Problem 85 - 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.] $$ 3 L+8 K=48 \text { and } K=24 \cdot L^{-1} $$

Step-by-Step Solution

Answer

The intersection point is \( (L, K) = (8, 3) \).

Step 1: Substitute K into the Isocost Equation

Start by substituting the isoquant expression for capital, \(K = 24 \cdot L^{-1}\), into the isocost equation \(3L + 8K = 48\). This gives us: \[ 3L + 8 \cdot (24 \cdot L^{-1}) = 48 \] which simplifies to \[ 3L + \frac{192}{L} = 48. \]

Step 2: Clear the Fraction by Multiplying by L

To eliminate the fraction, multiply every term by \(L\): \[ L \cdot 3L + L \cdot \frac{192}{L} = 48L. \] This simplifies to: \[ 3L^2 + 192 = 48L. \]

Step 3: Rearrange into a Quadratic Equation

Rearrange the terms of the equation into the standard quadratic form \(ax^2 + bx + c = 0\): \[ 3L^2 - 48L + 192 = 0. \]

Step 4: Solve the Quadratic Equation

We solve \(3L^2 - 48L + 192 = 0\) using the quadratic formula \(L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 3\), \(b = -48\), and \(c = 192\). Calculate \[ L = \frac{48 \pm \sqrt{(-48)^2 - 4 \cdot 3 \cdot 192}}{6}. \] This simplifies to \[ L = \frac{48 \pm \sqrt{2304 - 2304}}{6} = \frac{48}{6} = 8. \] So, \(L = 8\).

Step 5: Calculate K Using L = 8

Substitute \(L = 8\) back into the isoquant equation for \(K\) to find \(K\): \[ K = 24 \cdot (8)^{-1} = 24 \cdot \frac{1}{8} = 3. \] So, \(K = 3\).

Key Concepts

Isocost Line

An isocost line represents all combinations of two inputs that result in the same total cost in production. Essentially, it shows us how much of two resources—commonly labor (L) and capital (K)—can be used without exceeding a specific budget. Think of it as a budget line in economics, but for inputs in production rather than goods.

The general form of the isocost line equation is given as:

• \( wL + rK = C \)

where:

• \( w \) is the wage rate or cost of labor per unit.
• \( r \) is the rental rate or cost of capital per unit.
• \( C \) is the total budget or cost constraint.

In our specific example, the isocost equation is \( 3L + 8K = 48 \), which implies that the company spends 3 units of whatever currency on each unit of labor and 8 units on each unit of capital to not exceed a budget of 48 units. By examining isocost lines, firms can determine the most cost-effective combination of labor and capital to utilize in production.

Isoquant Curve

An isoquant curve represents all the different combinations of two inputs that result in the same level of output. It is similar to the "indifference curve" concept in consumer theory but applied to outputs.

Here, the isoquant is described by the equation \( K = aL^b \), where:

• \( K \) is the amount of capital.
• \( L \) is the amount of labor.
• \( a \) and \( b \) are constants that define the particular production function.

In the exercise, the isoquant is defined by the equation \( K = 24L^{-1} \), meaning that as labor increases, capital required for the same production decreases at a specific rate. This inverse relationship indicates a substitution effect between labor and capital, where increases in labor mean less capital is needed, assuming constant output.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form:

• \( ax^2 + bx + c = 0 \)

It appears most frequently in the context of projects involving physics, engineering, and economics as it models parabolic patterns.

When dealing with systems like isocosts and isoquants, we often transform these into quadratic equations to find solutions. In our example, we manipulated the initial equation into the form:

• \( 3L^2 - 48L + 192 = 0 \)

This equation was solved using the quadratic formula:\[L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where values of \( a \), \( b \), and \( c \) are from the coefficients of the quadratic equation. Solving gives specifics like labor values needed to meet certain production and cost conditions.

Intersection Point

The intersection point of curves like an isocost line and an isoquant curve represents a combination of inputs that satisfy both cost constraints and a specific output level. Finding this point is crucial in production optimization.

In our model, we began by substituting the isoquant equation into the isocost equation, resulting in a quadratic equation. Solving this gave us a specific value for labor \( L = 8 \). Then, using this labor value, we substituted back into the isoquant equation to find the corresponding capital, resulting in \( K = 3 \).

Thus, the intersection (\( L, K \)) is \( (8, 3) \), meaning optimal production occurs with 8 units of labor and 3 units of capital within the assessed constraints of cost and output.