# Monetary Policy: New-Keynesian Model

#### Background

The basic New-Keynesian (NK) Model is a model of the business cycle model that accounts for the well-documented upward-sloping relationship between real economic activity and inflation over short time horizons. The basic elements of the NK model can be expressed with four equations that reflect how the economy evolves at a quarterly frequency: $$y_t = E_t y_{t+1} - \frac{1}{\sigma}(r_{t} -\bar{r}) + g_t$$ $$\pi_t - \pi^T = \beta (E_t \pi_{t+1} - \pi^T ) + \kappa(y_t -\bar{y}) + u_t$$ $$i_{t} = r_{t} + E_t\pi_{t+1}$$ $$i_{t} = \bar{r} + \pi^T + \phi_{\pi} (\pi_t- \pi^T) + \phi_{y}(y_t-\bar{y}),$$ where $$y_t$$ denotes log real GDP, $$\pi_t$$ denotes inflation, $$r_t$$ is the real interest rate, and $$i_t$$ is the nominal interest rate. The first equation is the dynamic IS specification that links the demand for goods in the current quarter with national income expected in the following quarter and to the current real interest rate. The second equation is a standard NK Phillips curve: the deviation in inflation form the central bank's target $$\pi^T$$ is dependent on the expected future deviation of inflation from target and to the output gap $$y_t - \bar{y}$$. The third equation is the Fisher equation that defines the real interst rate as the nominal interest rate less the inflation rate.

The model has two exogenous variables $$g_t$$ and $$u_t$$ that relfect exogenous variation in the demand for goods and exogenous variation in the rate of inflation. $$g_t$$ and $$u_t$$ are AR(1) processes: $$g_t = \rho_g g_{t-1} + \epsilon^g_t$$ $$u_t = \rho_u u_{t-1} + \epsilon^u_t,$$ $$\epsilon^g_t$$ and $$\epsilon^u_t$$ are white noise processes with standard deviations $$\sigma_g$$ and $$\sigma_u$$. For more information about AR(1) processes, check out my AR(1) simulation page.

In the simulation exercise below, some parameters of the model are fixed while others may be chosen by the user. The parameters that are fixed for the simulation are:

Parameter Description Value
$$1/\sigma$$ Household intertemporal elasticity of substitution 1
$$\bar{r}$$ Steady state real rate of interest (annualized) 0.02
$$\beta$$ Household subjective discount rate $$\exp(-0.02/4)$$
$$\bar{y}$$ Log natural rate of output 0

The parameters that must be chosen by the user are:

Parameter Description
$$\pi^T$$ Central bank inflation target
$$\kappa$$ Inflation sensitivity to the output gap
$$\phi_{\pi}$$ Policy rule weight on inflation deviation from target
$$\phi_y$$ Policy rule weight on output gap
$$\rho_g$$ Autocorrelation of exogenous demand component
$$\sigma_g$$ Standard deviation of demand shock
$$\rho_{u}$$ Autocorrelation of exogenous inflation component
$$\sigma_u$$ Standard deviation of inflation shock

#### Instructions

Choose the number of periods for your simulation. Choose values for the monetary policy parameters $$\pi^T$$, $$\phi_{\pi}$$, and $$\phi_{y}$$. Then choose values for the parameters of the exogenous demand and inflation processes. Select which type of simulation that you wish to perform and which types of shocks to include in your simulation:

1. Stochastic simulation: Random values for the exogenous processes selected will be generated.
2. Impulse response: A one-standard deviation shock to the exogenous process selected. If you select Monetary policy, then the simulation will be computed for a one-percent shock (annualized) to the monetary policy rule with an autocorrelation of 0.5.

Note that the policy parameters that you choose have to induce a stable equilibrium. This means that the following condition must be satisfied: $$\kappa(\phi_{\pi}-1)+(1-\beta)\phi_y > 0.$$ You will receive a warning if the values that you input produce an unstable result. You will not be allowed to destroy the simulated economy.

Finally, all simulated values are plotted in percentage units. The inflation, real interest, and nominal interest rates are expressed as annualized differences from steady state values.

Parameters
Simulation