My problem is easy, but i need help.
My problem is
Q=3K^.35 L^.87
EL=?
Ek=?
CRS=? CRS is 1.22> 1 I know, but not the rest
than part b is when K=30, L=83, what is MPk=? MPL=?
Please help
To solve this problem, we need to understand the different variables and their meanings.
Q represents the total output or quantity produced.
K represents the amount of capital used in the production process.
L represents the amount of labor used in the production process.
EL represents the elasticity of labor demand.
EK represents the elasticity of capital demand.
CRS represents the degree of returns to scale. A value greater than 1 indicates increasing returns to scale, a value equal to 1 represents constant returns to scale, and a value less than 1 suggests decreasing returns to scale.
Now, let's solve the problem step by step.
Step 1: Find the elasticity of labor demand (EL).
Given the production function: Q = 3K^0.35 L^0.87
To calculate the elasticity of labor demand (EL), we differentiate the production function with respect to L and multiply it by L/Q, which represents the proportionate change in L and Q.
EL = (dQ/dL) * (L/Q)
Taking the partial derivative with respect to L, we get:
dQ/dL = 3 * 0.87 * K^0.35 * L^(-0.13)
Substituting the derivative back into the equation, we have:
EL = (3 * 0.87 * K^0.35 * L^(-0.13)) * (L/Q)
EL = 2.61 * (K^0.35 / Q) * L^(0.87 - 0.13)
Finally, we can simplify the equation:
EL = 2.61 * (K^0.35 / Q) * L^0.74
Step 2: Find the elasticity of capital demand (EK).
Given the production function: Q = 3K^0.35 L^0.87
To calculate the elasticity of capital demand (EK), we differentiate the production function with respect to K and multiply it by K/Q, which represents the proportionate change in K and Q.
EK = (dQ/dK) * (K/Q)
Taking the partial derivative with respect to K, we get:
dQ/dK = 3 * 0.35 * K^(-0.65) * L^0.87
Substituting the derivative back into the equation, we have:
EK = (3 * 0.35 * K^(-0.65) * L^0.87) * (K/Q)
EK = 1.05 * (K^(-0.65) / Q) * L^(0.87)
Finally, we can simplify the equation:
EK = 1.05 * (K^(-0.65) / Q) * L^0.87
Step 3: Determine the returns to scale (CRS).
We are given that CRS is 1.22, which indicates increasing returns to scale.
Step 4: Calculate the marginal product of capital (MPK) and the marginal product of labor (MPL) when K = 30 and L = 83.
MPK is defined as the change in output resulting from a one-unit increase in capital, holding labor constant.
MPL is defined as the change in output resulting from a one-unit increase in labor, holding capital constant.
To calculate MPK, we differentiate the production function with respect to K:
MPK = dQ/dK
MPK = 3 * 0.35 * K^(-0.65) * L^0.87
To calculate MPL, we differentiate the production function with respect to L:
MPL = dQ/dL
MPL = 3 * 0.87 * K^0.35 * L^(-0.13)
Substituting the given values K = 30 and L = 83 into the equations will allow us to calculate MPK and MPL.
I hope this explanation helps you solve the problem.