(b) Impact of furlough scheme on income inequality (16 points)

To analyze the impact on income inequality, we compare Scenario A (with furlough scheme) and Scenario B (without furlough scheme).

Scenario A:

To calculate the Gini coefficient using the Lorenz curve, connecting the points (0, 0), (12, 0), (90, 54), and (100, 100), one can follow these steps:

1. Identify the area under the Lorenz curve (A):

This is the area between the line of perfect equality (the diagonal line from (0, 0) to (100, 100)) and the Lorenz curve.

2. Identify the total area under the line of perfect equality (B):

This is the area of the triangle formed by points (0, 0), (100, 0), and (100, 100).

B = 1/2 x base x height = 1/2 x 100 x 100 = 5000

3. Calculate area A under the Lorenz curve:

To find the area A under the Lorenz curve, one can break it down into segments based on the points given.

- From (0, 0) to (12, 0): This segment contributes no area.

- From (12, 0) to (90, 54): This can be calculated as a trapezoid.

- From (90, 54) to (100, 100): This can also be calculated as a trapezoid.

Area from (12, 0) to (90, 54):

Area = 1/2 x (Base₁ + Base₂ ) x Height

Here: Base₁ = 0, Base₂ = 54, Height = (90 - 12) = 78.

A₁ = 1/2 x (0 + 54) x 78 = 1/2 x 54 x 78 = 2106

Area from (90, 54) to (100, 100):

A₂ = 1/2 x (Base₁ + Base₂ ) x Height

Here: Base₁ = 54, Base₂ = 100, Height = (100 - 90) = 10.

A₂ = 1/2 x (54 + 100) x 10 = 1/2 x 154 x 10 = 770

Total Area A under the Lorenz curve:

A = A₁ + A₂ = 2106 + 770 = 2876

4. Calculate the Gini coefficient:

The Gini coefficient G is calculated using the formula:

G = B - A/B = 5000 - 2876/5000 = 2124/5000 = 0.4248

Thus, the Gini coefficient is approximately 0.42.

Scenario B:

To calculate the Gini coefficient for Scenario B with the Lorenz curve connecting the points (0, 0), (20, 0), (90, 48), and (100, 100), one can follow these steps:

1. Identify the area under the Lorenz curve (A):

This is the area between the line of perfect equality (the diagonal line from (0, 0) to (100, 100)) and the Lorenz curve.

2. Identify the total area under the line of perfect equality (B):

This is the area of the triangle formed by the points (0, 0), (100, 0), and (100, 100).

B = 1/2 x base x height = 1/2 x 100 x 100 = 5000

Calculate area A:

To find the area A under the Lorenz curve, you can break it down into segments based on the points given.

- From (0, 0) to (20, 0): This segment contributes no area.

- From (20, 0) to (90, 48): This can be calculated as a trapezoid.

- From (90, 48) to (100, 100): This can also be calculated as a trapezoid.

Area from (20, 0) to (90, 48):

Area₁ = 1/2 x (Base₁ + Base₂ ) x Height

Here: Base₁ = 0, Base₂ = 48, Height = (90 - 20) = 70.

A₁ = 1/2 x (0 + 48) x 70 = 1/2 x 48 x 70 = 1680

From (90, 48) to (100, 100):

A₂ = 1/2 x (Base₁ + Base₂ ) x Height

Here, Base₁ = 48, Base₂= 100, Height = (100 - 90) = 10.

A₂ = 1/2 x (48 + 100) x 10 = 1/2 x 148 x 10 = 740

Total Area A Under the Lorenz Curve

A = A₁ + A₂= 1680 + 740 = 2420

4. Calculate the Gini coefficient:

The Gini coefficient G is calculated using the formula:

G = B - A/B

Substituting in our values:

G = 5000 - 2420/5000 = 2580/5000 = 0.516

Thus, the Gini coefficient for Scenario B is approximately 0.52.

Effect on Inequality:

- In Scenario A, the Gini coefficient is 0.42, and in Scenario B, it is 0.52.

- Income inequality increases in both scenarios compared to the pre-pandemic level (Gini of 0.36).

- However, the furlough scheme in Scenario A mitigates the increase in income inequality compared to Scenario B.