Production Cost Analysis And Estimation Applied Problems

Production Cost Analysis And Estimation Applied Problemswilliam Is The

Production Cost Analysis and Estimation Applied Problems William is the owner of a small pizza shop and is considering increasing its products and reducing costs. The shop owns four ovens costing $1,000 each and employs workers paid $500 per week. The analysis involves identifying fixed and variable inputs, understanding returns to scale, and optimizing production efficiency. Additionally, a second problem concerns a shoe company's cost functions and optimal output decisions, including cost estimation and profit maximization strategies.

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Introduction

Effective production cost analysis is essential for small businesses seeking to optimize output, increase efficiency, and reduce costs. William's pizza shop exemplifies typical small-scale operations where understanding cost structures, returns to scale, and productivity thresholds directly impact profitability. Similarly, the Paradise Shoes Company illustrates how fixed and variable costs influence decision-making regarding expansion and profit maximization. This paper discusses these scenarios in detail, highlighting fundamental economic principles such as fixed and variable inputs, returns to scale, marginal cost, and optimal output levels.

Identification of Fixed and Variable Inputs

In William's pizza shop, fixed inputs are those that do not change with the level of production in the short run. The four ovens, costing a total of $1,000, represent fixed inputs because their cost remains constant whether the shop produces a few or many pizzas. On the other hand, variable inputs include labor — the workers paid $500 weekly — because the number of workers employed likely varies with production needs or output levels. The number of pizzas produced per week depends on these inputs, with labor being variable as additional workers can be hired or dismissed based on production goals.

Returns to Scale and Production Efficiency

Analyzing the production function, as employment of workers increases, the shop may initially experience increasing returns to scale due to specialization and improved efficiency. For example, hiring additional workers might lead to more pizzas per worker because tasks become divided. However, beyond a certain point, the shop may experience diminishing returns due to congestion, limited kitchen space, or over-utilization of fixed inputs like ovens. Identifying the range where returns shift from increasing to diminishing is essential for optimizing workforce size.

The most efficient number of workers, in terms of pizzas produced per worker, is achieved when marginal productivity peaks. As additional workers contribute less to output, the shop should aim to employ the number of workers at which marginal productivity begins to decline. The reason marginal productivity declines when more workers are hired beyond a certain level is that of the Law of Diminishing Marginal Returns — after a point, adding more workers leads to overcrowding and reduced incremental output.

Cost Minimization and Marginal Cost Analysis

To determine the number of workers minimizing the marginal cost of pizza production, we compare the marginal cost of labor to the additional pizzas produced each worker can produce. Assuming each worker's weekly wage is $500, the marginal cost per pizza depends on the additional output generated. When the marginal productivity per worker declines, the cost per pizza rises, signaling the optimal employment level for minimizing costs.

Impact of Expansion and Economies of Scale

Expanding the business could lead to economies of scale, whereby the average costs per pizza decrease due to increased production and more efficient utilization of fixed inputs. For instance, purchasing ovens in bulk or optimizing labor schedules reduces average costs. However, beyond a certain scale, the firm might face diseconomies of scale, where increased complexity, management challenges, or resource limitations cause average costs to rise. Constant returns to scale occur when proportional increases in inputs lead to proportional increases in output, maintaining unit costs.

Applied Cost and Production Problems – Paradise Shoes Company

The company's total variable cost (TVC) function is given as TVC = 3450 + 20Q + 0.008Q², where Q represents pairs of shoes produced weekly. The demand function is Q = 4100 – 25P.

At an output of 1,000 pairs:

- The TVC is calculated as TVC = 3450 + 20(1000) + 0.008(1000)².

- This yields TVC = 3450 + 20,000 + 8,000 = $31,450.

At 1,200 pairs:

- The TVC is TVC = 3450 + 20(1200) + 0.008(1200)².

- Calculating gives TVC = 3450 + 24,000 + 11,520 = $38,970.

The fixed cost component is $3,450, and variable costs increase with output. To derive the marginal cost (MC) curve, differentiate TVC with respect to Q:

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

This expression indicates that marginal costs increase linearly with Q, beginning at $20 when Q is zero.

The incremental cost of producing an additional 200 pairs (from 1,000 to 1,200) is the difference in total variable costs:

Incremental Cost = $38,970 - $31,450 = $7,520.

Profit maximization involves setting marginal revenue (MR) equal to marginal cost (MC). The demand function (Q = 4100 – 25P) rearranged to P = (4100 – Q)/25 allows for calculating price at various output levels. The total revenue (TR) is P*Q; the marginal revenue (MR) is derived accordingly. Equating MR and MC yields the profit-maximizing output and price.

At Q = 1,200, the price is P = (4100 – 1200)/25 = 2900/25 = $116.

The total revenue at Q = 1200 is TR = PQ = $116 1200 = $139,200.

Total costs are VC + fixed costs = $38,970 + $3,450 = $42,420.

Profit = TR – TC = $139,200 – $42,420 = $96,780.

Decision on Further Expansion

Considering the costs and revenues, expanding production beyond 1,200 pairs involves assessing whether additional units will still be profitable. Because marginal costs increase with output, and demand decreases as price drops, further expansion should only occur if the new price remains above average total cost and yields positive profit margins. Based on the calculations, continuing expansion appears profitable up to the point where MR = MC, which is near or slightly above 1,200 pairs, suggesting further expansion could be advantageous if demand sustains higher output levels.

Conclusion

In small business production analysis, identifying fixed and variable inputs helps optimize resource allocation. Understanding returns to scale and the point of diminishing marginal returns aids in workforce and capacity planning. For the pizza shop, balancing employment levels to maximize productivity while minimizing costs is essential. The shoe company’s cost structure demonstrates how variable costs impact decision-making related to expansion and profit maximization. Overall, systematic cost analysis and marginal cost assessments are critical for sound managerial decisions in production and expansion strategies.

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