Ashford 4 Week 3 Assignment: Production Cost Analysis And Es

Ashford 4 Week 3 Assignmentproduction Cost Analysis And Estimatio

Analyze two applied problems involving production cost analysis and estimation, addressing fixed and variable inputs, returns to scale, marginal costs, and profit maximization strategies based on given cost and demand functions. Provide detailed calculations, explanations, and insights based on economic principles.

Paper For Above instruction

Introduction

Production cost analysis is fundamental in understanding how a business can optimize its output and profitability by examining internal costs, input productivity, and market demand. This essay addresses two practical problems involving cost estimation, returns to scale, and profit maximization: William’s pizza shop and Paradise Shoes Company. Through detailed calculations and economic analysis, we aim to identify optimal input levels, examine scales of production, and make strategic recommendations based on the data provided.

Problem 1: William’s Pizza Shop

William owns a small pizza shop with four ovens costing a total of $1,000, and employs workers earning $500 weekly each. The relationship between workers employed and pizzas produced per week allows an analysis of production inputs, returns to scale, and efficiency.

Fixed and Variable Inputs

In William’s production function, the fixed input is the ovens, as their total cost ($1,000) remains constant regardless of the number of pizzas produced, assuming the ovens are either fully operational or not used at all. The labor input—paid at $500 per week—is a variable cost because the number of workers can be adjusted to modify output.

Returns to Scale and Efficiency

To analyze returns, suppose the number of workers employed varies and their corresponding pizza output is recorded. If increasing workers leads to proportionally higher output, the firm experiences increasing returns to scale; if output increases proportionally, constant returns; and if output increases less than proportionally, diminishing returns.

Assuming data shows that initially, adding workers increases pizzas produced at an increasing rate, but after a certain point, the additional output per worker diminishes, indicating diminishing returns. The most efficient number of workers is where the marginal product per worker peaks, often at the point where the marginal cost per pizza is minimized.

Most Efficient Workforce Level and Marginal Cost

If the weekly production data indicates that 4 workers produce 200 pizzas, and adding a 5th worker increases output to 220, the marginal product of the fifth worker is 20 pizzas. The marginal cost (MC) of producing an additional pizza can be derived as:

MC = (Change in total cost) / (Change in quantity) = (Additional wages + Ovens’ cost if applicable) / Additional pizzas produced.

Since ovens are fixed at $1,000, the marginal cost primarily depends on wages; hence, with workers paid $500/week, and if adding one worker increases output by 20 pizzas, then:

MC = $500 / 20 = $25 per pizza, suggesting the highest efficiency occurs at the level where this cost is minimized. Diminishing marginal productivity occurs because, after a certain number of workers, additional workers have less capital (ovens) to work with, leading to overcrowding or inefficiencies.

Expansion and Economies of Scale

Expanding the business by increasing the number of workers or adding more ovens could lead to economies of scale initially, where average costs decrease as output increases due to specialization and more efficient utilization of resources. However, beyond a certain point, diseconomies of scale may occur owing to factors such as overcrowding or management inefficiencies, thereby increasing average costs.

Returns to Scale Summary

Constant returns to scale occur if doubling inputs results in doubling output, while diseconomies of scale occur if output increases less than proportionally with increased inputs. The data or production function details determine these phases.

Problem 2: Paradise Shoes Company

The company’s total variable cost (TVC) function is:

TVC = 3450 + 20Q + 0.008Q²

with Q representing pairs of shoes produced weekly. Its demand equation is Q = 4100 – 25P, where P is the price per pair.

Calculating Marginal Cost (MC)

The marginal cost is derived as the derivative of total variable cost with respect to quantity:

MC = d(TVC)/dQ = 20 + 0.016Q

This expression indicates that marginal cost increases with output level due to the quadratic cost component.

Estimating Incremental Costs for Increasing Output

Moving from 1,000 to 1,200 pairs, the additional cost (incremental cost) is:

Incremental TVC = TVC at Q=1200 – TVC at Q=1000

At Q=1000, TVC = 3450 + 20×1000 + 0.008×1,000,000 = 3450 + 20,000 + 8,000 = 31,450

At Q=1200, TVC = 3450 + 20×1200 + 0.008×1,440,000 = 3450 + 24,000 + 11,520 = 38,970

Incremental cost = 38,970 – 31,450 = 7,520

This indicates that producing an additional 200 pairs costs approximately $7,520.

Optimal Price and Output for Profit Maximization

Profit maximization occurs where marginal revenue (MR) equals marginal cost (MC). The demand equation can be rearranged to find P as a function of Q:

P = (4100 – Q)/25

The marginal revenue (MR) for a linear demand is:

MR = P + (dP/dQ) × Q = (4100 – Q)/25 – (Q/25) = (4100 – 2Q)/25

Setting MR = MC gives:

(4100 – 2Q)/25 = 20 + 0.016Q

Multiplying both sides by 25:

4100 – 2Q = 500 + 0.4Q

4100 – 500 = 2Q + 0.4Q

3600 = 2.4Q

Q ≈ 1500 pairs

Substituting back into demand to find price:

P = (4100 – 1500)/25 = 2600/25 = $104 per pair

Thus, the profit-maximizing output is approximately 1,500 pairs, with a price of $104 per pair.

Further Expansion Analysis

To determine whether to expand beyond 1,200 pairs, compare the marginal cost at Q=1,200:

MC = 20 + 0.016×1200 = 20 + 19.2 = $39.2

Since the market price at 1,500 pairs is $104 — significantly higher than $39.2 — expanding further appears profitable, assuming demand remains strong and the firm can accommodate additional capacity without significant increases in fixed costs or decreasing market share.

However, considerations such as capacity constraints, market saturation, and diminishing demand elasticity must also be evaluated before further expansion.

Conclusion

Both cases exemplify core economic principles: the importance of understanding fixed and variable costs, the dynamics of returns to scale, and the critical role of marginal analysis in profit maximization. William’s pizza shop should optimize workforce levels considering diminishing returns, while Paradise Shoes should expand production to the point where marginal revenue equals marginal cost, maximizing profitability. Strategic decisions should also account for economies of scale and potential diseconomies, ensuring sustainable growth and cost control.

References

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