The Real Business Cycle Model and Dynamic Stochastic General Equilibrium (DSGE) Models

The Real Business Cycle (RBC) Model

Introduction to RBC and DSGE Models

The Real Business Cycle (RBC) model is a foundational framework in modern macroeconomics, forming part of the broader class of Dynamic Stochastic General Equilibrium (DSGE) models. These models are used for policy analysis and forecasting, especially in applied macroeconomics. RBC models emphasize microfoundations, rational expectations, and the role of technology shocks in driving economic fluctuations.

• Dynamic: Models are intertemporal, considering how decisions today affect outcomes in the future.

• Stochastic: The economy is subject to random shocks, particularly productivity shocks.

• General Equilibrium: Multiple markets (goods, capital, labor) are analyzed simultaneously, with few exogenous variables.

Business Cycles

Business cycles refer to high- to medium-frequency fluctuations in economic activity, typically measured by GDP. These cycles are characterized by irregular timing and amplitude, making them difficult to predict. Business cycles affect not only GDP but also other macroeconomic variables.

• Expansion: Periods of increasing economic activity.

• Recession: Periods of declining economic activity.

• Irregularity: Cycles are not uniform in duration or magnitude.

Origins and Assumptions of RBC Models

RBC models emerged in response to dissatisfaction with Keynesian macroeconomics, particularly the lack of microfoundations and explicit expectations. The Lucas critique highlighted the importance of rational expectations. Kydland and Prescott (1982) demonstrated that models with microfoundations and rational expectations could fit empirical data well.

• Perfectly competitive markets and rational expectations.

• Technology shocks are the primary source of economic fluctuations.

• Monetary neutrality and Ricardian equivalence are assumed.

• Endogenous labor supply (constant in the long run).

• Stochastic productivity (real shocks).

Structure of DSGE Models

Model Setup

DSGE models are structural, involving economic agents such as households, firms, central banks, and governments. The setup includes describing technological and resource constraints, exogenous influences, and institutional settings.

• Households: Maximize utility subject to constraints.

• Firms: Maximize profits using production functions.

• Central Banks: May follow rules or minimize loss functions for monetary policy.

• Government: May conduct fiscal policy.

The Household Problem

Households maximize an intertemporal utility function, considering consumption, labor supply, and capital accumulation. The utility function is:

• Utility Function:

$\max_{C_{j,t}, L_{j,t}, K_{j,t+1}} E_t \sum_{t=0}^{\infty} \beta^t \left( \frac{C_{j,t}^{1-\sigma}}{1-\sigma} - \frac{L_{j,t}^{1+\phi}}{1+\phi} \right)$

• Et: Expectation operator

• β: Intertemporal discount factor

• σ: Relative risk aversion (reciprocal of intertemporal elasticity of substitution)

• φ: Marginal disutility of labor (inverse of Frisch elasticity)

Households face the following budget constraint:

$C_{j,t} + I_{j,t} = W_t L_{j,t} + R_t K_{j,t} + \Pi_t$

• Ij,t: Investment

• Wt: Wage rate

• Rt: Return on capital

• Πt: Profits

The law of motion for capital:

$K_{t+1} = (1 - \delta) K_t + I_t$

• δ: Depreciation rate

Types of Variables

• State Variables: $K_t$, $A_t$

• Control Variables: $C_t$, $L_t$

• Other Variables: $w_t$, $R_t$

Solving the Household Problem

The household problem is solved using the Lagrangian method, leading to first-order conditions (FOCs):

• Consumption FOC: $C_{j,t}^{-\sigma} = \lambda_{j,t}$

• Labor FOC: $-L_{j,t}^{\phi} + \lambda_{j,t} W_t = 0$

• Capital FOC: $-\lambda_{j,t} + \beta E_t \lambda_{j,t+1} [(1 - \delta) + E_t R_{t+1}] = 0$

These can be rewritten as:

• Intratemporal Choice: $C_{j,t}^{\sigma} L_{j,t}^{\phi} = W_t$

• Intertemporal Choice (Euler Equation): $C_{j,t}^{-\sigma} = \beta E_t [C_{j,t+1}^{-\sigma} (1 + R_{t+1} - \delta)]$

The Firm Problem

Firms operate in a perfectly competitive market and use a Cobb-Douglas production function:

$Y_{j,t} = A_t K_{j,t}^{\alpha} L_{j,t}^{1-\alpha}$

• Yt: Output

• At: Total factor productivity

• α: Capital share in production

Profit maximization leads to:

• Capital FOC: $R_t = \alpha \frac{Y_{j,t}}{K_{j,t}}$

• Labor FOC: $W_t = (1 - \alpha) \frac{Y_{j,t}}{L_{j,t}}$

Productivity Process

Productivity is modeled as an exogenous stochastic process:

$A_t = \exp(\varepsilon_{A,t}) A_{t-1}^{\rho_A}$

$\varepsilon_{A,t} \sim N(0, \sigma^2_{\epsilon})$

• ρA: Autoregressive parameter (|ρA| < 1 for stationarity)

• εA: Exogenous shock

Equilibrium Condition

The equilibrium condition for the representative household and firm is:

$Y_t = C_t + I_t$

The Non-linear RBC Model

System of Equations

The non-linear RBC model consists of eight equations:

• Labor supply: $C_t^{\sigma} L_t^{\phi} = W_t$

• Euler equation: $C_{j,t}^{-\sigma} = \beta E_t [C_{j,t+1}^{-\sigma} (1 + R_{t+1} - \delta)]$

• Law of motion of capital: $K_{t+1} = (1 - \delta) K_t + I_t$

• Production function: $Y_t = A_t K_t^{\alpha} L_t^{1-\alpha}$

• Demand for capital: $K_t = \alpha \frac{Y_t}{R_t}$

• Demand for labor: $L_t = (1 - \alpha) \frac{Y_t}{W_t}$

• Equilibrium condition: $Y_t = C_t + I_t$

• Productivity process: $A_t = \exp(\varepsilon_{A,t}) A_{t-1}^{\rho_A}$

Simplified RBC Model (Three Equations)

The model can be collapsed into three key equations:

• $C_t^{-\sigma} = \beta E_t \left[ C_{t+1}^{-\sigma} \left( \alpha A_{t+1} K_{t+1}^{\alpha-1} L_{t+1}^{1-\alpha} + 1 - \delta \right) \right]$

• $C_t^{\sigma} L_t^{\phi} = (1 - \alpha) A_t K_t^{\alpha} L_t^{-\alpha}$

• $K_{t+1} = (1 - \delta) K_t + A_t K_t^{\alpha} L_t^{1-\alpha} - C_t$

Steady State Analysis

At the steady state, endogenous variables are constant. Key steady-state relationships include:

• $A^* = 1$

• $R^* = \frac{1}{\beta} - (1 - \delta)$

• $Y^* = \left( \frac{R^*}{R^* - \delta \alpha} \right)^{\frac{\sigma}{\sigma + \phi}} \left( W^* \left( \frac{W^*}{1 - \alpha} \right)^{\phi} \right)^{\frac{1}{\sigma + \phi}} K^* L^*$

• $L^* = \left( \frac{R^*}{\alpha} \right)^{\frac{1}{\alpha - 1}}$

• $W^* = (1 - \alpha) \left( \frac{K^*}{L^*} \right)^{\alpha}$

• $\delta = \frac{I^*}{K^*}$

Additional relationships for consumption, labor, capital, and investment are derived from these steady-state conditions.

Non-linear vs Linear Models

RBC models are systems of non-linear difference equations, which generally lack closed-form solutions. Standard solution techniques involve log-linear approximation around the steady state and applying algorithms for solving linear rational expectations models (e.g., Blanchard-Kahn algorithm).

• Log-linearization: Simplifies the system for computational analysis.

• Software tools: RISE can linearize models automatically, or linearization can be done manually for greater transparency.

Practical Application: RISE Software

Model Implementation in RISE

RISE is a software tool used for simulating and estimating DSGE models. Models are written in specific file formats (.rs, .rz, .dsge) and loaded into RISE for analysis.

• Model file extensions: .rs, .rz, .dsge

• Building a model object: m=rise(modelFileName) or m=dsge(modelFileName)

• Documentation: Methods can be listed using methods(xxx.empty(0))

• Stochastic shocks: All shocks are iid N(0,1)

RISE Model Blocks

RISE models are structured using specific blocks:

• Compulsory: @endogenous (or var), @exogenous (or varexo), @model, @parameters, @observables (or varobs), @planner objective

• Optional: log vars, steady state model, parameterization, parameter restrictions

RISE Model Syntax

• Equations end with a semicolon.

• Timing notation: C, C{0}, C(0), C{t}, C(t) for Ct; C{−1}, C(−1), C{t−1}, C(t−1) for Ct−1; C{1}, C(1), C{+1}, C(+1), C{t+1}, C(t+1) for Ct+1.

• Steady state notation: C{stst}, C(stst), steady state(C), C{steady state}.

• # for definitions, ! for endogenous probabilities, ? for inequality restrictions.

Summary Table: Key Equations of the RBC Model

Equation	Description
$C_t^{\sigma} L_t^{\phi} = W_t$	Labor supply (intratemporal choice)
$C_{j,t}^{-\sigma} = \beta E_t [C_{j,t+1}^{-\sigma} (1 + R_{t+1} - \delta)]$	Euler equation (intertemporal choice)
$K_{t+1} = (1 - \delta) K_t + I_t$	Law of motion of capital
$Y_t = A_t K_t^{\alpha} L_t^{1-\alpha}$	Production function
$K_t = \alpha \frac{Y_t}{R_t}$	Demand for capital
$L_t = (1 - \alpha) \frac{Y_t}{W_t}$	Demand for labor
$Y_t = C_t + I_t$	Equilibrium condition
$A_t = \exp(\varepsilon_{A,t}) A_{t-1}^{\rho_A}$	Productivity process

Example: Policy Analysis Using RBC Models

Suppose a central bank considers lowering the inflation target. RBC and DSGE models can simulate the effects on consumption, investment, output, and inflation over time, providing valuable insights for policy decisions.

Additional info: The notes also introduce practical aspects of model calibration and simulation using Matlab and RISE, emphasizing the importance of computational tools in modern macroeconomic analysis.