Using calculus to find marginal revenue equals marginal cost

Chapter 8: Production Costs: Where Less Is More

cost-minimizing input combination for a specific quantity of output. If you determine the cost-minimizing input combination for every output level, you have the points comprising the total cost function. Thus, the total cost function is derived from production theory.

Figure 8-3 illustrates a series of production isoquants and isocost curves. Because a production isoquant represents a fixed quantity of output, the cost of producing that quantity of output is minimized at the point where the isocost curve is tangent to the isoquant. For the production isoquant qa, production cost is minimized by employing K a of capital and L a of labor. (See Chapter 6 for a more detailed explanation.) This results in the cost level C a . So, one point on your total cost curve corresponds to output qa and total cost C a. (See Figure 8-1.) Other points on your total cost curve are derived through the remaining production isoquants and isocost curve combinations.

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Figure 8-3:

Production isoquants and total cost.

Your firm’s production isoquant is

where q is the quantity of output produced, K is the quantity of capital employed in machine-hours, N is the quantity of land employed in acres, and L is the quantity of labor employed in hours. Capital and land are both fixed inputs. The quantity of capital is fixed at 400 machine-hours, and the quantity of land is fixed at 625 acres. Your input prices are $50 per machine-hour of capital, $40 per acre of land, and $25 per hour for labor.

To convert the production isoquant into a total cost equation, take the following steps:

130 Part II: Considering Which Side You’re On in the Decision-Making Process

1. Using the given input prices, write the equation for total cost.

2. Substitute 400 for K and 625 for N in the equation for total cost.

Because capital and land are fixed, they’re your fixed inputs, and they determine your fixed costs.

3. Substitute 400 for K and 625 for N in the production isoquant.

Total cost needs to be expressed as a function of total product, q, instead of labor, L. You use the production isoquant to make this substitution.

4. Rearranging the equation to solve for L gives you

5. Squaring both sides of the equation leaves you with

6. Substitute the equation in Step 5 for L in the total cost function.

By using your production isoquant, you generate an equation for total cost that’s expressed as a function of the quantity of output produced.

Hoping for less by minimizing per-unit costs

You can also use calculus to determine various minimum costs, such as the minimum average total cost, minimum average variable cost, and minimum marginal cost. To determine each of these minimums, you simply take the derivative of the appropriate function, set it equal to zero, and determine the quantity of output that minimizes that concept.