Foundations of Dynamic Stochastic General Equilibrium Model

Author

Affiliation

Hirotaka Fukui

Kobe University, Graduate School of Economics

Modified

November 24, 2025

Important

All rights reserved. All errors in this document are the responsibility of the author. Although every effort has been made to ensure the accuracy of this material, I cannot guarantee that it is free of errors or misprints. Use at your own risk. If you find any errors, please get in touch with me.

Starting Point

In a dynamic stochastic general equilibrium framework, the optimality conditions and market clearing conditions for households, firms, and policymakers can be summarized as a system of nonlinear expectational equations:

\[ \mathbb{E} F(x_{t+1}, y_{t+1}, x_t, y_t, z_t; \theta) = 0 \]

where

$x_t \in \mathbb{R}^{n_x}$ denotes the state (predetermined) variables
$y_t \in \mathbb{R}^{n_y}$ represents the jump (control) variables
$z_t \in \mathbb{R}^{n_z}$ stands for the exogenous processes, which typically evolve according to

\[ z_{t+1} = \Lambda(\theta) z_t + \Sigma(\theta) \varepsilon_{t+1}, \quad \varepsilon_t \sim \mathcal{N}(0, I). \]

$\theta$ is the parameter vector collecting structural parameters of the model.

The function $F(\cdot) = 0$ is a compact representation of a set of equilibrium conditions, including the Euler equations, labor supply conditions, price-setting equations, capital accumulation dynamics, policy rules, etc.

Example: A structure incorporating the Euler equation of a representative household

An illustrative framework embedding the Euler equation characterizing the intertemporal optimization behavior of a representative household.

The representative household maximizes expected lifetime utility:

\[ \max_{c_{t}, k_{t+1}, n_{t}} \mathbb{E}_{0} \sum^{\infty}_{t=0} \beta^{t} U(c_{t}, 1-n_{t}) \]

subject to the intertemporal budget constraint:

\[ c_{t}+k_{t+1} = (1-\delta)k_{t}+w_{t}n_{t}+r_{t}k_{t} \]

First-Order Conditions (Interior Solution)

From the optimallity conditions, we obtain:

$\lambda_{t} = U_{c}(c_{t}, 1-n_{t})$
$\lambda_{t} = -U_{n}(c_{t}, 1-n_{t})/w_{t}$
${t} = {t}[ {t+1} (r{t+1}+1-) ] $

From equations (1) and (3), we obtain the canonical Euler equation, which characterizes the intertemporal trade-off in consumption.

By substituting equations (1) through (3) into the system and rearranging, the model can be expressed in the form of a moment condition:

\[ \mathbb{E}_{t}\Biggl[ F(x_{t+1}, y_{t+1}, x_{t}, y_{t}, z_{t}; \theta) \Biggr] = 0 \]

This formulation embeds the Euler condition within a broader structural framework suitable for estimation or simulation.

Balanced growth Path

In the deterministic version of the model, where stochastic shocks are turned off, the growth trend is governed by the following law of motion:

\[ z_{t+1} = \Lambda(\theta)z_{t}+\Sigma(\theta)\epsilon_{t+1} \]

In this setting, many endogenous variables evolve proportionally to the growth trend, typically taking the form:

\[ x_{t} = \tilde{x} \cdot z_{t} \]

This formulation implies that the dynamics of key macroeconomic variables are scaled by the underlying growth process, reflecting balanced growth path behavior in the absence of uncertainty.

Definition 1 (Balanced Growth Path) A path $\{C_t, K_t, Y_t\}_{t=0}^{\infty}$ is called a Balanced Growth Path (BGP) if there exists a constant $g \in \mathbb{R}$ such that \[ \begin{align} \ln \frac{C_{t+1}}{C_t} = \ln \frac{K_{t+1}}{K_t} = \ln \frac{Y_{t+1}}{Y_t} = g \quad \forall t \ge 0. \end{align} \]

Equivalently, $C_t = C_0 e^{gt}$, $K_t = K_0 e^{gt}$, and $Y_t = Y_0 e^{gt}$.

Theorem 1 (Balanced Growth Path: Existence and Uniqueness) Consider a discrete-time economy described by the following system: \[ \begin{align} K_{t+1} &= F(K_t, L_t) - C_t - \delta K_t, \\ L_{t+1} &= (1+n)L_t, \\ U_0 &= \sum_{t=0}^{\infty} \beta^t u(C_t), \end{align} \] where $K_t$ is capital, $L_t$ is labor, $C_t$ is consumption, $\delta \in (0,1)$ is the depreciation rate, and $\beta \in (0,1)$ is the discount factor.

Then the following two statements are equivalent:

There exists at least one Balanced Growth Path in the economy.
The production function $F$ and the utility function $u$ satisfy the following monotonic scalar transformation conditions: \[ \begin{align} F(aK, aL) &= a\, F(K,L), \quad \forall a>0, \\ u(aC) &= \phi(a)\, u(C), \quad \forall a>0, \end{align} \] where $\phi:\mathbb{R}_+ \to \mathbb{R}_+$ is continuous and strictly monotone increasing.

Moreover, if $\phi(a)$ is strictly monotone, then the Balanced Growth Path is unique.

Proof (General Framework).

Production

Aggregate output is produced according to the following production function:

\[ Y_{t} = A_{t}F(K_{t}, N_{t}L_{t}) \]

where:

$A_{t}$ denotes the level of technology,
$K_{t}$ represents the aggregate capital,
$N_{t}, L_{t}$ is the population and the employment rate, respectively.

Assumption T1 (Constant Returns to Scale)

The production function $F: \mathbb{R}^2_{+} \rightarrow \mathbb{R}_{+}$ is assumed to be continuous, quasi-concave, and homogeneous of degree one in reproducible factors $(K, NL)$, satisfying constant returns to scale (CRS).

Exogenous Growth

Technological progress and population growth follow deterministic exponential trends:

\[ \begin{align} A_{t+1} & = (1+g_{A})A_{t}, \\ N_{t+1} & = (1+g_{N})N_{t} \end{align} \]

Assumption T2

The growth rates $g_{A}, g_{N} \geq 0$ are constant and exogenously given.

Capital Accumulation

Capital evolves according to the standard law of motion:

\[ K_{t+1} = (1-\delta)K_{t}+I_{t} \]

Preferences

The representative household maximizes lifetime utility given by:

\[ U = \sum^{\infty}_{t=0} \beta^t u(C_{t}, 1-L_{t}) \]

with the period utility function specified as a generalized separable CRRA form:

--- title: Foundations of Dynamic Stochastic General Equilibrium Model engine: julia format: html: code-tools: true --- ::: {.callout-important} All rights reserved. All errors in this document are the responsibility of the author. Although every effort has been made to ensure the accuracy of this material, I cannot guarantee that it is free of errors or misprints. Use at your own risk. If you find any errors, please get in touch with me. ::: # Starting Point In a dynamic stochastic general equilibrium framework, the optimality conditions and market clearing conditions for households, firms, and policymakers can be summarized as a system of nonlinear expectational equations: $$ \mathbb{E} F(x_{t+1}, y_{t+1}, x_t, y_t, z_t; \theta) = 0 $$ where - $x_t \in \mathbb{R}^{n_x}$ denotes the **state (predetermined) variables** - $y_t \in \mathbb{R}^{n_y}$ represents the **jump (control) variables** - $z_t \in \mathbb{R}^{n_z}$ stands for the **exogenous processes**, which typically evolve according to $$ z_{t+1} = \Lambda(\theta) z_t + \Sigma(\theta) \varepsilon_{t+1}, \quad \varepsilon_t \sim \mathcal{N}(0, I). $$ - $\theta$ is the parameter vector collecting structural parameters of the model. The function $F(\cdot) = 0$ is a compact representation of a set of equilibrium conditions, including the Euler equations, labor supply conditions, price-setting equations, capital accumulation dynamics, policy rules, etc. ## Example: A structure incorporating the Euler equation of a representative household An illustrative framework embedding the Euler equation characterizing the intertemporal optimization behavior of a representative household. The representative household maximizes expected lifetime utility: $$ \max_{c_{t}, k_{t+1}, n_{t}} \mathbb{E}_{0} \sum^{\infty}_{t=0} \beta^{t} U(c_{t}, 1-n_{t}) $$ subject to the intertemporal budget constraint: $$ c_{t}+k_{t+1} = (1-\delta)k_{t}+w_{t}n_{t}+r_{t}k_{t} $$ ### First-Order Conditions (Interior Solution) From the optimallity conditions, we obtain: 1. $\lambda_{t} = U_{c}(c_{t}, 1-n_{t})$ 2. $\lambda_{t} = -U_{n}(c_{t}, 1-n_{t})/w_{t}$ 3. $\lambda_{t} = \beta \mathbb{E}_{t}[ \lambda_{t+1} (r_{t+1}+1-\delta) ] $ From equations (1) and (3), we obtain the canonical Euler equation, which characterizes the intertemporal trade-off in consumption. By substituting equations (1) through (3) into the system and rearranging, the model can be expressed in the form of a moment condition: $$ \mathbb{E}_{t}\Biggl[ F(x_{t+1}, y_{t+1}, x_{t}, y_{t}, z_{t}; \theta) \Biggr] = 0 $$ This formulation embeds the Euler condition within a broader structural framework suitable for estimation or simulation. # Balanced growth Path In the deterministic version of the model, where stochastic shocks are turned off, the growth trend is governed by the following law of motion: $$ z_{t+1} = \Lambda(\theta)z_{t}+\Sigma(\theta)\epsilon_{t+1} $$ In this setting, many endogenous variables evolve proportionally to the growth trend, typically taking the form: $$ x_{t} = \tilde{x} \cdot z_{t} $$ This formulation implies that the dynamics of key macroeconomic variables are scaled by the underlying growth process, reflecting balanced growth path behavior in the absence of uncertainty. ::: {#def-BGP} ## Balanced Growth Path A path $\{C_t, K_t, Y_t\}_{t=0}^{\infty}$ is called a **Balanced Growth Path (BGP)** if there exists a constant $g \in \mathbb{R}$ such that $$ \begin{align} \ln \frac{C_{t+1}}{C_t} = \ln \frac{K_{t+1}}{K_t} = \ln \frac{Y_{t+1}}{Y_t} = g \quad \forall t \ge 0. \end{align} $$ Equivalently, $C_t = C_0 e^{gt}$, $K_t = K_0 e^{gt}$, and $Y_t = Y_0 e^{gt}$. ::: ::: {#thm-BGP} ## Balanced Growth Path: Existence and Uniqueness Consider a discrete-time economy described by the following system: $$ \begin{align} K_{t+1} &= F(K_t, L_t) - C_t - \delta K_t, \\ L_{t+1} &= (1+n)L_t, \\ U_0 &= \sum_{t=0}^{\infty} \beta^t u(C_t), \end{align} $$ where $K_t$ is capital, $L_t$ is labor, $C_t$ is consumption, $\delta \in (0,1)$ is the depreciation rate, and $\beta \in (0,1)$ is the discount factor. Then the following two statements are equivalent: 1. There exists at least one Balanced Growth Path in the economy. 2. The production function $F$ and the utility function $u$ satisfy the following **monotonic scalar transformation conditions**: $$ \begin{align} F(aK, aL) &= a\, F(K,L), \quad \forall a>0, \\ u(aC) &= \phi(a)\, u(C), \quad \forall a>0, \end{align} $$ where $\phi:\mathbb{R}_+ \to \mathbb{R}_+$ is continuous and strictly monotone increasing. Moreover, if $\phi(a)$ is strictly monotone, then the Balanced Growth Path is unique. ::: ::: {.proof} ## General Framework ### Production Aggregate output is produced according to the following production function: $$ Y_{t} = A_{t}F(K_{t}, N_{t}L_{t}) $$ where: - $A_{t}$ denotes the level of technology, - $K_{t}$ represents the aggregate capital, - $N_{t}, L_{t}$ is the population and the employment rate, respectively. #### Assumption T1 (Constant Returns to Scale) The production function $F: \mathbb{R}^2_{+} \rightarrow \mathbb{R}_{+}$ is assumed to be continuous, quasi-concave, and homogeneous of degree one in reproducible factors $(K, NL)$, satisfying constant returns to scale (CRS). ### Exogenous Growth Technological progress and population growth follow deterministic exponential trends: $$ \begin{align} A_{t+1} & = (1+g_{A})A_{t}, \\ N_{t+1} & = (1+g_{N})N_{t} \end{align} $$ #### Assumption T2 The growth rates $g_{A}, g_{N} \geq 0$ are constant and exogenously given. ### Capital Accumulation Capital evolves according to the standard law of motion: $$ K_{t+1} = (1-\delta)K_{t}+I_{t} $$ ### Preferences The representative household maximizes lifetime utility given by: $$ U = \sum^{\infty}_{t=0} \beta^t u(C_{t}, 1-L_{t}) $$ with the period utility function specified as a generalized separable CRRA form: :::