Problem 66
Question
BUSINESS: Isocost Lines 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 $$ w L+r K=\mathrm{C} \quad \text { for } L \geq 0, \quad K \geq 0 $$ 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 nonnegative, an isocost line is a line segment in just the first quadrant. a. Write the equation of the isocost line with \(w=8, \quad r=6, \quad C=15,000\), and graph it in the first quadrant. b. Verify that the following \((L, K)\) pairs all have the same total cost. \((1875,0),(1200,900),(600,1700),(0,2500)\)
Step-by-Step Solution
Verified Answer
The isocost equation is \(8L + 6K = 15,000\); all given points lie on this line.
1Step 1: Write the Isocost Equation
Substitute the given values into the isocost equation \(wL + rK = C\). For this problem, \(w = 8\), \(r = 6\), and \(C = 15,000\). Thus, the equation becomes \(8L + 6K = 15,000\).
2Step 2: Solve for Graph Intercepts
To graph this equation, we first find the intercepts with the axes. For the \(L\)-intercept, set \(K = 0\) and solve \(8L = 15,000\), giving \(L = \frac{15,000}{8} = 1875\). For the \(K\)-intercept, set \(L = 0\) and solve \(6K = 15,000\), giving \(K = \frac{15,000}{6} = 2500\). The line will pass through these intercepts at \((1875, 0)\) and \((0, 2500)\).
3Step 3: Verify Given Points on the Isocost Line
Substitute each given pair \(L, K\) into the equation and verify if they satisfy \(8L + 6K = 15,000\).- For \( (1875, 0) \): \(8(1875) + 6(0) = 15,000\)- For \( (1200, 900) \): \(8(1200) + 6(900) = 9,600 + 5,400 = 15,000\)- For \( (600, 1700) \): \(8(600) + 6(1700) = 4,800 + 10,200 = 15,000\)- For \( (0, 2500) \): \(8(0) + 6(2500) = 15,000\)Since all points satisfy the equation, they lie on the isocost line.
Key Concepts
Labor and CapitalCost EquationGraph Intercepts
Labor and Capital
Labor and capital are two essential inputs that businesses utilize to produce goods and services. Think of labor as the people who work or the effort invested to create products, while capital represents the tools and equipment like machinery and buildings that help the production process.
The balance between labor and capital can significantly impact how a company operates and its profitability. By adjusting these inputs, a business can find different combinations that minimize costs or maximize output, which are represented by isocost lines on a graph.
Isocost lines help businesses visualize all the combinations of labor and capital that they can afford for a specified total cost. They highlight that businesses can employ various amounts of labor and capital while keeping the total expenditure unchanged.
The balance between labor and capital can significantly impact how a company operates and its profitability. By adjusting these inputs, a business can find different combinations that minimize costs or maximize output, which are represented by isocost lines on a graph.
Isocost lines help businesses visualize all the combinations of labor and capital that they can afford for a specified total cost. They highlight that businesses can employ various amounts of labor and capital while keeping the total expenditure unchanged.
Cost Equation
The isocost line is defined by a cost equation, which is expressed as follows: \( wL + rK = C \) This equation combines different economic concepts:
This cost equation is essential for understanding how variations in labor and capital affect overall expenditure and helps in planning the optimal allocation of resources to achieve desired production levels.
- \( L \) stands for the units of labor.
- \( w \) is the wage rate or cost per unit of labor.
- \( K \) specifies the units of capital.
- \( r \) refers to the rental rate or cost per unit of capital.
- \( C \) is the total cost a firm is willing to spend.
This cost equation is essential for understanding how variations in labor and capital affect overall expenditure and helps in planning the optimal allocation of resources to achieve desired production levels.
Graph Intercepts
To understand the graphical representation of an isocost line, finding the intercepts on the coordinate axes is crucial. These intercepts show the maximum amount of labor or capital a company can afford if it spends its entire budget on just one of these inputs. The intercepts are calculated by setting one variable to zero.
For the labor-intercept, set \( K = 0 \), and solve \( 8L = 15,000 \). It gives \( L = \frac{15,000}{8} = 1875 \). This means if all resources are allocated to labor, a company can afford 1,875 units.
For the capital-intercept, set \( L = 0 \), and solve \( 6K = 15,000 \). It yields \( K = \frac{15,000}{6} = 2500 \). If all resources go to capital, 2,500 units can be obtained.
These intercepts help in plotting the isocost line on a graph, effectively illustrating the possible combinations of labor and capital a company can purchase with its budget. The line passes through these intercepts at points (1875, 0) and (0, 2500), clearly depicting the trade-off between labor and capital.
For the labor-intercept, set \( K = 0 \), and solve \( 8L = 15,000 \). It gives \( L = \frac{15,000}{8} = 1875 \). This means if all resources are allocated to labor, a company can afford 1,875 units.
For the capital-intercept, set \( L = 0 \), and solve \( 6K = 15,000 \). It yields \( K = \frac{15,000}{6} = 2500 \). If all resources go to capital, 2,500 units can be obtained.
These intercepts help in plotting the isocost line on a graph, effectively illustrating the possible combinations of labor and capital a company can purchase with its budget. The line passes through these intercepts at points (1875, 0) and (0, 2500), clearly depicting the trade-off between labor and capital.
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