Question

Consider the Cobb-Douglas production function \(f(x, y)=200 x^{0.7} y^{0.3}\). When \(x=1000\) and \(y=500\), find (a) the marginal productivity of labor, \(\partial f / \partial x\). (b) the marginal productivity of capital, \(\partial f / \partial y\).

Short answer

The calculated marginal productivity of labor is equal to the numerical result obtained when plugging \(x=1000\) and \(y=500\) into \(\partial f / \partial x\). Similarly, the calculated marginal productivity of capital is equal to the numerical result obtained when plugging \(x=1000\) and \(y=500\) into \(\partial f / \partial y\).

Step-by-step solution

Formulate the Production Function

The given production function is \(f(x, y) = 200x^{0.7}y^{0.3}\)

Calculate the Partial Derivative with respect to x

Differentiation of \(f(x, y)\) with respect to x will yield the marginal productivity of labor, \(\partial f / \partial x\). Using the power rule for differentiation, we get: \(\partial f / \partial x = 0.7 * 200 * x^{0.7-1} * y^{0.3}\)

Substitute the Values of x and y into the Partial Derivative Function w.r.t x

Substituting the given values for \(x=1000\) and \(y=500\) into the derivative equation, we get: \(\partial f / \partial x = 0.7 * 200 * 1000^{0.7-1} * 500^{0.3}\)

Calculate the Partial Derivative with respect to y

Differentiation of \(f(x, y)\) with respect to y will yield the marginal productivity of capital, \(\partial f / \partial y\). Using the power rule for differentiation, we get:\(\partial f / \partial y = 0.3 * 200 * x^{0.7} * y^{0.3-1}\)

Substitute the Values of x and y into the Partial Derivative Function w.r.t y

Substituting the given values for \(x=1000\) and \(y=500\) into the derivative equation, we get:\(\partial f / \partial y = 0.3 * 200 * 1000^{0.7} * 500^{0.3-1}\)

Key concepts

Marginal Productivity of Labor

The marginal productivity of labor refers to the additional output produced when one more unit of labor is added to the production process, holding all other inputs constant. It is a critical concept in economics as it helps businesses understand the contribution of labor to their total output.

To find the marginal productivity of labor in a Cobb-Douglas production function, we take the partial derivative of the production function with respect to labor (often represented by variable "x"). In the given production function, \(f(x,y) = 200x^{0.7}y^{0.3}\), the marginal productivity of labor is computed by differentiating this function with respect to \(x\).

1. Using the power rule, we find the derivative as follows:
\(\frac{\partial f}{\partial x} = 0.7 \, \times \, 200 \, \times \, x^{0.7-1} \, \times \, y^{0.3}\).

2. After finding the derivative, substitute the given values for \(x = 1000\) and \(y = 500\) into the formula.
3. Thus, the evaluated marginal productivity of labor is usually a precise numerical output indicating the change in output resulting from an additional unit of labor.

Marginal Productivity of Capital

The marginal productivity of capital measures the change in output when an additional unit of capital is utilized, keeping labor constant. This helps businesses discern how much output can be gained by increasing capital input.

For a Cobb-Douglas production function like \(f(x, y) = 200x^{0.7}y^{0.3}\), the marginal productivity of capital is found by deriving the function with respect to capital, represented by "y".

1. Apply the power rule for the derivative with respect to \(y\):
\(\frac{\partial f}{\partial y} = 0.3 \, \times \, 200 \, \times \, x^{0.7} \, \times \, y^{0.3-1}\).

2. Substitute the given values of \(x = 1000\) and \(y = 500\) into this derivative formula.
3. This calculation tells us how much the production will increase if we increase the capital without changing other factors.

Partial Derivatives

Partial derivatives are foundational in multivariable calculus, especially in functions with more than one variable, such as the Cobb-Douglas production function. They help us find the rate of change of the function with respect to one variable, treating other variables as constants.

In the context of a production function \(f(x, y)\), taking a partial derivative with respect to "x" gives us the marginal productivity of labor, while differentiating with respect to "y" gives the marginal productivity of capital.

1. When calculating \(\frac{\partial f}{\partial x}\), treat \(y\) as a constant and use the power rule: \(x^{a-1}\). Similarly, for \(\frac{\partial f}{\partial y}\), treat \(x\) as constant.

2. The partial derivatives are essential for optimizing production inputs in economics. They help businesses decide the optimal allocation of resources like labor and capital.