Pls help me to sole micro problem.Production function is Y=K^0.5L^0.5 and w=$1 (wage) and r=$2 (rent) P=$5
I need to find the long term optimum amount for K and L. Also long term optimun output and a cost function.
Thank you.
To find the long-term optimum amount for capital (K) and labor (L), we need to maximize the production function subject to the constraints of the wage rate (w) and the rental rate (r).
The optimal input combination occurs when the marginal product of capital (MPk) divided by the rental rate (r) equals the marginal product of labor (MPl) divided by the wage rate (w).
Mathematically, this can be represented as:
MPk/r = MPl/w
First, let's calculate the marginal product of capital (MPk) and the marginal product of labor (MPl):
MPk = 0.5*K^(-0.5)*L^0.5
MPl = 0.5*K^0.5*L^(-0.5)
Now, substituting these values into our equation:
0.5*K^(-0.5)*L^0.5 / r = 0.5*K^0.5*L^(-0.5) / w
Let's plug in the given values for r ($2) and w ($1):
0.5*K^(-0.5)*L^0.5 / 2 = 0.5*K^0.5*L^(-0.5) / 1
Simplifying the equation:
K^(-0.5)*L^0.5 = 4*(K^0.5)*(L^(-0.5))
Now, we can cross-multiply to eliminate the fractions:
K^(-0.5)*L^0.5 = 4*K^0.5*L^(-0.5)
Next, raise both sides of the equation to the power of 2 to eliminate the exponents:
K^(-1)*L = 16*K*L^(-1)
Rearranging the terms:
K^(-1)*L / K*L^(-1) = 16
Multiplying both sides by K*L:
(L^2)/(K^2) = 16
Taking the square root of both sides:
L / K = 4
So, the optimal input combination is:
L = 4K
To find the long-term optimum output (Y), substitute the optimal input combination into the production function:
Y = K^(0.5)*(4K)^(0.5)
Simplifying:
Y = 2K
Now, let's derive the cost function. The total cost (C) is given by the sum of the cost of labor (Cw) and the cost of capital (Cr):
C = Cw + Cr
The cost of labor is the wage rate (w) multiplied by the amount of labor (L):
Cw = w * L
Substituting the optimal input combination (L = 4K):
Cw = w * 4K
Similarly, the cost of capital is the rental rate (r) multiplied by the amount of capital (K):
Cr = r * K
Substituting the given rental rate (r = $2):
Cr = $2 * K
Combining the two cost components:
C = Cw + Cr = w * 4K + $2 * K
To further simplify, substitute the given wage rate (w = $1):
C = $1 * 4K + $2 * K = $4K + $2K = $6K
Hence, the long-term cost function is:
C = $6K
To summarize:
- Optimal amount of capital (K): Cannot be determined without additional information.
- Optimal amount of labor (L): L = 4K
- Optimal output (Y): Y = 2K
- Cost function (C): C = $6K
Please note that the specific value of K cannot be determined without additional information, as it depends on various factors such as market demand, cost considerations, and technological constraints.