in real terms In particular we simply set the price of final goods in both period P1 P2 1 This was without loss of generality because we simply expressed all prices in the model above the interest rate r1 in units of the consumption good ie saving one unit of apples more generally consumption goods today pays back 1 r1 units of apples tomorrow In order to talk meaningfully about inflation we reintroduce nominal prices For instance we write the household budget constraint in nominal terms dollars as P1C1 P2C2 1 i1 P1Π1 P2Π2 1 i1 6 where i1 is the nominal interest rate interest rate in terms of dollars rather than apples Similarly the firms problem becomes Ω max K2 P1Y1 K2 P2 1 i1 A2K2 7 For reasons that will become clear momentarily we also introduce period t 0 a preperiod in which a reference price P0 is determined The idea is that this reference price P0 is determined some time before period 1 before households and firms know economic conditions like A1 A2 etc Definition The inflation rate in this economy is π2 P2 P1 P1 π1 P1 P0 P0 Note that the main role of preperiod 0 and the reference price P0 is to enable us to define shortrun inflation π1 the other role is to motivate price stickiness in Section 4 below Definition The real interest rate is the nominal interest rate adjusted for inflation 1 r1 1 i1 P2P1 1 i1 1 π2 8 The relation between real and nominal interest rates can be written in a simpler fashion using the approximation that r1π2 is negligible if both r1 and π2 are small this is a very commonly 3 used approximation you may have come across in other courses 1 i1 1 r11 π2 i1 r1 π2 9 This equation is known as the Fisher equation after economist Irving Fisher 22 Money Supply and Money Demand A satisfactory treatment of nominal variables variables in terms of dollars also requires an analysis of money demand and money supply demand and supply of dollars We simply assume that the governmentcentral bank sets the money supply More on this in section 6 below To discuss briefly the issue of money demand we introduce the concept of velocity which you have already encountered in EC1B1 Definition Velocity which we denote by Vt is the average number of times a piece of money turns over in a year So consider an economy with nominal GDP ie the total amount of purchases in terms of dollars equal to PtYt Velocity answers the question How large is the required stock of money in the economy Suppose money can be used only once ie velocity is Vt 1 Then clearly the required stock of money is Mt PtYt Suppose money can be used twice Vt 2 Then Mt PtYt2 More generally if money can be used Vt times MtVt PtYt 10 This is equation is known as the quantity equation and you have already seen it in EC1B11 There are different theories of velocity or equivalently money demand Quantity theory see EC1B1 Vt V fixed Other theories of money demand covered in EC1B1 for example the one Lecture 8 Mon etary Policy in Modern Economies in which velocity is given by log Vt φit vt where φ is a parameter it is the nominal interest rate and vt is a money demand shock BaumolTobin model2 Consider a consumer who spends PC dollars on consumption each year Denote by T the amount of time in fractions of year between trips to the bank 1Arguably a more useful way of thinking about this equation is just as an accounting identity that defines Vt That is suppose you observe an economy with nominal GDP PtYt and stock of money Mt Then you conclude that velocity is Vt PtYtMt 2The description here is purposely brief and only sketches the model If you want to read more on this you dont have to good references are chapter 104 of Kurlat and Chapter 4 of DoepkeLehnertSellgren available here httpsfacultywcasnorthwesternedumdo738bookhtm 4 Then 1T is the number of trips per year If the consumer spaces out his consumption expenditure equally over the year then he spends PCT dollars between trips to the bank If we assume that his money holdings decline linearly mt PcTt at a point 0 t T then his average money holdings over this period are M PCT²2T PCT2³ In this theory the velocity is Vt 2T This follows from setting Yt C market clearing without capital Mt M and the quantity equation 10 In the BaumolTobin model T is then a choice variable that depends on the costs of holding money such as the interest rate Cashinadvance CIA models This model simply assumes that purchasing goods requires some cashonhand which has to be put aside in advance The CIA constraint is PtCt Mt Hence with Yt Ct velocity is assumed to be Vt 1 MankiwWeinzierl restrict themselves to the simplest such theory the quantity theory with constant velocity They use slightly different notation In particular they write the quantity equation top of p7 as MₜV PₜCₜ V 1φ In their theory Mt is only the currency used for buying consumption goods Ct as opposed to all output Yt Ct It Since V is constant they simply redefine Mₜ MₜV Mₜφ and write Mt PₜCₜ 11 they also argue that the cost of holding money is negligible which implies that a this can be ignored when writing budget constraints the more conventional way would be to build this in as in a CIA model and b that velocity V 1φ is large 3 Flexible Prices Monetary Neutrality We are interested in examining the equilibrium of the economy in particular the role played by money and inflation We first consider the case with flexible final goods prices Later we will examine the case with sticky prices which is the defining assumption of a New Keynesian Model ³More formally using that mt PcT t average money holdings are M 1T ₀ᵀ mtdt 1T PC T² T²2 PCT2 Definition Neutrality of Money means that a change in monetary variables like nominal interest rates or the stock of money affects only nominal variables in the economy such as prices and nominal wages but has no effect on real inflationadjusted variables like employment real GDP and real consumption The purpose of this section is to argue that with flexible prices monetary neutrality holds in the above economy More concretely this will simply mean that real variables C₁C₂I₁Y₁Y₂ do not depend on policy tools of the central bank that is the money supply M₂ or the nominal interest rate i₁ Only the nominal variables in particular P₁ and P₂ do In contrast when we examine sticky prices below monetary neutrality will not hold anymore As already noted the problem of a firm is 7 The problem of a household is to maximize utility 1 subject to 6 The key to showing monetary neutrality is to realize that we can simply rewrite the problems of firms and households in real terms As a result everything will be exactly as in the real model in section 1 To this end consider for instance the budget constraint of the household 6 Dividing through by the price level in period one P₁ we have C₁ P₂C₂ 1 i₁P₁ Π₁ P₂Π₂ 1 i₁P₁ Using the definition of the real interest rate 8 we can immediately see that this constraint is simply the budget constraint of the household in real terms 2 Similarly defining the firms value in real terms as W ΩP₁ the firms problem can be written as a maximization over real variables 3 With this insight all real variables are found in the exact same way as above In particular the equilibrium real variables are as in 5 It is easy to see that monetary neutrality holds in this economy that is monetary policy instruments money supply interest rates do not affect these real variables Instead the nominal variables are given by Claim The nominal variables in this economy are P₁ 1 1βA₂σ A₂ A₁K₁ M₂ 1 i₁ P₂ 1 1βA₂σ A₂ A₂A₁K₁ M₂ M₁ 1βA₂σ A₂ M₂ 1 i₁ 12 Derivation The Euler equation in terms of nominal variables is UC1βUC2 1 i1P2P1 C2C1 β1 i1σ P1P2σ 13 Using 11 we can eliminate prices and obtain C2C11σ β1 i1σ M1M2σ M1 1β C2C11σ 1 M21 i1 1β βA21σ M21 i1 where the second equality follows because the Euler equation 13 can still be written in terms of real variables as C2C1 β1 r1σ βA2σ Rearranging this is the last equation in 12 The expression for P1 then follows from P1 M1C1 and the expression for C1 in 5 Similarly the expression for P2 follows from P2 M2C2 and the expression for C2 in 5 Note from 12 that the nominal price levels P1 and P2 are proportional to the money supply M2 which means that an increase in the money supply of 10 will increase price levels by 10 A decrease in the interest rate i1 increases the price level P1 while not affecting P2 which means lower longrun inflation 1 π2 P2P1 1 i1A2 or deflation In summary printing money is inflationary and interest rates are deflationary in the longrun There is a simple and direct intuition for why interest rate cuts are deflationary that uses the Fisher equation 9 ie i1 r1 π2 Monetary neutrality is equivalent to the real interest rate not changing in response to monetary policy in the model above the real interest rate is always just 1 r1 A2 In that case the Fisher equation can simply be written as π2 i1 r1 If the central bank lowers nominal interest rates i1 it must be that inflation π2 falls so as to keep the real interest rate constant In the blogosphere this idea is sometimes discussed under the name Fisherism or NeoFisherism As we just discussed this model has the feature that real variables in 5 are determined completely independently of the nominal variables in 12 Therefore real and nominal variables could be analyzed separately This property is called classical dichotomy 4 Sticky Prices We will now change one single assumption in the above model namely that prices are perfectly flexible in the short run period one We will argue that this changes results dramatically But first lets discuss why we think that prices may be sticky 41 Stickiness in Reality and Economic Modeling I like the following quote While I regard the evidence for such stickiness as overwhelming the assumption of at least temporarily rigid nominal prices is one of those things that works beautifully in practice but very badly in theory Paul Krugman see httpswebmitedu krugmanwwwislmhtml Possible reasons for stickiness in reality Menu costs the assumption is that a firm has to pay a fixed cost whenever it wants to change prices eg a restaurant has to print a new menu Hence the firm only changes its price when the payoff from so doing is large enough to justify paying the fixed cost Rational inattention the assumption is that acquiring information is costly and that therefore firms optimally choose to not pay attention to whats going on all the time in particular not to whats happening to monetary policy Both of these stories have similarly realistic implications but are quite hard to model In particular the models are too complicated to embed in simple representative agent dynamic models like the one here In practice therefore sticky prices are often modeled with a technique called Calvopricing The assumption is that with an exogenous probability say α a firm may change its price but with probability 1 α it is forced to keep its previous price This assumption is obviously not satisfactory and it is often ridiculed the Calvofairy that tips you on the shoulder and says now youre allowed to change your price The Calvo model is an example of whats called timedependent sticky prices This is in contrast to state dependent models such as the menu cost and inattention models explained above Time dependent models are considered to be less satisfactory by most economists With this discussion in mind we will make an even more simplistic and extreme assumption than Calvo pricing the price level in period one P1 is completely fixed and cannot be changed It equals the reference price from preperiod t 0 ie P1 P0 The key friction is that the price at t 1 is set before households and firms know the economic conditions at t 1 and hence it cannot respond to any changes to these economic conditions as it would with flexible prices This assumption will break monetary neutrality In contrast the price level in period two P2 is flexible 42 Equilibrium with Sticky Prices First note that a model with a completely fixed price level is necessarily a model in which market clearing doesnt hold for one or more markets In our case it is the market for final 8 goods in period one that is not necessarily in equilibrium and in particular demand for goods which we denote by Y1D can be smaller than the supply of goods Y1S A1 K1 To see this suppose we start out in a case where the final goods market in period one is in equilibrium Y1D Y1S Then a negative shock hits that decreases demand Y1D for instance we will show below that a negative shock to future A2 has that feature We will show below that mot surprisingly demand Y1D is a downwardsloping function of P1 If P1 were flexible the price would adjust downwards so as to equate demand and supply again But if P1 is sticky or as here completely fixed this is not possible and there will have to be an excess supply of final goods in period one Y1D Y1S We therefore replace the market clearing condition with the inequality Y1 A1 K14 In fact the case where the inequality is strict will be the interesting case in which there is a role for government policy In this sense this version of the New Keynesian model is a model of disequilibrium Claim The equilibrium with sticky prices is given by cf MankiwWeinzierl equations 3136 with g2 0 C1 1βA2σ A2 M21 i1 P1 C2 A2 M21 i1 P1 I1 M21 i1 P1 Y1 1 1βA2σ A2 M21 i1 P1 Y2 A2 M21 i1 P1 1 r1 A2 P2 1 i1A2 P1 14 Derivation Consumption still satisfies 4 However now we cannot use the market clearing condition Y1 A1 K1 anymore because it may not hold This implies that the real value of the firm is W Y1 A1 K1 In the derivation of 5 in Lecture notes 5 we substituted W A1 K1 into 4 This step now breaks down because now W Y1 but we do not know the value of Y1 Instead we proceed as follows First note that from the firms first order condition the 4We can never have the case Y1 A1 K1 because equilibrium quantities are always determined by the short side of the market that is Y1 minY1D Y1S ie Y1 Y1D if Y1D Y1S and Y1 Y1S if Y1S Y1D real interest rate is still given by 1 r1 A2 see the comment in the next paragraph on this somewhat surprising result Hence because i1 is chosen by the monetary policy authority and P1 is fixed P2 is determined from the definition of the real interest rate 8 Rearranging gives the last equation in 14 Consumption in period two C2 is then immediately pinned down from equilibrium in the money market P2 C2 M2 Rearranging gives the expression for C2 Next use the equation for C2 and C2 A2 K2 to get the expression for I1 K2 Finally we can find the expression for C1 from the Euler equation C2C1 β1 r1σ βA2σ C1 1βA2σ C2 and the expression for C2 Therefore we have expressions for C1 and I1 and hence Y1 C1 I1 1βA2σ A2 1 M21 i1 P1 Comment on equilibrium real interest rate and investment As can be seen from the last equation in 14 the equation for P2 the equilibrium real interest rate 1r1 1i1P1P2 satisfies 1 r1 A2 This looks like monetary policy does not affect real interest rates just like in the model with flexible prices in which this was a result of monetary neutrality This result is somewhat surprising given that we made a big deal out of monetary neutrality not holding with sticky prices What is going on In a nutshell this result is a bit of an artefact of a particular assumption we made in order to get nice analytical formulae namely that production is linear and features constant returns to capital Y2 A2 K2 Intuitively this assumption makes investment demand perfectly elastic as a function of the real interest rate and results in investment responding very strongly which pushes up period 2 output Y2 C2 and pushes down the price level P2 until the real interest rate equals 1 r2 A2 With the more realistic but analytically less tractable assumption of diminishing returns to capital monetary policy instead does affect the real interest rate r1 as you may have expected The result that the equilibrium real interest rate is unaffected by monetary policy is therefore due to a special assumption made purely for tractability and should not be taken too seriously5 To make the point that this result is special to the constant returns assumption change 5Also note that the seeming zero passthrough from i1 to r1 is an equilibrium outcome The right way to think about it is that there definitely is passthrough from the former to the latter and that monetary policy is very much nonneutral In fact the whole issue is that it is too nonneutral in the sense that investment demand is infinitely elastic and responds extremely strongly to changes in monetary policy and this is what pushes the equilibrium real rate to adjust to equal 1 r1 A2 the assumption that production in the second period is Y₂ FK₂ A₂K₂α with 0 α 1 Fortunately one does not have to resolve the whole model to make this point instead one can separately analyze one block of the model namely the determining the real interest rate r₁ investment I₁ and secondperiod price level P₂ We will show below that these become 1 r₁ αα A₂ 1 i₁P₁ M₂1α 15 K₂ I₁ α M₂ 1 i₁P₁ 16 P₂ 1 αα A₂ 1 i₁P₁α M₂1α 17 As expected when α 1 the real interest rate equals 1 r₁ A₂ again and investment I₁ and the price level P₂ boil down to the expressions in 14 More importantly whenever α 1 ie there are diminishing returns to capital the real interest rate r₁ in 15 now depends on monetary policy i₁ and M₂ In particular a nominal rate cut a decrease in i₁ results in a decrease in r₁ The same is true when the central bank prints more money ie M₂ increases Note that with flexible prices the same would not be true ie monetary neutrality still holds even when α 1 Lets derive expressions 15 to 17 To pin down r₁ I₁ K₂ and P₂ it is sufficient to work with the following three equations i the firms optimality condition for investment ii the quantity equation in period 2 and iii the Fisher equation α A₂ K₂α1 1 r₁ M₂ P₂ C₂ P₂ A₂ K₂α 1 r₁ 1 i₁ P₂ P₁ Combining the first and third equations we have α A₂ K₂α1 1 i₁ P₁ P₂ Substituting for P₂ from the second equation we have an equation for K₂ α A₂ K₂α1 1 i₁ P₁ M₂ A₂ K₂α 6 The derivation of this equation is as above and in Lecture 5 firms choose K₂ to maximize profits Ω maxK₂ P₁Y₁ K₂ P₂ 1 i₁ A₂ K₂α which is the same problem as 7 but with the new production function AK₂α in place of AK₂ The corresponding optimality condition condition is α A₂ K₂α1 1 r₁ with 1 r₁ 1 i₁ P₂ P₁ Rearranging gives 16 Substituting back into the first equation gives 15 Finally using the third equation yields 17 5 Policy in the New Keynesian Model Sticky prices introduce a friction that implies that the welfare theorems break down Therefore unlike in the RBC model of section 1 there is a role for policy Policy can broadly be split into two categories 1 Monetary policy central bank sets interest rate or money supply 2 Fiscal policy government spending tax cuts stimulus checks cash for clunkers trade in old car for new more fuelefficient car and get cash Below we will consider a recession and what policy is called for We will show that in normal times monetary policy is sufficient to correct the friction implied by sticky prices However in abnormal times fiscal policy may be useful stimulus package 6 Monetary Policy 61 What Is the Central Banks Policy Instrument The Money Supply or the Interest Rate Before we continue a brief detour seems appropriate In the above model we have assumed that the policy maker central bank sets the nominal interest rate in period 1 i1 but the money supply in period 2 M2 look at equations 14 and you will see the ratio of the two enters When the media report on changes in central bank policy they often just say that the central bank has raised or lowered interest rates So why dont we follow this strategy here in both periods The short answer is that because this is a two period model there is only one interest rate namely the one between periods one and two i1 So we cannot choose i2 as a policy instrument Standard New Keynesian models are modeled as infinite horizon economies which implies that this problem doesnt occur theres always a tomorrow and the policy instrument is simply the sequence it t0 More generally though it should be noted that the two are really equivalent Showing the equivalence of i1 M2 and M1 M2 as policy instruments in the present framework will be part of the next problem set You should also read 12 the blog post by Mankiw on the question What Is the Feds Policy Instrument The Money Supply or the Interest Rate here httpgregmankiwblogspotcom200605islmmodelhtml Williamson Ch12 on how monetary policy is conducted in practice In particular you should know the concepts of open market operations and helicopter drops 62 A Recession and the Response of Monetary Policy We will now consider the effect of a recession as above triggered by a decrease in future productivity A₂ First compare the effects of this drop in the flexible price and sticky price models Assume that the IES is σ 1 Mankiw and Weinzierls canonical case We have Flexible prices C₁ C₂ I₁ Y₁ Y₂ Sticky prices C₁ C₂ Ī₁ Y₁ Y₂ the last by more than in the flexible price case So the recession is worse with sticky prices and therefore there is a role for policy The optimal monetary policy is simply the one that undoes all the distortions induced by price stickiness thereby equating the equilibrium allocations under sticky prices 14 to that under flexible prices 5 Most directly this policy is the one that delivers full employment Y₁ A₁ K₁ in 14 or M₂ 1 i₁ P₁ 1 1 1 β A₂σ A₁ K₁ 18 Under the assumption σ 1 it can be seen that the optimal policy response to a drop in A₂ is to decrease shortterm interest rates i₁ or to increase future money supply M₂ Again this policy undoes all the distortions due to sticky prices Therefore there is no need for fiscal policy which we havent even introduced into the model yet Note that this policy only works because prices P₁ are sticky If instead prices were flexible the increase in aggregate demand due to an increase in M₂ or decrease in i₁ would be immediately offset by an increase in P₁ You can verify this from the first equation in 12 63 Feasibility of Monetary Policy Zero Lower Bound One important question is whether the optimal policy above is always feasible One constraint on policy is the socalled zero lower bound Definition The zero lower bound ZLB is the requirement that nominal interest rates cannot be negative i₁ 0 The reason for this constraint is that money is an alternative asset to bonds Money always pays an interest rate of zero Therefore if the interest rate on bonds would go negative no one would lend If they did they would get less money back than they lent and they would be better off putting their money in their mattress As we will see if interest rates are at the zero lower bound monetary policy will be impotent Because of this feature the zero lower bound is sometimes referred to as a liquidity trap see for instance the paper by Krugman 1998 Its Baaack Japans Slump and the Return of the Liquidity Trap However the term zero lower bound is more descriptive so we will use that one In order to analyze the effect the ZLB has on the economy some more notation is useful In particular we denote prerecession variables with a ˆ For instance productivity drops from Â₂ to A₂ Â₂ We further assume that the prerecession economy features full employment that is Ŷ₁ A₁ K₁ no hat needed on K₁ because it is fixed For now also that the money supply in period two is fixed M₂ M₂ We can therefore write the relative frop in output as Y₁ Ŷ₁ 1 1 β A₂σ 1 1 β Â₂σ A₂ Â₂ 1 î₁ 1 i₁ where we use that prices are fixed in the shortrun P₁ P₁ and M₂ M₂ Since Ŷ₁ A₁ K₁ Y₁ 1 1 β A₂σ 1 1 β Â₂σ A₂ Â₂ 1 î₁ 1 i₁ A₁ K₁ This can be written in terms of the nominal interest rate as 1 i₁ 1 1 β A₂σ 1 1 β Â₂σ A₂ Â₂ 1 î₁ A₁ K₁ Y₁ The interest rate that delivers Y₁ A₁ K₁ is then 1 i₁ 1 1 β A₂σ 1 1 β Â₂σ A₂ Â₂ 1 î₁ We need 1 i₁ 0 Therefore it can be seen that the ZLB is more likely to bind if the prerecession interest rate î₁ is already close to zero Furthermore the RHS of this equation is an increasing function of A₂ Hence there is a cutoff A₂conventional such that i₁ 0 if and only if A2 A2conventional That is productivity cannot drop too far otherwise the ZLB binds For instance consider the case where the prerecession interest rate is already at zero ˆi1 0 In that case a productivity drop A2 ˆA2 always implies that the interest rate that delivers full employment Y1 A1K1 is negative Therefore monetary policy has no power 64 Fiscal vs Monetary Policy We have shown that if the ZLB binds monetary policy looses its power For now keep the assumption that M2 is fixed and ask can fiscal policy undo the distortions due to sticky prices We introduce fiscal policy in the exact same way as MankiwWeinzierl that is we denote government spending by Gt lumpsum taxes by Tt and assume that the government has a present value budget constraint P1G1 P2G2 1 i1 P1T1 P2T2 1 i1 The resource constraints are now C1 I1 G1 Y1 C2 G2 Y2 19 and the household budget constraint is P1C1 P2C2 1 i1 P1Π1 T1 P2Π2 T2 1 i1 We will also denote by gt GtAtKt the share of government purchases in full employment Claim the equilibrium with sticky prices satisfies these are equations 31 to 36 in the 15 paper C₁ 1βA₂σ A₂ M₂1i₁P₁ C₂ A₂ M₂1i₁P₁ I₁ 11g₂ M₂1i₁P₁ Y₁ 1 1βA₂σ A₂1g₂1g₂ M₂1i₁P₁ G₁ Y₂ A₂ 11g₂ M₂1i₁P₁ P₂ 1i₁A₂ P₁ 20 Derivation The expression for C₁ and C₂ are derived in the same was as in the derivation of 14 Output in period 2 Y₂ can then be found from 19 which becomes C₂ Y₂1g₂ or Y₂ C₂1g₂ Similarly the expression for investment follows because I₁ K₂ Y₂A₂ Finally output in period 1 is found from substituting the expressions for I₁ and C₁ into 19 The equilibrium has the feature that consumption C₁ and C₂ are not affected by changes in government spending As MankiwWeinzierl put it the governmentspending multiplier here is precisely one Here as in that model an increase in government spending puts idle resources to work and raises income Consumers meanwhile see their income rise but recognize that their taxes will rise by the same amount to finance that new higher level of government spending As a result consumption and investment are unchanged and the increase in income precisely equals the increase in government spending Therefore there is room for government intervention By increasing G₁ the government can achieve full employment Y₁ A₁K₁ However note that the shortfall in private consumption C₁ I₁ A₁K₁ is made up by public consumption ie G₁ fills the gap C₁ I₁ G₁ A₁K₁ That is fiscal policy restores the firstbest level of GDP Y₁ but not consumption C₁ Therefore this is a secondbest policy Summarizing in normal times when the ZLB doesnt bind monetary policy is sufficient to restore the flexible price equilibrium In abnormal times when the ZLB binds there is a role for fiscal policy