Illustrate the impact of an increase in upsilon on the

Question

Question 1. Plot the equations

X² ± y² = 1/2

x² + y² = 1

x² + y² = 2

1/4 y = x(1- x)
1 y = x(1 - x)
2y = x -(1- x)

Take both x and y to be between 0 and 1.

Question 2. Consider the equations:

x² + y² = α, 0< α <1.
βy = x(1 - x), 0<β.

Given the parameter values, these intersect in the positive orthant. Depict what happens to the point of intersection as:

1. α increases and

2. β increases.

Question 3. Consider a production process using input x to produce output y. The relationship between inputs and outputs is somewhat odd: for technological reasons the input - output levels must satisfy two conditions:

x² + y² = α, 0 < α < 1.

βy = x(1 - x), 0 < β.

This pair of equations has a unique nonnegative solution (x,y ≥ 0). Graph these functions in (x,y) space. At this solution, calculate ∂x/∂α,∂y/∂α, ∂x/∂k and ∂y/∂k.

4. In relation to you graph, interpret the signs of the derivatives (they are positive or negative).

Question 4. Plot the equations

y - 1 log(x + 1) = 0

y - 2 log(x + 1) = 0

y - 3 log(x + 1) = 0

xy = 1, 2, 3.

Question 5. Consider the equation pair (Υ > 0):

y - ylog(x +1) = 0.

xy = Υ

Plot both functions on one graph. Illustrate the impact of an increase in Υ on the solution to this pair of equations.