NK-DSGE Model

Main references

  • 이종화, and 김진일. (2021). "동태적 거시경제학", 박영사.
  • Gali, J. (2008). "Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework." Princeton University Press, Princeton.

Introduction

The New-Keynesian Dynamic Stochastic General Equilibrium(NK-DSGE) comprises of the following two relationships

  • Dynamics IS(investment-savings) curve: relationship between interest rates and output gaps
  • New-Keynesian phillips curve: relationship between inflation and output gap.

NK-DSGE model encompass the price stickiness to explain the implication of monetary policies, especially the liquidity effects.

  • In perfectly flexible prices, monetary policy can only affect nominal variables, while leaving the real variables unchanged. The incorporation of nominal frictions enables the monetary policy to alter real interest rates and aggregate demand.
  • Sticky prices create a short-run trade-off between inflation and output. Central banks can exploit this by adjusting policy rates to stabilize demand shocks.
  • The liquidity effect—the inverse relationship between money supply and nominal interest rates—is amplified in sticky-price models. When prices adjust slowly, an increase in money supply lowers real interest rates more persistently, boosting demand.

Here, we assume 'Calvo contracts' to explain the price rigidity.

  • Each firm randomly faces a chance to adjust its prices in every term, following an independent poisson distribution.
  • For instance, while the prices of portion of firms remain fixed, the other portion of firms can adjust their prices without any additional costs.
  • This generates forward-looking behavior: firms set prices based on expected future marginal costs and demand conditions.

Derivation of NK-DSGE

Household

Assume that we have infinitely many consumption goods of , where . Then the aggregated consumption is note that

  • : elasticity of substitution among differentiated goods
    • elasticity of substitution equals on every goods .
    • as , : consumption goods are perfect substitutes.

Solution for General Utility Function

For the general utility function , the utility maximization problem (UMP) is defined as
Using the Lagrangian, we have and the first order conditions are
thus, we have the following results

  • labor supply equation: optimization between consumption and labor
  • Euler equation:

Solution under CRRA Function

Additionally, we assume CRRA utility function:

  • : constant relative risk aversion coefficient
  • : the (inverse) elasticity of leisure (and thus labor supply) with respect to wage or marginal utility changes.

Then, we have and the log-linearization gives us

and where

Note that we can also drive the following equation from the Euler equation, using the relationships of and , where the last equation is from the Fisher equation this representation will be used to drive the stochastic discount factors in profit maximization problems.

Individual Consumption Ratio

Lastly, we drive the optimal ratio for individual consumption:

From UMP, we have

and from the budget constraint, we can drive

\begin{align} & \int_{0}^{1} P_{t}(i) C_{t}(i) \,di + Q_{t} B_{t} \leq B_{t-1} + W_{t} L_{t} + T_{t} + F_{t} \\ & \Rightarrow \int_{0}^{1} P_{t}(i) C_{t}(i) \,di \leq B_{t-1} - Q_{t} B_{t} + W_{t} L_{t} + T_{t} + F_{t} =: {Z}_{t} \\ & \Rightarrow \int_{0}^{1} P_{t}(i) A_{t} P_{t}(i)^{-\varepsilon} \,di = A_{t} \int_{0}^{1} P_{t}(i)^{1-\varepsilon} \,di \leq {Z}_{t} \\ & \Rightarrow A_{t} \leq \frac{{Z}_{t}}{\int_{0}^{1} P_{t}(i)^{1-\varepsilon} \,di} \end{align}$$thus at optimal consumption, we have $$C_{t}(i)= A_{t}P_{t}(i)^{-\varepsilon}= \frac{{Z}_{t}}{\int_{0}^{1} P_{t}(i)^{1-\varepsilon} \,di} P_{t}(i)^{-\varepsilon}$$ Plugging this back into the aggregate consumption, $$\begin{align} C_{t}&= \left(\int_{0}^{1} C_{t}(i)^{1- \frac{1}{\varepsilon}} \ di\right)^{\frac{\varepsilon}{\varepsilon-1}} \\ &= \left( \int_{0}^{1}\left\{ \frac{{Z}_{t}}{\int_{0}^{1} P_{t}(j)^{1-\varepsilon} \,dj} P_{t}(i)^{-\varepsilon} \right\}^{\frac{\varepsilon-1}{\varepsilon}} di \right)^{\frac{\varepsilon}{\varepsilon-1}}\\ &= \left( \frac{{Z}_{t}}{\int_{0}^{1} P_{t}(j)^{1-\varepsilon} \,dj} \right) \left( \int_{0}^{1}P_{t}(i)^{1-\varepsilon} \ di \right)^{\frac{\varepsilon}{\varepsilon-1}}\\ &= Z_{t} \left( \int_{0}^{1}P_{t}(i)^{1-\varepsilon} \ di \right)^{\frac{1}{\varepsilon-1}} \end{align}$$and since $$Z_{t}=B_{t-1} - Q_{t} B_{t} + W_{t} L_{t} + T_{t} + F_{t} =P_{t}Y_{t}=P_{t}C_{t}$$we have $$P_{t}=\left( \int_{0}^{1}P_{t}(i)^{1-\varepsilon} \ di \right)^{\frac{1}{1-\varepsilon}}$$which is the price of a unit of $C_{t}$. Therefore, the individual consumption ratio to the aggregated consumption is $$\begin{align} C_{t}(i)&= \frac{{Z}_{t}}{\int_{0}^{1} P_{t}(i)^{1-\varepsilon} \,di} P_{t}(i)^{-\varepsilon}\\[4pt] &= C_{t}P_{t}^{\varepsilon-1}P_{t}P_{t}(i)^{-\varepsilon}\\[4pt] &= C_{t}\left( \frac{P_{t}(i)}{P_{t}} \right)^{-\varepsilon} \end{align}

Firms

Let the production function be and the price adjustments follow the Calvo model: where the specific terms are specified as follows:

  • Stochastic discount factor:
  • demand function at for the unadjusted price from :
  • nominal cost function for firms: note that we will denote its partial derivative as follows and the marginal cost will be denoted as

Now, the first order conditions are

\begin{align} & \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t}\left[ \frac{\Lambda_{t,t+k}}{P_{t+k}} \left( Y_{t+k|t} + P_{t}^{*} \frac{\partial Y_{t+k|t}}{\partial P_{t}^{*}} - \frac{\partial \Psi(Y_{t+k|t})}{\partial Y_{t+k|t}} \frac{\partial Y_{t+k|t}}{\partial P_{t}^{*}} \right) \right]=0\\ & \Rightarrow \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t}\left[ \frac{\Lambda_{t,t+k}}{P_{t+k}} \left( Y_{t+k|t} + P_{t}^{*} (-\varepsilon) \frac{(P_{t}^{*})^{-\varepsilon-1}}{(P_{t+k})^{-\varepsilon}}C_{t+k} - \frac{\partial \Psi(Y_{t+k|t})}{\partial Y_{t+k|t}} (-\varepsilon) \frac{(P_{t}^{*})^{-\varepsilon-1}}{(P_{t+k})^{-\varepsilon}}C_{t+k} \right) \right]=0\\ & \Rightarrow \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t}\left[ \frac{\Lambda_{t,t+k}}{P_{t+k}} \left( Y_{t+k|t} + (-\varepsilon) \left(\frac{P_{t}^{*}}{P_{t+k}}\right)^{-\varepsilon}C_{t+k} - \frac{\partial \Psi(Y_{t+k|t})}{\partial Y_{t+k|t}} (-\varepsilon) \frac{1}{P_{t}^{*}} \left(\frac{P_{t}^{*}}{P_{t+k}}\right)^{-\varepsilon}C_{t+k} \right) \right]=0\\ & \Rightarrow \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t}\left[ \frac{\Lambda_{t,t+k}}{P_{t+k}} \left( Y_{t+k|t} -\varepsilon Y_{t+k|t} + \varepsilon \Psi_{y_{t+k|t}} \frac{1}{P_{t}^{*}} Y_{t+k|t} \right) \right]=0\\ & \Rightarrow \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t}\left[ \frac{\Lambda_{t,t+k}}{P_{t+k}} Y_{t+k|t} \left\{ P_{t}^{*} + \frac{\varepsilon}{1 -\varepsilon} \Psi_{y_{t+k|t}} \right\} \right]=0\\ & \Rightarrow \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t}\left[ \frac{\Lambda_{t,t+k}}{P_{t+k}} Y_{t+k|t} \left\{ P_{t}^{*} + M \Psi_{y_{t+k|t}} \right\} \right]=0 \end{align}$$where $M= \frac{\varepsilon}{1-\varepsilon}$ denotes the frictionless mark-up. By rearranging the terms, we have $$\begin{align} & \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t} \left[\frac{\Lambda_{t,t+k}}{P_{t+k}} Y_{t+k|t}\cdot P_{t}^{*}\right] = \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t} \left[\frac{\Lambda_{t,t+k}}{P_{t+k}} Y_{t+k|t} \cdot M \Psi_{y_{t+k|t}}\right] \\ & \Rightarrow \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t} \left[\beta^{k} \left(\frac{C_{t+k}}{C_{t}}\right)^{-\theta} \frac{P_{t}^{*}}{P_{t+k}} \left(\frac{P_{t}^{*}}{P_{t+k}}\right)^{-\varepsilon} C_{t+k}\right] = M \cdot \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t} \left[\beta^{k} \left(\frac{C_{t+k}}{C_{t}}\right)^{-\theta} \frac{1}{P_{t+k}} \left(\frac{P_t^*}{P_{t+k}}\right)^{-\varepsilon} C_{t+k} \Psi_{y_{t+k|t}}\right]\\ & \Rightarrow \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t} \Big[\beta^{k} (C_{t+k})^{1-\theta} (C_{t})^{\theta} (P_{t}^{*})^{1-\varepsilon} (P_{t+k})^{\varepsilon-1}\Big] = M \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t} \Big[\beta^{k} (C_{t+k})^{1-\theta} (C_{t})^{\theta} (P_{t}^{*})^{-\varepsilon} (P_{t+k})^{\varepsilon-1} \Psi_{y_{t+k|t}}\Big]\\ & \Rightarrow (C_{t})^{\theta}(P_{t}^{*})^{1-\varepsilon} \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t} \Big[\beta^{k} (C_{t+k})^{1-\theta} (P_{t+k})^{\varepsilon-1}\Big] = (C_{t})^{\theta}(P_{t}^{*})^{-\varepsilon} M \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t} \left[\beta^{k} (C_{t+k})^{ 1-\theta} (P_{t+k})^{\varepsilon} \frac{\Psi_{y_{t+k|t}}}{P_{t+k}} \right]\\ & \Rightarrow \frac{P_{t}^{*}}{P_{t-1}} \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t} \Big[\beta^{k} (C_{t+k})^{1-\theta} (P_{t+k})^{\varepsilon-1}\Big] = \frac{M}{P_{t-1}} \sum_{k=0}^{\infty} \psi^{k} \mathbb{E}_{t} \Big[\beta^{k} (C_{t+k})^{1-\theta} (P_{t+k})^{\varepsilon} MC_{t+k|t}\Big] \end{align}

Remark that at the steady state, we have Using Taylor expansion, the left hand side would be

and the right hand side would be

Therefore we have where is the mark-up at the Steady-State().

Price Index

Given the culminated price index where is the elasticity of substitution between the consumption goods, the Calvo contracts give us the following price index and inflation formula.

By the log-linearization, we have and the first order Taylor expansion gives us where the steady state is

Thus, by subtracting the steady state, we have

General Equilibrium

Dynamic IS curve

  • Euler equation: where , , and .

  • market clearing condition: and since aggregated consumption is CES function of each consumption we have .

Combining the two, we have

  • Dynamic IS: thus output is the function of nominal interest rates, expected inflation, and expected output.

Supply function

  • aggregated labor (homogeneous labor supply from each household):
  • individual productive function:
  • individual consumption ratio:
  • individual output ratio: from the market clearing condition for , we have Now, combining the aggregated labor equation and individual output ratio, we have

and by the log-linearization

where means the dispersion of the prices. note that and it is ommited as its first order taylor approximation is zero.

Therefore, we have

  • Supply function:

Demand function

  • demand function at for the unadjusted price from : and by the market clearing conditions, by the log-linearization,

thus

  • demand function:

Marginal cost

From the definition of marginal costs, we have as . By taking logs on the both sides, we have Then, for the given price , the marginal cost becomes where the last equality comes from the demand function .

  • marginal cost:

Inflation

From the firm's problem, we have the following price index: as , we have

thus therefore

NK Phillips curve

  • labor supply function: and by the market clearing conditions,
  • production function:
  • marginal cost:

Thus plugging labor supply function and production function into marginal cost, we have under the flexible price, we have where is the natural output.

Thus, the deviation of marginal cost from its steady state is Finally, the phillips curve is where .

Summarization

  • dynamic IS: and from steady state, as in the natural output, the expected inflation is zero and . Thus the deviation from steady state is and if , we have
  • NK-Philips Curve:
    • , where is the calvo coefficient

Equilibrium

Recursive Solutions

Given the structural equations

  • NKPC:
  • DIS:
  • policy rule:

we can expand each equations to construct matrix equations.

For the policy rule, and for DIS, where .

Lastly, for NKPC, we have Combining the results,

Blanchard and Kahn's conditions

Proposition (Blanchard and Kahn's conditions).

The solution exists if the eigenvalue of smaller than , which condition identical to

Proof.The proof follows Bullard & Mitra (2002).

First, we drive the characteristic equation of the :

thus, the eigenvalue of is situated inside the unit circle if the following two conditions hold:

  • : note that .
  • :

External Shocks

External Monetary Policy Shocks

Technology Shocks

Optimal Monetary Polices

Interactive Graph
Table Of Contents
NK-DSGE Model
Introduction
Derivation of NK-DSGE
Household
Solution for General Utility Function
Solution under CRRA Function
Individual Consumption Ratio
Firms
Price Index
General Equilibrium
Dynamic IS curve
Supply function
Demand function
Marginal cost
Inflation
NK Phillips curve
Summarization
Equilibrium
Recursive Solutions
Blanchard and Kahn's conditions
External Shocks
External Monetary Policy Shocks
Technology Shocks
Optimal Monetary Polices