Macro Study Notes
Xinyu Zhou
Module 5
Why does money matter? The New Keynesian model introduces nominal rigidities to explain how monetary policy affects real activity in the short run, while preserving long-run neutrality. This is the modern workhorse for monetary policy analysis.
The classical dichotomy states that real variables (output, employment, real wages, real interest rates) are determined by preferences and technology independently of nominal variables (money supply, price level).
Representative household maximizes:
$\displaystyle \mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t u(c_t, l_t)$
Subject to: $P_t c_t + Q_t B_{t+1} = B_t + W_t l_t - T_t$.
Labor supply (intratemporal): $-\frac{u_{l,t}}{u_{c,t}} = \frac{W_t}{P_t} \equiv w_t$
Euler equation (intertemporal): $u_{c,t} = \beta \mathbb{E}_t\!\left[u_{c,t+1} \frac{P_t}{P_{t+1}}\frac{1}{Q_t}\right]$
With separable CRRA utility $u(c,l) = \frac{c^{1-\sigma}}{1-\sigma} - \frac{l^{1+\varphi}}{1+\varphi}$ and linear production $y = zl$:
A strong form of the classical dichotomy holds: all real variables are independent of nominal variables.
Let $i_t \equiv -\ln Q_t$, $\pi_{t+1} \equiv \ln(P_{t+1}/P_t)$, $\rho \equiv -\ln\beta$:
Consumption growth depends on the real interest rate $r_t = i_t - \mathbb{E}_t\pi_{t+1}$. Higher real rates depress current consumption relative to future consumption.
For nominal rigidities to matter, firms must have price-setting power. This is introduced via differentiated products and monopolistic competition (Dixit-Stiglitz).
$c_t = \left(\int_0^1 c_t(j)^{\frac{\varepsilon-1}{\varepsilon}} dj\right)^{\frac{\varepsilon}{\varepsilon-1}}, \quad \varepsilon > 1$
Demand for variety $j$:
where the ideal price index is $P_t = \left(\int_0^1 P_t(j)^{1-\varepsilon} dj\right)^{\frac{1}{1-\varepsilon}}$.
A firm with constant marginal cost $W/z$ maximizes profit subject to the demand curve:
The real wage is $w_t = \frac{\varepsilon-1}{\varepsilon} z_t < z_t$ — firms pay workers less than their marginal product. Output and employment are below the perfectly competitive level.
The flexible-price (natural) level of output is:
$\log y_t^n = \psi_{yz} \log z_t = \frac{1+\varphi}{\sigma+\varphi} \log z_t - \frac{1}{\sigma+\varphi}\ln\frac{\varepsilon}{\varepsilon-1}$
In the NK model, actual output fluctuates around this natural level — the gap is the output gap $x_t \equiv \log y_t - \log y_t^n$.
The Calvo (1983) model of price stickiness: each period, a random fraction $\theta$ of firms cannot change their price, while $1-\theta$ can re-optimize.
Fraction $\theta$ of firms keep old prices, $1-\theta$ set the optimal reset price $P_t^*$:
$P_t = \left[\theta P_{t-1}^{1-\varepsilon} + (1-\theta)(P_t^*)^{1-\varepsilon}\right]^{\frac{1}{1-\varepsilon}}$
Log-linearized around zero-inflation steady state:
A firm that can re-optimize at $t$ maximizes expected discounted profits over the period its price remains fixed:
$\displaystyle \max_{P_t^*} \mathbb{E}_t \sum_{k=0}^{\infty} (\theta\beta)^k \frac{u_{c,t+k}}{u_{c,t}} \frac{P_t}{P_{t+k}} \left[P_t^* - \frac{W_{t+k}}{z_{t+k}}\right] y_{t+k|t}$
Log-linear FOC:
$\hat{p}_t^* = (1-\theta\beta)\sum_{k=0}^{\infty} (\theta\beta)^k \mathbb{E}_t[\hat{w}_{t+k} - \hat{z}_{t+k} + \hat{p}_{t+k}]$
The reset price is a discounted sum of expected future nominal marginal costs.
Combining the price level dynamics, the optimal reset price, and the relationship between real marginal cost and the output gap yields:
where $\kappa$ is the slope of the Phillips curve with respect to the output gap $x_t$.
Iterating the NKPC forward:
Current inflation is the discounted sum of expected future output gaps — it is inherently forward-looking. Expected future booms ($x_{t+k} > 0$) raise current inflation; expected future busts lower it.
Using household labor supply and firm production:
$\hat{w}_t - \hat{z}_t = (\sigma + \varphi)\hat{y}_t - (1+\varphi)\hat{z}_t$
At the natural (flexible-price) level: $\hat{y}_t^n = \frac{1+\varphi}{\sigma+\varphi}\hat{z}_t$.
The central bank raises the nominal rate when inflation is above target or output is above potential. Combined with the Fisher equation, this determines the real interest rate.
The optimal policy is to undo the distortions from price stickiness, making allocations equal the flexible-price outcome ("divine coincidence"). This means stabilizing both inflation and the output gap:
$\pi_t = 0, \quad x_t = 0 \quad \text{at all times}$
In the baseline NK model, this "divine coincidence" holds — stabilizing inflation stabilizes the output gap. With real imperfections (e.g., cost-push shocks), a trade-off may emerge.
The interest rate that delivers full employment is:
$1 + i_t = \frac{1}{\beta}\left(\frac{A_2}{\mathbb{E}_t A_2}\right)^{1+1/\sigma} \cdot (1+\hat{i})$
The ZLB is more likely to bind when:
When the ZLB binds, monetary policy becomes impotent — this is the liquidity trap. Fiscal policy may need to play the stabilizing role.
A fundamental issue in monetary policy design: should policy follow a rule or be at the policymaker's discretion?
Central bank minimizes loss: $L(u, \pi) = u^2 + \gamma\pi^2$, subject to the Phillips curve $u - \bar{u} = -\alpha(\pi - \mathbb{E}\pi)$.
Central bank commits to $\pi = 0$. Under a credible commitment rule, $\mathbb{E}\pi = 0$, and unemployment is $u = \bar{u}$. This yields the best sustainable outcome.
Private sector sets $\mathbb{E}\pi$ first. Central bank then chooses $\pi$. Even if the CB announces $\pi = 0$, once expectations are set, the CB is tempted to exploit the Phillips curve to get lower $u$ with some inflation.
Central bank's best response: $\pi = \frac{\alpha}{\gamma + \alpha^2}\bar{u} + \frac{\alpha^2}{\gamma+\alpha^2}\mathbb{E}\pi$.
In equilibrium $\pi = \mathbb{E}\pi$:
Result: Inflation is positive under discretion, but unemployment is the same ($u = \bar{u}$). The inflation bias arises because the CB cannot credibly commit to low inflation.