Advanced Macroeconomics An Easy Guide Real business cycles

CHAPTER 14 Real business cycles So far we have mostly talked about dynamics , the process of capital accumulation , generational issues , etc . However , a lot of focuses on the short term the departures from the trend that we ve been mostly concerned about . This is , of course , particularly dent in recession times ! Some of the biggest questions in revolve around this how can we understand and the , cyclical evolution of the economy ?

What can we ( or should we ) do about ?

These are obviously important questions , and they are very much at the heart of the development of as a discipline , as we discussed in the first chapter of the book . In fact , business cycles is where the distinction between schools of thought became more evident giving credence to the idea that economists never agree with each other . Many of your policy will derive from which view of the world you have , Essentially , one school is ingrained in the perspective where there is scope for on the cycle and that doing so is . Its modern version is the New approach originated in the 19805 in response to the empirical and methodological challenges from the 19705 , The second approach is quite skeptical about what policy can or should do , as it views the cycle as the result of optimal adjustments to real shocks . Its modern version was born , also in the , with the Real Business Cycle ( framework , which argued that a perfectly competitive economy , with no distortions or aggregate imbalances of the type , but subject to shocks , could largely replicate the frequency data for economies . Recent years have seen a great deal of methodological convergence , with both views adopting , to a large extent , the dynamic stochastic general equilibrium ( framework that essentially implements the with whatever exogenous shocks and market imperfections that you may feel are relevant , Because this model can be to work as a perfectly competitive economy , or as one with more characteristics , it has become the new workhorse of . This has allowed for a more conversation in recent decades . In light of that , and because we have covered much of the ground when we studied the , we will start by describing the framework , which started the trend that turned the into the workhorse of modern . This framework , from a theory standpoint , is , conceptually , a simple extension of the to a context with stochastic shocks . How to cite this book chapter , and , A . 2021 . Advanced An Easy Guide . 14 . Real business cycles , London Press . DOI License .

206 REAL BUSINESS CYCLES Yet we will see that it has a sharp message . In particular , it delivers the ultimate statement to the extent that business cycles are optimal responses to productivity shocks , policy are worse than ineffective they are bad , because they deviate the economy from its poral optimum . From this benchmark , we will then turn to the theories , The basic model The basic model is simply the ( in discrete time ) with two additions stochastic shocks to productivity ( to generate in output ) and a labour supply choice ( to generate in employment ) Fluctuations come from optimal individual responses to the stochastic shocks . The basic model , first introduced by and Prescott ( 1982 ) is built around a typical framework of , There are three differences with what we ve seen so far in the book . First , we introduce uncertainty in the form of exogenous productivity shocks , without which ( as we ve seen ) no emerge . Second , we also introduce a choice of how much labour will be supplied in other words , there is a choice . This is what will enable us to say something about in employment . Finally , models typically use discrete time . This is so because the objective is to compare the simulated data from the model with that of the real data , which is always discrete , and also because the models quickly become too complicated for analytical solutions . One has to resort to numerical methods of solution , and computers can more easily handle discrete data . The consumers problem works as follows ( subject to the household budget constraint in which individuals own the capital stock and labour endowment , and rent those out to the firms , Cy , the production function , the labour endowment equation , and a productivity shock process , is consumption , indicates leisure , is the rate of return on capital ( net of depreciation ) is the capital stock , I , is the amount of labour devoted to market activities Finally , is a productivity parameter which is subject to random shocks , The rest are parameters which should be relatively

REAL BUSINESS CYCLES 207 The importance of labour supply As we ve pointed out , one of the literature main departures from the standard is the presence of a labour supply choice . This is crucial to generate in employment , which are a pervasive feature of actual business cycles , Let us consider this choice within the context of the basic model . With log utility , the consumers objective function can be thought of as , Notice that the household has two control variables , consumption and leisure . We have seen before the solution to the problem of optimal allocation of consumption it is the familiar equation ( Leaving aside uncertainty , for the moment , and using the log assumption , we can rewrite this as , The choice , in contrast , is static it takes place in each period with no implication for the next period . The equates the marginal of additional consumption to the marginal cost of lost leisure ( I ( Again using the log utility assumption , we get ( To simplify things further , assume for the moment that is exogenous think of a small open economy . In this setup , we can use ( into ( 147 ) to obtain hi , WA ( This means that leisure will be relatively high in those periods when the wage is relatively low in other words , a higher wage increases the supply of labour . We can also see the impact of the interest rate a high interest rate will lead to a higher supply of labour . The intuition is that it is worth working more in the present the higher the interest rate , as it provides a higher return in terms of future leisure . These responses of the labour supply , driven by substitution in labour and leisure , are at the very heart of the in employment in models . The labour supply Let pause for a minute to explore a bit deeper the shape of this labour supply curve . Consider the case when wages and income are constant , a case that would be akin to analysing the effect of permanent shocks to these variables . Lets see the form of the labour supply in this case .

208 REAL BUSINESS CYCLES Since consumption is constant at income level ( substituting this into ( we obtain I I ) 1412 ) Using the log for consumption and allowing for ( allows us to simplify ) hi hi ( This is a strong result that says that leisure is independent of the wage level . You may think this is too strong but , surprisingly , it fits the data very well , at least when thinking about labour supply in the very , very long run ?

It does seem that for humans income and substitution just cancel out ( or maybe you prefer a more setup in which people just cant work more than eight hours per day , or where the disutility of labour beyond eight hours a day becomes unbearable if repeated every day ) Does this mean that labour supply does not move at all ?

Not really The above was derived under the assumption of the constancy of the wage . This is akin to assuming that any change in wages is permanent , which induces a very large response in the shadow value of consumption that works to offset the labour supply effect of the change in wages ( totally cancelling it in the log case ) But if the wage moves for a very short period of time we can assume the shadow value of consumption to remain constant , and then changes in the wage will elicit a labour supply response . Thus , while the elasticity of labour supply may be zero , it is positive in the short run . The basic mechanics In its essence , the story goes as follows consider a positive productivity shock that hits the , making it more productive . As a result of that shock , wages ( and interest rates ( go up , and individuals want to work more as a result . Because of that , output goes up . It follows that the elasticity of labour supply ( and the closely related elasticity of substitution ) are crucial parameters for models . One can only obtain large in employment , as needed to match the data , if this elasticity is sufficiently high . What is the elasticity of labour supply in this ( 148 ) as implying a labour supply curve ( a relation between and , basic model ?

Consider the case when , in which consumption is a constant . We can read ( Am , 1415 ) REAL BUSINESS CYCLES 209 where A is the ( constant ) marginal utility of consumption Let assume a slightly more general , form for the utility of leisure ( hi ( plugging this in ( gives ?

Aw ) 01 Aw ) which can be used to compute the labour supply Aw , This equation has a labour supply elasticity in the short run equal to If we assume that ( logarithmic utility in leisure ) and that and A are such that ( think about an workday ) this gives you em . This doesnt seem to be enough to replicate the employment observed in the data . On the other hand , it seems to be quite high if compared to micro data on the elasticity of labour supply Do you think a decrease of 10 in real wages ( because of , for instance ) would lead people to work 20 fewer hours ?

The indivisible labour solution The model thus delivers an elasticity of labour supply that is much higher than what micro suggests , posing a challenge when it comes to matching in employment . One proposed solution for the conundrum is to incorporate the fact that labour decisions are often indivisible . This means that people may not make adjustments so much on the intensive margin of how many hours to work in your job , but more often on the extensive margin of whether to work at all . This implies that the aggregate elasticity is large even when the individual elasticity is small . Hansen ( 1985 ) models that by assuming that there are costs of going to work . This can actually make labour supply very responsive for a range of wage levels . The decision variables are both days of work 11 13 , and , then , the hours of work each day We assume there is a commuting cost in terms of utility , which you pay if you decide to work on that day , regardless of how many hours you work ( this would be a sort of commuting time )

210 REAL BUSINESS CYCLES The objective function is now ( where we leave aside the term ) to simplify notation , and abuse notation to have ( be a function of hours worked , rather than leisure , entering negatively in the utility function . The budget constraint is affected in that now wage income is equal to , It is easy to see that we have the same , which is unchanged because the terms in consumption in both and budget constraint are still the same , and ( because the term in , is multiplied by , in both and budget constraint , so that , cancels out . What changes is that now we have an extra with respect to ( Assume , so that , is constant Then ( to ( which gives the optimal amount of hours worked ( when the agent decides to work ) Then ( 14122 ) to ( If ( then ( otherwise This gives rise to a labour supply as shown in Figure The important point is that this labour supply curve is infinitely elastic at a certain wage . The intuition is that on the margin at which people decide whether to work at all or not , the labour supply will be very sensitive to changes in wages ?

Figure The Hansen labour supply WA ( elastic segment REAL BUSINESS CYCLES ! I model at work models typically can not be solved analytically , and require numerical methods . We discuss an example ofa calibration approach to assess the success of the model in describing actual economic . Having discussed the basic intuition behind generating output and employment from real shocks to the economy , let us now talk a little bit more generally about how models are usually handled . The main challenge is that even simple are impossible to solve analytically , so the alternative is to use numerical methods . How is this done ?

In a nutshell , the strategy is to solve for the of the model which , in addition to the equations determining the nature of the stochastic shocks , will describe the dynamic path of the variables of interest . This will often imply a step in which the are around the balanced growth path , since it is easiest to analyse the properties of a linear system . We then need to provide numbers for the computer to work with this is done by calibrating the parameters . ber what we have discussed of calibration when talking about growth empirics this approach was pretty much pioneered by the literature . Because the model is calibrated on the basis of that are brought from outside the model , the procedure provides somewhat of an independent test of the relevance of the model . With this in hand , the model is simulated , typically by considering how the variables of interest responds after being exposed to a stochastic draw of exogenous productivity shocks . The results are then compared to the data . In both cases , the simulated and the real data , we work with the business cycle component , the data . This is usually done using the , which is a statistical procedure to out the trend component of a time series . What output of the model is then compared to the data ?

Mostly second moments variances and of the variables of interest . A model is considered successful if it matches lots of those empirical moments , Calibration An example Let us consider the basic model , and the calibration proposed by Prescott ( 1986 ) which is the actual of this approach and where Prescott tackles the issue of assigning parameters to the of the model . For example , at the time , he took as good a capital share of To estimate the production function , he starts with a we ve used repeatedly ( Remember that the interest rate has to equal the marginal product of capital , which means that we have an equation for the return on capital 11 . Now lets put numbers to this . What is a reasonable rate of depreciation ?

Let use ( itself to it out . If we assume that the rate of depreciation is 10 per year ( becomes REAL BUSINESS CYCLES ?

This value for the capital output ratio is considered reasonable , so the 10 rate of depreciation seems to be a reasonable guess . How about the discount factor ?

It is assumed equal to the interest rate . This is not as restrictive as it may seem , but we can skip that for now . This implies a yearly discount rate of about ( the real interest rate ) so that ( again , per year ) As for the elasticity of substitution , he argues that is a good approximation , and uses the share of leisure equal to , as we had anticipated ( this gives a labour allocation of half , which is reasonable if we consider that possible working hours are 16 per day ) Finally , the productivity shock process is derived from ( as cussed in Chapter when we talked about growth accounting ) which , in the case of the at the time , yielded , This is a highly persistent process , in which shocks have very effects . The calibration for the standard deviation of the disturbance is . So , endowed with all these parameters , we can pour them into the and run the model over time in fact , multiple times , with different random draws for the productivity shock . This will give a time series for the economy in the theoretical model . We will now see how the properties of this economy compare to those of the real economy . Let start with some basic results taken directly from Prescott paper . Figure shows log and its trend . The trend is computed as a ( of this as a smoothed , but not fixed , line tracing the data ) It is not a great way to compute the business cycle ( particularly at the edges of the data set ) but one that has become quite popular . Once the trend is computed , the cycle is easily estimated as the difference between the two and is showing in figure . Figure also shows the variation over the cycle in hours wor ed . As you can see , there is a large positive correlation between the two . Real business cycle papers will typically include a table with the properties of the economy , stood as the volatility of the variables and their over time . Table and show this from Prescott original paper for both the real data and the ca model . As you can see , things work surprisingly well in the sense that most characteristics of the economy match . The volatility of output and the relative volatility of and investment appear to be the optimal response to the supply shocks . The only caveat is that hours do not seem to move as much as in the data . This is why Prescott implemented Hansen extension . Figure shows how labour and output move in the Hansen economy ( they seem to match better the pattern in Figure ) The Appendix to this chapter ( at the end of the book ) will walk you through an actual example so that you learn to numerically solve and simulate an el yourself !

REAL BUSINESS CYCLES Figure The output Actual and Trend Logs of . Gross National Product I Log Actual ' 19471950 1960 1970 19801982 Source of basic data data bank Figure The cycle in the Deviations From Trend of Gross National Product and Nonfarm Employee Hours in the United States Quarterly , Hours , 19471950 1960 1970 19801982 Source data data bank

REAL BUSINESS CYCLES Table The data for the cycle , from Prescott ( 1986 ) Cyclical Behavior of the Economy Deviations From Trend of Key Variables . Cross Correlation of With Standard Deviation 95 ) Gross National Product Personal Consumption Expenditures Services Goods Fixed investment Expenditures Nonresidential Investment Structures Equipment Capital Stocks Total Nonfarm Inventories Nonresidential Structures Nonresidential Equipment Labor Input Nonfarm Hours Average Weekly Hours in . Productivity ( Source data data bank Table The variables in the Prescott model , from Prescott ( 1986 ) Cyclical Behavior of the Economy Cross Correlation of With Standard Deviation ( Gross National Product ( Consumption ( Investment 549 ( Inventory Stock 220 14 60 (

REAL BUSINESS CYCLES Capital Stock ( Hours ( Productivity ( Real Interest Rate ( Annual ) These are the means of 20 simulations , each ofwhich was 116 periods song . The numbers parentheses are standard errors . Source and Prescott 1984 Figure The correlation of output and hours in the Hansen model Deviations From Trend of and Hours Worked in Hansen Indivisible Labor Economy Source Gray Hansen , Department of Economics . University of California , Santa Barbara Quarters Assessing the contribution 120 The approach led to a methodological revolution in all macro models from then on have been expected to be framed as a dynamic stochastic model with fully agents and rational expectations . Whether or not you buy it as an explanation for business cycle in general , and the associated critique of policy interventions , the approach can be useful in understanding at least some aspects of actual .

REAL BUSINESS CYCLES Prescott ( 1986 ) the claim in favour of the model Economic theory implies that , given the nature of the shocks to technology and peoples willingness and ability to and substitute , the economy will display like those the US . economy His claim is that his model economy matches remarkably well the actual data and , to the extent that it doesn , its probably because the measurement does not really capture what the theory says is important hence the title of Theory Ahead of Business Cycle Measurement , The startling policy implications of these are highlighted as follows Costly efforts at stabilization are likely to be counterproductive . Economic are optimal responses to in the rate of technological change . In other words , business cycle policy is not only useless , but harmful . One should focus on the of the average rate of technological change . The macro literature has vigorously pursued and refined the path opened by Prescott . Lots of changes have been considered to the original , such as different sources for the shocks ( for instance , government spending ) or the inclusion of a number of market distortions ( eg . taxation ) On the other hand , many objections have been raised to the basic message of . Do we really see such huge shifts in technology on a quarterly basis ?

Or , is the residual capturing something else ?

Remember , it is the measure of our . Do we really believe that are driven by people willingness to reallocate labour ?

If these are optimal , why do they feel so painful ?

How about the role of monetary policy , for which the model has no role ?

Finally , it seems that the features of the that are obtained are very close to the nature of the stochastic process that is assumed for the shocks how much of an explanation is that ?

and Prescott eventually received the Nobel Prize in Economics in 2004 , partly for this contribution . We will return to other contributions by the pair when we discuss monetary policy . More importantly , the approach led to two developments . First , it generated a fierce counterattack to . The rational expectations revolution had stated that policy was ineffective and Prescott said it was wrong and harmful . Second , by validating the model , this calibrated stochastic version of the became the workhorse of , so that the approach won the methodological contest . In macro these days , people are pretty much expected to produce a model with fully agents ( with rational expectations ) that is amenable to the of calibration . Even the folks who believe in business cycles are compelled to frame them in such models , though including some price rigidity , monetary disturbance , and so on , as we will see soon . In addition , even if you believe that the approach is a better description of business cycles in general , it may still be the case that a simple framework can explain some tant economic episodes . For instance , take a look at Figure 145 , which depicts , employment , consumption , and investment data for the US , over . The sharp drop we see is , of course , the economic response to the pandemic , which fits well the supply shock paradigm . The response looks a lot like the kind of logic we have seen in this chapter a shock to productivity in this case , the threat posed by a virus radically changes the and leads to a postponement of labour supply and , consequently , to drops in employment and output . Notice that consumption in this case is not smoother than output even though investment is the most volatile variable , as the model predicts . Why is consumption more volatile in this context ?

et al . 2020 ) provide an explanation , They make the realistic assumption that during the pandemic , people attached a risk of contagion to the act of consuming ( consumption means going out to a restaurant , shopping mall , etc . and , therefore , reduced consumption more than they would have done if only adjusting for the change in wealth , The example illustrates that some specific changes in the setup may be required on occasion to adjust empirical

REAL BUSINESS CYCLES Figure Trajectories of macro variables in response to 100 Index ( 2019 ) 85 Date Consumption Employment Investment This particular example illustrates that real shocks actually exist , and shows , more broadly , that you do not have to be an true believer to accept that the logic they illuminate has practical What have we learned ?

The approach to business cycle is conceptually very straightforward take the basic model , add productivity shocks ( and a choice ) and you will get business cycle . It highlights the importance of substitution and labour supply as important potential driving factors behind these , and can provide a useful lens with which to understand , at the very least , in some circumstances ( as illustrated by the case of the pandemic ) The approach also has a very sharp message in terms of policy you should not pursue cyclical policy . If are simply the optimal response of a economy to real shocks , policy would only add noise to the process , and make adjustments harder . As we will see , the approach has a very different , more message The contraposition of these two traditions and particularly the role they assign to policy intervention is very much at the heart of policy debates . But we also learned that , underpinning this policy divergence , is a substantial degree of ological convergence All of mainstream modern , to a first approximation , speaks

REAL BUSINESS CYCLES the language that was first introduced by the approach that of dynamic , stochastic general ( models . What next ?

Readers who are interested in the approach can go to ( 2008 ) The of , which , as the title indicates , provides a simple and practical introduction to solving models , The paper by Prescott ( 1986 ) is also worth reading , as it provides a veritable manifesto of the original approach . Those who want to dig deeper into the type of recursive methods that have become ubiquitous in , to solve models in the methodological tradition inaugurated by the approach , should look into and Sargent ( 2018 ) You will actually see many of the themes that we discuss in this book , but presented at a whole other level of formal rigor . Notes You will often see I used to refer to leisure and to labour , but we are going to stick with for labour , for consistency . Think about as standing for holidays . One way to think about this is asking yourself the following question how many times higher are real wages today than , say , 300 years ago ?

And how many more hours do we work ?

Here is a weird little prediction from this model note that consumption is constant regardless of the employment decision . This means that unemployed and employed workers have the same tion . But , since work generates disutility , that means that unemployed workers are better off ! For a discussion on the of this debate , see Chetty et al . 2011 ) Remember that and Prescott ( 2002 ) argue for I , but this was later An alternative story is provided by ( 2020 ) In his , utility changes and ple , due to , require fewer goods to obtain the same utility . The result is a sharp fall in optimal consumption as in et al . 2020 ) References Chetty , 81 Weber , A , 2011 ) Are micro and macro labor supply ties consistent ?

A review of evidence on the intensive and extensive margins . American Economic Review , 101 ( 2020 ) The ( tech . National Bureau of Economic Research , Hansen , 1985 ) Indivisible labor and the business cycle . Journal Economics , 16 ( Prescott , 1982 ) Time to build and aggregate . Sargent , I . 2018 ) Recursive theory . MIT Press , 2008 ) The of , Cambridge , Massachusetts , London Harvard . Prescott , 2002 ) Barriers to riches . MIT Press . Prescott , 1986 ) Theory ahead of measurement , Conference Series on Public Policy , 25 , 2020 ) Should we hibernate in a lockdown ?

Economics Bulletin , 40 (