Choose all dot points (1 -4) that has a balance point of R.
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If you project S onto the (x,y)-plane, it casts a "shadow" corresponding to the trapezoidal region
T = {(x,y) : 0 ≤ x ≤ 10 - y and -4 ≤ y ≤ 4}
Let z = f(x, y) = √(16 - y²) and z = g(x, y) = -√(16 - y²), each referring to one half of the cylinder to either side of the plane z = 0.
The surface element for the "positive" half is
dS = √(1 + (∂f/∂x)² + (∂f/dy)²) dx dy
dS = √(1 + 0 + 4y²/(16 - y²)) dx dy
dS = √((16 + 3y²)/(16 - y²)) dx dy
The the surface integral along this half is
You'll find that the integral over the "negative" half has the same value, but multiplied by -1. Then the overall surface integral is 0.
The curves of the labor market are illustrations of demand and supply, and the curves are all linear functions
<h3>4.1: The graph of the curves of the labor market</h3>The equations are given as:
D = 36 - 2W
S₁ = 9 + W ---- domestic (i.e. without immigrants)
S₂ = 10 + 2W ---- total (i.e. including immigrants)
See attachment for the graph
<h3>4.2: The equilibrium wage rate before immigration</h3>This is the point of intersection between the curves of S₁ and D.
From the graph, we have:
(W, D) = (9,18)
Hence, the equilibrium wage rate before immigration is $9, and the number of workers that would be hired is 18
<h3>4.3: The equilibrium wage rate after immigration</h3>This is the point of intersection between the curves of S₂ and D.
From the graph, we have:
(W, D) = (6.5,23)
Hence, the equilibrium wage rate after immigration is $6.5, and the number of workers that would be hired is 23
<h3>4.3.1: Number of domestic workers hired</h3>Substitute 6.5 for W in S₁ = 9 + W
S₁ = 9 + 6.5
S₁ = 15.5
Remove decimal
S₁ = 16
Hence, the number of domestic workers that would be hired is 16
<h3>4.3.1: Number of immigrants hired</h3>Substitute 6.5 for W in S₂ = 10 + 2W
S₂ = 10 + 2 * 6.5
S₂ = 23
Subtract S₁ = 16 from S₂ = 23
Immigrants = 23 - 16
Immigrants = 7
Hence, the number of immigrants that would be hired is 7
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