A basic new keynesian model dsge model

Here is a basic New Keynesian DSGE model:

Model Structure

The model consists of three main blocks:

1. Household Sector: represents the behavior of households, including consumption, labor supply, and financial decisions.
2. Firm Sector: represents the behavior of firms, including production, pricing, and investment decisions.
3. Monetary Policy: represents the behavior of the central bank, including the setting of interest rates and the implementation of monetary policy.

Variables and Parameters

Household Sector

• C: consumption
• L: labor supply
• W: wage
• R: interest rate
• P: price level
• π: inflation rate
• β: discount factor (0 < β < 1)
• ψ: consumption Euler equation coefficient (0 < ψ < 1)
• ρ: labor supply elasticity (0 < ρ < 1)

Firm Sector

• Y: output
• K: capital
• L: labor
• P: price level
• π: inflation rate
• α: capital share of output (0 < α < 1)
• β: discount factor (0 < β < 1)
• ρ: investment elasticity (0 < ρ < 1)

Monetary Policy

• R: interest rate
• π: inflation rate
• Eπ: expected inflation rate
• σ: monetary policy shock standard deviation
• τ: monetary policy reaction coefficient (0 < τ < 1)

Equations

Household Sector

• Consumption Euler Equation: C_t = Et [βC{t+1} + (1-ψ)(W_t - P_t)]
• Labor Supply: L_t = ρ(W_t - P_t)
• Budget Constraint: W_t L_t + R_t K_t = Ct + (1-δ)K{t-1}

Firm Sector

• Production Function: Y_t = K_t^α L_t^(1-α)
• Investment: K_{t+1} = (1-δ)K_t + I_t
• Price Setting: P_t = E_t [πt + P{t-1}]
• Inflation: π_t = Pt - P{t-1}

Monetary Policy

• Monetary Policy Rule: R_t = Et [R{t+1}] + τ(π_t - Eπ_t) + σε_t
• Inflation Targeting: Eπ_t = π_t

Shock Processes

• Monetary Policy Shock: ε_t ~ N(0, σ^2)
• Technology Shock: ε_t ~ N(0, σ^2)

Solving the Model

To solve the model, we need to find the equilibrium values of the variables (C, L, Y, K, P, π, R) for each period t. We can do this using a numerical method, such as the Kalman filter or a linearized solution.

Linearized Model

To linearize the model, we can use a first-order Taylor series expansion around the steady-state values of the variables. This gives us a system of linear equations that can be solved using standard linear algebra techniques.

Steady-State Values

The steady-state values of the variables are:

• C: consumption = Y
• L: labor supply = 1
• W: wage = Y
• R: interest rate = 0
• P: price level = 1
• π: inflation rate = 0
• K: capital = Y
• Y: output = Y

These values are used as the reference point for the linearized model.

Key Features

The basic New Keynesian DSGE model has several key features:

• Rational Expectations: households and firms form expectations about future variables based on their own information and the information available to them.
• Monetary Policy: the central bank sets interest rates to achieve its inflation target.
• Price Stickiness: firms set prices in advance, which leads to a delay in the adjustment of prices to changes in demand and supply.
• Investment: firms invest in capital to increase their production capacity.
• Labor Supply: households supply labor to firms in exchange for wages.

This is a basic model, and there are many ways to extend and modify it to capture additional features of the economy.