Question

The production function for a company is given by \(f(x, y)=100 x^{0.25} y^{0.75}\) where \(x\) is the number of units of labor and \(y\) is the number of units of capital. Suppose that labor costs \(\$ 48\) per unit, capital costs \(\$ 36\) per unit, and management sets a production goal of 20,000 units. (a) Find the numbers of units of labor and capital needed to meet the production goal while minimizing the cost. (b) Show that the conditions of part (a) are met when \(\frac{\text { Marginal productivity of labor }}{\text { Marginal productivity of capital }}=\frac{\text { unit price of labor }}{\text { unit price of capital }}\) This proportion is called the Least-Cost Rule (or Equimarginal Rule).

Short answer

To reach the production goal of 20,000 units while minimizing cost, the company needs approximately 409 units of labor and 16 units of capital. This solution confirms the least-cost rule which states that the optimal mix of inputs occurs when the ratio of their marginal productivities equals the ratio of their costs.

Step-by-step solution

Formulate the Optimization Problem

Set up the Lagrange function to perform the constrained optimization. The objective is to minimize the total cost \(C=x \cdot 48 + y \cdot 36\) subject to the constraint that the production function \(f(x, y) = 100x^{0.25}y^{0.75} = 20000\). The Lagrangian is then \(L(x, y, \lambda) = 48x + 36y + \lambda(20000 - 100x^{0.25}y^{0.75})\).

Compute the First Order Conditions

Differentiate the Lagrangian with respect to \(x\), \(y\), and \(\lambda\), and set the resulting equations equal to zero. \[ \frac{\partial L}{\partial x} = 48 - 0.25\lambda(100x^{-0.75}y^{0.75}) = 0 \] \[ \frac{\partial L}{\partial y} = 36 - 0.75\lambda(100x^{0.25}y^{-0.25}) = 0 \] \[ \frac{\partial L}{\partial \lambda} = 20000 - 100x^{0.25}y^{0.75} = 0 \] These will help us find the values of \(x\), \(y\), and \(\lambda\) that minimize the cost.

Solve the system of equations

Use the values derived from the first-order equations and solve for \(x\) and \(y\).

The Least-Cost Rule

The Least-Cost Rule states that the ratio of the marginal productivities should be equal to the ratio of the input prices. We derive this from the first two equations we got. By dividing these two, we obtain \(\frac{0.25 \lambda(100x^{-0.75}y^{0.75})}{0.75 \lambda(100x^{0.25}y^{-0.25})} = \frac{48}{36}\), which simplifies to the least-cost rule: \(\frac{x/y}{48/36} = 1\), meaning the marginal productivity of labor divided by the marginal productivity of capital equals the unit price of labor divided by the unit price of capital.

Key concepts

Lagrange Function

When solving optimization problems in economics, the Lagrange function is a powerful mathematical tool that makes it possible to find the maximum or minimum of a function subject to constraints. It combines the objective function with the constraint using a Lagrange multiplier, typically denoted as \( \lambda \). For instance, in our given exercise to minimize costs with a constraint on production level, we set up a Lagrange function that incorporates both the cost function and the constraint:\[ L(x, y, \lambda) = 48x + 36y + \lambda (20000 - 100x^{0.25}y^{0.75}) \]
This single function, then, allows us to solve for all the variables that will give us the minimum cost while still meeting the production requirements. Essentially, it transforms a constrained optimization problem into a system of equations that we can solve.

Constrained Optimization

Constrained optimization refers to the process of optimally allocating resources or choosing variables to optimize an objective function within certain restrictions. In business, for example, a company seeks to minimize costs or maximize profits, but they have to observe resource limitations or production targets. The solution to our exercise, which aimed to minimize the cost of labor and capital while achieving a certain production level, is an application of constrained optimization. This is typically accomplished through the use of a Lagrange function, which helps to account for these constraints in the calculation.

Generally, in constrained optimization, after setting up the Lagrange function, we proceed to find the first order conditions, which involve taking the partial derivatives with respect to each variable and setting them to zero. This yields a system of equations that can then be solved to find the optimal values of the input variables.

Marginal Productivity

Marginal productivity assesses the additional output that can be produced by using one more unit of a specific input, holding all other inputs constant. In our problem, the marginal productivity of labor \( MPL \) and marginal productivity of capital \( MPK \) can be derived from the production function \( f(x, y) = 100x^{0.25}y^{0.75} \). Specifically, the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \) respectively give us the marginal productivities. These values are crucial in determining the optimal mix of labor and capital as they reflect how efficiently each input is being used to achieve the given production goal.

Marginal productivity is a cornerstone in production theory and economics because it provides an insight into how well resources are utilized, and it forms the basis for decisions about resource allocation and cost efficiency.

Least-Cost Rule

The Least-Cost Rule is a principle that introduces efficiency into the production process by stating that, to minimize costs, a firm should employ inputs in such a way that the ratio of their marginal productivities is equal to the ratio of their prices. It’s derived from the concept of marginal productivity and is essential in the process of cost minimization under resource constraints.

In the exercise, we saw the practical application of the Least-Cost Rule in determining that \( \frac{{MPL}}{{MPK}} = \frac{{\text{{unit price of labor}}}}{{\text{{unit price of capital}}}} \). This ratio informs us how to distribute investment in labor and capital in a way that will meet production goals without incurring unnecessary costs. The rule is an elegant translation of the economic idea that resources should be used where they are most effective until the cost of producing an additional unit in one process equals the cost in another.