Macroeconomic Models: IS-LM, Labor Markets, and Growth

Posted on Feb 25, 2026 in Business Administration and Management (BAM)

Question 1: Goods and Financial Markets – Extended IS-LM

Problem Statement:

Consider a closed economy in the short run with the following behavioral equations:

  • C = 400 + 0.5(Y – T)
  • I = 200 + 0.1Y – 2000(r + x)
  • G = 400
  • T = 200 + 0.2Y

The Central Bank sets a real interest rate target of r = 0.05 and the initial risk premium is x = 0.03.

(a) Obtain the IS relation and calculate equilibrium

First, we simplify disposable income: YD = Y – (200 + 0.2Y) = 0.8Y – 200.

Substituting this back into the demand function:

Y = 400 + 0.5(0.8Y – 200) + 200 + 0.1Y – 2000(r + x) + 400

Y = 400 + 0.4Y – 100 + 200 + 0.1Y – 2000(r + x) + 400

Grouping autonomous terms and Y terms:

Y = 900 + 0.5Y – 2000(r + x) → 0.5Y = 900 – 2000(r + x)

The IS equation is: Y = 2[900 – 2000(r + x)], where 2 is the multiplier.

Substituting r = 0.05 and x = 0.03 (r + x = 0.08):

Y = 2[900 – 2000(0.08)] = 2[900 – 160] = 2[740] = 1480

To calculate the Budget Balance (T – G):

T = 200 + 0.2(1480) = 200 + 296 = 496.

Balance = 496 – 400 = +96 (Budget Surplus).

(b) Financial Crisis and Risk Premium Transmission

A financial crisis causes the risk premium to rise to x’ = 0.08. An increase in the risk premium (x) represents a shock in the financial market that directly impacts the goods market. Although the Central Bank keeps the policy rate (r) at 5%, the borrowing rate faced by firms and consumers (r + x) increases from 8% to 13%.

  • Transmission Mechanism: Higher borrowing costs make investment projects less profitable, leading to a direct reduction in investment (I). This drop in demand triggers a multiplier process: lower production leads to lower disposable income, which reduces consumption, and lower sales, which further reduces investment through the 0.1Y component.
  • Calculation: Δ(r + x) = 0.05. ΔY = Multiplier × [-2000 × Δ(r + x)] = 2 × [-2000 × 0.05] = 2 × [-100] = -200.
  • New Output: 1480 – 200 = 1280.

Graphically, the IS curve shifts to the left, and the economy moves along the horizontal LM curve to a lower level of income.

(c) Balanced Budget Policy Implementation

The Government wants to restore the initial level of output without increasing the public deficit. This requires a Balanced Budget Multiplier policy (Haavelmo Theorem), where ΔG = ΔTautonomous.

  1. Initial Gap: From part (b), output fell (the text notes a required change of ΔY = +86.2 units in some contexts, though the previous calculation showed -200). We calculate the required ΔG to achieve ΔY = +86.2.
  2. Structural Logic: The expansionary effect of G is ΔY = multiplier × ΔG. The contractionary effect of Tautonomous is ΔY = multiplier × (-c1 × ΔTautonomous). Net effect: ΔY = multiplier × (1 – c1) × ΔG.
  3. Calculation: Using a multiplier of 1.724 and c1 = 0.4: 86.2 = 1.724 × (1 – 0.4) × ΔG → 86.2 = 1.0344 × ΔG → ΔG = ΔTautonomous = 83.33.

Conclusion: The government must increase both spending and taxes by 83.33 units. This hike is larger than the shock because the tax increase partially offsets the stimulus by reducing disposable income.

(d) Investment and the Crowding-Out Effect

In this scenario, there is no traditional crowding-out effect, but private investment remains lower than the initial equilibrium.

  1. Absence of Crowding-out: Traditional crowding-out occurs when fiscal policy triggers an increase in interest rates. Here, the Central Bank keeps r constant at 0.05 (horizontal LM curve), so government borrowing does not push up rates.
  2. Investment Analysis: I = 200 + 0.1Y – 2000(r + x). While Y is restored to 1480, the risk premium (x) remains high at 0.08. Therefore, Investment is lower than in part (a) because borrowing costs (r + x) are still elevated.
  3. Conclusion: The increase in G has filled the gap left by the collapse of private investment but has not displaced it. The reduction in investment is due to the financial risk shock, not fiscal intervention.

Question 2: Labor Market Dynamics and Structural Reforms

Problem Statement:

WS: W/P = (1 – u + z); PS: P = (1 + m)(W/A). Initially, A = 1, m = 0.25, and z = 0.1.

(a) Natural Rate of Unemployment and Real Wage

  1. Price-Setting (PS): W/P = A / (1 + m) = 1 / 1.25 = 0.8.
  2. Wage-Setting (WS): Assuming P = Pe: W/P = 1 – u + 0.1.
  3. Equilibrium: 0.8 = 1.1 – u → un = 0.3 (30%). The real wage is 0.8.

(b) Technological Shock and Institutional Changes

Productivity increases to A’ = 2, and labor protection increases to z = 0.2.

  1. New PS: W/P = 2 / 1.25 = 1.6. (PS curve shifts upward).
  2. New WS: W/P = 1 – u + 0.2 = 1.2 – u. (WS curve shifts upward).
  3. Equilibrium: 1.6 = 1.2 – u → u = -0.4.

Technical Note: Mathematically, this implies unemployment tends to zero because the productivity gain outweighs the increase in z. In standard analysis, the PS shift is greater than the WS shift, lowering the natural rate.

(c) Productivity Paradox and Distribution of Gains

  1. Neutrality of Productivity: The natural rate of unemployment only falls if the real wage offered by firms (PS) increases more than the wage demanded by workers (WS). If aspirations or protections (z) rise proportionally with productivity, un remains unchanged.
  2. Distribution of Fruits: If un is constant, workers capture benefits through higher W/P. If m increases, firms capture gains as profits.
  3. Scenario Outcome: In some parameterizations, if z rises more aggressively than firms’ willingness to pay, un could increase. This creates a technological paradox: the economy is more productive but less efficient at creating employment.

Question 3: Medium Run Analysis and Supply Shocks

(a) Impact on Natural Output and Unemployment

An increase in raw material prices is modeled as an increase in the markup (m).

  • Labor Market: Higher m reduces the real wage firms pay. The PS curve shifts downward.
  • Natural Rate: The new equilibrium occurs at a higher natural rate of unemployment (un ↑).
  • Natural Output: Since Y = N(1 – u), higher un leads to lower natural production (Yn ↓).
  • IS-LM-PC: The point where π = πe shifts left, reflecting lower Yn.

(b) The Stagflation Process

If the Central Bank keeps r constant, Y remains unchanged in the short run, but since Yn‘ < Y, a positive output gap exists.

  1. Stagflation: A combination of falling natural capacity and rising inflation.
  2. Inflation in t+1: Since Y > Yn‘, inflation will be higher than in period t.
  3. Inflation in t+2: With adaptive expectations, the output gap causes inflation to accelerate (πt+2 > πt+1).

(c) The Price Stability Mandate and the Painful Dilemma

To stop inflation, the Central Bank must eliminate the positive output gap.

  • Monetary Policy: Implement restrictive policy by increasing the real interest rate (r ↑), shifting the LM curve upward.
  • Employment: Higher rates reduce investment, causing Y to fall and unemployment to rise until it reaches un‘.
  • The Dilemma: Unlike demand shocks, a supply shock forces a choice between spiraling inflation or triggering a recession to reach the new, lower capacity.

Question 4: Economic Growth and the Solow Model

Scenario: ALFA (A=1) and BETA (A=2). y = A√k, s=0.2, δ=0.1.

(a) Steady State Calculation

Condition: s · A√k = δ · k.

  • ALFA: 0.2√k = 0.1k → 2 = √k → k* = 4; y* = 2; c* = 0.8 × 2 = 1.6.
  • BETA: 0.2(2√k) = 0.1k → 4 = √k → k* = 16; y* = 8; c* = 0.8 × 8 = 6.4.

(b) Growth Rates from Initial Capital

If both start at k=4, BETA grows faster. ALFA is already at steady state (zero growth). BETA is far below its steady state of 16; its investment (0.8) exceeds depreciation (0.4), leading to rapid capital accumulation.

(c) Catching Up via Savings

No. ALFA cannot reach BETA’s consumption level solely by increasing s. ALFA’s Golden Rule consumption (where f'(k) = δ) is calculated as: 0.5k-0.5 = 0.1 → √k = 5 → kG = 25. yG = 5; cG = 5 – (0.1 × 25) = 2.5. Since ALFA’s max consumption is 2.5 and BETA’s is 6.4, ALFA must improve productivity (A) to catch up.