📊 AD-AS Dynamics & The Phillips Curve

The Aggregate Demand-Aggregate Supply (AD-AS) model extends macroeconomic analysis to variable general price levels ($P$), explaining the determination of inflation, output gaps, and economic fluctuations.

---

1. 📉 Aggregate Demand (AD) Curve

The AD curve traces combinations of $(P,Y)$ derived from the simultaneous intersection of IS and LM curves at varying price levels.

       IS-LM Plane                                       AD-AS Plane
   i ▲                  LM(P_1)                    P ▲
     │                  / LM(P_0)                    │
     │                 /  /                          │  P_1 •          \ AD
     │        • E_1   /  /                           │      │           \
     │         \     /  /                            │  P_0 •────────────•
     │          \   /  /                             │      │            │ \
     │           \ /  /                              │      │            │  \
     └────────────┴──┴──────► Y                      └──────┴────────────┴───► Y
                 Y_1 Y_0                                   Y_1          Y_0

1.1 Why AD Slopes Downward

1. Pigou's Wealth Effect: Lower price level ($P\downarrow$) increases real purchasing power of money balances ($M/P\uparrow$), stimulating consumption ($C\uparrow$).
2. Keynes' Interest-Rate Effect: Lower price level reduces transaction money demand, lowering interest rate ($i\downarrow$) and stimulating investment ($I\uparrow$).
3. Mundell-Fleming Exchange-Rate Effect: Lower interest rates cause domestic currency depreciation ($e\downarrow$), boosting net exports ($NX\uparrow$).

---

2. ⚡ Aggregate Supply (AS) Mechanics

• Short-Run Aggregate Supply ($SRAS$): Upward-sloping due to sticky wages/prices and misperceptions:$$Y=Y_n+\alpha (P-P^e)(\alpha >0)$$where $Y_n$ is natural output (potential GDP) and $P^e$ is the expected price level.
• Long-Run Aggregate Supply ($LRAS$): Vertical at potential GDP $Y_n$, determined strictly by technology and factor endowments (Classical neutrality).

  Price Level (P)
    ▲                  LRAS
    │                   │          SRAS
 P* │───────────────────┼─────────/
    │                   │ \      /
    │                   │  \    /
    │                   │   \  /
    │                   │    \/
    │                   │    /\
    │                   │   /  \ AD
    └───────────────────┴──┴────┴────────► Real Output (Y)
                       Y_n

2.1 Shocks & Stagflation

An Adverse Supply Shock (e.g. 1970s oil embargo) shifts the $SRAS$ curve leftward/upward:

• Output drops below potential ($Y<Y_n⟹$ recession/unemployment).
• Price level surges ($P\uparrow ⟹$ inflation).
• Stagflation: The coexistence of stagnant economic output and high inflation.

---

3. 🔄 The Phillips Curve & Inflation Dynamics

A.W. Phillips (1958) observed an empirical inverse tradeoff between nominal wage inflation and unemployment. Milton Friedman and Edmund Phelps (1968) demonstrated that this tradeoff exists only in the short run.

3.1 Expectations-Augmented Phillips Curve

$${\pi }_t={\pi }_t^e-\beta (u_t-u_n)+{\nu }_t(\beta >0)$$

where ${\pi }_t$ is inflation, ${\pi }_t^e$ is expected inflation, $u_n$ is the Natural Rate of Unemployment (NAIRU), and ${\nu }_t$ is a supply shock.

       Inflation (π)
           ▲
           │          LRPC (Vertical at NAIRU)
           │           │
       π_1 │───────────┼──────────•  SRPC (Higher π^e)
           │           │         / \
       π_0 │───────────•────────/───• SRPC (Lower π^e)
           │           │\      /     \
           │           │ \    /       \
           └───────────┴──┴──┴─────────┴─────► Unemployment (u)
                      u_n

• Short-Run Phillips Curve ($SRPC$): Downward sloping for fixed ${\pi }^e$.
• Long-Run Phillips Curve ($LRPC$): Perfectly vertical at $u=u_n$. There is no long-run tradeoff between inflation and unemployment.

---

4. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Inflation-Output Gap Phillips Dynamics

Problem: An economy's Phillips curve is given by ${\pi }_t={\pi }_t^e-0.5(u_t-5\mathrm{\%})+{\nu }_t$. Potential output is related to unemployment via Okun's law: $(Y_t-Y_n)/Y_n=-2(u_t-5\mathrm{\%})$. The natural unemployment rate is $u_n=5\mathrm{\%}$.

1. If agents form adaptive expectations (${\pi }_t^e={\pi }_{t-1}$), calculate the inflation rate in year $t$ if the government maintains an output boom $Y_t=1.04Y_n$ with zero supply shocks (${\nu }_t=0$) starting from initial inflation ${\pi }_0=2\mathrm{\%}$.
2. What happens to inflation if the boom is maintained for 3 consecutive years?

Step-by-Step Rigorous Solution:

1. Calculate unemployment rate under the boom:

   $$\frac{Y_t-Y_n}{Y_n}=4\mathrm{\%}=-2(u_t-5\mathrm{\%})⟹u_t-5\mathrm{\%}=-2\mathrm{\%}⟹u_t=3\mathrm{\%}$$

2. Year 1 Inflation (${\pi }_1$):

   $${\pi }_1={\pi }_0-0.5(3\mathrm{\%}-5\mathrm{\%})=2\mathrm{\%}-0.5(-2\mathrm{\%})=2\mathrm{\%}+1\mathrm{\%}=3.0\mathrm{\%}$$

3. Year 2 & Year 3 Inflation (Accelerating Inflation):

   • Year 2: Expected inflation adapts to prior year ${\pi }_2^e={\pi }_1=3\mathrm{\%}$.$${\pi }_2=3\mathrm{\%}+1\mathrm{\%}=4.0\mathrm{\%}$$
   • Year 3: ${\pi }_3^e={\pi }_2=4\mathrm{\%}⟹{\pi }_3=4\mathrm{\%}+1\mathrm{\%}=5.0\mathrm{\%}$. Economic Insight: Attempting to permanently peg output above potential triggers accelerating hyper-inflation (the Accelerationist Hypothesis), proving why central banks must anchor long-run inflation expectations.