Module 4

Short-Run Fluctuations

While growth theory explains long-run trends, short-run macroeconomics explains business cycles — why output, employment, and inflation deviate from trend. We build from the goods market to the full dynamic AS-AD model.

Goods Market & the IS Curve

Consumption Function

The key behavioral assumption: consumption depends on disposable income.

$C = c_0 + c_1(Y - T), \quad c_0 > 0, \;\; 0 < c_1 < 1$

$c_1$ is the marginal propensity to consume (MPC) — how much consumption increases for each additional unit of disposable income.

Equilibrium Output & the Multiplier

In a closed economy: $Y = C + I + G$. Substituting $C = c_0 + c_1(Y - T)$:

Equilibrium Output: $\quad Y = \frac{1}{1 - c_1}(c_0 - c_1 T + I + G)$

The multiplier $\frac{1}{1-c_1} > 1$: an increase in autonomous spending raises output by more than one-for-one.

Intuition: First round: $\Delta G$ increases demand by $\Delta G$ and output by $\Delta G$. This raises income by $\Delta G$, which raises consumption by $c_1\Delta G$ (second round). This additional spending raises income further, etc. After $n$ rounds: $1 + c_1 + c_1^2 + \cdots + c_1^{n-1} \to \frac{1}{1-c_1}$.

Endogenous Investment & the IS Curve

Make investment depend on output $Y$ and interest rate $i$:

$I = I(Y, i) = b_0 + b_1 Y - b_2 i, \quad b_0, b_1, b_2 > 0$

Investment rises with output (accelerator effect) and falls with the interest rate (cost of borrowing).

Solving goods market equilibrium gives the IS curve:

IS Curve: Negative relationship between output $Y$ and interest rate $i$ in goods market equilibrium. Higher $i$ reduces investment, lowering equilibrium output.

Financial Markets & the LM Curve

Money Demand

Money demand depends positively on nominal income (transactions motive) and negatively on the interest rate (opportunity cost):

$M^d = P \cdot Y \cdot L(i)$

where $L(i)$ is decreasing in $i$.

LM Curve

Money supply $M^s = \bar{M}$ set exogenously by the central bank. Equilibrium $M^s = M^d$ gives:

LM Curve: Positive relationship between $Y$ and $i$ in money market equilibrium. Higher $Y$ increases money demand, requiring a higher $i$ to restore equilibrium.

Modern central banks set interest rates directly (Taylor rule), making the LM curve horizontal — but the IS-LM framework remains useful for intuition.

IS-LM Model

The IS-LM model jointly determines output and the interest rate in the short run, assuming fixed prices.

Fiscal Policy in IS-LM

Increase in $G$: IS shifts right. Both $Y$ and $i$ rise. Higher $i$ crowds out some private investment.
Tax cut: IS shifts right through higher disposable income and consumption.
Crowding out: The rise in $i$ from fiscal expansion partially offsets the initial increase in output.

Monetary Policy in IS-LM

Increase in money supply: LM shifts down. $Y$ rises, $i$ falls.
Monetary transmission: Lower $i$ stimulates investment, raising output through the multiplier.
Liquidity trap: At very low interest rates, money demand becomes perfectly elastic — LM is flat, monetary expansion has no further effect on $i$.

Labour Market & the Phillips Curve

Wage and Price Setting

Wage setting: $W = P^e F(u, z)$, wages depend on expected prices, unemployment (negatively), and institutional factors $z$.

Price setting: $P = (1 + m)W$, firms set prices as a markup over wages.

Combining gives the natural rate of unemployment $u_n$:

$F(u_n, z) = \frac{1}{1+m}$

Phillips Curve

The relationship between inflation and unemployment:

Phillips Curve: $\quad \pi_t = \pi_t^e + (m + z) - \alpha u_t$

With anchored expectations $\pi_t^e = \bar{\pi}$, this becomes:

$\pi_t - \bar{\pi} = -\alpha(u_t - u_n)$

Inflation is above expected inflation when unemployment is below the natural rate.

Expectations-Augmented Phillips Curve

Modern formulation: $\pi_t = \pi_t^e - \alpha(u_t - u_n)$

If $\pi_t^e = \pi_{t-1}$ (adaptive expectations): $\pi_t - \pi_{t-1} = -\alpha(u_t - u_n)$

The change in inflation is negatively related to the unemployment gap. To reduce inflation, unemployment must be above $u_n$ for a sustained period.

IS-LM-PC Model

The IS-LM-PC model integrates the short run (IS-LM) with the medium run (Phillips Curve), showing how the economy adjusts over time.

Three Equations

IS: $Y = C(Y-T) + I(Y, i) + G$
LM: $i = \bar{i}$ (central bank sets rate)
PC: $\pi_t - \pi_t^e = -\alpha(u_t - u_n)$

Using Okun's Law $u_t - u_n = -\beta(Y_t - Y_n)$ to replace unemployment with output gap:

$\pi_t - \pi_t^e = \alpha\beta(Y_t - Y_n) \equiv \kappa(Y_t - Y_n)$

Adjustment Dynamics

If $Y > Y_n$, inflation rises. This prompts the central bank to raise $i$, shifting IS left, reducing $Y$. This continues until $Y = Y_n$.

In the medium run: $Y = Y_n$, $u = u_n$, $\pi$ is stable at $\bar{\pi}$. Money is neutral in the medium run.

Dynamic AS-AD Model

The DAS-DAD model generalizes IS-LM-PC to include dynamics of inflation expectations and explicit monetary policy rules.

Five Equations

Demand for Goods (DAD): $Y_t = \bar{Y} - \alpha(r_t - \rho) + \varepsilon_t$
Output depends negatively on the real interest rate gap.
Real Interest Rate (Fisher): $r_t = i_t - E_t\pi_{t+1}$
Phillips Curve (DAS): $\pi_t = E_{t-1}\pi_t + \phi(Y_t - \bar{Y}) + v_t$
Adaptive Expectations: $E_t\pi_{t+1} = \pi_t$
Monetary Policy (Taylor Rule): $i_t = \pi_t + \rho + \theta_\pi(\pi_t - \pi^*) + \theta_Y(Y_t - \bar{Y})$

Reduced Form: DAD and DAS

Substituting the Taylor rule into the IS curve yields the DAD equation:

DAD: $\quad Y_t = \bar{Y} - \frac{\alpha\theta_\pi}{1+\alpha\theta_Y}(\pi_t - \pi^*) + \frac{1}{1+\alpha\theta_Y}\varepsilon_t$

DAS: $\quad \pi_t = \pi_{t-1} + \phi(Y_t - \bar{Y}) + v_t$

The DAD is downward-sloping in $(Y, \pi)$ space. The DAS is upward-sloping.

Shocks in DAS-DAD

Demand Shocks ($\varepsilon_t$)

A positive demand shock shifts DAD right. On impact, both $Y$ and $\pi$ rise. As inflation expectations adjust upward (DAS shifts up), output gradually returns to $\bar{Y}$ while inflation remains elevated. Once the shock disappears, DAD returns to original position, output falls below potential, and inflation is gradually squeezed out.

Supply Shocks ($v_t$)

A positive supply shock (e.g., oil price increase) shifts DAS up. On impact, $\pi$ rises and $Y$ falls — stagflation. If transitory, DAS eventually shifts back. The output-inflation trade-off depends on the DAD slope: steeper DAD means supply shocks cause larger inflation changes and smaller output changes.

Change in Inflation Target $\pi^*$

If the central bank reduces $\pi^*$, the DAD shifts left. On impact: output falls, inflation begins to decline (disinflationary recession). As inflation expectations fall, DAS shifts down, and output gradually recovers. The sacrifice ratio measures the cumulative output loss per percentage point of disinflation.

Monetary Policy Implications

Trade-off Between Output and Inflation Volatility

The slope of DAD determines the division of supply shock effects:

$\displaystyle \frac{d\pi}{dY}\bigg|_{DAD} = -\frac{1+\alpha\theta_Y}{\alpha\theta_\pi}$

Large $\theta_\pi$ makes DAD flat: small $\pi$ variance, large $Y$ variance.
Large $\theta_Y$ makes DAD steep: small $Y$ variance, large $\pi$ variance.

The Fed (dual mandate) may prefer a different $\theta_Y$ vs $\theta_\pi$ mix than the ECB (price stability mandate).

The Taylor Principle

Taylor Principle: For stable inflation dynamics, the nominal interest rate must increase more than one-for-one with inflation: $\quad \frac{di}{d\pi} = 1 + \theta_\pi > 1 \;\Leftrightarrow\; \theta_\pi > 0$

If $\theta_\pi > 0$, the real interest rate rises when inflation rises, which dampens demand and stabilizes inflation. If $\theta_\pi < 0$, the real rate falls when inflation rises (the real rate goes the wrong way) — inflation becomes unstable and can spiral out of control.