An Economist Estimated The Cost Function Of A Single Product

An Economist Estimated That The Cost Function Of A Single Product Firm

An economist estimated that the cost function of a single-product firm is: C(Q) = 60 + 20Q + 15Q² + 5Q³. Based on this information, determine the following:

  • a. The fixed cost of producing 10 units of output.
  • b. The variable cost of producing 10 units of output.
  • c. The total cost of producing 10 units of output.
  • d. The average fixed cost of producing 10 units of output.
  • e. The average variable cost of producing 10 units of output.
  • f. The average total cost of producing 10 units of output.
  • g. The marginal cost when Q = 10.

Paper For Above instruction

Introduction

Understanding the cost structure of a firm is fundamental in microeconomics, as it influences decision-making related to production, pricing, and output levels. The given cost function, C(Q) = 60 + 20Q + 15Q² + 5Q³, reflects the total costs associated with producing a quantity Q of a single product. This paper aims to analyze various cost components at Q=10 units, including fixed costs, variable costs, total costs, average costs, and marginal costs, providing a comprehensive understanding of the firm's cost dynamics.

Fixed Cost at Q=10

The fixed cost corresponds to costs that do not vary with the quantity produced. In the given cost function, fixed costs are represented by the constant term, which is 60. At Q=10, therefore, the fixed cost remains unchanged at 60, as it does not depend on the level of output.

Variable Cost at Q=10

The variable cost (VC) includes all costs that vary with output. It can be derived by subtracting the fixed costs from the total cost at the specified output level. The variable cost function is thus VC(Q) = C(Q) - Fixed Costs. At Q=10:

VC(10) = (60 + 2010 + 1510² + 5*10³) - 60 = 60 + 200 + 1500 + 5000 - 60 = 6,700 - 60 = 6,640

Total Cost at Q=10

The total cost (TC) when producing 10 units is obtained directly by substituting Q=10 into the cost function:

C(10) = 60 + 2010 + 1510² + 5*10³ = 60 + 200 + 1500 + 5000 = 6,760

Average Fixed Cost at Q=10

Average fixed cost (AFC) is calculated by dividing fixed costs by the quantity produced:

AFC = Fixed Cost / Q = 60 / 10 = 6

Average Variable Cost at Q=10

Average variable cost (AVC) is the variable cost divided by the quantity:

AVC = VC / Q = 6,640 / 10 = 664

Average Total Cost at Q=10

Average total cost (ATC) is the total cost divided by the quantity:

ATC = TC / Q = 6,760 / 10 = 676

Marginal Cost at Q=10

The marginal cost (MC) is derived as the derivative of the total cost function with respect to quantity Q:

MC = dC(Q)/dQ = d/dQ [60 + 20Q + 15Q² + 5Q³] = 20 + 30Q + 15Q²

At Q=10:

MC(10) = 20 + 3010 + 1510² = 20 + 300 + 1500 = 1820

Conclusion

Analyzing the cost structure at Q=10 units reveals that the fixed costs are $60, the variable costs are significantly higher at $6,640, leading to a total cost of $6,760. The average fixed, variable, and total costs per unit highlight the cost efficiency and scale effects of production. The marginal cost of producing the tenth unit is $1,820, indicating the cost of producing an additional unit at this production level. These insights are crucial for managerial decision-making concerning production and pricing strategies in competitive markets.

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