📖 IS-LM 模型导论

Economics

I. Lesson Focus

This lesson introduces the IS-LM model, which simultaneously determines equilibrium output and interest rates by combining goods-market equilibrium (IS) and money-market equilibrium (LM). Candidates must understand the slopes and shift factors of both curves, the transmission mechanisms of fiscal and monetary policy, the concept of crowding-out, and the special case of the liquidity trap.

II. The Problem

Suppose an economy suffers from weak aggregate demand due to falling business investment (goods market disequilibrium) while the central bank wants to adjust interest rates to stimulate activity without triggering inflation. Policymakers need a single interest rate and output level at which both the market for goods and services and the money market clear simultaneously. The IS-LM framework solves this joint-equilibrium problem and is a core Keynesian tool for analyzing the short-run effects of fiscal and monetary policy on output and interest rates.

III. The IS Curve: Goods-Market Equilibrium

The IS (Investment-Saving) curve shows combinations of real output (Y) and the real interest rate (r) at which the goods market is in equilibrium: planned aggregate expenditure equals actual output.

$$ Y = C + I + G + (X - M) $$

Consumption is usually $C = C_0 + c(Y - T)$; investment is interest-sensitive: $I = I_0 - b r$.

Substituting and rearranging yields the IS equation (simplified form):

$$ Y = \frac{1}{1-c}[C_0 - cT + I_0 + G + (X-M) - b r] $$

The IS curve is downward-sloping: higher r reduces investment, lowers aggregate demand, and reduces equilibrium output.

Shifts in the IS curve: - Fiscal expansion (↑G or ↓T) shifts IS right. - Rise in autonomous consumption or investment (↑C₀ or ↑I₀) shifts IS right. - Increase in net exports shifts IS right.

IV. The LM Curve: Money-Market Equilibrium

The LM (Liquidity-Money) curve shows combinations of Y and r at which the money market is in equilibrium: real money supply equals real money demand.

$$ \frac{M}{P} = L(Y, r) $$

Money demand is typically $L = kY - h r$ (transactions demand rises with Y; speculative demand falls with r).

With fixed prices, the LM equation is:

$$ r = \frac{k}{h} Y - \frac{1}{h} \left( \frac{M}{P} \right) $$

The LM curve is upward-sloping: higher Y raises transactions demand for money; with fixed supply, r must rise to restore equilibrium.

Shifts in the LM curve: - Increase in nominal money supply (↑M) shifts LM right (down). - Rise in price level (↑P) reduces real money supply and shifts LM left (up). - Increase in money demand (higher k) shifts LM left.

V. IS-LM Equilibrium

The intersection of IS and LM simultaneously determines equilibrium interest rate $r^$ and output $Y^$. At this point both markets clear.

Disequilibrium triggers automatic adjustment: - If current Y lies above the intersection, excess supply of goods causes inventories to rise; firms cut production and Y falls. - If current r lies below equilibrium, excess demand for money leads agents to sell bonds, driving bond prices down and r up until equilibrium is restored.

VI. Fiscal and Monetary Policy Effects

Fiscal Policy (IS shifts): - Expansionary fiscal policy (↑G or ↓T) shifts IS right. The new equilibrium moves up along the LM curve: Y rises and r rises (crowding-out effect). - Size of crowding-out depends on LM slope: steeper LM (interest-insensitive money demand) produces larger crowding-out.

Monetary Policy (LM shifts): - Expansionary monetary policy (↑M) shifts LM right. The new equilibrium moves down along the IS curve: Y rises and r falls. - Liquidity trap (horizontal LM segment): monetary policy cannot lower r further and becomes completely ineffective.

Policy Mix: Appropriate combination of fiscal and monetary expansion can raise Y while keeping r unchanged.

Worked Cases

Case 1: Basic Equilibrium Calculation

Given: $C = 200 + 0.75(Y - T)$, $I = 300 - 20r$, $G = 150$, $T = 100$, $X-M = 20$. Money market: $\frac{M}{P} = 0.4Y - 30r$, and $\frac{M}{P}=150$.

IS curve derivation:
$Y = 200 + 0.75(Y-100) + 300 - 20r + 150 + 20$
$Y = 620 + 0.75Y - 20r$
$0.25Y = 620 - 20r$
$Y = 2480 - 80r$

LM curve:
$150 = 0.4Y - 30r$
$Y = 375 + 75r$

Equilibrium: $2480 - 80r = 375 + 75r$
$2105 = 155r$ → $r^ = 13.58\%$
$Y^ = 375 + 75×13.58 ≈ 1393.5$

Case 2: Expansionary Fiscal Policy

Government spending rises to $G=200$. New IS: $Y = 2580 - 80r$.
Intersect with original LM: $2580 - 80r = 375 + 75r$
$2205 = 155r$ → $r^ = 14.23\%$, $Y^ ≈ 1442$
Output rises by 48.5 while r increases by 0.65 percentage points, illustrating partial crowding-out.

Case 3: Expansionary Monetary Policy and Liquidity Trap

Money supply rises so $\frac{M}{P}=180$; new LM: $Y = 450 + 75r$.
Intersect with original IS: $2480 - 80r = 450 + 75r$
$2030 = 155r$ → $r^ = 13.1\%$, $Y^ ≈ 1432.5$
If the economy is in a liquidity trap (LM horizontal at r = 2%), further LM shifts only increase idle balances; Y remains unchanged and monetary policy is ineffective.

Traps

Key Formulas

• IS (simplified): $Y = \frac{A - b r}{1-c}$ where A = autonomous spending
• LM: $r = \frac{k}{h}Y - \frac{1}{h}\left(\frac{M}{P}\right)$
• Equilibrium: Solve IS = LM simultaneously for $r^$ and $Y^$
• Fiscal multiplier in IS-LM < simple closed-economy multiplier $\frac{1}{1-c}$
• Monetary transmission: ↑M → ↓r → ↑I → ↑Y
• Crowding-out intensity positively related to LM slope, negatively related to IS slope

Practice Questions

Q1. The negative slope of the IS curve is primarily caused by:
A. Higher interest rates increasing money demand
B. Higher interest rates reducing investment and aggregate demand
C. Higher output increasing transactions demand for money
D. Higher prices reducing real money supply

Q2. All else equal, an increase in government purchases shifts:
A. IS left and lowers equilibrium interest rates
B. IS right and raises equilibrium interest rates
C. LM right and lowers equilibrium output
D. LM left and raises equilibrium interest rates

Q3. If the LM curve is vertical, then:
A. Monetary policy is completely ineffective
B. Fiscal policy has no crowding-out
C. Fiscal expansion affects only interest rates, not output
D. Money demand is completely interest-inelastic

Q4. Which of the following shifts the LM curve to the right?
A. An increase in the price level
B. An increase in the money supply
C. An increase in the marginal propensity to consume
D. An increase in government taxes

Q5. In the liquidity-trap region:
A. Monetary policy is effective and fiscal policy is ineffective
B. The fiscal multiplier is maximized and monetary policy is completely ineffective
C. The IS curve is horizontal
D. Investment is completely insensitive to interest rates

Q6. An economy has IS: $Y=2000-100r$ and LM: $Y=500+150r$. The equilibrium interest rate is closest to:
A. 5%
B. 6%
C. 7.5%
D. 10%

Q7. The central bank conducts expansionary monetary policy while the government simultaneously pursues contractionary fiscal policy. The most likely outcome is:
A. Output rises and interest rates definitely rise
B. Output falls and interest-rate direction is uncertain
C. Output is unchanged and interest rates fall
D. Both output and interest-rate directions are uncertain

Q8. In the IS-LM model, crowding-out is smallest when:
A. The LM curve is vertical
B. The IS curve is very flat
C. The LM curve is very flat
D. Investment is completely interest-insensitive

Answers

Takeaways

• The IS curve is downward-sloping (goods-market equilibrium); the LM curve is upward-sloping (money-market equilibrium).
• Their intersection gives simultaneous equilibrium values of r and Y.
• Expansionary fiscal policy shifts IS right → higher Y and higher r (partial crowding-out).
• Expansionary monetary policy shifts LM right → higher Y and lower r.
• In a liquidity trap, monetary policy loses effectiveness while fiscal policy becomes fully effective.
• Coordinated policy can increase output without changing the interest rate.