Business Cycles Theory and Policy

Come studiareCerca un argomento oppure apri un capitolo dall’indice. I titoli ti portano direttamente alla spiegazione corretta.

7. Solving the linear rational-expectations model

Introduction

Solving a forward-looking linear model means finding expectations and decision rules consistent with every equilibrium condition. Unlike a static linear system, arbitrary algebraic solutions are not equally admissible: the selected process must be stable and respect the predetermined state variables.

The method of undetermined coefficients conjectures policy functions linear in the beginning-of-period capital stock and observed technology. Substitution into the log-linearized Euler, labor, production and resource equations identifies coefficient restrictions. The stable root is selected to obtain the economically relevant recursive equilibrium.

The same linear rational-expectations system can be expressed in Harald Uhlig's matrix-based toolkit and implemented numerically in Matlab or Dynare. These computational environments solve for response coefficients, assess dynamic stability and generate model-implied statistical moments.

Integrated theoretical explanation

Log-linear equilibrium equations are not yet a complete economic solution because they include expectations about variables at future dates. A recursive equilibrium expresses current decisions as functions of states known when those decisions are made. In the RBC benchmark, the key states are installed capital from the previous period and the current technology state. Installed capital is predetermined by earlier investment; technology is exogenous but observed within the period. Consumption, hours and the choice of new capital adjust to these states according to decision rules that must satisfy both current and expected future equilibrium conditions.

The method of undetermined coefficients starts by proposing linear policy functions. For example, write k̂ₜ = λkk k̂ₜ₋₁ + λkz ẑₜ, with analogous expressions for consumption, output and labor. The coefficients are initially unknown, but linearity suggests that any stable first-order solution can be represented in this form. These equations have an immediate economic interpretation: one coefficient describes the effect of inheriting an additional unit of capital relative to the steady state; the other describes the effect of a higher productivity state when inherited capital is held fixed.

Substitute the proposed rules into production, the labor condition, the resource constraint and the forward-looking Euler condition. Because the technology state itself obeys ẑₜ₊₁ = ρẑₜ + uₜ₊₁, rational expectations can be computed using the conditional mean of future shocks. Collect all terms involving k̂ₜ₋₁ and all terms involving ẑₜ. For the identities to hold for every possible state combination, each corresponding coefficient must satisfy a separate algebraic restriction. The resulting nonlinear equations in the λ coefficients are solved together, and the economically admissible stable branch is selected.

Stability matters because the model contains a predetermined asset and forward-looking choices. An algebraic root that causes the state vector to explode under small disturbances may violate the transversality conditions of the household problem and fail to describe a reasonable equilibrium around steady state. A stable solution lets the economy converge after temporary shocks when the underlying exogenous process is stationary. This is not merely a preference for smooth-looking graphs: it is the dynamic consistency requirement connecting optimal saving, expected returns and the time path of accumulated capital.

The calibrated decision coefficients make the economic responses concrete. In the reported numerical example, k̂ₜ ≈ 0.95k̂ₜ₋₁ + 0.15ẑₜ, while ĉₜ ≈ 0.53k̂ₜ₋₁ + 0.45ẑₜ and ŷₜ ≈ 0.05k̂ₜ₋₁ + 1.97ẑₜ. Labor responds approximately as n̂ₜ ≈ −0.48k̂ₜ₋₁ + 1.52ẑₜ, and investment has a much stronger technology loading. These are local responses based on the particular calibration, not general coefficients that should be transplanted into any economy.

Uhlig's toolkit provides a matrix representation of the same system. Endogenous state variables, non-state controls and exogenous shocks are separated into vectors. Static equations link contemporaneous and lagged states to controls, while expectational equations include future values. Solving the matrix system produces coefficient matrices P and Q for the endogenous state transition and R and S for the remaining control variables. This approach scales to models with more variables than can comfortably be solved by hand and gives numerical diagnostics, including candidate roots, stability information and model moments.

The practical computer workflow follows the same conceptual sequence as the hand derivation: choose parameter values, calculate the deterministic steady state, declare the ordering of variables, construct equation matrices, verify their dimensions and call the solver. Output matrices must be interpreted against the declared variable order; swapping consumption and investment in a software vector can make otherwise correct numbers economically meaningless. The examples also show how a calibration can be used to obtain autocorrelations and standard deviations after filtering simulated data.

Dynare is an alternative environment for specifying and solving dynamic stochastic general-equilibrium systems. Rather than treating a software package as a source of economic assumptions, the researcher supplies the equations, calibrated or estimated parameters, shock process and requested analyses. The quality of the outcome still depends on whether the equations are correctly timed and whether the steady state and equilibrium structure are coherent. Numerical software is an implementation of a mathematical model, not a substitute for specifying the model or interpreting its results.

Formal framework and essential relationships

{\hat{k}}_{t} = λ_{k k} {\hat{k}}_{t - 1} + λ_{k z} {\hat{z}}_{t}

The endogenous state evolves according to inherited capital and the current productivity state. Its response coefficients emerge by substituting the conjectured laws into the equilibrium system.

{\hat{c}}_{t} = λ_{c k} {\hat{k}}_{t - 1} + λ_{c z} {\hat{z}}_{t}, {\hat{y}}_{t} = λ_{y k} {\hat{k}}_{t - 1} + λ_{y z} {\hat{z}}_{t}

Consumption and output are controls that move immediately in response to the states. Similar equations characterize labor and investment.

Endogenous variable	Loading on k̂ₜ₋₁	Loading on ẑₜ
Capital, k̂ₜ	0.95	0.15
Consumption, ĉₜ	0.53	0.45
Output, ŷₜ	0.05	1.97
Hours, n̂ₜ	−0.48	1.52
Investment, îₜ	−1.40	6.52

The policy-function coefficients are displayed in rounded form for readability. A numerical matrix implementation can report additional decimal places, so minor rounding differences do not indicate a different model.

𝐱_{t} = P 𝐱_{t - 1} + Q 𝐳_{t}, 𝐲_{t} = R 𝐱_{t - 1} + S 𝐳_{t}

Uhlig’s solution uses a matrix law for states x and controls y, with exogenous technology z. The transition eigenvalues and estimated matrices show whether the selected equilibrium is stable.

Explanations, derivations and empirical applications

Linear System Of Equations

0 = E_{t} [β α \frac{\overline{y}}{\overline{k}} ({\hat{y}}_{t + 1} - {\hat{k}}_{t}) + {\hat{c}}_{t} - {\hat{c}}_{t + 1}]

0 = - {\hat{c}}_{t} + {\hat{y}}_{t} - {\hat{n}}_{t}

0 = - \overline{y} {\hat{y}}_{t} + \overline{c} {\hat{c}}_{t} + \overline{k} {\hat{k}}_{t} - (1 - δ) \overline{k} {\hat{k}}_{t - 1}

0 = - {\hat{y}}_{t} + {\hat{z}}_{t} + (1 - α) {\hat{n}}_{t} + α {\hat{k}}_{t - 1}

{\hat{z}}_{t} = ρ {\hat{z}}_{t - 1} + u_{t}

Method Of Undetermined Coefficients

Installed capital and current productivity are the state variables. Conjecture linear decision rules:

{\hat{y}}_{t} = λ_{y k} {\hat{k}}_{t - 1} + λ_{y z} {\hat{z}}_{t}

{\hat{k}}_{t} = λ_{k k} {\hat{k}}_{t - 1} + λ_{k z} {\hat{z}}_{t}

{\hat{c}}_{t} = λ_{c k} {\hat{k}}_{t - 1} + λ_{c z} {\hat{z}}_{t}

{\hat{n}}_{t} = λ_{n k} {\hat{k}}_{t - 1} + λ_{n z} {\hat{z}}_{t}

Eliminate output and labor to reduce the model to capital, consumption and technology:

0 = - {\hat{k}}_{t} + a_{1} {\hat{k}}_{t - 1} - a_{2} {\hat{c}}_{t} + a_{3} {\hat{z}}_{t}

0 = E_{t} [{\hat{c}}_{t} + a_{4} {\hat{k}}_{t} - a_{5} {\hat{c}}_{t + 1} + a_{6} {\hat{z}}_{t + 1}]

{\hat{z}}_{t} = ρ {\hat{z}}_{t - 1} + u_{t}

a_{1} = \frac{\overline{y}}{\overline{k}} + 1 - δ, a_{3} = \frac{\overline{y}}{α \overline{k}}, a_{2} = \frac{\overline{c}}{\overline{k}} + (1 - α) a_{3}

a_{4} = 0, a_{6} = β \frac{\overline{y}}{\overline{k}}, a_{5} = 1 + (1 - α) a_{6}

Substitute the conjectured capital and consumption rules and use the conditional expectation of productivity:

E_{t} {\hat{z}}_{t + 1} = ρ {\hat{z}}_{t}

0 = (- λ_{k k} + a_{1} - a_{2} λ_{c k}) {\hat{k}}_{t - 1} + (- λ_{k z} - a_{2} λ_{c z} + a_{3}) {\hat{z}}_{t}

0 = (λ_{c k} + a_{4} λ_{k k} - a_{5} λ_{c k} λ_{k k}) {\hat{k}}_{t - 1} + (λ_{c z} + a_{4} λ_{k z} - a_{5} λ_{c k} λ_{k z} - a_{5} λ_{c z} ρ + a_{6} ρ) {\hat{z}}_{t}

Next Steps

The forward-looking consumption rule must be evaluated using next period’s endogenous capital state:

E_{t} {\hat{c}}_{t + 1} = λ_{c k} {\hat{k}}_{t} + λ_{c z} ρ {\hat{z}}_{t}

0 = {\hat{c}}_{t} + (a_{4} - a_{5} λ_{c k}) {\hat{k}}_{t} + (- a_{5} λ_{c z} + a_{6}) ρ {\hat{z}}_{t}

Substitute the capital and consumption rules again, then collect the coefficients on inherited capital and current technology separately.

Comparing Coefficients

0 = - λ_{k k} + a_{1} - a_{2} λ_{c k}, 0 = λ_{c k} + a_{4} λ_{k k} - a_{5} λ_{c k} λ_{k k}

0 = λ_{k k}^{2} - (a_{1} - \frac{a_{2} a_{4}}{a_{5}} + \frac{1}{a_{5}}) λ_{k k} + \frac{a_{1}}{a_{5}}

λ_{k k} = \frac{1}{2} [B \pm \sqrt{B^{2} - 4 a_{1} / a_{5}}], B = a_{1} - \frac{a_{2} a_{4}}{a_{5}} + \frac{1}{a_{5}}

λ_{c k} = \frac{a_{1} - λ_{k k}}{a_{2}}

Choose the economically admissible stable root. If the absolute product of the two roots exceeds one, at most one can lie strictly inside the unit circle.

0 = - λ_{k z} - a_{2} λ_{c z} + a_{3}

0 = λ_{c z} + a_{4} λ_{k z} - a_{5} λ_{c k} λ_{k z} - a_{5} λ_{c z} ρ + a_{6} ρ

λ_{c z} = \frac{(a_{4} - a_{5} λ_{c k}) a_{3} + a_{6} ρ}{a_{4} a_{2} - a_{5} λ_{c k} a_{2} + a_{5} ρ - 1}

λ_{k z} = a_{3} - a_{2} λ_{c z}

These restrictions follow from matching the coefficient on current productivity, independently of the coefficient on lagged capital.

Calibrated policy functions

Variable	Loading on inherited capital	Loading on productivity
Capital	0.95	0.15
Consumption	0.53	0.45
Output	0.05	1.97
Hours	−0.48	1.52
Investment	−1.40	6.52

Rounded illustrative coefficients describe local responses in log deviations.

Uhlig’s Toolkit

The matrix representation distinguishes equations that hold within a period from equations that contain expectations of future decisions. This separation allows a general numerical solution with state and control vectors.

0 = A x_{t} + B x_{t - 1} + C y_{t} + D z_{t}

0 = E_{t} [F x_{t + 1} + G x_{t} + H x_{t - 1} + J y_{t + 1} + K y_{t} + L z_{t + 1} + M z_{t}]

z_{t + 1} = N z_{t} + u_{t + 1}

The vectors distinguish endogenous states x, other endogenous variables y and exogenous states z. Matrix dimensions and rank conditions must be compatible with the equation and variable counts; the selected solution must respect the predetermined states and stability requirements.

Solution

A stable recursive representation maps predetermined capital and current productivity into optimal control decisions. Its coefficients and matrix roots must satisfy the dynamic equilibrium equations and the timing of endogenous states.

The matrix solution has the recursive form xₜ = Pxₜ₋₁ + Qzₜ for endogenous states and yₜ = Rxₜ₋₁ + Szₜ for other endogenous variables. The matrices summarize how predetermined states and exogenous disturbances determine current allocations. Once they are known, impulse responses and stochastic simulations follow from repeated application of these transition laws.

Log-linearized System Of Equations

0 = E_{t} [β α \frac{\overline{y}}{\overline{k}} ({\hat{y}}_{t + 1} - {\hat{k}}_{t}) + {\hat{c}}_{t} - {\hat{c}}_{t + 1}]

0 = - {\hat{c}}_{t} + {\hat{y}}_{t} - {\hat{n}}_{t}

0 = - \overline{y} {\hat{y}}_{t} + \overline{c} {\hat{c}}_{t} + \overline{k} {\hat{k}}_{t} - (1 - δ) \overline{k} {\hat{k}}_{t - 1}

0 = - {\hat{y}}_{t} + {\hat{z}}_{t} + (1 - α) {\hat{n}}_{t} + α {\hat{k}}_{t - 1}

{\hat{z}}_{t} = ρ {\hat{z}}_{t - 1} + u_{t}

0 = \overline{k} {\hat{k}}_{t} - \overline{i} {\hat{i}}_{t} - (1 - δ) \overline{k} {\hat{k}}_{t - 1}

Uhlig’s Toolkit

0 = A x_{t} + B x_{t - 1} + C y_{t} + D z_{t}

0 = E_{t} [F x_{t + 1} + G x_{t} + H x_{t - 1} + J y_{t + 1} + K y_{t} + L z_{t + 1} + M z_{t}]

z_{t + 1} = N z_{t} + u_{t + 1}

x_{t} = {\hat{k}}_{t}, z_{t} = {\hat{z}}_{t}, y_{t} = {[\begin{matrix} {\hat{c}}_{t} & {\hat{y}}_{t} & {\hat{n}}_{t} & {\hat{i}}_{t} \end{matrix}]}^{⊤}

The first block contains labor, resource, production and accumulation equations; the second contains the Euler equation; the final block is the exogenous transition.

Applying The Toolkit

For the four-control representation, the expectational matrices in the Euler block are:

G = - β α \frac{\overline{y}}{\overline{k}}, J = [\begin{matrix} - 1 & β α \overline{y} / \overline{k} & 0 & 0 \end{matrix}], K = [\begin{matrix} 1 & 0 & 0 & 0 \end{matrix}]

F = H = L = M = 0

A = [\begin{matrix} 0 \\ \overline{k} \\ 0 \\ \overline{k} \end{matrix}], B = [\begin{matrix} 0 \\ - (1 - δ) \overline{k} \\ α \\ - (1 - δ) \overline{k} \end{matrix}]

C = [\begin{matrix} - 1 & 1 & - 1 & 0 \\ \overline{c} & - \overline{y} & 0 & 0 \\ 0 & - 1 & 1 - α & 0 \\ 0 & 0 & 0 & - \overline{i} \end{matrix}], D = [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}]

Rows correspond respectively to labor equilibrium, resources, production and capital accumulation; controls are ordered as consumption, output, labor and investment.

Numerical implementation

This implementation adds the gross capital return to the control vector, so its matrices differ in dimension from the four-control representation above. Keep this ordering consistent when interpreting numerical output. The variable named “interest” is the gross return r + 1 − δ.

% Quarterly RBC calibration
alpha = 0.36;
delta = 0.0225;
betta = 0.99;
rho_z = 0.95;
sigma_eps = 0.007;
N_bar = 1/3;
Z_bar = 1;

% Deterministic steady state
R_bar = 1/betta;  % Gross return, rental return plus 1-delta
YK_bar = (R_bar + delta - 1)/alpha;
K_bar = (YK_bar/Z_bar)^(1/(alpha-1))*N_bar;
I_bar = delta*K_bar;
Y_bar = YK_bar*K_bar;
C_bar = Y_bar-I_bar;
phi = (1-alpha)*Y_bar/(N_bar*C_bar);

% State: capital. Controls: consumption, output, labor,
% gross return, investment. Exogenous state: technology.
VARNAMES = char('capital','consumption','output','labor', ...
                'interest','investment','technology');
AA = [0; -K_bar; 0; 0; 0];
BB = [0; (1-delta)*K_bar; alpha; 0; -alpha*YK_bar];
CC = [-C_bar, Y_bar, 0, 0, -I_bar;
       0, 0, 0, 0, I_bar;
       0, -1, 1-alpha, 0, 0;
      -1, 1, -1, 0, 0;
       0, alpha*YK_bar, 0, -R_bar, 0];
DD = [0; 0; 1; 0; 0];
FF = 0; GG = 0; HH = 0;
JJ = [-1, 0, 0, 1, 0];
KK = [1, 0, 0, 0, 0];
LL = 0; MM = 0;
NN = rho_z;
Sigma = sigma_eps^2;
[l_equ,m_states] = size(AA);
[~,n_endog] = size(CC);
[~,k_exog] = size(DD);

% Analysis options for the toolkit
PERIOD = 4;
GNP_INDEX = 3;
IMP_SELECT = 1:7;
DO_SIMUL = 0;
DO_MOMENTS = 1;
HP_SELECT = 1:(m_states+n_endog+k_exog);
% With the toolkit on the MATLAB path, invoke its driver:
% do_it;

The coefficient matrices are reconstructed from the equilibrium equations. The rounded discount factor is set to 0.99, so ratios calculated from the code can differ slightly from calibration targets stated with fewer decimal places.

Calling The Toolkit

In a computational implementation, the researcher must make the model file and solver accessible to the numerical environment, supply the equilibrium matrices and run the routine. The economic content is in the model equations and parameter choices, not in the computer’s directory path. Output should be checked for consistency with the stated variable ordering and for stable roots.

Toolkit Output

The endogenous state is capital; the exogenous state is technology. Candidate transition roots and the selected stable root are:

Root	Absolute value	Selection
0.94574	0.94574	Stable root
1.0681	1.0681	Excluded unstable root

{\hat{k}}_{t} = 0.94574 {\hat{k}}_{t - 1} + 0.14719 {\hat{z}}_{t}

Control	Loading on inherited capital (R)	Loading on productivity (S)
Consumption	0.53404	0.45562
Output	0.050594	1.9678
Labor	−0.48345	1.5122
Gross return	−0.030642	0.063510
Investment	−1.4117	6.5418

These coefficients correspond to the five-control numerical representation and the stated variable ordering.

Cross-correlations of HP-filtered model series with output. Column j reports corr(vₜ₊ⱼ, Yₜ). Negative j compares past v with current output; positive j compares future v with current output.

Variable	-5	-4	-3	-2	-1	0	1	2	3	4	5
Capital	-0.48	-0.42	-0.32	-0.16	0.05	0.34	0.53	0.63	0.67	0.66	0.62
Consumption	-0.26	-0.14	0.02	0.24	0.52	0.87	0.77	0.66	0.54	0.42	0.31
Output	-0.02	0.11	0.27	0.47	0.71	1.00	0.71	0.47	0.27	0.11	-0.02
Labor	0.07	0.19	0.34	0.52	0.74	0.98	0.65	0.38	0.16	-0.01	-0.14
Gross return	0.11	0.23	0.37	0.54	0.74	0.97	0.61	0.33	0.10	-0.07	-0.19
Investment	0.04	0.17	0.32	0.51	0.73	0.99	0.67	0.41	0.19	0.02	-0.10
Technology	-0.01	0.11	0.28	0.48	0.72	1.00	0.71	0.46	0.26	0.10	-0.03

Variable	Standard deviation of HP-filtered log deviations
Capital	0.0046439
Consumption	0.0049190
Output	0.017925
Labor	0.013857
Gross return	0.00058974
Investment	0.059646
Technology	0.0091035

Values are in log-deviation units. Multiplication by 100 expresses their approximate percentage-point scale.

Dynare

Dynare is a computational framework for specifying, approximating and simulating dynamic stochastic general-equilibrium models. It automates parts of the numerical solution but still requires correctly timed equilibrium equations, a valid steady state, suitable parameter values and an explicit shock process. Its output should be interpreted as the solution of the economic model supplied by the researcher.

Synthesis

Stable rational-expectations solutions generate recursively usable decision rules, implementable via manual coefficient matching or computational toolkits.

Torna all’inizio ↑

7. Solving the linear rational-expectations model

Introduction

Integrated theoretical explanation

Formal framework and essential relationships

Explanations, derivations and empirical applications

Linear System Of Equations

Method Of Undetermined Coefficients

Next Steps

Comparing Coefficients

Calibrated policy functions

Uhlig’s Toolkit

Solution

Log-linearized System Of Equations

Uhlig’s Toolkit

Applying The Toolkit

Numerical implementation

Calling The Toolkit

Toolkit Output

Dynare

Synthesis

Un dubbio da chiarire?Studiamolo insieme.

Un dubbio da chiarire?
Studiamolo insieme.