Real Business Cycle Models

Transcription

1 Phd Macro, 2007 (Karl Whelan) 1 Real Business Cycle Models The Real Business Cycle (RBC) model introduced in a famous 1982 paper by Finn Kydland and Edward Prescott is the original DSGE model. 1 The early rational expectations models of Lucas and Sargent had followed in the Keynesian tradition of making monetary and fiscal policies a central feature of business cycle theory. However, those models were generally not deemed to be empirically successful. RBC modelers took a different direction: Their approach takes the microfoundations orientation of rational expectations modelling to its logical extreme. It views the economy as determined by a set of optimizing agents with rational expectations operating with access to fully-functioning markets. While many now question the specific assumptions underlying the early RBC models, the methodology has endured. Today s DSGE model builders may view aspects of the economy differently (e.g. the role of monetary policy) but they share the goal of motivating all of their model s relationships on the basis of optimizing behaviour by households and firms. In these notes, we will set out a basic RBC model and discuss how the model s first-order conditions can be turned into a system of linear difference equations of the form we know how to solve. We will also discuss some of the arguments for and against RBC models. The RBC Model Economy The most basic RBC models assume perfectly functioning competitive markets. This means that the outcomes generated by decentralized decisions by firms and households can be replicated as the solution to a social planner problem. In the RBC model that we will examine, the social planner wants to maximize E t [ i=0 ( C 1 η )] β i t+i 1 η an t+i where C t is consumption, N t is hours worked, and β is the representative household s rate of time preference. The economy faces constraints described by (1) Y t = C t + I t = A t K α t 1N 1 α t (2) K t = I t + (1 δ)k t 1 (3) 1 Time to Build and Aggregate Fluctuations, Econometrica, November 1982, Volume 50, pages This paper was cited in the 2004 Nobel prize award given to Kydland and Prescott.

2 Phd Macro, 2007 (Karl Whelan) 2 and a process for the technology term A t. To simplify some aspects of the derivations below, I will assume that A t does not trend over time, so our model economy has an average growth rate of zero (changing this so the economy grows on average changes very little of what follows). Specifically, we will assume that the technology term follows an AR(1) process of the form log A t = (1 ρ)log A + ρ log A t 1 + ǫ t (4) Objections to the RBC Approach Before discussing how to solve these models, how to simulate them, and their properties, it is worth acknowledging that a lot of people have a negative reaction to them. Some of the objections regularly aired are as follows: Perfect Markets and Rational Expectations: The idea that the economy can characterized as a perfectly competitive market equilibrium solution describing the behaviour of a set of completely optimizing rational individuals does not appeal to all. Surely we know that markets are not always competitive and don t always work well, and surely people are not always completely rational in their economic decisions? Monetary and Fiscal Policy: RBC models exhibit complete monetary neutrality, so there is no role at all for monetary policy, something which many people think is crucial to understanding the macroeconomy. We haven t put government spending in this model, but if we did, the model would exhibit Ricardian equivalence, so the effects of fiscal policy would be different from what most people imagine them to be. Skepticism about Technology Shocks: RBC models give primacy to technology shocks (the ǫ t shocks) as the source of economic fluctuations. But what are these shocks? Is it really credible that all economic fluctuations, including recessions, are just an optimal response to technology shocks? What are these technology shocks anyway if we were experiencing these big shocks wouldn t we be reading about them in the newspapers? Wouldn t recessions have to be periods in which their is outright technological regress, so that (for some reason) firms are using less efficient technologies than previously is this credible?

3 Phd Macro, 2007 (Karl Whelan) 3 Discussion of these Objections There is some substance to most (though not all) of these criticisms. However, they should not be seen as arguments against the usefulness of the RBC approach: The RBC model should be seen as a benchmark against which more complicated models can be assessed. If a model with optimizing agents and instantaneous market clearing can explain the business cycle, then can market imperfections such as sticky prices really be seen as crucial to understanding macroeconomic fluctuations? To explain why these imperfections might be important, one needs to understand the limitations of the RBC model. The framework provided by RBC models can also be extended to account for a wide range of factors not considered in the basic model discussed here. For instance, if we allow for imperfect competition, then the decentralized market outcome cannot be characterized as the outcome of a social planner problem. So, one needs to derive the FOCs from separate modelling of the decisions of firms and households. But this is easily done. In addition, other frictions such as sticky prices which would allow for monetary policy to have a role can also be added. Technology shocks are probably a bit more important than one might think at first. Growth theory teaches us that increases in TFP are the ultimate source of long-run growth in output per hour. Is there any particular reason to think that these increases have to take place in a steady trend-like manner? Put this way, random fluctuations in TFP growth doesn t seem so strange. Also, at least in theory, RBC models could generate recessions without outright declines in technology. This can happen if the elasticity of output with respect to technology is greater than one; then potentially, a slowdown in TFP growth below trend could trigger a recession. A Technical Comment Before getting on with solving the model, I ll make a brief technical comment about maximization techniques for problems like this; don t worry if you don t understand this. Technically, the best way to solve optimization problems like this that feature random variables is called stochastic dynamic programming. I don t have the time to teach these methods in this class, so instead I will use Lagrangian methods, which I know you have seen. In doing

4 Phd Macro, 2007 (Karl Whelan) 4 this, I will assume that one can differentiate things of the form E t F (x) to give E t F (x). One might wonder if this is ok, but it is. To see this consider the problem of choosing a value of x to maximize the expected value of F (a, x) where a is random and can take on the value a k with probability p k. This expected value is given by This is maximized by setting N G (x) = E t p k F (a k, x) (5) k=1 G N (x) = E t p k F (a k, x) = E t F (x) = 0 (6) k=1 So, the FOC for maximizing E t F (x) is just E t F (x) = 0. The Social Planner s Problem The Lagrangian for this problem can be written as [ { C 1 η }] ( ) L = E t β i t+i 1 η an t+i + λ t A t+i Kt+i 1N α t+i 1 α + (1 δ)k t+i 1 C t+i K t+i i=0 (For convenience, I have substituted the capital accumulation constraint into the resource constraint.) Now this equation might look a bit intimidating. It involves an infinite sum, so technically there is an infinite number of first-order conditions for current and expected future values of C t, K t and N t. To see that the problem is less hard than this makes it sound, note that the time-t variables appear in this sum as C 1 η ( ) ( ) t 1 η an t + λ t A t Kt 1N α t 1 α C t K t + βe t λ t+1 A t+1 Kt α Nt+1 1 α + (1 δ)k t After that, the time-t variables don t ever appear again. So, the FOCs for these variables consist of differentiating equation (8) with respect to the relevant variables and setting the derivatives equal to zero. After that, the time t + n variables appear exactly as the time t variables do, except that they are in expectation form and they are multiplied by the discount rate β n. But the FOCs for the time t + n variables will be identical to those for the time t variables. So differentiating equation (8) is enough to give us the equations that describe the optimal dynamics at all times for this economy. These FOCs are L C t : C η t λ t = 0 (9) (7) (8)

5 Phd Macro, 2007 (Karl Whelan) 5 L K t L N t L λ t : βe t ( α Y ) t+1 λ t+1 + (1 δ)λ t+1 λ t = 0 (10) K t : a + (1 α) λ t Y t N t = 0 (11) : A t K α t 1N 1 α t + (1 δ)k t 1 C t K t = 0 (12) One trick that helps make this system a bit easier to understand is to define the marginal value of an additional unit of capital next year as R t+1 = α Y t+1 K t + 1 δ (13) This will be the gross interest rate for this economy (i.e. the interest rate will be R t+1 1). With this definition in hand, equation (10) can be written as λ t = βe t (λ t+1 R t+1 ) (14) and this can be combined with the equation (9) to give ( ) C η t = βe t C η t+1 R t+1 (15) or, alternatively (( ) η ) Ct βe t R t+1 = 1 (16) C t+1 The Full Set of Model Equations The RBC model can then be defined by the following six equations (three identities describing resource constraints and three FOCs) Y t = C t + I t (17) Y t = A t K α t 1N 1 α t (18) K t = I t + (1 δ)k t 1 (19) (( ) η ) Ct 1 = βe t R t+1 (20) Y t N t = C t+1 as well as the process for the technology variable. R t a 1 α Cη t (21) = α Y t K t δ (22)

6 Phd Macro, 2007 (Karl Whelan) 6 Those of you who have been following me thus far might notice something funny about the system given by equations (17) to (22). They are not a set of linear difference equations, but a mix of both linear and nonlinear equations: Haven t I been saying that DSGE modelling is all about sets of linear difference equations? In general, these nonlinear systems cannot be solved analytically. However, it turns out their solution can be very well approximated by turning them into a set of linear equations in the deviations of the logs of these variables from their steady-state values. This approach gives us a system that expresses variables in terms of their percentage deviations from the steady-state paths in which all real variables are growing at the same rate. The steady-state path is relevant because the stochastic economy will, on average, tend to fluctuate around the values given by this path. This method can be thought of as giving a system of variables that represents the business-cycle component of the model. Log-Linearization: What s The Deal? The deal is as follows. The particular economy that we are looking has a zero-growth steady-state path, so for each real variable X t there is a corresponding X that is constant on the steady-state path. We will use lower-case letters to define log-deviations of variables from their steady-state values. x t = log X t log X (23) The key to the log-linearization method is that every variable can be written as X t = X X t X = X e xt (24) Then, the big trick is that a first-order Taylor approximation of e xt is given by e xt 1 + x t (25) So, we can write variables as X t X (1 + x t ) The second trick is that when one has variables multiplying each other such as X t Y t X Y (1 + x t )(1 + y t ) (26) you set terms like x t y t = 0 because we are looking at small deviations from steady-state and multiplying these small deviations together one gets a term close to zero. So, the log-linear approximation for these terms is X t Y t X Y (1 + x t + y t ) (27)

7 Phd Macro, 2007 (Karl Whelan) 7 That s it. Substitute these approximations for the variables in the model, lots of terms end up canceling out, and when you re done you ve got a system in the deviations of logged variables from their steady-state values. The paper on the reading list by Harald Uhlig discusses this stuff in a bit more detail and provides more examples. Log-Linearization: Example 1 I will not go through the details of the log-linearization of all six of our model equations from (17) to (22). Instead, I will describe the log-linearization of (17) and (18) and then leave the other four equations for you to go through in your problem set. So, start with equation (17). This equation can also written as Y e yt = C e ct + I e it (28) Using the first-order approximation, this becomes Y (1 + y t ) = C (1 + c t ) + I (1 + i t ) (29) Note, though, that the steady-state terms must obey identities so Y = C + I (30) Canceling these terms on both sides, we get Y y t = C c t + I i t (31) which we will write as y t = C Y c t + I Y i t (32) This is the log-linearized version of the equation that we will work with. Log-Linearization: Example 2 Now consider equation (18). This can be re-written in terms of steady-states and logdeviations as Y e yt = (A e at )(K ) α e αkt (N ) 1 α e (1 α)nt (33) Again, use the fact the steady-state values obey identities so that Y = A (K ) α (N ) 1 α (34)

8 Phd Macro, 2007 (Karl Whelan) 8 So, canceling gives e yt = e at e αkt e (1 α)nt (35) Using first-order Taylor approximations, this becomes (1 + y t ) = (1 + a t )(1 + αk t )(1 + (1 α) n t ) (36) Ignoring cross-products of the log-deviations, this simplifies to y t = a t + αk t + (1 α) n t (37) The Full Log-Linearized System Once all the equations have been log-linearized, we have a system of seven equations of the form y t = C Y c t + I Y i t (38) y t = a t + αk t + (1 α) n t (39) k t = I K i t + (1 δ)k t 1 (40) n t = y t ηc t (41) c t = E t c t+1 1 η E tr t+1 (42) ( α Y ) r t = R (y t k t 1 ) (43) K a t = ρa t 1 + ǫ t (44) We are nearly ready to put the model on the computer. However, notice that three of the equations have coefficients that are values relating to the steady-state path. These need to be calculated. Steady-State Coefficients This turns out to be pretty easy. Start with the steady-state interest rate. We have (( ) η ) Ct 1 = βe t R t+1 C t+1 Because we have no trend growth in technology in our model, the steady-state features consumption, investment, and output all taking on constant values. Thus, in steady-state, we have C t = C t+1 = C, so (45) R = β 1 (46)

9 Phd Macro, 2007 (Karl Whelan) 9 Next, we have So, in steady-state, we have R t = α Y t K t δ (47) Y K = β 1 + δ 1 (48) α Together with the steady-state interest equation, this tells us that α Y ( β 1 ) R K = αβ + δ 1 = 1 β (1 δ) (49) α which is the steady-state value required for equation (43). Next, we use the identity K t = I t + (1 δ)k t 1 (50) and the fact that in steady-state we have K t = K t 1 = K, to give I K = δ (51) which is required in equation (40). This can then be combined with the previous steadystate ratio to give and obviously I I Y = K Y K = αδ β 1 + δ 1 C Y = 1 αδ β 1 + δ 1 which gives us the steady-state ratios in equation (38). (52) (53) The Final System Using these steady-state identities, our system becomes ( ) ( ) αδ αδ y t = 1 β 1 c t + + δ 1 β 1 i t (54) + δ 1 y t = a t + αk t + (1 α) n t (55) k t = δi t + (1 δ)k t 1 (56) n t = y t ηc t (57) c t = E t c t+1 1 η E tr t+1 (58) r t = (1 β (1 δ))(y t k t 1 ) (59) a t = ρa t 1 + ǫ t (60)

10 Phd Macro, 2007 (Karl Whelan) 10 This is written in the standard format that allows us to apply solution algorithms such as the Binder-Pesaran program to obtain a reduced-form solution, and thus simulate the model on the computer. 2 One thing to recall is that all of these coefficients can be interpreted as elasticities. Also, impulse response functions can be interpreted as describing the percentage responses to one percent shocks. RBC Models Generate Cycles Figure 1 simulates our model. It uses parameter values intended for analysis of quarterly time series: α = 1 3, β = 0.99, δ = 0.015, ρ = 0.95, and η = 1.3 RATS programs to generate charts of this type are available on the website, so if you wanted, you could work with the model yourself. The figure demonstrates the main successful feature of the RBC model: It generates actual business cycles and they don t look too unrealistic. In particular, reasonable parameterizations of the model can roughly match the magnitude of observed fluctuations in output, and the model can match the fact that investment is far more volatile than consumption. Deficiencies of RBC Models Despite this success, RBC models have still come in for some criticism. One reason is that they have not quite lived up to the hype of their early advocates. Part of that hype stemmed from the idea that RBC models contained important propagation mechanisms for turning technology shocks into business cycles: Increases in technology induced extra output through higher capital accumulation and by inducing people to work more. In other words, some of the early research suggested that even in a world of iid technology levels, one would expect RBC models to still generate business cycles. However, Figure 2 shows that output fluctuations in this model follow technology fluctuations quite closely: This shows that these additional propagation mechanisms are quite weak. Cogley and Nason (1995) note another fact about business cycles that the RBC model 2 Most of the textbook treatments of RBC models discuss solving it with the method of undetermined coefficients as discussed in the paper on the reading list by John Campbell. This involves guessing the form of the solution and then plugging that solution back into the equation to determine the coefficients. Campbell s paper is great it has lots of useful insights into the RBC model but this generally requires far too much algebra to be practical. 3 This last value might look a bit funny because technically C1 η t+i is not defined at this value. However, 1 η this function tends towards log C t as η tends to one, and that is what is meant by this.

11 Phd Macro, 2007 (Karl Whelan) 11 does not match. Output growth is positively autocorrelated (not very autocorrelation coefficient of 0.34 but significantly so in a statistical sense). But RBC models do not generate this pattern: See Figure 3. They can only do so if one simulates a technology process that has a positively autocorrelated growth rate. Cogley and Nason relate this back to the IRFs generated by RBC models. Figure 4 shows the responses of output, consumption, investment, and hours to a unit shock to ǫ t. Figure 5 highlights that the response of output to the technology shock pretty much matches the response of technology itself. Cogley-Nason argue that one needs instead to have humped-shaped responses to shocks a growth rate increase needs to be followed by another growth rate increase if a model is to match the facts about autocorrelated output growth. The responses to technology shocks do not deliver this. In addition, while we don t have other shocks in the model (e.g. government spending shocks), Cogley-Nason point out that RBC models don t generate hump-shaped responses for these either. A more recent critique relates to the response of hours to the technology shock. The VAR paper by Gali that we discussed a few weeks ago argues that the data point to technology shocks actually having a negative effect on hours worked, not a positive one. This argues against the model s interpretation of labor market fluctuations. Others have also argued that the model s assumption that hours worked are highly responsive to real wages does not seem to match the microeconomic evidence on labour supply. There are currently a number of branches of research aimed at fixing the deficiencies of the basic RBC approach. Some of them involve putting extra bells and whistles on the basic market-clearing RBC approach: Examples include variable utilization, lags in investment projects, habit persistence in consumer utility. The second approach is to depart more systematically from the basic RBC approach by adding rigidities such as sticky prices and wages. Some papers do both. I plan to spend some of the next few weeks looking at some of these models.

12 15 Figure 1 RBCs Generates Cycles with Volatile Investment (Simulation) Output Investment Consumption

13 2.7 Figure 2 But the Cycles Rely Heavily on Technology Fluctuations (Simulation) Output Technology

14 2.0 Figure 3 RBCs Do Not Generate Autocorrelated Growth (Simulation) Output Growth

15 9.6 Figure 4 RBC Model: Impulse Responses to Technology Shock Output Consumption Investment Hours

16 2.25 Figure 5 RBC Model: Impulse Responses to Technology Shock Output Technology