Adaptive Beliefs in RBC models

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1 Adaptive Beliefs in RBC models Sijmen Duineveld May 27, 215 Abstract This paper shows that waves of optimism and pessimism decrease volatility in a standard RBC model, but increase volatility in a RBC model where a news shock increases contemporaneous output. The waves of optimism and pessimism are introduced with an Adaptive Beliefs System (Brock & Hommes, 1997, 1998) which have also been introduced in monetary models by DeGrauwe (211) and Massaro (213). In the Adaptive Beliefs System (ABS) used here the representative agent's beliefs about the future productivity level are a weighted average of an optimistic and a pessimistic forecasting rule. In the Rational Expectations (RE) version of the standard growth model higher expected productivity reduces hours worked, and therefore output, due to the wealth eect. With Bounded Rationality a positive productivity shock also results in higher expected productivity, and thus reduces current output as the agent expects to be richer in the future and reduces his current hours worked relative to the RE case. However, the opposite is true for models where higher expected productivity increases contemporaneous labour supply. For example, in the Pigou cycle model of Jaimovich & Rebelo (29) higher expected productivity results in higher current labour supply and output. Introducing BR in that model thus results in higher output volatility. It is shown that the amplication of output depends the strength of the wealth eect, just as the wealth eect is crucial for the eect of a news shock on current output. 1 Introduction As DeGrauwe (21) notes the tight incorporation of Rational Expectations (RE) in macroeconomics has resulted in very little attention being paid to models with some form of Bounded Rationality. Within the Bounded Rationality literature the Learning Approach (Evans & Honkapohja, 21) has so far received the most attention, but these types of models generally converge to the I am grateful to the University of Augsburg Research Center for Global Business Management for its nancial support. University of Augsburg, saduineveld@hotmail.com 1

2 RE solution and have therefore similar dynamics, although some more recent adaptive learning models use misspecied statistical models to form expectations (see for example Branch & McGough, 211). This paper is also related to the 'news shock' and 'noisy signal' literature where agents receive an inaccurate or noisy signal about future values of exogenous variables, most of all the productivity level. For example Jaimovich and Rebelo (29) analyze news shocks about future productivity levels (which may or may not materialize) in a RBC model. Lorenzoni (29) on the other hand analyzes a New Keynesian model where agents receive a noisy signal about the current level of (aggregate) productivity which also inuences the expectations of future productivity levels. These noisy signals and news shocks are in eect Bounded Rationality as well. However, more recent macroeconomic papers have used another type of Bounded Rationality that can result in much more volatile or even unstable systems. This literature uses Adaptive Belief Systems (Brock & Hommes, 1997) in a macroeconomic environment (Massaro, 213; DeGrauwe, 21). The papers of both DeGrauwe and Massaro however concentrate on monetary models. As far as the author knows this research is the rst attempt to implement an Adaptive Beliefs System (ABS) in a Real Business Cycle (RBC) model. Its main purpose is to analyze this type of Bounded Rationality as a propagation mechanism. For this purpose Bounded Rationality is analyzed in two RBC models. One model is a standard growth model and the other the Pigou cycle model of Jaimovich and Rebelo (abbreviated JR) (29). The main dierence between the two models is that in the standard growth model a higher expected productivity reduces current hours worked and results in lower output. In the Pigou cycle model however an expected increase in productivity results in an increase in hours worked and capital utilization, and therefore output also increases. This dierence between the Standard Growth and Pigou cycle model is crucial for the eect of Bounded Rationality on business cycle amplication, because with Bounded Rationality (BR) a positive productivity shock results in higher expected productivity. Adaptive Beliefs Systems are introduced here with respect to expected productivity levels. Instead of the heterogeneous agent models employed by Massaro (213) and DeGrauwe (21) a representative agent is used here. Otherwise the Adaptive Beliefs System is standard. The representative agent expects future productivity levels to be a weighted average of two extreme forecasting rules, an optimistic and a pessimistic forecasting rule. The weights of each rule depend on the recent performance of each rule: when there have recently been more positive (negative) shocks, then the agent expects more positive (negative) shocks in the near future. The Bounded Rationality in this model is conned to the exogenous productivity process without any feedback loop that endogenously reinforces positive or negative shocks. So unlike the models in DeGrauwe or Massaro the model used here has a stable steady state. The exogenous waves of optimism and pessimism can be measured by the weights of each forecasting rule. To put 2

3 some discipline on these waves of optimism and pessimism, the weights of the optimistic and pessimistic rule are calibrated to consumer condence, just as Jaimovich & Rebelo (29) calibrated their model on professional forecasts of GDP. There has been a discussion in the adaptive learning literature on whether it suces to use one-period ahead forecasts in combination with one period Euler equations as in Evans & Honkapohja (26) or whether one has to use forecasts at all horizons and an optimal policy that satises Euler equations at all forward iterations (Preston, 25). In the model used here there is a representative agent which is only bounded rational with respect to the future productivity level, and rational otherwise. This means that all next period's state variables, other that the productivity level, are known with certainty. Moreover, at longer forecasting horizons the Perceived Law of Motion (PLM) of the state variables would coincide with the Actual Law of Motion (ALM) of state variables if the Bounded Rational expectations would be correct. Since the Bounded Rationality only applies to an exogenous variable the model can be solved with the recursive methods also employed in RE models, meaning only one period ahead forecasts and one period Euler equations. The paper is organized as follows. The next section describes the two RBC models and the Adaptive Beliefs System of Bounded Rationality. Section 3 then describes the calibration. Section 4 describes the results in terms of Business Cycle statistics and Impulse Response Functions, and also includes some robustness check. Finally some conclusions are dwawn in section 5. 2 Models As mentioned in the introduction Bounded Rationality is introduced in two RBC models, a Standard Growth model and the Pigou cycle model of Jaimovich & Rebelo (29). Since these models have been described in detail by others, the description here will be brief. 2.1 Standard Growth model Consumers optimize expected life-time utility: U = Ẽ t= β t C1 ν t 1 ν N 1+ 1 η t (1) η with utility time-separable in consumption, C, and hours worked, N. Production, Y, is given by the standard Cobb-Douglas production function, where α measures the labour share of income and Z is Total Factor Productivity (TFP). The income is used for either investment, I, or consumption: 3

4 Combining these two equations yields: Y t = Z t Kt 1 α Nt α (2) Y t = C t + I t (3) Z t K 1 α t N α t = C t + I t (4) With investment and the next periods capital stock, K t+1, dened by: K t+1 = I t + (1 δ) K t (5) The FOCs with respect to C t, N t and K t+1 are then: λ t = C ν t (6) λ t αz t Kt 1 α Nt α 1 = N 1 η t (7) { [ λ t = βẽt λt+1 (1 α) Zt+1 Kt+1 α N t+1 α + 1 δ ]} (8) where λ t is the Lagrange multipliers on constraint 4, and Ẽt indicates a bounded rational expectation. The true law of motion for Total Factor Productivity is: with ɛ t N (, 1). log (Z t ) = ρ log (Z t 1 ) + σ z ɛ t 2.2 Jaimovich & Rebelo model This model is described in detail in Jaimovich & Rebelo (29) and was created to generate comovement in output, consumption, investment, hours worked and the real wage as observed in the data. Other RBC models can generate this comovement in response to contemporaneous shocks to Total Factor Productvity, but fail to do so in case of news shocks or investment specic shocks. To generate the comovement the model has 3 features at which it departs from more standard models. Firstly, it uses preferences that include the form used by Greenwood, Hercowitz, and Human (1988) (when φ = ) and the form chosen by King, Plosser, and Rebelo (1988) (when φ = 1) as special cases. Changing parameter φ then allows one to govern the strength of the short run wealth-eect on labour supply. Secondly, it allows for capital utilization which can amplify the business cycle and also inuences the marginal product of labour. Third, there are investment adjustment costs which ensure that investment is carried forward in time when investment levels are expected to rise in the future. 4

5 Consumers optimize expected life-time utility: ( ) U = Ẽ β t Ct ψnt θ 1 σ X t 1 1 σ t= (9) where C is consumption, and N is hours worked, and X is given by: X t = C φ t X 1 φ t 1 (1) which makes the preferences non-time-separable in consumption and hours. Production is given by the Cobb-Douglas production function with capital utilization, u, and is used for consumption and investment, I: Combining these two equations yields: Y t = A t (u t K t ) 1 α N α t (11) Y t = C t + I t (12) A t (u t K t ) 1 α N α t = C t + I t (13) Investment and the next periods capital stock, K t+1, are related by: ( )] It K t+1 = I t [1 ϕ + [1 δ (u t )] K t (14) I t 1 ( ) I where ϕ t I t 1 are the investment adjustment costs and δ (u t ) is the deprication as a function of the capital utilization rate. The FOCs for the social planner with respect to C t, X t, N t, u t, K t+1 and I t : λ t = ( C t ψnt θ ) ν X t µt φc φ 1 t X 1 φ t 1 (15) ( C t ψnt θ ) ν [ ] X t + µ = β Ẽ t µ t+1 (1 φ) C φ t+1 X φ t (16) λ t αz t (u t K t ) 1 α N α 1 t = ( C t ψn θ t X t ) ν θψn θ 1 t X t (17) η t δ (u t ) K t = λ t (1 α) Z t u α t Kt 1 α Nt α (18) 5

6 { η t = βẽt λt+1 (1 α) Z t+1 u 1 α t+1 K α t+1 N t+1 α + η t+1 [1 δ (u t+1 )] } (19) ( It λ t = η t [1 ϕ [ +Ẽt I t 1 βη t+1 ϕ ( It+1 I t ) ϕ ( It I t 1 ) ( It I t 1 ) It ) 2 ] I t 1 ]... (2) where µ t, λ t and η t are the Lagrange multipliers on the constraints (1), (13) and (14) respectively, and Ẽt indicates a bounded rational expectation. 2.3 Bounded Rationality The Bounded Rationality framework is the Adaptive Beliefs systems used by Brock & Hommes (1997) and applied to the expected productivity level. Expected Productivity The expected productivity level of the representative agent is a weighted average of an optimistic and a pessimistic forecasting rule. The expected productivity for the pessimistic and optimistic rule are: Ẽ pes t z t+1 = ρz t g t Ẽ opt t z t+1 = ρz t + g t where z t = log (Z t )and g t = σ z so the bias is one standard deviation. The expected productivity level for the representative agent is: Ẽ t z t+1 = ω pes Ẽ pes t z t+1 + ω opt Ẽ opt t z t+1 = ρz t ω pes g + (1 ω pes ) g The last two terms are the bias which is the weighted average of the pessimistic and the optimistic rule. The weights are determined in the same way as Brock & Hommes (1998). The current period tness measure is: τ pes (Ẽpes ) 2 t = t 1 z t z t = ( gt ɛ t ) 2 τ opt (Ẽopt ) 2 t = t 1 z t z t = (gt ɛ t ) 2 where ɛ t is the productivity shock in period t. The total tness measure for each rule i = [pes, opt] is then: 6

7 here M i t = τ i,t χm i t 1 where χ is a measure of the memory strength. The weights of each forecasting rule is then: ω pes t = ω opt t = pes eγmt pes γm e t opt + eγmt opt γm e t pes opt eγmt + eγmt where γ is a measure of the intensity of choice (DeGrauwe, 21) as it determines how responsive the weights are to changes in tness measure. If γ = then each rule has weight.5, and if γ = then the ω i,t is either or 1. To simplify the notation the bias can be rewritten as: b t = (1 ω pes t ) g ω opt g which is a function of the shock in this period, ɛ t, and the two state variables M pes opt t 1 and Mt 1. Thus we can write the bias in period t as: b t = B ( ɛ t, M pes t 1, M t 1) opt. The expected productivity level can then be written as: t Ẽ t z t+1 = ρz t + b t The Impulse Response Function (IRF) is given for the expectation of future productivity level when there is a shock to TFP of one standard deviation. It gives the expected productivity level in the future at the time the productivity shock occurs (t = ), meaning E i z t for t =, 1,.., 34. 7

8 .12 Figure 1: Expected TFP at time of shock E RE (zt ) E BR (zt ) t The agent is only Bounded Rational with respect to future productivity level, but does take the inuence of productivity on other variables into account. However, the one period ahead values for the other state variables are known with certainty. The state variables are capital stock K t, productivity level Z t and last period's tness measures M pes opt t 1 and Mt 1, plus X t 1 and I t 1 in case of the JR model. All these variables, except capital and productivity, have a one period lag, so their values in the next period are known. The agent also knows his own policy functions, so next period's capital stock, K t+1, is also known at period t. If we write C t = Ω ( C K t, Z t, M pes t 1, M t 1) opt for the policy function of consumption, then the one period ahead Bounded Rational expectation for consumption is: Ẽ t C t+1 = Ω C ( K t+1, ẼtZ t+1, M pes t ), M opt For the Standard Growth model the expected value of all variables can be derived from this policy function and the expected values of the state variables Forecasts of the state variables further in the future can be derived using the recursive structure. In case of the Jaimovich and Rebelo model the only dierence is that the agent has two policy functions and more state variables, but otherwise expected values can be calculated in a similar way. t 8

9 3 Calibration Table 1: Parameters in Standard Growth model α.64 β.985 ν 2 θ 3 δ.25 ρ.95 σ z.4 χ.95 γ 1, 3.1 Calibration Standard Growth model The parameters for the Standard Growth model are set at relatively standard values, and will also be used in the JR model. Only the labour supply elasticity η is set at 3 compared to 2.5 in the Jaimovich & Rebelo model. The standard deviation of the stochastic process is set at the same value as in the Jaimovich and Rebelo model such that the expected future value of TFP are the same in both models (see section 3.3for more on this). 3.2 Calibration Jaimovich & Rebelo In their paper Jaimovich and Rebelo (29) calibrated the model such that empirical business cycle statistics were matched when there is only investmentspecic technical progress. Since only Total Factor Productivity shocks are used here, the model would be too volatile if the same parameters would be used. Therefore the parameters for capital utilization and investment adjustment costs are adjusted. The elasticity of δ (u ss ) is set higher at δ (u ss ) u ss /δ (u ss ) =.3 instead of.15 in JR, meaning that capital utilization becomes more costly, and volatility is reduced. The chosen functional form for depreciation is: δ (u t ) = ζ 1 u 2 t + ζ 2 u t + ζ 3. Regarding investment adjustment costs the second derivative is set at a lower value of ϕ (1) =.4, which results in ϱ =.2, given the chosen ( ) ( I function form: ϕ t I 2. I t 1 = ϱ t I t 1 1) This functional form ensures that ϕ (1) = and ϕ (1) = as in JR. Other parameters that are changed compared to the setting in JR are the depreciation rate, risk aversion parameter and the parameters governing the exogenous TFP process. The depreciation rate in steady state, δ (u ss ), is set to.25 which results in a standard yearly depreciation rate of roughly 1%, where JR had 5% per year. The risk aversion parameter, ν, is set at 2, which is more standard than the logarithmic value of 1 used in JR. The persistence parameter ρ in the stochastic process is set to.95 and the standard deviation is set to σ z =.4, such that the standard deviation in detrended output (under 9

10 Table 2: Parameters in adjusted Jaimovich & Rebelo model α.64 β.985 ν 2 θ 1.4 ψ 1 δ (u ss ).25 δ (u ss ) u ss /δ (u ss ).3 ϕ (1).4 φ.1 ρ.95 σ z.4 χ.95 γ 1, Rational Expectations) is roughly 1.6%. used in JR. That also Other parameters are set at values 3.3 Calibration of Bounded Rationality and stochastic process The parameters γ and χ of the bounded rationality are set such that the volatility in the weights of each forecasting rule ω i roughly matches the standard deviation and auto-correlation coecient in Consumer Condence 1, which are.15 and.7, respectively. The parameters γ and χ only depend on the parameters of the stochastic process. In order to use the same BR parameters γ and χ in both the Standard Growth and the JR model, the stochastic process in both models are also the same (parameters σ z and ρ). The parameter ρ is set at its standard value of.95, while σ z is adjusted downward such that the standard deviation in output in the Jaimovich and Rebelo model roughly matches its empirical value. Since the JR model is more volatile than the standard growth model the volatility in output will be too low in the standard growth model. However, the qualitative eect of Bounded Rationality in the standard growth model should not be aected by this too low standard deviation of Total Factor Productivity (and thus output). 4 Results The results are obtained by a rst order pertubation of around the steady state, with all variables in logs (except those of the Bounded Rationality). 2. All series 1 Mnemonic USCNFCONQ in Datastream, data from For the Standard Growth model the results were almost identical using projection methods, more specically Galerkin projection with a rst order Chebyshev polynomial. 1

11 Table 3: Business cycle statistics Standard Growth model Data RE BR Standard deviation Y Standard deviation N Standard deviation C Standard deviation I Correlation Y and N Correlation Y and C Correlation Y and I Correlation Y and Z Correlation C and N Stand. dev. Consumer Condence Autocorr. Consumer Condence are obtained after detrending with the Hodrick-Prescott lter with λ = Results Standard Growth model The Business Cycle statistics of the Standard Growth model can be found in Table 3. The empirical data are taken from Branch & McGough (211), and are supplemented with the statistics on investment as reported by Jaimovich & Rebelo (29). The standard deviation and auto-correlation on Consumer Condence series are calculated from the series USCNFCONQ in Datastream from The rst thing to note is that the standard deviation of most variables are too low in this model. As mentioned above, the reason for this is that the same stochastic process for Total Factor Productivity is used as in the Jaimovich & Rebelo model such that the Bounded Rationality parameters can be set at the same values to allow for a fair comparison regarding the inuence of Bounded Rationality on amplifying the model. Thus, the focus will here be not such much on the absolute value of the statistics, but the change of business cycle statistics due to Bounded Rationality. The standard deviation of output becomes a factor 4 smaller with BR, while the standard deviation of hours, consumption and investment all increase. The relative volatility of these latter variables become unrealistically high. However, the correlation between output and hours worked becomes very close to empirically observed values, while the correlations between output and consumption and output and TFP match the data very poorly, just as the correlation between consumption and hours worked. The conclusion is that Bounded Rationality reduces the volatility in output but increases the volatility in other variables, worsening the overall t with the data. The correlations between variables also t the data less good with Bounded Rationality. It is well-know that in a Standard Growth model a positive news shocks reduces output in the period the news arrives. With Bounded Rationality a 11

12 positive TFP shock also results in higher expected productivity and thus reduces current output compared to the RE case. It is therefore not surprising that Bounded Rationality reduces output in this model. This is demonstrated in the Impulse Response Functions in Figure 2. Figure 2: Response to TFP shock in Standard Growth model 6 x Output RE 2 BR News x Consumption RE BR News Investment 4 x 1 3 Hours worked.1.1 RE.2 BR News RE BR News The IRFs show the response to a one standard deviation shock in Total Factor Productivity. For the Rational Expectations and Bounded Rational models these shocks arrive in period 1. In the line designated News there is a news shock in period 1 that TFP will increase with one standard deviation in period 6, so 5 periods in advance. The initial response in output with the Bounded Rationality is clearly similar to the initial response in output with a news shock as output falls in both cases, although with Bounded Rationality output falls despite an increase in TFP. The explanation is the wealth-eect: as consumers expect higher productivity in the future, they increase their consumption, and also their leisure, and so reduce their labour supply. The reduction in labour supply results in lower output. This wealth-eect on labour supply is present with both news shocks and with Bounded Rationality as is clear from the plots of hours worked and output. In all three cases there is an increase in future consumption is expected due to higher expected future productivity, so it is little surprising that consumption increases in all 3 cases. The response of investment is the result of the changes in consumption and output. With Bounded Rationality and News shocks output falls, but consumption increases, so investment has to fall as well. 12

13 Table 4: Business cycle statistics Jaimovich & Rebelo model Data RE BR Standard deviation Y Standard deviation N Standard deviation C Standard deviation I Standard deviation U Correlation Y and N Correlation Y and C Correlation Y and I Correlation Y and Z Correlation C and N Stand. dev. Consumer Condence Autocorr. Consumer Condence Results Jaimovich & Rebelo model The results of the business cycle statistics for the Jaimovich & Rebelo model can be found in Table 4. Clearly the introduction of Bounded Rationality increases the absolute volatility in output, hours, consumption and utilization, but reduces the volatility in investment. The relative volatility of consumption and utilization increases. These changes in the relative volatility make that the Bounded Rational model ts the data less good. The correlation coecients match the data very poorly in the RE version of the model as the correlations are much higher than empirically found in the data. With Bounded Rationality these correlations match the data even less good as they all approach 1. It may therefore be concluded that the Rational Expectations version of the JR model is able to generate a Pigou cycle when a TFP shock occurs, but that the correlations are unrealistically high with these type of shocks. Of course, it should be noted that the model was calibrated on other types of shocks by Jaimovich & Rebelo. Introducing Bounded Rationality with respect to TFP does strengthen the internal propagation mechanisms in this Pigou cycle model, but that comes at the cost of a less good t to business cycle statistics. The increase in volatility in output might be expected as Bounded Rationality has a similar eect as a news shock. When a positive news shock arrives in the Jaimovich & Rebelo model output increases on impact. With Bounded Rationality a positive TFP shock also results in higher expected productivity and output increases more than with Rational Expectations. To understand this better the IRFs are shown in Figure 3. As in the case of the Standard Growth model the IRFs show the response to a one standard deviation shock in Total Factor Productivity. For the Rational Expectations and Bounded Rational models these shocks arrive in period 1. In the line designated News there is a news shock in period 1 that TFP will increase with one standard deviation in period 6, so 5 periods in advance. It 13

14 Figure 3: Response to TFP shock in Jaimovich & Rebelo model Output Consumption RE BR News RE.2 BR News Investment.12 Hours worked RE BR News RE.2 BR News is clear from these gures that the eect of Bounded Rationality works in the same direction as a News shock. For example, output increases when there is a news shock, and with Bounded Rationality output is also higher compared to the RE case. There are several reasons that result in an increase in output when there is positive news about the future. First of all the wealth eect on labour supply is relatively small, since the parameter φ is small, so preferences are close to the GHH case, which has no wealth-eect at all. Second, capital utilization allows for an increase in the marginal product of labour (higher benets of working). Third, investment adjustment costs are necessary to increase the capital utilization rate (see Jaimovich & Rebelo, 29, for a more detailed discussion). As in the Standard Growth model higher expectations of future productivity increase consumption, since the agent tries to smooth consumption and expects to be be wealthier in the future. However, contrary to the Standard Growth model this is accompanied by an increase in hours worked, because the utilization rate increases, which raises the marginal product of labour. Thus, higher expected productivity, due to either Bounded Rationality or a news shock, increases hours worked. Finally it should be noted that with the parameter settings used here investment drops when an increase in TFP is announced 5 periods in advance, so the pure co-movement Pigou cycle (where output, consumption, hours and investment all increase) to a news shock has disappeared. This is also reected in the Bounded Rational case where investment is lower than in the RE case. 14

15 Table 5: Robustness BR amplication output φ St.dev.(Y BR ) > St.dev.(Y RE ) >, news shock (2 periods) >, news shock (3 periods) >, news shock (4 periods) >, news shock (5 periods) -.53 Y RE Y RE Y RE Y RE Robustness of amplication To test whether the increase in volatility due to Bounded Rationality is a robust result, the short run wealth-eect on labour supply (φ), is changed 3. The reason to use the wealth-eect on labour supply is that labour supply (in combination with utilization) determines whether output increases or not when productivity is expected to rise in the near future. If φ = the preferences take the form used by Greenwood, Hercowitz, and Human (1988) and there is no short run wealth eect on labour supply. When φ = 1 preferences take the form chosen by King, Plosser, and Rebelo (1988). It is tested at what value of φ Bounded Rationality stops amplifying the Business Cycle (measured by volatility in output). It is also reported at which value of φ a news shocks no longer results in an initial increase in output. The news shocks arrive in period 1, and announce that TFP will increase 2 till 5 periods later. Bounded Rationality stops amplifying output when φ >.67, which is roughly the same value at which a news shock, announced 4-5 periods in advance, no longer generates an initial increase in output. This is an indication of the relationship between news shocks and the amplication due to Bounded Rationality. 5 Conclusion The paper shows that the eect of Bounded Rationality in the form of Adaptive Beliefs Systems on the volatility in an RBC model depends on the nature of the model. In the RBC models explored here there is at least a loose relation between the initial response of output to positive news shocks and the ampli- cation of output with Bounded Rationality. However, in both the Standard Growth model and the Jaimovich and Rebelo model adding Bounded Rationality made the t to empirical business cycle statistics worse. 3 Jaimovich & Rebelo (29) also check when the co movement between variables disappears when varying adjustment costs, labour supply elasticity and capital utilization, but we are mainly interested in amplication of output due to Bounded Rationality and this depends almost solely on the wealth-eect on labour supply. 15

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