Problem 86 - Brief Applied Calculus

Question

The intersection of an isocost line \(w L+r K=C\) and an isoquant curve \(K=a L^{b}\) (see pages 18 and 32 ) gives the amounts of labor \(L\) and capital \(K\) for fixed production and cost. Find the intersection point \((L, K)\) of each isocost and isoquant. [Hint: After substituting the second expression into the first, multiply through by \(L\) and factor.] $$ 5 L+4 K=120 \text { and } K=180 \cdot L^{-1} $$

Step-by-Step Solution

Answer

The intersection point is \((L, K) = (12, 15)\).

Step 1: Substitute the Isoquant Equation

Start by substituting the isoquant equation \(K = 180 \cdot L^{-1}\) into the isocost equation \(5L + 4K = 120\). This results in:\[5L + 4(180 \cdot L^{-1}) = 120\]

Step 2: Simplify the Equation

Multiply through by \(L\) to eliminate the fraction:\[5L^2 + 720 = 120L\]

Step 3: Rearrange and Simplify Further

Move all terms to one side to form a quadratic equation:\[5L^2 - 120L + 720 = 0\]

Step 4: Factor the Quadratic Equation

Attempt to factor the quadratic equation:\[(L - 12)(L - 12) = 0\]This results in \(L = 12\).

Step 5: Find the Corresponding K Value

Substitute \(L = 12\) back into the isoquant equation to find \(K\):\[K = 180 \cdot 12^{-1} = 15\]

Step 6: Verify the Solution

Check if the found \((L, K)\) satisfies both original equations.For the isocost line: \(5(12) + 4(15) = 60 + 60 = 120\), which satisfies the equation.For the isoquant curve: \(K = 180 \cdot (12)^{-1} = 15\), which is consistent.

Key Concepts

Isocost Line

Imagine you're managing a budget to produce goods. You have two inputs: labor and capital. The isocost line represents all possible combinations of these inputs that cost the same total amount. It is akin to a budget constraint in economics. The equation for an isocost line is often represented as \( wL + rK = C \), where:

• \( w \) is the wage rate for labor.
• \( r \) is the rental rate for capital.
• \( C \) is the total cost you are willing to spend.

To pick the best combination, you look at how much labor \( L \) and how much capital \( K \) you can afford for a given cost \( C \). This line will shift if there is a change in total cost or input prices. By understanding the isocost line, you learn to maximize production without overspending.

Isoquant Curve

The isoquant curve helps you visualize how labor and capital can be combined to produce the same amount of output. An isoquant is like a contour line on a map; it shows different combinations that lead to the same level of production. The mathematical expression \( K = aL^{b} \) defines this curve, where:

• \( K \) is the amount of capital.
• \( L \) is the amount of labor.
• \( a \) and \( b \) are constants determined by technology or production conditions.

The slope of an isoquant indicates how easily labor can be substituted for capital while maintaining the same output. If the curve is steep, more labor is required to substitute for a reduced amount of capital. Conversely, if it is flat, labor and capital can replace each other more easily.

Quadratic Equation

A quadratic equation is essential in finding the intersection between an isocost line and an isoquant curve. The typical form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here:

• \( a \), \( b \), and \( c \) are constants.
• \( x \) represents the variable you want to solve.

In the given problem, you substitute the isoquant equation into the isocost equation. After simplifying and manipulating the equation, you arrive at a quadratic form: \( 5L^2 - 120L + 720 = 0 \). Solving this quadratic helps determine the amount of labor that will balance your inputs for a set cost and output. Using methods like factoring, you can find the roots of this equation, which gives you the desired labor quantity.

Labor and Capital

In economics, labor and capital are the fundamental inputs in the production process. They are the building blocks of any production function.

• Labor: Refers to the human workforce involved in production. It consists of the effort and time put in by workers.
• Capital: Encompasses tools, machinery, and buildings required for production. Unlike labor, capital is a physical asset.

Balancing labor and capital is crucial as businesses aim to produce efficiently. The intersection of an isocost line and an isoquant curve precisely identifies the optimal combination of labor and capital for a specified budget and output level. This balance is the sweet spot where cost minimization and output maximization occur. Understanding how to effectively utilize labor and capital is key to achieving cost-effective production.