🔄 Inflation, Unemployment & Business Cycles

Macroeconomies do not expand in smooth deterministic trajectories; they experience cyclical expansions, peaks, recessions, and troughs accompanied by fluctuations in labor market tightness and general price levels.

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1. 👥 The Labor Market & Unemployment Typology

The labor force comprises all individuals aged $16+$ who are either employed ($E$) or actively seeking employment ($U$).

$$\text{Labor Force }LF=E+U$$$$\text{Unemployment Rate }u=\frac{U}{LF}\times 100$$$$\text{Labor Force Participation Rate }LFPR=\frac{LF}{\text{Working-Age Population}}\times 100$$

1.1 The Three Types of Unemployment

Unemployment Type	Underlying Economic Cause	Policy Cure
Frictional	Time delays in matching workers to job vacancies (search costs, transitions).	Job search portals, career counseling.
Structural	Mismatch between worker skills/locations and employer requirements (automation, globalization).	Vocational retraining, moving subsidies.
Cyclical	Deficiency of aggregate demand during economic recessions ($Y<Y_n$).	Countercyclical monetary/fiscal stimulus.

1.2 The Natural Rate of Unemployment (NAIRU)

$$u_n=u_{frictional}+u_{structural}$$

When the actual unemployment rate equals the natural rate ($u=u_n$), cyclical unemployment is zero and the economy operates at potential output ($Y=Y_n$).

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2. 📉 Okun's Law (Output-Unemployment Elasticity)

Arthur Okun (1962) established the empirical relationship between real GDP growth and changes in unemployment:

$$\frac{\mathrm{\Delta }Y}{Y}=g_n-\beta \mathrm{\Delta }u(\beta \approx 2.0)$$

where $g_n$ is the long-run potential GDP growth rate ($\approx 3\mathrm{\%}$).

In gap form:

$$\frac{Y-Y^{\ast }}{Y^{\ast }}=-\beta (u-u_n)$$

A $1\mathrm{\%}$ increase in cyclical unemployment ($u-u_n=+1\mathrm{\%}$) is associated with a $2\mathrm{\%}$ loss of potential output.

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3. 💸 Inflation Mechanics & Quantity Theory of Money

Inflation is a sustained increase in the economy's general price level ($P$).

• Demand-Pull Inflation: "Too much money chasing too few goods" ($AD$ shifts rightward beyond capacity).
• Cost-Push Inflation: Supply-side cost shocks ($SRAS$ shifts leftward, e.g. energy price spikes).

3.1 The Quantity Theory of Money (Fisher Equation)

$$M\cdot V=P\cdot Y$$

where $M$ is money supply, $V$ is money velocity, $P$ is price level, and $Y$ is real GDP.

Taking log-derivatives:

$$\frac{\mathrm{\Delta }M}{M}+\frac{\mathrm{\Delta }V}{V}=\frac{\mathrm{\Delta }P}{P}+\frac{\mathrm{\Delta }Y}{Y}⟹\mu +\mathrm{\Delta }v=\pi +g$$

Assuming constant velocity ($\mathrm{\Delta }v=0$):

$$\pi =\mu -g$$

Friedman's Maxim: "Inflation is always and everywhere a monetary phenomenon, in the sense that it is and can be produced only by a more rapid increase in the quantity of money than in output."

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4. 📈 Business Cycle Theories

                   ┌─────────────────────────────────────────────────────────┐
                   │               THEORIES OF BUSINESS CYCLES               │
                   ├────────────────────────────┬────────────────────────────┤
                   │  New Keynesian Theory      │ Real Business Cycle (RBC)  │
                   │                            │                            │
                   │ • Driven by demand shocks  │ • Driven by real supply &  │
                   │ • Sticky nominal wages     │   Total Factor Productivity│
                   │   and prices (Calvo)       │   (TFP) technology shocks  │
                   │ • Sub-optimal fluctuations │ • Fluctuations represent   │
                   │ • Justifies active fiscal  │   optimal Pareto-efficient │
                   │   and monetary policy      │   dynamic market responses │
                   └────────────────────────────┴────────────────────────────┘

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5. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Okun's Law and Quantity Theory Integration

Problem: An economy has potential output growing at $g=3.0\mathrm{\%}$ per year, natural rate of unemployment $u_n=5.0\mathrm{\%}$, and constant velocity of money ($\mathrm{\Delta }v=0$). The central bank expands the money supply at $\mu =7.0\mathrm{\%}$ per year.

1. Calculate the long-run equilibrium inflation rate ${\pi }^{\ast }$.
2. In year $t$, an economic shock causes actual GDP to grow by only $1.0\mathrm{\%}$. Calculate the resulting change in the unemployment rate $\mathrm{\Delta }u$ using Okun's Law ($\beta =2.0$).
3. If the initial unemployment rate was $5.0\mathrm{\%}$, what is the new unemployment rate?

Step-by-Step Rigorous Solution:

1. Calculate Long-Run Inflation Rate:

   $${\pi }^{\ast }=\mu -g=7.0\mathrm{\%}-3.0\mathrm{\%}=4.0\mathrm{\%}/\text{year}$$

2. Apply Okun's Law:

   $$\frac{\mathrm{\Delta }Y}{Y}=g-\beta \mathrm{\Delta }u⟹1.0\mathrm{\%}=3.0\mathrm{\%}-2.0\mathrm{\Delta }u$$$$2.0\mathrm{\Delta }u=3.0\mathrm{\%}-1.0\mathrm{\%}=2.0\mathrm{\%}⟹\mathrm{\Delta }u=+1.0\mathrm{\%}$$

3. New Unemployment Rate:

   $$u_t=u_{t-1}+\mathrm{\Delta }u=5.0\mathrm{\%}+1.0\mathrm{\%}=6.0\mathrm{\%}$$

   Economic Finding: A $2\mathrm{\%}$ growth shortfall below potential raises the unemployment rate by $1.0$ percentage point.