📈 The IS-LM Framework & Goods-Money Balance

Developed by John Hicks (1937) to formalize Keynes' General Theory, the IS-LM model determines simultaneous short-run equilibrium output ($Y$) and interest rates ($i$) by integrating the Goods Market (IS: Investment-Saving) with the Money/Financial Assets Market (LM: Liquidity Preference-Money Supply).

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1. 📦 The IS Curve (Goods Market Equilibrium)

The IS curve represents combinations of $(Y,i)$ where goods market demand equals aggregate supply:

$$Y=C(Y-T)+I(i)+G$$

Assuming linear specifications:

$$C=C_0+c_1(Y-T),I=I_0-d_{1}i$$$$Y=C_0+c_{1}Y-c_{1}T+I_0-d_{1}i+G$$$$Y(1-c_1)=A_0-d_{1}i⟹Y=\frac{A_0}{1-c_1}-\frac{d_1}{1-c_1}i\text{where }{A}_0=C_0-c_{1}T+I_0+G$$

1.1 Properties of the IS Curve

• Slope: Downward-sloping ($\frac{di}{dY}=-\frac{1-c_1}{d_1}<0$). A higher interest rate raises the cost of borrowing, depressing business investment $I(i)$ and aggregate demand.
• Shifts: Autonomous increases in government spending ($\mathrm{\Delta }G>0$), tax cuts ($\mathrm{\Delta }T<0$), or consumer/business confidence shift the IS curve rightward by $\mathrm{\Delta }Y=\frac{1}{1-c_1}\mathrm{\Delta }G$.

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2. 💵 The LM Curve (Money Market Equilibrium)

The LM curve represents combinations of $(Y,i)$ where real money supply equals real money demand (Liquidity Preference):

$$\frac{M^s}{P}=L(Y,i)=kY-hi(k>0,h>0)$$

Solving for interest rate $i$:

$$i=\frac{k}{h}Y-\frac{1}{h}\frac{M^s}{P}$$

2.1 Properties of the LM Curve

• Slope: Upward-sloping ($\frac{di}{dY}=\frac{k}{h}>0$). Higher income $Y$ increases transaction demand for money; with fixed money supply $M/P$, the interest rate must rise to clear the money market.
• Shifts: An increase in real money supply ($\mathrm{\Delta }M>0$) shifts the LM curve rightward/downward by $\mathrm{\Delta }Y=\frac{1}{k}\mathrm{\Delta }(M/P)$.

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3. ⚡ General Equilibrium & Policy Interventions

  Interest Rate (i)
    ▲                                    LM
    │                                   /
    │                   IS             /
 i* │────────────────────•────────────/
    │                   / \          /
    │                  /   \        /
    │                 /     \      /
    │                /       \    /
    └───────────────┴─────────┴──┴────────► Real Output (Y)
                   Y*

3.1 Fiscal Expansion & Crowding Out

When government increases expenditure ($\mathrm{\Delta }G>0$), IS shifts right. Output expands to $Y_1$, which raises money demand, driving up interest rates from $i_0$ to $i_1$. The higher interest rate crowds out private investment:

$$\text{Crowding-Out Effect}=I(i_0)-I(i_1)>0$$

Policy Regime              LM Slope            Fiscal Multiplier          Crowding Out %
────────────────────────────────────────────────────────────────────────────────────────
Classical Case             Vertical (h → 0)    k_G = 0                    100% (Complete)
Intermediate Normal Case   Upward-sloping      0 < k_G < 1/(1-c_1)        Partial
Liquidity Trap / ZLB       Horizontal (h → ∞)  k_G = 1/(1-c_1)            0% (Zero!)

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4. 🎯 Olympiad-Level Worked Master Problem

Master Problem: IS-LM Mathematical Equilibrium

Problem: An economy is described by the following structural equations:

• Goods Market: $C=200+0.8(Y-T)$, $I=400-2000i$, $G=300$, $T=250$.
• Money Market: $\frac{M}{P}=800$, $L(Y,i)=0.4Y-4000i$.

1. Derive the mathematical equations for the IS and LM curves.
2. Calculate the general equilibrium output $Y^{\ast }$ and interest rate $i^{\ast }$.
3. Calculate the degree of investment crowding out if government spending rises to $G^{\prime }=400$.

Step-by-Step Rigorous Solution:

1. Derive the IS Equation:

   $$Y=200+0.8(Y-250)+400-2000i+300=700+0.8Y-2000i$$$$0.2Y=700-2000i⟹Y=3500-10000i\text{(IS Curve)}$$

2. Derive the LM Equation:

   $$800=0.4Y-4000i⟹0.4Y=800+4000i⟹Y=2000+10000i\text{(LM Curve)}$$

3. Solve for Equilibrium $(Y^{\ast },i^{\ast })$:

   $$3500-10000i=2000+10000i⟹20000i=1500⟹i^{\ast }=0.075=7.5\mathrm{\%}$$$$Y^{\ast }=2000+10000(0.075)=2000+750=\mathrm{\$}2750$$

   Baseline Investment: $I_0=400-2000(0.075)=400-150=\mathrm{\$}250$.

4. Fiscal Expansion ($\mathrm{\Delta }G=100⟹G^{\prime }=400$): New IS Curve: $0.2Y=800-2000i⟹Y=4000-10000i$. Equating with LM:

   $$4000-10000i=2000+10000i⟹20000i=2000⟹i^{\prime }=0.10=10.0\mathrm{\%}$$$$Y^{\prime }=2000+10000(0.10)=\mathrm{\$}3000$$

   New Investment: $I^{\prime }=400-2000(0.10)=400-200=\mathrm{\$}200$.

   $$\text{Private Investment Crowded Out}=I_0-I^{\prime }=250-200=\mathrm{\$}50$$