Problem 67 - Applied Calculus

Question

An isocost line (iso means "same") shows the different combinations of labor and capital (the value of factory buildings, machinery, and so on) a company may buy for the same total cost. An isocost line has equation $$ w L+r K=C \quad \text { for } \quad L \geq 0, \quad K \geq 0 $$ where \(L\) is the units of labor costing \(w\) dollars per unit, \(K\) is the units of capital purchased at \(r\) dollars per unit, and \(C\) is the total cost. Since both \(L\) and \(K\) must be non-negative, \(a n\) isocost line is a line segment in just the first quadrant. a. Write the equation of the isocost line with \(w=10, r=5, C=1000,\) and graph it in the first quadrant. b. Verify that the following \((L, K)\) pairs all have the same total cost. $$ (100,0), \quad(75,50), \quad(20,160), \quad(0,200) $$

Step-by-Step Solution

Answer

The isocost line is \(K = 200 - 2L\), and all given pairs have a total cost of 1000.

Step 1: Write the Isocost Line Equation

Substitute the given values into the equation of the isocost line: \(w = 10\), \(r = 5\), and \(C = 1000\). This gives us \(10L + 5K = 1000\).

Step 2: Rearrange for Graphing

To graph this equation, rearrange for \(K\) in terms of \(L\): \(5K = 1000 - 10L\), which simplifies to \(K = 200 - 2L\). This represents a line with a y-intercept at \( (0, 200) \) and a slope of \(-2\).

Step 3: Graph the Isocost Line

Plot the isocost line using the equation \(K = 200 - 2L\). It passes through \((0, 200)\) and \((100, 0)\), covering the first quadrant where \(L \geq 0\) and \(K \geq 0\).

Step 4: Verify the Total Cost for (L, K) Pairs

Check each pair \((L, K)\) to see if they satisfy \(10L + 5K = 1000\):- For \((100, 0)\): \(10 \times 100 + 5 \times 0 = 1000\).- For \((75, 50)\): \(10 \times 75 + 5 \times 50 = 1000\).- For \((20, 160)\): \(10 \times 20 + 5 \times 160 = 1000\).- For \((0, 200)\): \(10 \times 0 + 5 \times 200 = 1000\).Each pair satisfies the equation, confirming they all have the same total cost.

Key Concepts

Linear Equations

Linear equations form the backbone of many mathematical concepts, especially in economic analysis. An isocost line is essentially a linear equation that shows the trade-off between inputs - namely labor (L) and capital (K) - for producing a good at a constant total cost (C). In our exercise, the linear equation derived is \(10L + 5K = 1000\). This represents a straight line when plotted on a graph with labor on the x-axis and capital on the y-axis.
The isocost line's slope is calculated from the rearranged equation \(K = 200 - 2L\), where -2 is the slope. This negative slope indicates that as one increases labor, capital must decrease to maintain the same cost. Both labor and capital are restricted to non-negative values, ensuring the line remains in the first quadrant. This visual representation helps in understanding how changes in labor affect the available capital when the total cost remains unchanged.

Economic Concepts

In economics, the isocost line helps businesses understand cost constraints. It shows the various combinations of resources a company can utilize within a fixed budget. The line reflects all market prices and demonstrates the opportunity costs businesses face.
Key economic concepts illustrated by isocost lines include:

• Budget constraint: Firms can only spend within their total budget \(C\) on factors like labor \(L\) and capital \(K\).
• Opportunity cost: Choosing more of one input (e.g., labor) means sacrificing the amount of another (capital) within the same cost, guided by the slope -2.
• Factor pricing: The prices \(w\) (for labor) and \(r\) (for capital) determine the steepness of the isocost line, showing how different marketplace conditions can alter cost strategies.

Understanding these concepts helps businesses make informed decisions about resource allocation.

Cost Analysis

Cost analysis using isocost lines involves identifying optimal input combinations that minimize cost or maximize output. By positioning the isocost line tangent to an isoquant – another fundamental curve displaying equal output combinations – companies can find their most efficient production points.
Let's review the provided \(L, K\) pairs to confirm their cost consistency with our isocost equation \(10L + 5K = 1000\): for instance, at \(L = 75, K = 50\), we calculate \(10 \times 75 + 5 \times 50 = 750 + 250 = 1000\), verifying that this combination satisfies the total cost. Similarly, other pairs, such as \(L = 100, K = 0\) and \(L = 0, K = 200\), also meet the equation's requirements.
In economic practice, such analysis assists firms in understanding their fixed cost profile, enabling them to map their input choices to desired financial outcomes efficiently.