Lecture 15 Real Business Cycle Model. Noah Williams

Transcription

1 Lecture 15 Real Business Cycle Model Noah Williams University of Wisconsin - Madison Economics 702/312

2 Real Business Cycle Model We will have a shock: change in technology. Then we will have a propagation mechanism: intertemporal labor substitution and capital accumulation. We will have fluctuations as an equilibrium outcome. Basic idea: intertemporal substitution. When productivity is high, want to work more, produce more. When it is low, the reverse. Changes in productivity drive output. Capital accumulation makes the impact of a shock for several periods.

3 A Competitive Equilibrium This economy has a unique competitive equilibrium. This economy satisfies the conditions that assure that both welfare theorems hold. Why is this important? Practical: We can solve instead the Social Planner s Problem associated with it. Normative: Business cycles in the model are efficient. Fluctuations are the optimal response to a changing environment. They are not sufficient for inefficiencies or for government intervention. In this model the government can only worsen the allocation.

4 Equilibrium Conditions Euler equation under uncertainty: u C (C t, 1 N t) = βe t [u C (C t+1, 1 N t+1) (1 + z t+1f K (K t+1, N t+1) δ)] Labor market optimality: Goods market clearing: u l (C t, 1 N t ) u C (C t, 1 N t ) = z tf N (K t, N t ) K t+1 = z t F(K t, N t ) + (1 δ)k t C t Need to specify evolution of TFP z t in order to form expectations

5 Evolution of the Technology z t changes randomly over time. Ignore growth and just think of fluctuations around a trend. We assume it follows the process: log z t = ρ log z t 1 + ε t ε t N (0, σ 2 ) This process is called AR(1): an autoregression of order 1. The parameter ρ governs how persistent are the changes in TFP. If ρ = 1 they are permanent. If 0 < ρ < 1 they are persistent but eventually die out.

6 Examples of TFP Processes ρ=0 ρ=0.7 ρ=

7 Solving the Model In general the model does not have a known paper and pencil analytic solution. Analysis of the model requires some approximations (such as linearization) or numerical analysis. Based on numerical solution of the model, run Monte Carlo simulation to characterize distribution of equilibrium outcomes. Modern macroeconomics is quantitative

8 Solving the Model in a Special Case There is one known case where we can work out an explicit solution. Set δ = 1 (full depreciation) use our Cobb-Douglas production, and log utility: u(c, 1 N ) = (1 a) log C + a log(1 N ). Specialize the key equilibrium conditions: ac t = (1 α) z t Kt α Nt α (1 a)1 N t 1 C t = βe K t+1 = z t K α t N 1 α t [ αzt+1 K α 1 t+1 N 1 α t+1 C t+1 C t ]

9 Make the following guesses: C t = (1 s)y t, N t = N Constant saving rate s, constant labor supply N. Substitute into conditions: N 1 α a(1 s)z t Kt α (1 a)(1 N = (1 α) z t Kt α N α. ) [ 1 αz t+1 Kt+1 α 1 (1 s)z t Kt α = βe N 1 α (1 s)z t+1 Kt+1 α [ ] α = βe (1 s)k t+1 [ α = βe (1 s)sz t Kt α s = βα N 1 α N 1 α N 1 α ] ]

10 Implications This special case is then similar to the Solow model: constant savings rate. Constant labor supply (no growth). Difference is random shocks. Now K t+1 = sy t, so Taking logs: N 1 α Y t+1 = z t+1 Kt+1 α = z t+1 (sy t ) α N 1 α. log Y t+1 = µ + log z t+1 + α log Y t = µ + ρ log z t + α log Y t + ε t+1. where µ = α log s + (1 α) log N

11 Implications: Output Persistence log Y t+1 = µ + ρ log z t + α log Y t + ε t+1. Output and technology together follow a (vector) AR(1). Can simplify further, using: So then: log z t = log Y t µ α log Y t 1 log Y t+1 = (1 ρ)µ + (ρ + α) log Y t αρ log Y t 1 + ε t+1. Output follows an AR(2) process. Output is persistent because of the TFP shocks and because of capital accumulation.

12 Output and TFP Co-movements 4 Ouput (black) and TFP (red), ρ = Ouput (black) and TFP (red), ρ =

13 Simulations from a Quantitative Version We have seen the qualitative behavior of the model, showing that the real business cycle model is consistent with the data. Apart from the special case we studied, to fully solve the model we need to use numerical methods. Calibrate the model: choose parameters to match some key economic data. Example: set β so that steady state real interest rate matches US data. Program up on computer and simulate: use random number generator to draw technology shocks, feed them through the model. Compute correlations and volatilities and compare to US data.

14 Calibrating an RBC Model This problem will show how to choose some parameters of a RBC model to match the data, a process known as calibration. Suppose preferences are given by: β t (1 + n) t [(1 a) log c t + a log(1 N t)] t=0 here n > 0 is the population growth rate and c t and N t are per capita consumption and hours. Suppose labor-augmenting technology grows at rate g so A t = (1 + g) t. Thus the aggregate resource constraint is: c t + I t = (1 + g) (1 α)t k α t N 1 α t, where I t is per capita investment an k t is the per capita capital stock. Finally the law of motion for the capital in per capita terms is: (1 + g)(1 + n)k t+1 = (1 δ)k t + I t 1 Working directly with the social planner s problem, find the first order condition for hours worked and also find the Euler equation for the optimal consumption allocation.

15 We will use that for Cobb-Douglas production F K = αy /K, F N = (1 α)y /N. The Lagrangian is L = β t (1 + n) {(1 t a) log c t + a log(1 N t) t=0 λ t [ ct + (1 + g)(1 + n)k t+1 (1 δ)k t (1 g) (1 a)t k α t N 1 α t where we have already substituted for investment. Now the FOC are 1 a = λ t c t a y t = (1 α)λ t 1 N t N [ t ] (1 + g)(1 + n)λ t = βλ t+1(1 + n) 1 δ + α yt+1 k t+1 Now let us consolidate these three equations by eliminating the λs, ] } a = (1 α) 1 a y t (1) 1 N t c t N [ t ] (1 + g) ct+1 = β 1 δ + α yt+1 (2) c t k t+1

16 2. This model has a balanced growth path (BGP) in which hours worked N t is constant and all other per capita variables grow at the constant rate g, i.e. k t+1 = (1 + g)k t and so on. Using the two relations derived in part (a) and the law of motion for capital, find three equations relating the hours N, the capital/output ratio k/y, the consumption/output ratio c/y, and the investment/capital ratio I /k to each other and the parameters of the model. First, we divide the law of motion for capital by k t, and use k t+1/k t = 1 + g to obtain (1 + g) 2 (1 + n) = 1 δ + I (3) k Then using (1), Finally from (2), a 1 N = (1 α) 1 a N y c [ (1 + g) 2 = β 1 δ + α y ] k (4) (5)

17 3. Suppose α = 0.4, n = and g = , which are estimated from US data. 1 Given a value of I /k = in the data, find a value of δ consistent with this in the BGP. Using (3), we obtain δ = Given a value of k/y = 3.32 and your value of δ find a value of β from the BGP relations. Now using (5) and the previously obtained value of δ, we can calculate β = Given a value of N = 0.31 and y/c = 1.33 find a value of a from the BGP relations. Finally from (4), a =

18 Figure Small shocks and large cycles Abel/Bernanke, Macroeconomics, 2001 Addison Wesley Longman, Inc. All rights reserved

19 0.32 Labor

20 3.19 Capital

21 0.64 Output

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30 Assessment of the Basic Real Business Model It accounts for a substantial amount of the observed fluctuations. Accounts for the covariances among a number of variables. Has some problems accounting for hours worked, consumption volatility. Are fluctuations in TFP really productivity fluctuations? Factor utilization rates vary over the business cycle. During recessions, firms reduce the number of shifts. Similarly, firms are reluctant to fire trained workers. Neither is well-measured show up in the Solow residual. There is no direct evidence of technology fluctuations. Is intertemporal labor supply really so elastic? All employment variation in the model is voluntary, driven by intertemporal substitution. Deliberate monetary policy changes appear to have real effects.

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