Respuesta :
Explanation:
1. A) To find the utility-maximizing quantities of goods X and Y, we need to maximize the utility function subject to the budget constraint.
The budget constraint is given by:
PxX + PyY = M
Where:
Px = Price of good X = 8 birr
Py = Price of good Y = 4 birr
M = Total income = 80 birr
Substituting the values into the budget constraint equation, we have:
8X + 4Y = 80
To maximize the utility function U(X, Y) = 2XY + X, we can use the method of Lagrange multipliers.
The Lagrangian function is given by:
L(X, Y, λ) = U(X, Y) - λ(PxX + PyY - M)
Taking partial derivatives with respect to X, Y, and λ, and setting them equal to zero, we get the following equations:
∂L/∂X = 2Y + 1 - λPx = 0
∂L/∂Y = 2X - λPy = 0
∂L/∂λ = PxX + PyY - M = 0
Solving these equations simultaneously will give us the utility-maximizing quantities of X and Y.
B) To find the Marginal Rate of Substitution (MRS) at equilibrium, we need to find the ratio of the marginal utilities of X and Y.
MRS = Mux/Muy = Px/Py
Substituting the values, we have:
Mux/Muy = 8/4 = 2
Therefore, the MRS at equilibrium is 2.
2. A) The average product of labor (APL) function can be found by dividing the total product (Q) by the labor input (L).
APL = Q/L
Given the production function Q = 8KL - 0.5K^2 - 0.2L^2, and fixed capital input K = 10, we can substitute the values to find the APL function.
APL = (8KL - 0.5K^2 - 0.2L^2) / L
Simplifying further, we have:
APL = 8K - 0.5K^2/L - 0.2L
B) To find the level of labor at which the total output of cut-flowers reaches the maximum, we need to find the maximum point of the production function.
Taking the derivative of the production function with respect to L and setting it equal to zero, we can find the critical point.
∂Q/∂L = 8K - 0.4L = 0
Solving for L, we have:
8K = 0.4L
L = 20K
Therefore, at the level of labor L = 20K, the total output of cut-flowers reaches the maximum.
C) To find the maximum achievable amount of cut-flower production, we substitute the value of L = 20K into the production function.
Q = 8K(20K) - 0.5K^2 - 0.2(20K)^2
Simplifying further, we have:
Q = 160K^2 - 0.5K^2 - 0.8K^2
Q = 159.7K^2
Therefore, the maximum achievable amount of cut-flower production is 159.7K^2.
3. A) The total cost (TC) function is given by:
TC = 4Q^3 - 6Q^2 + 4Q + 10
To find the expression for Total Fixed Cost (TFC), we need to find the cost when output (Q) is zero.
TFC = TC(Q=0) = 4(0)^3 - 6(0)^2 + 4(0) + 10
TFC = 10
Therefore, the expression for Total Fixed Cost (TFC) is 10.
To find the expression for Total Variable Cost (TVC), we subtract TFC from TC.
TVC = TC - TFC
TVC = 4Q^3 - 6Q^2 + 4Q + 10 - 10
TVC = 4Q^3 - 6Q^2 + 4Q
B) To derive the expressions for Average Fixed Cost (AFC), Average Variable Cost (AVC), Average Cost (AC), and Marginal Cost (MC), we need to divide the corresponding costs by the quantity (Q).
AFC = TFC/Q
AFC = 10/Q
AVC = TVC/Q
AVC = (4Q^3 - 6Q^2 + 4Q)/Q
AVC = 4Q^2 - 6Q + 4
