In This Assignment, You Will Define And Calculate The Remain ✓ Solved

In this assignment, you will define and calculate the remain

In this assignment, you will define and calculate the remaining six major cost elements of a business, when given the total costs and the quantity produced, as well as use the computed costs to determine a minimum cost output level for that business. In addition, you will also clearly explain how the average total cost of a new output level is affected by its share of fixed costs and variable costs.

Paper For Above Instructions

Introduction

This paper defines and calculates six major cost elements of a firm when total cost (TC) and output quantity (Q) are known. It then uses those computed costs to identify a minimum-cost output level. Finally, the paper explains how changes in the relative shares of fixed and variable cost influence average total cost (ATC). The definitions and methods follow standard microeconomic theory (Mankiw, 2014; Pindyck & Rubinfeld, 2017).

Core cost definitions and formulas

When total cost (TC) and quantity produced (Q) are given, the standard derived cost elements are:

  • Fixed Cost (FC): the portion of TC that does not vary with Q. If FC is not provided directly, it can be inferred as TC when Q = 0 or from the persistent intercept of a TC function (Varian, 2014).
  • Variable Cost (VC): VC = TC − FC. VC changes with Q and captures costs that vary with production scale (Nicholson & Snyder, 2012).
  • Average Total Cost (ATC): ATC = TC / Q. This measures cost per unit and equals AFC + AVC (Mankiw, 2014).
  • Average Fixed Cost (AFC): AFC = FC / Q. AFC declines as Q increases because fixed costs are spread over more units (Pindyck & Rubinfeld, 2017).
  • Average Variable Cost (AVC): AVC = VC / Q. AVC reflects per-unit variable costs and may rise or fall with Q depending on marginal technology (Varian, 2014).
  • Marginal Cost (MC): MC = dTC / dQ (or the discrete change in TC divided by the change in Q). MC is the additional cost of producing one more unit and determines optimal output in cost minimization (Samuelson & Nordhaus, 2010).

Illustrative numerical example (constructed TC function)

To make the calculations concrete, consider a common quadratic total cost function:

TC(Q) = 200 + 10Q + 2Q²

Here FC = 200 (the constant term), and VC(Q) = 10Q + 2Q². Use these to compute the remaining elements:

  • FC = 200 (by inspection of TC at Q = 0) (Varian, 2014).
  • VC = TC − FC = (200 + 10Q + 2Q²) − 200 = 10Q + 2Q².
  • ATC = TC / Q = 200/Q + 10 + 2Q.
  • AFC = FC / Q = 200/Q.
  • AVC = VC / Q = 10 + 2Q.
  • MC = dTC/dQ = 10 + 4Q (the derivative of TC with respect to Q) (Pindyck & Rubinfeld, 2017).

Determining the minimum-cost output level

In standard analysis, ATC is minimized where MC = ATC (or where the derivative of ATC = 0). Using the formulas above:

Set MC = ATC → 10 + 4Q = 200/Q + 10 + 2Q

Simplify: 4Q − 2Q = 200/Q → 2Q = 200/Q → 2Q² = 200 → Q² = 100 → Q = 10 (taking the economically meaningful positive root) (Nicholson & Snyder, 2012).

Thus the minimum average total cost occurs at Q = 10 units. Evaluate costs at Q = 10:

  • TC(10) = 200 + 10(10) + 2(10)² = 200 + 100 + 200 = 500.
  • ATC(10) = 500 / 10 = 50.
  • AFC(10) = 200 / 10 = 20; AVC(10) = 10 + 2(10) = 30; note ATC = AFC + AVC = 20 + 30 = 50.
  • MC(10) = 10 + 4(10) = 50, which equals ATC(10) confirming the minimum point (Mankiw, 2014).

How shares of fixed and variable costs affect ATC for a new output level

ATC = AFC + AVC; therefore, changes in the relative shares of FC and VC affect ATC through two channels:

  1. Spreading effect of fixed costs: AFC = FC/Q decreases as Q increases. When FC is a large share of TC, increasing Q yields significant reductions in ATC because AFC falls rapidly (Pindyck & Rubinfeld, 2017).
  2. Behavior of variable costs: AVC depends on marginal technology and input prices. If AVC is increasing in Q (rising marginal costs), increases in Q may raise AVC and push ATC up, potentially outweighing the AFC decline (Varian, 2014).

Use two output comparisons to illustrate. We already computed Q = 10 (ATC = 50). Consider Q = 20:

  • TC(20) = 200 + 10(20) + 2(20)² = 200 + 200 + 800 = 1,200.
  • ATC(20) = 1,200 / 20 = 60.
  • AFC(20) = 200 / 20 = 10; AVC(20) = 10 + 2(20) = 50; ATC = 10 + 50 = 60.

Compared with Q = 10, AFC fell from 20 to 10 (a beneficial spreading effect) but AVC rose from 30 to 50 (higher per-unit variable cost). Because AVC increased more than AFC decreased, ATC increased from 50 to 60. This demonstrates that when variable costs grow with output (e.g., due to diminishing returns), the net effect of raising Q can be an increase in ATC despite lower AFC (Samuelson & Nordhaus, 2010).

Practical interpretation and managerial implications

Managers should decompose total cost into FC and VC and compute AFC, AVC, ATC, and MC to find cost-minimizing production. If FC is large relative to TC, increasing output often reduces ATC through lower AFC; capacity utilization becomes critical (Besanko & Braeutigam, 2011). Conversely, if variable costs rise rapidly with output, expansion may increase ATC. The minimum ATC point (where MC = ATC) provides a useful benchmark for long-run planning and pricing (Mankiw, 2014; Varian, 2014).

Conclusion

Given TC and Q, the six core derived cost elements are FC, VC, ATC, AFC, AVC, and MC. Calculations using a representative TC function (TC = 200 + 10Q + 2Q²) show how to compute each cost measure and how to find the ATC-minimizing output (Q = 10) where MC equals ATC. The effect of a new output level on ATC depends on the balance between the spreading of fixed costs (reducing AFC) and the behavior of variable costs (affecting AVC). Managers must analyze both components to determine whether scaling production will lower unit costs or raise them (Pindyck & Rubinfeld, 2017).

References

  • Mankiw, N. G. (2014). Principles of Economics (7th ed.). Cengage Learning. (See discussion on costs and average/marginal relationships.)
  • Pindyck, R. S., & Rubinfeld, D. L. (2017). Microeconomics (9th ed.). Pearson. (Standard treatment of cost functions, ATC, AVC, and MC.)
  • Varian, H. R. (2014). Intermediate Microeconomics: A Modern Approach (9th ed.). W. W. Norton & Company. (Derivations of cost curves and optimization.)
  • Nicholson, W., & Snyder, C. (2012). Microeconomic Theory: Basic Principles and Extensions (11th ed.). Cengage Learning. (Detailed mathematical exposition of cost curves.)
  • Samuelson, P. A., & Nordhaus, W. D. (2010). Economics (19th ed.). McGraw-Hill Education. (Introductory insight into fixed and variable costs.)
  • Besanko, D., & Braeutigam, R. (2011). Microeconomics (4th ed.). Wiley. (Managerial perspective on cost structure and capacity utilization.)
  • Stiglitz, J. E., & Walsh, C. E. (2006). Economics (4th ed.). W. W. Norton & Company. (Discussion of cost allocation and market implications.)
  • Investopedia. (2021). Average Total Cost (ATC) Definition. Retrieved from https://www.investopedia.com/terms/a/average-total-cost.asp (Practical definition and examples.)
  • Khan Academy. (n.d.). Costs of production. Retrieved from https://www.khanacademy.org/economics-finance-domain/microeconomics/firm-economic-profit (Accessible summaries of AFC, AVC, ATC, and MC.)
  • OECD. (2017). Measuring Productivity — Measurement of costs and productivity. Retrieved from https://www.oecd.org/sdd/productivity-stats/ (Context on cost measures and firm productivity.)

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