Advanced Macroeconomics An Easy Guide (New) Keynesian theories of fluctuations A primer

CHAPTER 15 ( New ) theories of A primer thinking starts from a different viewpoint , at least compared with that of the approach , regarding the functioning of markets . In this perspective , output and employment cate that labour markets , good markets , or both , are not working , leading to unnecessary ment . The idea is that , at least in some circumstances , the economy is ( rather than ) so that the challenge is to increase expenditure , If that could be done , then supply will respond automatically . This is Keynes Principle of Effective Demand ) As a result , models focus on aggregate demand management as opposed to policies . Later on in the book we will discuss specifically the role of and monetary policy in aggregate demand , but in this chapter we need to understand the framework under which this aggregate demand matters . Of course there is a lot of controversy among economists as to how is it possible that a situation where markets fail to clear may persist over time . Why is there unemployment ?

Can unemployment be involuntary ?

If it is involuntary , why don people offer to work for less ?

Why are prices rigid ?

Why can adjust their prices ?

How essential is price in comparison with distortions on the labour market ?

And , in this setup , do consumers satisfy their budget constraints ?

These are difficult questions that have led to a large amount of literature trying to develop models with features in a equilibrium framework with rational expectations . This line of work that has been dubbed New , emerged as a reaction to the challenge posed by the New Classical approach . Over time , and as New Classical thinking evolved into the approach , the literature coalesced around the models with the New literature ing on these models while adding to them one or many market imperfections . In any event , this is a very broad expanse of literature that we will not be able to review here , We will thus focus on three steps . First , we will revisit the standard model . This model captures the essence ( or so most economists think ) of the approach , by imposing the assumption of price , which gives rise to the possibility of aggregate demand management . This simple approach , however , begs the question of what could explain those . Our second step therefore will be to provide a brief discussion of possible for them . There may be many reasons for why a nominal price adjustment is incomplete client relationships , staggered price adjustment , contracts , asymmetric information , menu costs , etc . Not all of How to cite this book chapter , and , A . 2021 . Advanced An Easy Guide . 15 . New ) theories A primer , London Press . DOI License .

220 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER these lead to aggregate price , but we will not get into these details here . We will instead focus on a model where asymmetric information is the reason for incomplete nominal adjustments . This model highlights , in particular , the role of expectations in determining the reach of aggregate demand management ?

Last but not least , we will see how these combine to give rise to the modern New models , which reinterpret the insights in a rather different guise , and constitute the basis of most of policy analysis these days . We include at the end of the book an appendix that takes you through the first steps needed so that you can run a model ofyour own ! With at least some analytical framework that makes sense of the paradigm and its interpretation , in later chapters we will discuss the mechanisms and policy levers for demand management , with an emphasis on monetary policy and policy . We revisit the basic Version of the model that should be familiar from undergraduate the model . In 1937 , provided a theoretical framework that can be used to analyse the General ory Keynes had published the previous year . Keynes book had been relatively hard to crack , so the profession embraced Hicks simple representation that later became known as the , model and went on to populate intermediate macro textbooks ever since ( Hicks won the Nobel Prize in in 1972 for this work ) While much maligned in many quarters ( particularly because of its static nature and lack of ) this simple model ( and its cousin , the model ) is still very much in the heads of policy makers . The model is a general equilibrium framework encompassing three markets goods , money and bonds , though only two are usually described as the third will clear automatically if the other two do ( remember Law from micro ! It is standard to represent the model in terms of interest rates and output , and to look at the equilibrium in the money and goods market . The corresponding equations are A money market equilibrium locus called the curve ( i , and a goods market equilibrium called the IS curve ) A , Fiscal , where Fiscal stands for government expenditures and for the real exchange rate , or , alternatively , Fiscal , Finally , a relationship between nominal and real interest rates .

( NEW ) THEORIES OF FLUCTUATIONS A PRIMER Classical version of the model In the classical version of the model , all prices are , and so are real wages Thus the labour market clears the amount of labour as in Figure . With full employment of labor and capital , output is determined by the supply constraint and becomes an exogenous variable , which we will indicate with a bar ( i ) The IS , can then be used to determine so that savings equals investment ( I , thus the name of the curve ) The nominal interest rate is just the real rate plus exogenous expectations ( equivalent to the expected growth rate of prices ) With and i , then the determines the price level given a stock of nominal money , i , which is an alternative way of writing the quantity equation of money . In short , the structure of the model is such that the labor market determines the real wage and output . The IS determines the real and nominal interest rate , and the money market determines the price level . This is typically interpreted as a description of the long run , the situation to which the at any given moment The idea is that prices eventually adjust so that supply ends up determining output That is why we ignored aggregate demand when discussing growth . There we concentrated on the evolution of the supply capacity In the classical version of the model ( or in the long run ) that supply capacity determines what is produced . Figure The classical model

222 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER The version of the model However , as Keynes famously , in the long run , we will all be dead . In the short run , the assumption is that prices are or rigid , and do not move to equate supply and demand , so now the IS and curves jointly determine i as in Figure . Notice that Yis determined without referring to the labour market , so the level of labour demand may generate involuntary unemployment . It is typical in this model to think of the effects of monetary and policy by shifting the IS and curves , and you have seen many such examples in your intermediate macro courses . If you don quite remember it , you may want to get a quick refresher from any undergraduate macro textbook you prefer . We will later show how we can think more carefully about these policies , in a dynamic , context . An interpretation The Fed Is the model useful ?

Yes , because policy makers use it . For example , when the Fed talks about ing or contracting the economy it clearly has a framework in mind . It is true that the Fed does not typically operate on the money stock , but one way of thinking about how the Fed behaves is to think of it as determining the interest rate and then adjusting the money supply to the chosen rate ( money becomes somewhat endogenous to the interest rate ) In our model , i becomes exogenous and endogenous as in Figure ( A , Fiscal , Figure The model

( NEW ) THEORIES OF FLUCTUATIONS A PRIMER 223 Figure The model with an exogenous interest rate , IS ?

A mam ' I IS I We can represent using the same , i space , but the curve is now horizontal as the Fed sets the ( nominal ) interest rate . Alternatively , we can think about it in the , space , since is the new endogenous variable . Here we would have the same old curve , but now the IS curve becomes vertical in the , space . Both represent the same idea if the Fed wants to expand output , it reduces the interest rate , and this requires an expansion in the quantity of money . As a side note , you may have heard of the possibility of a liquidity trap , or alternatively , that policy may hit the zero interest lower bound . What does this mean ?

We can think about it as a situation in which interest rates are so low that the demand for money is elastic to the interest rate . In other words , because nominal interest rates can not be negative ( after all , the nominal return on cash is set at zero ) when they reach a very low point an increase in the supply of money will be hoarded as cash , as opposed to leading to a greater demand for goods and services . In that case , the framework tells us that ( conventional ) monetary policy is ineffective , Simply put , interest rates can not be pushed below zero ! This opens up a series of policy debates . There are two big questions that are associated with this ) Is monetary policy really ineffective in such a point ?

It is true that interest rate policy has lost its effectiveness by hitting the zero boundary , but that doesn necessarily mean that the demand for money is elastic . The Fed can still pump money into the economy ( what came to be known as quantitative easing ) by purchasing government ( and increasingly private ) bonds , and this might still have an effect ( maybe through expectations ) In this scenario , can policy be effective ?

These are debates we come back to in full force in our discussions of and monetary policy . From to Another way to understand the assumption on price rigidity in generating a role for aggregate demand management is to go from the representation to one in which is one of the endogenous variables . The curve implies that an increase in prices leads to a decrease in the supply of real money balances , which shifts to the left . Since IS is not affected , that means that a higher leads to a lower level of output . This is the aggregate demand ( AD ) curve in Figure . An increase in aggregate demand ( through monetary or policy ) will shift the AD curve to the right . The effect that this will have on equilibrium output will depend on the effect of this on prices , which in turn depends on the aggregate supply ( AS ) of goods and services . The classical case is one in

224 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER Figure model , which this supply is independent of the price level a vertical AS curve . This is the case where prices are fully . The case we considered , in contrast , is one in which AS is horizontal ( is fixed ) and hence the shift in AD corresponds fully to an increase in output . This is an economy that is not supply constrained . In the intermediate case , where prices adjust but not completely , AS is positively sloped , and shifts in aggregate demand will have at least a partial effect on output . The AS curve is the mirror image of the Phillips curve the empirical observation of a tradeoff between and , We assume you are familiar with the concept from your intermediate macro courses , and we will get back to that when we discuss the modern New approach . of incomplete nominal adjustment We go over a possible explanation for the incomplete adjustment of prices or , more broadly , for why the AS curve may be . We study the Lucas model of imperfect information , which illustrates how we can solve models with rational expectations . We have now reestablished the idea that , if prices do not adjust automatically , aggregate demand management can affect output and employment . The big question is , what lies behind their ure to adjust ?

Or , to put it in terms of the framework , we need to understand why the AS curve is positively sloped , and not vertical , Old arguments were built on things such as ( adaptive ) expectations , money illusion , and the like . This in turn rubbed more economists the wrong way . How can rational individuals behave in such a way ?

Their discomfort gained traction when the Phillips curve tradeoff seemed to break down empirically in the late and early . The model is also useful from a methodological point of view it shows how a rational expectations model is solved . In other words , it shows how we use the model to compute the expectations that are an essential piece of the model itself Lets see !

( NEW ) THEORIES OF FLUCTUATIONS A PRIMER 225 The Lucas island model The challenge to orthodoxy , and hence the initial push from which modern were built , took the shape of the pioneering model by Lucas ( 1973 ) part of his contribution . He derived a AS curve in a model founded at the individual level , and where individuals had rational expectations . This model also more explicit the role of expectations in constraining aggregate demand policy . The key idea was that of imperfect information individuals can observe quite accurately the prices of the goods they produce or consume most often , but they not really observe the aggregate price level . This means that , when confronted with a higher demand for the good they produce , they are not quite sure whether that an increase in its relative price a case in which they should respond by increasing their output or simply a general increase in prices a case in which they should not respond with quantities , but just adjust prices , We will see that rational expectations implies that individuals should split the difference and attribute at least part of the increase to relative prices . How much so will depend on how often general price increases occur ) This yields the celebrated Lucas supply curve , a supply curve in which output increases when the price increases in excess of its expected level . The model is one with many agents ( Lucas original places each person on a ent island , which is why the model is often referred to as the Lucas island model ) Each agent is a that every period sees a certain level of demand . The basic question is to figure out if an increase in demand is an increase in real demand , which requires an increase in tion levels , or if it is simply an increase in nominal demand , to which the optimal response is just an increase in prices , The tension between these two alternatives is what will give power to the model . In order to solve the model we will start with a with perfect information and , once this benchmark case is solved , we will move to the case of asymmetric information , which is where all the interesting action is . The model with perfect information The representative producer of good i has production function , so that her feasible consumption is , Utility depends ( positively ) on consumption and ( negatively ) on labour effort . Let assume the , If is known ( perfect information ) the problem is easy the agent has to maximize her utility ( 1513 ) with respect to her supply of the good ( which is , at the same time , her supply of labour ) Replacing ( and ( 1512 ) in ( 1513 ) gives ,

226 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER The first order condition for is , 1515 ) which can be written as a labour supply curve ( 1516 ) or , if expressed in logs ( denoted in lower case letters ) as , 1517 ) As expected , supply ( production ) increases with the relative price of the good . Next , we need to think about demand for every good i , and about aggregate demand . The former takes a very simple form . It can be derived from basic utility but no need to do so here as the form is very intuitive . Demand depends on income , relative prices , and a taste shock in log format it can be written as ' where is average income and is the average price level . The taste shock , is assumed to affect relative tastes , hence it averages to zero across all goods . It is also assumed to be normally distributed , for reasons that will soon be clear , with variance . How about aggregate demand ?

We will assume that there is an aggregate demand shifter , a policy variable we can control , which in this case will be It can be anything that shifts the AD curve within the framework developed above , but to ideas we can think about monetary policy To introduce , it consider a money demand function in log form . We assume that is also normally distributed , with mean ( and variance . Equilibrium To find the equilibrium we make demand equal to supply for each good . This is a model with market clearing and where all variables , particularly , are known . from which we obtain the individual price ( pi ( and from which we can obtain the average price . Averaging ( we get ( 1522 ) which implies

( NEW ) THEORIES OF FLUCTUATIONS A PRIMER 227 You may it strange , but it is not remember that output is in logs . Replacing the solution for output in ( 15119 ) we get that , that prices respond fully to monetary shocks . In other words , the world with perfect information is a typical classical version with prices and where aggregate demand management has no effect on real variables and full impact on nominal variables . Lucas supply curve When there is imperfect information , each producer observes the price of her own good , but can not observe perfectly what happens to other prices . She will have to make her best guess as to whether a change in her price represents an increase in relative prices , or just a general increase in the price level . In other words , labour supply will have to be determined on the basis of expectations . Because we assume rational expectations , these will be determined by the mathematical expectation that is consistent with the model in other words , individuals know the model and form their expectations rationally based on this knowledge . Denote relative prices as ( then the analog to ( is now , It so happens that if the distribution of the shocks and is jointly normal , then so will be , and , Since . and pi are jointly normally distributed , a result from statistics tells us that the conditional expectation is a linear function ( More , in this case , we have what is called a signal extraction problem , in which one variable of interest ( is observed with noise . What you observe ( is the sum of the signal you re interested in ( ri ) plus noise you don really care about ( It turns out that , with the assumption of normality , the solution to this problem is ( where , and are the variances of relative price and general price level , respectively . They are a complicated function of and . This expression is very intuitive if most of the variance comes from the signal , your best guess is that a change in , indicates a change in relative prices . Substituting in ( yields , over all the individual supply curves , and i ( we have that ( which is actually a Phillips curve , as you know from basic macro courses .

228 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER This became known as the Lucas supply curve . Note that this is a supply curve , in which output increases when the price increases in excess of its expected level . Why is it so ?

Because when facing such an increase , imperfectly informed producers rationally attribute some of that to an increase in relative prices , It also says that labour and output respond more to price changes if the relative relevance of nominal shocks is smaller , Why is this ?

Because the smaller the incidence of nominal shocks , the more certain is the producer that any price shock she faces is a change in real demand . Solving the model We know from the framework that , with a supply curve , aggregate demand shocks affect equilibrium output . How do we see that in the context of this model ?

Plugging ( 1529 ) into the aggregate demand equation ( yields ( 1530 ) that can be used to solve for the aggregate price level and income 17 15 ( lam la ( 1532 ) Now , rational expectations means that individuals will figure this out in setting their own expectations . In other words , we can take the expectations of ( 1531 ) to ( mE ( 1533 ) which implies , in turn , that ( 1534 ) Using this and the fact that ( we have that ( In short , the model predicts that changes in aggregate demand ( monetary policy ) will have an effect on output , but only to the extent that they are unexpected . This is a very powerful conclusion in the sense that systematic policy will eventually lose its effects people will it out , and come to expect it . When they do , they change their behaviour accordingly , and you won be able to exploit it . This is at the heart of the famous Lucas critique as the policy maker acts , the aggregate supply curve will change as a result of that , and you can think of them as stable relationships independent of policy . As we can see , the imperfect information approach highlights the role of expectations in mining the effectiveness of macro policy . This insight is very general , and lies behind a lot of modern policy making targeting as a way of coordinating expectations , the problem of time , etc . In fact , we will soon see that this insight is very much underscored by the modern New approach .

( NEW ) THEORIES OF FLUCTUATIONS A PRIMER 229 Imperfect competition and nominal and real We show that , with imperfect competition and nominal , there is a role for aggregate demand policy Imperfect competition means that can set prices , and that output can deviate from the social optimum . Nominal mean that prices fail to adjust automatically . The two combined mean that output can be increased ( in the short run ) and that doing so can be desirable . We discuss how real amplify the impact of nominal . The Lucas model was seen at the time as a major strike against policy thinking . After all , while it illustrates how we can obtain a AS curve from a fully approach with rational agents , it also fails to provide a for systematic macro policy The New sian tradition emerged essentially as an attempt to reconcile rational expectations and the possibility and desirability of systematic policy Consider the desirability in the Lucas model ( or the approach that essentially came out of that tradition ) business cycles are the result of optimal responses by individuals to disturbances that occur in this economy , and aggregate demand policy can only introduce noise . If the market is socially optimal , then any due to aggregate demand shocks is a departure from the optimum , and thus undesirable . The New view departs from that by casting imperfect petition in a central role . The key to justifying policy intervention is to consider the possibility that the level of output is suboptimal , and imperfect competition yields exactly that . In addition , this is consistent with the general impression that are bad and booms are good . Besides the issue of desirability , we have argued that the Lucas model also implies that systematic policy is powerless rational agents with rational expectations it out , and start adjusting prices accordingly The second essential foundation of New thinking is thus the existence and importance of barriers to price adjustment . Note that this is also related to imperfect competition since price adjustment can only matter if are , which requires some monopoly power . It is not enough to have imperfect competition to have these , however , as will also want to adjust prices rather than output in response to nominal shocks . We thus have to understand how barriers , that are most likely rather small at the micro level , and which have become known in the literature by the term menu costs , can still have large effects . Do we really think that in the real world the costs of adjusting prices are large enough to lead to sizeable consequences in output ?

It turns out that the key lies once again with imperfect competition . Consider the of a decrease in aggregate demand on the behaviour of monopolist , illustrated in Figure . ing the behaviour of all other as given , this will make any given want to set a lower price . If there were no costs of adjustment , the would go from point A in Figure to point If the doesn adjust at all , it would go to point . It follows that its gain from adjusting would be the shaded triangle . If the menu cost is greater than that , the would choose not to adjust . But what is the social cost of not adjusting ?

It is the difference in consumer surplus corresponding to a decrease in quantity from point to point . This is given by the area between the demand curve and the marginal cost curve , between and This is much bigger than the shaded triangle ! In other words , the social loss is much bigger than the loss from not adjusting , and it follows that small menu costs can have large social Another type of rigidity emphasised by New are real ( as distinct from the nominal kind ) These correspond to a low sensitivity of the desired price to aggregate output . If the

230 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER Figure Welfare of imperfect competition Price Quantity desired price doesn change much with a change in output , the incentive to adjust prices will be lower . Think about the slope of the marginal cost curve in Figure 155 ) If there are no costs of adjustment ( nominal ) that doesn matter , of course but the real amplify the effect of the nominal ones . These real could come from many sources , such as the labour market . If labour supply is relatively inelastic ( think about low levels of labour mobility , for instance ) we would have greater real . This actually sets the stage for us to consider our next topic in the study of cyclical labour markets and unemployment . In sum , a combination of imperfect competition , nominal ( menu costs ) and real ties implies that aggregate demand policy is both desirable and feasible . We will now turn to a very brief discussion of how this view of the world has been embedded into dynamic tic general equilibrium ( models such as those introduced by the tradition to give birth to the modern New view of New models We express the modern New ( model in its canonical ( Version , combining the New IS curve , the New Phillips curve , and a policy rule . We show the and Versions ofthe model . New models embody the methodological consensus underpinning modern . It has become impossible to work in any Central Bank , for instance , out coming across a New model . But modern , state models are very complicated . If you thought that models were already intricate , consider the celebrated model by and ( 2003 ) originally developed as an empirical model of the Euro area . It contains , in addition to productivity shocks , shocks to adjustment costs , the equity premium ,

( NEW ) THEORIES OF FLUCTUATIONS A PRIMER wage markup , goods markup , labor supply , preferences , and the list goes on and on , Another is that there is little consensus as to which model is best to . So what we do here is consider a few of the key ingredients in models , and explain how they combine into what is often called the canonical New model . The canonical New model We develop the model in continuous time , which is simpler and allows for the use of phase grams , so that we can readily put the model to work and develop some intuition about its operation and dynamics . Later , we turn to discrete time , and write down the version of the model that is most commonly used in practical and policy applications . The demand side of the canonical New model is very simple . We start from our model of consumer , which by now we have seen many times . You will recall the equation of the representative consumer . where , is consumption , is the elasticity of substitution in consumption , and is the rate of time discounting , In a closed economy with no investment , all output , is consumed . Therefore , and , i , 1539 ) where we have used the , i , and ' is the nominal interest rate , taken to be exogenous and constant for the time being . If we the output gap as , where , is the natural or long run level of output , then the output gap evolves according to ( 15 41 ) where is the percentage growth rate of the natural level of output , assumed constant for now . Finally , letting letters denote , using the equation ( we have , i , 1542 ) where is the natural or interest rate , which depends on both preferences and productivity growth . It is the interest rate that would prevail in the absence of distortions , and corresponds to a situation in which output is equal to potential . This last equation , which we can think of as a dynamic New IS equation ( or ) the demand side of the model . The equation says that output is rising when the real interest rate is above its ( or natural ) level . Contrast this with the conventional IS equation , which says that the level of output ( as opposed to the rate of change of output in the equation above ) is above its level when the real interest is below its ( or neutral ) level .

232 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER The differs from traditional IS in other important ways . First , it is derived from founded , household behaviour . Second , the relationship between interest rates and output emerges from the behaviour of consumption , rather than investment , as was the case in the old IS . Intuitively , high interest rates are linked to low output now because people decide that it is better to postpone consumption , thereby reducing aggregate demand , Turn now to the supply side of the model . We need a description of how prices are set in order to capture the presence of nominal . There are many different models for that , which are typically as or . models are those in which adjustment is triggered by the state of the economy Typically , decide to adjust ( and pay the menu cost ) if their current prices are too far from their optimal desired level . models , in contrast , are such that get to adjust prices with the passage of time , say , because there are term contracts . This seems slightly less compelling as a way of understanding the underpinnings of price adjustment , but it has the major advantage of being easier to handle . We will thus focus on dependent models , which are more widely used , There are several models , but the most popular is the model . 1983 ) assumes that the economy is populated by a continuum of . Each of them is a point in the , interval , thus making their total equal to one . The key innovation comes from the technology each sets its output price in terms of domestic currency and can change it only when it receives a signal . The probability of receiving such a signal periods from now is assumed to be independent of the last time the got the signal , and given by , at . If the signal is stochastically independent across , we can appeal to the law of large numbers to conclude that a share of will receive the signal per unit of time . By the same principle , of the total number of that set their price at time , a share ' will not have received the signal at time Therefore , ae ( is the share of that set their prices at time and have not yet received a signal at time I . Next , let , be the ( log of the ) price set by an individual ( when it gets the signal ) and the ( log of the ) price level , as the arithmetic average of all the prices , still outstanding at time , weighted by the share of with the same , aj ' 1545 ) It follows that the price level is sticky , because it is a combination of prices ( which , because they are , can not jump suddenly ) How is , set ?

Yun ( 1996 ) was the to solve the full problem of that must set prices , understanding that it and all competitors will face stochastic signals . Getting to that solution is involved , and requires quite a bit of algebra ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER 233 Here we just provide a reduced form , and postulate that the optimal price , set by an individual depends on the contemporaneous price level , the expected future paths of the ( log of ) expected relative prices , and of the ( log of ) the output gap co , where , recall , is the consumers discount rate and is a sensitivity parameter ?

So the relative price the chooses today depends on a discounted , average of all future tive prices ( and all output gaps , This is intuitive . For instance , if the output gap is expected to be positive in the future , then it makes sense for the to set a higher ( relative ) price for its good to take advantage of buoyant demand , Note from this expression that along any path in which the future and are continuous of time ( which we now assume ) is also , and necessarily , a continuous function of time . We can therefore use rule to differentiate the expressions for , and , with respect to time , obtaining ) pi a ( and , Combining the two we have , girl . Differentiating the expression for the rate , again with respect to time , yields ir , Finally , combining the last two expressions we arrive at it , pit , where 01211 , This is the canonical New Phillips curve , In the traditional Phillips curve , the rate of was an increasing function of the output gap . By contrast , in the the change in the rate of is a decreasing function of the output gap ! Notice , also that while , is a sticky variable , its rate of change It , is not it is intuitive that Ir , should be able to jump in response to expected changes in relevant variables , Solving this equation forward we obtain 00 II , I So the rate today is the present discounted value of all the future expected output gaps , The more overheated the economy is expected to be in the future , the higher is today To complete the supply side of the model we need to specify why the output gap should by anything other than zero that is , why can and are willing to supply more output than their maximizing level . The standard story , adopted , for instance , by Yun ( 1996 ) has two components . Output is produced using labour and can hire more ( elastically supplied ) labour in the run to enlarge production when desirable . When demand rises ( recall the previous section of this

234 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER chapter ) facing prices will it advantageous to supply more output , up to a point . This curve and the dynamic curve , taken together , fully describe this model economy . They are a pair of linear differential equations in two variables , and xi , with i i as an exogenous policy variable . In this model there is no between keeping low and stabilising output . If i , then xi is an equilibrium . and Gali ( 2007 ) term this the divine coincidence . The steady state is 72 ( from xi ) 1554 ) from ) 1555 ) where denote the steady state . If , in addition , we assume i , then ?

In matrix form , the dynamic system is xi , ii . i , 1556 ) 1557 ) It is straightforward to see that Det ( and ( It follows that one of the of is positive ( or has positive real parts ) and the other is negative . This means the system exhibits saddle path stability , in other words that for each there is a value of xi from which the system will converge asymptotically to the steady state . But remember that here both and it are jump variables ! This means that we have a continuum of convergent equilibria , because we can initially choose both , and xi . The graphical representation of this result is as follows . When drawn in , xi space , the Phillips curve is , while the IS schedule is horizontal , as you can see in the phase diagram in Figure . If , there exists a no ?

such that both variables converge to the steady state in a trajectory . The converse happens if . Along a converging path , and output do move together , as in the standard Phillips curve analysis . To see that , focus for instance on the quadrant of the diagram . There , both output and are below their long run levels , so that a depressed economy produces unusually low . As output rises toward its run resting point , so does . But the important point is that there exists an infinity of such converging paths , one for each ( initial condition ! An exogenous path for the nominal interest , whichever path that may be , is not enough to pin down the rate of ( and the output gap ) uniquely . What is the intuition for this indeterminacy or ?

To see why may occur , suppose agents believe that output that is low today will gradually rise towards steady state . According to the , New Phillips curve , a path of low output implies a path of low . But with the nal interest rate exogenously , low expected increases the real rate of interest and lowers consumption and output . The initial belief is thus . where A Taylor rule in the canonical New model In Chapter 19 we further discuss interest rate policy and interest rate rules . Here we simply introduce the and used rule the Taylor rule , named after Stanford economist John

( NEW ) THEORIES OF FLUCTUATIONS A PRIMER 235 Figure Indeterminacy in the model Taylor , who first proposed it as a description of the behaviour of monetary policy in the US . In Taylor ( 1993 ) the rule takes the form i , where 42 , and ( tax are two coefficients chosen by the monetary authority The choice of If requires it be equal to the normal or natural real rate of interest in the steady state . In what follows we will often assume , so that when , rises above the ( implicit ) target of , the nominal interest rises more than proportionately , and the real interest goes up in an effort to reduce Similarly , so that when the output gap is positive , rises from its normal level Using the Taylor rule in the equation ( yields , so that the rate of increase of the output gap is increasing in its own level and also increasing in ( because ) The resulting dynamic system can be written as ( Now Det ( and ( It follows that ) lis sufficient to ensure that both of are positive ( or have positive real parts ) Because both at and , are jump variables , the steady state is now unique . After any permanent unanticipated shock , the system just jumps to the steady state and remains there ! As you can see in the phase diagram in Figure , the , schedule ( the ) now slopes down . All four sets of arrows point away from the steady state point which is exactly what you need to guarantee uniqueness of equilibrium in the case of a system of two jumpy variables ! where

236 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER Figure Active interest rule in the model ! Go back to the expression Det ( which reveals that if ) and is not too large , then Det ( Since , in addition , we would have a case of one positive and one negative eigenvalue , so that , again , multiplicity of equilibria ( in fact , of equilibria ) would occur . So there is an important policy lesson in all of this , In the canonical New model , interest rate policy has to be sufficiently activist ( aggressively , one might say ) in order to uniqueness of equilibrium in particular , to ensure that the rate of and the output gap are pinned down , In the literature , policy rules where ( 12 , are usually called active policy rules , and those where 42 , are referred to as passive policy rules , In this very simple model , which boils down to a system of linear differential equations , the ness result is simple to derive and easy to understand , In more complex versions of the model , instance , in models which need to be around the steady state or in circumstances in which the zero lower bound on the nominal interest rate binds , dynamics can be considerably more complicated , and the condition 42 , in the Taylor rule need not be to guarantee uniqueness of equilibrium , For a more detailed treatment of these issues , see et al . et al . et al . 2002 ) 2011 ) and Gali ( 2015 ) Before ending this section , let us put this model to work by analysing a shock . Let imagine a monetary tightening implemented through a transitory exogenous increase in the interest rate ( think of the interest moving to If , with the policy shifter ) or , alternatively , imagine that at time , the natural rate of interest suddenly goes down from to , where , because the trend rate of growth of output , has temporarily dropped . After , either goes back to zero , or the natural rate of interest goes back to and remains there forever . How will and output behave on impact , and in the time interval between times and ?

The phase diagram in Figure below shows the answer to these questions in a simple and intuitive manner . Notice that either of these changes imply a leftward shift of the equation . So , during the transition the dynamics are driven by the original it and new equations which intersect at point .

( NEW ) THEORIES OF FLUCTUATIONS A PRIMER 237 Figure A reduction in the natural rate The system must return to A exactly at time On impact , and the output jump to a point , such as , which is pinned down by the requirement that between and dynamics be those of the system with steady state at point That is , during tie temporary shock the economy must travel to the , with rising and the output gap narrowing , in anticipation of the positive reversion of the shock at Between and the negative shock , intuitively enough , causes output and to be below their initial ( and , after ) levels . If initially , so ?

and , as drawn below , then during the duration of the shock the economy experiences and a recession ( a negative output gap ) What happens to the nominal interest rate ?

Between and , both and the output gap are below their target levels of zero So , the monetary authority relaxes the policy stance in response to both the lower and the negative output gap . But that relaxation is not enough to keep the economy from going into recession and . We return to this issue in Chapter 22 Back to discrete time The canonical New model has a natural counterpart in discrete time , which is more broadly used for practical applications In discrete time the Phillips curve becomes ( see Gali ( 2015 ) for the detailed derivation ) II , where i is the discount factor , is the expectations operator ( with expectations computed as of time ) and the output gap is again in logs To derive the IS curve , again start from the equation , which in logs can be written as ,

238 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER where we have already used , Ifwe recall the fact that for small , and , log ( and log ( equation ( becomes ' Finally , subtracting from both sides yields , where we have used the i . and the fact that . If the natural rate of output is not constant , so that ) A , 1565 ) becomes , A ( i , or , i , where , and again the variable is the natural , or , interest rate , which can move around as a result of preference shocks ( changes in ) or productivity growth ( A ) To close the model , we can again appeal to an interest rule of the form it in . 1570 ) As before , policy makers in charge of interest rate setting respond to deviations in expected from the target ( here equal to zero ) and to deviations of output from the full employment or natural rate of output . Taylor argued that this rule ( with ) and an target of is a good description of how monetary policy actually works in many countries , in particular , of how the Fed has behaved in modern times ( since the ) There is an active research program in trying to compute optimal Taylor rules and also to estimate them from data . In practice , no central bank has formally committed exactly to such a rule , but the analysis of monetary policy has converged onto some variant of a Taylor rule on interest rate rules more broadly as the best way to describe how central banks operate . Substituting the interest rate rule into the equation ( in the simple case of constant ) yields , 11 ) i ) This equation plus the constitute a system of two difference equations in two unknowns . As in the case of continuous time , it can be shown that an interest rule that keeps ' constant does not guarantee uniqueness of equilibrium . But , it again turns out that if , a rule does ensure that both of the characteristic matrix of the system are larger than one . Since both Ir , and are jumpy variables , that guarantees a unique outcome the system simply jumps to the steady state and stays there . To analyse formally the dynamic properties of this system , rewrite the ( as ,

( NEW ) THEORIES OF FLUCTUATIONS A PRIMER 239 Next , use this ( to yield , 12 , i ) and ( together constitute the canonical New model in discrete time . In matrix form , the dynamic system is , where ( 11 , i ) 1575 ) Now Det ( and ( where A , and A , are the . For both 11 and to be larger than one , a necessary and sufficient condition is that Det ( which , using the expressions for the determinant and the trace , is equivalent to ( This condition clearly obtains if ( So , the policy implication is the same in both the ous time and the discrete time of the model an activist policy rule is required , in which the interest rate to changes in the ( expected ) rate of , in order to ensure uniqueness of . For a classic application of this model to monetary policy issues , see et al . 1999 ) In later chapters of this book we use the model to study a number of issues , some of which require the to the acronym shocks ! As we saw in our discussion of models , the business cycle properties we obtain from the model will depend on the properties we assume for those shocks The kind of model that is used in practice will add many bells and whistles to the cal version , in the description of the behaviour of firms , households , and . In doing so , it will open the door to a lot of different shocks as well . It will then try to either calibrate the model parameters , or estimate them using Bayesian techniques , and use the model to evaluate policy . You will in Appendix a basic illustration of this kind of model , so you can do it for yourself ! These models will nevertheless keep the essential features of a firm grounding on household and firm , which is a way to address the Lucas Critique , and also of the consequent importance of expectations . We discuss the issues in greater detail in Chapter 17 and thereafter .

240 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER What have we learned ?

We have gone over the basics of the view of the business cycle , from its old version to the modern canonical New model . We saw the key role of imperfect price ment , leading to an aggregate supply curve , under which aggregate demand shocks have real consequences . We showed how imperfect competition and nominal ( and real ) are crucial for that . We saw how the equation of consumption gives rise to the modern New sian IS curve , while the model of price setting gives rise to the New Phillips curve . Finally , we saw how we need to specify a policy rule ( such as the Taylor rule ) to close the model . There is no consensus among as to whether the or classical ( view is correct , This is not surprising since they essentially involve very different world views in terms of the functioning of markets . Are market failures ( at least relatively ) pervasive , or can we safely leave them aside in our analysis ?

This is hardly the type of question that can be easily settled by the type of evidence we deal with in the social sciences . Having said that , it important to stress the methodological convergence that has been achieved in , and that has hopefully been conveyed by our discussion in the last two chapters . Nowadays , essentially all of macro deals with models with rational agents , the ence being in the assumptions about the shocks and that are present ( or absent ) and driving the . By providing a framework that allows policy makers to cater the model to what they believe are the constraints they face , means that the controversy about the fundamental discrepancies can be dealt , in a more way within a framework . Imagine the issue of price rigidity , which is by of price adjustment , If you believe in no price , It has a value , if you think there are you just change the value . And nobody is going to for the value of , are they ?

Worst case scenario , you just run it with both parameters and look at the output . No wonder then that the models have become a workhorse , for example , in Central Banking , What next ?

Any number of macro textbooks cover the basics of the model , in its version . The textbook by ( 2018 ) covers the topics at the graduate level , and is a great introduction to the fundamentals behind the New view For the canonical , modern New approach , the book by Gali ( 2015 ) is the key reference . Notes This is whats behind Keynes ( and misquoted ) statement ( from . 16 of the General Theory ) that To dig holes in the ground , paid for out of savings , will increase , not only employment , but the real national dividend of useful goods and ( Note , however that he immediately goes on to say that it is not reasonable , however , that a sensible community should be content to remain dependent on such fortuitous and often wasteful mitigation when once we understand the upon which effective demand ) Similarly , in . 10 he states If the Treasury were to old bottles with banknotes , bury them at suitable depths in disused which are then up to the surface with town rubbish , and leave it to private enterprise on

( NEW ) THEORIES OF FLUCTUATIONS A PRIMER principles of to dig the notes up again ( the right to do so being obtained , of course , by tendering for leases of the territory ) there need be no more unemployment and , with the help of the repercussions , the real income of the community , and its capital wealth also , would probably become a good deal greater than it actually is . It would , indeed , be more sensible to build houses and the like but if there are political and practical difficulties in the way of this , the above would be better than As we will see , this is not exactly a model it was actually the opening shot in the expectations revolution . The New approach , however , is the direct descendant of that revolution , by incorporating the rational expectations assumption and championing the role of aggregate demand policy under those conditions . See Hicks ( 1937 ) This is a simplifying assumption of certainty equivalence behaviour . Note that this uses the law of expectations , which states that ( you can not be systematically wrong in your guess . The mathematical intuition is as follows because the is in point A , the derivative of its income with respect to price is set at zero , and any gain from changing prices , from the perspective , will be of second order . But point A does not correspond to a social optimum , because of imperfect competition , and that means that the effects of a change in prices on social welfare will be of first order . Somewhat confusingly , people often refer to the modern New view of and to models as synonyms . However , it is pretty obvious that models are dynamic , stochastic , and too ! We prefer to keep the concepts separate , so we will always refer to New models , See et al . appendix , for a full derivation in continuous time . Implicit in this equation is the assumption that discount future at the household rate of discount , rule ?

Why , of course , you recall it from calculus that how you differentiate an integral . If you need a refresher , here it is take a function ( Ia ' the derivative of respect to is ( a ( Intuitively , there are three components of the , marginal impact of changing on those of increasing the upper and lower limits of the integral ( which are given by evaluated at those limits ) and that of changing the function at every point between those limits ( which is given by ) All the other stuff is what you get from your chain rule . Whatever i was initially , in drawing the below we assume the intercept does not change in response to the shock that is , it does not fall as the natural interest rate drops temporarily References , Monetary policy and multiple equilibria . ican Economic Review , 91 ( The perils of Taylor rules , Theory , 96 ( 2002 ) Avoiding liquidity traps , 110 (

242 ( NEW ) THEORIES OF FLUCTUATIONS A PRIMER , I . 2007 ) Real wage and the New model . journal of Money , Credit and Banking , 39 , A . 1983 ) Staggered prices in a framework . Journal of Monetary , 12 ( Gali , 1999 ) The science of monetary policy A New tive . Journal Literature , 37 ( Gali , I . 2015 ) and the business cycle An introduction to the New framework and its applications . Princeton University Press . Hicks , 1937 ) Keynes and the classics A suggested interpretation . Lucas , 1973 ) Some international evidence on . The American Economic Review , 63 ( 2018 ) Advanced . Hill . 2003 ) An estimated dynamic stochastic general equilibrium model of the Euro area . ofthe European Economic Association , Taylor , I . 1993 ) Discretion versus policy rules in practice . Conference Series on Public Policy ( Vol . 39 , 2011 ) Interest and prices Foundations ofa theory policy . Princeton Press . Yun , 1996 ) Nominal price rigidity , money supply , and business cycles . journal of Monetary Economics , 37 (