New Keynesian Model

Macro

Compared with Real Business Cycle, the New Keynesian Model adds Nominal Rigidity to the RBC model (other remain the same). This directly leads to the situation that output and employment are not at the optimal level. This model allows for Involuntary Unemployment.

Model

The model is very similar to Real Business Cycle, and also other similar models in Macroeconomics. It includes household, firms, and central bank.

Some assumptions:

Household

• consume baskets of goods and supply labour to firms
• save risk-free government bond (not capital here)
• owns firms and receive dividends (if they make profits)

Problem

${\max }_{C_t,N_t,B_t}{\mathbb{E}}_0{\sum }_{t=0}^{\infty }{\beta }^{t}U(C_t,L_t),$

subject to

$${\int }_0^1{P}_{i,t}{C}_{i,t}di+Q_t{B}_t\le B_{t-1}+W_t{L}_t+{\text{Profits}}_t,\text{and no - ponzi}$$

• ${\int }_0^1{P}_{i,t}{C}_{i,t}di$: since in the market there’s continuum of varieties indexed by $[0,1]$. $C$ here is the quantity. Overall it means the total expenditure you spend on the goods in the market.

Why we don’t construct it as discrete? It’s because in this setting, each firm is a monopolistic competitor. Each firm produces a unique variety and has some pricing power, but it depends on the elasticity of substitution $ϵ$.

And people got utility from consuming this basket of goods, we have to introduce Constant Elasticity of Substitution (CES).

The optimality for household:

Optimal consumption and bond holdings (Euler):

$Q_t=\beta {\mathbb{E}}_t\left[\frac{U_{C,t+1}^{\prime }}{U_{C,t}^{\prime }}\frac{P_t}{P_{t+1}}\right]$

Static labor vs consumption optimization:

$$-\frac{U_{L,t}^{\prime }}{U_{C,t}^{\prime }}=\frac{W_t}{P_t}$$

Optimal expenditure allocation

$$C_{i,t}={\left(\frac{P_{i,t}}{P_t}\right)}^{-ϵ}{C}_t$$

This could also be considered as demand function with constant price elasticity.

See derivation from the MN4.

Log Linearize

• **Why $i_t=-\log Q_t$ ?

The reason why we have $i_t=-\log Q_t$ is that, recall $Q_t$ means the price of a bond that pays 1 unit of money at maturity. If you pay $Q_t$ today, you get 1 tomorrow. So the nominal interest rate satisfies:

$$\frac{1}{Q_t}=1+i_t$$

By using the equation: $\log (1+i_t)\approx i_t$ for small rates, we have

$$i_t\approx -\log Q_t$$

Thus we have the new Euler Equation:

$$c_t={\mathbb{E}}_t\left(c_{t+1}\right)-\frac{1}{\sigma }\left(\underset{-\log Q_t}{\underbrace{i_t}}-\underset{\log P_{t+1}/P_t}{\underbrace{E_t\left[{\pi }_{t+1}\right]}}-\underset{-\log \beta }{\underbrace{\rho }}\right)$$

Firms

• Produce differentiated goods, so there’s no longer a representative firm: firm produce a variety
• Choose the price at which they sell their variety.

A specific firm only produces identical good, but still faces Monopolistic Competition

Under this setting, all firms have same production function and same productivity:

$$Y_{i,t}=A_t{L}_{i,t}^{1-\alpha }$$

Flexible Prices

The sticky price applies here: firm can choose price only with some probability. (Calvo Price)

Notice that firms. are owned by households, so they discount future profits using the households’ stochastic discount factor.

$$\begin{array}{r}{Q}_{t,t+1}=\beta \left[\frac{U_{C,t+1}^{\prime }}{U_{C,t}^{\prime }}\frac{P_t}{P_{t+1}}\right]\Rightarrow Q_{t,t+k}={\beta }^k\left[\frac{U_{C,t+k}^{\prime }}{U_{C,t}^{\prime }}\frac{P_t}{P_{t+k}}\right]\end{array}$$

Here, $Q_{t,t+k}$ means: paying $Q_t$ today, got 1 dollar at $t+k$.

Thus firms maximize the profit:

$${\mathbb{E}}_t\sum _{k=0}^{\infty }{Q}_{t,t+k}\left(P_{i,t+k}{Y}_{i,t+k}-W_{t+k}{L}_{i,t+k}\right)$$

Finally we could get from symmetric equilibrium, we could only take the derivative of a representative firm $i$. We could get $P_{i,t}=P_t$.

$$P_t^{FLEX}=\frac{ϵ}{ϵ-1}{W}_t\left(\frac{1}{1-\alpha }\right){\left(\frac{1}{A_t}\right)}^{\frac{1}{1-\alpha }}{C}_t^{\frac{\alpha }{1-\alpha }}$$

Price equals a constant markup $\frac{ϵ}{ϵ-1}$ over the marginal cost. This is the standard Monopolistic Competition result. For example, as $ϵ\to \infty$, the goods becomes perfect substitute, the markup goes down to $1$ so we are back to the Perfect Competition.

Wages

Recall

$$P_t=\frac{ϵ}{ϵ-1}{W}_t\left(\frac{1}{1-\alpha }\right){\left(\frac{1}{A_t}\right)}^{\frac{1}{1-\alpha }}{C}_t^{\frac{\alpha }{1-\alpha }}$$ $$\frac{W_t}{P_t}=\frac{ϵ-1}{ϵ}\left(\underset{MP{L}_t}{\underbrace{A_t^{\frac{1}{1-\alpha }}{C}_t^{-\frac{\alpha }{1-\alpha }}(1-\alpha )}}\right)$$

So we could conclude wage is marked down because of market power in the production side. Lower wage reduces labor supply.

How about if price is not flexible, i.e., the Calvo Price.

Central Bank

Set the nominal interest rate on government bonds

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The following part would have further discussion on the NK model, and try to discuss some properties. One is the Heterogeneity in New Keynesian Model