Macroeconomic Equilibrium: IS-LM and WS-PS Analysis

Posted on Feb 4, 2026 in Economy

IS-LM Model in a Closed Economy

QUESTION 1:

Consider a closed economy defined by the following equations:

  • Consumption (C): 200 + 0.4(Y – T)
  • Investment (I): 150 + 0.1Y – 1000(r + x)
  • Government Spending (G): 250
  • Taxes (T): 100 + 0.2Y
  • Real Interest Rate (r): 0.05 (Set by the Central Bank)
  • Risk Premium (x): 0.02

Equilibrium Output and the IS Relation

To find the IS relation, we start from the goods market equilibrium condition (Y = Z):

Y = [200 + 0.4(Y – (100 + 0.2Y))] + [150 + 0.1Y – 1000(r + x)] + 250

  • Calculating Disposable Income (YD):
    YD = Y – 100 – 0.2Y = 0.8Y – 100
  • Substituting into the demand equation:
    Y = 200 + 0.4(0.8Y – 100) + 150 + 0.1Y – 1000(r + x) + 250
    Y = 200 + 0.32Y – 40 + 150 + 0.1Y – 1000(r + x) + 250
  • Grouping Y terms and autonomous components:
    Y = 560 + 0.42Y – 1000(r + x)
    Y – 0.42Y = 560 – 1000(r + x) → 0.58Y = 560 – 1000(r + x)
  • Final IS Equation:
    Y = (1 / 0.58) [560 – 1000(r + x)]
  • Multiplier: 1 / 0.58 ≈ 1.724
  • Equilibrium Output: Substituting r = 0.05 and x = 0.02:
    Y = 1.724 [560 – 1000(0.07)] = 1.724 [560 – 70] = 1.724 [490] = 844.8

Investment Shocks and Multiplier Effects

b) Business expectations worsen, and autonomous investment falls from 150 to 100. Calculate the new equilibrium.

If autonomous investment (I) decreases by 50 units:

  • Numerical calculation: ΔY = Multiplier · ΔI = 1.724 · (-50) = -86.2
  • New Output: Y’ = 844.8 – 86.2 = 758.6
  • Explanation: The drop in investment reduces aggregate demand. Through the multiplier effect, production falls by more than the initial drop in investment (86.2 vs. 50) because the lower income subsequently reduces consumption and induced investment. Graphically, the IS curve shifts to the left.

Fiscal Policy Intervention

c) The Government decides to intervene to restore the initial level of output using Government Spending (G), while the Central Bank keeps the interest rate constant. Calculate the required change in G.

  1. Identification of policy: An expansionary fiscal policy is required.
  2. Calculation: Since the multiplier for G is the same as for I (1.724), the government must increase spending by exactly the amount that investment fell to neutralize the shock.
    ΔG = 50.
  3. Graphical Interpretation: The IS curve shifts back to the right, returning to the original intersection point with the horizontal LM curve at Y = 844.8.

Labor Market Equilibrium: WS-PS Model

QUESTION 2: Labor Market (WS-PS)

Equations: WS: W/P = 1 – 2u + z; PS: P = (1 + m)W.

Data: m = 0.25; z = 0.05. Labor force L = 2000. Production function Y = N.

Natural Rate of Unemployment and Real Wage

  1. Real Wage (PS): From P = (1 + m)W, the real wage is W/P = 1 / (1 + m) = 1 / 1.25 = 0.8.
  2. Natural Rate (Un): We equate the PS and WS relations:
    0.8 = 1 – 2u + 0.05 → 2u = 1.05 – 0.8 → 2u = 0.25 → Un = 0.125 (12.5%)
  • Price Setting (PS): Firms set prices as a markup over costs. This implies a constant real wage: W/P = 0.8.
  • Wage Setting (WS): Workers and firms negotiate wages based on W = Pe(1 – 2u + z). In the medium run, P = Pe. Equating both yields un = 12.5%. The real wage is determined by the degree of competition (m), while un ensures wage demands match what firms pay.

Labor Market Reform and Unemployment Benefits

b) The government tightens eligibility for unemployment benefits (reducing z to 0.02). Explain the effect and calculate the new un.

  • Analysis: A reduction in z decreases workers’ bargaining power. For any given level of unemployment, they will accept a lower nominal wage. The WS curve shifts downward.
  • Calculation: 0.8 = 1 – 2u + 0.02 → 2u = 1.02 – 0.8 → 2u = 0.22 → Un = 0.11 (11%)
  • Reasoning: Since being unemployed is now less protected, the “outside option” for workers is reduced. The natural rate of unemployment decreases because the labor market becomes more “flexible.”

Natural Level of Output

c) What happens to the natural level of output (Yn)?

Given Y = N and u = 1 – Y/L:

  • Initial: Yn = L(1 – un) = 2000(1 – 0.125) = 1750
  • Final: Yn’ = 2000(1 – 0.11) = 1780
  • Conclusion: The labor market reform increases the economy’s natural productive capacity.

Medium-Run Dynamics and Supply Shocks

QUESTION 3: The Medium Run (IS-LM-PC Model)

The economy is at its natural level. Suddenly, energy prices rise, leading firms to increase their markup (m).

Impact on the Phillips Curve

An increase in the price of energy inputs leads firms to increase their markup (m) to maintain profitability.

  • The real wage offered by firms 1/(1+m) falls, shifting the PS curve downward. This leads to a higher natural rate of unemployment (un increases).
  • Effect on PC: Since un is higher, the natural level of output (Yn) falls. The Phillips Curve shifts upward, meaning that at the previous level of production, there is now “excessive” demand relative to the new capacity, causing inflation.

Inflationary Pressures

b) If the Central Bank does not react immediately, what happens to inflation over time?

  • If output remains at the old natural level Yn, it is now higher than the new natural level (Y > Yn’).
  • This creates a positive output gap. According to the Phillips Curve, the change in inflation will be positive.
  • Result: Inflation will accelerate year after year.

Adjustment to Medium-Run Equilibrium

c) Explain the adjustment required to reach the new medium-run equilibrium.

To stabilize inflation, the Central Bank must perform a restrictive monetary policy by raising the real interest rate (r).

  • Monetary Policy: Increasing r is represented by an upward shift of the LM curve.
  • Transmission: Higher interest rates reduce investment and aggregate demand. Output falls along the IS curve until it reaches the new, lower Yn’.
  • Final Equilibrium: At Yn’, inflation is stable again, but the economy suffers from stagflation: lower production and higher structural unemployment than before the shock.