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Kuna swali hapa kiongozi msaada kdgo. A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the constrained factor demand functions. 2. Derive the function C=c(q) when W1=W2=100?
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Kuna swali hapa kiongozi msaada kdgo. A production function for a perf...


Constrained Factor Demand Functions:

To derive the constrained factor demand functions, we need to maximize the production function subject to the cost constraint. The Lagrange function is used to solve this optimization problem.

1. Set up the Lagrange function:
L = Q - λ(C - W1X1 - W2X2)

2. Take partial derivatives with respect to X1, X2, and λ:
∂L/∂X1 = 0 => A(1/2)X1^(-1/2)X2^(1/2) - λW1 = 0
∂L/∂X2 = 0 => A(1/2)X1^(1/2)X2^(-1/2) - λW2 = 0
∂L/∂λ = 0 => C - W1X1 - W2X2 = 0

3. Solve the system of equations to find the constrained factor demand functions:
From the first two equations, we get:
X1 = (AW1^2)/(4λ^2) and X2 = (AW2^2)/(4λ^2)

Function C=c(q) when W1=W2=100:

1. Substitute X1 and X2 back into the cost function C:
C = W1X1 + W2X2
C = 100(AW1^2)/(4λ^2) + 100(AW2^2)/(4λ^2)
C = 25(AW1^2 + AW2^2)/(λ^2)

2. Substitute the value of λ using the production function:
From the third equation, C = Q
25(AW1^2 + AW2^2)/(AW1W2) = Q
C = 25(Q/W1W2) = c(q)

Therefore, the cost function C=c(q) when W1=W2=100 is C = 25Q.
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Kuna swali hapa kiongozi msaada kdgo. A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the constrained factor demand functions. 2. Derive the function C=c(q) when W1=W2=100?
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