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A dynamic aggregate demand and supply framework where inflation adjusts over time, combining a monetary policy rule with adaptive inflation expectations.

Derivation

Step-by-step mathematical derivation with typeset equations and expandable detail.

Sections

A dynamic framework with inflation

The dynamic AD-AS model extends the static IS-LM/AD-AS framework by making inflation (rather than the price level) the central variable, incorporating a Taylor-type monetary policy rule, and allowing expectations to evolve over time. The model consists of five equations: a dynamic IS curve relating the output gap to the real interest rate, a Fisher equation linking nominal and real rates, a Phillips curve connecting inflation to the output gap and expected inflation, an adaptive expectations rule for inflation, and a Taylor rule describing how the central bank sets the nominal interest rate.

All variables are expressed as deviations from their natural or target values. The output gap $\tilde{Y}_t = Y_t - \bar{Y}_t$ measures the deviation of actual output from potential. Inflation $\pi_t$ is the rate of price change. The nominal interest rate $i_t$ is the central bank's policy instrument. By combining these building blocks, we derive two reduced-form relationships: the Dynamic Aggregate Demand (DAD) curve and the Dynamic Aggregate Supply (DAS) curve, which jointly determine the output gap and inflation in each period.

\tilde{Y}_t = -\alpha(r_t - \bar{r}) + \varepsilon_t

Dynamic IS curve: the output gap falls when the real interest rate $r_t$ exceeds the natural rate $\bar{r}$ , with $\varepsilon_t$ as a demand shock.

r_t = i_t - E_t \pi_{t+1}

Fisher equation: the ex ante real interest rate equals the nominal rate minus expected inflation.

i_t = \bar{r} + \pi_t + \phi_\pi(\pi_t - \pi^*) + \phi_Y \tilde{Y}_t

Taylor rule: the central bank raises the nominal rate above its neutral level when inflation exceeds target $\pi^*$ or when the output gap is positive.

Deriving the DAD curve

The Dynamic Aggregate Demand curve is derived by substituting the Taylor rule and Fisher equation into the dynamic IS curve to eliminate the interest rate. Start with the IS curve $\tilde{Y}_t = -\alpha(r_t - \bar{r}) + \varepsilon_t$ . Replace $r_t$ using the Fisher equation: $r_t = i_t - E_t\pi_{t+1}$ . Then substitute the Taylor rule for $i_t$ . After simplification, the result is a negative relationship between the output gap $\tilde{Y}_t$ and current inflation $\pi_t$ : higher inflation triggers a stronger monetary policy response (via the Taylor rule), raising the real interest rate and depressing demand.

The slope of the DAD curve depends on the Taylor rule coefficients $\phi_\pi$ and $\phi_Y$ . A more aggressive response to inflation ( $\phi_\pi$ large) makes the DAD curve steeper in $(\tilde{Y}, \pi)$ space, meaning a given inflation increase produces a larger output contraction. Demand shocks $\varepsilon_t$ shift the DAD curve: a positive demand shock shifts it rightward (higher output at every inflation rate). The Taylor principle ( $\phi_\pi > 0$ ) ensures that the central bank raises the real interest rate when inflation rises, which is necessary for the DAD curve to slope downward and for the model to be stable.

\tilde{Y}_t = -\frac{\alpha \phi_\pi}{1 + \alpha \phi_Y}(\pi_t - \pi^*) + \frac{1}{1 + \alpha \phi_Y}\varepsilon_t

DAD curve: the output gap is negatively related to the inflation gap $\pi_t - \pi^*$ . Higher inflation triggers tighter policy, reducing demand.

\frac{\partial \tilde{Y}_t}{\partial \pi_t} = -\frac{\alpha \phi_\pi}{1 + \alpha \phi_Y} < 0

DAD slope: strictly negative when $\phi_\pi > 0$ (Taylor principle holds), confirming the downward-sloping demand relationship.

Deriving the DAS curve

The Dynamic Aggregate Supply curve comes from combining a Phillips curve with an expectations formation mechanism. The Phillips curve is $\pi_t = E_{t-1}\pi_t + \phi \tilde{Y}_t + v_t$ , where $\phi > 0$ links the output gap to inflation pressure and $v_t$ is a supply shock. Under adaptive expectations, $E_{t-1}\pi_t = \pi_{t-1}$ : agents forecast inflation by extrapolating the most recent observation. Substituting this into the Phillips curve yields the DAS curve.

The DAS curve is an upward-sloping relationship in $(\tilde{Y}, \pi)$ space: higher output gaps push inflation above last period's level. The curve shifts upward one-for-one with last period's inflation (built-in inertia) and with supply shocks $v_t$ . Over time, the DAS curve moves: if inflation was high last period, the DAS curve sits higher this period, reflecting the ratcheting effect of adaptive expectations. The intersection of DAD and DAS at each date determines the contemporaneous output gap and inflation rate, and the adaptive updating of expectations carries the system forward dynamically.

\pi_t = \pi_{t-1} + \phi\, \tilde{Y}_t + v_t

DAS curve: inflation equals last period's inflation plus a term proportional to the output gap and a supply shock. Upward-sloping in $(\tilde{Y}, \pi)$ space.

\text{DAD} \cap \text{DAS}: \quad (\tilde{Y}_t,\, \pi_t) \text{ solved jointly each period}

Short-run equilibrium: the output gap and inflation are determined by the intersection of the DAD and DAS curves, with the DAS curve shifting over time as $\pi_{t-1}$ updates.

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