In an economy with a production function given by 𝑌 = 𝐴𝐾^0.5𝐿^0.5, where 𝑌 represents GDP, 𝐾 represents capital stock, and 𝐿 represents labor, and the parameter 𝐴 is equal to 12. If the capital stock is 256 and labor is 625, both paid their marginal products, what will be the value of the real wage of labor (W/P)?
Question
In an economy with a production function given by 𝑌 = 𝐴𝐾^0.5𝐿^0.5, where 𝑌 represents GDP, 𝐾 represents capital stock, and 𝐿 represents labor, and the parameter 𝐴 is equal to 12. If the capital stock is 256 and labor is 625, both paid their marginal products, what will be the value of the real wage of labor (W/P)?
Solution
The production function is given by Y = AK^0.5L^0.5. The marginal product of labor (MPL) is the additional output produced when one more unit of labor is added, holding all other inputs constant.
To find the MPL, we take the derivative of the production function with respect to L:
MPL = dY/dL = 0.5 * A * K^0.5 * L^-0.5
Given that A = 12, K = 256, and L = 625, we can substitute these values into the MPL equation:
MPL = 0.5 * 12 * 256^0.5 * 625^-0.5
Solving this gives us the MPL, which is the real wage of labor (W/P) when labor is paid its marginal product.
Let's calculate it:
MPL = 0.5 * 12 * (256^0.5) * (625^-0.5) MPL = 6 * 16 * (1/25) MPL = 6 * 16/25 MPL = 3.84
So, the real wage of labor (W/P) is 3.84.
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